First order logic and identity

Question

Can anyone explain to me why $x=x$ is a truth of logic?

I know that a truth of logic is a theorem based on an empty set of premises.

For example:

$F\alpha \implies (\exists x)(Fx)$

From existential generalization.

Where $\alpha,\; x,\; F$ are an ambiguous name, variable and a predicate, respectively.

This is a theorem of logic.

How do you prove identity is a truth of logic in a logical system without identity?

Do we need extra axioms/rules of inference?

Thanks

See First-order logic with equality: it is an axiom. If there are no axioms for equality we cannot prove the basic properties: reflexivity, symmetry, transitivity, substitution. — Mauro ALLEGRANZA, Dec 14 '22 at 09:44
See e.g. the post Rules of First Order Logic with Equality as well as Equality and its axioms and Open axioms of equality and Some questions about the properties of equality — Mauro ALLEGRANZA, Dec 14 '22 at 09:53
How shall an object (whatever it is) not be identical to ITSELF ? Does "$x=x$" really need a proof ? Barely to imagine ! It must be an axiom. — Peter, Dec 14 '22 at 10:04
Mauro, i am referring to Suppes, page 104 and the rule of indiscernibles. — ryaron, Dec 14 '22 at 10:22
Compare with List of rules (page 99); rewritten in that form, the rule is: "from no premise derive $t=t$". — Mauro ALLEGRANZA, Dec 14 '22 at 10:33
In the same way, the Substitution rule is: "from premises $S$ and $t_1=t_2$, derive $T$, where $T$ results from $S$ by replacing..." — Mauro ALLEGRANZA, Dec 14 '22 at 10:34

score 1 · Accepted Answer · answered Dec 14 '22 at 10:22

See also Patrick Suppes' Introduction, page 104: Rule governing Identity.

We need it, with Substitution, in order to prove the basic properties: reflexivity, symmetry, transitivity.

See also The Logic of Identity: if we assume the so-called Identity of Indiscernibles principle as definition of identity:

$∀F(Fx↔Fy)→x=y$

we may easy derive all the above properties.

From the principle we get: $(Fx↔Fx)→x=x$, form which, by tautology $P ↔ P$ and Modus Ponens, we get: $x=x$.

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