Search Results for Logic


  1. Tarski biography
    • These appointments were of great significance since Alfred took a course on logic given by Lesniewski who quickly saw his genius and persuaded him to change from biology to mathematics.
    • Tarski taught logic at the Polish Pedagogical Institute in Warsaw from 1922 to 1925 then in that year he was appointed Docent in mathematics and logic at the University of Warsaw.
    • which is among the most important papers ever written on mathematical logic.
    • Not only does this paper provide a mathematically rigorous articulation of several ideas that had been developing in earlier mathematical logic, it also presents foundations on which later logic could be built.
    • This work on logical consequence has had a profound influence and has been discussed by many authors; see for example [Tarski on truth and logical consequence (Stanford, Ca., 1986).','Reference ',5)">5], [Modern Logic 7 (2) (1997), 109-130.
    • Symbolic Logic 53 (1) 1988), 51-79.
    • Formal Logic 37 (1) (1996), 125-151.','Reference ',36)">36], [J.
    • Logic 25 (6) (1996), 617-677.','Reference ',53)">53], [Bull.
    • Symbolic Logic 3 (2) (1997), 216-241.','Reference ',56)">56], and [Hist.
    • Logic 2 (1981), 11-20.','Reference ',70)">70].
    • In 1937 he published another classic paper, this time on the deductive method, which presents clearly his views on the nature and purpose of the deductive method, as well as considering the role of logic in scientific studies.
    • His seminars at Berkeley fast became a power-house of logic.
    • Tarski made important contributions in many areas of mathematics: set theory, measure theory, topology, geometry, classical and universal algebra, algebraic logic, various branches of formal logic and metamathematics.
    • He produced axioms for 'logical consequence', worked on deductive systems, the algebra of logic and the theory of definability.
    • In 1968 Tarski wrote another famous paper Equational logic and equational theories of algebras in which he presented a survey of the metamathematics of equational logic as it then existed as well as giving some new results and some open problems.
    • is one of the finest pieces of expository writing in all of mathematical logic.
    • His work includes Geometry (1935), Introduction to Logic and to the Methodology of Deductive Sciences (1936), A decision method for elementary algebra and geometry (1948), Cardinal Algebras (1949), Undecidable theories (1953), Logic, semantics, metamathematics (1956), and Ordinal algebras (1956).
    • Introduction to Logic and to the Methodology of Deductive Sciences is an introduction written at the level of an undergraduate course in logic and axiomatics.
    • It is only when we see Tarski's papers collected in one place that we can begin to appreciate the scope and profundity of his influence on modern mathematical thought and, in particular, on modern mathematical logic.
    • Mathematical logic as we know it today is almost inconceivable without Tarski's contributions.
    • He was made honorary editor of Algebra Universalis and served as President of the Association for Symbolic Logic from 1944 to 1946 and the International Union for the History and Philosophy of Science in 1956-57.

  2. Curry biography
    • He was given a topic in the theory of differential equations by George Birkhoff but he began reading books on logic which seemed to him far more interesting that his research topic.
    • He asked various faculty members at Harvard, and Norbert Wiener at MIT, if they thought he might change to undertake research in logic.
    • He again approached various faculty members at Harvard, and Norbert Wiener at MIT, asking whether they thought that he might write his doctoral dissertation on logic.
    • Wiener's reply was typical - avoid logic unless you have something to say, but now you certainly have something to say! .
    • Curry now made his final change in direction and decided to give up his doctoral studies on differential equations and to write a doctoral dissertation on logic.
    • Some papers published during the early years of his research include The universal quantifier in combinatory logic (1931), Some additions to the theory of combinators (1932), Apparent variables from the standpoint of combinatory logic (1933), and Some properties of equality and implication in combinatory logic (1934).
    • The Association for Symbolic Logic was founded in 1936 with Curry as one of the founders.
    • His retiring presidential address The combinatory foundations of mathematical logic was published in the Journal of Symbolic Logic in 1942.
    • After giving a very clear exposition of the fundamentals of combinatory logic, showing its close relationship to the λ-calculus developed by Church, Curry went on to describe his recent work.
    • He had examined simplified methods of deriving the paradoxes (such as those of Richard and Russell) in systems of logic which are inconsistent, and had also developed a method of introducing into combinatory logic undefined notions of generality, such as quantification or formal implication, in such a way that a consistency theorem like that of Church and Rosser could still be derived.
    • His major texts include Combinatory Logic (1958) (with Robert Feys), and Foundations of Mathematical Logic (1963).
    • Curry began working on Combinatory Logic in 1950 when he was awarded a Fulbright Grant that enabled him to work with Robert Feys at Louvain.
    • E J Cogan, reviewing the book, gives a nice description of combinatory logic:- .
    • Combinatory logic is concerned with certain basic notions of the foundations of mathematics which are usually used in an intuitive and unanalysed way.
    • The part of combinatory logic which is concerned with questions of a fundamental nature which, like substitution, involve variables, is called the theory of combinators.
    • In Foundations of Mathematical Logic Curry develops the topic from an algebraic basis using Gentzen's methods.
    • In 1966 he accepted the position of Professor of Logic, History of Logic, and Philosophy of Science at Amsterdam.
    • The authors of [To H B Curry : essays on combinatory logic, lambda calculus and formalism (London-New York, 1980), vii-xi.','Reference ',3)">3] make some nice comments about Curry and his wife:- .
    • And this has undoubtedly been an important contribution to the enthusiasm of many of those of us working in combinatory logic.
    • There are always many parties and other, less formal gatherings, and we conjecture that Virginia's cooking has also played a role in the growth of interest in combinatory logic.

  3. Schroder biography
    • Dipert [Modern Logic 1 (2-3) (1990/91), 117-139.','Reference ',12)">12] speculates that his reasons for going to Zurich may not have been entirely academic ones since he was a very enthusiastic mountain climber and made a number of difficult ascents without a guide during his time in Switzerland.
    • Ernst Schroder's important work is in the area of algebra, set theory and logic.
    • However, he never considered himself to be a logician, as Peckhaus points out [Bulletin of Symbolic Logic 5 (1999), 433-450.','Reference ',22)">22]:- .
    • What was the connection between logic and algebra in Schroder's research? ..
    • In fact Schroder started out being interested in mathematical physics, and his move towards logic was simply an attempt to deepen its foundations.
    • Now one sees Schroder moving towards logic with Lehrbuch der Arithmetik und Algebra fur Lehrer und Studierende Ⓣ published by Teubner in 1873.
    • He wrote his first work on mathematical logic Der Operationskreis des Logikkalkuls Ⓣ, influenced by George Boole and Hermann Grassmann, in 1877.
    • He was the first to use the term 'propositional calculus' and seems to be the first to use the term 'mathematical logic'.
    • In fact he compares algebra and Boole's logic saying:- .
    • In Vorlesungen uber die Algebra der Logik Ⓣ, a large work published between 1890 and 1905 (it was edited and completed by Eugen Muller after his death), Schroder gave a detailed account of algebraic logic, provided a source for Alfred Tarski to develop the modern algebraic theory and gave an extensive bibliography of the history of logic.
    • Brady writes [From Pierce to Skolem: A Neglected Chapter in the History of Logic (Elsevier, 2000).','Reference ',3)">3]:- .
    • Schroder's concept of solving a relational equation was a precursor of Skolem functions, and he inspired Lowenheim's formulation and proof of the famous theorem that every sentence with an infinite model has a countable model, the first real theorem of modern logic.
    • Schroder said his aim was (see for example [Bulletin of Symbolic Logic 5 (1999), 433-450.','Reference ',22)">22]):- .
    • to design logic as a calculating discipline, especially to give access to the exact handling of relative concepts, and, from then on, by emancipation from the routine claims of natural language, to withdraw any fertile soil from "cliche" in the field of philosophy as well.
    • Brady writes [From Pierce to Skolem: A Neglected Chapter in the History of Logic (Elsevier, 2000).','Reference ',3)">3]:- .
    • Schroder considered quantifiers (or, at least, sums and products equivalent to quantifiers for a fixed domain) in first- and higher-order logic.
    • He understood that there are notions such as countability that are beyond relative calculus (and also beyond first-order predicate logic).
    • Dipert [Modern Logic 1 (2-3) (1990/91), 117-139.','Reference ',12)">12] gives an interesting account of Schroder's character which he compares with that of Peirce:- .
    • When I started to trace the later development of logic, the first thing I did was to look at Schroder's 'Vorlesungen uber die Algebra der Logik', ..
    • [whose] third volume is on the logic of relations (Algebra und Logik der Relative, 1895).
    • The three volumes immediately became the best-known advanced logic text, and embody what any mathematician interested in the study of logic should have known, or at least have been acquainted with, in the 1890s.
    • Schroder participated in the development of mathematical logic as an independent discipline in the second half of the nineteenth century.
    • As a result he was an outsider, at a disadvantage in chosing terminology, in outlining his argumentation, and in judging what mathematical logic could accomplish.

  4. Scholz biography
    • This marked an important change in the direction of Scholz's research for at this time his interests turned towards mathematical logic.
    • In [Mathematicians under the Nazis (Princeton University Press, 2003).','Reference ',3)">3] Segal suggests that Scholz's love of structure was also important in his move into mathematical logic:- .
    • Of course for someone like Scholz, who had trained in theology and then philosophy, mathematical logic involved a deep understanding of mathematics which he had never studied.
    • Right from the time he arrived at Munster, Scholz worked towards building a school of mathematical logic there.
    • He gave lecture courses on mathematical logic and also lectured on the great philosophers.
    • In 1931 he published Geschichte der Logik Ⓣ, a short but erudite study, which looks at the history of results in logic leading to the study of mathematical logic.
    • pioneered the view that ancient and medieval logic was not something totally different (for better or for worse) from what modern logicians are doing by mathematical means.
    • The author was one of the first to see clearly that there is no better aid than modern logic to make clear what Aristotle, the Stoics, the Schoolmen and also a few post-Renaissance figures like Leibniz, were really after.
    • Logic 6 (1) (1941), 32-34.','Reference ',4)">4]:- .
    • in his concluding summary that the philosophy which he intended is neither more nor less than mathematical logic, axiomatics ..
    • Apparently his view is that foundational research must first attain a more advanced stage, and at the same time the minds of philosophers must, in the school of mathematical logic and axiomatics, be turned to the spirit of clearness and precision - whose compatibility with philosophical profundity is stressed by Scholz in opposition to a frequent opinion - before a valuable speculative synthesis can be hoped for.
    • As we indicated above, Scholz's aim was to establish a world centre of mathematical logic at Munster.
    • By this time his research team at Munster were being referred to as "the Munster school of mathematical logic." The Technical University of Karlsruhe held Ernst Schroder's papers and Scholz was also successful in adding these to the growing collection of resources of his school [Philos.
    • Its title was changed again in 1943 to the Chair of Mathematical Logic and Foundational Questions in Mathematics.
    • In some ways the Nazi laws against the Jews helped Scholz establish Munster as an important centre for logic since the leading researchers in the other centres of Berlin and Gottingen were forced out.
    • However, Scholz was able to play the system to the advantage of mathematical logic by keeping on good terms with Nazis like Bieberbach.
    • Scholz's connections with Bieberbach had led earlier to funds being provided for a series of monographs on mathematical logic which had started in 1937.
    • What Scholz has understood is doubtless this, to obtain from the German State huge amounts of publication money for this logic production.
    • We fundamentally reject this logic which praises the English empiricists and sensory philosophers such as the Englishmen Locke, Berkeley, Hume, and by now find it really time to speak for once about the "Great Germans".
    • Scholz remained as head of the Institute for Mathematical Logic and Foundational Research at Munster until he retired in 1952.
    • The authors state (i) that mathematical logic is intended to provide a precise(r) formulation of the notion of consequence on which mathematical theories are based, (ii) that they consider those mathematical theories which fall within the scope of two-valued logic 'where every proposition is true or false'.

  5. Church biography
    • Symbolic Logic 1 (4) (1995), 486-488.','Reference ',4)">4]:- .
    • Princeton in the 1930's was an exciting place for logic.
    • His work is of major importance in mathematical logic, recursion theory, and in theoretical computer science.
    • Logic 18 (4) (1997), 211-232.','Reference ',10)">10] is in three parts and in the last of these Manzano:- .
    • Church's Theorem, showing the undecidability of first order logic, appeared in A note on the Entscheidungsproblem published in the first issue of the Journal of Symbolic Logic.
    • Symbolic Logic 3 (2) (1997), 154-180.','Reference ',11)">11].
    • Church was a founder of the Journal of Symbolic Logic in 1936 and was an editor of the reviews section from its beginning until 1979.
    • In fact he published a paper A bibliography of symbolic logic in volume 4 of the Journal and he saw the reviews section as a continuation and expansion of this work.
    • in symbolic logic, wherever and in whatever language published ..
    • Symbolic Logic 4 (2) (1998), 172-180.','Reference ',5)">5] highlights Church's guiding role in defining the boundaries of the discipline of symbolic logic through this editorial work and testifies to his unflagging industry and conscientiousness and his high editorial standards.
    • The aim of comprehensive coverage, which in 1936 had seemed quite practical, became less so as the years went by and by 1975 the rapid expansion in symbolic logic publications forced Church to give up this aspect and begin to provide only selective coverage.
    • Upon his retirement, Princeton was unwilling to continue accommodating the small staff working on the reviews for the Journal of Symbolic Logic.
    • Church wrote the classic book Introduction to Mathematical Logic in 1956.
    • This was a revised and very much enlarged edition of Introduction to mathematical logic which Church published twelve years earlier in 1944.
    • the first half of an introductory course in mathematical logic given to graduate students in mathematics [at Princeton in 1943].
    • Logic 18 (4) (1997), 211-232.','Reference ',10)">10] that the 1956 edition of the book:- .
    • defined the subject matter of mathematical logic, the approach to be taken and the basic topics addressed.
    • Symbolic Logic 4 (2) (1998), 129-171.','Reference ',3)">3].
    • Church considered this topic for about 40 years during the latter part of his career, beginning with his paper A formulation of the logic of sense and denotation in 1951.
    • Although most of Church's contributions are directed towards mathematical logic, he did write a few mathematical papers of other topics.

  6. Ackermann biography
    • This formalism formed the basis of Bourbaki's logic and set theory.
    • Among Ackermann's later work are consistency proofs for set theory (1937), full arithmetic (1940) and type free logic (1952).
    • Symbolic Logic 23 (2) (1958), 215-216.','Reference ',5)">5]:- .
    • Symbolic Logic 34 (3) (1969), 481-488.','Reference ',2)">2].
    • Symbolic Logic 34 (3) (1969), 481-488.','Reference ',2)">2].
    • In 1957 Ackermann published Philosophische Bemerkungen zur mathematischen Logik und zur mathematischen Grundlagenforschung Ⓣ and its English translation Philosophical observations on mathematical logic and on investigations into the foundations of mathematics.
    • This paper, written for non-experts in the subject, gives an excellent overview of how Ackermann viewed mathematical logic.
    • Symbolic Logic 23 (3) (1958), 342-343.','Reference ',3)">3]:- .
    • An objection to mathematical logic is that it is not the same as the philosophical logic which forms the foundation of our thought and which alone is necessary for thinking.
    • Ackermann remarks that the traditional modes of inference are included in mathematical logic, besides many others, like the statement logic or the logic of relations.
    • One has the illusion of getting by with "Aristotelian logic" in mathematics just as long as mathematical reasonings are insufficiently analyzed.
    • A further objection to mathematical logic is that it is incomplete, in the sense that, by Godel's result of 1931 (the text says 1932), intuitively correct theorems cannot be proved within a given system.
    • Mathematical logic is of use in clarifying the concepts of "necessity" and "possibility," the distinction between "analytic" and "synthetic." Discussing the "triviality" of logic, the author presents the decision problem.
    • Ackermann presents intuitionism, which constructs a mathematics with a minimum of logic, and the Frege-Russell analysis of the number concept.
    • In these concepts "there is nothing foreign to logic." He concludes that "in the theory of natural numbers we have a domain which is capable of an intuitive foundation with a minimum of logic in the sense of [Brouwer] and also of attainment purely by logical definitions if we presuppose an extensive logic." .

  7. Poretsky biography
    • Although the main events of P S Poretsky's life and work have been known through obituaries such as [Izvestiya Fiziko-matematicheskogo obshchestva pri (Imperatorskom) kazanskom universitete, (2) 16 (1908), 3-7.',6)">6], it is only recently that further material on his life and work has come to light and this is described in [Modern Logic 3 (1) (1992), 93-94.',4)">4] and in more detail in [History of Science and Technology (Russian) 4 (2005), 64-73.',5)">5].
    • V A Bazhanov, the author of these papers, states in his summary of [Modern Logic 3 (1) (1992), 93-94.',4)">4]:- .
    • These include: various documents related to Poretsky's lectures on mathematical logic for mathematics department students at Kazan University which were intended to be given for three semesters in the autumn of 1887 and all of 1888 but were delivered only during the 1888 Spring semester, a complete mathematical logic program compiled by Poretsky, materials related to Poretsky's father and family, Poretsky's Magister's (master's) dissertation and the decision of the physics-mathematics faculty council to award him the doctorate in astronomy rather than the Magister, a complete list of the sources he used (including Boole, Jevons, Schroder, and Peano), biographical data and materials regarding his illness and subsequent dismissal from Kazan University.
    • Poretsky became interested in logic through Alexander V Vasiliev soon after arriving in Kazan in 1876.
    • In session 1887-1888, Poretsky lectured on mathematical logic, the first time that such a course had been given in Russia.
    • I think teaching mathematical logic is very useful ..
    • Mathematical logic is a branch of the general science of operations, and in this respect deserves attention from mathematicians.
    • The basic concepts of mathematical logic to a great extent clarifies the fundamental theorems of mathematical theory.
    • Poretsky worked on mathematical logic for the rest of his life, extending and augmenting results of George Boole, Stanley Jevons, Ernst Schroder and John Venn [Dictionary of Scientific Biography (New York 1970-1990).',1)">1]:- .
    • In papers published from 1880 to 1908, Poretsky systematically studied and solved many problems of the logic of classes and of propositions.
    • He published major works on methods of solution of logical equations, and on the reverse mode of mathematical logic.
    • He applied his logic calculus to the theory of probability.
    • He continued to undertake research into mathematical logic for the remaining eighteen years of his life.
    • Scientists in Russia and the Soviet Union made a significant contribution to the development of mathematical logic - both its classical and non-classical areas.
    • Of course, the separation of logic into "classical" and "non-classical" is quite arbitrary.
    • Thus, A N Kolmogorov left outstanding results in both the classical and the non-classical areas of modern logic.
    • But who in Russia was the pioneer of mathematical logic? ..
    • He was the first in Russia not only to be engaged in research on mathematical logic, and the first to deliver a course on mathematical logic (the Kazan University), but also achieved - thanks to his understanding of the subject and his development of original methods - word-wide visibility and recognition.

  8. Bernays biography
    • E Specker writes [Logic Colloquium \'78, Mons, 1978, Stud.
    • Logic Foundations Math.
    • He wrote later (see [Logic Colloquium \'78, Mons, 1978, Stud.
    • Logic Foundations Math.
    • To be sure, the paper was of definite mathematical character, but investigations inspired by mathematical logic were not taken quite seriously - they were thought of as amusing, half-way part of recreational mathematics.
    • Many things I had in the paper have therefore not been recorded accordingly in descriptions of the development of mathematical logic.
    • Bernays wrote a second habilitation in which he established the completeness of propositional logic; this was in fact a study of Russell and Whitehead's Principia Mathematica, and uses ideas from Schroder.
    • To some people it seemed that he was even exploiting his logic assistant.
    • He visited Princeton in session 1935-36 and gave courses on mathematical logic and set theory.
    • The work attempted to build mathematics from symbolic logic.
    • Akihiro Kanamori writes about this famous book [Bulletin of Symbolic Logic 15 (1) (2009), 43-69.','Reference ',8)">8]:- .
    • Going into many reprintings and an eventual second edition thirty years later, this monumental work provided a magisterial exposition of the work of the Hilbert school in the formalization of first-order logic and in proof theory and the work of Godel on incompleteness and its surround, including the first complete proof of the Second Incompleteness Theorem.
    • Recent re-evaluation of Bernays' role actually places him at the centre of the development of mathematical logic and Hilbert's program.
    • Between 1937 and 1954 Bernays wrote a whole series of articles in the Journal of Symbolic Logic which attempted to achieve this goal.
    • Bernays' virtues as a writer on philosophy of mathematics are evident: contact with actual mathematics, especially mathematical logic, combined with a familiarity with the issues that concern philosophers and sensitivity to the difficulties that philosophical positions are prone to.
    • He was also elected president of the International Academy of the Philosophy of Science, and made an honorary chair of the German Society for Mathematical Logic and Foundational Research in the Exact Sciences.
    • He was on the editorial board of several journals, Dialectica, the Journal of Symbolic Logic, and the Archiv fur mathematische Logik und Grundlagenforschung.
    • Finally, we give details of his personality as given in [Logic Colloquium \'78, Mons, 1978, Stud.
    • Logic Foundations Math.

  9. Lewis biography
    • His first publications, which appeared at this time, include Professor Santayana and Idealism (1912), Implication and the Algebra of Logic (1912), Realism and Subjectivism (1913), Interesting Theorems in Symbolic Logic (1913), A New Algebra of Implications and Some Consequences (1913), The Calculus of Strict Implication (1913), The Matrix Algebra for Implications (1914), and A Too Brief Set of Postulates for the Algebra of Logic (1915).
    • In 1918 he published the book A Survey of Symbolic Logic which he wrote so that his students at Berkeley might have a class textbook.
    • The Preface to A Survey of Symbolic Logic is given at THIS LINK.
    • fills an important hiatus in the literature of logistics and mathematical logic.
    • He treats the history of symbolic logic in an impartial and comprehensive way, slighting neither the founders of the classical theory nor the principal innovators of the present day.
    • After a good resume of the classical theory of equations and inequations, he proceeds to a parallel development of the foundations of the logic of propositions, propositional functions, and classes on the Boole-Peirce-Schroder basis and on that of the 'Principia', exhibiting both the formal identity of the two systems and the inadequacy of Peirce's enumerative method of defining universal and particular propositions in terms respectively of iterated logical multiplication and iterated logical addition.
    • During these years at Harvard he published a number of important books such as: Mind and the World-Order: Outline of a Theory of Knowledge (1929), in which he presented his ideas which grew out of his investigations in the field of exact logic and its application to mathematics; (with Cooper Harold Langford) Symbolic Logic (1932), which develops a modal system of "strict implication" for interpreting the logical force of "if .
    • He writes in [Notre Dame Journal of Formal Logic XI (2) (1970), 129-140.','Reference ',17)">17]:- .
    • Let us end by quoting the summary of his achievements from [Notre Dame Journal of Formal Logic XI (2) (1970), 129-140.','Reference ',17)">17]:- .
    • He is the principal founder of the modern symbolic treatment of modal logic and theory of entailment.
    • He contributed to the history of logic.
    • His 'Survey' was a pioneer textbook, and both of his logic books combined pedagogical value with important contributions to the subject.
    • His contributions to philosophy of logic and to epistemology are important and influential.
    • He has heightened the awareness of the interrelations between logic, epistemology, and value theory; and has made a strong case for cognitivism in valuation.
    • C I Lewis's Survey of Symbolic Logic .

  10. Adian biography
    • And in 1957 an event happened which completely changed life for both him and his teacher, the Department of Mathematical Logic was created in the Steklov Mathematical Institute (MIAN), and Novikov was invited to lead it.
    • In 1965, at the invitation of A A Markov, Adian also took a second position, in the Department of Mathematical Logic at MSU.
    • His work there continues to ensure a close and fruitful collaboration of the department with the Department of Mathematical Logic at MIAN.
    • The Department of Mathematical Logic in the Faculty of Mechanics and Mathematics at MSU went through a similar period of turbulence, for similar reasons, when the head of the department, A A Markov, fell sick at the end of the 1970s.
    • Adian has always devoted much attention to strengthening the Department of Mathematical Logic at MIAN, to training researchers in the Department of Mathematical Logic at MSU, and to developing new connections between these two related groups.
    • His students are prominent researchers in algebra, mathematical logic, and computational complexity theory.
    • After finishing at MSU, the strongest of them transferred to positions in the Department of Mathematical Logic at MIAN, which under his leadership became one of the most prominent and respected research centres in logic.
    • In the Department of Mathematical Logic at MSU Adian has for many years led a seminar on algorithmic problems of algebra and logic, in addition to sharing leadership of the department's main seminar with V A Uspenskii.
    • Several times he has also given mandatory lecture courses in mathematical logic for the first and fourth years, and special lecture courses on algorithmic problems of algebra and on infinite periodic groups.
    • Adian is in essence the creator and leader of a whole research school in mathematical logic and algorithmic problems of algebra.
    • As long ago as the end of 1950s, S M Nikol'skii invited Adian, at the suggestion of Novikov, to edit the section on mathematical logic in Referativnyi Zhurnal: Matematika, the Russian mathematical review journal, because there was then a huge backlog of articles to be reviewed.
    • In the shortest possible time Adian rectified the situation there with respect to logic by mobilizing almost all his colleagues for the thankless task of writing reviews (for only a paltry fee).
    • At about the same time, he drew attention to the fact that a remarkable textbook on mathematical logic written by Novikov had not been published, and that undergraduate and graduate students had to read a typescript.
    • He declined Novikov's offer that he should be coauthor, and about half a year later the first edition of Novikov's textbook on mathematical logic appeared.
    • dissertations in mathematical logic, algebra, number theory, geometry, and topology, first as the vice-chairman and later, after the death of Vinogradov, as the chairman.

  11. Jevons biography
    • An important influence on Jevons while he was studying in London was De Morgan, not in terms of Jevons thoughts on economics but certainly in terms of his thoughts on logic and probability.
    • He was appointed to a second post in 1865 when he became a part-time professor of logic and political economy at Queen's College, Liverpool.
    • Then in 1866 he was appointed to a chair of political economy at Manchester and also to a professorship in logic and mental and moral philosophy.
    • He published Pure Logic in 1864, developed the 'logical piano' which was exhibited at the Royal Society in 1870, and he published The Theory of Political Economy in 1871.
    • Jevons's main contributions outside economics are in mathematical logic.
    • It was Boole, particularly with his book The Laws of Thought (1854), who strongly influenced Jevons' ideas on mathematical logic.
    • Logic 12 (1) (1991), 15-35.','Reference ',12)">12].
    • The article [Mathesis 7 (3) (1991), 351-362.','Reference ',11)">11] discusses the differences between Boole's and Jevons' concepts of logic.
    • Grattan-Guinness [Studies in the History of Statistics and Probability II (London, 1977), 180-212.','Reference ',10)">10] suggests that the main difference between their approach was that, although both believed they were studying the laws of thought, Boole had a more algebraic concept of logic while Jevons argued that mathematics proceeds from logic.
    • The 'logical piano', a machine designed by Jevons and constructed by a Salford watchmaker, had 21 keys for operations in equational logic.
    • Although its principal value was as an aid to the teaching of the new logic of classes and propositions, it actually solved problems with superhuman speed and accuracy ..
    • His most important book in logic was Principles of science (1874).
    • This work made important contributions to probability as well as to logic.
    • Logic 19 (2) (1998), 83-99.','Reference ',18)">18] takes a somewhat less positive attitude.
    • His mechanical reductionism was directed towards this project, and Jevons tried to found mathematics on logic through the development of a theory of number.

  12. Quine biography
    • From this time up to the break in his career for war service Quine's research was mainly on logic although always with a philosophical motivation.
    • One of the most important papers he published during this period was New Foundations for Mathematical Logic in the American Mathematical Monthly in 1937.
    • I taught Mathematical Logic and Set Theory as well as a general course of Logic in Philosophy.
    • challenged received notions of knowledge, meaning and truth, and exceeded even the extreme empiricism of logical positivism by arguing that logic and mathematics, like factual statements, are open to revision in the light of experience.
    • The totality of our so-called knowledge or beliefs, from the most casual matters of geography and history to the profoundest laws of atomic physics or even of pure mathematics and logic, is a man-made fabric which impinges on experience only along the edges.
    • We have commented above about Quine's work in mathematical logic.
    • Symbolic logic represented for him the framework for the language of science.
    • I do not do anything with computers, although one of my little results in mathematical logic has become a tool of the computer theory, the Quine McCluskey principle.
    • I arrived at it not from an interest in computers, but as a pedagogical device, a slick way of introducing that way of teaching mathematical logic.
    • Among Quine's publications are works on logic, metaphysics, the philosophy of language, and the philosophy of logic.
    • His 22 books include A System of Logistic (1934), Mathematical Logic (1940), Elementary Logic (1941), On What There Is (1948), From a Logical Point of View (1953), Word and Object (1960), Set Theory and Its Logic (1963), Philosophy of Logic (1970), The Time of My Life: an autobiography (1985), Quiddities (1990), and From Stimulus to Science (1995).

  13. Janovskaja biography
    • Logic 22 (3) (2001), 129-133.','Reference ',8)">8]):- .
    • She returned to Moscow in 1943 and was appointed Director of the Mathematical Logic Seminar at Moscow State University.
    • Janovskaja worked on the philosophy of mathematics and logic.
    • Her work in mathematical logic became very important in the development of the subject in Soviet Union.
    • In her writings on philosophy of mathematics and philosophy of logic, she took the offensive against the idealist philosophy of the bourgeois West, represented in her mind by Gottlob Frege, and against the so-called Machism, that is, conventionalism, represented by Rudolf Carnap and his Principle of Tolerance, according to which in logic one is free to choose one's rules.
    • Janovskaja published two major studies of the history of mathematical logic in the USSR between 1917 and 1957.
    • An important aspect of Janovskaja's work was in translating into Russian and editing works of high international repute in mathematical logic.
    • For example in 1947 she translated into Russian and published Hilbert and Ackermann's Grundzuge der theoretischen Logik Ⓣ, and, in 1948, Tarski's Introduction to logic and to the methodology of deductive sciences.
    • Logic 22 (3) (2001), 129-133.','Reference ',8)">8]:- .
    • She zealously reinforced mathematical logic as a self-sufficient and respectable science having nothing to do with either idealism or fideism in mathematics or philosophy of mathematics.
    • The article contains an extended general discussion of the mathematical method, and of such concepts of mathematical logic as the theory of plurality, the theory of algorithms, the law of full mathematical induction, recursive functions and Turing machines.
    • The article contains an appraisal of the work of Descartes in the light of contemporary standards of logic and rigour.
    • In 1959 she became the first Head of the new Department of Mathematical Logic at Moscow State University.
    • Logic 22 (3) (2001), 129-133.','Reference ',8)">8]:- .

