nLab
semantics

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Context

Type theory

natural deduction metalanguage, practical foundations

  1. type formation rule
  2. term introduction rule
  3. term elimination rule
  4. computation rule

type theory (dependent, intensional, observational type theory, homotopy type theory)

syntax object language

computational trinitarianism = propositions as types +programs as proofs +relation type theory/category theory

logiccategory theorytype theory
trueterminal object/(-2)-truncated objecth-level 0-type/unit type
falseinitial objectempty type
proposition(-1)-truncated objecth-level 1-type/h-prop
proofgeneralized elementprogram
conjunctionproductproduct type
disjunctioncoproduct ((-1)-truncation of)sum type (bracket type of)
implicationinternal homfunction type
negationinternal hom into initial objectfunction type into empty type
universal quantificationdependent productdependent product type
existential quantificationdependent sum ((-1)-truncation of)dependent sum type (bracket type of)
equivalencepath space objectidentity type
equivalence classquotientquotient type
inductioncolimitinductive type, W-type, M-type
higher inductionhigher colimithigher inductive type
completely presented setdiscrete object/0-truncated objecth-level 2-type/preset/h-set
setinternal 0-groupoidBishop set/setoid
universeobject classifiertype of types
modalityclosure operator monadmodal type theory, monad (in computer science)

homotopy levels

semantics

Semantics

Idea

Semantics is the interpretation of the syntax of a theory in a model.

Examples

Categorical semantics of dependent type theory

Notably the semantics of type theories is given by categories. For instance

For more on this see at categorical semantics and at relation between type theory and category theory.

category: logic

Revised on December 20, 2012 01:37:55 by Urs Schreiber (82.169.65.155)
Edit | Back in time (5 revisions) | See changes | History | Views: Print | TeX | Source | Linked from: function, logic, inhabited set, product, internal logic, context, homotopy n-type, essentially algebraic theory, homotopy type, internal hom, negation, type theory, coherent logic, sequent calculus, dependent product, computer science, implication, type, preset, element, propositions as types, syntax - semantics duality, proposition, inhabited object, theory, dependent type, applications of (higher) category theory, W-type, type theory - contents, first-order hyperdoctrine, tripos, identity type, display map, calculus of constructions, signature (in logic), Horn theory, algebraic model for modal logics, type-theoretic definition of category, pure type system, the logic S4(m), modal logic, normal modal logic, term, variable, cartesian logic, hyperdoctrine, categorical semantics, lambda-calculus, syntactic category, cartesian theory, axiom, homotopy type theory, type of propositions, theorem, unit type, type of types, axiom UIP, Boolean hyperdoctrine, contents of contents, syntax, product type, quotient type, stack semantics, cohesive homotopy type theory, interval type, homotopy level, function type, bracket type, dependent product type, Coq, empty type, inductive type, function extensionality, Agda, higher inductive type, existential quantifier, universal quantifier, Kripke-Joyal semantics, sub-(infinity,1)-category - internal formulation, positive type, proofs as programs, proof, HoTT methods for homotopy theorists, univalence axiom, intensional type theory, type-theoretic model category, judgmental equality, dependent type theory, contractible type, extensional type theory, sum type, semantics, Oberwolfach HoTT-Coq tutorial, h-set, decidability, extension (semantics), h-proposition, equivalence in homotopy type theory, canonical form, polymorphism, coherent hyperdoctrine, negative type, type formation, beta-reduction, eta-conversion, dependent sum type, substitution, twisted smooth cohomology in string theory, fixed-point combinator, relation between type theory and category theory, M-type, categorical model of dependent types, sequent, first-order theory, Haskell, logical framework, Burali-Forti's paradox, Gallina specification language, judgment, Russell's paradox, context extension, function application, numeral, definition, program, elementary function arithmetic, Brouwer-Heyting-Kolmogorov interpretation, second-order arithmetic, computational trinitarianism, monad (in computer science), computation, programming language, natural deduction, geometry of physics, metalanguage, theory of presheaf type, Practical Foundations for Programming Languages, term in context, contradiction, inconsistency, modal type, function type natural deduction - table, type of h-sets, Bishop set, dependent product natural deduction - table, antecedent, succedent, consequent, looping combinator, deductive system, dependent sum natural deduction - table, judgements for types and terms - table, product natural deduction - table, dependent sum, substitution natural deduction - table, h-groupoid, coinductive type, practical foundation of mathematics, modal type theory, differential homotopy type theory, interpretation, axiom K, observational type theory, meaning explanation, natural numbers type