tag:blogger.com,1999:blog-4987609114415205593.post7902557987993220705..comments2024-03-28T07:29:53.593+00:00Comments on M-Phi: Identity, Indiscernibility and Individuation CriteriaJeffrey Ketlandhttp://www.blogger.com/profile/01753975411670884721noreply@blogger.comBlogger2125tag:blogger.com,1999:blog-4987609114415205593.post-64714455118909737482012-08-06T02:38:12.288+01:002012-08-06T02:38:12.288+01:00Hi Johannes,
"... this criterion does not ha...Hi Johannes,<br /><br />"... this criterion does not have the form (IC)." <br /><br />Right. Not immediately, ... but, recall the result that, if $=$ is definable in $\mathcal{M}$, then it is definable using the HB indiscernibility formula $x \sim_L y$. So, if $=$ for the K-entities is definable at all, an identity criterion should cast into the form (IC).<br /><br />So, in this case, i.e., the natural numbers with their blunt linear ordering $(\mathbb{N}, \leq)$, as you say, we can choose the "individuating conditions" $I_K$ to be $\{(\leq,2)\}$. <br /><br />"So, it seems that you're assuming that there is some underlying background theory T, right? "<br /><br />Well, as usual, I'm thinking of some structure $\mathcal{M}$, for which identity is HB-definable, rather than a theory, T; though one could take $T = Th(\mathcal{M})$. Maybe it's ok to think of a theory though.<br /><br />Yes, roughly the idea is that identity criteria should be expressed in the form (IC), using some conjunction of HB-clauses. Then, perhaps, particular HB-clauses are somehow relevant to the "essence" of the entities in question.<br /><br />For set theory, the HB-clause is the extensionality condition. For other kinds of entities, some appropriate HB-clause defined using a predicate should do the trick. E.g., "x and y have the same location" or "x and y have the same parts", etc.<br /><br />Cheers, JeffJeffrey Ketlandhttps://www.blogger.com/profile/01753975411670884721noreply@blogger.comtag:blogger.com,1999:blog-4987609114415205593.post-87428757765958456282012-08-06T01:30:02.917+01:002012-08-06T01:30:02.917+01:00Hey Jeff,
great post with some interesting ideas!...Hey Jeff,<br /><br />great post with some interesting ideas! Just a thought about your suggestion that all individuation criteria have the form <br /><br />(IC) $K(x) \wedge K(y) \rightarrow [\bigwedge \{x \sim_{R,i} y \mid (R,i) \in I_K \} \rightarrow x = y$.<br /><br />As a syntactic claim about the form of individuation criteria this seems to be false. Take the antisymmetry of the $\leq$ relation on the natural numbers $\mathbb{N}$. This gives rise to the following individuation criterion:<br /><br />$\mathbb{N}(x)\wedge\mathbb{N}(y)\rightarrow (x\leq y\land y\leq x)\rightarrow x=y$.<br /><br />Note that this criterion does not have the form (IC). However, it is equivalent to the criterion<br /><br />$\mathbb{N}(x)\wedge\mathbb{N}(y)\rightarrow x\sim_{\leq,2}y\rightarrow x=y$,<br /><br />but only within the theory of total orders (or PA, or whatever). So, it seems that you're assuming that there is some underlying background theory $T$, right? And then your conjecture is that any (reasonable) individuation criterion $\phi$ in a given theory $T$ is equivalent (in the theory) to a criterion of the form (IC) $K(x) \wedge K(y) \rightarrow [\bigwedge \{x \sim_{R,i} y \mid (R,i) \in I_K \} \rightarrow x = y$. Does that sound about right to you? Then your conjecture would ammount to something like a normal form lemma (or so) for individuation criteria, which would be nice. (Although I'm not sure that it's true but that's a different issue).Johanneshttps://www.blogger.com/profile/18180095195423447146noreply@blogger.com