So most certainly “x=x” is universally valid, relative to the standard semantics, and in the sense just described, there is no counter-model.
Indeed. If we want such a counter-model, then we’ll need a different formalisation. This is what I provided above.
It looks as though it should be in some way possible to defend the idea that the Law of Identity is in some way “true in virtue of its meaning”.
I would be surprised if this were the case. I guess my argument above doesn’t aim to argue for the Law of Identity a priori, but rather as a way of representing that our variables don’t need to be more fine-grained given a particular context and a particular equivalence function. In other words, we adopt the Law of Identity because it is part of a formalisation (more properly, a class of formalisations) that is useful in an incredibly wide range of circumstances. At least part of why this is useful so widely because we can use it to formalise parts of our cognition and we use our cognition everywhere.
Indeed. If we want such a counter-model, then we’ll need a different formalisation. This is what I provided above.
I would be surprised if this were the case. I guess my argument above doesn’t aim to argue for the Law of Identity a priori, but rather as a way of representing that our variables don’t need to be more fine-grained given a particular context and a particular equivalence function. In other words, we adopt the Law of Identity because it is part of a formalisation (more properly, a class of formalisations) that is useful in an incredibly wide range of circumstances. At least part of why this is useful so widely because we can use it to formalise parts of our cognition and we use our cognition everywhere.