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- Work Organon Aristotle
- Work Logical Fragments Chrysippus & the Stoics
- Work Isagoge (Introduction) Porphyry
- Work On Aristotle’s Logic (translations & commentaries) Boethius
- Work Kitāb al-Qiyās (The Syllogism) al-Fārābī
- Work al-Shifāʾ: The Logic Ibn Sīnā (Avicenna)
- Work al-Ishārāt wa-l-Tanbīhāt (Pointers & Reminders) Ibn Sīnā (Avicenna)
- Work Miʿyār al-ʿIlm (The Criterion of Knowledge) al-Ghazālī
- Work Dialectica Peter Abelard
- Work Talkhīṣ Manṭiq Arisṭū (Epitome of Aristotle’s Logic) Ibn Rushd (Averroes)
- Work Summulae Logicales Peter of Spain
- Work Īsāghūjī al-Abharī
- Work Maṭāliʿ al-Anwār (The Rising-Points of Lights) Sirāj al-Dīn al-Urmawī
- Work Sharḥ al-Ishārāt (Commentary on the Pointers) Naṣīr al-Dīn al-Ṭūsī
- Work al-Risāla al-Shamsiyya al-Kātibī al-Qazwīnī
- Work Qisṭās al-Afkār (The Balance of Thoughts) Shams al-Dīn al-Samarqandī
- Work Ars Magna (The Great Art) Ramon Llull
- Work al-Radd ʿalā al-Manṭiqiyyīn (Refutation of the Logicians) Ibn Taymiyya
- Work Summa Logicae William of Ockham
- Work Taḥrīr al-Qawāʿid al-Manṭiqiyya (on al-Shamsiyya) Quṭb al-Dīn al-Rāzī
- Work Tahdhīb al-Manṭiq al-Taftāzānī
- Work al-Sullam al-Munawraq al-Akhḍarī
- Work Logical Writings (Generales Inquisitiones) Gottfried Wilhelm Leibniz
- Work Theory of Science (Wissenschaftslehre) Bernard Bolzano
- Work A System of Logic John Stuart Mill
- Work Formal Logic Augustus De Morgan
- Work An Investigation of the Laws of Thought George Boole
- Work Begriffsschrift Gottlob Frege
- Work Foundations of a General Theory of Sets Georg Cantor
- Work Grundlagen der Geometrie David Hilbert
- Work On the Foundations of Mathematics L. E. J. Brouwer
- Work Investigations in the Foundations of Set Theory Ernst Zermelo
- Work Principia Mathematica Whitehead & Russell
- Work Tractatus Logico-Philosophicus Ludwig Wittgenstein
- Work On Formally Undecidable Propositions Kurt Gödel
- Work The Concept of Truth in Formalized Languages Alfred Tarski
- Work The Logical Syntax of Language Rudolf Carnap
- Work Collected Papers Gerhard Gentzen
- Work An Unsolvable Problem of Elementary Number Theory Alonzo Church
- Work On Computable Numbers Alan Turing
- Work Semantical Considerations on Modal Logic Saul Kripke
- Work Categories for the Working Mathematician Saunders Mac Lane
- Work A Mathematical Introduction to Logic Herbert Enderton
- Work Intuitionistic Type Theory Per Martin-Löf
- Work An Introduction to Non-Classical Logic Graham Priest
- Work Homotopy Type Theory The Univalent Foundations Program
- Mind Aristotle 384–322 BCE
- Mind Diodorus Cronus c. 340–280 BCE
- Mind Chrysippus 279–206 BCE
- Mind Galen c. 129–216
- Mind Porphyry c. 234–305
- Mind Boethius c. 477–524
- Mind al-Kindī c. 801–873
- Mind al-Fārābī c. 872–950
- Mind Avicenna (Ibn Sīnā) 980–1037
- Mind al-Ghazālī 1058–1111
- Mind Peter Abelard 1079–1142
- Mind Zayn al-Dīn al-Sāwī d. c. 1145
- Mind Ibn Rushd (Averroes) 1126–1198
- Mind Fakhr al-Dīn al-Rāzī 1149–1209
- Mind al-Khūnajī 1194–1248
- Mind al-Abharī c. 1200–1265
- Mind Sirāj al-Dīn al-Urmawī 1198–1283
- Mind Naṣīr al-Dīn al-Ṭūsī 1201–1274
- Mind al-Kātibī al-Qazwīnī 1203–1277
- Mind Peter of Spain c. 1215–1277
- Mind Ramon Llull c. 1232–1316
- Mind Ibn Taymiyya 1263–1328
- Mind Shams al-Dīn al-Samarqandī fl. c. 1290
- Mind William of Ockham 1287–1347
- Mind Quṭb al-Dīn al-Rāzī c. 1290–1365
- Mind John Buridan c. 1301–1362
- Mind al-Taftāzānī 1322–1390
- Mind al-Sayyid al-Sharīf al-Jurjānī 1339–1414
- Mind Jalāl al-Dīn al-Dawānī 1426–1502
- Mind al-Akhḍarī c. 1512–1546
- Mind Mullā Ṣadrā c. 1571–1640
- Mind Gottfried Wilhelm Leibniz 1646–1716
- Mind Bernard Bolzano 1781–1848
- Mind Arthur Schopenhauer 1788–1860
- Mind Augustus De Morgan 1806–1871
- Mind John Stuart Mill 1806–1873
- Mind George Boole 1815–1864
- Mind John Venn 1834–1923
- Mind Charles Sanders Peirce 1839–1914
- Mind Ernst Schröder 1841–1902
- Mind Georg Cantor 1845–1918
- Mind Gottlob Frege 1848–1925
- Mind Giuseppe Peano 1858–1932
- Mind Alfred North Whitehead 1861–1947
- Mind David Hilbert 1862–1943
- Mind Ernst Zermelo 1871–1953
- Mind Bertrand Russell 1872–1970
- Mind Jan Łukasiewicz 1878–1956
- Mind L. E. J. Brouwer 1881–1966
- Mind C. I. Lewis 1883–1964