  14. Shepherdson biography
    • Towards the end of his career his interests turned towards the computer language PROLOG and also to fuzzy logic.
    • Three papers written nearly ten years after he retired were on the topic of fuzzy logic and were all written jointly with Petr Hajek.
    • These three papers are The Liar Paradox and Fuzzy Logic (2000), Rational Pavelka Predicate Logic Is A Conservative Extension of Lukasiewicz Predicate Logic (2000), and A note on the notion of truth in fuzzy logic (2001).
    • Shepherdson proposed the setting up of the British Logic Colloquium in 1970, and then became a co-founder of the Colloquium with Robin Gandy.
    • His idea in setting up this Colloquium was to carry on the tradition in British mathematical logic which had been so well served by the great names of Bertrand Russell, Frank Ramsey and Alan Turing.
    • He served on the Science Research Council from 1968 to 1971, on the Association for Symbolic Logic being a Member of the Executive Committee for European Affairs 1966-72.
    • He also served on the Committee on Logic in East Asia from 1967 to 1970.
    • John's service to the University was great [but] to mathematical logic in the U.K.
    • it was an order of magnitude greater: he trained a number of students who went on to be logicians, and he made Bristol a place for research in logic to which many young logicians were attracted; so much so that it now makes for a roll-call of those involved in the subject in the U.K.
    • Over his 40 years in the department, a tradition of logic in Bristol was established, again, for example through a long-running M.Sc.
    • in Logic and the Theory of Computation.

  15. Frege biography
    • Frege was one of the founders of modern symbolic logic putting forward the view that mathematics is reducible to logic.
    • He lectured on all branches of mathematics, in particular analytic geometry, calculus, differential equations, and mechanics, although his mathematical publications outside the field of logic are few.
    • His writings on the philosophy of logic, philosophy of mathematics, and philosophy of language are of major importance.
    • In effect, it constitutes perhaps the greatest single contribution to logic ever made and it was, in any event, the most important advance since Aristotle.
    • He stated in the Preface to the work that he wanted to prove the basic truths of arithmetic "by means of pure logic".
    • This aim makes Frege the first to fully develop the main thesis of logicism, that mathematics is reducible to logic.
    • Frege then went on to give his own definitions of the basic concepts of arithmetic based purely on logic, and from these he deduced, again using pure logic, the basic laws of arithmetic.
    • a series of brilliant philosophical articles in which he elaborated his philosophy of logic.
    • In the longer term, however, Frege has become a major influence on the development of philosophical logic and the man who seems to have been largely ignored by his contemporaries has been avidly read by many in the second half of the twentieth century, particularly after his works were translated into English.
    • In 1923 Frege came to the conclusion that the aim he had set himself throughout most of his career, namely to found arithmetic on logic, was wrong.
    • His revolutionary new logic was the origin of modern mathematical logic - a field of import not only to abstract mathematics, but also to computer science and philosophy.

  16. Kleene biography
    • It had been Oswald Veblen who had proposed that the development of logic required careful analysis by mathematicians.
    • It was certainly an exciting place to be undertaking research applying mathematical techniques to logic with visitors such as Kurt Godel - Kleene attended a course he gave at the Institute for Advanced Study.
    • Kleene received a doctorate from Princeton for his thesis entitled A Theory of Positive Integers in Formal Logic in 1934.
    • we shall be concerned primarily with the development of the system of logic based on a set of postulates proposed by A Church.
    • By carrying out the construction on the basis of a certain subset of Church's formal axioms, we show that this portion at least of the theory of positive integers can be deduced from logic without the use of the notions of negation, class, and description.
    • Chapter I is by far the best introduction to intuitionistic logic which is at present available for a mathematical logician.
    • Kleene's best known books are Introduction to Metamathematics (1952) and Mathematical Logic (1967).
    • The aim of this book is to provide a connected introduction to the subjects of mathematical logic and recursive functions in particular, and to the newer foundational investigations in general.
    • Since the number of such texts in the field of logic is quite small, this book by an outstanding authority in the field is especially welcome.
    • Other honours included election to the National Academy of Sciences (1969), election as President of the Association for Symbolic Logic (1956-58), president of the International Union of the History and the Philosophy of Science (1961) and of the Union's Division of Logic, Methodology and Philosophy of Science (1960-62).
    • He was editor of the Journal of Symbolic Logic for twelve years.
    • He was an avid climber and, until well into his seventies, led the biannual logic picnic at Madison (now the Kleene Memorial Logic Picnic) on hikes up the cliffs at Devil's Lake.

  17. Russell biography
    • Bertrand Russell published a large number of books on logic, the theory of knowledge, and many other topics.
    • Over a long and varied career, Bertrand Russell made ground-breaking contributions to the foundations of mathematics and to the development of contemporary formal logic, as well as to analytic philosophy.
    • His contributions relating to mathematics include his discovery of Russell's paradox, his defence of logicism (the view that mathematics is, in some significant sense, reducible to formal logic), his introduction of the theory of types, and his refining and popularizing of the first-order predicate calculus.
    • The significance of the paradox follows since, in classical logic, all sentences are entailed by a contradiction.
    • In the eyes of many mathematicians (including David Hilbert and Luitzen Brouwer) it therefore appeared that no proof could be trusted once it was discovered that the logic apparently underlying all of mathematics was contradictory.
    • A large amount of work throughout the early part of this century in logic, set theory, and the philosophy and foundations of mathematics was thus prompted.
    • Although first introduced by Russell in 1903 in the Principles, his theory of types finds its mature expression in his 1908 article Mathematical Logic as Based on the Theory of Types and in the monumental work he co-authored with Alfred North Whitehead, Principia Mathematica (1910, 1912, 1913).
    • Of equal significance during this same period was Russell's defence of logicism, the theory that mathematics was in some important sense reducible to logic.
    • The first is that all mathematical truths can be translated into logical truths or, in other words, that the vocabulary of mathematics constitutes a proper subset of that of logic.
    • The second is that all mathematical proofs can be recast as logical proofs or, in other words, that the theorems of mathematics constitute a proper subset of those of logic.
    • In much the same way that Russell wanted to use logic to clarify issues in the foundations of mathematics, he also wanted to use logic to clarify issues in philosophy.
    • As one of the founders of "analytic philosophy", Russell is remembered for his work using first-order logic to show how a broad range of denoting phrases could be recast in terms of predicates and quantified variables.

  18. Tichy biography
    • He remained at the Charles University, being appointed a lecturer in the Department of Logic in 1961.
    • His first book Logic for Students of Pedagogical Institutes (Czech) was published in the following year.
    • In 1993 he was offered the position of Head of the Department of Logic at the Faculty of Philosophy and Arts in the Charles University of Prague.
    • Perhaps his most important work was the book The foundations of Frege's logic (1988).
    • This book is not an introduction to Frege's general philosophy, or a running commentary on Frege's work, or a polemic against recent commentators on Frege, or an attempt to give Frege a place in the history of logic and philosophy.
    • As a result, the author discusses a wide range of living philosophical issues in the latter half of his book: Church's logic of sense and denotation, Montague's intensional logic, Gentzen's sequent calculus, and Hilbert's formal axiomatics, to name a few.
    • Perhaps his most enduring claim to fame lies in his theory called Transparent Intensional Logic, the culmination of his extensive work on semantics and logic.
    • On the cover of [Pavel Tichy\'s Collected Papers in Logic and Philosophy (Otago-Praha, Otago UP, Filosofia, 2004).','Reference ',1)">1], Pavel Tichy's Collected Papers in Logic and Philosophy, this appreciation of Tichy's work is given:- .
    • He developed what he called Transparent Intensional Logic, a semantic theory within which to analyse both natural and artificial languages.
    • The theory remains one of the most inspiring and controversial doctrines of contemporary philosophical logic, attracting passionate defenders and equally fierce opponents.

  19. Lesniewski biography
    • At that time Jan Łukasiewicz was teaching at Lwow, being promoted from Privatdozent to extraordinary professor in 1911, and he greatly influenced Lesniewski in the first course on mathematical logic which he gave there.
    • He began to study formal logic and began to make strenuous attempts to understand Russell's paradox which he had learnt through Jan Łukasiewicz.
    • There he began to get more involved in the study of mathematical logic.
    • During this time Łukasiewicz and Lesniewski founded the Warsaw School of Logic.
    • From then until 1939 he published a series of twelve papers giving his theories of logic and mathematics.
    • These lectures are given in [Lesniewski\'s lecture notes in logic (Dordrecht, 1988).','Reference ',5)">5].
    • ','Reference ',15)">15] argues that the importance of Lesniewski's work is in providing an alternative to the classical approach to logic and the foundations of mathematics.
    • Lesniewski's contributions to logic concentrate on the structure of a sentence, and he argues for the traditional idea of a sentence as consisting of a subject, an object and a copula.
    • The three major logical systems which Lesniewski developed were: Protothetic, a theory of propositions and propositional functors, similar in power to a theory of propositional types, providing an extended propositional calculus with quantified functional variables; Ontology, which is an axiomatised theory of common names based on protothetic which may be characterised as a cross between traditional term logic and modern type theory, containing, besides singular terms, also empty and plural terms and a host of other interesting features; and Mereology, which is an axiomatic extension of ontology for a theory of classes quite different from set theory providing a formal theory of part and whole similar to the calculus of individuals.
    • Stanislaw Lesniewski was one of the co-founders of the Polish School of Logic and an author of a new and wholly original system of the foundations of logic and mathematics.
    • He was also the forerunner and originator of many ideas included as a matter of course in modern textbooks of logic and the foundations of mathematics.
    • Although Lesniewski played a considerable role during the period of development of modern mathematical logic and of the foundations of mathematics, his systems are not as well known as they deserve to be and the fact remains that his systems are not generally accepted as a tool in the foundational practice.

  20. Moisil biography
    • Algebra was not the only new research topic for Moisil during these years, for he became interested in logic after reading a paper by Jan Łukasiewicz.
    • It was a period in which he alternated his old interests in continuous mathematics with applications to mechanics and physics with his new interests in discrete mathematics, mainly in algebra and logic.
    • His first paper on algebra and logic was Recherches sur l'algebra de la logique Ⓣ (1935).
    • By non-chrysippian logic he means multivalued logic (non-aristotelian).
    • Moisil feels that the strictest standpoint in classical (formal) logic is represented by Chrysippus rather than Aristotle.
    • goal was to algebrize Łukasiewicz's logic.
    • Boolean algebras, algebraic models of classical logic, are particular cases of the new structures.
    • Moisil invented L-M algebras in order to create an algebraic structure playing the same role with respect to multiple-valued logic as Boolean algebras play with respect to classical, bivalent logic.
    • Extensive use is also made of n-valued logic.
    • Rings and ideals (Romanian) (1954); Time sequential operation of circuits with ideal relays (Romanian) (1960); Essays old and new on non-classical logic (Romanian) (1965); Algebraic theory of schemes with contacts and relays (Romanian) (1965); and Actual functioning of relay switching circuits.

  21. Sleszynski biography
    • The year 1909 was significant in another way, for it was the one in which he published his translation of Louis Couturat's famous book The algebra of logic.
    • This text had a major influence on the development of mathematical logic in Russia since it became the main textbook used by students of the subject over many years.
    • In 1919 he was promoted from extraordinary professor to become the full Professor of Logic and Mathematics the Jagellonian University.
    • Sleszynski's main work was on continued fractions, least squares and axiomatic proof theory based on mathematical logic.
    • Sleszynski assumes that the part of traditional logic created by Aristotle is a theory of relations which may hold between two classes.
    • We should mention another interesting work by Sleszynski, namely On the significance of logic for mathematics (Polish) published in 1923.
    • McCall writes in [Polish Logic, 1920-1939: Papers by Ajdukiewicz Andothers (Oxford University Press US, 1967).','Reference ',1)">1]:- .
    • Much indeed can be learned from the rich collection of [Sleszynski's] papers on various subjects in the realm of formal logic, and of mathematical logic and its history ..
    • Introduction to mathematical logic, complete proof, mathematical proof, exposition of the theory of propositions, the Boolean calculus, Grassmann's logic, Schroder's algebra, Poretsky's seven laws, Peano's doctrine, Burali-Forti's doctrine - these are some of the themes pursued in this work, from which I personally have learned a great deal and thanks to which I have got a clear idea of many an unclear thing.

  22. Smullyan biography
    • He wanted to learn about groups, rings and fields, the foundations of mathematics and mathematical logic.
    • Returning to New York from San Francisco, Smullyan studied mathematics and logic on his own and it was at this time that he began to compose chess puzzles.
    • Smullyan published Languages in which self reference is possible in the Journal of Symbolic Logic in 1957.
    • Smullyan's publications have been quite remarkable with the two outstanding books on retrograde analysis chess problems [The Chess Mysteries of the Sherlock Holmes (New York, 1979).','Reference ',2)">2] and [The Chess Mysteries of the Arabian Knights (New York, 1981).','Reference ',3)">3], a whole series of marvellous popular puzzle books such as [What is the name of this book? (New York, 1978).','Reference ',1)">1] and [Satan, Cantor, and Infinity and other mind-boggling puzzles (New York, 1992).','Reference ',4)">4], and some books on the foundations of mathematics and mathematical logic which are in many ways in a class of their own.
    • Beginning with fun-filled monkey tricks and classic brain-teasers with devilish new twists, Professor Smullyan spins a logical labyrinth of even more complex and challenging problems as he delves into some of the deepest paradoxes of logic and set theory, including Godel's revolutionary theorem of undecidability.
    • The most original, most profound and most humorous collection of recreational logic and mathematics problems ever written.
    • We have mentioned one of his books on mathematical logic above.
    • He published another text First-order logic in 1968:- .
    • This book deals primarily with the proofs of, and the interconnections between, various formulations of the completeness theorem for first-order logic.
    • for the general mathematician, philosopher, computer scientist and any other curious reader who has at least a nodding acquaintance with the symbolism of first-order logic, and who can recognize the logical validity of a few elementary formulas.
    • A standard one-semester course in mathematical logic is more than enough for the understanding of this volume.
    • I watched him teach a graduate level logic course, as he lurched to the blackboard (where he writes in a serviceable hand and in complete sentences) and paced about his desk, fidgeting and chuckling.

  23. MacColl biography
    • This move seems to have been well thought out for, not only was Boulogne a prosperous town with much to offer the well-educated, it also had the advantage of having close links with England so MacColl could in many ways have the best of both worlds [History and Philosophy of Logic 22 (2) (2001), 81-98.',2)">2]:- .
    • In 1883 he wrote to C S Peirce saying something about his financial problems (see [History and Philosophy of Logic 22 (2) (2001), 81-98.',2)">2]):- .
    • HUGH M'COLL, BA (LONDON UNIVERSITY) Gives Lessons in MATHEMATICS, CLASSICS, ENGLISH, LOGIC, with all the other subjects of the University Course, and prepares Young Gentlemen for the Naval and military Examinations.
    • Most of MacColl's original contributions to mathematics and logic were through papers, discussions and books after he moved to France.
    • He wrote to Bertrand Russell in 1905 about this work (see for example [History and Philosophy of Logic 22 (2) (2001), 81-98.',2)">2]):- .
    • This I thought would be my final contribution to logic or mathematics, ..
    • He published a series of nine papers entitled Symbolic Logic in The Athenaeum between 1903 and 1904, and a series of eight articles Symbolic Reasoning in Mind between 1880 and 1906.
    • His most famous book, however, was Symbolic Logic and Its Applications (1906).
    • For reaching those who do not follow closely the development of symbolic logic, publication in book form is almost essential; and it is much to be hoped that this book will be widely read.
    • The present work is not quite in line with those of other current writers on symbolic logic; but it has merits which most of their work do not have, and it serves in any case to prevent the subject from getting into a groove.
    • MacColl has other interests outside mathematics and logic.
    • He also wrote Man's Origin, Destiny and Duty (1909) in which he discusses his [Nordic Journal of Philosophical Logic 3 (1), 175-196.',4)">4]:- .

  24. Mostowski biography
    • He entered Warsaw University after graduating from the Stefan Batory Gymnasium and it was at this time that he became especially interested in the foundations of mathematics, particularly mathematical logic and set theory.
    • was awarded in February 1939 for his thesis On the Independence of Finitenesss Definitions in a System of Logic, officially directed by Kuratowski but in practice directed by Tarski who was a young lecturer at that time.
    • Mostowski wrote an important Polish text Mathematical Logic which was published in 1948.
    • This is an exceptionally good handbook of mathematical logic.
    • It covers a great deal of material and shows many applications of mathematical logic to mathematical problems ..
    • The author is eager to persuade a mathematician that logic can be useful in his work.
    • The symbolism of mathematical logic is used throughout, but with moderation, and ample motivation is given in the text appealing to the intuition on the infinite sets.
    • Lectures on the development of mathematical logic and the study of the foundations of mathematics in 1930-1964 which reports on a 16 lecture course given by Mostowski in the summer of 1964 in Vaasa, Finland, gives the best view of what he saw as the most important developments in his subject during his career.
    • The topics include completeness, incompleteness, decidability, and undecidability theorems; computability, recursive functions, hierarchies, and functionals; intuitionistic logic and its interpretations; constructive mathematics, foundations of set theory, including Cohen's independence proofs; and finally model theory, ending in a special chapter on direct and reduced products.
    • He was the editor of the Mathematical, Astronomical and Physical series of the Bulletin of the Polish Academy of Sciences, on the editorial board of several journals including Fundamenta Mathematicae, Dissertationes Mathematicae, the Journal of Symbolic Logic and Studia Logica.
    • He was one of the founders and editors of the Annals of Mathematical Logic.

  25. Padoa biography
    • (Before Padoa became a teacher at the High School "Cristoforo Colombo" in Genoa, Vitali had taught there from 1904 to 1923.) In 1932 Padoa was appointed as a lecturer in Mathematical Logic at the University of Genoa and he held this post until 1936 when he reached the age limit for lecturers and was forced to retire.
    • This was certainly not the first time that Padoa had attempted to leave school teaching and become a university lecturer; he had applied unsuccessfully for a lectureship in Mathematical Logic in 1901, a lectureship in Descriptive Geometry in 1909, and a lectureship in Theoretical Philosophy in 1912.
    • After he was appointed at the staff of the University of Genoa, he taught courses on 'Mathematical Logic' (1932-34), 'Ideographic Logic' (1934-37), and 'Descriptive Geometry' (1935-36).
    • He belonged to Peano's school of mathematical logic, popularising this type of work.
    • Various scientists from all countries who have adopted the ideographic logic believe that it is refined and completed today.
    • Continuing his work as a collaborator and adviser, M Padoa has led everyone to appreciate the simplicity and power of the ideographic language - which has given birth to a new development in deductive logic and a further analysis of the different branches of mathematics - and to consult profitably and easily the many books in which it is applied.
    • This admirable exposition of the mathematical logic of Peano and his school was given in the form of lectures delivered under the auspices of the University of Geneva, and was published in the 'Revue de Metaphysique et de Morale' for 1912.
    • Cooper Harold Langford writes [The Journal of Symbolic Logic 2 (1) (1937), 57.','Reference ',8)">8]:- .
    • Logic, Padoa says, is not in a particularly fortunate position.
    • The paper contains an outline of the fundamental ideas introduced into logic by Peano, together with examples illustrating how the distinctions Peano drew remove confusions and paradoxes.

  26. Davis biography
    • Davis took a course on symbolic mathematical logic in his first year and a reading course on real variable theory with Emil Post in his second year.
    • In his third year he began taking a reading course with Post on mathematical logic but this stopped when Post had a breakdown.
    • in 1948 he knew he wanted to undertake research in mathematical logic.
    • He had already written a report as part of an advanced logic course he had taken in his final year and this was later incorporated into his Ph.D.
    • There he taught a logic course which had computability as one of the main topics.
    • The 1958 paper with Hilary Putnam was the result of their families sharing a house while at a five-week logic conference at Cornell in the summer of 1957.
    • Symbolic Logic 3 (4) (1958), 432-433.','Reference ',12)">12]:- .
    • We noted above that Computability and unsolvability started life as a lecture course while Davis's books Lecture notes on mathematical logic (1959) and Computability (1974) are actually lecture courses.
    • The road from Leibniz to Turing which was reprinted in the following year under the title Engines of logic.
    • Symbolic Logic 7 (1) (2001), 65-66.','Reference ',2)">2]:- .
    • This book surveys developments in mathematics and logic that led the way to the design and construction of modern digital computers.

  27. Cohen biography
    • Symbolic Logic 15 (4) (2009), 439-440.','Reference ',14)">14]:- .
    • Cohen, through these friendships, had also begun to take an interest in logic [Bull.
    • Symbolic Logic 15 (4) (2009), 439-440.','Reference ',14)">14]:- .
    • As a graduate student Cohen's connection with logic were his friendships with a lively group of students who became logicians; Michael Morley, Anil Nerode, Bill Howard, Ray Smullyan, and Stanley Tennenbaum.
    • For a while he lived in Tennenbaum's house and absorbed logic by osmosis, for there were no courses in logic in the Chicago mathematics department.
    • A dramatic aspect of the continuum hypothesis work is that Cohen was a self-taught outsider in logic.
    • In addition it presents also the main classical results in logic and set theory.
    • For epoch-making results in mathematical logic which have enlivened and broadened investigations in the foundation of mathematics.
    • Like many great mathematicians, his mathematical interests and contributions were very broad, ranging from mathematical analysis and differential equations to mathematical logic and number theory.
    • Symbolic Logic 14 (3) (2008), 351-378.','Reference ',11)">11]:- .

  28. Skolem biography
    • [Skolem] conducted the regular graduate courses in algebra and number theory, and rather infrequently lectured on mathematical logic.
    • Skolem was remarkably productive publishing around 180 papers on topics such as Diophantine equations, mathematical logic, group theory, lattice theory and set theory.
    • However, as Fenstadt explains in [Thoralf Skolem, Selected works in logic (Oslo, 1970).','Reference ',2)">2]:- .
    • Logic 1 (2) (1996), 107-117.','Reference ',9)">9], sees Skolem as being a pioneer in computer science:- .
    • Skolem's two systems could be considered as a programming language for defining objects and a programming logic for proving properties about the objects.
    • Hao Wang, in A survey of Skolem's work in logic which appears in [Thoralf Skolem, Selected works in logic (Oslo, 1970).','Reference ',2)">2] writes:- .
    • Wang also indicates how useful it is to read Skolem's original papers [Thoralf Skolem, Selected works in logic (Oslo, 1970).','Reference ',2)">2]:- .
    • Nordic Journal of Philosophical Logic .

  29. Ockham biography
    • Peter Lombard, a conservative theologian, wrote the text as a reaction against some who at the time were applying Aristotle's logic to theology.
    • Ockham lectured on logic and natural philosophy in a Franciscan school from 1321 to 1324 while he waited to return to university to study for his doctorate.
    • During these years he wrote many deep works on philosophy and logic.
    • The logic was clear to Ockham; Pope John XXII was no true pope and he denounced him with written charges.
    • Ockham had convinced other leading Franciscans of the logic of his arguments, and together they fled to Pisa on 26 May 1328.
    • One might argue that it was a pity that he became distracted from his work on philosophy and logic during these latter years.
    • In his studies of mathematical logic Ockham made important contributions to it which are significant today.
    • He considered a three valued logic where propositions can take one of three truth values.
    • And from the standpoint of the philosophy of the 1980s and 1990s, Ockham's interest in terminist logic, linguistic theory, and semiotics has placed him in the forefront of those medieval thinkers used as sources in contemporary philosophical discussion.

  30. Peano biography
    • In 1888 Peano published the book Geometrical Calculus which begins with a chapter on mathematical logic.
    • at once a landmark in the history of mathematical logic and of the foundations of mathematics.
    • In 1891 Peano founded Rivista di matematica, a journal devoted mainly to logic and the foundations of mathematics.
    • The first paper in the first part is a ten page article by Peano summarising his work on mathematical logic up to that time.
    • Of course it was the precision of his thinking, using the exactness of his mathematical logic, that gave Peano this clarity of thought.
    • Such a collection, which would be long and difficult in ordinary language, is made noticeably easier by using the notation of mathematical logic ..
    • As the days went by, I decided that this must be owing to his mathematical logic.
    • Although Peano is a founder of mathematical logic, the German mathematical philosopher Gottlob Frege is today considered the father of mathematical logic.

  31. Kuroda biography
    • In many ways Kuroda's work on the foundations of mathematics and mathematical logic had made it difficult for him to become a central figure in Japanese mathematics.
    • Over the following years he published papers such as Intuition and consistency in mathematics (Japanese) (1947), On the logic of Aristotle and the logic of Brouwer (Japanese) (1948), and Intuitionistische Untersuchungen der formalistischen Logik Ⓣ (1951).
    • Symbolic Logic 21 (2) (1956), 196.','Reference ',3)">3]:- .
    • illustrates intuitionistic points of view towards mathematics, and investigates intuitionistic logic as formalized by Heyting-Gentzen.
    • In fact he makes it clear in the 1951 paper that he shares L E J Brouwer's view that mathematics is an activity of thought that is independent of logic and based on immediate evidence that is intuitively clear.
    • Therefore, in formulating these systems, some special conditions to restrict the free application of logic are needed, for instance, simple or ramified type theory, introduced to logic first by Russell, or the restriction of the comprehension axiom of set theory.
    • Despite his many deep works on mathematical logic and the foundations of mathematics, Kuroda had a lifelong interest in algebraic number theory.

  32. Lob biography
    • It seems an unlikely palace for Lob to be able to study mathematics and logic at an advanced level, but one must remember that professional mathematicians were also interned in the camp and were keen to teach.
    • He remained at Leeds for 20 years being promoted to Reader, then to Professor of Mathematical Logic.
    • At Leeds Lob established the Leeds Logic Group, one of three international centres for Mathematical Logic in Britain.
    • Examples of papers by Lob in the Journal of Symbolic Logic in the 1950s are Concatenation as basis for a complete system of arithmetic (1953), Solution of a problem of Leon Henkin (1955), and Formal systems of constructive mathematics (1956).
    • Examples of papers he published in the 1970s are A model theoretic characterization of effective operations (1970), Hierarchies of number-theoretic functions (1970), A reduction theorem for predicate logic (1972) and Embedding first order predicate logic in fragments of intuitionistic logic (1976).

  33. Szmielew biography
    • She entered the University of Warsaw in 1935 and studied mathematical logic.
    • A fellow student of mathematical logic at that time was Andrzej Mostowski.
    • The authors of [Alfred Tarski: Life and Logic (Cambridge University Press, Cambridge, 2004).','Reference ',1)">1] related the events that followed:- .
    • He welcomed the change at first - at fifteen he was happy to have a secluded place to himself- but he was not happy when it became clear that the rapport between Alfred and Wanda went far beyond a shared interest in mathematical logic.
    • Dana Scott, reviewing Szmielew's paper in [The Journal of Symbolic Logic 24 (1) (1959), 59.','Reference ',10)">10], writes:- .
    • In 1956 Tarski visited Szmielew, Borsuk and Mostowski in Warsaw [Alfred Tarski: Life and Logic (Cambridge University Press, Cambridge, 2004).','Reference ',1)">1]:- .
    • The authors of [Alfred Tarski: Life and Logic (Cambridge University Press, Cambridge, 2004).','Reference ',1)">1] write that:- .
    • Szmielew's daughter Aleksandra, asked about her mother's character, said she was [Alfred Tarski: Life and Logic (Cambridge University Press, Cambridge, 2004).','Reference ',1)">1]:- .

  34. Venn biography
    • In 1862 he returned to Cambridge University as a lecturer in Moral Science, studying and teaching logic and probability theory.
    • He had already become interested logic, philosophy and metaphysics, reading the treatises of De Morgan, Boole, John Austin, and John Stuart Mill.
    • Venn extended Boole's mathematical logic and is best known to mathematicians and logicians for his diagrammatic way of representing sets, and their unions and intersections.
    • Venn wrote Logic of Chance in 1866 which Keynes described as:- .
    • Venn published Symbolic Logic in 1881 and The Principles of Empirical Logic in 1889.
    • probably his most enduring work on logic.
    • Venn's interest turned towards history and he signalled this change in direction by donating his large collection of books on logic to the Cambridge University Library in 1888.

  35. Lyndon biography
    • He began working on mathematical logic, moved to a study of relational algebras, then finally undertook research in homological algebra.
    • Combinatorial group theory was not Lyndon's first book, however, for he had published Notes on logic ten years earlier in 1966.
    • The book contains a concise and clear exposition of basic notions of mathematical logic.
    • More specifically, the author deals with the first-order logic with symbols for relations and functions.
    • As should be clear from this description, the book is intended as a text to enable a mathematically well-trained reader to acquaint himself quickly with the basic results of logic.
    • In the reviewer's opinion the book has set a new standard for texts in mathematical logic which will not be easy to supersede.
    • They include: Groups rings and dimension subgroups; Two investigations on the borderline of logic and algebra; Decision problems of finite automata design and related arithmetic; On Dehn's algorithm and the conjugacy problem; Projectivities of free products; Continuous model theory and set theory; Real length functions in groups; Automorphisms of the fundamental group of an orientable 2-manifold; Some algorithmic problems for semigroups; and Groups acting on trees.

  36. Hay biography
    • believed in teaching the logic of the subject rather than just following the theorem-proof format of the text (which of course was straight out of Euclid).
    • In fact the constraints were more severe than this, for ever since her teacher Rosenbaum had turned her on to mathematical logic, she had wanted to undertake research in that topic.
    • It was even harder to find a university with strong graduate schools in both experimental psychology and mathematical logic.
    • course she had started, but no university within travelling distance offered mathematical logic at doctoral level.
    • She published papers on mathematical logic, recursive function theory, and theoretical computer science.
    • She published Axiomatization of the infinite-valued predicate calculus in the Journal of Symbolic Logic in 1963 in which she gave a set of nine axiom schemes and two rules for the predicate calculus based on the infinite-valued sentential calculus of Łukasiewicz.
    • While Louise Hay was widely recognized for her contributions to mathematical logic and for her strong leadership as head of the Department of Mathematics, Statistics, and Computer Science at the University of Illinois at Chicago, her devotion to students and her lifelong commitment to nurturing the talent of young women and men secure her reputation as a consummate educator.

