Suppose I have a deductive system based on the language $L=\{Variables,\vee,\wedge,\rightarrow,\leftrightarrow,\neg\}$ consisting of propositional tautologies and rules of inference. How do I introduce non-logical axioms? More precisely, do non-logical axioms have to be formulas of $L$ exclusively, i.e. non-tautologous formulas expressed only with the logical symbols $\vee,\wedge,\rightarrow,\leftrightarrow,\neg$ of $L$, or can they be expressed in a stronger language involving additional symbols, like $\forall,\exists$, e.g. ZFC (I guess, in that case, they have to be stated separately)?
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1What do you mean by non-logical axioms? Something like the Peano Axioms? If so, yes, you need some language that specifies certain predicates, functions, and/or constants. E.g if you want to have $1+1=2$ you’ll need sone way to represent $1$, $2$, and $+$… only the $=$ is part of pure logic. – Bram28 May 07 '23 at 11:27
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See First-order arithmetic for non-logical axioms. Example: $\forall x\ (0\neq S(x))$ – Mauro ALLEGRANZA May 07 '23 at 11:35
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"expressed only with the logical symbols ∨,∧,→,¬ of L, or can they be expressed in a stronger language comprising additional symbols, like ∀,∃, " Quantifiers are "logical symbols". – Mauro ALLEGRANZA May 07 '23 at 11:49
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A mathematical theory add some non-logical symbols, like the binary predicate $\in$ for set theory and the function symbol $s$ for arithmetic and states non-logical axioms regarding them. – Mauro ALLEGRANZA May 07 '23 at 11:50
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See also this post as well as this one. – Mauro ALLEGRANZA May 08 '23 at 06:52