Kind of a weird question but where does the $\in$ symbol come from exactly and where do we imbue this symbol with any kind of meaning?
As far as I can tell it isn't a symbol that is part of the alphabet when it comes to propositional logic or first order logic. It just sort of pops out of nowhere when you get to something like the axioms of ZFC or set theory where we start saying things like $a \in S$ without really discussing what this symbol means or how you use it appropriately.
We all know it means "element of" but is there a more formal basis for its definition? Is there some kind of rule or axiom that shows how this syntax is meant to be used somewhere? I am coming at this from the perspective of syntax and semantics if that helps. Please note I am not asking about the history of the symbol $\in$.
For example let's say I start throwing around $a★b$ or $a★S$ in my syntax, you'd go, whoa, wait a second, I don't know what that symbol means or what we're allowed to do with it or how we're supposed to use it. Where would I "point you" to show these things?
Best Answer
Propositional logic contains only propositional connectives $\land, \lor,\ldots$ and propositional variables. You are correct that it does not have the symbol $\in$.
First-order logic (sometimes called predicate logic) has a number of basic symbols: variables, propositional connectives, quantifiers, and (usually) $=$ for equality. However, one of the most important things about first-order logic is that you can add optional symbols to represent functions, predicates/relations, and constants.
For instance, the axioms of group theory can be stated within the first order language containing the extra symbols $\{\times, {}^{-1}, e\}$ (indeed, you actually only need $\times$).
When we are dealing with logic we need to be careful to not confuse the syntax of our system with its semantics. Syntactically, $\in$ is just a symbol that we can take or leave in any particular language. This symbol does not need to refer to set membership; it doesn't even need to be a binary relation! I could interpret $\in$ as anything that the logic lets me: a constant, a relation of any arity, or a function of any arity.
The "meaning" of the symbol $\in$ isn't really a question for logic. We have, as a mathematical community, decided that $\in$ denotes set membership. It is very convenient in mathematics to be able to talk about collections of objects, so the symbol is used a lot. However, the question of what $\in$ "really means" is similar to what $0$ "really means": these are questions that are more of metaphysics. There may be some people who claim that they have no notion of "set" or "collection". To these skeptics, talking about ZFC might not be too helpful. To people who do have a notion of "set" we can ask the question of whether particular claims are try about their idea of set. Most of the mathematical community have agreed that sets (whatever they are) satisfy ZFC.