  37. Boole biography
    • In 1854 he published An investigation into the Laws of Thought, on Which are founded the Mathematical Theories of Logic and Probabilities.
    • Boole approached logic in a new way reducing it to a simple algebra, incorporating logic into mathematics.
    • It began the algebra of logic called Boolean algebra which now finds application in computer construction, switching circuits etc.
    • I am now about to set seriously to work upon preparing for the press an account of my theory of Logic and Probabilities which in its present state I look upon as the most valuable if not the only valuable contribution that I have made or am likely to make to Science and the thing by which I would desire if at all to be remembered hereafter ..
    • Boole's system of logic is but one of many proofs of genius and patience combined.
    • That the symbolic processes of algebra, invented as tools of numerical calculation, should be competent to express every act of thought, and to furnish the grammar and dictionary of an all-containing system of logic, would not have been believed until it was proved.

  38. Jungius biography
    • where he commented on the Dialectic of Peter Ramus, as well as writing on logic and composing poetry.
    • In general, however, he preferred to concentrate on mathematics and logic.
    • The Logica Hamburgensis (1638) of Jungius, composed for the use of pupils at the Akademisches Gymnasium, presented late medieval theories and techniques of logic.
    • Joachim Jungius was a German polymath somewhat in the Leibnizian tradition of encyclopaedic range as well as originality in several domains of knowledge, including logic, mathematics, and natural philosophy.
    • shifted: from his early training in the late scholasticism of Francisco Suarez and his like, to astronomy, logic and mathematics, educational reform, medicine, and chemical philosophy (in his case corpuscularism), to an ambitious program to organise, systematise, and taxonomise - as well as further to contribute to (based on a mathematical paradigm) - the sum total of human knowledge.
    • Kangro's most fortunate discovery, in his own estimation, was a fragment that reflects Jungius's fundamental ideas on logic applied to science.
    • These three methods replaced the prevalent ancient form of logic based on the syllogism.

  39. Herbrand biography
    • He had already set himself the goal of creating a school of mathematical logic in France to rival that in Gottingen.
    • For his doctoral thesis he studied mathematical logic under the supervision of Ernest Vessiot who at that time was the Director of the Ecole Normale.
    • On the one hand mathematical logic was a surprising choice, given the lack of interest in that topic in France in this period.
    • Vessiot played an important role here, convincing analysts somewhat sceptical about the importance of mathematical logic.
    • He made contributions to mathematical logic where Herbrand's theorem on the theory of quantifiers appears in his doctoral thesis.
    • Herbrand's theorem establishes a link between quantification theory and sentential logic which is important in that it gives a method to test a formula in quantification theory by successively testing formulas for sentential validity.
    • Thus, the lifelong interest he had in mathematical logic was perhaps born of the wondrous hope of anticipating the future by means of finite calculi and combinations.

  40. Peirce Charles biography
    • By the age of twelve Charles was reading standard university level texts on logic, and in the following year he began reading Immanuel Kant's Critique of Pure Reason.
    • He gave the Harvard lectures on The Logic of Science in the spring of 1865 and the Lowell Institute lectures on The Logic of Science; or Induction and Hypothesis in the latter part of 1866.
    • Although his work had been wide ranging in the sciences, he had always been interested in philosophy and logic and, in 1879, he was appointed as Lecturer in Logic in the Department of Mathematics at Johns Hopkins University.
    • He then extended his father's work on associative algebras and worked on mathematical logic, topology and set theory.
    • In 1877 and 1878 Peirce published six essays on Illustrations of the Logic of Science in the Popular Science Monthly.

  41. Rasiowa biography
    • Her thesis, presented in 1950, was on algebra and logic Algebraic treatment of the functional calculus of Lewis and Heyting and these topics would be the main areas of her research throughout her life.
    • She led the Foundations of Mathematics Section from 1964 and the Mathematical Logic Section after its creation in 1970.
    • Her main research was in algebraic logic and the mathematical foundations of computer science.
    • In algebraic logic she continued work by Post, Stone, Tarski and Łukasiewicz [Bull.
    • Of course Rasiowa's work on algebraic logic was in precisely the right area to make her a natural contributor to theoretical computer science.
    • Her contribution to theoretical computer science stems from her conviction that there are deep relations between methods of algebra and logic on the one side and essential problems of foundations of computer science on the other.
    • It was partly through her endeavours that the Polish Society for Logic and the Philosophy of Science was set up.

  42. Post biography
    • While studying at the College of the City of New York he studied mathematics but there is little sign that at this stage he was particularly attracted towards logic.
    • thesis was on mathematical logic, and we shall discuss it further in a moment, but first let us note that Post wrote a second paper as a postgraduate, which was published before his first paper, and this was a short work on the functional equation of the gamma function.
    • The final, and perhaps the most remarkable, new idea which Post introduced in his thesis was to give a framework for systems of logic as inference systems based on a finite process of manipulation of symbols.
    • Such a system of logic that Post proposed produces, in today's terminology, a recursively enumerable set of words on a finite alphabet.
    • it is fundamentally weak in its reliance on the logic of Principia Mathematica ..
    • He also made a mathematical study of Łukasiewicz's three-valued logic.
    • but he said there are limitations and symbolic logic is:- .

  43. Aristotle biography
    • Aristotle was not primarily a mathematician but made important contributions by systematising deductive logic.
    • According to a tradition which arose about two hundred and fifty years after his death, which then became dominant and even today is hardly disputed, Aristotle in these same years lectured - not once, but two or three times, in almost every subject - on logic, physics, astronomy, meteorology, zoology, metaphysics, theology, psychology, politics, economics, ethics, rhetoric, poetics; and that he wrote down these lectures, expanding them and amending them several times, until they reached the stage in which we read them.
    • What do these works contain? There are important works on logic.
    • Aristotle believed that logic was not a science but rather had to be treated before the study of every branch of knowledge.
    • Aristotle's name for logic was "analytics", the term logic being introduced by Xenocrates working at the Academy.
    • Aristotle believed that logic must be applied to the sciences [Aristotle (Oxford, 1982).','Reference ',6)">6]:- .

  44. Chrysippus biography
    • Chrysippus was one of the first to organise propositional logic as an intellectual discipline.
    • Diogenes Laertius in [Lives of eminent philosophers (New York, 1925).',3)">3] lists 118 works on logic by Chrysippus, and of these 118 there are seven books occupying 15 rolls of papyri concerning the Liar Paradox.
    • One claim which Chrysippus made in the area of logic was to reject that the impossible does not follow from the possible.
    • Logic 13 (2) (1992), 133-148.',8)">8] which also examines more generally his views on modal logic.
    • He believed that logic and physics are necessary to differentiate between good and evil.
    • To him the value of physics and logic is mainly for this purpose.

  45. Johnson biography
    • During these nineteen years of holding temporary positions he published three papers on Boolean logic and one on probability.
    • However he taught logic and mathematical economics, publishing important works in both areas.
    • In the area of logic he published papers such as The logical calculus (1892) and Analysis of thinking (1918), both of which appeared in Mind.
    • He is most famed, however, for Logic (1921, 1922, 1924), a work that one of his students persuaded him to publish.
    • Logic is in three volumes, the fourth on probability was never finished, but the parts which were written were published in Mind after his death.
    • Logic won him considerable fame leading to his election as a fellow of the British Academy (1923) and to the award of honorary degrees from universities such as Manchester (1922) and Aberdeen (1926).

  46. Wittgenstein biography
    • Wittgenstein left his aeronautical research in Manchester in 1911 to study mathematical logic with Russell in Trinity College, Cambridge.
    • During this period at Cambridge, Wittgenstein continued to work on the foundations of mathematics and also on mathematical logic.
    • Despite Russell's attempts to stop him, Wittgenstein went to Skjolden in Norway and this proved an extremely fruitful period during which lived in isolation working on his ideas on logic and language that would form the basis of his great work the Tractatus Logico-Philosophicus.
    • During these four years of active service Wittgenstein had written his great work in logic, the Tractatus, and the manuscript was found in his rucksack when he was taken prisoner.
    • Although Wittgenstein had not wished to return to academic life during this period he was not completely isolated from the study of mathematical logic, the foundations of mathematics, and philosophy.
    • In the following years Wittgenstein lectured there on logic, language, and the philosophy of mathematics.

  47. Shelah biography
    • Symbolic Logic 38 (4) (1973), 648-649.','Reference ',1)">1]:- .
    • He then returned to the Hebrew University of Jerusalem where he became a professor in 1974 before being appointed to the A Robinson Chair for Mathematical Logic in 1978, a position he continues to hold.
    • for his many fundamental contributions to mathematical logic and set theory and their applications within other parts of mathematics.
    • Saharon Shelah has for many years been the leading mathematician in the foundations of mathematics and mathematical logic.
    • Symbolic Logic 54 (2) (1989), 633-635.','Reference ',7)">7]:- .
    • This may seem to be a rather strange invariant to consider, but it was quite natural, as set-theoretic language and concerns permeate logic.

  48. Robinson biography
    • As an undergraduate at the Hebrew University Robinson has been interested in both algebra and mathematical logic.
    • By now Robinson was a world leading authority in aerodynamics yet he continued with his interest in mathematical logic.
    • Robinson is best known, however, for his work on mathematical logic.
    • He was deeply concerned with most forms of human culture and creativity, and on all he could converse with the fascinating combination of logic, insight and knowledge that characterised his mathematics.
    • When one considers the wealth, profundity and diversity of his interests and the continuous interplay in his thinking of pure mathematics, applied mathematics, logic and philosophy one is constantly reminded of Leibniz to whom he felt a natural affinity and for whom he had the deepest admiration.
    • Today I should like to offer a partial answer: It cannot be a wholly bad worlds in which an Abraham Robinson could live and think; in which his wife and friends are able to cherish his memory; and in which his life's work will be remembered as long as logic, mathematics and philosophy matter to mankind.

  49. Couturat biography
    • He remained in Caen until 1899 when he moved to Paris, again taking leave of absence, to continue his research on the logic of Leibniz.
    • logic was not only the heart and soul of his system, but the centre of his intellectual activity, the source of all his discoveries, ..
    • The topics covered in the correspondence include: the foundations of geometry, extension versus intension in logic, the Russell paradox, the axiom of choice, the controversies with Poincare, logic, Leibniz, Peano, Kant, arithmetical induction, mathematical existence, politics, international language, and some personal matters.
    • In 1905 Couturat became Henri-Louis Bergson's assistant at the College de France, working on the history of logic during the academic year 1905-06.
    • Outside of France he is generally known for his Leibniz studies, but he was distinctly a philosopher in his own right, with a central interest in mathematical logic.

  50. Gentzen biography
    • M E Szabo writes in [Studies in Logic (Amsterdam, 1969).','Reference ',2)">2]:- .
    • As we have mentioned, Gentzen's work was on logic and the foundations of mathematics.
    • He introduced the notion of 'logical consequence' which provided a logic closer to mathematical reasoning than the systems proposed by Frege, Russell and Hilbert.
    • In a paper published in Mathematische Zeitschrift in 1935 Gentzen introduced two new versions of predicate logic now called the N-system and the L-system.
    • In the following year he gave a consistency proof in terms of an N-type logic for the system S of arithmetic with induction.
    • he was hoping to be able to return to Gottingen and devote himself fully to the study of mathematical logic and the foundations of mathematics.

  51. Kalicki biography
    • Financed by a British Council Scholarship for two years, Kalicki studied at the University of London, receiving his doctorate in mathematical logic in July 1948.
    • Kalicki worked on logical matrices and equational logic and published 13 papers on these topics from 1948 until his death five years later.
    • In 1950 he defined an algebra binary connective D on truth tables in Note on truth-tables published in the Journal of Symbolic Logic.
    • The next paper in the same part of the Journal of Symbolic Logic is also by Kalicki.
    • Two years later another paper by Kalicki in the Journal of Symbolic Logic is A test for the equality of truth-tables which gives a necessary and sufficient condition for the equivalence of two truth-tables.
    • He had unbounded confidence that almost anybody could be brought to an understanding of the fundamental results of modern logic if only he, Kalicki, could find the right words.

  52. Fabri biography
    • He made a brilliant start to his career, teaching logic, philosophy and natural philosophy in Arles.
    • He was professor of logic in Aix-en-Provence for a year from 1638 during which time he became the leader of a group of scientists and became a life-long friend of Pierre Gassendi.
    • For six years from 1640, Fabri was professor of logic and mathematics at the College de la Trinite in Lyon.
    • There he taught logic, mathematics, natural philosophy, metaphysics and astronomy [Dictionary of Scientific Biography (New York 1970-1990).
    • The logic lectures he gave in Lyon were published by his student Pierre Mousnier in 1646 as Philosophiae tomus primus Ⓣ, and the lectures he gave on natural philosophy were published in the same year as Tractatus physicus de motu locali Ⓣ.

  53. Dodgson biography
    • He published The Game of Logic in 1887 and Symbolic Logic Part I in 1896.
    • Symbolic Logic Part I contains the statement that:- .
    • By 1896 Dodgson had developed a mechanical test of validity for a large part of the logic of terms, an achievement usually credited to Leopold Lowenheim [in 1915].
    • As early as 1894 Dodgson used truth tables for the solution of specific logic problems.

  54. Malcev biography
    • Malcev's first publications were on logic and model theory and resulted from work he had begun entirely on his own.
    • Malcev also studied Lie groups and topological algebras, producing a synthesis of algebra and mathematical logic.
    • In 1960, Malcev was appointed to a chair in mathematics at the Mathematics Institute at Novosibirsk and to be chairman of the Algebra and Logic Department at Novosibirsk State University.
    • Malcev founded the Siberian section of the Mathematics Institute of the Academy of Sciences, a logic-algebraic school with many members, and directed the world famous seminar Algebra i Logika.
    • We mentioned some of the prizes he received above, such as the State Prize in 1946, but another important honour which he received in 1964 was a Lenin Prize for his series of papers on the applications of mathematical logic to algebra.

  55. Poincare biography
    • The first point to make is the way that Poincare saw logic and intuition as playing a part in mathematical discovery.
    • It is by logic we prove, it is by intuition that we invent.
    • Logic, therefore, remains barren unless fertilised by intuition.
    • But to explain what "intuition" was in mathematics, Poincare fell back on saying it was the part which did not follow by logic:- .
    • something other than pure logic is necessary.

  56. Ramsey biography
    • In this work he accepted the claim by Russell and Whitehead made in the Principia Mathematica that mathematics is a part of logic.
    • Ramsey published Mathematical Logic in the Mathematical Gazette in 1926.
    • He also criticises Hilbert in Mathematical Logic saying that he had attempted to reduce mathematics to:- .
    • His second paper on mathematics On a problem of formal logic was read to the London Mathematical Society on 13 December 1928 and published in the Proceedings of the London Mathematical Society in 1930.
    • Being a friend of Keynes certainly did not stop Ramsey attacking Keynes' work, however, and in Truth and probability , which Ramsey published in 1926, he argues against Keynes' ideas of an a priori inductive logic.

  57. Karp biography
    • Her doctoral thesis was on mathematical logic.
    • Karp was a mathematical logician but, as noted in [Infinitary Logic : In memoriam Carol Karp (Berlin - Heidelberg - New York, 1975).','Reference ',1)">1], her work was closely related to algebra:- .
    • She lectured on this later work as described in [Infinitary Logic : In memoriam Carol Karp (Berlin - Heidelberg - New York, 1975).','Reference ',1)">1]:- .
    • Karp did give lectures at Maryland in the Fall of 1970 on infinitary logic and recursion theory.
    • Again quoting from [Infinitary Logic : In memoriam Carol Karp (Berlin - Heidelberg - New York, 1975).','Reference ',1)">1]:- .

  58. Specker biography
    • In 1953 he published The axiom of choice in Quine's New Foundations for Mathematical Logic.
    • Specker's contributions to mathematics are reviwed in [Logic and algorithmic, Zurich, 1980 (Univ.
    • Geneve, Geneva, 1982), 11-24.','Reference ',5)">5] where his 32 publications up to 1979 are divided into 10 categories: topology, recursive analysis, combinatorial set theory, type theory, axiomatic set theory, Ramsey's theorem, arithmetic, logic of quantum mechanics, algorithms, and miscellaneous.
    • Examples of his later papers are The fundamental theorem of algebra in recursive analysis (1969), Die Entwicklung der axiomatischen Mengenlehre Ⓣ (1978), (with H Kull) Direct construction of mutually orthogonal Latin squares (1987), Application of logic and combinatorics to enumeration problems (1988).
    • Ernst Specker has made decisive contributions towards shaping directions in topology, algebra, mathematical logic, combinatorics and algorithms over the last 40 years.

  59. Albert biography
    • Buridan taught natural philosophy and logic at the University of Paris during the first half of Albert's time there.
    • His books on logic are his best, particularly when he examined logical paradoxes.
    • The Perutilis Logica is a logic handbook consisting of six parts: Propositions; Properties of Terms; Type of Propositions; Consequences and Syllogisms; Fallacies; and Insolubles and Obligations.
    • Albert of Saxony's teachings on logic and metaphysics were extremely influential.
    • Although Buridan remained the predominant figure in logic, Albert's 'Perutilis logica' was destined to serve as a popular text because of its systematic nature and also because it takes up and develops essential aspects of the Ockhamist position.

  60. Kalmar biography
    • There he became interested in mathematical logic and this was a field in which he was to make major contributions.
    • This was not all he founded at Szeged, for he also set up the Cybernetic Laboratory and the Research Group for Mathematical Logic and Automata Theory.
    • He worked on mathematical logic solving certain cases of the decision problem for the first order predicate calculus, simplified results of Bernays, and worked on ideas of Post, Godel and Church.
    • He was acknowledged as the leader of Hungarian mathematical logic.
    • His special fields of interest in computer science included programming languages, automatic error correction, non-numerical applications of computers and the connection between computer science and mathematical logic.

  61. Bachmann Friedrich biography
    • His first area of research was Mathematical Logic and the Foundations of Mathematics, and in 1933 he was awarded his PhD by the Westfalische Wilhelms University of Munster for his thesis Untersuchungen zur Grundlegung der Arithmetik mit besonderer Beziehung auf Dedekind, Frege, und Russell Ⓣ.
    • His thesis advisor at Munster had been Heinrich Scholz (1884-1956) who had been appointed there as a full professor of Philosophy in 1928 and later became professor of Mathematical Logic and Foundations of Mathematics.
    • After further work on logic, including starting to prepare the correspondence between Gottlob Frege and Bertrand Russell for publication, Bachmann turned to questions on geometry.
    • Although he did not like publicity much, his profound knowledge of philosophy and logic allowed him to play an important role when a chair in the department of philosophy was dedicated to logic and filled by Paul Lorenzen from 1956 to 1962, and later by Kurt Schutte from 1963 to 1966, broadening the scope of mathematics at Kiel beyond the 'Mathematisches Seminar' proper.

  62. Hamilton William biography
    • He continued to study logic and moral philosophy at Glasgow before entering the University of Edinburgh in 1806 to study medicine.
    • Further philosophical articles in the Edinburgh Review on topics such as the philosophy of the conditioned, perception, and logic enhanced his reputation.
    • In 1836 Hamilton became professor of logic and metaphysics at the University of Edinburgh, giving his inaugural lecture on 21 November.
    • Hamilton was one of the first in a series of British logicians to create the algebra of logic and introduced the 'quantification of the predicate'.

  63. Lukasiewicz biography
    • During this time Łukasiewicz and Lesniewski founded the Warsaw School of Logic.
    • Łukasiewicz published his famous text Elements of mathematical logic in Warsaw in 1928 (the English translation appeared in 1963): [Dictionary of Scientific Biography (New York 1970-1990).','Reference ',1)">1]:- .
    • viewing mathematical logic as an instrument of enquiry into the foundations of mathematics and the methodology of empirical science, Łukasiewicz succeeded in making it a required subject for mathematics and science students in Polish universities.
    • He worked on mathematical logic, wrote essays on the principle of non-contradiction and the excluded middle around 1910, developed a three value propositional calculus (1917) and worked on many valued logics.

  64. Shatunovsky biography
    • The school of mathematics at Odessa was not quite the quality of the St Petersburg school, but nevertheless it was excellent with leading mathematicians such as the geometer Benjamin Fedorovich Kagan, Ivan Vladislavovich Sleszynski whose main work was on continued fractions, least squares and axiomatic proof theory based on mathematical logic, S P Yaroshenko, and the mathematical historian I Yu Timchenko.
    • Anellis writes [Modern Logic 6 (1) (1996), 7-36.',2)">2]:- .
    • And in life, as in mathematics, Shatunovsky was a brilliant thinker, sometimes paradoxical, but always using invincible logic.
    • In mathematics, as in any other spheres of thought, Shatunovsky was striking in his originality, poignancy and unshakable logic.

  65. Al-Tusi Nasir biography
    • These topics included logic, physics and metaphysics while he also studied with other teachers learning mathematics, in particular algebra and geometry.
    • However, al-Tusi did some of his best work while moving round the different strongholds, and during this period he wrote important works on logic, philosophy, mathematics and astronomy.
    • In logic al-Tusi followed the teachings of ibn Sina.
    • Logic 16 (2) (1995), 257-268.','Reference ',33)">33] Street describes this as follows:- .

  66. Von Neumann biography
    • Such situations are devoid of logic and the fact that the Neumann's were opposed to Kun's government did not save them from persecution.
    • In his youthful work, he was concerned not only with mathematical logic and the axiomatics of set theory, but, simultaneously, with the substance of set theory itself, obtaining interesting results in measure theory and the theory of real variables.
    • It represented for him a synthesis of his early interest in logic and proof theory and his later work, during World War II and after, on large scale electronic computers.
    • When von Neumann realised he was incurably ill, his logic forced him to realise that he would cease to exist, and hence cease to have thoughts ..

  67. Heyting biography
    • This work had another beneficial effect as far as Heyting was concerned for it brought him to the attention of Heinrich Scholz who held the chair of mathematical logic in Munster.
    • Although Heyting's version of intuitionist logic differed somewhat from that of Brouwer, it is clear that one of his main aims was to make Brouwer's ideas more accessible and better known.
    • There were others interested in intuitionist logic working on similar problems of formalisation at the same time as Heyting.
    • Logic 15 (2) (1994), 149-172.','Reference ',4)">4] shows the major influence that Heyting has had on the study of the foundations of mathematics and in so doing shows the importance of Heyting's contributions.

  68. Avicenna biography
    • He also studied logic and metaphysics, receiving instruction from some of the best teachers of his day, but in all areas he continued his studies on his own.
    • The first is a scientific encyclopaedia covering logic, natural sciences, psychology, geometry, astronomy, arithmetic and music.
    • He also wrote on psychology, geology, mathematics, astronomy, and logic.
    • To grasp the intelligible both reason and logic are required.

  69. Lowenheim biography
    • Despite war service in France, Hungary and Serbia between August 1915 and December 1916, he published a series of important papers on mathematical logic during the eleven years from 1908 to 1919, extending work by Charles Peirce, Schroder, and Whitehead.
    • We will explain below some of the ideas he introduced into mathematical logic over this period, but at this point we will continue to sketch events in his life.
    • In fact he lost unpublished manuscripts on logic, geometry, music and the history of art.
    • In this paper Lowenheim proved the remarkable result that for any set of sentences of standard predicate logic, if there is an interpretation in which they are true in some domain, there is also an interpretation that makes them true in a countable subset of the original domain.

  70. MacLane biography
    • Up to this time he had worked on mathematical logic but that was not a topic that was attractive to those making appointments to mathematics departments.
    • Offered the option of giving the advanced course at Harvard on logic or algebra, he opted for algebra and he began to move in that direction.
    • He worked on and off throughout his career on mathematical logic, no surprise for a student of Bernays, and he did some early work on planar graphs.
    • Mac Lane was the author of seven books: (with Garrett Birkhoff) A Survey of Modern Algebra (1941); Homology (1963); (with Garrett Birkhoff) Algebra (1967); Categories for the Working Mathematician (1971); Mathematics, Form and Function (1985); (with Ieke Moerdijk) Sheaves in Geometry and Logic: A First Introduction to Topos Theory (1992); and Saunders Mac Lane: A Mathematical Autobiography (2005).

  71. Jourdain biography
    • One of the highlights was a course on mathematical logic given by Russell which strongly influenced Jourdain.
    • Inspired by Russell, Jourdain worked mainly in mathematical logic.
    • Other papers which he wrote on mathematical logic and the foundations of set theory include On the question of the existence of transfinite numbers which was published in the Proceedings of the London Mathematical Society in 1907.
    • Jourdain also applied logic to physics in papers such as On some Points in the Foundation of Mathematical Physics which was published in The Monist in 1908.

  72. Ladd-Franklin biography
    • Charles Peirce's only academic post was the courses on logic he gave at Johns Hopkins from 1879 and 1884 and it was these courses which led Ladd to write a doctoral thesis entitled The Algebra of Logic.
    • Also in 1901, she became the associate editor for logic and philosophy in Baldwin's Dictionary of Philosophy and Psychology, a position she held until 1905.
    • The year before this Johns Hopkins relented and allowed her to teach one course in logic and philosophy.

  73. De Bruijn biography
    • He continued to hold this position until June 1944 [Applied Logic Series 28 (Kluwer Academic Publishers, Dordrecht, 2003).','Reference ',1)">1]:- .
    • He began publishing papers on combinatorics relevant to his work during this period such as The problem of optimum antenna current distribution (1946), A combinatorial problem (1946), On the zeros of a polynomial and of its derivative (1946), and A note on van der Pol's equation (1946) [Applied Logic Series 28 (Kluwer Academic Publishers, Dordrecht, 2003).','Reference ',1)">1]:- .
    • He lists his interests (on his website) as: Geometry, Number Theory, Classical and Functional Analysis, Applied Mathematics, Combinatorics, Computer Science, Logic, Mathematical Language, Brain Models.
    • The Preface [Applied Logic Series 28 (Kluwer Academic Publishers, Dordrecht, 2003).','Reference ',1)">1] records:- .

  74. Reichenbach biography
    • He unconditionally rejected speculative metaphysics and theology because their claims could not be substantiated either a priori, on the basis of logic and mathematics, or a posteriori, on the basis of sense-experience.
    • In the two papers be examines using Łukasiewicz's three-valued logic in quantum mechanics.
    • Among other works, published after he emigrated, are Elements of Symbolic Logic (1947) and The Rise of Scientific Philosophy (1951).
    • This textbook, not primarily intended for mathematicians, emphasizes more the interpretation and application of formal logic than the construction of the formal system itself.

  75. Gorbunov biography
    • The theory of quasivarieties is a branch of algebra and mathematical logic that deals with a fragment of the first-order logic, the so-called universal Horn logic.
    • Viktor's name was and will always be affiliated with the Siberian School of Algebra and Logic, which was founded by Anatoly Ivanovich Malcev.

  76. Foster biography
    • at that time group theory was perhaps a household word, and I saw an opportunity to introduce some group theory into mathematical logic.
    • thesis Formal Logic in Finite Terms to Princeton University in 1930 and the degree was awarded in the following year.
    • He wrote a paper Formal logic in finite terms, based on his thesis, which was published in 1931 in the Annals of Mathematics.
    • Foster, as a student of Church, naturally began his research career working in mathematical logic.

  77. Robinson Raphael biography
    • His doctoral dissertation was on complex analysis, but he also worked on logic, set theory, geometry, number theory, and combinatorics.
    • A typical paper on logic was Finite sequences of classes which appeared in 1945.
    • The methods and aims of this work are probably more easily intelligible and more interesting to the 'ordinary' mathematician than those of any other branch of mathematical logic.

  78. Subbotovskaya biography
    • Her mathematical studies in algebraic logic and her musical studies were not entirely distinct since she was interested in the mathematical structure of music.
    • Bella worked with A A Lyapunov on problems of optimization while she also undertook research on algebraic logic aiming at a doctorate.
    • She continued research for her doctorate and published Comparison of bases for the realization by formulas of functions of an algebra of logic in 1963.

  79. Lamy biography
    • The translators, who have not been identified with certainty, attributed the original to "Messieures du Port Royal." Lamy was not a member of the Port Royal group, but he had benefited from some of their thinking about language and logic, and readers easily accepted the new rhetoric as the counterpart of the well-known Port-Royal Logic and grammars, thus greatly increasing its sale.
    • The influence of Ramism is slight, that of Cartesian method strong, and there are many points of contact with the Port-Royal Logic.

  80. Vacca biography
    • Our work on the logic of Leibniz was almost completed (at least we thought so) when we had the pleasure, at the International Congress of Philosophy (August, 1900), of making the acquaintance of Mr Giovanni Vacca, at that time mathematical assistant at the University of Turin, who had examined, the year before, the manuscripts of Leibniz preserved in Hannover, and had extracted from them several formulae of logic inserted in the "Formulaire de Mathematiques" of Mr Peano.
    • However Vacca continued his mathematical work and gave a course at the University of Genoa on mathematical logic.

  81. Saccheri biography
    • When Count Gravere's theses were examined and published, Saccheri took the opportunity to publish the course on logic that he had been delivering at the College in Turin.
    • This gives him [Saccheri] the right to an eminent place in the history of modern logic.
    • Nevertheless the first seventy pages (apart from a few isolated phrases), up to Proposition 32 inclusive, constitute an ensemble of logic and of geometric acumen which may be called perfect.

  82. Burali-Forti biography
    • In 1893-94 Burali-Forti gave an informal series of lectures on mathematical logic at the University of Turin.
    • This new mathematical logic of Peano, which had far-reaching implications and tremendous importance, achieving its culmination in the 'Principia Mathematica' of Whitehead and Russell, was then but little known.
    • The sound mastery of the methods and language of ideographic logic made Burali-Forti one of the first and most active collaborators on the 'Formulaire de Mathematiques'.

  83. Turing biography
    • The year 1933 saw the beginnings of Turing's interest in mathematical logic.
    • A M Turing read a paper on "Mathematics and logic".
    • The major publication which came out of his work at Princeton was Systems of Logic Based on Ordinals which was published in 1939.

  84. Cantelli biography
    • from 1900 to 1915 [there was] an interest in the foundational problems of probability, a conviction that logic would play a role in resolving these problems, and varying conceptions of logic, all of which predate and serve as a historical foundation for this 'golden age'.
    • Cantelli's first publications on probability examined the foundations of the subject in connection with logic, for example in Sui fondamenti del calcolo delle probabilita Ⓣ (1905).

