I made the following observation to Chris on Facebook which he encouraged me to post here.
My point was basically just that, in reply to the statement “If we don’t have such a model to reject, the statement will be tautological”, it is in fact true relative to the standard semantics for first-order languages with equality that there is indeed no model-combined-with-an-interpretation-of-the-free-variables for which “x=x” comes out false. That is to say, relative to the standard semantics the formula is indeed a “logical truth” in that sense, although we usually only say “tautology” for formulas that are tautologies in propositional logic (that is, true under every Boolean valuation, a truth-valuation of all subformulas starting with a quantifier and all subformulas which are atomic formulas which then gets extended to a truth-valuation of all subformulas using the standard rules for the propositional connectives). So most certainly “x=x” is universally valid, relative to the standard semantics, and in the sense just described, there is no counter-model.
I take it that Chris’ project here is in some way to articulate in what sense the Law of Identity could be taken as a statement that “has content” to it. It sounds as though the best approach to this might be to try to take a look at how you would explain the semantics of statements that involve the equality relation. 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”.
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.
I made the following observation to Chris on Facebook which he encouraged me to post here.
My point was basically just that, in reply to the statement “If we don’t have such a model to reject, the statement will be tautological”, it is in fact true relative to the standard semantics for first-order languages with equality that there is indeed no model-combined-with-an-interpretation-of-the-free-variables for which “x=x” comes out false. That is to say, relative to the standard semantics the formula is indeed a “logical truth” in that sense, although we usually only say “tautology” for formulas that are tautologies in propositional logic (that is, true under every Boolean valuation, a truth-valuation of all subformulas starting with a quantifier and all subformulas which are atomic formulas which then gets extended to a truth-valuation of all subformulas using the standard rules for the propositional connectives). So most certainly “x=x” is universally valid, relative to the standard semantics, and in the sense just described, there is no counter-model.
I take it that Chris’ project here is in some way to articulate in what sense the Law of Identity could be taken as a statement that “has content” to it. It sounds as though the best approach to this might be to try to take a look at how you would explain the semantics of statements that involve the equality relation. 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”.
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.