 Abelard, Peter
(Pierre Abélard) (10791142)
 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 HeloiseAbelard and Heloise were the original celebrity
couple.
 Ackermann,
Wilhelm (18961962)
 German logician and student of Hilbert. Gave
the first direct consistency proof of a nontrivial mathematical
theory, and contributed to research on the decision problem. Coauthor
(with Hilbert) of Grundzüge der theoretischen Logic
(1928).
 Aristotle
(384322 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 (l6l21694)
 Leading 17thcentury French theologian, who, through correspondence,
exerted significant intellectual influence on Descartes and Leibniz. His
Logique de PortRoyal (written together with Pierre Nicole)
was one of the most important logic textbooks of the early modern period.
 Austin, J. L.
(19111916)
 British philosopher and main proponent of "ordinary language philosophy"
and critic of logical positivism. He initiated the theory
of speech acts.
 Bernays,
Paul (18881977)
 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 twovolume Grundlagen der Mathematik (1934,
1939, with David Hilbert).
 Bernstein,
Felix (18781956)
 German mathematician, student of Cantor and Hilbert, who worked mostly in applied mathematics. He
did, however, make significant contributions to set theory, including
the CantorBernstein Theorem, which says that two infinite sets X,
Y are equivalent (i.e., can be put into 11 correspondence)
if there are are 11 functions of X to Y, and of Y to X.
 Bolzano,
Bernhard (17811848)
 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 11 correspondence with
a proper subset of itself.
 Boole,
George (18151864)
 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
(19401996)
 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. Coauthor, with Richard
Jeffrey, of the classic text Computability and Logic.
 Brouwer,
L. E. J. (18811966)
 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 doublenegation rule.
 Buridan, Jean
(13001358)
 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 (18451918)
 German mathematician who founded set theory as a discipline, and showed
that whereas the rational numbers are countable (can be put in 11
correspondence with the integers), the real numbers are not.
 Carnap, Rudolf
(18911970)
 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., 18321898)
 British logician, mathematician, photographer, and novelist, especially
remembered for Alice's
Adventures in Wonderland (1865) and its sequel, Through
the LookingGlass (1871).
 Church,
Alonzo (19031995)
 US logician who invented lambda calculus, which is the basis of functional
programming languages, showed that firstorder logic is undecidable,
and proposed what is now known as the ChurchTuring
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 (18061871)
 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 (17071783)
 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 (18941970)
 German mathematician who made important contributions to set theory.
 Fitch, Frederic (19081987)
 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 (18481925)
 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 (19061978)
 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)
 US logician and student of Tarski who gave an
elegant, improved proof of Gödel's completeness
theorem for firstorder logic.
 Hilbert,
David (18621943)
 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
(19182001)
 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 (19091994)
 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 (18491913)
 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 (16461716)
 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 (18781957)
 German logician working in the algebra of logic, proved the decidability
of firstorder logic with only oneplace predicate symbols and the
LöwenheimSkolem Theorem, according to which
every satisfiable set of sentences (in a countable language) has a countable
model.
 Malcev,
Anatoly (19091967)
 Russian/Soviet mathematician who worked in algebra and model theory.
He proved that, among others, the theory of finite groups is undecidable.
 Montague,
Richard (19301971)
 US logician and linguist, and student of Tarski,
who made seminal contributions to the formal semantics of natural
language.
 Mostowski,
Andrzej (19131975)
 Polish logician and student of Tarski who worked
in set theory and model theory.
 Ockham,
William of (12851347/49)
 Franciscan philosopher. theologian, and political writer, a late scholastic
thinker regarded as the founder of a form of nominalismthe 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.
 Padoa,
Alessandro (18681937)
 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 (18581932)
 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 (18391914)
 US logician and philosopher who originated American pragmatism, founded
the discipline of semiotics, and introduced quantifiers independently
of Frege.
 Post,
Emil Leon (18971954)
 US logician who gave first published proof of the completeness and
truthfunctional completeness of propositional logic, and pioneered
the theory of computability.
 Ramsey,
Frank Plumpton (19031930)
 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(a_{1}), F(a_{2}),
F(a_{3}), ..., where a_{1}, a_{2},
a_{3}, ... are all the elements of the domain of discourse.
Exercise 10.22 and Ramsey's World show the limitations of this view
 Reichenbach,
Hans (18911953)
 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 (19181974)
 German/Israeli/US mathematician who made significant contributions
to model theory and pioneered nonstandard analysis.
 Robinson, John Alan (1930)
 Computer scientist who pioneered automatedtheorem
proving, proposing the resolution calculus in his 1965 paper,
"A MachineOriented Logic Based on the Resolution Principle." Resolution
is the basis of logic programming languages such as Prolog.
 Robinson,
Julia Bowman (19191985)
 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 (18721970)
 British logician and philosopher, best known for his work in mathematical
logic (Principia
Mathematica 191013, with Alfred North Whitehead)
and for his social and political campaigns, including his advocacy of
both 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 (18891942)
 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 BernaysSchönfinkel class), is decidable.
 Schröder,
Ernst (18411902)
 German mathematician who made significant contributions to set theory
and algebraic logic.
 Sheffer, Henry Maurice (18821964)
 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 (18871963)
 Norwegian logician known especially for the Löwenheim
Skolem Theorem and Skolem's Paradox: It follows from the LöwenheimSkolem
Theorem that if set theory has a model, it has a countable model;
yet set theory proves that there are uncountable sets.
 Socrates (c.
470c. 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 (19021983)
 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 BanachTarski
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 (19121954)
 British mathematician who proved, independently of Church, that the
decision problem for firstorder 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 ChurchTuring Thesis.
Committed suicide after being arrested for homosexuality.
 Venn,
John (18341923)
 British logician who worked in the tradition of Boole,
(re)invented Venn Diagrams
(see also Euler).
 Wiener,
Norbert (18941964)
 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.
 Whitehead,
Alfred North (18611947)
 British mathematician and philosopher, who collaborated with
Bertrand Russell on Principia Mathematica.
 Wittgenstein,
Ludwig (18891951)
 Austrian/British philosopher, who was one of the most influential figures
in AngloAmerican philosophy during the second quarter of the 20th century
and who produced two original and influential systems of philosophical thoughthis
logical theories and later his language philosophy. Proved the decidability
of propositional logic in his Tractatus Logico
Philosophicus (1921).
 Zorn,
Max (19061993)
 German/US mathematician working in topology and algebra, who proved
Zorn's Lemma, an equivalent of the Axiom
of Choice.