  85. Holder biography
    • A thorough study of the methods of ratiocination employed in mathematics, mechanics, and the exact natural sciences has led Professor Holder to the conviction that the deductive method there employed is made up of series of concatenated conclusions of quite characteristic form, and that consequently these sciences have a peculiar method and logic of their own.
    • Finally, and above all, there is required an unusual ability in order to realize the goal which Dr Holder has set before himself, namely to bring to self-conscious expression the logic of mathematical inference.
    • Dr Holder (who is professor of mathematics at Leipzig) desires his work to be regarded as primarily a contribution to the logic of mathematics, and modestly leaves to other logicians the problem of incorporating his findings in more comprehensive treatises.

  86. Enriques biography
    • He became fascinated by other subjects while at the high school, such as logic, epistemology, pedagogy, and the history of science.
    • Books on psychology and logic, physiology, and comparative psychology, a critique of knowledge etc., sit on my coffee table where I savour them with delight trying to extract the essence for what concerns my problem ..
    • He held a subtle position, according to which knowledge is inseparable from the means of knowing, logic from psychology.

  87. Aldrich biography
    • In 1691 he published Artis logicae compendium a treatise on logic which was to be the main text on the topic for 150 years in England.
    • Even when Richard Whately published Elements of logic in 1826 it still took Aldrich's work as his starting point, but then this much more modern text took over the role which Artis logicae compendium had held for so long; see [Hist.
    • Logic 5 (1) (1984), 1-18.',4)">4] for further details.

  88. Kaluznin biography
    • In 1959 Kaluznin became Head of the Department of Algebra and Mathematical Logic, a department created as a result of his own initiative.
    • Having always been perceived as an "alien", Kaluznin was forced to leave his position as Head of the Department of Algebra and Mathematical Logic, though he retained his professorship at Kiev State University until 1985.
    • Kaluznin made several applications of the wreath product to mathematical logic and mathematical chemistry.

  89. Keynes biography
    • His father, John Nevile Keynes, was a lecturer at the University of Cambridge where he taught logic and political economy.
    • John Nevile published Formal Logic four months after John Maynard was born.
    • He came top in logic, psychology, and the essay, while his worst subjects were mathematics and economics.

  90. Matiyasevich biography
    • They met for the first time at the International Congress for Logic, Methodology and Philosophy of Sciences held in Bucharest, Romania in 1971 where Matiyasevich gave the lecture On recursive unsolvability of Hilbert's tenth problem in which he described the joint work he had been carrying out with Julia Robinson.
    • In 1980 Matiyasevich was appointed head of the Laboratory of Mathematical Logic at the Leningrad Department of the Steklov Institute.
    • Valentina Harizanov writes [Modern Logic 5 (3) (1995), 345-355.','Reference ',5)">5]:- .

  91. Kothe biography
    • A meeting with Innsbruck philosopher Alfred Kastil, of the school of Franz Brentano, brought philosophy again into the foreground Since I was fascinated by epistemology and logic, in particular the paradoxes of set theory, it seemed best to give up chemistry and to study mathematics together with philosophy instead.
    • Symbolic Logic 40 (2) (1975), 241.','Reference ',2)">2]:- .
    • The order of the book follows a remorseless logic.

  92. Coutts biography
    • His second year was slightly more successful since he passed Natural Philosophy and English at the first attempt in April 1906 but failed Logic and Metaphysics.
    • He sat the examination in Logic and Metaphysics at the resit in October 1906 but, despite improving his performance from 25% to 40%, he failed again.
    • He passed Logic and Metaphysics at the third attempt in April 1907, achieving a comfortable pass at 58%.

  93. Nikodym biography
    • He presented popular lectures on radio such as: Logic and intuition in science, On infinity, On paradoxes in logic, What good is algebra?, On different kinds of spaces, The mystery of gravitation, On the importance of theory and these were published in 1946 as a book Let's look deeply inside the mind (Spojrzmy w glebiny mysli).
    • His last book The Mathematical Apparatus for Quantum-Theories, based on the Theory of Boolean Lattices published in 1966 by Springer-Verlag contains, on almost thousand pages, the mathematical formalism for quantum mechanics or more precisely a detailed study of the Boolean subalgebras of the logic of closed subspaces of a complex Hilbert space.

  94. Maior biography
    • He lectured both to theology students and to arts students, giving couses on logic to the latter group of students.
    • He was interested in mathematics and logic and applied these to physics, writing an important text on the infinite Propositum de infinito in 1506 [Dictionary of Scientific Biography (New York 1970-1990).
    • Maior's importance for physical science derives from his interest in logic and mathematics and their application to the problems of natural philosophy.

  95. Lakatos biography
    • His work was influenced by Popper and by Polya and he went on to write his doctoral thesis Essays in the Logic of Mathematical Discovery submitted to Cambridge in 1961.
    • However, in 1976, two years after his death, the work did appear as a book: J Worrall and E G Zahar (eds.), I Lakatos : Proofs and Refutations : The Logic of Mathematical Discovery .
    • The point is to lay bare the inner workings of mathematical growth and change as a historical process, as a process with its own laws and its own 'logic', one which is best understood in its rational reconstruction, of which the actual history is perhaps only a parody.

  96. Descartes biography
    • He studied there taking courses in classics, logic and traditional Aristotelian philosophy.
    • The work describes what Descartes considers is a more satisfactory means of acquiring knowledge than that presented by Aristotle's logic.

  97. Kuratowski biography
    • One was Łukasiewicz, a professor of philosophy who worked on mathematical logic.
    • From the very first lecture I was enchanted by the clarity, logic, and polish of his exposition and the material he presented.

  98. Carmichael biography
    • Also in 1930, Carmichael published The Logic of Discovery.
    • The book under review therefore is especially noteworthy in the fact that it belongs to that limited class of philosophical essays which defines its terms, states its postulates, and then proceeds to the development of its theme with an inevitable logic.

  99. Alison biography
    • In 1879 he entered Edinburgh University, and during the sessions from 1879 to 1884 gained distinction in Latin, Greek, logic, and mathematics.
    • He was also medallist in natural philosophy and next to medallist in logic and mathematics.

  100. Macbeath biography
    • Later in 1923, Alec was appointed senior lecturer in Logic and Metaphysics at the University of Glasgow.
    • The family moved to Belfast in 1925 when Alec was appointed as professor of logic and metaphysics at Queen's University, Belfast.

  101. Alexiewicz biography
    • Żyliński, however, who worked on number theory, algebra, logic and the foundations of mathematics, survived the war and taught in Gliwice from 1946 to 1951.
    • The affect of these changes on mathematics at Poznan was to combine the two Chairs of Mathematics with the Chair of Logic.

  102. Forder biography
    • Forder is extremely widely read in mathematical logic and philosophy, pure mathematics, relativity, quantum mechanics and astrophysics, and on these subjects I have heard him speak with knowledge, and authority, and with marked originality.
    • is plainly, in his way, a rather remarkable man, since he combines so much experience of comparatively elementary teaching with a real understanding of and enthusiasm for the logic of his subject ..

  103. Stott biography
    • Examples of these books are (i) Logic Taught By Love (1890), (ii) Lectures on the Logic of Arithmetic (1903), (iii) The preparation of the child for science (1904), and (iv) Philosophy and the fun of algebra (1909).

  104. Barrow biography
    • At Felstead Barrow learnt Greek, Latin, Hebrew and logic in preparation for University.
    • He tried to classify the different branches of mathematics arguing that algebra is not part of true mathematics and should be considered to be logic while [Isaac Barrow His Life and Times (London, 1944).','Reference ',5)">5]:- .

  105. Hammer biography
    • Most of Peter Hammer's scientific production has its roots in the work of George Boole on propositional logic.
    • Among the main research topics which have received his attention, one finds an impressive array of methodological studies dealing with combinatorial optimization, some excursions into logistics and game theory, numerous contributions to graph theory, to the algorithmic aspects of propositional logic, to artificial intelligence and, more recently, to the development of innovative data mining techniques.

  106. Wrinch biography
    • The two became friends and Russell's ideas on mathematical logic were a major influence on Wrinch throughout her career.
    • Also at Cambridge she attended lectures by W E Johnson on logic.

  107. Al-Kindi biography
    • Al-Kindi "was the most leaned of his age, unique among his contemporaries in the knowledge of the totality of ancient scientists, embracing logic, philosophy, geometry, mathematics, music and astrology.
    • Logic Rev.

  108. Leibniz biography
    • As he progressed through school he was taught Aristotle's logic and theory of categorising knowledge.
    • At Jena the professor of mathematics was Erhard Weigel but Weigel was also a philosopher and through him Leibniz began to understand the importance of the method of mathematical proof for subjects such as logic and philosophy.

  109. Dantzig biography
    • In the first place the logical exactitude is not always such as would be expected in a book in which so much use is made of symbolic logic.
    • Some of his papers in the area of significes are Mathematics, logic, and empirical science (Dutch) (1946), General Procedures of Empirical Science (1947), Signifies, and its relations (1949), Some Informal Information on "Information" (1953-55), and Mannoury's Impact on Philosophy and Significs (1956-58).

  110. Eudemus biography
    • Another work by Eudemus was on logic, in fact he may well have written two logic books and he also wrote On Discourse.

  111. Gerbert biography
    • At the monastery, Gerbert learnt literature, theology, history, and philosophy, but would not have studied any mathematics and only a very little logic.
    • The reason he wanted to go to Rheims was that Garamnus, perhaps Europe's leading logician, was there and he had offered to teach Gerbert logic in exchange for Gerbert teaching him music and mathematics.

  112. Peter biography
    • Two years later she became an editor of the recently founded Journal of Symbolic Logic.
    • The first publication was in Hungarian and was reviewed by John Kemeny who writes [The Journal of Symbolic Logic 13 (3) (1948), 141-142.','Reference ',7)">7]:- .

  113. Wallis biography
    • He also studied logic at this school but mathematics was not considered important in the best schools of the time, so Wallis did not come in contact with that topic at school.
    • His non-mathematical works include many religious works, a book on etymology and grammar Grammatica linguae Anglicanae (Oxford, 1653) and a logic book Institutio logicae (Oxford, 1687).

  114. Bayes biography
    • We do know that in 1719 Bayes matriculated at the University of Edinburgh where he studied logic and theology.
    • the first occurrence of a probability logic result involving conditional probability.

  115. Bloch biography
    • It's a matter of mathematical logic.
    • [He committed] a crime of logic, performed in the name of absolute rationalism, as dangerous as any spontaneous passion.

  116. Kaestner biography
    • He was concerned with philosophical questions in mathematics and other areas such as logic.
    • However Kastner was quite unenthusiastic about logic, but this is not surprising for a mathematician of this period who was interested in geometry.

  117. Kneebone biography
    • Kneebone's main research interests were in the philosophy of mathematics and he wrote papers such as Philosophy and Mathematics (1947), Induction and Probability (1949), Abstract Logic and Concrete Thought (1955), and The Philosophical Basis of Mathematical Rigour (1957).
    • These are: (with John Greenlees Semple) Algebraic Projective Geometry (1952); (with John Greenlees Semple) Algebraic Curves (1959); Mathematical Logic and the Foundations of Mathematics (1963); and (with Brian Rotman) The Theory of Sets and Transfinite Numbers (1966).

  118. Bradwardine biography
    • It is during this period at Oxford that almost all of his works on logic, mathematics, and philosophy were written.
    • However, he questioned the logic of his own arguments as he felt perhaps the existence of geometry already assumes that atomism is false.

  119. Hardie Robert biography
    • Hardie then returned to Edinburgh where he was appointed as an Assistant in Logic and Metaphysics at the University in 1889.
    • Hardie remained an assistant until 1892 when he was promoted to Lecturer in Logic and Metaphysics.

  120. Menger biography
    • This book reprints all the articles (in German) along with chapters (in English) surveying the important developments in economics, logic, topology and geometry that were reported in the 'Ergebnisse'.
    • I rank your achievement among the greatest of modern logic and send you my heartiest congratulations.

  121. Bolzano biography
    • The first two volumes cover his ideas on the philosophy of logic, the third volume presents a theory of scientific discovery, while the final volume presents his methodology of writing textbooks [DVT---Dejiny Ved Tech.
    • Bolzano's theory of science (Wissenschaftslehre) contains a great amount of very valuable information concerning the development of logic from its beginnings in Aristotle till the post-Kantian period.

  122. Godel biography
    • It became slowly obvious that he would stick with logic, that he was to be Hahn's student and not Schlick's, that he was incredibly talented.
    • However after Schlick, whose seminar had aroused Godel's interest in logic, was murdered by a National Socialist student in 1936, Godel was much affected and had another breakdown.

  123. Foulis biography
    • In 2007 he published papers such as Effects, observables, states, and symmetries in physics and (with Richard J Greechie) Quantum logic and partially ordered abelian groups.
    • I have recently been appointed Visiting Professor at Florida Atlantic University, where I am participating in a seminar on quantum logic and where I am a member of the Ph.D.

  124. Boethius biography
    • Up to the 12th century his writings and translations were the main works on logic in Europe becoming known collectively as the Logica vetus (meaning the old logic).

  125. Tacquet biography
    • He then studied mathematics, logic and physics at Louvain from 1631 until 1635.
    • Logic 13 (1) (1992), 43-58.

  126. Silva biography
    • But Silva was not only publishing research articles, he was also publishing articles on mathematical education during these years such as Mathematical logic and education (in three parts all published in 1941), On the way to establish Taylor's formula (1942), The theory of the logarithms in school education (1942), and Reason? (1942).
    • Silva began working on a doctoral thesis in logic entitled Towards a general theory of homomorphisms.

  127. Ito biography
    • I attempted to describe Levy's ideas, using precise logic that Kolmogorov might use.
    • The beauty in mathematical structures, however, cannot be appreciated without understanding of a group of numerical formulae that express laws of logic.

  128. Bernoulli Daniel biography
    • First however Daniel was sent to Basel University at the age of 13 to study philosophy and logic.
    • The next chair to fall vacant at Basel that Daniel applied for was the chair of logic, but again the game of chance of the final selection by drawing of lots went against him.

  129. Gardner biography
    • We certainly do not want to even list the titles of over sixty works so we will give a selection: Logic Machines and Diagrams (1958); The Annotated Alice (1960); Relativity for the Million (1962); The Ambidextrous Universe: Mirror Asymmetry and Time-Reversed Worlds (1964); Mathematical Carnival: A New Round-up of Tantalizers and Puzzles from "Scientific American" (1975); The Incredible Dr Matrix (1976); Aha! Insight (1978); Science: Good, Bad, and Bogus (1981); Aha! Gotcha: Paradoxes to Puzzle and Delight (1982); The Whys of a Philosophical Scrivener (1983); Codes, Ciphers and Secret Writing (1984); Entertaining Mathematical Puzzles (1986); Time Travel and Other Mathematical Bewilderments (1987); Perplexing Puzzles and Tantalizing Teasers (1988); Fractal Music, Hypercards and More (1991); My Best Mathematical and Logic Puzzles (1994); Classic Brainteasers (1995); Calculus Made Easy (1998); A Gardner's Workout: Training the Mind and Entertaining the Spirit (2001); Mathematical Puzzle Tales (2001); and Bamboozlers (2008).

  130. Appel biography
    • His thesis advisor was Roger Lyndon and he was awarded the degree in 1959 for his thesis Two Investigations on the Borderline of Logic and Algebra.
    • He was told to find something that connected logic and algebra and he worked on the problem for months.

  131. Ramus biography
    • His teaching was aimed at attacking Aristotle and in particular Aristotle's logic.
    • Using this approach Ramus worked on many topics and wrote a whole series of textbooks on logic and rhetoric, grammar, mathematics, astronomy, and optics.

  132. Novikov biography
    • In 1957 Novikov set up a new department at the Steklov Institute, namely the Department of Mathematical Logic, and he was appointed as the first head of that department.
    • He began to study mathematical logic and the theory of algorithms just before 1940.

  133. Besicovitch biography
    • He was taught by Markov at the University of St Petersburg where he originally intended to work in mathematical logic but he changed topics to study analysis since the library was not good enough in the logic area.

  134. Thue biography
    • contains many notions and ideas about trees, term rewriting and word problems which are surprisingly modern and have later come to play important roles in mathematics, logic, and computer science.
    • With the death of Professor Thue on 7 March this year, the Academy of Science has lost one of its most illustrious members, a mathematical genius who united the gift of extraordinary originality with a rare perspicacity and sense of logic.

  135. Nunes biography
    • This appointment was only as a substitute for the professor, but on 15 January 1530 he was appointed to the chair of Logic.
    • He taught logic for at least a year but it appears that his classes were not very successful with students so he moved to the chair of metaphysics on 4 April 1532.

  136. Newman biography
    • in recognition of his distinguished contributions to combinatory topology, Boolean algebras and mathematical logic.
    • The design brought into play his knowledge of formal logic.

  137. Snedecor biography
    • and even in the world in recognizing the nature of one of the major 20th century revolutions or evolutions, the recognition that interpretation of data is a remarkably difficult activity requiring integration of mathematics and some logic of analogy.
    • Although "no mathematics beyond elementary algebra is required" and "the attempt is made to present the logic of the science with only so much mathematical symbolism as is necessary for clarity," the "lay reader" has to learn, and learn quickly, a formidable amount of what he must regard as jargon.

  138. Reidemeister biography
    • Led by Reidemeister, the group of mathematicians at Vienna spent a year studying the deep ideas on logic and mathematics in the Tractatus.
    • The sixth (on geometry and logic) begins with Hjelmslev's idea of representing the points and lines of the Euclidean (or non-Euclidean) plane by involutory transformations that leave them invariant ..

  139. Levi Beppo biography
    • Before going to Parma he had already published over forty articles on topics ranging from algebraic geometry to logic, particularly working on the axiom of choice.
    • He wrote articles on logic, differential equations, complex variable, as well as on the border between analysis and physics.

  140. Brouwer biography
    • Brouwer was somewhat like Nietzsche in his ability to step outside the established cultural tradition in order to subject its most hallowed presuppositions to cool and objective scrutiny; and his questioning of principles of thought led him to a Nietzschean revolution in the domain of logic.
    • He in fact rejected the universally accepted logic of deductive reasoning which had been codified initially by Aristotle, handed down with very little change into modern times, and very recently extended and generalised out of all recognition with the aid of mathematical symbolism.

  141. Ball biography
    • There he studied mathematics, logic, and moral philosophy.
    • He also studied philosophy under George Croom Robertson (1842-1892), the professor of philosophy of mind and logic.

  142. Kramer biography
    • Her examination of Omar Khayyam and algebra, Newton and calculus, Fermat and probability, Lewis Carroll and logic and Einstein and relativity provides an intriguing book for non-mathematicians and a valuable reference source for teachers and students.
    • One cannot easily think of a topic within layman's comprehension which is not presented in considerable detail, including analysis, algebra, logic and foundations.

  143. Lax Anneli biography
    • She was attracted to the constructions and logic of geometry, completing advanced problems with ease [Humanistic Mathematics Network Journal 21 (California State University Press, 1999).','Reference ',6)">6]:- .
    • These two significant encounters with geometry convinced her that logic was what made mathematics satisfying and pleasing.

  144. Manin biography
    • He has also written famous papers on formal groups, the arithmetic of rational surfaces, cubic hypersurfaces, noncommutative algebraic geometry, instanton vector bundles and mathematical logic.
    • Other books by Manin include Cubic forms: algebra, geometry, arithmetic (Russian) (1972), A course in mathematical logic (1977), Computable and noncomputable (Russian) (1980), Quantum groups and noncommutative geometry (1988), Topics in noncommutative geometry (1991), Frobenius manifolds, quantum cohomology, and moduli spaces (1999).

  145. Walker John biography
    • Walker's grandfather had been a fellow of Trinity and published classical texts as well as elementary mathematics and logic texts.

  146. Grieve biography
    • He began his Honours studies in 1905-06 but also took Logic at Ordinary level in that year.

  147. Gentle biography
    • degree, broadened his field of study by including a number of extra subjects, such as logic and psychology.

  148. Kolchin biography
    • Although the articles in this volume are in the main devoted to commutative algebra, linear algebraic group theory, and differential algebra, the diversity of subjects covered - complex analysis, algebraic K-theory, logic, stochastic matrices, differential geometry, ..

  149. Wiener Norbert biography
    • from Harvard at the age of 18 with a dissertation on mathematical logic supervised by Karl Schmidt.

  150. Harish-Chandra biography
    • Although he was convinced that the mathematician's very mode of thought prevented him from comprehending the essence of theoretical physics, where, he felt, deep intuition and not logic prevailed, and sceptical of any mathematician who presumed to attempt to understand it, he was even more impatient with those mathematicians in whom a sympathy for theoretical physics was lacking, a failing he attributed in particular to the French school of the 1950s.

  151. Kreisel biography
    • In 1946 Kreisel returned to Cambridge to undertake research, studying mathematical logic.

  152. Dougall Charles biography
    • In 1887-88 he was placed 11th in the Senior Division of Logic and Rhetoric.

  153. Castelli biography
    • So their Highnesses asked Cosimo Boscaglia [a Platonist and Professor of Logic and Philosophy at Pisa], who answered that in truth their existence could not be denied.

  154. Mitchell James biography
    • He won medals or prizes in a wide range of subjects: mathematics, natural philosophy, Latin, Greek, logic, and psychology.

  155. Arnauld biography
    • Arnauld's next work was Port-Royal Logic which was another book of major importance.

  156. Llull biography
    • Llull used logic and mechanical methods involving symbolic notation and combinatorial diagrams to relate all forms of knowledge.

  157. Edgeworth biography
    • Three years later, however, he was lecturing on logic at King's College, London.

  158. Chrystal biography
    • The logic of the subject, which, both educationally and scientifically speaking, is the most important part of it, is wholly neglected.

  159. Golab biography
    • Professor Golab dealt with different fields of mathematics such as geometry, topology, algebra, analysis, logic, functional and differential equations, the theory of numerical methods and various applications of mathematics.

  160. Trahtman biography
    • At Bar-Ilan University, Trahtman taught courses in discrete mathematics, theory of sets, algebra, analytical geometry, mathematical logic, finite automata, formal languages, rings and modules, and differential equations.

  161. Zeno of Sidon biography
    • It is believed that, among the areas he studied, he contributed to logic, atomic theory, biology, ethics, literary style, oratory, poetry, the theory of knowledge, and to mathematics.

  162. Adamson biography
    • As the reviews [The Mathematical Gazette 83 (496) (1999), 168-169.','Reference ',1)">1] and [Studia Logica: An International Journal for Symbolic Logic 69 (3) (2001), 433-435.','Reference ',9)">9] both point out, Adamson's "best laid plans" can be foiled by students who turn to the second part for the proofs of the results as they are reading the first part.

  163. Forsythe biography
    • Computer science must also concern itself with such theoretical subjects supporting this technology as information theory, the logic of the finitely constructible, numerical mathematical analysis, and the psychology of problem solving.

  164. Zaremba biography
    • Starting with the works of G Hamel, this question has been studied by many specialists in mechanics, mathematics and logic.

  165. Shafarevich biography
    • I tried to remember how I turned to mathematics and could not recall the logic or reason for it, though I think that to work on history then would really have been hard.

  166. Jeffreys biography
    • Harold Jeffreys on Logic and Scientific Inference .

  167. Capelli biography
    • Battaglini [Aldo Ursini, Paolo Agliano, Roberto Magari, Logic and algebra (CRC Press, 1996), 283-316.','Reference ',11)">11]:- .

  168. Comrie biography
    • In his second year he studied Natural Philosophy, Logic, English Literature, and Latin 2.

  169. Novikov Sergi biography
    • V A Uspenskii, a pupil of Kolmogorov, organised a seminar during Novikov's first year as a student in which problems in set theory, mathematical logic, and functions of a real variable were studied.

  170. Yang Hui biography
    • Firstly he explains the logic behind the problem, secondly he gives a numerical solution to the problem, and thirdly he shows how the method he has presented can be modified to solve similar problems.

  171. Johnstone biography
    • In the following session he took courses in Logic, Psychology and Chemistry.

  172. Hirzebruch biography
    • He was allowed to enter the Westfalische Wilhelms University of Munster in November 1945 where he studied mathematics, physics and mathematical logic.

  173. Sturm biography
    • The author describes how Tarski showed in 1940 that Sturm's method of proof could be used in mathematical logic to prove the completeness of elementary algebra and geometry.

  174. Bernoulli Nicolaus(I) biography
    • In 1722 he left Italy and returned to his home town to take up the chair of logic at the University of Basel.

  175. Kennedy-Fraser biography
    • In 1905-06 he studied Logic and Chemistry at Ordainary level and Intermediate Mathematics at Honours level.

  176. Macdonald William biography
    • In his first year 1868-9 he studied English Literature, Greek 1, Latin 1, and Mathematics 1; in 1869-70 he studied Logic, Greek 2, Latin 2, and Mathematics 2; in 1870-71 he studied Moral Philosophy, Political Economy and Mathematics 3; in 1871-72 he studied Natural Philosophy and Chemistry; finally in 1872-73 he studied Natural Philosophy and Mathematics 3.

  177. Cipolla biography
    • The author describes the notion off the null class and discusses its introduction and the symbols used to represent it, in the history of symbolic logic.

  178. Belanger biography
    • So the Ecole Centrale des Arts et Manufactures devised a teaching plan, attempting to satisfy the condition without sacrificing in proofs the tight logic without which mathematics becomes an often misleading semi-science, not compromising clarity by excessive brevity, but by choosing those parts of analytic geometry and the infinitesimal calculus which every engineer must know, and especially those which are necessary for the study of mechanics viewed from the point of view of its practical application to industrial work.

  179. Hemchandra biography
    • He was instructed in religion, Indian philosophy, the sacred scriptures, logic and grammar.

  180. Nash-Williams biography
    • Increasingly, interactions between infinite combinatorics and mathematical logic are coming to light.

  181. Sinan biography
    • occupied himself with topics within his competence, such as the science of Euclid, the Almagest Ⓣ, astronomy, the theories of meteorological phenomena, logic, metaphysics, and the philosophical systems of Socrates, Plato, and Aristotle.

  182. Ascoli biography
    • On the one hand he was always attentive to the latest developments in analysis while, on the other hand, he had his roots in the classic treatment of the great masters, and found nourishment in the history of mathematics, in logic, and in methodology.

  183. Higman biography
    • He also met Max Newman in Cambridge and Newman's interest in the interaction between group theory and logic had a lasting influence on him.

  184. Kolmogorov biography
    • These included his versions of the strong law of large numbers and the law of the iterated logarithm, some generalisations of the operations of differentiation and integration, and a contribution to intuitional logic.

  185. Gibb biography
    • In 1902-03 he passed Chemistry and Education, then Logic and Metaphysics, and Rhetoric and English Literature in the following year.

  186. Chatelet Albert biography
    • The first chapter presents an introduction to logic, followed by sections on algebras of sets, lattices, mappings, and operations.

  187. Blum biography
    • Especially striking is the interplay of various mathematical disciplines such as algebraic number theory, algebraic geometry, logic, and numerical analysis, to mention a few.

  188. Fefferman biography
    • At that time I had a wonderful professor of mathematical logic: Carol Karp, who was interested in what you might you say if you could speak in infinitely long sentences.

  189. Pieri biography
    • he produced two research papers on algebraic geometry, two book reviews, four papers on logic and foundations of arithmetic, and about seven on foundations of geometry, including two major works axiomatising Euclidean and complex projective geometry.

  190. Griffiths Brian biography
    • if the object of teaching is to communicate, rather than to give aesthetic satisfaction to the expositor, then we must be prepared to put pedagogical techniques above mere logic ..

  191. Stringham biography
    • He attended the following courses in 1879-80: 'Elliptic Functions' by W Story, 'Higher Plane Curves' by W Story, 'Calculus of Variations' by T Craig, 'Spherical Harmonics' by T Craig, 'Symbolic Logic' by C S Peirce, 'Quaternions' by W Story, and 'Theory of Numbers' by J J Sylvester.

  192. Cariolaro biography
    • The courses he taught at this University were: Calculus 1, Calculus 2, Linear Algebra, Mathematical Reasoning, Logic and Problem Solving.

  193. Gleason biography
    • Chapters I to VI cover elementary logic and set theory; Chapters VII to X deal with the various "number systems" from the natural integers to the complex numbers; Chapter XI briefly returns to set theory (countable sets, cardinal numbers and the axiom of choice); finally, the last four chapters deal, respectively, with limits of complex sequences, infinite series and products, metric spaces, and the elementary theory of holomorphic functions of one variable (Cauchy integral excluded, but the logarithmic function is defined and studied).

  194. Levi biography
    • He wrote a commentary on the Bible and the Talmud; and in all branches of science, especially in logic, physics, metaphysics, mathematics, and medicine, he has no equal on earth.

  195. Ulam biography
    • We had a good professor in high school, Zawirski, who was a lecturer in logic at the university.

  196. Penrose biography
    • was a course on mathematical logic by Steen.

  197. Ceva Giovanni biography
    • He entered the university of Pisa in 1670 and there he studied under Donato Rossetti (1633-1686), the professor of logic, who was a strong supporter of atomic theories.

  198. Bremermann biography
    • He studied pattern recognitions as part of the rapidly developing area of artificial intelligence, introducing fuzzy logic methods to try to obtain more powerful methods than could be achieved by a rule based approach.

  199. Bollobas biography
    • This book is an in-depth account of graph theory, written with such a student in mind; it reflects the current state of the subject and emphasizes connections with other branches of mathematics, like optimization theory, group theory, matrix algebra, probability theory, logic, and knot theory.

  200. Lueroth biography
    • He also has some fine results on logic, a topic he worked on in collaboration with his friend Ernst Schroder.

  201. Feigenbaum biography
    • The machine came with a paper by Shannon on Boolean logic which fascinated Feigenbaum with his self-learning attitude.

  202. Konig Julius biography
    • Clearly retirement had been undertaken so that he could spend more time on things which he wanted to do - is this not the reason why many academics take early retirement? He spent the last part of his life working on his own approach to set theory, logic and arithmetic, which was published in 1914, the year after his death.

  203. Klugel biography
    • You might believe that the whole of the science is jeopardized by an uncertain logic.

  204. Magnitsky biography
    • Even in this practical manual, the ancient Liberal Art of Arithmetic, that is, mathematics as a form of logic and expression prevailed.

  205. Wilder biography
    • How does culture (in its broadest sense) determine a mathematical structure, such as a logic? .

  206. Lichnerowicz biography
    • And then, at some point something clicks and you feel, before having any proof, you have made a major advance and then you return to your desk, you check and finally expose the things you have discovered to precise logic.

  207. Hahn biography
    • [Hahn] maintains that logic and mathematics are essentially tautological and say nothing about the external world.

