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I've been wondering about this. The textbook ''Language, Proof and Logic'' defines a first-order validity as the following:

''A sentence of FOL is a first-order validity if it is a logical truth when you ignore the meanings of the names, function symbols, and predicates other than the identity symbol.''

The sentence ∀x x = x contains no predicates, other than the identity symbol, so it has to be a first-order validity, right? Or am I getting it wrong? Is it simply a logical truth (that isn't a first-order validity)?

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  • Yes it is a FO valid formula i e a predicate logic with equality logical truth Oct 19 at 16:47
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    Yes, though bear in mind that we can choose to do first order logic with identity, in which case your formula is a logical truth, or we can choose to do first order logic without identity, in which case the identity predicate requires interpretation the same as any other predicate, and your formula is not a logical truth, though it will still be true under 'normal' interpretations.
    – Bumble
    Oct 19 at 18:22

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