# Who are Boole, Fitch, and Tarski?

Here’s a list of the logicians that show up in Barwise and Etchemendy’s Language, Proof, and Logic and in the exercise files for Tarski’s World. Most names are linked to websites with more information.

Abelard, Peter (Pierre Abélard) (1079-1142)
French theologian and philosopher best known for his work on the problem of universals. He is also known for his poetry and for his celebrated love affair with Heloise–Abelard and Heloise were the original celebrity couple.
Ackermann, Wilhelm (1896-1962)
German logician and student of Hilbert. Gave the first direct consistency proof of a non-trivial mathematical theory, and contributed to research on the decision problem. Co-author (with Hilbert) of Grundzüge der theoretischen Logic (1928).
Aristotle (384-322 BCE)
Ancient Greek philosopher and founder of logic as an independent discipline. His theory of categories and syllogisms as presented in On Interpretation and the Prior Analytics shaped the field of logic up until the 19th century.
Arnauld, Antoine (l6l2-1694)
Leading 17th-century French theologian, who, through correspondence, exerted significant intellectual influence on Descartes and Leibniz. His Logique de Port-Royal (written together with Pierre Nicole) was one of the most important logic textbooks of the early modern period.
Austin, J. L. (1911-1960)
British philosopher and main proponent of “ordinary language philosophy” and critic of logical positivism. He initiated the theory of speech acts.
Bernays, Paul (1888-1977)
Swiss logician and philosopher, assistant and collaborator of Hilbert. Gave the first proof of the completeness of the propositional calculus in 1918, and made significant contributions to proof theory, set theory, and the philosophy of mathematics. Principal author of the two-volume Grundlagen der Mathematik (1934, 1939, with David Hilbert).
Bernstein, Felix (1878-1956)
German mathematician, student of Cantor and Hilbert, who worked mostly in applied mathematics. He did, however, make significant contributions to set theory, including the Cantor-Bernstein Theorem, which says that two infinite sets X, Y are equivalent (i.e., can be put into 1-1 correspondence) if there are are 1-1 functions of X to Y, and of Y to X.
Bolzano, Bernhard (1781-1848)
Bohemian mathematician and theologian who was an important figure in the move to rid the calculus of infinitesimals. He gave a detailed proof proof of the binomial theorem in 1816, and a proof of the intermediate value theorem in 1817, using methods that would be rediscovered by Cauchy four years later. He also worked on ealry set theory, and suggested the following definition: a set is infinite if it is in 1-1 correspondence with a proper subset of itself.
Boole, George (1815-1864)
British mathematician who first proposed to study logic as by mathematical methods, and thus founded mathematical logic as a field of research. His approach was algebraic, and his “algebra of logic” was the main approach to logic of the 19th century. Author of An Investigation into the Laws of Thought (1854).
Boolos, George (1940-1996)
US philosopher who developed provability logic (logic with a sentence connective “it is provable that”) and made many other contributions to logic and the philosophy of mathematics. Co-author, with Richard Jeffrey, of the classic text Computability and Logic.
Brouwer, L. E. J. (1881-1966)
Dutch mathematician and philosopher of mathematics who proposed an alternative system of mathematics. In intuitionistic mathematics, all proofs have to be carried out constructively, and mathematical objects themselves are based on constructive processes. Intuitionism rejects the principle of the excluded middle and the double-negation rule.
Buridan, Jean (1300-1358)
Aristotelian philosopher and logician, who also worked on mechanics and optics. He was a student of William of Ockham and later taught at the University of Paris. Author of Consequentie (1493)
Cantor, Georg (1845-1918)
German mathematician who founded set theory as a discipline, and showed that whereas the rational numbers are countable (can be put in 1-1 correspondence with the integers), the real numbers are not.
Carnap, Rudolf (1891-1970)
German/US philosopher of Logical Positivism, and founding member of the Vienna Circle. He contributed significantly to logic, probability theory, philosophy of langauge and philosophy of science.
Carroll, Lewis (pseud., 1832-1898)
British logician, mathematician, photographer, and novelist, especially remembered for Alice’s Adventures in Wonderland (1865) and its sequel, Through the Looking-Glass (1871).
Church, Alonzo (1903-1995)
US logician who invented lambda calculus, which is the basis of functional programming languages, showed that first-order logic is undecidable, and proposed what is now known as the Church-Turing Thesis, i.e., the thesis that the intuitively computable functions are exactly the recursive functions.
Cooper, Robin
Linguist who wrote a paper with Jon Barwise on generalized quantifiers.
DeMorgan, Augustus (1806-1871)
British mathematician and logician working in the tradiation of Boole. His contributions to logic include the formulation of DeMorgan’s laws and work leading to the development of the theory of relations.