  208. Pitman biography
    • In a course called Advanced Logic at the University of Melbourne, Professor W R Boyce-Gibson, a very able philosopher and an excellent lecturer, devoted two or three lectures to [statistics].

  209. Grothendieck biography
    • He worked on the theory of topoi which are highly relevant to mathematical logic.

  210. Burgess biography
    • In addition to taking Honours courses in Mathematics and Natural Philosophy over the next two sessions, Burgess also took the Ordinary course in Logic and Metaphysics.

  211. MacMillan Chrystal biography
    • Having already taken the Honours courses in Mathematics and Natural Philosophy in 1894-95 and 1895-96 respectively, she graduated in April 1900 with Second Class Honours in Moral Philosophy and Logic as well as First Class Honours in Mathematics and Natural Philosophy.

  212. Fine biography
    • It must be acknowledged that Fine's arrogance about his own accomplishments undoubtedly made his errors of logic all the more intolerable to his opponents.

  213. Castelnuovo biography
    • Probability is a science of recent formation; hence in it, better than in other branches of mathematics, one can see the relationship between the empirical contribution and the one given by reasoning, and between the process of inductive and deductive logic used in it.

  214. Conway biography
    • At this stage he was working on mathematical logic but things were not going well.

  215. Fredholm biography
    • Fredholm was not what is usually called a brilliant speaker He talked slowly in a monotone voice and it could happen that he got embroiled in computational mistakes at the blackboard But this had little importance In fact, his lectures revealed an unusual mastery of his subject and he had the ability of communicating to his students a feeling for the unity and logic of physical theory which is so apparent in his own written work.

  216. Radon biography
    • The course he studied was a broad one including mathematics, physics, chemistry, logic, philosophy, and he also included some lecture courses on music.

  217. Maxwell biography
    • At the age of 16, in November 1847, Maxwell entered the second Mathematics class taught by Kelland, the natural philosophy (physics) class taught by Forbes and the logic class taught by William Hamilton.

  218. Goodstein biography
    • Goodstein worked on mathematical logic, in particular ordinal numbers, recursive arithmetic, analysis, and the philosophy of mathematics.

  219. Riccioli biography
    • He taught logic, physics, and metaphysics at the Jesuit College in Parma from 1629 to 1632.

  220. Ingarden biography
    • Probably at that time he developed his broad intellectual interests, exceeding his future profession of a theoretical physicist, and including linguistics, epistemology, logic, history of science as well as biology, biophysics and informatics, to name just a few.

  221. Cassels James biography
    • At Edinburgh University Cassels studied Ordinary Mathematics, Natural Philosophy, and Chemistry in session 1910-11; Ordinary Logic, Intermediate Honours Mathematics and Experimental Physics in 1911-12; Intermediate Honours Mathematics, and Natural Philosophy (Dynamics and Thermodynamics), Advanced Mathematics, and Experimental Physics in session 1912-13; Honours Advanced Mathematics, Function Theory, Electrostatics, Dynamics (Advanced), and Experimental Physics (Advanced) in session 1913-14.

  222. Green Sandy biography
    • His voice was never raised; logic and clarity sufficed.

  223. Halmos biography
    • Virginia had been born on 21 December 1915 in Omaha, Nebraska and had studied at Vassar College followed by graduate study in logic and the foundations of mathematics at Brown University.
    • Professor Halmos may look like one mathematician, but in reality be is an equivalence class and has worked in several fields including algebraic logic and ergodic theory; this afternoon his representative from Hilbert space will speak to us.
    • These include Finite dimensional vector spaces (1942), Measure theory (1950), Introduction to Hilbert space and theory of spectral multiplicity (1951), Lectures on ergodic theory (1956), Entropy in ergodic theory (1959), Naive set theory, Algebraic logic (1962), A Hilbert space problem book (1967) and Lectures on Boolean algebras (1974).

  224. Kasner biography
    • in 1896 having studied a range of subjects including mathematics, astronomy, logic, physics, and political science.

  225. Yushkevich biography
    • Yushkevich's work was characterised by an exceptional skill in analysing historical sources, irreproachable logic, carefully considered assessments and historical judgements, and a striking ability to illuminate specific problems by placing them in general historical setting.

  226. Brash biography
    • In session 1906-07 he studied Chemistry, Political Economy, German, and Logic and Metaphysics.

  227. Porphyry biography
    • In particular his commentary on Aristotle's Categories led to the later developments of logic.

  228. De Morgan biography
    • He introduced De Morgan's laws and his greatest contribution is as a reformer of mathematical logic.

  229. Hobbes biography
    • He did not much care for logic, yet he learned it, and thought himself a good disputant.

  230. Vailati biography
    • Vailati worked on mathematical logic, working closely with Peano on this topic, and also on the history and methodology of science.

  231. Whitney biography
    • A final section on other topics includes nine papers on logic, geometry, and the mathematics of physical quantities, for the last of which he received a Lester Ford Award.

  232. Sitter biography
    • The expanding universe of de Sitter must be regarded as something more than an inexorable conclusion drawn from the strictest kind of logic with which the human mind is familiar.

  233. Bassi biography
    • She is said to have studied anatomy, natural history, logic, metaphysics, philosophy, chemistry, hydraulics, mechanics, algebra, geometry, ancient Greek, Latin, French, and Italian.

  234. Thabit biography
    • logic, psychology, ethics, the classification of sciences, the grammar of the Syriac language, politics, the symbolism of Plato's Republic ..

  235. Pullar biography
    • In 1879-80, his second year of study, Pullar took the courses Logic, Greek 2, and Latin 2.

  236. Black biography
    • In this work he looked at two main ideas, one being the nature of and the observability of vagueness and the other one being the relevance vagueness might have for logic.

  237. Bruno Giordano biography
    • He attended lectures on humanities, logic and dialectics in Naples and it was at this time that he was influenced by one of his teachers towards Averroism.

  238. Karsten biography
    • However his luck changed when the professor of logic at Rostock died leaving a professorial vacancy.

  239. Browne biography
    • These notes were Sets, Logic and Mathematical Thought (1957), Introduction to Linear Algebra (1959), Elementary Matrix Algebra (1969), and Algebraic Structures (1974).

  240. Routledge biography
    • In 1956 he published Logic on electronic computers: a practical method for reducing expressions to conjuctive normal form.

  241. Zylinski biography
    • Eustachy Żyliński worked in number theory, algebra, logic and foundations of mathematics.

  242. Fine Nathan biography
    • As a mathematician Fine had wide interests publishing on many different topics including number theory, logic, combinatorics, group theory, linear algebra, partitions and functional and classical analysis.

  243. Bacon biography
    • His initial studies covered the trivium of grammar, logic, and rhetoric.

  244. Wenninger biography
    • Fortunately there was one man there, Thomas Greenwood, in the philosophy department, who was willing to give me a course in English, in symbolic logic.

  245. Schwartz Jacob biography
    • A brief list of some of the areas to which Schwartz has made major contributions gives some notion of his breadth: spectral theory of linear operators, von Neumann algebras, macro economics, the mathematics of quantum field theory, parallel computation, computer time-sharing, high-level programming languages, compiler optimization, transformational programming, computational logic, motion planning in robotics, and, most recently, multimedia.

  246. Pairman biography
    • In her first year at Edinburgh University she studied Mathematics, Natural Philosophy, Chemistry, and Logic.

  247. Al-Samarqandi biography
    • He wrote works on theology, logic, philosophy, mathematics and astronomy which have proved important in their own right and also in giving information about the works of other scientists of his period.

  248. Boscovich biography
    • He remained at the Collegium Romanum, however, and continued to teach mathematics and logic there while undertaking deep scientific research in addition to studying theology.

  249. Kahler biography
    • His speculative considerations are illustrated by suggestive examples from set theory, mathematical logic, abstract algebra and differential, algebraic and analytic geometry.

  250. Cohn biography
    • It is Cohn's merit to provide a coherent treatment of this subject which at the same time leads the reader to a wide range of interesting and important research problems, related to questions in algebra, geometry and logic.

  251. Dirac biography
    • reflects Dirac's very characteristic approach: abstract but simple, always selecting the important points and arguing with unbeatable logic.

  252. Fraenkel biography
    • Symbolic Logic 13 (1) (1948), 56.','Reference ',28)">28]:- .

  253. Plato biography
    • He also contributed to logic and legal philosophy, including rhetoric.

  254. Kemeny biography
    • A teaching innovation which Kemeny introduced was in developing a Finite Mathematics course including topics that are no surprise to us today: logic, probability and matrix algebra.

  255. Crofton biography
    • This ninth edition of Encyclopaedia Britannica was edited by T S Baynes, Professor of Logic and Metaphysics at the University of St Andrews.

  256. Kline biography
    • Accordingly the book shows how various developments in mathematics proper in turn influenced developments in logic, astronomy, philosophy, painting, music, religious thought, literature, and the social sciences.

  257. De Moivre biography
    • Despite the Edict, the Protestant Academy at Sedan was suppressed in 1682 and de Moivre, forced to move, then studied logic at Saumur until 1684.

  258. Kronecker biography
    • Intuitionism stresses that mathematics has priority over logic, the objects of mathematics are constructed and operated upon in the mind by the mathematician, and it is impossible to define the properties of mathematical objects simply by establishing a number of axioms.

  259. Gysel biography
    • According to Gysel himself, mathematics is the least popular and least enjoyable subject for many pupils, because it demands strict, implacable logic and a selfless devotion to the subject matter.

  260. Schrodinger biography
    • I was a good student in all subjects, loved mathematics and physics, but also the strict logic of the ancient grammars, hated only memorising incidental dates and facts.

  261. Lacroix biography
    • The authors, breaking with the intuitionism that had dominated eighteenth-century French treatises, updated the logic of geometry manuals.

  262. Drysdale biography
    • In the following session, from October 1896 to March 1897 he studied Logic, Psychology and Chemistry at the Ordinary level, then from May to July of 1897 he studied Latin, also at the Ordinary level.

  263. McMullen biography
    • His work there involved VLSI design problems: graph theory, logic synthesis, boolean minimization, and sparse matrix processing.

  264. Born biography
    • The list of courses he took in session 1901-02 was certainly impressive, including mathematics, astronomy, physics, chemistry, logic, philosophy, and zoology.

  265. Euwe biography
    • He is logic personified, a genius of law and order.

  266. Beattie biography
    • He took the Preliminary Examinations of the Educational Institute of Scotland, passing English, History, Geography, Latin, Arithmetic, Algebra, Euclid I II III, Mechanics, Logic, and Natural Philosophy.

  267. Tanaka biography
    • These discussions helped us understand how to seek the logic and concepts behind the material we read.

  268. Apaczai biography
    • if the time during which we now make efforts to ram [students'] heads almost to surfeit with grammar and in some cases with rhetoric or logic, were used for lectures on the interesting subjects of mathematics and physics, we would give them a source of unspeakable joy for all their lives ..

  269. Philip biography
    • In the following year, 1870-71, he took Logic, Latin 2, Mathematics 2, Anatomy, Physiology and Hygiene, Natural History and Comparative Anatomy.

  270. Lefebure biography
    • The professor cannot be considered a benevolent reviewer, his voice is harsh, his tone is dismissive, also he questioned in a very firm stiff manner, and showed no mercy for any errors of logic.

  271. Caramuel biography
    • from the University of Alcala having written a dissertation on Infinite Logic, after which he entered the Cistercian Order at the Monasterio de la Espina near Medina de Rioseco, Valladolid.

  272. Biancani biography
    • The Jesuit College, established in Padua in 1542, had become an important educational establishment by 1590 offering a three-year philosophy degree; logic was taught in year one, natural philosophy and physical science in year two, and metaphysics and natural philosophy in year three.

  273. Lambert biography
    • In [Filosofia e geometria : Lambert interprete di Euclide, Il Filarete : Pubblicazioni della Facolta di Lettere e Filosofia dell\'Universita degli Studi di Milano 183 (Florence, 1999).','Reference ',3)">3] Basso highlights Lambert's understanding of the main concepts of the deductive-geometric methodology, namely axioms, postulates, theorems, problems, constructions, and logic.

  274. Hayes biography
    • My courses with her included Calculus, Celestial Mechanics, Logic, and Astronomy.

  275. Nash biography
    • In September 1948 Nash entered Princeton where he showed an interest in a broad range of pure mathematics: topology, algebraic geometry, game theory and logic were among his interests but he seems to have avoided attending lectures.

  276. Thomae biography
    • Logic 8 (1) (1987), 25-44.

  277. Budan de Boislaurent biography
    • He undertook various teaching duties at the College, including logic and physics in his last two years there.

  278. Hutton James biography
    • At the University of Edinburgh Hutton was taught mathematics by Maclaurin and logic and metaphysics by John Stevenson.

  279. Tao biography
    • There are also appendices on mathematical logic and the decimal system.

  280. Fiorentini biography
    • There are three parts, dealing with logic (i.e., first order predicate calculus), set theory and algebraic structures, respectively.

  281. Truesdell biography
    • While at Princeton working on his thesis, Truesdell attended Alonzo Church's graduate course Introduction to Mathematical Logic and the notes that he took became Church's book published with this title in 1944.

  282. Orszag biography
    • This book is not the usual "mathematical methods for engineers" text, which could contain, according to the taste of its author, almost any elementary, or intermediate level topic in analysis, linear algebra, numerical analysis, game theory, systems theory, functional analysis, probability, statistics, or even logic, generally with little cohesion among the different parts of such a mixture.

  283. Lonie biography
    • 1837-38 Greek Provectior, Mathematics 1, Logic .

  284. Rado biography
    • He worked on logic and theoretical computer science, particularly Turing machines, publishing On non-computable functions in 1962 and Computer studies of Turing machine problems in 1965.

  285. Carslaw biography
    • The breadth of his course in comparison to courses of today is shown by the fact that he also studied Latin, Greek, Moral Philosophy and Logic.

  286. Temple biography
    • to establish the consistency of set theory, abstract arithmetic and propositional logic and the method used is to construct a new and fundamental theory from which these theories can be deduced.

  287. Macintyre biography
    • The vocabulary covers a fairly wide field in pure mathematics, but applied mathematics, statistics and mathematical logic are not included.

  288. Halsted biography
    • After this brief period of study, he became Sylvester's first student at Johns Hopkins University where he studied for his doctorate which was awarded in 1879 for his dissertation Basis for a Dual Logic .

  289. Kantorovich biography
    • The absence of any orator's abilities neighboured his deep logic and special mastery in polemics.

  290. Albertus biography
    • While in Paris Albertus began the task of presenting the entire body of knowledge, natural science, logic, rhetoric, mathematics, astronomy, ethics, economics, politics and metaphysics.

  291. Cusa biography
    • He was interested in geometry and logic and had clearly made a study of at least parts of Euclid's Elements and works of Thomas Bradwardine and Campanus of Novara.

  292. Whitehead biography
    • At the time they began collaborating, Whitehead was working on his article Memoir on the algebra of symbolic logic while Russell was close to finishing the first draft of his Principles of mathematics.

  293. Lorgna biography
    • Spallanzani served as professor of logic, metaphysics, and Greek, then as professor of physics at the University of Modena before gaining worldwide recognition as a physiologist while holding a chair at the University of Pavia.

  294. Mihoc biography
    • Probability is a science of recent formation; hence in it, better than in other branches of mathematics, one can see the relationship between the empirical contribution and the one given by reasoning, and between the process of inductive and deductive logic used in it.

  295. Bernoulli Jacob biography
    • Jacob Bernoulli's first important contributions were a pamphlet on the parallels of logic and algebra published in 1685, work on probability in 1685 and geometry in 1687.

  296. Lawson biography
    • In session 1887-88 he took the classes Greek 2, Latin 2, and Mathematics 1; in 1888-89 he took Logic, English Literature, and Mathematics 2; in session 1889-90 he took Natural Philosophy, Moral Philosophy and Political Economy, English Literature, Advanced Metaphysics, and Mathematics 3; in session 1890-91 he took Natural Philosophy, Chemistry, Natural History, and Physiology.

History Topics

  1. Bolzano's manuscripts references
    • J Berg, Bolzano's Logic (Stockholm, 1962).
    • Y Bar-Hillel, Bolzano's propositional logic, Arch.
    • J Berg, Is Russell's antinomy derivable in Bolzano's logic?, Bolzano - Studien.
    • J Berg, A requirement for the logical basis of scientific theories implied by Bolzano's logic of variation, in Impact of Bolzano's epoch on the development of science (Prague, 1982), 415-425.
    • K Berka, Bernard Bolzano - historian of logic (Czech), DVT---Dejiny Ved Tech.
    • Logic 12 (3) (1983), 299-318.
    • P Simons, Bolzano, Tarski, and the limits of logic, Bolzano - Studien.
    • W Stelzner, Compatibility and relevance: Bolzano and Orlov, in The Third German-Polish Workshop on Logic & Logical Philosophy, Dresden, 2001, Logic Log.
    • Logic 2 (1981), 11-20.

  2. Bolzano's manuscripts references
    • J Berg, Bolzano's Logic (Stockholm, 1962).
    • Y Bar-Hillel, Bolzano's propositional logic, Arch.
    • J Berg, Is Russell's antinomy derivable in Bolzano's logic?, Bolzano - Studien.
    • J Berg, A requirement for the logical basis of scientific theories implied by Bolzano's logic of variation, in Impact of Bolzano's epoch on the development of science (Prague, 1982), 415-425.
    • K Berka, Bernard Bolzano - historian of logic (Czech), DVT---Dejiny Ved Tech.
    • Logic 12 (3) (1983), 299-318.
    • P Simons, Bolzano, Tarski, and the limits of logic, Bolzano - Studien.
    • W Stelzner, Compatibility and relevance: Bolzano and Orlov, in The Third German-Polish Workshop on Logic & Logical Philosophy, Dresden, 2001, Logic Log.
    • Logic 2 (1981), 11-20.

  3. Bolzano publications.html
    • This volume contains a biography of Bolzano together with details of the topics on which he worked: mathematics, logic, theology, philosophy and aesthetics.
    • This volume in the series of posthumous writings of the Collected works of Bernard Bolzano contains transcriptions of some early manuscripts which present his views in the 1810's on the foundations of logic, mathematics and physics.
    • Among the other contributions, a manuscript on the basic concepts of logic adumbrates fundamental themes in Bolzano's major work, the Wissenschaftslehre of 1837.
    • Most of the items on philosophy relate to logic.
    • Bernard Bolzano, Grundlegung der Logik (German), [Foundations of logic] Selected paragraphs from Wissenschaftslehre, Band I und II.
    • The first two volumes of Bolzano's Wissenschaftslehre published in 1837 are concerned with logic.
    • Bolzano's manuscript giving his ideas on logic and semantics which became the basis of his major publication Wissenschaftslehre which appeared in 1837.

  4. Set theory references
    • Symbolic Logic 2 (1) (1996), 1-71.
    • Symbolic Logic 53 (1) (1988), 2-6.
    • G H Moore, The Origins of Zermelo's axiomatisation of set theory, Journal of Philosophical Logic 7 (1978), 307-329.

  5. Set theory references
    • Symbolic Logic 2 (1) (1996), 1-71.
    • Symbolic Logic 53 (1) (1988), 2-6.
    • G H Moore, The Origins of Zermelo's axiomatisation of set theory, Journal of Philosophical Logic 7 (1978), 307-329.

  6. Mathematical games references
    • 1 : Symbolic logic and The game of logic (New York, 1958).

  7. Jaina mathematics references
    • I Schneider, The contributions of the sceptic philosophers Arcesilas and Carneades to the development of an inductive logic compared with the Jaina-logic, in Proceedings of the Symposium on the 1500th Birth Anniversary of Aryabhata I, New Delhi, 1976, Indian J.

  8. U of St Andrews History
    • His works on Logic, Physics and Natural Philosophy and Metaphysics were studied in the second, third and fourth years respectively.
      Go directly to this paragraph
    • John Maior, renowned for his work in philosophy, logic and in particular on infinity, lectured in theology at St Andrews from 1531 until 1534 when he became the Provost of St Salvator's, a post he held until his death at the age of 80 in 1550.
      Go directly to this paragraph

  9. Jaina mathematics references
    • I Schneider, The contributions of the sceptic philosophers Arcesilas and Carneades to the development of an inductive logic compared with the Jaina-logic, in Proceedings of the Symposium on the 1500th Birth Anniversary of Aryabhata I, New Delhi, 1976, Indian J.

  10. Ledermann interview
    • He said, "Yes, I remember you, you are a mathematician, and now this mathematical logic, which you are studying, would you say because of this, the traditional logic of Aristotle is now invalid?" Now I knew what he meant.

  11. Christianity and Mathematics
    • The first of these were arguments based purely on logic, for example that the "day" is defined by the passage of the Sun in the sky so to talk, as in Genesis, of several days passing before the Sun was created makes no sense.
    • He encouraged logic, mathematics and science to be seen as contributing to Christianity rather than being opposed to it.

  12. Word problems
    • It required computability theory and developments in mathematical logic to even make the questions precise, but these areas were to not only provide explicit questions, they also provided solutions to the questions.
    • In the 1930s Kurt Godel investigated how symbolic manipulation in formal logic could be simulated by functions on the natural numbers.

  13. Mathematical games references
    • 1 : Symbolic logic and The game of logic (New York, 1958).

  14. Topology history references
    • J Dieudonne, The beginnings of topology from 1850 to 1914, in Proceedings of the conference on mathematical logic 2 (Siena, 1985), 585-600.

  15. 20th century time references
    • I Prigogine, The rediscovery of time, in Logic, methodology and philosophy of science, VIII, Moscow, 1987 (Amsterdam, 1989), 29-46.

  16. Classical time references
    • I Prigogine, The rediscovery of time, in Logic, methodology and philosophy of science, VIII, Moscow, 1987 (Amsterdam, 1989), 29-46.

  17. Real numbers 2 references
    • Logic 8 (1) (1987), 25-44.

  18. Amusements.html
    • Every puzzle that is worthy of consideration can be referred to mathematics and logic.

  19. Squaring the circle
    • Although some, such as Aristotle, seemed to fail to understand the logic of Hippocrates argument, there seems little doubt that Hippocrates was perfectly aware that his methods had failed to square the circle.

  20. Real numbers 3 references
    • Logic 8 (1) (1987), 25-44.

  21. Harriot's manuscripts
    • Now that the Harriot papers have seen the light of day, Harriot stands clear as a key figure at the time when the new science of logic, reason, mathematics, and experiment was coming into being.

  22. Infinity references
    • J E Fenstad, Infinities in mathematics and the natural sciences, in Methods and applications of mathematical logic, Campinas, 1985, Contemp.

  23. Infinity
    • Fenstad, in [Methods and applications of mathematical logic, Campinas, 1985, Contemp.

  24. Measurement
    • Although we might think there is an inescapable logic in dividing it in a systematic manner, this ignores the way that measuring grew up with people measuring shorter lengths using other parts of the human body.

  25. Mathematical games

  26. Set theory

  27. Calculus history
    • However when Berkeley published his Analyst in 1734 attacking the lack of rigour in the calculus and disputing the logic on which it was based much effort was made to tighten the reasoning.
      Go directly to this paragraph

  28. function concept
    • If logic were the teacher's only guide, he would have to begin with the most general, that is to say, the most weird functions.

  29. Classical time references
    • I Prigogine, The rediscovery of time, in Logic, methodology and philosophy of science, VIII, Moscow, 1987 (Amsterdam, 1989), 29-46.

  30. Real numbers 2 references
    • Logic 8 (1) (1987), 25-44.

  31. Bolzano's manuscripts
    • The first volume in the new series Bernard Bolzano-Gesamtausgabe published by Friedrich Frommann Verlag and edited by Eduard Winter, Jan Berg, Friedrich Kambartel, Jaromir Louzil, and Bob van Rootselaar, contains a biography of Bolzano together with details of the topics on which he worked: mathematics, logic, theology, philosophy and aesthetics.

  32. Greek astronomy
    • Alexandria in the second century AD saw the publication of Ptolemy's remarkable works, the 'Almagest' and the 'Handy Tables', the 'Geography', the 'Tetrabiblos', the 'Optics', the 'Harmonics', treatises on logic, on sundials, on stereographic projection, all masterfully written, products of one of the greatest scientific minds of all times.

  33. Real numbers 3 references
    • Logic 8 (1) (1987), 25-44.

  34. Infinity references
    • J E Fenstad, Infinities in mathematics and the natural sciences, in Methods and applications of mathematical logic, Campinas, 1985, Contemp.

  35. 20th century time references
    • I Prigogine, The rediscovery of time, in Logic, methodology and philosophy of science, VIII, Moscow, 1987 (Amsterdam, 1989), 29-46.

  36. Topology history references
    • J Dieudonne, The beginnings of topology from 1850 to 1914, in Proceedings of the conference on mathematical logic 2 (Siena, 1985), 585-600.

Famous Curves

No matches from this section

Societies etc

  1. Gödel Lecturer
    • The Godel Lecture is an invited address delivered each year at the Association for Symbolic Logic Annual Meeting.
    • 1993 Angus Macintyre, Logic of Real and p-adic Analysis: Achievements and Challenges.
    • 1999 Stephen A Cook, Logic and computatonal complexity.
    • 2006 Per Martin-Lof, The two layers of logic.

  2. National Academy of Sciences of Italy
    • Spallanzani served as professor of logic, metaphysics, and Greek, then as professor of physics at the University of Modena before gaining worldwide recognition as a physiologist while holding a chair at the University of Pavia.
    • A physicist who made contributions to chemistry, he worked at Bologna and Pisa where he taught logic, then physics.
    • He worked at the universities of Reggio and Modena as professor of logic, metaphysics and Greek, then at the University of Pavia as professor of natural history.
    • An entomologist who was professor of logic at Pisa, becoming professor of natural history there in 1801.

  3. Karp Prize
    • It is awarded for an outstanding paper or book in the field of symbolic logic.
    • The award is made by the Association for Symbolic Logic every five years.
    • for his fundamental work connecting logic with geometric group theory.

  4. BMC 1951
    • Goodstein, R LMathematical logic .
    • Shepherdson, J CMathematical logic .
    • Turing, A MMathematical logic .

  5. International Congress Speaker
    • Stephen Cole Kleene, Mathematical Logic: Constructive and Non-Constructive Operations.
    • Alonzo Church, Logic, Arithmetic, and Automata.
    • Anatoly Ivanovich Malcev, On Some Questions on the Border of Algebra and Logic.

  6. BMC Committee
    • Symbolic logic, Algebraic methods in analysis, Valuation theory, Topological groups.
    • 1) that an evening session should be devoted to symbolic logic, .

  7. Minutes for 1997
    • Prof Hugh Woodin (Berkeley), mathematical logic.
    • Special Sessions for 50th BMC It was agreed that the special sessions for Manchester will be on Mathematical Logic and Dynamical Systems, and that Prof A J Macintyre (Oxford) and Dr S M Rees (Liverpool) should be invited to organise the special sessions with the help of Manchester staff in these areas.

  8. Minutes for 1997
    • Prof Hugh Woodin (Berkeley), mathematical logic.
    • Special Sessions for 50th BMC It was agreed that the special sessions for Manchester will be on Mathematical Logic and Dynamical Systems, and that Prof A J Macintyre (Oxford) and Dr S M Rees (Liverpool) should be invited to organise the special sessions with the help of Manchester staff in these areas.

  9. BMC 2000
    • Friedman, H The mathematical meaning of mathematical logic .
    • Zilber, B Logic of classical analytic functions and diophantine number theory .

  10. Minutes for 1997
    • Prof Hugh Woodin (Berkeley), mathematical logic.
    • Special Sessions for 50th BMC It was agreed that the special sessions for Manchester will be on Mathematical Logic and Dynamical Systems, and that Prof A J Macintyre (Oxford) and Dr S M Rees (Liverpool) should be invited to organise the special sessions with the help of Manchester staff in these areas.

  11. BMC 1974
    • Scott, D SSheaves and logic .

  12. BMC 1986
    • Pitts, A MMathematical logic and category theory .

  13. BMC 1991
    • Macintyre, A J Logic and analytic functions .

  14. BMC 1977
    • Paris, J BAn application of logic to number theory .

  15. Minutes for 2002
    • Suggested themes: number theory, analysis, groups, algebra / groups, topology, logic/etc, history/etc.

  16. Minutes for 1963
    • Consideration of the possibility of combining the Stochastic Analysis and Logic Colloquia with the BMC was held over, because it could not be arranged for 1964.

  17. Minutes for 1954
    • It was agreed that one day should be devoted to each of Probability with Logic and Foundations, Analysis with Number Theory, and Topology.

  18. Czech Academy of Sciences
    • The Institute is concerned mainly with mathematical analysis (differential equations, numerical analysis, functional analysis, theory of functions, mathematical physics), probability theory and mathematical statistics, mathematical logic, theoretical computer science and graph theory, numerical algebra, topology (general and algebraic) and theory of teaching mathematics.

  19. Bulgarian Academy of Sciences
    • As third and final example we mention that in 1980 the Centre for Mathematics and Mechanics of the Bulgarian Academy of Sciences organised a conference on mathematical logic dedicated to the memory of A A Markov (1903-1979).

  20. Sylvester Medal
    • for his distinguished contributions to combinatory topology, Boolean algebras and mathematical logic.

  21. IMU Nevanlinna Prize
    • All mathematical aspects of computer science, including complexity theory, logic of programming languages, analysis of algorithms, cryptography, computer vision, pattern recognition, information processing and modelling of intelligence.

  22. Royal Irish Academy
    • Let us note a further mathematical connection in that in 1946 the Academy established a professorship of Mathematical logic and appointed Jan Lukasiewicz.

  23. AMS Steele Prize
    • In 1994 the last of these three categories was put onto a five year cycle of topics: analysis, algebra, applied mathematics, geometry and topology, and discrete mathematics/logic.

  24. EMS Founder Members
    • student of Logic at Edinburgh .

  25. Swedish Academy of Sciences
    • International awards include the Crafoord Prize established in 1980 and awarded every year (for research in mathematics, astronomy, geology, and biology) and the Rolf Schock Prizes awarded every second year (for logic and philosophy, mathematics, the visual arts, and music).

  26. Wolf Prize
    • for his many fundamental contributions to mathematical logic and set theory, and their applications within other parts of mathematics.

  27. MAA Chauvenet Prize
    • Are Logic and Mathematics Identical?, Science 138 (1962), 788-794.

  28. Serbian Academy of Sciences
    • In the 1970s logic became the most important research area, having previously not been studied by Institute members.

