Saturday, August 26, 2006

Sommers-Englebretsen Term Logic, Part III

In Part I, I gave a rough-and-ready characterization of the basics of SETL. In Part II, I looked at some basic issues. We can handle categorical assertions; singular terms; relations; singular identities; propositions about propositions; and propositions about the domain of discourse. In this post I will look at two issues that are more complicated, namely, modality and irreducible identity between variables. (As noted previously, sources and relevant readings will follow in a later post.)

(1) Modality. Basic SETL has no modalities (necessity, possibility); and the extension for handling modalities is still incomplete. It appears that SETL can actually handle certain forms of modality fairly easily, with only a small extension. De re modality turns out to be nothing other than ordinary term-logic arguments with modalized terms. (Indeed, this is what distinguishes 'de re modality' from 'de dicto modality', at least using the terms in this sense: de re modalities modalize the terms of the proposition, whereas de dicto modalities modalize the proposition itself. Thus we could symbolize "All S is possibly P" as (-S+◊P) and add rules of inference relevant to such statements. 'De dicto modality' is considerably trickier. In any case, I will look at extension to modality a little more closely in a later post.

(2) Identity between variables. William Purdy has argued, with considerable force, that while SETL can handle just about anything modern predicate logic can, there is one thing that the latter appears to handle more easily: identity in cases where both sides of the identity are variables (in the predicate logic). SETL can easily handle identity in cases where at least one side is not a variable. Sommers and Englebretsen often talk as if this ended the matter; SETL can handle identity. But Purdy pointed out that they are always speaking of what I called above 'singular' identity; and there is one form of identity in modern predicate logic that turns out to be fairly important that is not 'singular' identity -- the case already noted, where both sides of the identity would be variables in the predicate logic. He rigorously argued that PCS, a formal language like modern predicate logic in many ways, but like SETL (and unlike modern predicate logic) able to put a singular term in the predicate, and having the theory of identity associated with SETL, turns out to be equivalent to a subset of modern predicate logic, but fails to handle well-formed formula involving identity between two variables (when those formula are not reducible in predicate logic to a form not involving variable identity).

The argument is interesting and important. And, as it's quite advanced, I can't be certain I've adequately understood. I'm not sure it is completely adequate, however. One thing that appears to be missing from PCS that is clearly found in SETL is what I've called category nominalization. Now, since every variable has an associated domain of discourse (namely, the domain of discourse for propositions with that variable), at least some identity between variables should be expressible in SETL. So, for instance, if (x)(y)(x=y) is the identity in question, what is naturally relevant to the truth of this identity is the domain of discourse; the identity basically says, for every member in the domain relevant to x and every member of the domain relevant to y, x is identical to y. Which yields:

±/TERM1/±/TERM2/

Where /TERM1/ is the nominalization of the domain associated with x and /TERM2/ is the nominalization of the domain associated with y. This, and equivalent propositions, are expressible in SETL. So at least some variable identity is expressible, because nominalized categories or domains can do the work of variables -- and, indeed, this is not surprising, because you can't have variables without domains; it would be like having variables that are incapable of having values, which is to say, variables that are not variables at all. And when you translate variables directly from predicate logic to SETL, e.g., (x)(Px), you translate using domain nominalization, namely,

-/P/+P

/P/ is the domain associated with x in (x)(Px), assuming that this one sentence is the only relevant sentence. Any weirdness up to this point is due to the fact that you can be entirely arbitrary about the domain associated with a variable; which means that for a lot of domains in which variable identity would be important there would be no ready-made terms for handling the identity (we'd have to neologize). In fact, Purdy notes that the limitation of PCS is that it can't handle unnamed elements in the domain of discourse. But there seems to be no reason why nominalized domains or categories can't cover unnamed elements (indeed, it would seem that they must).

Thus Purdy seems to be right that the theory of identity associated with SETL doesn't on its own allow for irreducible variable identities. But SETL does allow for category nominalization, which does appear to allow SETL to handle unnamed elements; and given this, it looks like at least some irreducible variable identities could be handled in SETL. In any case, this is an issue that needs further investigation.

So, having looked at some of these key issues, we can know proceed to discussion of inference in SETL; which we will begin in the next post.

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