The discipline that asks whether the argument holds before asking whether it is true.
Logic is the study of valid inference — the rules that determine whether a conclusion follows from its premises, regardless of what the premises are about. It began with Aristotle’s syllogistic in the fourth century BC and remained essentially Aristotelian for over two thousand years. Then, in the space of a single century, everything changed. Frege invented predicate logic and tried to reduce arithmetic to it. Cantor built set theory and discovered that infinity comes in sizes. Russell found a paradox at the heart of Frege’s system and spent a decade trying to repair it. Hilbert proposed that all of mathematics could be formalised, proved consistent, and made decidable. Gödel showed it could not. Turing defined computability and proved the halting problem undecidable. Tarski gave formal semantics to the concept of truth. Kripke gave semantics to possibility and necessity. The department assembles the central figures of this tradition — from the syllogism to the incompleteness theorems — as working interlocutors. Many are cross-posted from Mathematics, Philosophy, Computing, and Rhetoric, but here they are gathered as logicians, practitioners of the art that underlies all the others.
The founder of logic as a discipline. His Organon — the Categories, De Interpretatione, Prior Analytics, Posterior Analytics, Topics, and Sophistical Refutations — defined the study of valid inference for over two thousand years. The syllogism is his invention: All A are B; All B are C; therefore All A are C. He also established the theory of the categories (substance, quantity, quality, relation, etc.) and the foundations of metaphysics (being qua being, substance and accident, potentiality and actuality). Cross-posted from Philosophy.
Can help you study: The Organon, syllogistic logic, the categories, the square of opposition, demonstration and dialectic, the foundations of metaphysics, and the argument that logic is the instrument that every science requires before it can begin.
A broader Aristotle simulacrum covering logic, rhetoric, and metaphysics together — for questions that cross the boundaries of the specialised Aristotle baselines. The syllogism, the categories, the enthymeme, the four causes, substance and accident, potentiality and actuality — all available in one interlocutor. Cross-posted from Philosophy.
Can help you study: Syllogistic logic, rhetoric, metaphysics, the four causes, substance, the categories, and any question that requires Aristotle whole rather than Aristotle in parts.
Third head of the Stoic school and the most important logician between Aristotle and Frege. He invented propositional logic — the logic of if...then, either...or, and not — which Aristotle’s syllogistic could not express. His five indemonstrable argument forms (modus ponens, modus tollens, disjunctive syllogism, and two conjunctive forms) are the foundation of all propositional reasoning. He wrote over 700 works; almost none survive. We reconstruct his logic from later reports, especially Diogenes Laertius and Sextus Empiricus.
Can help you study: Stoic propositional logic, the five indemonstrables, conditionals, the liar paradox, modal logic in antiquity, the Stoic theory of lekta (sayables), and the argument that propositions, not terms, are the fundamental units of logic.
French dialectician, theologian, and the most original logician of the twelfth century. His Sic et Non (Yes and No) assembled 158 questions on which the Church Fathers contradict each other — not to undermine authority but to teach students to reason through contradiction. He advanced the problem of universals beyond anything since antiquity, arguing that universals are not things but ways of predicating — a position close to nominalism. He also developed propositional analysis independently of the Stoic tradition.
Can help you study: Dialectic, Sic et Non, the problem of universals, nominalism, propositional analysis, medieval logic, and the argument that doubt is the beginning of inquiry.
English Franciscan friar, philosopher, and logician whose nominalism and principle of parsimony (“Ockham’s Razor”) transformed medieval philosophy. He argued that universals are not real entities but mental signs — that only individuals exist. His Summa Logicae is the most comprehensive logic textbook of the Middle Ages, covering terms, propositions, and syllogisms with a rigour that anticipates modern formal logic. His supposition theory — analysing how terms stand for things in different propositional contexts — is a genuine contribution to semantics.
Can help you study: Nominalism, Ockham’s Razor, supposition theory, the Summa Logicae, the problem of universals, mental language (oratio mentalis), and the argument that entities are not to be multiplied beyond necessity.
German polymath who dreamed of reducing all reasoning to calculation. His characteristica universalis was a proposed universal symbolic language in which every concept could be expressed; his calculus ratiocinator was the machine that would manipulate those symbols. He invented binary arithmetic, built a calculating machine, and independently co-invented the calculus. The dream of mechanical reasoning that Boole, Frege, and Turing eventually realised begins with Leibniz. “Calculemus!” — let us calculate.
Can help you study: The characteristica universalis, the calculus ratiocinator, binary arithmetic, the principle of sufficient reason, monads, the identity of indiscernibles, and the dream of reducing all thought to computation.
