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In all of the established propositional logics that I’m aware of, a propositional atom is treated as a meta-variable. In certain first-order proof systems, this does not hold for those same logics since the Deduction Theorem and the Generalization Rule need to work together properly to avoid invalidities like A→∀xA.

Are there well-known propositional logics in which propositional atoms are not also interpretable/usable as meta-variables? How do those logics deal with things like Uniform Substitution? Also, is there a specific philosophical reason for propositional logics to be axiomatized so that atoms serve as meta-variables? I am aware that propositional atoms that serve as meta-variables are mathematically easier to understand.

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  • If by "propositional atoms" you mean propositional letters I am not quite sure what "in certain first-order proof systems, this does not hold" means. Those proof systems do not have propositional letters, only predicate letters and object variables. And what exactly would you like propositional letters to be if not "meta-variables"? Something for which you can substitute different things in different places, as with nonterminal symbols?
    – Conifold
    Commented May 21 at 22:44
  • @Conifold some first-order proof systems still have propositional atoms, especially since nullary predicates are functionally equivalent to propositional atoms.
    – PW_246
    Commented May 21 at 23:06
  • @Conifold with respect to the article you suggested, I’d actually be talking about terminal symbols.
    – PW_246
    Commented May 21 at 23:09
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    Atoms: p1,p2,p3,... Meta-variables (aka: schema): A,B,... In general, the use is simple: we use meta-variable to formalize rules of inference: "form A and A to B, derive B" (Modus Ponens) because in this way we provide general recipes to manufacure multiple instances. The same for axioms. IF we use atoms for axioms: p1 to (p2 to p1), we have to add a Substitution Rule of inference, because at the level of object language p1 to (p1 to p1) is a different formula from p2 to (p1 to p2). This is the context... But I'm not sure to understand your specific concern with predicate logic. Commented May 22 at 5:44
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    Some texts distinguish bewteen symbols that represent particular simple statements (i.e. atomic sentences), and statement variables that are placeholders in which statements, both simple and compound, may be substituted. This is done to facilitate the symbolising of argument forms. For example, Copi uses upper case letters A, B, C... for simple statements and lower case p, q, r... for variables. (Copi, Cohen, Rodych, Introduction to Logic, 15th edition (2019), chapter 8, p. 286.)
    – Bumble
    Commented May 22 at 7:07

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Hmm, I see where you're coming from, but I think you might be overthinking this a bit. It seems you're kinda hung up on the whole meta-variable thing. Look, in propositional logic, we're dealing with these atomic statements – like the building blocks of our arguments. We wanna see how these statements connect, how they can make a whole argument true or false. We’re not really getting into the nitty-gritty of what those atoms actually represent – that’s more of a first-order logic thing.

So, yeah, atoms act like meta-variables in the sense that we can plug in any darn statement we want, as long as it fits the logic rules. That makes things super convenient, right? Uniform Substitution works because we're not really messing with the meaning of the atom itself, just replacing it with something that functions the same way logically.

As for those fancy first-order proof systems, yeah, things can get kinda hairy with quantifiers and such. But that doesn't necessarily mean propositional logic is flawed or doing something wrong. It's just playing a different game, focusing on the relationships between statements, not the internal structure of those statements.

So, to answer your question – I haven't really come across propositional logics where atoms aren't treated sorta like meta-variables. It's kinda built into the whole concept, you know? Maybe there's some crazy theoretical system out there, but it would probably be super niche and not really used in everyday logic-ing

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