In which sense "closure" is a closure?

Question

In predicate and first-order logic, if $\phi$ is a sentence, then $\forall X . \phi$ is said to be the (universal) closure of $\phi$. Is the use of the word "closure" incidental, or is there a connection with closure as meant in topology, or in lattice theory, or whatever? In the latter case, what is the exact connection?

Answer 1

A formula is called open if it has free variables. The reason for that name is that such a formula doesn't get a truth value as soon as you specify a structure; you have to also specify values for the free variables. The meaning of the formula in a structure is still "open" in the sense of being subject to further specification. In contrast, a formula without free variables is said to be closed. (Closed formulas are also called sentences, but that's irrelevant to your question.) One way to convert an open formula to a closed one is to quantify all its free variables, so that they're not free any more. The closed formula obtained by closing an open formula $\alpha$ in this way is called a closure of $\alpha$ --- the universal closure or the existential closure, depending on the type of quantifier that was used.

So the terminology "closure" here arises from a different notion of "closed" than what one finds in algebra or topology, where "closed" means that set contains the results of certain constructions (algebraic constructions or limiting processes) performed on its members.

Answer 2

Actually, there is a topological meaning to closure of logical formulas, if one represents formulas by string diagrams. A description of such a string diagram calculus, interpreting Peirce's existential graphs for relational calculus, can be found here.

Here the lines of a string diagram that meet the boundary of its ambient rectangle correspond to the free variables of a formula; they are essentially synonymous with the incoming and outgoing lines of Feynman diagrams. (We can think of them as open lines if we confine attention to the interior of the rectangle.) Internal lines of the diagram (again in the Feynman diagram sense) correspond to bound variables. Existential quantification corresponds to "capping" an incoming or outgoing line, so that it acquires an internal node as boundary point and becomes an internal line. Thus existential closure as mentioned in Andreas's answer corresponds to a topological closure of a line of the diagram.

I can imagine this connection was known already to Peirce, but I'd have to scour his writings to be sure.

Answer 3

The notion of an algebra with an existential quantifier $\exists$ has been algebraized to the notion of a monadic algebra and a monadic algebra is a Boolean algebra with a closure operator on it such that the closed sets form a Boolean subalgebra.

A closure operator on a poset $X$ is a mapping $C:X\rightarrow X$ such that $x\leq C(x)=C(C(x))$ whenever $x\in X$ and $x\leq y\rightarrow C(x)\leq C(y)$.

It was shown by Kuratowski that the closure operators for the topologies on a set $X$ are precisely the closure operators $C:P(X)\rightarrow P(X)$ such that $C(0)=0$ and $C(R\cup S)=C(R)\cup C(S)$, but closure operators appear in all areas of mathematics.

A monadic algebra is a pair $(B,\exists)$ such that $B$ is a Boolean algebra and $\exists:B\rightarrow B$ is a closure operator on $B$ such that $\exists(x\vee y)=\exists x\vee\exists y$ and $(\exists x)\wedge y=0$ if and only if $x\wedge\exists y=0$. Monadic algebras are useful in algebra since the monadic algebras are in a one-to-one correspondence with the relatively complete Boolean subalgebras of a Boolean algebra. Furthermore, Monadic algebras satisfy the rule $C(R\cup S)=C(R)\cup C(S)$, so monadic algebras can be thought of as special kinds of $``$topological$"$ closure operators on Boolean algebras.

Finally, I should mention that the monadic algebras of the form $(P(X),\exists)$ are precisely the algebras where there exists some equivalence relation $E$ on $X$ so that $\exists R=E[R]$ for all $R\subseteq X$. In this case, monadic algebras are the trivial topological closure operators.