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-proposition, mere proposition
proofgeneralized elementprogram
cut rulecomposition of classifying morphisms / pullback of display mapssubstitution
cut elimination for implicationcounit for hom-tensor adjunctionbeta reduction
introduction rule for implicationunit for hom-tensor adjunctioneta conversion
logical 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, (idemponent) monadmodal type theory, monad (in computer science)
linear logic(symmetric, closed) monoidal categorylinear type theory/quantum computation
proof netstring diagramquantum circuit
(absence of) contraction rule(absence of) diagonalno-cloning theorem
synthetic mathematicsdomain specific embedded programming language

homotopy levels




In logic, a proposition is intended to be interpreted semantically as having a truth value. In modern logic, it’s cleanest to start by specifying a context and considering the propositions in that context.


If (in a given context Γ\Gamma) we have a type AA, then we may extend Γ\Gamma to a context ΔΓ,x:A\Delta \coloneqq \Gamma, x\colon A (assuming that the variable xx is not otherwise in use). We may then think of any proposition in Δ\Delta as a predicate PP in Γ\Gamma with the free variable xx of type AA; this generalises to more complicated extensions of contexts (say by several variables).

If PP is a predicate with free variable xx of type AA and tt is a term of type AA, then we get a proposition P[t/x]P[t/x] by substituting tt for every instance of xx in PP. Conversely, any proposition QQ may be interpreted as a predicate Q[x^]Q[\hat{x}] in which the free variable xx simply doesn’t appear. (We have Q[x^][t/x]=QQ[\hat{x}][t/x] = Q for every term tt.)

There is a more traditional approach of viewing a predicate as a function from terms to propositions, a propositional function. Then P[t/x]P[t/x] is written P(t)P(t), while PP itself from above is written P(x)P(x) (since a variable is a term). In this approach, less care is usually taken with the context, so that Q[x^]Q[\hat{x}] may be conflated with QQ (since Q[x^](x)=QQ[\hat{x}](x) = Q, or this would be so if xx were a term in Γ\Gamma instead of only in Δ\Delta).

In category-theoretic logic

In categorial logic/categorical semantics, we have a category 𝒞\mathcal{C} and a class of monomorphisms? (often all monomorphisms) \mathcal{M} in 𝒞\mathcal{C}. Then a context is an object of 𝒞\mathcal{C} and a proposition in the context Γ\Gamma is an \mathcal{M}-subobject of Γ\Gamma. We also have a class of display maps (often all morphisms in 𝒞\mathcal{C}) such that \mathcal{M} is closed under pullbacks both along display maps and along sections of display maps. These two ways of pulling back propositions in one context to propositions in another context correspond (respectively) to forming Q[x^]Q[\hat{x}] and P[t/x]P[t/x].

More specifically, if 𝒞\mathcal{C} is a finitely complete category, then the objects of 𝒞\mathcal{C} may equivalently be viewed as contexts and as types in the internal language of 𝒞\mathcal{C}; a morphism from Γ\Gamma to AA is a term of type AA in context Γ\Gamma. The extension of Γ\Gamma by a variable xx of type AA is the product Γ×A\Gamma \times A, and the display map to Γ\Gamma is simply the projection. Every term t:ΓAt\colon \Gamma \to A defines a section of this display map, and we may literally construct Q[x^]Q[\hat{x}] and P[t/x]P[t/x] as pullbacks.

If 𝒞\mathcal{C} is even a topos, then a proposition QQ in Γ\Gamma may be identified with a term whose type is the subobject classifier Ω\Omega, and the predicate Q[x^]Q[\hat{x}] is the composite Γ×AΓΩ\Gamma \times A \to \Gamma \to \Omega. Given a term t:ΓAt\colon \Gamma \to A and a predicate P:Γ×AΩP\colon \Gamma \times A \to \Omega, the proposition P[t/x]P[t/x] is the composite ΓΓ×AΩ\Gamma \to \Gamma \times A \to \Omega. Internalising a bit (by currying), we may view QQ as a global element 1Ω Γ1 \to \Omega^\Gamma and PP as a morphism AΩ ΓA \to \Omega^\Gamma, recovering the view that predicates are proposition-valued ‘functions’ (morphisms).

In general, we may intuitively think of an object AA in the slice category 𝒞/Γ\mathcal{C}/\Gamma as the ‘set’ (object) of possible values of terms tt of type AA in context Γ\Gamma, and think of a predicate PP with a free variable of type AA (in the same context) as being the ‘subset’ (subobject) on those tt for which the statement P(t)P(t) is true.

In type theory

In type theory under the propositions as types paradigm, every type represents the proposition that it is inhabited. Hence the types which have at most one term may be identified with propositions (“propositions as some types”). In homotopy type theory these are the (-1)-types. The reflection that sends types to their underlying proposition qua (-1)-truncation is the n-truncation modality for n=(1)n = (-1), also called bracket type-formation.

Propositional and predicate logic

In propositional logic, we fix a single context (considered the empty context?) and consider the logic of propositions in that context. In predicate logic, we fix the empty context but work also in extensions of that context by free variables. Predicate logic uses quantifiers as a way to move between contexts, more specifically to move from a predicate PP in a given context Γ\Gamma (which is a proposition in some extension of Γ\Gamma) to a proposition in Γ\Gamma. The free variables in the predicate still appear in the written form of the proposition, but they are now bound variables and are not free in the proposition's context; some logicians prefer to systematically replace bound variables with numbered placeholders (especially when defining Gödel number?s and the like).

Last revised on May 18, 2019 at 14:02:20. See the history of this page for a list of all contributions to it.