  29. European Mathematical Society Prize
    • In mathematical logic, he found a striking example of a combinatorial unprovable statement.

  30. Minutes for 1977
    • 5 Analysis, 3 Topology, 4 Algebra, 2 Number Theory, 1 Geometry, 1 Combinatorics, 1 Logic, 1 Probability.

  31. Minutes for 1949
    • Symbolic logic, Algebraic methods in analysis, Valuation theory, Topological groups.

  32. Minutes for 1957
    • (b) that one day should be devoted to each of (i) Analysis and Statistics (ii) Algebra and Logic (iii) Geometry and Topology.

  33. BMC Press release
    • This year these lectures will be on the mathematics of the nature of matter ('string theory'), logic and the interaction of mathematics with computational problems.

  34. BMC 1979

  35. Minutes for 1950
    • 1) that an evening session should be devoted to symbolic logic, .

  36. BMC 1998

  37. Minutes for 2000
    • Logic: .

  38. BMC 1955
    • Newman, M H AOrdinal logic .

  39. References for Gottingen
    • V Peckhaus, Ernst Zermelo in Gottingen, History and Philosophy of Logic 11 (1990), 19-58.


  1. References for Tarski
    • P Simons, Philosophy and logic in Central Europe from Bolzano to Tarski : Selected essays (Dordrecht, 1992).
    • I H Anellis, Tarski's development of Peirce's logic of relations, in Studies in the logic of Charles Sanders Peirce (Bloomington, IN, 1997), 271-303.
    • C N Bach, Tarski's 1936 account of logical consequence, Modern Logic 7 (2) (1997), 109-130.
    • Symbolic Logic 53 (1) (1988), 36-50.
    • Symbolic Logic 53 (1) (1988), 20-35.
    • Symbolic Logic 53 (1) 1988), 51-79.
    • Logic 10 (2) (1989), 165-179.
    • Symbolic Logic 51 (4) (1986), 913-941.
    • S R Givant, Tarski's development of logic and mathematics based on the calculus of relations, in Algebraic logic, Budapest, 1988 (Amsterdam, 1991), 189-215.
    • Logic 19 (4) (1998), 227-234.
    • Formal Logic 37 (1) (1996), 125-151.
    • Symbolic Logic 51 (4) (1986), 866-868.
    • Symbolic Logic 51 (4) (1986), 883-889.
    • Symbolic Logic 53 (1) (1988), 2-6.
    • Logic 25 (6) (1996), 567-580.
    • Symbolic Logic 51 (4) (1986), 890-898.
    • J D Monk, The contributions of Alfred Tarski to algebraic logic, J.
    • Symbolic Logic 51 (4) (1986), 899-906.
    • Logic 19 (3) (1998), 153-160.
    • J Pla i Carrera, Alfred Tarski and contemporary logic I (Catalan), Butl.
    • Logic 25 (6) (1996), 617-677.
    • Symbolic Logic 3 (2) (1997), 216-241.
    • G Schurz, Tarski and Carnap on logical truth-or : what is genuine logic?, in Alfred Tarski and the Vienna Circle, Vienna, 1997 (Dordrecht, 1999), 77-94.
    • Symbolic Logic 61 (2) (1996), 653-686.
    • P Simons, Bolzano, Tarski, and the limits of logic, Bolzano-Studien, Philos.
    • Symbolic Logic 53 (1) (1988), 80-91.
    • Symbolic Logic 51 (4) (1986), 907-912.
    • Symbolic Logic 5 (2) (1999), 175-214.
    • Logic 2 (1981), 11-20.
    • Symbolic Logic 53 (1) (1988), 7-19.
    • Symbolic Logic 51 (4) (1986), 869-882.
    • Symbolic Logic 52 (4) (1987), vii.

  2. References for Schroder
    • G Brady, From Pierce to Skolem: A Neglected Chapter in the History of Logic (Elsevier, 2000).
    • J Gasser, A Boole Anthology : Recent and Classical Studies in the Logic of George Boole (Springer-Verlag, 2000).
    • A N Kolmogorov, A P Yushkevich, A Shenitzer, H Grant and O B Sheinin, Mathematics of the 19th Century : Mathematical Logic, Algebra, Number Theory, Probability Theory (Birkhauser, 2001).
    • I H Anellis, Schroder material at the Russell archives, Modern Logic 1 (2-3) (1990/91), 237-245.
    • R R Dipert, Individuals and extensional logic in Schroder's Vorlesungen uber die Algebra der Logik, Modern Logic 1 (2-3) (1990/91), 140-159.
    • R R Dipert, The life and work of Ernst Schroder, Modern Logic 1 (2-3) (1990/91), 117-139.
    • F Ferrante, The origins of thought in Ernst Schroder's Introduction to lessons on algebra of logic (1890), Metalogicon 9 (2) (1996), 105-137.
    • F Ferrante, 'Folgerichtigkeit' - the basic conception of logical thought in Ernst Schroder's introduction to [his] Lessons on algebra of logic (1890), Metalogicon 8 (1) (1995), 33-40.
    • L Gruszecki, Ernst Schroder's algebra of logic, Zeszyty Nauk.
    • N Houser, The Schroder-Peirce correspondence, Modern Logic 1 (2-3) (1990/91), 206-236.
    • S G Ibragimov, On forgotten works of Ernst Schroder lying between algebra and logic (Russian), Istor.-Mat.
    • V Peckhaus, Wozu Algebra der Logik? Ernst Schroders Suche nach einer universalen Theorie der Verknupfungen, Modern Logic 4 (4) (1994), 357-381.
    • V Peckhaus, Ernst Schroder und die 'pasigraphischen Systeme' von Peano und Peirce, Modern Logic 1 (2-3) (1990/91), 174-205.
    • V Peckhaus, 19th Century Logic between Philosophy and Mathematics, Bulletin of Symbolic Logic 5 (1999), 433-450.
    • V Peckhaus, The influence of Hermann Gunther Grassmann and Robert Grassmann on Ernst Schroder's algebra of logic, in Hermann Gunther Grassmann (1809-1877): visionary mathematician, scientist and neohumanist scholar, Boston Stud.
    • V Peckhaus, Schroder's Logic, in D M Gabbay and John Woods (eds.), Handbook of the History of Logic.
    • 3: The Rise of Modern Logic: From Leibniz to Frege (North Holland, 2004), 557-609.
    • C Thiel, Ernst Schroder and the distribution of quantifiers, Modern Logic 1 (2-3) (1990/91), 160-173.
    • Logic 2 (1981), 21-23.

  3. References for Frege
    • Logic 19 (3) (1998), 137-151.
    • R Born, Frege, in Philosophy of science, logic and mathematics in the twentieth century (London, 2001), 124-156.
    • Logic 16 (2) (1995), 245-255.
    • R R Dipert, Peirce, Frege, the logic of relations, and Church's theorem, Hist.
    • Logic 5 (1) (1984), 49-66.
    • B S Hawkins, Jr., Peirce and Frege, a question unanswered, Modern Logic 3 (4) (1993), 376-383.
    • Symbolic Logic 58 (1993), 579-601.
    • Logic 18 (1) (1997), 17-31.
    • W Kneale, Gottlob Frege and mathematical logic, in A J Ayer et al., The Revolution in Philosophy (New York, 1956).
    • I Max, Freges 'selbstverstandliche Voraussetzung' und die Behandlung von Existenzprasuppositionen durch die free logic, in Frege conference, 1984, Schwerin, 1984 (Berlin, 1984), 240-245.
    • Logic 11 (1) (1990), 5-17.
    • Logic 8 (1) (1987), 25-44.
    • Logic 22 (2) (2001), 57-73.
    • Logic 2 (1981), 21-23.
    • C Thiel, From Leibniz to Frege : mathematical logic between 1679 and 1879, in Logic, methodology and philosophy of science, VI, Hannover, 1979 (Amsterdam-New York, 1982), 755-770.
    • Logic 18 (4) (1997), 201-209.

  4. References for MacColl
    • M Astroh, I Grattan-Guinness and S Read, A survey of the life of Hugh MacColl (1837-1909), History and Philosophy of Logic 22 (2) (2001), 81-98.
    • F Cavaliere, L'opera di Hugh MacColl alle origini delle logiche non-classiche, Modern Logic 6 (4) (1996), 373-402.
    • S E Cuypers, The Metaphysical Foundations of Hugh MacColl's Religious Ethics, Nordic Journal of Philosophical Logic 3 (1), 175-196.
    • J G Hibben, Review: Symbolic Logic and Its Applications by Hugh MacColl, The Philosophical Review 16 (2) (1907), 190-194.
    • S H Olsen, Hugh MacColl-Victorian, in M Astroh and S Read (eds.), Proceedings of the Conference 'Hugh MacColl and the Tradition of Logic' at Greifswald, 1998 (Nordic Journal of Philosophical Logic, 1999), 197-229.
    • V Peckhaus, Hugh MacColl and the German Algebra of Logic, in M Astroh and S Read (eds.), Proceedings of the Conference 'Hugh MacColl and the Tradition of Logic' at Greifswald, 1998 (Nordic Journal of Philosophical Logic, 1999), 141-173.
    • S Rahman, Ways of understanding Hugh MacColl's concept of Symbolic Existence, in M Astroh and S Read (eds.), Proceedings of the Conference 'Hugh MacColl and the Tradition of Logic' at Greifswald, 1998 (Nordic Journal of Philosophical Logic, 1999), 35-58.
    • S Rahman, Hugh MacColl: eine bibliographische erschliessung seiner hauptwerke und notizen zu ihrer rezeptionsgeschichte, History and Philosophy of Logic 18 (3) (1997), 165-183.
    • S Rahman and J Redmond, Hugh MacColl and the birth of logical pluralism, Handbook of the History of Logic - British Logic in the Nineteenth Century 4 (North Holland, Amsterdam, 2008), 533-604.
    • S Read, Hugh MacColl and the Algebra of Strict Implication, in M Astroh and S Read (eds.), Proceedings of the Conference 'Hugh MacColl and the Tradition of Logic' at Greifswald, 1998 (Nordic Journal of Philosophical Logic, 1999), 59-83.
    • B Russell, Review: Symbolic Logic and Its Applications by Hugh MacColl, Mind, New Series 15 (58) (1906), 255-260.

  5. References for Hilbert
    • A Church, Review: Grundzuge der Theoretischen Logik (3rd ed), by D Hilbert and W Ackermann, The Journal of Symbolic Logic 15 (1) (1950), 59.
    • F H Fischer, Review: Grundzuge der Theoretischen Logik (4th ed), by D Hilbert and W Ackermann, The Journal of Symbolic Logic 25 (2) (1960), 158.
    • R L Goodstein, Review: Principles of Mathematical Logic, by D Hilbert and W Ackermann, The Mathematical Gazette 35 (314) (1951), 293-294.
    • S C Kleene, Review: Grundlagen der Mathematik Vol 2, by D Hilbert and P Bernays, The Journal of Symbolic Logic 5 (1) (1940), 16-20.
    • G T Kneebone, Review: Grundlagen der Mathematik I (2nd edition), by D Hilbert and P Bernays, The Journal of Symbolic Logic 35 (2) (1970), 321-323.
    • G T Kneebone, Review: Principles of Mathematical Logic, by D Hilbert, W Ackermann and Robert E Luce, Philosophy 27 (103) (1952), 375-376.
    • G T Kneebone, Review: Grundlagen der Mathematik II (2nd ed), by D Hilbert and Bernays, The Journal of Symbolic Logic 39 (2) (1974), 357.
    • G Lolli, Hilbert and logic (Italian), in The ideas of David Hilbert (Italian), Catania, 1999, Matematiche (Catania) 55 (suppl.
    • W H McCrea, Review: Principles of Mathematical Logic, by D Hilbert, W Ackermann, The British Journal for the Philosophy of Science 2 (8) (1952), 332-333.
    • W V Quine, Review: Grundzuge der Theoretischen Logik, by D Hilbert and W Ackermann, The Journal of Symbolic Logic 3 (2) (1938), 83-84.
    • Symbolic Logic 5 (1) (1999), 1-44.
    • Jan von Plato, Review: David Hilbert's Lectures on the Foundations of Geometry 1891-1902, edited by Michael Hallett and Ulrich Majer, The Bulletin of Symbolic Logic 12 (3) (2006), 492-494.
    • Logic 18 (4) (1997), 201-209.
    • G Zubieta R, Review: Principles of Mathematical Logic, by D Hilbert, W Ackermann, The Journal of Symbolic Logic 16 (1) (1951), 52-53.

  6. References for Church
    • C A Anderson and M Zeleny, Logic, meaning and computation : Essays in memory of Alonzo Church, Logic, meaning and computation (Dordrecht, 2001).
    • C A Anderson, Alonzo Church's contributions to philosophy and intensional logic, Bull.
    • Symbolic Logic 4 (2) (1998), 129-171.
    • Symbolic Logic 1 (4) (1995), 486-488.
    • Symbolic Logic 4 (2) (1998), 172-180.
    • In honor of Alonzo Church's 75th birthday with some remarks from the History of logic of A Dumitriu, Internat.
    • Logic Rev.
    • A Irving, Alonzo Church (1903-1995), Modern Logic 5 (1995), 408-410.
    • D Kaplan and T Burge, Remembering Alonzo Church, Logic, meaning and computation (Dordrecht, 2001), xi--xiii.
    • Logic 18 (4) (1997), 211-232.
    • Symbolic Logic 3 (2) (1997), 154-180.
    • UCLA philosopher, mathematician Alonzo Church dead at 92, Modern Logic 5 (4) (1995), 410-412.
    • ULCA Philosopher, Mathematician Alonzo Church dead at 92, History of Logic Newsletter 19 (Sept 1995), 1-2.

  7. References for Janovskaja
    • I H Anellis, Yanovskaya's 'ghost', Modern Logic 6 (1) (1996), 77-84.
    • I H Anellis, Sof'ya Aleksandrovna Yanovskaya's contributions to logic and history of logic, Modern Logic 6 (1) (1996), 7-36.
    • Logic 8 (1) (1987), 45-56.
    • I G Bashmakova, S S Demidov and V A Uspenskii, Sof'ya Aleksandrovna Yanovskaya (Russian), Modern Logic 6 (4) (1996), 357-372.
    • V Bazhanov, Restoration : S A Yanovskaya's path in logic, Hist.
    • Logic 22 (3) (2001), 129-133.
    • D P Gorskii, Sof'ja Aleksandrovna Janovskaja (Russian), in Studies in systems of logic dedicated to the memory of S A Janovskaja (Izdat.
    • B A Kushner, Sof'ja Aleksandrovna Janovskaja : a few reminiscences, Modern Logic 6 (1) (1996), 67-72.
    • A A Markov, A S Kuzichev and Z A Kuzicheva, Sof'ya Aleksandrovna Yanovskaya's work in the field of mathematical logic, Modern Logic 6 (1) (1996), 3-6.
    • B Rosenfeld, Reminiscences of S A Yanovskaya, Modern Logic 6 (1) (1996), 73-76.
    • B A Trakhtenbrot, In memory of S A Yanovskaya (1896-1966) on the centenary of her birth, Modern Logic 7 (2) (1997), 160-187.

  8. References for Leibniz
    • H Ishiguro, Leibniz's philosophy of logic and language (Cambridge, 1990).
    • Logic 12 (4) (1983), 143-147.
    • F Duchesneau, Leibniz and the philosophical analysis of science, in Logic, methodology and philosophy of science VIII (Amsterdam-New York, 1989), 609-624.
    • Z A Kuzicheva, Leibniz' logical program and its role in the history of logic and cybernetics (Russian), Voprosy Kibernet (Moscow) 78 (1982), 3-36.
    • W Lenzen, Concepts vs predicates : Leibniz's challenge to modern logic, in The Leibniz renaissance (Florence, 1989), 153-172.
    • Leibniz's logic, Topoi 9 (1) (1990), 29-59.
    • H A Lindemann, Leibniz and modern logic (Spanish), An.
    • Formal Logic 18 (2) (1977), 235-242.
    • J Pieters, Origines de la decouverte par Leibniz du calcul infinitesimal, in Publications from the Center for Logic 2 (Louvain-la-Neuve, 1981), 1-22.
    • Symbolic Logic 19 (1954), 1-13.
    • Logic 12 (1) (1991), 1-14.

  9. References for Lesniewski
    • J T J Srzednicki and Z Stachniak (eds.), Lesniewski's lecture notes in logic (Dordrecht, 1988).
    • Formal Logic 20 (4) (1979), 934-946.
    • W Marciszewski, The place of Stanislaw Lesniewski in contemporary logic-philosophical thought (Polish), Wiadom.
    • R Poli and M Libardi, Lesniewski's conception of logic, in The Lvov-Warsaw School and Contemporary Philosophy 1995 (Dordrecht, 1998), 139-152.
    • V F Rickey, A survey of Lesniewski's logic, in Lesniewski's systems protothetic (Dordrecht, 1998), 23-41.
    • V F Rickey, A survey of Lesniewski's logic, in On Lesniewski's systems, Krakow, 1976, Studia Logica 36 (4) (1977/78), 407-426.
    • Logic 15 (2) (1994), 227-235.
    • Logic 3 (2) (1982), 165-191.
    • S J Surma, On the work and influence of Stanislaw Lesniewski, in Logic Colloquium 76, Oxford, 1976 (Amsterdam, 1977), 191-220.
    • Formal Logic 7 (1966), 361-364.
    • J Wolenski, Lesniewski's logic and the concept of being, in Stanislaw Lesniewski aujourd'hui, Grenoble, October 8-10, 1992 (Neuchatel, 1996), 93-101.

  10. References for Boole
    • Logic 12 (1) (1991), 15-35.
    • I Grattan-Guinness, Psychology in the foundations of logic and mathematics : the cases of Boole, Cantor and Brouwer, Hist.
    • Logic 3 (1) (1982), 33-53.
    • T Hailperin, Boole's abandoned propositional logic, Hist.
    • Logic 5 (1) (1984), 39-48.
    • W Kneale, Boole and the Revival of Logic, Mind 57 (1948), 149-175.
    • W Kneale, Boole and the revival of logic, Mind 57 (1948), 149-175.
    • L de Ledesma, and L M Laita, George Boole : From differential equations to mathematical logic, in Proceedings of the Mathematical Meeting in Honor of A Dou (Madrid, 1989), 341-351.
    • J Richards, Boole and Mill : differing perspectives on logical psychologism, in History and philosophy of logic 1 (Tunbridge Wells, 1980), 19-36.
    • G C Smith, Boole's annotations on 'The mathematical analysis of logic', Hist.
    • Logic 4 (1) (1983), 27-38.

  11. References for Leshniewski
    • J T J Srzednicki and Z Stachniak (eds.), Lesniewski's lecture notes in logic (Dordrecht, 1988).
    • Formal Logic 20 (4) (1979), 934-946.
    • W Marciszewski, The place of Stanislaw Lesniewski in contemporary logic-philosophical thought (Polish), Wiadom.
    • R Poli and M Libardi, Lesniewski's conception of logic, in The Lvov-Warsaw School and Contemporary Philosophy 1995 (Dordrecht, 1998), 139-152.
    • V F Rickey, A survey of Lesniewski's logic, in Lesniewski's systems protothetic (Dordrecht, 1998), 23-41.
    • V F Rickey, A survey of Lesniewski's logic, in On Lesniewski's systems, Krakow, 1976, Studia Logica 36 (4) (1977/78), 407-426.
    • Logic 15 (2) (1994), 227-235.
    • Logic 3 (2) (1982), 165-191.
    • S J Surma, On the work and influence of Stanislaw Lesniewski, in Logic Colloquium 76, Oxford, 1976 (Amsterdam, 1977), 191-220.
    • Formal Logic 7 (1966), 361-364.
    • J Wolenski, Lesniewski's logic and the concept of being, in Stanislaw Lesniewski aujourd'hui, Grenoble, October 8-10, 1992 (Neuchatel, 1996), 93-101.

  12. References for Russell
    • (1908) Mathematical Logic as Based on the Theory of Types, American Journal of Mathematics 30 (1908), 222-262.
    • in B Russell, Logic and Knowledge (London, Allen and Unwin, 1956), 59-102, and in J van Heijenoort, From Frege to Godel (Cambridge, Mass., Harvard University Press, 1967), 152-182.
    • (1956) Logic and Knowledge: Essays, 1901-1950 (London: George Allen and Unwin, New York: The Macmillan Company).
    • (1994) Foundations of Logic, 1903-05, The Collected Papers of Bertrand Russell, Vol.
    • I Grattan-Guinness, Dear Russell, Dear Jourdain: A Commentary on Russell's Logic, Based on His Correspondence with Philip Jourdain (New York: Columbia University Press, 1977).
    • A Church, Comparison of Russell's Resolution of the Semantical Antinomies With That of Tarski, Journal of Symbolic Logic 41 (1976), 747-760.
    • K Godel, Russell's Mathematical Logic, in P A Schilpp (ed.), The Philosophy of Bertrand Russell, 3rd ed., (New York: Tudor, 1951), 123-153.
    • P W Hylton, Logic in Russell's Logicism, in D Bell and N Cooper (eds), The Analytic Tradition: Philosophical Quarterly Monographs, Vol.
    • W V Quine, On the Theory of Types, Journal of Symbolic Logic 3 (1938), 125-139 and in A D Irvine, Bertrand Russell: Critical Assessments, Vol 2, (London: Routledge, 1996).
    • F P Ramsey, Mathematical Logic, Mathematical Gazette 13 (1926), 185-194.

  13. References for Bolzano
    • J Berg, Bolzano's Logic (Stockholm, 1962).
    • Y Bar-Hillel, Bolzano's propositional logic, Arch.
    • J Berg, Is Russell's antinomy derivable in Bolzano's logic?, Bolzano - Studien.
    • J Berg, A requirement for the logical basis of scientific theories implied by Bolzano's logic of variation, in Impact of Bolzano's epoch on the development of science (Prague, 1982), 415-425.
    • K Berka, Bernard Bolzano - historian of logic (Czech), DVT---Dejiny Ved Tech.
    • Logic 12 (3) (1983), 299-318.
    • P Simons, Bolzano, Tarski, and the limits of logic, Bolzano - Studien.
    • W Stelzner, Compatibility and relevance: Bolzano and Orlov, in The Third German-Polish Workshop on Logic & Logical Philosophy, Dresden, 2001, Logic Log.
    • Logic 2 (1981), 11-20.

  14. References for Fraenkel
    • Symbolic Logic 22 (3) (1957), 299.
    • Symbolic Logic 22 (2) (1957), 214-215.
    • Symbolic Logic 28 (2) (1963), 168-169.
    • Symbolic Logic 29 (1964), 1-30.
    • T Frayne, Review: Set Theory and logic, by A A Fraenkel, J.
    • Symbolic Logic 34 (1) (1969), 112-113.
    • Symbolic Logic 20 (2) (1955), 164-165.
    • E Mendelson, Review: Set Theory and logic, by A A Fraenkel, American Scientist 55 (1) (1967), 72A, 74A.
    • Symbolic Logic 29 (3) (1964), 141.
    • Symbolic Logic 13 (1) (1948), 56.

  15. References for Bernays
    • A Kanamori, Bernays and set theory, Bulletin of Symbolic Logic 15 (1) (2009), 43-69.
    • C Parsons, Paul Bernays' later philosophy of mathematics, in Logic Colloquium 2005 (Assoc.
    • Logic, Urbana, IL, 2008), 129-150.
    • E Specker, Paul Bernays, in Logic Colloquium '78, Mons, 1978, Stud.
    • Logic Foundations Math.
    • G Takeuti, Work of Paul Bernays and Kurt Godel, in Logic, Methodology and Philosophy of Science VI, Hannover, 1979, Stud.
    • Logic Foundations Math.
    • P Weingartner, Nachruf auf Paul Bernays, in Ontology and logic, Proc.
    • R Zach, Completeness before Post: Bernays, Hilbert, and the development of propositional logic, Bulletin of Symbolic Logic 5 (1999) 331 .

  16. References for Davis
    • Symbolic Logic 7 (1) (2001), 65-66.
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  106. References for Pieri
    • E A Marchisotto, In the shadow of giants: The work of Mario Pieri in the foundations of mathematics, History and Philosophy of Logic 65 (1995), 107-119.

  107. References for Lehto
    • Logic and a stroke of luck, Helsinki University Bulletin (2) (2008).

  108. References for Al-Tusi Nasir
    • Logic 16 (2) (1995), 257-268.

  109. References for Levi
    • C H Manekin, The Logic of Gersonides : An Analysis of Selected Doctrines (Kluwer Academic, Dordrecht, 1992).

  110. References for Chrysippus
    • Logic 13 (2) (1992), 133-148.

  111. References for Von Neumann
    • H Araki, Some of the legacy of John von Neumann in physics: theory of measurement, quantum logic, and von Neumann algebras in physics, The legacy of John von Neumann (Providence, R.I., 1990), 119-136.

  112. References for Pedoe
    • A Church, Review: The Gentle Art of Mathematics, by Daniel Pedoe, The Journal of Symbolic Logic 31 (4) (1966), 675.

  113. References for Kalicki
    • Logic 2 (1981), 41-53.

  114. References for Kline
    • T Drucker, Morris Kline (1908-1992), Modern Logic 3 (2) (1993), 156-158.

  115. References for Artin
    • Formal Logic 5 (1964), 1-9.

  116. References for Tacquet
    • Logic 13 (1) (1992), 43-58.

  117. References for Lambert
    • Z A Kuziceva, Symbolic logic in the writings of J H Lambert (Russian), Istor.-Mat.

  118. References for Zermelo
    • Logic 11 (1) (1990), 19-58.

  119. References for Moisil
    • G Georgescu, A Iogulescu and S Rudeanu, Grigore C Moisil (1906-1973) and his school in algebraic logic, Int.

  120. References for Birkhoff Garrett
    • In memoriam: Garrett Birkhoff [1911 - 1996], Modern Logic 7 (1) (1997), 81-82.

  121. References for Weyl
    • Symbolic Logic 1 (2) (1995), 145-169.

  122. References for Laplace
    • M A Gomez Villegas, The problem of inverse probability : Bayes and Laplace (Spanish), in Current perspectives in logic and philosophy of science (Spanish) (Madrid, 1994), 385-396.

  123. References for Foster
    • I H Anellis, In memoriam Robin O Gandy (1919-1995) and Alfred L Foster (1904-1994), Modern Logic 6 (1) (1996), 85-87.

  124. References for Sleszynski
    • S McCall, Polish Logic, 1920-1939: Papers by Ajdukiewicz Andothers (Oxford University Press US, 1967).

Additional material

  1. Hilbert reviews
    • The Bulletin of Symbolic Logic 12 (3) (2006), 492-494.
    • The critical work on the foundations of Analysis during last century has led to investigations in fundamental logic during the last decades.
    • From the Foreword by Hilbert: This book deals with theoretical logic (also called mathematical logic, logic calculus or algebra of logic).
    • Theoretical logic ..
    • is an application of formal methods of mathematics to the field of logic.
    • It addresses logic as a similar formula based language in which it has long been customary to express mathematical relationships.
    • This is the second edition of the well-known introduction to symbolic logic which first appeared ten years ago.
    • The changes made have increased the value of what was already a first-rate book indispensable to the serious student of mathematical logic.
    • The many-mansioned discipline known as "symbolic logic" has for a long time ceased to be a simple affair, and introductions to it vary according to the special interests to which they cater.
    • The chief emphasis of an introductory work may thus fall upon symbolic logic as a calculus devised for solving problems not capable of being handled by traditional formal logic; upon modern symbolism as an instrument for distinguishing and exhibiting abstract logical forms; upon the reducibility of mathematics to general logic, and hence upon providing material preparatory to reading Principia Mathematica; or upon the syntactical and semantic studies of recent years, and upon the import of these studies for the issues in the foundations of mathematics and in the general philosophy of the formal sciences.
    • The Journal of Symbolic Logic 3 (2) (1938), 83-84.
    • As those familiar with the first edition will recall, the four chapters correspond to four increasingly comprehensive levels of logic.
    • In Chapter 4 logic receives its full generality: predication and quantification are al- lowed not only with respect to individuals but also with respect to predicates and propositions.
    • The Journal of Symbolic Logic 15 (1) (1950), 59.
    • Though it is generally regarded as the youngest of the mathematical disciplines, symbolic logic was conceived two and a half centuries ago by the same mind which fathered the differential and integral calculus, for it was Leibniz who first formulated the idea of a mathematical calculus by which the truth or falsehood of a proposition might be evaluated with the same facility as the answer to an addition sum.
    • This is from a review of the English translation with title Principles of Mathematical Logic (1950).
    • This is from a review of the English translation with title Principles of Mathematical Logic (1950).
    • This well-known book is described by its American editor as "a classic text in the field of mathematical logic," and most justly so.
    • Hilbert and Ackermann's Principles has a very special place in the literature of mathematical logic because it treats, with unsurpassed excellence of presentation, of just those parts of symbolic logic which are clear of the realm of controversy and which are therefore of permanent value to mathematicians and philosophers.
    • This is from a review of the English translation with title Principles of Mathematical Logic (1950).
    • This is a translation of the very well-known Grundzuge der theoretischen Logik of Hilbert and Ackermann, one of the classic texts of mathematical logic.
    • The Journal of Symbolic Logic 16 (1) (1951), 52-53.
    • This is from a review of the English translation with title Principles of Mathematical Logic (1950).
    • The Journal of Symbolic Logic 25 (2) (1960), 158.
    • A new section on intuitionistic sentential logic is added, as well as one on Ackermann's "strenge Implikation".
    • The emphasis in geometry, just as in any other part of mathematics, may be laid upon the logic of the argument, and proceed from abstraction to abstraction: or, again, it may be laid upon the quality of the subject-matter itself.
    • partly consists in transferring the formalist view, now usual in geometry, to logic and arithmetic.
    • The Journal of Symbolic Logic 35 (2) (1970), 321-323.
    • The style of Grundlagen der Mathematik, and the spirit in which it is written, are very different from what is now usual in systematic expositions of logic and proof theory, and the book has to a very high degree the virtues of a more reflective and less specialized age than that of today.
    • The Journal of Symbolic Logic 5 (1) (1940), 16-20.
    • the subjects of Hilbert's investigations are demonstrations, supposed to be set out in full detail in the symbolism of mathematical logic; in particular, he discusses arithmetical demonstrations, set out in this way without any word of the ordinary language, and moving forwards by definite rules, which correspond to the laws of logic.
    • The Journal of Symbolic Logic 39 (2) (1974), 357.

