# Must all logical axioms be proven to be universally valid?

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.

• Yes all logical axioms are valid. Oct 3 at 6:02
• @MauroALLEGRANZA Do we need to prove every of these logical axioms to be valid, or we can just assume some of them are valid? Oct 3 at 9:42
• 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... Oct 3 at 9:47
• 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 $\forall x \varphi(x) \to \varphi [x/t]$. Oct 3 at 11:49