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
In logic, the universal quantifier “” is the quantifier used to express, given a predicate with a free variable of type , the proposition
which is intended to be true if and only if is true for every possible element of .
Note that it is quite possible that may be provable (in a given context) for every term of type that can actually be constructed in that context yet cannot be proved. Therefore, we cannot define the quantifier by taking the idea literally and applying it to terms.
We work in a logic in which we are concerned with which propositions entail which propositions (in a given context); in particular, two propositions which entail each other are considered equivalent.
Let be an arbitrary context and a type in so that is extended by a free variable of type . We assume that we have a weakening rule that allows us to interpret any proposition in as a proposition in . Fix a proposition in , which we think of as a predicate in with the free variable . Then the universal quantification of is any proposition in such that, given any proposition in , we have
It is then a theorem that the universal quantification of , if it exists, is unique. The existence is typically asserted as an axiom in a particular logic, but this may be also be deduced from other principles (as in the topos-theoretic discussion below).
Often one makes the appearance of the free variable in explicit by thinking of as a propositional function and writing instead; to go along with this one usually conflates and . Then the rule appears as follows:
In terms of semantics (as for example topos-theoretic semantics; see the next section), the weakening from to corresponds to pulling back along a product projection , where is the interpretation of the type , and is the interpretation of . In other words, if a statement read in context is interpreted as a subobject of , then the statement read in context is interpreted by pulling back along the projection, obtaining a subobject of .
As observed by Lawvere, we are not particularly constrained to product projections; we can pull back along any map . (Often we have a class of display maps and require to be one of these.) Alternatively, any pullback functor can be construed as pulling back along an object , i.e., along the unique map corresponding to an object in the slice , since we have the identification .
In terms of the internal logic in some ambient topos ,
the predicate is a (-1)-truncated object of the over-topos ;
a truth value is a (-1)-truncated object of itself.
Universal quantification is essentially the restriction of the direct image right adjoint of a canonical étale geometric morphism ,
where is the functor that takes an object to the product projection , where is the dependent sum (i.e., forgetful functor taking to ) that is left adjoint to , and where is the dependent product operation that is right adjoint to .
The dependent product operation restricts to propositions by pre- and postcomposition with the truncation adjunctions
to give universal quantification over the domain (or context) :
Dually, the extra left adjoint , obtained from the dependent sum by pre- and post-composition with the truncation adjunctions, expresses the existential quantifier. (The situation with the universal quantifier is somewhat simpler than for the existential one, since the dependent product automatically preserves -truncated objects (= subterminal objects), whereas the dependent sum does not.)
The same makes sense, verbatim, also in the (∞,1)-logic of any (∞,1)-topos.
This interpretation of universal quantification as the right adjoint to context extension is also used in the notion of hyperdoctrine.
Let Set, let be the set of natural numbers and let be the subset of even natural numbers. This expresses the proposition .
Then the dependent product
is the set of sections of the bundle . But since over odd numbers the fiber of this bundle is the empty set, there is no possible value of such a section and hence the set of sections is itself the empty set, so the statement “all natural numbers are even” is, indeed, false.
basic symbols used in logic
The observation that substitution forms an adjoint pair/adjoint triple with the existential/universal quantifiers is due to
William Lawvere, Adjointness in Foundations, (tac:16), Dialectica 23 (1969), 281-296
William Lawvere, Quantifiers and sheaves, Actes, Congrès intern, math., 1970. Tome 1, p. 329 à 334 (pdf)
and further developed in
comprehension schema as an adjoint functor_, Proceedings of the AMS Symposium on Pure Mathematics XVII (1970), 1-14.
Last revised on May 31, 2022 at 02:16:07. See the history of this page for a list of all contributions to it.