I just noticed there is a similarity between logic operations on propositions and the operations of set theory. It seems:
$$\begin{array}{llll}
\textrm{disjunction} & (-)\vee (-)& \textrm{corresponds to union}& (-)\cup (-)\\
\textrm{conjunction} & (-) \wedge (-)& \textrm{correspons to intersection} & (-)\cap (-)\\
\textrm{negation} & \sim (-) & \textrm{correspons to taking complements} & c(-),
\end{array}
$$
and I conjecture:
$$\begin{array}{lll}
\textrm{conditional} & (-)\rightarrow (-) & \textrm{corresponds to inclusion} & \subset\\
\textrm{biconditional} & (-)\leftrightarrow (-) & \textrm{corresponds to equality} & =
\end{array}$$
How far does it go? I believe there is some kind of functor between some category whose objects are propositions and the category of sets, is that right?
Thanks
Best Answer
"How far does it go ?" : as far as it can get !
Here's a general idea about how set theory is a semantics for classical propositional logic (note that if we change the formal system we're looking at without changing our assumptions on sets, for instance if we're studying intuitionistic logic from a classical point of view, then we have to take another semantics, in this specific case, topological spaces can be appropriate) :
Suppose you have a set of propositional variables $V$, a "global" set $E$, and a function $[-] :V\to \mathcal{P}(E)$. Then you can build a function that goes from the set $\mathrm{Form}$ of formulas to $\mathcal{P}(E)$ by expanding $[-]$ according to the rules you displayed : if $\varphi, \psi$ are formulas and you already defined $[\varphi]$ and $[\psi]$, then define $[\varphi \land \psi] = [\varphi]\cap [\psi]$, similarly for $\lor, \neg$, and define $[\varphi\implies \psi]$ as $\{x\in E\mid x\in [\varphi]\implies x\in[\psi]\}$.
These rules allow you to define $[\varphi]$ for any formula $\varphi$ by induction, going from the variables and gaining complexity.
Then you can prove the following things : if $\varphi$ is a theorem of classical logic, then $[\varphi] = E$, which tells you that the set-operations behave according to the logical ones, but also you can prove : if for any $E$ and any $[-] :V\to \mathcal{P}(E)$, $[\varphi] = E$, then $\varphi$ is a theorem of classical logic ! This tells you that actually the logical operations behave just like set-theoretic operations as well.
There's actually a lot more you can say about this sort of thing (for instance : what happens if you add quantifiers ? Or in another direction what happens if we replace $\mathcal{P}(E)$ by some other type of object ? If we completely change the type of object, what kind of logic do we get ? etc. etc.)
If you absolutely want to use the words "functor" and "category" you can, but at this level they're not the most relevant thing.