Self-taught English mathematician who demonstrated that the laws of logic can be expressed as algebraic equations. His The Laws of Thought (1854) reduced AND, OR, and NOT to operations on truth values (1 and 0), creating Boolean algebra — the mathematical foundation of every digital circuit, every database query, and every search engine in the world. He held the first chair of mathematics at Queen’s College, Cork, and died at forty-nine from a fever contracted after walking to a lecture in the rain.
Can help you study: Boolean algebra, The Laws of Thought, algebraic logic, truth values, AND/OR/NOT as mathematical operations, and the argument that logic is a branch of mathematics.
American philosopher, logician, and semiotician who independently invented quantificational logic, developed a diagrammatic logic (existential graphs) that in some respects surpasses Frege’s notation, and introduced abduction — inference to the best explanation — as the third mode of reasoning alongside deduction and induction. He is the founder of pragmatism (which he later renamed pragmaticism to distinguish it from James’s version) and semiotics. He died in poverty, largely unrecognised.
Can help you study: Abduction, existential graphs, semiotics, pragmaticism, the classification of signs, the logic of discovery, and the argument that there are three modes of inference, not two.
German mathematician who created set theory and proved that infinity comes in sizes — that the real numbers are uncountably infinite while the natural numbers are merely countably so. His diagonal argument is one of the most beautiful proofs in mathematics. His continuum hypothesis remains independent of the standard axioms. Kronecker called him a corrupter of youth; Hilbert called his work a paradise from which no one shall expel us. Cross-posted from Mathematics.
Can help you study: Set theory, cardinality, transfinite numbers, the diagonal argument, the continuum hypothesis, ordinals, and the argument that completed infinities are legitimate mathematical objects.
German logician and philosopher who invented modern predicate logic — the Begriffsschrift (concept-script) of 1879, which replaced Aristotle’s syllogistic with a notation powerful enough to express all of mathematics. His Grundgesetze der Arithmetik attempted to derive arithmetic from pure logic. Russell’s paradox showed that the system was inconsistent. He also introduced the distinction between sense (Sinn) and reference (Bedeutung), which founded the philosophy of language. Cross-posted from Mathematics.
Can help you study: Predicate logic, the Begriffsschrift, the foundations of arithmetic, sense and reference, the concept of a function, Russell’s paradox and its aftermath, and the logicist programme.
A dedicated logic simulacrum built from Frege’s logical works — the Begriffsschrift, the Grundgesetze, and the philosophical papers on concept, object, function, and sense. Where the cross-posted Frege covers the full range of his contributions, this simulacrum focuses on the technical logic: quantification, function-argument structure, the concept-object distinction, and the programme of reducing arithmetic to logic.
Can help you study: The Begriffsschrift, predicate logic, quantification, function-argument structure, the concept-object distinction, the Grundgesetze, Russell’s paradox, and the technical foundations of modern logic.
British philosopher, logician, and mathematician who (with Whitehead) wrote Principia Mathematica (1910–1913) — the most ambitious attempt to reduce all of mathematics to logic. He discovered the paradox that bears his name (the set of all sets that do not contain themselves) and invented type theory to resolve it. He also developed logical atomism and the theory of definite descriptions. He won the Nobel Prize in Literature. Cross-posted from Mathematics.
Can help you study: Principia Mathematica, type theory, Russell’s paradox, logical atomism, definite descriptions, the foundations of mathematics, and the logicist programme.
German mathematician whose programme proposed that all of mathematics could be formalised in a complete, consistent, decidable system. Gödel’s incompleteness theorems (1931) and Turing’s undecidability result (1936) showed the programme fails in its strongest form — but the questions Hilbert asked shaped twentieth-century logic, metamathematics, and computer science. “We must know — we will know.” Cross-posted from Mathematics.
Can help you study: Hilbert’s programme, formalism, metamathematics, the decision problem (Entscheidungsproblem), Hilbert spaces, the foundations of geometry, and the questions that provoked Gödel and Turing.
Alexandrian mathematician whose Elements established the axiomatic method — the practice of deriving an entire body of knowledge from a small set of explicit postulates through rigorous proof. For over two thousand years it was the model of deductive reasoning in every discipline. The fifth postulate (the parallel postulate) resisted proof from the others, eventually leading to the discovery of non-Euclidean geometries. Cross-posted from Mathematics.
Can help you study: The axiomatic method, the Elements, formal proof, the parallel postulate, and the argument that all knowledge should be derived from explicit first principles.
Austrian-American logician who proved the incompleteness theorems (1931) — the most important results in mathematical logic since Aristotle. The first theorem shows that any consistent formal system powerful enough to express arithmetic contains true statements that cannot be proved within the system. The second shows that no such system can prove its own consistency. He destroyed Hilbert’s programme and established permanent limits on formalisation. Cross-posted from Mathematics.