- Mind Stanisław Leśniewski 1886–1939
- Mind Thoralf Skolem 1887–1963
- Mind Ludwig Wittgenstein 1889–1951
- Mind Rudolf Carnap 1891–1970
- Mind Emil Post 1897–1954
- Mind Haskell Curry 1900–1982
- Mind Alfred Tarski 1901–1983
- Mind Alonzo Church 1903–1995
- Mind Kurt Gödel 1906–1978
- Mind Jacques Herbrand 1908–1931
- Mind Willard Van Orman Quine 1908–2000
- Mind Gerhard Gentzen 1909–1945
- Mind Saunders Mac Lane 1909–2005
- Mind Stephen Cole Kleene 1909–1994
- Mind Alan Turing 1912–1954
- Mind Ruth Barcan Marcus 1921–2012
- Mind J. A. Robinson 1930–2016
- Mind Paul Cohen 1934–2007
- Mind Saul Kripke 1940–2022
- Mind Per Martin-Löf 1942–
- Mind Jean-Yves Girard 1947–
- Concept Proposition A declarative statement that is either true or false, the basic bearer of truth in logic.
- Concept Validity A property of arguments: the conclusion is true in every interpretation that makes all the premises true.
- Concept Soundness A calculus is sound when everything it proves is valid: it never derives a falsehood from truths.
- Concept Completeness A calculus is complete when every valid statement can be proved within it (Gödel’s completeness theorem, for first-order logic).
- Concept Consistency A theory is consistent if it proves no contradiction, never both a statement and its negation.
- Concept Quantifier A symbol binding a variable over a domain: the universal ∀ (“for all”) and the existential ∃ (“there exists”).
- Concept Syllogism Aristotle’s form of deductive inference: two premises sharing a middle term yield a conclusion.
- Concept Modus ponens The fundamental rule of detachment: from A and “A implies B”, infer B.
- Concept Truth table A table giving a compound proposition’s truth value for every assignment of truth to its atoms.
- Concept Tautology A proposition true under every interpretation: a logical truth.
- Concept Excluded middle The classical law that every proposition is either true or false, rejected by intuitionistic logic.
- Concept Natural deduction Gentzen’s proof system built from introduction and elimination rules that mirror ordinary reasoning.
- Concept Sequent calculus Gentzen’s calculus operating on sequents Γ ⊢ Δ, the spine of structural proof theory.
- Concept Cut elimination Gentzen’s Hauptsatz: every sequent proof can be rewritten without the cut rule, yielding the subformula property.
- Concept Model A structure assigning meaning to a language’s symbols, under which its sentences come out true or false.
- Concept Compactness A set of first-order sentences has a model iff every finite subset of it has a model.
- Concept Löwenheim–Skolem If a countable first-order theory has an infinite model, it has models of every infinite cardinality.
- Concept Diagonal (fixed-point) lemma In a strong enough theory, for any predicate ψ there is a sentence asserting its own ψ-ness: the engine of Gödel and Tarski.
- Concept Incompleteness Gödel: any consistent formal system strong enough for arithmetic contains true statements it cannot prove.
- Concept Gödel numbering A coding of a formal language’s symbols and proofs as natural numbers, letting arithmetic talk about itself.
- Concept Possible worlds Kripke’s semantics for modal logic: necessity is truth in all accessible worlds, possibility in some.
- Concept λ-abstraction Church’s notation for a function defined by an expression: the atom of functional computation.
- Concept Curry–Howard The correspondence under which propositions are types and proofs are programs: logic and computation, identified.
- Concept Decidability A problem is decidable if an algorithm settles every instance in finite time; logic abounds in undecidable ones.
- Concept Cardinality The size of a set, generalized to the infinite by Cantor: ℵ₀, the continuum, and beyond.
- Concept The ZFC axioms Zermelo–Fraenkel set theory with Choice: the standard axiomatic foundation of modern mathematics.
- Concept Boolean algebra The algebraic structure of classical propositions: meet, join, and complement obeying the distributive laws.
- Concept Univalence Voevodsky’s axiom: equivalent types are equal, the heart of homotopy type theory.
- Theorem The Syllogism (Barbara) Valid deduction reduced to form: two universal premises sharing a middle term force a universal conclusion.