Dodgson, Charles Lutwidge
see Lewis Carroll
Euler, Leonhard (1707-1783)
Swiss/German mathematician who made significant contributions to number theory, analysis, mechanics, and astronomy. The base of the natural logarithm e is named after him. He also invented Venn diagrams a century before Venn did.
Finsler, Paul (1894-1970)
German mathematician who made important contributions to set theory.
Fitch, Frederic (1908-1987)
US philosopher who, in his classic textbook Symbolic Logic (1952) gave a particularly elegant formulation of natural deduction, which is the basis of the system used in LPL.
Frege, Gottlob (1848-1925)
German mathematician and logician, who worked in the philosophy of mathematics and mathematical logic. Frege was the first to realize the importance of relations in logical languages, and developed a formal language with quantifiers (the first of its kind). The fundamental ideas in his Begriffsschrift (1879) made the modern development of logic in the 20th century possible. He also attempted to prove the axioms and theorems of mathematics in a purely logical system (Grundgesetze der Mathematik, 2 vols, 1893, 1903); however, Russell showed that his system was inconsistent.
Gödel, Kurt (1906-1978)
Austrian/US mathematician and logician. Proved the completeness of predicate calculus (1929) and the famous Incompleteness Theorems (1930), the first of which states that for any any sufficiently strong formal system of mathematics there are propositions that cannot be proved or disproved on the basis of the axioms of that system.
Henkin, Leon (1921-2006)
US logician and student of Tarski who gave an elegant, improved proof of Gödel‘s completeness theorem for first-order logic.
Hilbert, David (1862-1943)
German mathematician who, among many other significant contributions to all areas of mathematics, reduced geometry to a series of axioms and contributed substantially to the establishment of the foundations of mathematics. Grundzüge der theoretischen Logik (1928, with Wilhelm Ackermann), Grundlagen der Mathematik, 2 vols. (1934, 1939, with Paul Bernays).
Horn, Alfred (1918-2001)
US mathematician who studied a class of sentences now known as Horn sentences (conjunctive normal forms where each conjunct contains at most one negated atomic sentence). The theory of Horn sentences forms the basis of logic programming.
Kleene, Stephen Cole (1909-1994)
US mathematician and student of Church who developed recursive function theory, and also made contributions to Brouwer‘s intuitionism. Computer scientists know his name from the “Kleene star”, a notation for describing regular languages: {a, b, c}* is the set of all finite strings of a‘s, b‘s, and c‘s. Author of influential textbook Introduction to Metamathematics.
König, Julius (1849-1913)
Hungarian mathematician who worked mostly in algebra and analysis. In the last years of his life he became interested in set theory and logic, and wrote a book on New Foundations for Logic, Arithmetic, and Set Theory. Although it did not have a wide impact at the time, his ideas were later introduced into the logical mainstream by his student John von Neumann.
Leibniz, Gottfried Wilhelm (1646-1716)
German philosopher, mathematician, and political adviser, important both as a metaphysician and as a logician and distinguished also for his invention of the differential and integral calculus independently of Newton.
Leibniz formulated the Principle of the Identity of Indiscernibles, viz., that different objects must differ in the properties that they have. Together with the Indiscernibility of Identicals (=Elim), this gives, in a sense, a definition of identity. Exercise 11.10 and Leibniz’s Sentences deal with identity.
Löwenheim, Leopold (1878-1957)
German logician working in the algebra of logic, proved the decidability of first-order logic with only one-place predicate symbols and the Löwenheim-Skolem Theorem, according to which every satisfiable set of sentences (in a countable language) has a countable model.
Malcev, Anatoly (1909-1967)
Russian/Soviet mathematician who worked in algebra and model theory. He proved that, among others, the theory of finite groups is undecidable.
Montague, Richard (1930-1971)
US logician and linguist, and student of Tarski, who made seminal contributions to the formal semantics of natural language.
Mostowski, Andrzej (1913-1975)
Polish logician and student of Tarski who worked in set theory and model theory.
Ockham, William of (1285-1347/49)
Franciscan philosopher. theologian, and political writer, a late scholastic thinker regarded as the founder of a form of nominalism–the school of thought that denies that universal concepts such as father have any reality apart from the individual things signified by the universal or general term.
Italian logician and student of Peano, who investigated inependence and definability. Padoa’s Method is a way to show that a term cannot be defined in an axiomatic system.
In Exercise 13.54, you are asked to construct worlds in which one of four sentences is false, whereas the other are true. This is an application of Padoa’s Method for proving independence: the four sentences in Padoa’s Sentences are independent, i.e., no one of them can be proved from the others.
Peano, Giuseppe (1858-1932)
Italian mathematician and a founder of symbolic logic whose interests centered on the foundations of mathematics and on the development of a formal logical language.
Peirce, Charles Sanders (1839-1914)
US logician and philosopher who originated American pragmatism, founded the discipline of semiotics, and introduced quantifiers independently of Frege.
