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I am trying to study the foundations of mathematics from the bottom-up (propositional logic then predicate logic then the axioms of set theory.) Currently, I'm considering the following Hilbert-style axiomatization of propositional logic, but several issues are confusing me. $$A \rightarrow(B \rightarrow A)$$ $$(A\rightarrow(B\rightarrow C))\rightarrow((A\rightarrow B) \rightarrow(A\rightarrow C))$$ $$(\neg A \rightarrow\neg B) \rightarrow(B \rightarrow A)$$ Modus ponens is the only inference rule: $$A \rightarrow B , A \vdash B$$

From what I can tell, these are not actually axioms but axiom schemas, with symbols $A$, $B$, and $C$ being characters in a "meta-language" for which actual propositions in the object language would be substituted. I also am guessing that the connectives $\rightarrow$ and $\neg$ represent both themselves as logical operations in the object language and symbols referencing themselves in the meta-language.

1) Doesn't there need to be some sort of substitution rule for using the meta-language to reference the object language; that is, plugging in a proposition $P$ which has a definite but unknown truth value into the meta-symbol $A$ which can take in both true propositions and false propositions? Obviously, you just "plug it in" but...shouldn't there be a more rigorous/explicit definition? Ideally, I'm imagining a substitution rule for axiom schema that is similar (or the same as?) a substitution rule of inference for replacing terms in the object language which satisfy logical equivalence.

2) Consider the axiom $P \rightarrow(Q \rightarrow P)$, generated from the first above axiom schema by plugging in the atomic propositions $P$ and $Q$ for $A$ and $B$ respectively. Are these atomic propositions an as-of-yet undefined part of the propositional logic or are they outside of it? Could you say that the above axiomization is the syntatics of this propositional logic while specific atomic propositions belong to the semantics? For that matter, are the logical values True and False part of the propositional logic, some meta-languague describing it, or the propositional logic's semantics? Could you say the meta-language provides the system with semantics?

3) Modus ponens utilizes the symbols "$,$" and "$\vdash $" which are not connectives defined in propositional calculus; in fact "," seems to require some notion of sets even though the axioms of set theory are not yet available. I thought propositional logic could serve as the absolute bottom level of mathematical foundations and yet it seems that the notion of forming a set of objects (so you can apply an inference rule and generate a theorem) is required...but making statements about sets requires propositional calculus...how can this be reconciled?

This is my first time posting on math.stackexhange so suggestions on post etiquette and clarity are welcome. Thanks!

Mithrandir
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  • Which textbook are you using? If it has left you unclear over some point, which other textbook did you look at for help?? Elementary logic is blessed with numerous first-rate texts, which should resolve all these questions for you. – Peter Smith May 28 '20 at 21:12
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    I'm just passing through (been working all day on day-job stuff and about to call it quits for the day), but it sounds like you might want to look at §27: Formulations employing axiom schemata on pp. 148-151 of Introduction to Mathematical Logic by Alonzo Church (1956). As for the seemingly circular argument stuff, the part about "set of objects", you'll want to look up stuff about "metalanguage". Consider this: Do I need Peano's axioms for the natural numbers before I'm allowed to number the steps in a proof and then refer to the 4th step? – Dave L. Renfro May 28 '20 at 21:46
  • @PeterSmith My starting point was the book describing the Metamath theorem verification software. I was captivated by his description of a non-specialist's journey to pin down the foundations of math. I purchased Kleene's Introduction to Metamathematics but found it very daunting. I just started reading Kleene's Mathematical Logic which is more digestible at my level and am excited to continue. Ideally, I'm looking for textbooks at the upper undergrad level which don't shy away from subtle/complex ideas but have detailed explanations surveying the landscape of logic and its role in math. – Mithrandir May 28 '20 at 22:59
  • @PeterSmith Your profile led me to your site Logic Matters and I'm checking out the Teach Yourself Logic document. This is the type of thing I've been looking for, thank you so, so much for providing these resources!!! – Mithrandir May 28 '20 at 23:02
  • @DaveL.Renfro Thanks for this reference, I'm going to check it out! Thanks also for the insightful comment. Just as the steps in a proof necessarily follow some order even if the arithmetic of the associated numbers hasn't been developed, propositions can be grouped even if the nuances and constraints of set theory haven't been specified. Any recommendations on textbooks covering formal languages and meta-languages accessible to the late undergrad level are appreciated. – Mithrandir May 28 '20 at 23:26
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    It just occurred to me that somewhere I've written more than just a comment about these kinds of issues, and a quick google search (my name plus "metalogic") led me to the questions Foundation of Formal Logic and Highly Rigorous Logic Book. Regarding how far rigor like this can be pushed, for me it seems we start bumping into "non-rigorous" philosophical issues --- see How do I convince someone that $1+1=2$ may not necessarily be true?. – Dave L. Renfro May 29 '20 at 08:18
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    For 1) Church's classic textbook (thanks @DaveL.Renfro) is perfect; there you can find the two possibilities: (i) axiom schema, with the explanation in the meta-theory that every instance of a schema is an axiom, i.e. from $A \to (B \to A)$ we get $p \to (q \to p)$ and $(p \lor r) \to (q \to (p \lor r))$ and so on, and (ii) axioms, like $p \to (q \to p)$ and an additional rule of inference, the Rule of Substitution: "if $A,B$ are formulas and $p$ a prop variable, from $A$ infer $A[B/p]$". – Mauro ALLEGRANZA May 29 '20 at 09:25
  • To 3): obviously, the rules of inference are stated/described in the meta-language. Initially, we can avoid any "technical" symbols (like $\vdash$) that will be added later. MP is: "if $A,B$ are formulas, from $(A \to B)$ and $A$, infer $B$". – Mauro ALLEGRANZA May 29 '20 at 09:32

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