- Theorem Cantor’s Theorem No set can be mapped onto its own power set, so there are infinitely many distinct sizes of infinity.
- Theorem Löwenheim–Skolem A countable first-order theory with an infinite model has models of every infinite cardinality, including unintended ones.
- Theorem Completeness Theorem In first-order logic, semantic truth and formal provability coincide exactly: what is valid can be proved.
- Theorem Compactness Theorem A set of first-order sentences has a model iff every finite subset does: the engine of model theory.
- Theorem First Incompleteness Any consistent system strong enough for arithmetic contains a true sentence it cannot prove.
- Theorem Second Incompleteness No such system can prove its own consistency. Certainty about a system must come from outside it.
- Theorem Tarski’s Undefinability Arithmetical truth cannot be defined inside arithmetic itself. Truth always outruns its own language.
- Theorem Cut-Elimination (Hauptsatz) Every sequent proof can be rewritten without the cut rule: a proof never needs a detour through a lemma.
- Theorem Undecidability of the Halting Problem No algorithm can decide, for every program, whether it halts, nor whether a first-order sentence is valid.
- Theorem Consistency of Arithmetic Peano arithmetic’s consistency is provable using transfinite induction up to ε₀, exactly measuring its strength.
- Theorem Independence of the Continuum Hypothesis The continuum hypothesis can be neither proved nor refuted from ZFC: settled by Gödel and Cohen together.
- Theorem Lindström’s Theorem First-order logic is the strongest logic that has both compactness and the Löwenheim–Skolem property: a portrait of its limits.
- Theorem Curry–Howard Correspondence Proofs and programs are literally the same objects: to prove is to compute, and logic and computation are one.
- Paradox The Liar Consider the sentence that says of itself only that it is false.
- Paradox Russell’s Paradox Form the set of all sets that are not members of themselves.
- Paradox The Sorites (Heap) One grain is not a heap. Adding a single grain never turns a non-heap into a heap.
- Paradox Curry’s Paradox Consider the sentence: “If this very sentence is true, then everything is.”
- Paradox Berry’s Paradox Name “the least integer not nameable in fewer than twenty syllables,” in nineteen syllables.
- Paradox Grelling–Nelson Call a word “heterological” if it does not describe itself. Is “heterological” heterological?
- Field Basic Concepts The ground of everything: propositions and arguments, validity and soundness, and what it means for reasoning to be correct.
- Field Historical Context Logic did not spring forth fully formed. The long development from antiquity to the modern revolution, and why each step was taken.
- Field Propositional Logic Truth tables, the connectives, and rules of inference: the algebra of whole statements, and the natural first system to master.
- Field Ancient Logic Aristotle’s syllogistic and the propositional logic of the Stoics: the two founding streams of the Western tradition.
- Field Predicate Logic Quantifiers and first-order logic: the language that reaches inside propositions, to objects, properties, and relations.
- Field Medieval Logic The scholastic and Islamic logicians: supposition theory, modal syllogistic, and the bridge from the ancients to the moderns.
- Field The Modern Revolution Frege, Russell, Hilbert: symbolic logic and the logicist dream of deriving mathematics itself from logic.
- Field Set Theory & Foundations The ZF axioms, cardinality, and the great independence results: the standard foundation on which mathematics is built.
- Field Proof Theory Proofs as objects of study: natural deduction, the sequent calculus, and cut-elimination, the structure of demonstration itself.
- Field Mathematical Logic Model theory, computability, and recursion: logic turned on mathematics, and the limits Gödel and Tarski revealed.
- Field Modal Logic Necessity and possibility, made rigorous by Kripke’s possible-worlds semantics, with their kin: temporal, epistemic, and deontic logics.
- Field Non-Classical Logics Intuitionistic, many-valued, relevant, and fuzzy logics: systems that revise or reject the classical laws, each for a reason.
- Field Philosophical Logic Conditionals, the paradoxes, and the nature of logic itself: where logic turns its instruments back on its own foundations.
- Field Computational Logic Logic programming and automated reasoning, logic made to run: Prolog, resolution, SAT, and the theorem provers.
- Field Category Theory Objects and arrows, toposes and type theory: an abstract language for structure, and an alternative foundation for mathematics.
- Field Applications Logic at work in computer science, linguistics, and cognitive science: verification, the semantics of language, and human reasoning.
- Field Quantum Logic The non-distributive logic of quantum propositions, and the logical structure of quantum computation and information.
- Field Algebraic Logic Logic recast as algebra: Boolean, cylindric, and relation algebras, the algebraic shadow that every logic casts.
- Field Cutting-Edge Research Reverse mathematics, proof mining, homotopy type theory, descriptive complexity: where the field is being written now.
- Thread The Crisis of Foundations How the dream of a complete, certain foundation for mathematics was built, broken, and rebuilt, from Cantor’s paradise to Cohen’s verdict.
- Thread Proof Becomes Computation The slow discovery that a proof and a program are the same thing, from Gentzen’s rules to the proof assistants of today.
- Thread The Story of the Infinite Two and a half thousand years of refusing, then embracing, then taming the actual infinite, and finding its size beyond our reach.
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