  2. Harold Jeffreys on Logic and Scientific Inference
    • Harold Jeffreys on Logic and Scientific Inference .
    • We give below an extract from Jeffreys' book on Logic and Scientific Inference .
    • Logic and Scientific Inference .
    • As a matter of logic this is a commonplace.
    • The distinction between deductive logic and scientific inference may be illustrated by means of one of the classical instances of the former.
    • This type of argument, the syllogism, is one of those chiefly used in pure logic; indeed it was believed for ages that there was no other.
    • The notion of probability, which plays no part in logic, is fundamental in scientific inference.

  3. Poincaré on intuition in mathematics
    • Henri Poincare published Intuition and Logic in mathematics as part of La valeur de la science in 1905.
    • Intuition and Logic in Mathematics .
    • The one sort are above all preoccupied with logic; to read their works, one is tempted to believe they have advanced only step by step, after the manner of a Vauban who pushes on his trenches against the place besieged, leaving nothing to chance.
    • And yet nature is always the same; it is hardly probable that it has begun in this century to create minds devoted to logic.
    • Nothing is more shocking to intuition than this proposition which is imposed upon us by logic.
    • Pure logic could never lead us to anything but tautologies; it could create nothing new; not from it alone can any science issue.
    • In one sense these philosophers are right; to make arithmetic, as to make geometry, or to make any science, something else than pure logic is necessary.
    • All four are attributed to intuition, and yet the first is the enunciation of one of the rules of formal logic; the second is a real synthetic a priori judgment, it is the foundation of rigorous mathematical induction; the third is an appeal to the imagination; the fourth is a disguised definition.
    • This shows us that logic is not enough; that the science of demonstration is not all science and that intuition must retain its role as complement, I was about to say, as counterpoise or as antidote of logic.
    • Can logic give it to us? No; the name mathematicians give it would suffice to prove this.
    • In mathematics logic is called analysis and analysis means division, dissection.
    • Thus logic and intuition have each their necessary role.
    • Logic, which alone can give certainty, is the instrument of demonstration; intuition is the instrument of invention.
    • And first do you think these logicians have always proceeded from the general to the particular, as the rules of formal logic would seem to require of them? Not thus could they have extended the boundaries of science; scientific conquest is to be made only by generalization.
    • Is there room for a new distinction, for distinguishing among the analysts those who above all use this pure intuition and those who are first of all preoccupied with formal logic? .

  4. Borali-Forti preface
    • The Aristotelian logic, or its school, studies the forms of reasoning in everyday speech, and the terms of this which are needed to set out its laws.
    • Mathematical logic, studies the forms of reasoning of the deductive sciences and especially mathematics, and from this copy the symbols which serve to enunciate its laws.
    • While the terms of common language often take on different meaning or values depending on the position they occupy in the context, because of the frequent exceptions to which the rules of grammar are subject, the symbols of mathematical logic retain the same meaning everywhere, the laws not being subject to exceptions to that which it was agreed they satisfy.
    • Moreover, while the terms of the common language give rise to their own combinations depending on the special structure of each language, the symbols of mathematical logic are a universal writing system independent of any language.
    • The characteristics now mentioned clearly separate the school of logic from mathematical logic, and give mathematical logic a degree of scientific rigour that is impossible to achieve with the former.
    • The merit of having given the first germs of mathematical logic, is indisputably in that vast mind of the mathematician and philosopher Gottfried Leibniz (Dissertatio de arte combinatoria, Leipzig, 1666).
    • A solution of the problem of Leibniz has greatly contributed to the progress of mathematical logic.
    • This took scientific form, for the first time, more especially in the work of the Englishman George Boole (An investigation into the Laws of Thought, on Which are founded the Mathematical Theories of Logic and Probabilities, London, 1854) who has applied algebraic symbolism to logic.
    • Many writers have contributed to the progress of mathematical logic; refer, with its bibliography on this subject, to the massive and important work of Ernst Schroder, Algebra der Logik (1890-1910).
    • We note, however, explicitly - since results make it clear that mathematical logic belongs to the field of the mathematician who is a philosopher - that these writers are all mathematicians and their work has been published in journals of mathematics.
    • In American Universities mathematical logic is part of the official teaching through the work of George Bruce Halsted, Charles Sanders Peirce, and others.
    • Mathematical logic was introduced into Italy through the work of professor Giuseppe Peano (Calcolo Geometrico, secondo l'Ausdehnungslehre di H Grassmann, preceduto dalle operazioni della logica deduttiva, Turin, 1888) and Albino Nagy (Fondamenti del calcolo logico, Giornale di matematica 28 (1890), 1-35).
    • This book contains the elements of Mathematical Logic, and is developed from a course of science lectures I've given in the current school year at the University of Turin.

  5. A N Whitehead addresses the British Association in 1916, Part 2
    • It will be necessary to sketch in broad outline some relevant features of modern logic.
    • In doing so I shall try to avoid the profound general discussions and the minute technical classifications which occupy the main part of traditional logic.
    • It is characteristic of a science in its earlier stages - and logic has become fossilised in such a stage - to be both ambitiously profound in its aims and trivial in its handling of details.
    • The next section of logic is the algebraic stage.
    • The theory of the interconnection between the truth-values of the general propositions arising from any such aggregate of propositional functions forms a simple and elegant chapter of mathematical logic.
    • In this algebraic section of logic the theory of types crops up, as we have already noted.
    • The final impulse to modern logic comes from the independent discovery of the importance of the logical variable by Frege and Peano.
    • The essence of the process is, first to construct the notion in terms of the forms of propositions, that is, in terms of the relevant propositional functions, and secondly to prove the fundamental truths which hold about the notion by reference to the results obtained in the algebraic section of logic.
    • A more important question is the relation of induction, based on observation, to deductive logic.
    • I will now break off the exposition of the function of logic in connection with the science of natural phenomena.
    • Logic, properly used, does not shackle thought.
    • Also the mind untrained in that part of constructive logic which is relevant to the subject in hand will be ignorant of the sort of conclusions which follow from various sorts of assumptions, and will be correspondingly dull in divining the inductive laws.
    • The fundamental training in this relevant logic is, undoubtedly, to ponder with an active mind over the known facts of the case, directly observed.
    • Neither logic without observation, nor observation without logic, can move one step in the formation of science.
    • Logic is the olive branch from the old to the young, the wand which in the hands of youth has the magic property of creating science.

  6. Lewis's Survey
    • C I Lewis's Survey of Symbolic Logic .
    • C I Lewis published A Survey of Symbolic Logic in 1918.
    • The student who has completed some elementary study of symbolic logic and wishes to pursue the subject further finds himself in a discouraging situation.
    • He has, perhaps, mastered the contents of Venn's 'Symbolic Logic' or Couturat's admirable little book, 'The Algebra of Logic', or the chapters concerning this subject in Whitehead's 'Universal Algebra'.
    • These all concern the classic, or Boole-Schroder algebra, and his knowledge of symbolic logic is probably confined to that system.
    • The present book is an attempt to meet this need, by bringing within the compass of a single volume, and reducing to a common notation (so far as possible), the most important developments of symbolic logic.
    • A gossipy recital of results achieved, or a superficial account of methods, is of no more use in symbolic logic than in any other mathematical discipline.
    • My own contribution to symbolic logic, presented in Chapter V, has not earned the right to inclusion here; in this, I plead guilty to partiality.
    • Particularly in the last chapter, on "Symbolic Logic, Logistic, and Mathematical Method", it is not possible to give anything like an adequate account of the facts.

  7. R L Wilder: 'Cultural Basis of Mathematics II
    • I wonder what he would do about logic, however? .
    • In the index of Ball's first edition (1888) there is no mention of "logic;" but in the fourth edition (1908) "symbolic and mathematical logic" is mentioned with a single citation, which proved to be a reference to an incidental remark about George Boole to the effect that he "was one of the creators of symbolic or mathematical logic." Thus symbolic logic barely squeezed under the line because Boole was a mathematician! The index of Cajori's first edition (1893) contains four citations under "logic," all referring to incidental remarks in the text.
    • None of these citations is repeated in the second edition (1919), whose index has only three citations under "logic" (two of which also constitute the sole citations under "symbolic logic"), again referring only to brief remarks in the text.
    • Inspection of the text, however, reveals nearly four pages (407-410) of material under the title "Mathematical logic," although there is no citation to this subject in the index nor is it cited under "logic" or "symbolic logic." (It is as though the subject had, by 1919, achieved enough importance for inclusion as textual material in a history of mathematics although not for citation in the index!) .
    • Turning to the index of this book, I found so many citations to "logic" that I did not care to count them.
    • In particular, Bell devotes at least 25 pages to the development of what he calls "mathematical logic." Can there be any possible doubt that this subject, not considered part of mathematics in our culture in 1900, despite the pioneering work of Peano and his colleagues, is now in such "good standing" that any impartial definition of mathematics must be broad enough to include it? .
    • P W Bridgman, The logic of modern physics (Macmillan, New York, 1927).

  8. Bell books
    • This book deals with a thesis on the border lines of mathematics and logic, a region which has been extensively explored by the American school of mathematics during the last twenty years.
    • Bell traces the activities of "homo sap." in the evolution of deductive reasoning and the relation of this activity to what men mean when they use the word "truth." The book is an abbreviated intellectual history of western civilization in four phases as these are reflected in and symbolised by advances in fundamental logic and mathematics.
    • Gilded with literary iridescence and sparkling with original insights, the book takes the new logic seriously and nonchalantly transcends any inherent contradictions.
    • There is too much logic, too much stress on the logical basis of mathematics and too much on those parts of mathematics which depend on logical assumptions; too much also about the Greeks and their way of thinking.
    • For that large number of the somewhat general public to which Simon and Schuster cater in some of their publications it will be Dr Jekyll who writes the "heart-interest" material and the glittering generalities in an often loose style and it will be Mr Hyde who tries to expound the theory of algebraic ideals or of transfinite numbers or of symbolic logic, whereas for the professional mathematician the attributions for these respective parts will be inverted.
    • The Journal of Symbolic Logic 2 (2) (1937), 95.
    • It is unfortunate that these very readable and well written chapters, which give on the whole a correct picture of the importance which foundational criticism and with it the method of symbolic logic have acquired in modern mathematics, are nevertheless in certain particulars inaccurate or misleading.
    • The Journal of Symbolic Logic 5 (4) (1940), 152-153.
    • Professor Bell has written an extremely valuable and stimulating book, whether considered from the point of view of mathematics generally or from that of logic and foundations in particular.
    • But students of philosophy and of the history of formal logic will be well repaid for consulting it.
    • They will find in it not only an engrossing story of important stages in the construction of mathematical methods; they will also find a clear survey of the development of mathematical logic from its origins down to the present day and, what is perhaps even more valuable, an intelligible account of the influences which technical problems of mathematics have exercised upon the direction of logical inquiry.
    • Professor Bell's appraisals and accounts of the work that has been done on the foundations of mathematics and logic are not uniformly sober or adequate.
    • The Journal of Symbolic Logic 12 (2) (1947), 61-62.
    • It covers a remarkable amount of ground: rings, lattices, matrices, groups, Galois theory, number theory, the discovery of Neptune, the calculus, Fourier theory, questions of the infinite in mathematics, mathematical truth, and logic all get chapters.

  9. Fraenkel books
    • The Journal of Symbolic Logic 20 (2) (1955), 164-165.
    • Though devoid of any symbolic logic, its subject matter and quality are such as to make it vital to readers of this Journal.
    • The Journal of Symbolic Logic 28 (2) (1963), 168-169.
    • The authors employ a balanced blend of formal and informal methods of presentation, symbolic logic being introduced mainly in the statement of axioms and definitions, to ensure accuracy.
    • The Journal of Symbolic Logic 29 (3) (1964), 141.
    • Of all the many branches of mathematics, set theory has the closest links with logic and philosophy.
    • Accordingly, an author who attempts to give an adequate discussion of the foundations of set theory is confronted with a formidable task, for he must not only describe the purely technical aspects of the subject, but must also elucidate the role that logic and philosophy have played in its development.
    • The purpose of this book is not to present in a systematic way, a more or less extended part of set theory, but to present to readers without special mathematical training, problems at the borders of set theory and logic - like those of mathematical infinity or the set theory paradoxes.
    • Set Theory and logic (1966), by A A Fraenkel.
    • The word "logic" in the title indicates only that some attention is devoted to definitions and methods of proof.
    • There is no discussion of mathematical logic or of any technical logical problems.
    • The Journal of Symbolic Logic 34 (1) (1969), 112-113.
    • The book is designed to provide an easy introduction to the problem of infinity and a treatment of logical problems arising at several critical points in set theory for college freshmen in mathematics, logic, and philosophy.
    • The book concludes with remarks on the scope and applications of set theory, describing set theory as "a cornerstone in the foundations of many other branches of mathematics and the chief connecting link between mathematics and logic." The book is lucid and stimulating and should for the most part be rewarding reading for its intended audience.

  10. C Chevalley on Herbrand's thought

  11. C Chevalley: 'On Herbrand's thought
    • In June 1934 Claude Chevalley gave a lecture On Herbrand's thought at the Colloquium on Mathematical Logic which had been organised at the University of Geneva.
    • In 1928, he took first place in the Agregation, and was able to stay a fourth year at the Ecole, during which he completed his thesis, which contains the work in mathematical logic that he had been pursuing for two years.
    • He spent the academic year 1930-1931 in Germany: first in Berlin with von Neumann, where he continued his work in logic, notably comparing his results to those of Godel; then he went to Hamburg and finally to Gottingen.
    • Jacques Herbrand expressed himself rather little on the philosophical ideas relating to the problems of mathematical logic (see, however, the Introduction to his thesis).
    • The same considerations apply to logic.
    • Logic comprises a schema which is objective only insofar as it is purely formal.
    • Were one to give a sense to the symbols appearing in the formulas of logic, were one to consider them as representing operations of thought, one could cause logic to lose its objectivity.
    • This is why it is not surprising that different thinkers have different opinions on the value of the axioms of logic.
    • If the system of forms of classical logic is repugnant to Brouwer's thought, for example, this does not mean that this logic is denuded of value; it is an assertion about Brouwer's thought.
    • But in any case a human thought remains incongruous in any formalism; there is the same relationship between a formal logic and a mode of thought as between a mathematical equation and a physical phenomenon.
    • Just as mathematical physics permits us to penetrate further and further into the structure of matter, logic allows us to describe something nearer yet to man than his sensations: his intellectual thought.
    • This is the greatest demand one could make on a formal logic: it leads us to the very centre of the drama of Herbrand's thought, balancing between an investigation always more concrete and a formalism always more abstract.

  12. Shepherdson Tribute
    • I have the privilege of representing the mathematical logic community today, to honour the memory of one of the most admired people in the field.
    • When John began at Bristol in 1946, mathematical logic (as opposed to philosophy of mathematics) was a very young subject, in its teens.
    • Goodstein (who became in 1958 the first Professor in the UK to specialize in mathematical logic), was already active in logic pre-war, but his famous contributions came in the 1940's.
    • Robin Gandy, another crucial figure in the development of mathematical logic in the UK, and whom we associate with John in establishing our community, was about ten years older than John, but, like Robinson, had his research development slowed by the war, and did not finish his D.Phil.
    • John, in a long life, was to contribute in depth across the entire spectrum, and to subjects then unknown or just emerging, such as logic programming, computable mathematics, and finite automata.
    • It lasted for a couple of years, and transformed the intensity of life in logic at Oxford.
    • The extensive work on logic programming is further from my expertise, as is the late work with Hajek and Paris, but the style and the informal rigour are deeply appealing.
    • I did not know about the number of logicians already in John's orbit, achieving a ratio of logic to the rest perhaps never equalled anywhere! I had a very happy year at Bristol, sharing an office with Ken Appell, spending a lot of time talking to him about free groups.
    • There was a marvellous MSc course, sampling all the great new ideas bubbling up in logic.
    • Let us not forget his selfless contributions to initiating the ASL European Meetings, and the BLC, and to the Oxford Logic Guides.

  13. Kneebone books
    • Mathematical Logic and the Foundations of Mathematics (1963), by G T Kneebone.
    • The author has attempted to give a survey (at the junior-senior first-year graduate level) of the whole field of mathematical logic and foundations of mathematics up to the developments in the 1930s.
    • The book is based on a course of lectures given ay the University of London for an audience containing not only graduate students beginning to specialise in mathematical logic but undergraduates majoring in fields other than mathematics.
    • It seems to the reviewer that it is too technical for most non-mathematicians and it is questionable also whether a survey is the right way for a serious mathematics student to begin a study of mathematical logic.
    • The second group consists of postgraduate students who want an introduction to mathematical logic or the philosophy of mathematics that will give them a correct orientation, a view of the entire subject and its literature, and quickly enable them to work from primary sources.' At present this audience is not very well catered for.
    • Most existing books on mathematical logic (a notable exception being Beth's The Foundations of Mathematics) which attempt to be fairly comprehensive are either suitable only for the trained mathematician who can skim through pages of detailed calculation and extract the ideas behind them for himself or else represent a rather individual view of what is important rather than a balanced view of the whole field.
    • Mathematics is highly charged with significance and one of the important questions of foundations is to show how this comes about; and since logic merely provides a criterion of validity, not of significance, it fails to explain this feature of mathematics.
    • From this he draws the odd conclusion that formal logic is to count as applied mathematics.
    • The author has exceptional gifts as an expositor and gives a lucid account of mathematical logic and of the main movements in foundations.
    • The senior of the two authors of this little volume, Dr Kneebone, is well known for his books on logic and geometry, and his collaboration with a specialist in ordinary theory has produced a well-written introduction to axiomatic set theory and transfinite numbers.
    • The Journal of Symbolic Logic 37 (3) (1972), 614.

  14. Enriques' reviews
    • The first part of that work is concerned with the more general problems of the logic of science; the second part discusses the concepts of geometry, mechanics, physics, and biology.
    • After an introduction dealing with the presuppositions and limitations of the scientific concept of reality, the author proceeds to review the basic principles of logic, geometry, mechanics, and the mechanical theory of life.
    • The book contains six chapters treating in order the following topics: the general problem of knowledge and related matters; facts and theories and their interactions; the general problems of logic; the philosophical and psychological questions which are naturally raised in connection with the science of geometry; mechanics, its objective significance and the psychological development of its principles; the extension of mechanics into physics and the relation of the mechanical hypothesis to the phenomena of life.
    • It may be divided into five parts; the first is a general introduction and explanation of the author's position (which he calls Critical Positivism), the second deals with Logic and its applicability to the real world, the third deals with geometry, the fourth with the classical mechanics, and the last with electro-dynamics and the alterations which it has entailed in the mechanics of Newton.
    • The Historic Development of Logic.
    • Enriques does not claim completely to have recovered the spirit, but his noteworthy attempt should remind us how little has been done in the history of logic since Prantl.
    • While logicians dispute among themselves concerning the subject-matter of logic, whether it ought to be things, words, or ideas, Logos continues to be actualized in all forms of rational activity.
    • In order to determine the profounder bases of the principles employed by the mathematical sciences, Enriques has delved into the historical sources of logic, mathematics, and philosophy.
    • Enriques is well known for his mathematical, logical, philological, and historical knowledge which makes him pre-eminently equipped for the philosophical history of the sciences, as his 'Historic Development of Logic' shows.
    • Professor Enriques neatly summarizes the evidence from modern science against the Kantian doctrines of space, time, substance, causality, and logic.

  15. Peter's books
    • The Journal of Symbolic Logic 13 (3) (1948), 141-142.
    • The first two parts develop some of the fundamental ideas; the third part is devoted to mathematical logic.
    • This is followed by an introduction to formal logic, the propositional connectives, and quantifiers.
    • From there we pass to non-Euclidean geometry and mathematical logic and end with an account of Godel's construction of undecidable sentences; and all this is accomplished with a minimum of mathematical symbolism, A remarkable achievement.
    • Part III, "The Self-Critique of Pure Reason", deals with the many different kinds of geometries, the fourth dimension, and mathematical logic.
    • The Journal of Symbolic Logic 16 (4) (1951), 280-282.
    • The book is entirely elementary, and no knowledge of recursive functions or of mathematical logic is presupposed.
    • Only a minimum of knowledge of elementary number theory, analysis, and set theory including transfinite ordinals is presupposed and none of mathematical logic.
    • The Journal of Symbolic Logic 23 (3) (1958), 362-363.

  16. Lewis's papers
    • Implication and the Algebra of Logic.
    • The development of the algebra of logic brings to light two somewhat startling theorems: (1) a false proposition implies any proposition, and (2) a true proposition is implied by any proposition.
    • Such an attempt might be superfluous were it not that certain confusions of interpretation are involved, and that the expositors of the algebra of logic have not always taken pains to indicate that there is a difference between the algebraic and the ordinary meanings of implication.
    • As a result, symbolic logic appears to the uninitiated somewhat as an enfant terrible, which intimidates one with its array of exact demonstrations, and demands the acceptance of incomprehensible results.
    • The development of the algebra of logic has done more than emphasize the close relation of logic and mathematics.
    • And the proposition which states the particular implication relation - in more general form, because its variables have a wider range of meaning - is itself a mathematical proposition, in the algebra of logic.
    • Interesting Theorems in Symbolic Logic.
    • What these theorems reveal is the divergence of the meaning of "implies" in the algebra of logic from the "implies" of valid inference.

  17. Keynes: 'Probability' Introduction Ch I
    • In most branches of academic logic, such as the theory of the syllogism or the geometry of ideal space, all the arguments aim at demonstrative certainty.
    • The course which the history of thought has led Logic to follow has encouraged the view that doubtful arguments are not within its scope.
    • If logic investigates the general principles of valid thought, the study of arguments, to which it is rational to attach some weight, is as much a part of it as the study of those which are demonstrative.
    • But in the sense important to logic, probability is not subjective.
    • This relation is the subject-matter of the logic of probability.
    • As soon as we have passed from the logic of implication and the categories of truth and falsehood to the logic of probability and the categories of knowledge, ignorance, and rational belief, we are paying attention to a new logical relation in which, although it is logical, we were not previously interested, and which cannot be explained or defined in terms of our previous notions.
    • If the statement that an opinion was probable on the evidence at first to hand, but became untenable on further information, is not, solely concerned with psychological belief, I do not know how the element of logical doubt is to be defined, or how its substance is to be stated, in terms of the other indefinables of formal logic.
    • In any case a desire to reduce the indefinables of logic can easily be carried too far.

  18. Keynes: 'Probability' Introduction Ch II
    • I do not wish to become involved in questions of epistemology to which I do not know the answer; and I am anxious to reach as soon as possible the particular part of philosophy or logic which is the subject of this book.
    • The logic of knowledge is mainly occupied with a study of the logical relations, direct acquaintance with which permits direct knowledge of the secondary proposition asserting the probability-relation, and so to indirect knowledge about, and in some cases of, the primary proposition.
    • In all knowledge, therefore, there is some direct element; and logic can never be made purely mechanical.
    • Almost up to the end of the seventeenth century the traditional treatment of modals is, in fact, a primitive attempt to bring the relations of probability within the scope of formal logic.] .
    • Other parts of knowledge - knowledge of the axioms of logic, for example - may seem more objective.
    • Further, the difference between some kinds of propositions over which human intuition seems to have power, and some over which it has none, may depend wholly upon the constitution of our minds and have no significance for a perfectly objective logic.
    • Our logic is concerned with drawing conclusions by a series of steps of certain specified kinds from a limited body of premisses.

  19. Bolzano's publications
    • This volume contains a biography of Bolzano together with details of the topics on which he worked: mathematics, logic, theology, philosophy and aesthetics.
    • This volume in the series of posthumous writings of the Collected works of Bernard Bolzano contains transcriptions of some early manuscripts which present his views in the 1810's on the foundations of logic, mathematics and physics.
    • Among the other contributions, a manuscript on the basic concepts of logic adumbrates fundamental themes in Bolzano's major work, the Wissenschaftslehre of 1837.
    • Most of the items on philosophy relate to logic.
    • Bernard Bolzano, Grundlegung der Logik (German), [Foundations of logic] Selected paragraphs from Wissenschaftslehre, Band I und II.
    • The first two volumes of Bolzano's Wissenschaftslehre published in 1837 are concerned with logic.
    • Bolzano's manuscript giving his ideas on logic and semantics which became the basis of his major publication Wissenschaftslehre which appeared in 1837.

  20. Karl Menger on Hans Hahn
    • Hans Hahn wrote eight philosophical papers on Empiricism, logic, and mathematics between 1919 and 1934.
    • Secondly, it was Hahn who directed the interest of the Circle toward logic.
    • Carnap, having been a student of Frege's, was well-versed in logic, but was mainly concerned with the philosophy of science at the time he moved to Vienna.
    • Hahn, however, right after his return, began an intense study of symbolic logic with an eye to related philosophical problems.
    • To me," Hahn continued, "the Tractatus has explained the role of logic." In later writings, which are included in this volume, Hahn expounded his own (somewhat oversimplifying) view by describing logic as "a prescription for saying the same thing in various ways, and for extracting from what is said all that is (in a strict sense) connoted." .
    • How stimulating his lectures were I have tried to describe - in Chapter 21 of my book Selected Papers in Logic and Foundations, Didactics, Economics (Vol.

  21. Mac Lane books
    • These are Homology (1963); (with Garrett Birkhoff) Algebra (1967); Categories for the Working Mathematician (1971); Mathematics, Form and Function (1985); (with Ieke Moerdijk) Sheaves in Geometry and Logic: A First Introduction to Topos Theory (1992); and Saunders Mac Lane: A Mathematical Autobiography (2005).
    • The Journal of Symbolic Logic 53 (2) (1988), 643-645.
    • Instead, the reader is treated to a masterful guided tour of the subject, from the natural, rational, ordinal, and cardinal numbers, Euclidean and non-Euclidean geometries, real and complex numbers, functions, transformations, groups, and the calculus, to linear algebra, differential geometry, mechanics, complex analysis, topology, set theory, logic, and category theory.
    • Studia Logica: An International Journal for Symbolic Logic 49 (3) (1990), 424-426.
    • Sheaves in Geometry and Logic: A First Introduction to Topos Theory (1992), by Saunders Mac Lane and Ieke Moerdijk.
    • A startling aspect of topos theory is that it unifies two seemingly wholly distinct mathematical subjects: on the one hand, topology and algebraic geometry, and on the other hand, logic and set theory.
    • The Journal of Symbolic Logic 60 (1) (1995), 340-342.

  22. A N Whitehead addresses the British Association in 1916
    • The nature of induction, its importance, and the rules of inductive logic have been considered by a long series of thinkers, especially English thinkers, Bacon, Herschel, J S Mill, Venn, Jevons, and others.
    • There is a tendency in analysing scientific processes to assume a given assemblage of concepts applying to nature, and to imagine that the discovery of laws of nature consists in selecting by means of inductive logic some one out of a definite set of possible alternative relations which may hold between the things in nature answering to these obvious concepts.
    • King James said, 'No bishops, no king.' With greater confidence we can say, 'No logic, no science.' The reason for the instinctive dislike which most men of science feel towards the recognition of this truth is, I think, the barren failure of logical theory during the past three or four centuries.
    • To this hesitation I ascribe the barrenness of logic.
    • In the first place this last condemnation of logic neglects the fragmentary, disconnected character of human knowledge.
    • In fact, the true answer is embedded in the discussion of our main problem of the relation of logic to natural science.

  23. Carol R Karp: 'Languages with expressions of infinite length
    • The monograph is carefully and lucidly written and should make this field of logic accessible to a wider circle of logicians.
    • My interest in infinitary logic dates back to a February day in 1956 when I remarked to my thesis supervisor, Professor Leon Henkin, that a particularly vexing problem would be so simple if only I could write a formula that would say x = 0 or x = 1 or x = 2 etc.
    • Techniques for proving completeness theorems in logic and representation theorems for Boolean algebras combined to yield a completeness theorem: Valid formulas of denumerable length in which only finitely many variables can be quantified at a time are provable in a system very much like the ordinary first-order predicate calculus.
    • The present version of the material also owes much to the referee (not known to me) of an article submitted to the Journal of Symbolic Logic in 1962.

  24. Kuratowski: 'Introduction to Set Theory
    • In the first part of this book the reader will find a certain amount of information on mathematical logic.
    • The notation of mathematical logic is an indispensable tool of set theory and can be applied with great profit far beyond set theory.
    • The notation of mathematical logic is not devoid of general didactical values; by examples for concepts such as uniform convergence or uniform continuity it is possible to observe how much the definition of these concepts gains in precision and lucidity, when they are written in the symbolism of mathematical logic.

  25. Ernest Hobson addresses the British Association in 1910, Part 3
    • I am very far from wishing to minimise the high philosophic interest of the attempt made by the Peano-Russell school to exhibit Mathematics as a scheme of deductive logic.
    • Logic is, so to speak, the grammar of Mathematics; but a knowledge of the rules of grammar and the letters of the alphabet would not be sufficient equipment to enable a man to write a book.
    • After all, a mathematician is a human being, not a logic-engine.
    • I have already remarked that Euclid's treatment of the subject is not rigorous as regards logic.

  26. Aitken: 'Statistical Mathematics
    • The methods of logic and mathematics are then brought into play to develop the consequences of the axioms, producing an assemblage of theorems or propositions.
    • Probability as the Logic of Uncertain Inference.
    • One view is that probability may be regarded as a kind of extension of classical logic, an extension conveniently described as the "logic of uncertain inference." This view has been expounded by J M Keynes in A Treatise on Probability (London, 1921), especially in Part II, Chapters X-XVII, where references to earlier expositions are given.

  27. Pedoe's books
    • There are 9 chapters: mathematical games, chance and choice, where does it end (i.e., transfinite numbers), automatic thinking (logic, algebra of classes, etc.), two-way stretch, rules of play (elementary algebra, groups, etc.), an accountant's nightmare (infinite series), double talk (antinomies), what is mathematics.
    • The Journal of Symbolic Logic 31 (4) (1966), 675.
    • The topics treated are mathematical games, chance and choice, infinity, sets and logic, topology, groups-rings-fields, series, and more logic.

  28. Mandelbrot's Foreword to Dauben's Abraham Robinson
    • Abraham Robinson's life was extraordinary in many ways, if only because his professional work spanned three significant fields: airplane design, symbolic logic, and mathematical analysis.
    • The second recounts the progression of Robinson's works, from airplane design to mathematical logic to nonstandard mathematical analysis.
    • As we follow in near-chronological order the life of a single individual of amazing brilliance, stamina, and versatility, we are guided back and forth without artificiality through at least three widely disparate academic cultures (pure and applied mathematics and logic-philosophy); six countries, representing six distinct flavours of Western culture; and, within the United States, two very different institutions.
    • I fully agree with this characterization, but I also hear the voice of the devil's advocate who would turn the same evidence around, interpret Abby's achievements backward, and assert that a person who spent much of his life outside of mathematics departments, working on airplane design and symbolic logic, was ipso facto not really a mathematician.