Post, Emil Leon (1897-1954)
US logician who gave first published proof of the completeness and truth-functional completeness of propositional logic, and pioneered the theory of computability.
Ramsey, Frank Plumpton (1903-1930)
British mathematician, logician, and philosopher of science famous for his work in the foundation of mathematics and in combinatorics.
In his 1927 paper “Facts and Propositions,” Ramsey defended the view, following Wittgenstein, that “for all x, F(x)” is equivalent to the conjunction F(a1), F(a2), F(a3), …, where a1, a2, a3, … are all the elements of the domain of discourse. Exercise 10.22 and Ramsey’s World show the limitations of this view
Reichenbach, Hans (1891-1953)
German philosopher who was a leading representative of the Vienna Circle and founder of the Berlin school of logical positivism. He contributed significantly to logical interpretations of probability theory, theories of induction, and the philosophical bases of science.
Robinson, Abraham (1918-1974)
German/Israeli/US mathematician who made significant contributions to model theory and pioneered non-standard analysis.
Robinson, John Alan (1930-)
Computer scientist who pioneered automated-theorem proving, proposing the resolution calculus in his 1965 paper, “A Machine-Oriented Logic Based on the Resolution Principle.” Resolution is the basis of logic programming languages such as Prolog.
Robinson, Julia Bowman (1919-1985)
US logician who proved the undecidability of the field of rational numbers and took significant steps towards the solution of Hilbert’s Tenth Problem. Student and colleague of Tarski.
Russell, Bertrand (1872-1970)
British logician and philosopher, best known for his work in mathematical logic (Principia Mathematica 1910-13, with Alfred North Whitehead) and for his social and political writings, especially his advocacy of pacifism and nuclear disarmament. He received the Nobel Prize for Literature in 1950. Russell’s Paradox showed that one of Frege‘s logical systems was inconsistent, Russell subsequently tried to correct Frege’s mistake, which led to Russell’s logical work.
Schönfinkel, Moses (1889-1942)
Russian logician who, as a student of Hilbert and starting from Sheffer’s reduction of the propositional connectives to the Sheffer stroke, invented combinatory logic. Combinatory logic was later developed by Haskell Curry; Alonzo Church‘s lambda calculus is similar and takes ideas from Schönfinkel. Combinatory logic/lambda calculus is the basis of functional programming languages. Schönfinkel also made contributions to the decision problem. He and Bernays showed that satisfiability of prefix sentences of FOL with all existential quantifiers preceding all universal ones (the Bernays-Schönfinkel class), is decidable.
Schröder, Ernst (1841-1902)
German mathematician who made significant contributions to set theory and algebraic logic.
Sheffer, Henry Maurice (1882-1964)
American logician who showed that the connectives ‘or’ and ‘not’ can be defined using a single connective, the Sheffer stroke.
Sextus Empiricus (c. 200 CE)
Ancient Greek philosopher whose Outlines of Pyrrhonism is the main source of information about ancient skepticism. He discussed negation at length, so Exercise 4.32, an exercise about negation normal form, uses Sextus’ Sentences.
Skolem, Thoralf (1887-1963)
Norwegian logician known especially for the Löwenheim-Skolem Theorem and Skolem’s Paradox: It follows from the Löwenheim-Skolem Theorem that if set theory has a model, it has a countable model; yet set theory proves that there are uncountable sets.
Socrates (c. 470-c. 399 BCE)
“Father” of ancient Greek philosophy, teacher of Plato. Was sentenced to death for “corruption of youth.” He developed the philosophical method of dialectic, which examines views by pursuing their consequences: if they are tenable, they should not lead to false consequences.
Tarski, Alfred (1902-1983)
Polish/US mathematician and logician whose contributions to logic and metamathematics had lasting influence on the field in the 20th century. He is perhaps best known for the Banach-Tarski Paradox and his theory of truth, but made many fundamental contributions to logic, including his proof of the decidability of the theory of real numbers.
Turing, Alan Mathison (1912-1954)
British mathematician who proved, independently of Church, that the decision problem for first-order logic is unsolvable. In the process, he invented Turing Machines (a mathematical model of computers and the basis of much of computability theory). During WWII he worked at Bletchley Park and was instrumental in breaking the German Enigma code. Also proposed the Turing Test as a criterion of machine intelligence, and the Church-Turing Thesis. Committed suicide after being arrested for homosexuality.
Venn, John (1834-1923)
Wiener, Norbert (1894-1964)
US mathematician who studied with Russell and Hilbert and invented cybernetics. In his 1913 dissertation (he was 18!), he compared the logical systems of Schröder and Russell and Whitehead‘s Principia Mathematica, and proposed a way to define ordered pairs using sets, thus reducing the theory of relations to set theory.