When looking at the semantics of First Order Logic, I'm wondering whether it's possible for there to be an interpretation whereby α ≠ α for some term α.
Assuming that α is a constant here, I am unsure of how to define a possible case for this. For if it is a constant, then V(α) should refer to some object of the domain (D). How could it be that V(α) were to refer to some different object?
Assuming that α is a variable, there are two cases, I suppose, at which to look at:
- bound variables
- free variables
In the case of the bound variable, it is not possible for α ≠ α, as ∀ α(α ≠ α).
In the case of the free variable, a wff such as Fα says nothing, as far as I'm aware, about the identity.
I am probably overcomplicating things here massively and if that is so please let me know. I reckon the answer is much simpler and I have merely increased the scope of the question needlessly.
Thanks!
No.
$\forall x (x=x)$ is an axiom of First order logic with equality.
If $c$ is an individual constant of the language, from the above axiom we may derive the instance :
According to the rules of the semantics for FOL, an interpretation $\mathcal I$ assigns to each constant symbol $c$ a member $c^I$ of the universe (domain) $I$ of the interpretation.
Thus, $c=c$ must hold in every interpretation $\mathcal I$, because $c^I=c^I$.