proposition

basic constructions:

strong axioms

further

**natural deduction** metalanguage, practical foundations

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

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

logic | category theory | type theory |
---|---|---|

true | terminal object/(-2)-truncated object | h-level 0-type/unit 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

</table>

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

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

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

In categorial logic/categorical semantics, we have a category $\mathcal{C}$ and a class of monomorphisms? (often all monomorphisms) $\mathcal{M}$ in $\mathcal{C}$. Then a **context** is an object of $\mathcal{C}$ and a **proposition** in the context $\Gamma$ is an $\mathcal{M}$-subobject of $\Gamma$. We also have a class of display maps (often all morphisms in $\mathcal{C}$) 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 $Q[\hat{x}]$ and $P[t/x]$.

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

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

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

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 $n = (-1)$, also called bracket type-formation.

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 $P$ in a given context $\Gamma$ (which is a proposition in some extension of $\Gamma$) to a proposition in $\Gamma$. 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 April 27, 2017 at 03:26:31. See the history of this page for a list of all contributions to it.