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In first order logic, we select a set of sentences as the logical axioms. I would like to know if all the logical axioms are required to be proven to be universally valid in meta language. Or, we can select a formula that cannot be proven to be universally valid in meta language to be a logical axiom in a first order language? I raise this question because some logical axioms seem complicated to be proven.

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  • $\begingroup$ Yes all logical axioms are valid. $\endgroup$ Oct 3 at 6:02
  • $\begingroup$ @MauroALLEGRANZA Do we need to prove every of these logical axioms to be valid, or we can just assume some of them are valid? $\endgroup$
    – William
    Oct 3 at 9:42
  • $\begingroup$ There are many equivalent proof systems, each with his own axioms. For a specific proof system, we have to prove that ALL axioms are valid. Having said them, they are quite similar... $\endgroup$ Oct 3 at 9:47
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    $\begingroup$ The proof is straightforward using def of truth in a model for formulas: $\mathcal M,s \vDash \varphi$. You have e.g. to apply the semantical clause for $\forall$ to $\forall x \varphi(x) \to \varphi [x/t]$. $\endgroup$ Oct 3 at 11:49

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