  29. Karl Menger's books
    • The Journal of Symbolic Logic 10 (3) (1945), 103.
    • It consists in some resemblance in the author's method to that of combinatory logic, and in the reviewer's tentative conjecture that the treatment might be improved by more extensive adoption of some of the ideas of combinatory logic.
    • The Journal of Symbolic Logic 24 (3) (1959), 222-223.

  30. A N Whitehead: 'Autobiographical Notes
    • Logic and Science are the disclosure of relevant patterns, and also procure the avoidance of irrelevancies.
    • Also Sir William Rowan Hamilton's Quaternions of 1853, and a preliminary paper in 1844, and Boole's Symbolic Logic of 1859, were almost equally influential on my thoughts.
    • My whole subsequent work on Mathematical Logic is derived from these sources.
    • The other experience is that of a Congress on Mathematical Logic held in Paris in March, 1914.

  31. Mathematics at Aberdeen 1
    • As in the other universities of Europe, the teaching was dominated by the Logic, Theoretical Physics and Metaphysics of Aristotle.
    • He should teach Logic in the first year, Physics, Natural Philosophy and the Treatise on the Sphere in the second.
    • The first year was occupied by Latin, Elementary Greek and Logic; the second mainly Logic with the writing and declaiming of Latin and Greek.

  32. Mandelbrot: Foreword to 'Abraham Robinson' by Dauben
    • Abraham Robinson's life was extraordinary in many ways, if only because his professional work spanned three significant fields: airplane design, symbolic logic, and mathematical analysis.
    • The second recounts the progression of Robinson's works, from airplane design to mathematical logic to nonstandard mathematical analysis.
    • As we follow in near-chronological order the life of a single individual of amazing brilliance, stamina, and versatility, we are guided back and forth without artificiality through at least three widely disparate academic cultures (pure and applied mathematics and logic-philosophy); six countries, representing six distinct flavours of Western culture; and, within the United States, two very different institutions.
    • I fully agree with this characterization, but I also hear the voice of the devil's advocate who would turn the same evidence around, interpret Abby's achievements backward, and assert that a person who spent much of his life outside of mathematics departments, working on airplane design and symbolic logic, was ipso facto not really a mathematician.

  33. R L Wilder: 'Cultural Basis of Mathematics III
    • The restricted mathematics known as Intuitionism has won only a small following, although some of its methods, such as those of a finite constructive character, seem to parallel the methods underlying the treatment of formal systems in symbolic logic, and some of its tenets, especially regarding constructive existence proofs, have found considerable favour.
    • From the extension of the notion of number to the transfinite, during the latter half of the 19th century, there evolved certain contradictions around the turn of the century, and as a consequence the study of Foundations questions, accompanied by a great development of mathematical logic, has increased during the last 50 years.
    • P W Bridgman, The logic of modern physics (Macmillan, New York, 1927).

  34. Ernest Hobson addresses the British Association in 1910, Part 2
    • All attempts to characterise the domain of Mathematics by means-of a formal definition which shall not only be complete, but which shall also rigidly mark off that domain from the adjacent provinces of Formal Logic on the one side and of Physical Science on the other side, are almost certain to meet with but doubtful success; such success as they may attain will probably be only transient, in view of the power which the science has always shown of constantly extending its borders in unforeseen directions.
    • A strong tendency is manifested in many of the recent definitions to break down the line of demarcation which was formerly supposed to separate Mathematics from formal logic; the rise and development of symbolic logic has no doubt emphasised this tendency.

  35. Skolem: 'Abstract Set Theory
    • Further, there is a short remark on the possibility of finitist mathematics in a strict sense and finally some hints are given about the possibility of a set theory based on a logic with an Infinite number of truth values.
    • The possibility of set theory based on many-valued logic .
    • On the other hand it ought to be remarked that about the same time that Cantor exposed his ideas some other people were busy in developing what we today call mathematical logic.

  36. Gian-Carlo Rota: Alonzo Church
    • He never made casual remarks: they did not belong in the baggage of formal logic.
    • The person lecturing to us was logic incarnate.
    • His pauses, hesitations, emphases, his betrayals of emotion (however rare), and sundry other nonverbal phenomena taught us a lot more logic than any written text could.

  37. Hilbert's Grundlagen
    • David Hilbert (1862-1943) is one of the most outstanding representatives of mathematics, mathematical physics, and logic-oriented foundational sciences in general.
    • The first volume presents the motivation and philosophical foundation of Hilbert's and Bernay's original view on finitistic mathematics and their methodological standpoint for proof theory (§§1-2), a refined introduction to propositional and predicate logic (with equality) (§§ 3-5), and the consistency issues of a variety of (sub-) systems of number theory (including recursion theory, System (Z), and the elimination of the iota-operator) (§§ 6-8).
    • It has been a great impairment to science and foundational research that, although translations into Russian and French are available, there has never been an English edition of these two milestones in the development of modern mathematical logic.

  38. Gian-Carlo Rota: Alonzo Church
    • He never made casual remarks: they did not belong in the baggage of formal logic.
    • The person lecturing to us was logic incarnate.
    • His pauses, hesitations, emphases, his betrayals of emotion (however rare), and sundry other nonverbal phenomena taught us a lot more logic than any written text could.

  39. Students in 1711
    • Latin was studied in the first year, then, in their second year, they also studied logic.
    • Kenneth and Thomas will be once through their Logic by the end of this week ..
    • we must either read our Logic course with application, or be affronted when the rest answer better than we.

  40. E W Hobson: 'Mathematical Education
    • In accordance with the older and traditional treatment of Mathematical instruction in our schools, Geometry was treated in a purely abstract manner; the idea being that Euclid, as a supposed model of purely deductive logic, should be studied entirely with a view to the development of the logical faculty.
    • That the average boy or girl is not by nature appreciative of formal logic, or of the interest and meaning of abstract symbols, was thought to be a reason why the subjects so treated should be especially insisted on.
    • He is not interested in formal logic, therefore he must not be bored with learning a chain of theorems of which the object is not apparent to him.

  41. Isaac Todhunter: 'Euclid' Preface
    • After some hesitation on the point, all remarks relating to Logic have also been excluded.
    • Although the study of Logic appears to be reviving in this country, and may eventually obtain a more assured position than it now holds in a course of liberal education, yet at present few persons take up Logic before Geometry; and it seems therefore premature to devote space to a subject which will be altogether unsuitable to the majority of those who use a work like the present.

  42. Whyburn's books
    • The reviewer recalls a remark of a high school teacher in one of his classes, "I do not see what logic has to do with mathematics." Texts like the Chicago syllabus or 'Principles of Mathematics' by Allendoerfer and Oakley (McGraw-Hill) are specially aimed at this deficiency.
    • As a practical matter, therefore, I suspect that the text will be useful mainly as an exercise book for graduate students and instructors whose grasp of the formal logic of economic analysis is something less than certain.

  43. Education in St Andrews in 1849
    • Until about twenty years ago, when the late Principal Nicoll resigned the church living of St Leonard's in favour of the late Dr James Hunter, professor of logic, the office of principal of the United College and the pastorship of St Leonard's parish, had, from the union of the Colleges, been vested in one individual.
    • The eight professorships are devoted to the inculcating of Latin, Greek, Mathematics, Logic and Rhetoric, Medicine, Moral Philosophy, Natural Philosophy, and Civil History.

  44. Association 1904 Part 2.html

  45. G H Hardy's schedule of lectures in the USA
    • Hilbert's mathematical logic .
    • Hilbert's logic .

  46. Gordon Preston on semigroups
    • Mathematical logic had a hard fight to be recognised as a part of mathematics.
    • John Crossley, for example, left the permanent job of fellow of All Soul's, at Oxford, surely one of the most enviable jobs for anyone to have, to come to Monash as a professor, because he saw a chance here, which he did not see at most universities in Britain, that mathematical logic would be treated as the important part of mathematics that it is.

  47. Hans Hahn: 'The crisis in intuition
    • Because intuition turned out to be deceptive in so many instances, and because propositions that had been accounted true by intuition were repeatedly proved false by logic, mathematicians became more and more skeptical of the validity of intuition.
    • The task of completely formalizing mathematics, of reducing it entirely to logic, was arduous and difficult; it meant nothing less than a reform in root and branch ..

  48. Atiyah papers
    • In a different direction mathematics has a traditional link with logic and philosophy, a link which has acquired greater importance through the growth of computer science.
    • I will say nothing, for example, about the great events in the area between logic and computing associated with the names of people like Hilbert, Godel, and Turing.

  49. Keynes: 'Probability' Preface
    • The subject matter of this book was first broached in the brain of Leibniz, who, in the dissertation, written in his twenty-third year, on the mode of electing the kings of Poland, conceived of Probability as a branch of Logic.
    • A few years before, "un probleme," in the words of Poisson, "propose a un austere janseniste par un homme du monde, a ete l'origine du calcul des probabilites." In the intervening centuries the algebraical exercises, in which the Chevalier de la Mere interested Pascal, have so far predominated in the learned world over the profounder enquiries of the philosopher into those processes of human faculty which, by determining reasonable preference, guide our choice, that Probability is oftener reckoned with Mathematics than with Logic.

  50. Brink books
    • Throughout the book the emphasis is placed upon logic and method rather than on information.
    • This textbook for colleges and for nor mal and technical schools is noteworthy because of its completeness, its logic, its devices for the development of a high degree of technical skill, and its adapt ability to courses of various lengths and purposes.

  51. Gibson History 7 - Robert Simson
    • Simson's Opera Reliqua, published in 1776 from his MSS under the editorship of Professor Clow (Professor of Logic in Glasgow University), contains besides two short tracts on Logarithms and Limits the restoration of Apollonius's Determinate Section and Euclid's Porisms.
    • It is much to be regretted that he did not apply his profound knowledge of the Aristotelean logic to the somewhat crude reasonings of the founders of the modern algebra; had he devoted to this branch of mathematics a tithe of the labour he expended on the restoration of mutilated texts he would probably have had a more beneficial influence on the development of mathematics in Scotland.

  52. Survey of Modern Algebra
    • This has influenced us in our emphasis on the real and complex fields, on groups of transformations as contrasted with abstract groups, on symmetric matrices and reduction to diagonal form, on the classification of quadratic forms under the orthogonal and Euclidean groups, and finally, in the inclusion of Boolean algebra, lattice theory, and transfinite numbers, all of which are important in mathematical logic and in the modern theory of real functions.
    • There is also contact with the field of mathematical logic in the chapter on the algebra of classes and with the ideas of topology in the proof of the fundamental theorem of algebra.

  53. Papers about Lewis
    • J Corcoran, C I Lewis: History and Philosophy of Logic, Transactions of the Charles S Peirce Society 42 (1) (2006), 1-9.
    • S B Rosenthal, Logic and "Ontological Commitment": Lewis and Heidegger, The Journal of Speculative Philosophy, New Series 9 (4) (1995), 247-255.

  54. Hilbert quotes
    • With the exception of two books published alone (Grundlagen der Geometrie and Grundzuge einer allgemeinen Theorie der linearen Integralgleichungen), four more published with the collaboration of others (R Courant on mathematical physics, W Ackermann on logic, S Cohn-Vossen on geometry, and P Bernays on mathematical foundations), and about twenty papers, most of which would have meant duplication, Hilbert's monumental collected works were first published in three volumes between 1932 and 1935.
    • Hilbert's acceptance address was entitled "Natural Philosophy and Logic" .

  55. Thue speeches
    • In the ease and speed with which he grasped the most difficult questions, in creative imagination no less than in rigorous logic, he was far superior to his fellow students He acknowledged only one law: The resolute demand that his intellectual work must satisfy himself.
    • With the death of Professor Thue on 7 March this year, the Academy of Science has lost one of its most illustrious members, a mathematical genius who united the gift of extraordinary originality with a rare perspicacity and sense of logic.

  56. Horace Lamb addresses the British Association in 1904, Part 2
    • We have discussions on the principles of mechanics, on the foundations of geometry, on the logic of the most rudimentary arithmetical processes, as well as of the more artificial operations of the Calculus.
    • It is notorious that even in this realm of 'exact' thought discovery has often been in advance of strict logic, as in the theory of imaginaries, for example, and in the whole province of analysis of which Fourier's theorem is the type.

  57. R L Wilder: 'Cultural Basis of Mathematics I
    • 63-64) that "mathematical reality lies outside us, and that our function is to discover or observe it, and that the theorems which we prove, and which we describe grandiloquently as our 'creations' are simply our notes of our observations." On the other hand there is the point of view expressed by P W Bridgman [The logic of modern physics (Macmillan, New York, 1927).',3)">3] (p.
    • P W Bridgman, The logic of modern physics (Macmillan, New York, 1927).

  58. Andrew Forsyth addresses the British Association in 1905
    • He might, for instance, devote his attention to modern views of continuity, whether of quantity or of space; he might be heterodox or orthodox as to the so-called laws of motion; he might expound his notions as to the nature and properties of analytic functionality; a discussion of the hypotheses upon which a consistent system of geometry can be framed could be made as monumental as his ambition might choose; he could revel in an account of the most recent philosophical analysis of the foundations of mathematics, even of logic itself, in which all axioms must either be proved or be compounded of notions that defy resolution by the human intellect at the present day.

  59. Olds' teaching articles
    • When you have worked out a good definition for mathematics, test it by seeing whether it separates mathematics and physics, or mathematics and logic.

  60. Mathematics at Aberdeen 3
    • Less time was spent on Logic and Moral Philosophy, which were moved to the magistrand (fourth) year, thus recognizing the advantages of studying the rapidly expanding factual subjects first.

  61. Knorr's books
    • This book uses this perspective for the following three problems: what were the mathematical techniques used at various stages of the theory? What was its significance for pre-Euclidean geometry and for the ordering of the 'Elements'? What does its development give to the contemporary development of logic and thinking of philosophers of science? .

  62. George Temple's Inaugural Lecture II
    • 'Why is it that the ideas of the learned Englishman have such difficulty in getting acclimatised among us? It is doubtless because the education received by most enlightened Frenchmen predisposes them to relish precision and logic before every other quality.

  63. Harold Jeffreys: 'Scientific Inference' Preface
    • Discussions from the philosophical and logical point of view have tended to the conclusion that this principle cannot be justified by logic alone, which is true, and have left it at that.

  64. Halmos: creative art
    • Mathematics is abstract thought, mathematics is pure logic, mathematics is creative art.

  65. Comments by Charlotte Angas Scott
    • But while reading this brilliant exposition it is difficult to avoid cherishing a lurking regret, which is possibly very ungracious, that Klein could not himself spare time to arrange his work for publication; for though we have here in full measure the incisive thought and cultured presentation which together make even strict logic seem intuitive, yet at times we miss the minute finish and careful proportion of parts that we feel justified in expecting from him.

  66. May's papers
    • In the last few years there has been considerable discussion about the possibility of introducing into the secondary school curriculum topics such as logic, theory of sets, Boolean algebra, and the set theoretic approaches to relations, functions, and other topics of elementary mathematics.

  67. May's books
    • The book first introduces some of the ideas and notation of logic and sets, and then proceeds to use them for the remainder of its 600 pages.

  68. De Coste on Mersenne 1.html

  69. Harold Jeffreys on Probability
    • The logical demonstration is right or wrong as a matter of the logic itself, and is not a matter for personal judgment.

  70. George Salmon: from mathematics to theology
    • He usually sets out his arguments in the form of numbered lists, and frequently appeals to 'logic.' He is extremely thorough in his examination of all sides of an issue and offering a well-supported conclusion.

  71. Kurosh: 'Lectures on general algebra' Introduction
    • Eventually, the general theory of universal algebras came into being, as well as the even more general theory of models, which is interwoven with mathematical logic.

  72. Smith Teaching Papers
    • In spite of all that has been said in this country in opposition to mathematics in the past few years, the feeling of certainty still exists in the intellectual world that the science is not dead, is not dying, and is not stagnant; that it touches more lines of human interest today than ever before; and that its values have only been accentuated by the efforts made to relegate it to the position of formal grammar, formal rhetoric, and formal logic.

  73. Bell papers
    • For a philosophical discussion dealing, say, with mathematical logic, much more would be necessary.

  74. EMS obituary
    • As a student at St Andrews University he was placed in the Honours List of all his classes (which included Greek, Humanity, Rhetoric and English Literature, Moral Philosophy and Political Economy) obtaining first place in Mathematics, Natural Philosophy, Chemistry, Logic and Metaphysics, Anatomy and Physiology.

  75. Arthur Eddington's 1927 Gifford Lectures
    • I need not tell you that modern physics has by delicate test and remorseless logic assured me that my second scientific table is the only one which is really there - wherever "there" may be.

  76. Selected papers of Edward Marczewski' Preface
    • In that period he also published isolated papers dealing with real and complex analysis, applied mathematics and mathematical logic.

  77. Dudeney: 'Amusements in mathematics' Preface
    • Every puzzle that is worthy of consideration can be referred to mathematics and logic.

  78. Einstein: 'Geometry and Experience
    • It seems to me that complete clearness as to this state of things first became common property through that new departure in mathematics which is known by the name of mathematical logic or "Axiomatics." The progress achieved by axiomatics consists in its having neatly separated the logical-formal from its objective or intuitive content; according to axiomatics the logical-formal alone forms the subject-matter of mathematics, which is not concerned with the intuitive or other content associated with the logical-formal.

  79. Poincaré on non-Euclidean geometry
    • From these hypotheses he deduces a series of theorems between which it is impossible to find any contradiction, and he constructs a geometry as impeccable in its logic as Euclidean geometry.

  80. IMU Rolf Nevanlinna Prize
    • All mathematical aspects of computer science, including complexity theory, logic of programming languages, analysis of algorithms, cryptography, computer vision, pattern recognition, information processing and modelling of intelligence.

  81. Pólya's favourite quotes
    • Descartes: Good logic can be the worst enemy of good teaching.

  82. Mathematicians and Music
    • Other modes of expression and points of view were suggested by that great enthusiast to whom America owed much [Sylvester], him who called himself "the Mathematical Adam" because of the many mathematical terms he invented; for example, mathematic - to denote the science itself in the same way as we speak of logic, rhetoric or music, while the ordinary form is reserved for the applications of the science.

  83. De Coste on Mersenne
    • He studied with these learned men with great facility, not only literature, which because of its sweetness is called a Humanity, but also Logic, Physics, Metaphysics, Mathematics and some works on Theology, on all of which he thrived happily.

  84. Mathematics at Aberdeen 2
    • Details of the syllabus then taught at King's show that the Logic, Philosophy and Physics of Aristotle still formed a large part of the course.

  85. Serge Lang: 'Algebra
    • Most readers will already be acquainted with determinants, and we feel it is better for the organization of the whole book to allow ourselves such minor deviations from a total ordering of the logic involved.

  86. Cajori: 'A history of mathematics' Introduction
    • The interest which pupils take in their studies may be greatly increased if the solution of problems and the cold logic of geometrical demonstrations are interspersed with historical remarks and anecdotes.

  87. Centenary of John Leslie
    • Leslie had dared to speak of Hume with approval, stating that he was the first to treat of causation (cause and effect) in a truly philosophical manner, and had remarked that "the unsophisticated notions of mankind are in perfect unison with the deductions of logic, and imply nothing more at bottom in the relation of cause and effect than a constant and invariable sequence." On this the Edinburgh Presbytery charged him with "having laid a foundation for rejecting all the argument that is derived from the works of God, to prove either His Being, or His Attributes." The protest which was tendered by the ministers to the patrons of the chair, then the Provost and Town Council, stated that they were obliged by charter to act with the advice of the ministers.

  88. Samuel Wilks' books
    • Based on mathematics and on logic, its underlying principles and procedures are the same in each field to which it is applied.

  89. Bronowski and retrodigitisation
    • Louis Goodstein was Primrose's professorial colleague at Leicester from 1948 until retirement in 1977, being the first specialist in mathematical logic to hold a mathematical chair in the UK.

  90. Chrystal: 'Algebra' Preface
    • Among the treatises to which I am indebted in the matter of theory and logic, I should mention the works of De Morgan, Peacock, Lipschitz, and Serret.

  91. Wall's Creative mathematics
    • Much attention is given to matters of language and logic.

  92. Berge books
    • (ii) this elegant theory with its applications within mathematics to topology, logic, algebra, and combinatorial analysis will eventually become an undergraduate course at most modern universities.

  93. R A Fisher: 'Statistical Methods' Introduction
    • The author has attempted a fuller examination of the logic of planned experimentation in his book, The Design of Experiments.

  94. De Thou on François Viète
    • But at this time this generous man did not recognise the full merit of his adversary, and thus could not forebear to show resentment when he was corrected by him although he had not in fact fully examined the logic of his own argument.

  95. Science at St Andrews
    • This celebrated mathematician and philosopher was the Martin Luther among the men of science, refuting the logic of Aristotle, and siding with the practical geometer against Euclid.

  96. A D Aleksandrov's view of Mathematics
    • There is very little on foundations or logic.

  97. Von Neumann Silliman lectures
    • His publications included works on quantum theory, mathematical logic, ergodic theory, continuous geometry, problems dealing with rings of operators, and many other areas of pure mathematics.

  98. Mathematics at Aberdeen 4
    • From about 1765 the publicly defended thesis required for a degree had been replaced by the repetition of previously dictated answers to a few questions on Logic and Rhetoric.

  99. Smith's Teaching Books
    • Among the problems that he discusses are: Shall geometry continue to be taught as an application of logic, or shall it be treated solely with reference to its applications? Shall geometry be taught by itself, or shall it be mixed with algebra? Shall a textbook be used in which the basal propositions are proved in full, or shall one be employed in which the pupils are expected to invent the proofs for the basal propositions, as well as for the exercises? What form of terminology shall prevail? And shall geometry be made a strong elective subject to be taken only by those whose minds are capable of serious work? .

  100. Kline's books
    • has supplied substance to economic and political theories, has fashioned major painting, musical, architectural, and literary styles, has fathered our logic, and has furnished the best answers we have to fundamental questions about the nature of man and his universe.

  101. George Chrystal's Second Promoter's Address
    • My amiable and respected colleague Professor Campbell Fraser no longer holds the chair of Logic and Metaphysics, and no longer presides the Faculty of Arts as Dean.

  102. Al-Kashi's letter
    • He knows (Arabic) grammar well and he writes Arabic composition extremely well, and like-wise he is well posted in canon law; he has knowledge of logic, rhetoric, and elocution, and likewise of the Elements (of Euclid), and he himself cultivates the branches of mathematics, and this has reached the extent that one day while riding he wanted to determine the date, which was a Monday of [the month of] Rajab, between the fifth and the tenth in the year eight hundred and eighteen (A.

  103. Writings of Charles S Peirce' Preface
    • In choosing papers for A chronological edition, preference was in general given to the more significant writings in the philosophy of science, in logic, and in metaphysics.

  104. Philip Jourdain and Georg Cantor
    • The three men whose influence on "modern" pure mathematics - and indirectly modern logic and the philosophy which abuts on it - is most marked are Karl Weierstrass, Richard Dedekind, and Georg Cantor.

  105. Burali-Forti Russell letter
    • Previous studies have logic summers have greatly helped me establish the vector notation.

  106. Landau and Lifshitz Prefaces
    • The logic of the arrangement is that the topics dealt with here are closely akin also to those in fluid mechanics (Volume 6) and macroscopic electrodynamics (Volume 8).

  107. Leslie works
    • He admires its power but mislikes it for apparently subordinating the logic.

  108. Menger on teaching
    • Recent investigations have resolved it into an extensive spectrum of meanings, some pertaining to reality as investigated in science, others belonging to the realm of symbols studied in logic - two altogether different worlds.

  109. Finkel's Solution Book
    • - Galois's Theory of Equations; Lie's Theory as applied to Differential Equations; Riemann's Theory of Functions; The Calculus of Variations; Functions Defined by Linear Differential Equations; The Theory of Numbers; The Theory of Planetary Motions; Theory of Surfaces; Linear Associative Algebra; the Algebra of Logic; the Plasticity of the Earth; Elasticity; and the Elliptic and the Abelian Transcendants.

  110. Bertrand Russell on Euclid
    • Many more general criticisms might be passed on Euclid's methods, and on his conception of Geometry; but the above definite fallacies seem sufficient to show that the value of his work as a masterpiece of logic has been very grossly exaggerated.

  111. Slaught on Mathematics and Teaching
    • When I was a student in an academy devoted entirely to preparation for college, I met my first mathematical thrill in beginning the study of algebra under a teacher who was not a mere lesson hearer but who put due emphasis upon the reasons for every step in a solution and insisted upon our seeing the logic involved in every process.

  112. Truesdell's books
    • (with Subramanyam Bharatha) The concepts and logic of classical thermodynamics as a theory of heat engines.

  113. Reviews of Shafarevich's books
    • The systematic study of this new geometry leads us to 2-dimensional Lobachevsky geometry (also called non-Euclidean or hyperbolic geometry) which, following the logic of our study, is constructed starting from the properties of its group of motions.


  1. Quotations by Russell
    • Only mathematics and mathematical logic can say as little as the physicist means to say.
    • The fact that all Mathematics is Symbolic Logic is one of the greatest discoveries of our age; and when this fact has been established, the remainder of the principles of mathematics consists in the analysis of Symbolic Logic itself.

  2. A quotation by Boutroux
    • Logic is invincible because in order to combat logic it is necessary to use logic.

  3. Quotations by Church
    • Well it was not exactly a dissertation in logic, at least not the kind of logic you would find in Whitehead and Russell's Principia Mathematica for instance.

  4. Quotations by Dantzig
    • Not by logic, for logic has no existence independent of mathematics: it is only one phase of this multiplied necessity that we call mathematics.

  5. Quotations by Kac
    • Mark Kac and Stanislaw Ulam, Mathematics and Logic (1968) .

  6. A quotation by Dudeney
    • A good puzzle should demand the exercise of our best wit and ingenuity, and although a knowledge of mathematics and of logic are often of great service in the solution of these things, yet it sometimes happens that a kind of natural cunning and sagacity is of considerable value.

  7. Quotations by De Morgan
    • Every science that has thriven has thriven upon its own symbols: logic, the only science which is admitted to have made no improvements in century after century, is the only one which has grown no symbols.

  8. Quotations by Heaviside
    • Logic can be patient, for it is eternal.

  9. Quotations by Ampere
    • Ordinarily logic is divided into the examination of ideas, judgments, arguments, and methods.

  10. Quotations by Born
    • Science is not formal logic it needs the free play of the mind in as great a degree as any other creative art.

  11. Quotations by Hadamard
    • Logic merely sanctions the conquests of the intuition.

  12. Quotations by Polya
    • When introduced at the wrong time or place, good logic may be the worst enemy of good teaching.

  13. Quotations by Hoyle
    • I don't see the logic of rejecting data just because they seem incredible.

  14. Quotations by Boole
    • Mathematics had never had more than a secondary interest for him; and even logic .

  15. Quotations by Wiener Norbert
    • To see a difficult uncompromising material take living shape and meaning is to be Pygmalion, whether the material is stone or hard, stonelike logic.

  16. Quotations by Poincare
    • The geometer might be replaced by the "logic piano" imagined by Stanley Jevons; or, if you choose, a machine might be imagined where the assumptions were put in at one end, while the theorems came out at the other, like the legendary Chicago machine where the pigs go in alive and come out transformed into hams and sausages.

  17. Quotations by Wittgenstein
    • There can never be surprises in logic.

  18. Quotations by Ulam
    • Mark Kac and Stanislaw Ulam, Mathematics and Logic (1968) .

  19. Quotations by Weyl
    • Logic is the hygiene the mathematician practices to keep his ideas healthy and strong.

  20. Quotations by Courant
    • Its basic elements are logic and intuition, analysis and construction, generality and individuality.

  21. Quotations by Godel
    • Nothing new had been done in Logic since Aristotle! .

  22. A quotation by Arnauld
    • The Art of Thinking: Port-Royal Logic .

  23. Quotations by Boltzmann
    • the source of this kind of logic lies in excessive confidence in the so-called laws of thought.

  24. Quotations by Peirce Charles
    • Every other science, even logic, especially in its early stages in danger of evaporating into airy nothingness, degenerating, as the Germans say, into an arachnoid film, spun from the stuff that dreams are made of.

  25. A quotation by Duhem
    • The whole theory of electrostatics constitutes a group of abstract ideas and general propositions, formulated in the clear and concise language of geometry and algebra, and connected with one another by the rules of strict logic.


  1. Mathematical Chronology
    • Ibn Sina (usually called Avicenna) writes on philosophy, medicine, psychology, geology, mathematics, astronomy, and logic.
    • Nicholas of Cusa studies geometry and logic.
    • Boole publishes The Mathematical Analysis of Logic, in which he shows that the rules of logic can be treated mathematically rather than metaphysically.
    • Boole's work lays the foundation of computer logic.
    • Boole publishes The Laws of Thought on Which are founded the Mathematical Theories of Logic and Probabilities.
    • He reduces logic to algebra and this algebra of logic is now known as Boolean algebra.
    • His theory of topoi is highly relevant to mathematical logic, he had given an algebraic proof of the Riemann-Roch theorem, and provided an algebraic definition of the fundamental group of a curve.

  2. Chronology for 1850 to 1860
    • Boole publishes The Laws of Thought on Which are founded the Mathematical Theories of Logic and Probabilities.
    • He reduces logic to algebra and this algebra of logic is now known as Boolean algebra.

  3. Chronology for 1840 to 1850
    • Boole publishes The Mathematical Analysis of Logic, in which he shows that the rules of logic can be treated mathematically rather than metaphysically.
    • Boole's work lays the foundation of computer logic.

  4. Chronology for 1960 to 1970
    • His theory of topoi is highly relevant to mathematical logic, he had given an algebraic proof of the Riemann-Roch theorem, and provided an algebraic definition of the fundamental group of a curve.

  5. Chronology for 1300 to 1500
    • Nicholas of Cusa studies geometry and logic.

  6. Chronology for 900 to 1100
    • Ibn Sina (usually called Avicenna) writes on philosophy, medicine, psychology, geology, mathematics, astronomy, and logic.

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JOC/BS August 2001