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 “$\forall$” is the quantifier used to express, given a predicate $\phi$ with a free variable $x$ of type $T$, the proposition
which is intended to be true if and only if $\phi a$ is true for every possible element $a$ of $T$.
Note that it is quite possible that $\phi a$ may be provable (in a given context) for every term $a$ of type $T$ that can actually be constructed in that context yet $\forall x\colon T, \phi x$ 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 $\Gamma$ be an arbitrary context and $T$ a type in $\Gamma$ so that $\Delta \coloneqq \Gamma, x\colon T$ is $\Gamma$ extended by a free variable $x$ of type $T$. We assume that we have a weakening rule that allows us to interpret any proposition $Q$ in $\Gamma$ as a proposition $Q[\hat{x}]$ in $\Delta$. Fix a proposition $P$ in $\Delta$, which we think of as a predicate in $\Gamma$ with the free variable $x$. Then the universal quantification of $P$ is any proposition $\forall\, x\colon T, P$ in $\Gamma$ such that, given any proposition $Q$ in $\Gamma$, we have
It is then a theorem that the universal quantification of $P$, 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 $P$ explicit by thinking of $P$ as a propositional function and writing $P(x)$ instead; to go along with this one usually conflates $Q$ and $Q[\hat{x}]$. Then the rule appears as follows:
In terms of semantics (as for example topos-theoretic semantics; see the next section), the weakening from $Q$ to $Q[\hat{x}]$ corresponds to pulling back along a product projection $\sigma(T) \times A \to A$, where $\sigma(T)$ is the interpretation of the type $T$, and $A$ is the interpretation of $\Gamma$. In other words, if a statement $Q$ read in context $\Gamma$ is interpreted as a subobject of $A$, then the statement $Q$ read in context $\Delta = \Gamma, x \colon T$ is interpreted by pulling back along the projection, obtaining a subobject of $\sigma(T) \times A$.
As observed by Lawvere, we are not particularly constrained to product projections; we can pull back along any map $f \colon B \to A$. (Often we have a class of display maps and require $f$ to be one of these.) Alternatively, any pullback functor $f^\ast\colon Set/A \to Set/B$ can be construed as pulling back along an object $X = (f \colon B \to A)$, i.e., along the unique map $!\colon X \to 1$ corresponding to an object $X$ in the slice $Set/A$, since we have the identification $Set/B \simeq (Set/A)/X$.
In terms of the internal logic in some ambient topos $\mathcal{E}$,
the predicate $\phi$ is a (-1)-truncated object of the over-topos $\mathcal{E}/X$;
a truth value is a (-1)-truncated object of $\mathcal{E}$ itself.
Universal quantification is essentially the restriction of the direct image right adjoint of a canonical étale geometric morphism $X_\ast$,
where $X^\ast$ is the functor that takes an object $A$ to the product projection $\pi \colon X \times A \to X$, where $X_! = \Sigma_X$ is the dependent sum (i.e., forgetful functor taking $f \colon A \to X$ to $A$) that is left adjoint to $X^\ast$, and where $X_\ast = \Pi_X$ is the dependent product operation that is right adjoint to $X^\ast$.
The dependent product operation restricts to propositions by pre- and postcomposition with the truncation adjunctions
to give universal quantification over the domain (or context) $X$:
Dually, the extra left adjoint $\exists_X$, obtained from the dependent sum $X_!$ 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 $(-1)$-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 $\mathcal{E} =$ Set, let $X = \mathbb{N}$ be the set of natural numbers and let $\phi \coloneqq 2\mathbb{N} \hookrightarrow \mathbb{N}$ be the subset of even natural numbers. This expresses the proposition $\phi(x) \coloneqq IsEven(x)$.
Then the dependent product
is the set of sections of the bundle $2 \mathbb{N} \hookrightarrow \mathbb{N}$. 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
$\phantom{A}$symbol$\phantom{A}$ | $\phantom{A}$meaning$\phantom{A}$ |
---|---|
$\phantom{A}$$\in$ | $\phantom{A}$element relation |
$\phantom{A}$$\,:$ | $\phantom{A}$typing relation |
$\phantom{A}$$=$ | $\phantom{A}$equality |
$\phantom{A}$$\vdash$$\phantom{A}$ | $\phantom{A}$entailment / sequent$\phantom{A}$ |
$\phantom{A}$$\top$$\phantom{A}$ | $\phantom{A}$true / top$\phantom{A}$ |
$\phantom{A}$$\bot$$\phantom{A}$ | $\phantom{A}$false / bottom$\phantom{A}$ |
$\phantom{A}$$\Rightarrow$ | $\phantom{A}$implication |
$\phantom{A}$$\Leftrightarrow$ | $\phantom{A}$logical equivalence |
$\phantom{A}$$\not$ | $\phantom{A}$negation |
$\phantom{A}$$\neq$ | $\phantom{A}$negation of equality / apartness$\phantom{A}$ |
$\phantom{A}$$\notin$ | $\phantom{A}$negation of element relation $\phantom{A}$ |
$\phantom{A}$$\not \not$ | $\phantom{A}$negation of negation$\phantom{A}$ |
$\phantom{A}$$\exists$ | $\phantom{A}$existential quantification$\phantom{A}$ |
$\phantom{A}$$\forall$ | $\phantom{A}$universal quantification$\phantom{A}$ |
$\phantom{A}$$\wedge$ | $\phantom{A}$logical conjunction |
$\phantom{A}$$\vee$ | $\phantom{A}$logical disjunction |
$\phantom{A}$$\otimes$$\phantom{A}$ | $\phantom{A}$multiplicative conjunction$\phantom{A}$ |
$\phantom{A}$$\oplus$$\phantom{A}$ | $\phantom{A}$multiplicative disjunction$\phantom{A}$ |
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 February 10, 2021 at 15:48:53. See the history of this page for a list of all contributions to it.