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Second-order logic

Definition

In logic and mathematics, second-order logic is an extension of first-order logic, which itself is an extension of propositional logic. Second-order logic is in turn extended by higher-order logic and type theory.

Related concepts

Abstract logic Abstract object Ackermann set theory Aleph number Alfred North Whitehead Algebraic logic Alphabet (formal languages)Analytical hierarchy Ancestor Archimedean property Argument Arithmetic Arity Atomic formula Atomic model (mathematical logic)Atomic sentence Automata theory Automated proof checking Automated theorem proving Axiom Axiom of choice Axiom schema Axiomatic system Axiomatization of Boolean algebras Banach–Tarski paradox Bertrand Russell Bijection Binary operation Boolean algebra Boolean algebras canonically defined Boolean function Branching quantification Bulletin of Symbolic Logic Büchi-Elgot-Trakhtenbrot theorem Cantor's diagonal argument Cantor's paradox Cantor's theorem Cardinality Cartesian product Categorical theory Category (mathematics)Category of sets Category theory Cayley table Charles Sanders Peirce Church encoding Church–Turing thesis Class (set theory)Classical logic Co-NP Codomain Compactness theorem Complement (set theory)Complete theory Completeness (logic)Complexity class Computability theory Computable function Computable set Computably enumerable set Computational complexity theory Concept and object Concrete category Conservative extension Consistency Constructible universe Construction of the real numbers Constructive set theory Continuum hypothesis Countable set Countably infinite Courcelle's theorem Decidability (logic)Decision problem Deductive system Dependence logic Descriptive complexity Diagram (mathematical logic)Domain of a function Domain of discourse EXPTIME Element (mathematics)Elementary diagram Elementary equivalence Elementary function arithmetic Empty set Enumeration Equiconsistency Equivalence relation Euclid's Elements Euclidean geometry Existential quantification Expression (mathematics)Extension by definitions Extension by new constant and function names Extensionality Fagin's theorem Finitary relation Finite-valued logic Finite model theory Finite set First-order logic First-order predicate calculus First-order theory Fixed-point logic Forcing (mathematics)Formal grammar Formal language Formal proof Formal semantics (logic)Formal system Formation rule Formula (mathematical logic)Foundations of geometry Foundations of mathematics Free logic Free variables and bound variables Function (mathematics)Functional predicate Fuzzy set General set theory Generalized quantifier George Boolos Gottlob Frege Graffiti Graph theory Grothendieck universe Ground expression Ground formula Gödel's completeness theorem Gödel's incompleteness theorem Gödel's incompleteness theorems Gödel numbering Halting problem Harvard University Press Henkin quantifier Hereditary set Higher-order logic Hilary Putnam Hilbert's axioms Hilbert system Hintikka Historia Mathematica History of logic History of mathematical logic Image (mathematics)Inaccessible cardinal Independence-friendly logic Independence (mathematical logic)Inference Inference rules Infinite-valued logic Infinite set Information theory Inhabited set Injective Injective function Interpretation (logic)Interpretation (model theory)Interpretation function Intersection (set theory)Isomorphism Joel Spencer Jouko Väänänen Journal of Philosophy Journal of Symbolic Logic Kolmogorov complexity Kripke's theory of truth Kripke–Platek set theory Kurt Gödel Lambda calculus Large cardinal Law of excluded middle Least fixed point Lemma (mathematics)Leon Henkin Leonid Libkin Lindström's theorem List of Hilbert systems List of axioms List of first-order theories List of formal systems List of mathematical theories List of set identities and relations List of statements independent of ZFC Logic Logical biconditional Logical conjunction Logical connective Logical consequence Logical constant Logical disjunction Logical equality Logical equivalence Logical truth Logicism Löwenheim number Löwenheim–Skolem theorem Many-valued logic Map (mathematics)María Manzano Material conditional Mathematical induction Mathematical logic Mathematical object Mathematics Mereology Metalanguage Minimal axioms for Boolean algebra Model complete theory Model theory Monadic predicate calculus Monadic second-order logic Morse–Kelley set theory Moshe Y. Vardi NP (complexity)Naive set theory Natural deduction Negation Neukölln New Foundations Non-Euclidean geometry Non-logical symbol Non-standard model Non-standard model of arithmetic Nonfirstorderizability Omega language Open formula Operation (mathematics)Ordered field Ordinal analysis Ordinal number Oxford Oxford University Press PH (complexity)PSPACE P (complexity)P versus NP problem Paradoxes of set theory Partition of a set Peano arithmetic Peano axioms Philosophy of mathematics Plural quantification Power set Predicate (mathematical logic)Predicate logic Predicate variable Prentice Hall Prime model Primitive recursive arithmetic Primitive recursive function Principia Mathematica Proof of impossibility Proof theory Proposition Propositional calculus Propositional formula Propositional variable Quantifier (logic)Quantifier rank Reachability Real-closed field Real number Recursion Recursive set Recursively enumerable Regular language Relation (mathematics)Reverse mathematics Robinson arithmetic Rule of inference Russell's paradox S2S (mathematics)Satisfiability Saturated model Schröder–Bernstein theorem Second-order arithmetic Second-order propositional logic Self-verifying theories Semantic theory of truth Semantics of logic Sentence (mathematical logic)Sequent calculus Set (mathematics)Set theory Signature (logic)Singleton (mathematics)Skolem's paradox Skolem arithmetic Soundness Spectrum of a sentence Spectrum of a theory Springer-Verlag Square of opposition Stewart Shapiro Strength (mathematical logic)String (formal languages)Structure (mathematical logic)Substitution (logic)Substructure (mathematics)Supertask Supremum Surjective Surjective function Syllogism Symbol (formal)Syntax (logic)T-schema Tarski's axiomatization of the reals Tarski's axioms Tarski's theory of truth Tarski's undefinability theorem Tarski–Grothendieck set theory Tautology (logic)Term (logic)Term logic The Stanford Encyclopedia of Philosophy Theorem Theories of truth Theory (mathematical logic)Thoralf Skolem Three-valued logic Timeline of mathematical logic Transfer principle Transitive closure Transitive set True arithmetic Truth predicate Truth table Truth value Turing machine Type (model theory)Type theory Ultrafilter (set theory)Ultraproduct Uncountable set Undecidable problem Uninterpreted function Union (set theory)Uniqueness quantification Universal quantification Universal set Universe (mathematics)Upward Löwenheim–Skolem theorem Urelement Validity (logic)Variable (mathematics)Venn diagram Von Neumann universe Von Neumann–Bernays–Gödel set theory W. V. Quine Well-formed formula Willard Van Orman Quine ZFC Zermelo–Fraenkel set theory

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