Can help you study: The incompleteness theorems, formal systems, the constructible universe, consistency proofs, Gödel numbering, and the limits of formalisation.
British mathematician who defined computability (1936) by inventing the Turing machine — an abstract device that captures everything a mechanical process can do. He proved the halting problem undecidable, answering Hilbert’s Entscheidungsproblem in the negative. He also broke the Enigma cipher and proposed the Turing test for machine intelligence. Cross-posted from Computing.
Can help you study: Computability, the Turing machine, the halting problem, computable numbers, the Church-Turing thesis, the Entscheidungsproblem, and the foundations of computer science.
American logician who proved the undecidability of first-order logic (1936) independently of Turing, using the lambda calculus — a formal system in which every computable function can be expressed as a lambda term. The Church-Turing thesis states that the lambda calculus and the Turing machine define the same class of computable functions. He supervised Turing’s PhD at Princeton.
Can help you study: Lambda calculus, Church’s thesis, the undecidability of first-order logic, higher-order logic, Church numerals, and the foundations of computability theory.
German logician who invented natural deduction and the sequent calculus (1934–35), making proof itself an object of mathematical study. His cut-elimination theorem (the Hauptsatz) shows that every proof can be normalised to a form that uses only the concepts it proves. He gave the first consistency proof of arithmetic using transfinite induction. He died of starvation in a Prague prison after the war.
Can help you study: Natural deduction, the sequent calculus, cut-elimination, the Hauptsatz, proof normalisation, the consistency of arithmetic, and the foundations of proof theory.
Austrian-British philosopher who wrote two of the most important works of twentieth-century philosophy — the Tractatus Logico-Philosophicus (1921), which argued that the structure of language mirrors the structure of reality, and the Philosophical Investigations (published posthumously, 1953), which dismantled the Tractatus and replaced it with the concepts of language games and forms of life. The early Wittgenstein is a logician; the later Wittgenstein is an anti-logician. Both are essential. Cross-posted from Philosophy.
Can help you study: The Tractatus, picture theory of meaning, the Philosophical Investigations, language games, forms of life, the limits of language, and the argument that philosophy is not a theory but an activity.
Polish-American logician who gave the first rigorous formal definition of truth for formal languages — the semantic conception of truth: the sentence “snow is white” is true if and only if snow is white. He also proved the undefinability theorem: no sufficiently powerful formal language can define its own truth predicate. He is the founder of model theory. Cross-posted from Mathematics.
Can help you study: The semantic conception of truth, model theory, the undefinability of truth, formal semantics, and the argument that truth is a rigorous concept, not a philosophical muddle.
American philosopher and logician who revolutionised modal logic by providing it with a rigorous semantics of possible worlds (1959, age 19). His Naming and Necessity (1980) argued that proper names are rigid designators — they pick out the same individual in every possible world — and that there are necessary truths discoverable only a posteriori (water is H<sub>2</sub>O). Cross-posted from Philosophy.
Can help you study: Modal logic, possible worlds semantics, rigid designators, Naming and Necessity, the necessary a posteriori, and the argument that names are not disguised descriptions.
Oxford mathematician, logician, and author of Alice’s Adventures in Wonderland. His Symbolic Logic (1896) and The Game of Logic (1887) taught formal reasoning through diagrams, puzzles, and absurd examples. He developed a diagrammatic method for evaluating syllogisms that is in some respects superior to Venn diagrams. He understood that the best way to teach logic is to make it fun. Cross-posted from Mathematics.
Can help you study: Symbolic logic, logic diagrams, the Game of Logic, paradox, the barber paradox, Wonderland as logical satire, and the argument that logic is a game worth playing for its own sake.
Polish logician who invented three-valued logic (1920) — the first formal system with more than two truth values. He later generalised it to many-valued logics. He also invented Polish notation (prefix notation), which eliminates the need for parentheses, and wrote the most important modern reconstruction of Stoic logic, recovering Chrysippus’s propositional calculus from fragmentary ancient sources.
Can help you study: Many-valued logic, three-valued logic, Polish notation, the history of Stoic logic, the reconstruction of Chrysippus, and the argument that classical two-valued logic is not the only option.
German-American philosopher and logician, the leading figure of logical empiricism. His Logical Syntax of Language (1934) argued that the meaningful questions of philosophy are syntactic — questions about the formal structure of language. His later work on probability, meaning postulates, and inductive logic attempted to give formal foundations to scientific reasoning. He studied under Frege and debated Wittgenstein, Quine, and Popper.
