I've been told and read a number of times that implies, $\implies$, is the same as subset $\subseteq$. For example, in the wikipedia page on iff it states that "P only if Q", "if P then Q", and "P→Q" all mean that P is a subset, either proper or improper, of Q
I'm currently thinking in terms of propositional logic, and I don't understand how it makes sense to say that $P \implies Q$ means that $P$ is a subset of $Q$. To my current understanding, $P$ and $Q$ are both propositions, which means that they are in the set $\{T,F\}$. This set contains only two elements and neither of them are sets, so how can $P$ be a subset of $Q$?
I know that the two element Boolean algebra can be realised using $T = \{\emptyset\}$ and $F = \emptyset$, along with intersection union and complement. Then this would mean that I can represent $\implies$ using $\subseteq$, because $\emptyset \subseteq \{\emptyset\}$ etc. to give the correct truth table, but I don't see how that could relate $P \implies Q$ to $P \subseteq Q$.
I think when I'm told that $P \implies Q$ means that $P \subseteq Q$, this must be somehow taking the space of all propositions (perhaps for some specific set of axioms?) and then $P \subseteq Q$ in that space would mean that $P \implies Q$, is this correct? Can anyone elaborate on how this works and direct me to any related reading?
Best Answer
It's certainly not the case that implication "is" subsethood, since - as you say - that's mixing up types of objects.
However, note that in the wiki page, the relevant claim appears within the section on Euler diagrams: so it's not an absolute statement, but rather a statement about how Euler diagrams interpret sentences. And there's much less here than meets the eye: by definition, an Euler diagram for a collection of sentences is a diagram with regions corresponding to sentences where subsethood matches implication.
That said, there is a nontrivial interpretation of Euler diagrams. Specifically, to a sentence $P$ we can associate the set of "models" of $P$, $Mod(P)$, which you can think of as the ways in which $P$ can be true. Then we do indeed have a match up between impication and subsethood: the sentence "$P\rightarrow Q$" is provable from the laws of logic alone (= is a tautology) iff $Mod(P)\subseteq Mod(Q)$. See also these old answer of mine: 1, 2.
More generally, we have that $P\rightarrow Q$ is provable from some theory $T$ iff $Mod(T\cup\{P\})\subseteq Mod(T\cup\{Q\})$, where for $\Gamma$ a set of sentences we define $$Mod(\Gamma):=\bigcap_{\gamma\in\Gamma}Mod(\gamma)$$ (that is, $Mod(\Gamma)$ is the set of structures which are simultaneously models of all the sentences in $\Gamma$).