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In this answer by Henning Makholm, he states that the following statement can be proven in Peano Arithmetic: $$ \forall x:\forall y:x\cdot S(y)=(x\cdot y)+x \, . $$

The proof is very simple—since the claim to be proved is one of the axioms of PA, simply stating it with "(axiom)" next to it constitutes a proof.

My question is: which logic inference rule(s) tell us that an axiom constitutes a proof of itself?

Mauro ALLEGRANZA
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Joe
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    A proof of $X$ is a finite sequence of well-formed sentences (well-formed formulas with no free variables) in the language of the theory such that each sentence is either an axiom, a tautology, or a consequence of previous sentences using a valid rule of inference, and where the final sentence in the sequence is $X$. A proof of an axiom consists of a one-sentence proof listing the axiom. – Arturo Magidin Sep 19 '21 at 21:27
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    The answer depends on what logical system you are using, but all sane logical systems have the property that the axioms are provable. In natural deduction, the proofs of the axioms are the proof trees with exactly one node. – Rob Arthan Sep 19 '21 at 21:32
  • @ArturoMagidin: Thank you; that my answers my question. You can convert your comment to an answer, if you wish. – Joe Sep 19 '21 at 21:37
  • @ArturoMagidin So there's no such thing as a formal system that allows formulas with free variables? – David C. Ullrich Sep 19 '21 at 21:57
  • @DavidC.Ullrich: Of course, there is, as you know. Plenty of details to be filled out. – Arturo Magidin Sep 19 '21 at 22:00
  • @ArturoMagidin I was just curious why you specified "sentence" instead of "wff" in your definition... – David C. Ullrich Sep 19 '21 at 22:01
  • @DavidC.Ullrich: Most formal systems that are in use have a "universal generalization" rule that means that the wffs with free variables can be replaced with sentence in which all the free variables are universally quantified. – Arturo Magidin Sep 19 '21 at 22:02
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    @ArturoMagidin Of course in such a system evidently there are free variables in proofs, or the universal gemeralization rule would be useless. if anything such systems seem like counterexamples to the idea that the definition of proof should specify sentences - these systems are precisely why that word bothered me... – David C. Ullrich Sep 19 '21 at 22:08
  • @DavidC.Ullrich: Fair enough; too late to edit a comment, though. – Arturo Magidin Sep 19 '21 at 22:10
  • In a theory with axioms (like Peano arithmetic) the axioms are the starting points and they are not proved. They can be proved in some "more general" theory. – Mauro ALLEGRANZA Sep 20 '21 at 07:43
  • @MauroALLEGRANZA: Your comment seems to contradict Arturo's comment above. – Joe Sep 20 '21 at 22:52
  • Axioms are never "proven" (nor can they be, ultimately), so one says that the axiom is simply "given", not "proven". – Marcos Nov 16 '23 at 19:03
  • @Marcos: In the context of logic, the axioms of a certain formal system certainly are provable. See Arturo's comment above. – Joe Nov 24 '23 at 19:55
  • @Joe: No. If they are provable they are theorems or "valid conclusions". In the context of logic, you might have an axiom of "identity" that says "True is true.". This cannot be proven but you probably will want to keep it axiomatically. – Marcos Nov 25 '23 at 01:49
  • @Marcos: In any first-order theory, the axioms are theorems of the theory – simply stating the axiom amounts to proving it. Note that in formal logic, the term “proof” has a precise technical meaning. All of this is explained in Arturo’s comment above. – Joe Nov 25 '23 at 02:48

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As Arturo mentions in the comments, we actually don't need to use any inference rules to conclude that stating an axiom amounts to proving it. It is immediate from the definition of "proof" in predicate calculus:

A proof of $X$ is a finite sequence of well-formed sentences (well-formed formulas with no free variables) in the language of the theory such that each sentence is either an axiom, a tautology, or a consequence of previous sentences using a valid rule of inference, and where the final sentence in the sequence is $X$. A proof of an axiom consists of a one-sentence proof listing the axiom.

(Actually, as David C. Ullrich also points out in the comments, in most deductive systems, we don't need proofs to consists of well-formed sentences: we merely need each line to be a well-formed formula; however, this is a technical point that is irrelevant to the issue at hand.)

Joe
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