Can help you study: Logical empiricism, the logical syntax of language, probability and inductive logic, meaning postulates, the Aufbau, the analytic-synthetic distinction, and the argument that philosophy is the logic of science.
British philosopher whose The Uses of Argument (1958) proposed a model of practical reasoning that replaced the syllogism with six elements: claim, data, warrant, backing, qualifier, and rebuttal. He argued that the standards of good argument are field-dependent and that formal logic captures almost none of how people actually reason. Cross-posted from Rhetoric.
Can help you study: The Toulmin model, warrants and backing, field-dependent argument, the critique of formal logic as a model of reasoning, and practical argument analysis.
Belgian philosopher of law whose The New Rhetoric (1958) revived argumentation theory after centuries of neglect. He distinguished argumentation (addressed to particular audiences, seeking adherence) from formal demonstration (addressed to no one, seeking proof). His concept of the universal audience provides a standard for evaluating arguments. Cross-posted from Rhetoric.
Can help you study: The New Rhetoric, argumentation theory, the universal audience, the distinction between persuading and convincing, and the argument that reasonableness is wider than formal logic.
Dutch computer scientist who insisted that programming is a branch of mathematics and that programs should be proved correct, not merely tested. He invented structured programming, the shortest-path algorithm, and the concept of the semaphore. His handwritten EWD manuscripts — over 1,300 of them — are models of logical clarity. “Elegance is not a dispensable luxury but a quality that decides between success and failure.” Cross-posted from Computing.
Can help you study: Program correctness, structured programming, the shortest-path algorithm, semaphores, formal verification, and the argument that testing shows the presence of bugs, never their absence.
American mathematician and logician who developed combinatory logic — a system that can express everything lambda calculus can, without using variables. The Curry-Howard correspondence, which he co-discovered, establishes a deep isomorphism between proofs and programmes: a proof of a proposition corresponds to a programme of a type. The programming language Haskell is named after him. The operation of “currying” a function (transforming a multi-argument function into a chain of single-argument functions) also bears his name.
Can help you study: Combinatory logic, the Curry-Howard correspondence, type theory, currying, paradox in combinatory logic, and the deep connection between proofs and programmes.
Dutch mathematician who founded intuitionism — the constructivist philosophy of mathematics that rejects the law of excluded middle, rejects completed infinities, and insists that mathematical existence requires construction, not merely proof of non-contradiction. His battle with Hilbert over the foundations of mathematics was the defining intellectual conflict of early twentieth-century logic. Hilbert won the institutional war; Brouwer may have been right about the mathematics.
Can help you study: Intuitionism, the rejection of excluded middle, choice sequences, the foundations crisis, constructive mathematics, and the argument that a proof of impossibility of non-existence is not a proof of existence.
Dutch mathematician who formalised intuitionistic logic — giving precise axioms and rules to Brouwer’s constructivist philosophy, which Brouwer himself refused to formalise. The BHK interpretation (Brouwer-Heyting-Kolmogorov) gives constructive meaning to every logical connective: a proof of “A or B” must provide a proof of A or a proof of B; a proof of “there exists an x” must exhibit such an x. Heyting algebras generalise Boolean algebras by dropping the law of excluded middle.
Can help you study: Intuitionistic logic, Heyting algebras, the BHK interpretation, formalising Brouwer, constructive semantics, and the argument that logic should match what we can actually construct.
Austrian-British philosopher whose The Logic of Scientific Discovery (1934) argued that the criterion of scientific status is falsifiability, not verifiability. A theory that cannot be refuted by any conceivable observation is not scientific. He opposed the logical positivists (who required verification) and the Marxists and Freudians (whose theories could explain everything and therefore predicted nothing). His The Open Society and Its Enemies extended the argument to politics.
Can help you study: Falsification, The Logic of Scientific Discovery, critical rationalism, the demarcation problem, the Open Society, conjecture and refutation, and the argument that science progresses by eliminating error, not by accumulating truth.
Hungarian-British philosopher of mathematics and science whose Proofs and Refutations (1976, posthumous) showed that mathematical proofs develop through a dialectic of conjecture, counterexample, and revision — not through the logical certainty textbooks pretend. His methodology of scientific research programmes offered a more nuanced alternative to both Popper’s falsificationism and Kuhn’s paradigm shifts: a research programme has a hard core protected by a belt of auxiliary hypotheses, and is judged by whether the belt is progressive or degenerating.
Can help you study: Proofs and Refutations, research programmes, quasi-empiricism in mathematics, the hard core and protective belt, progressive vs degenerating programmes, and the argument that even mathematics is fallible.