Names and descriptions in modal logic
In the old days, it was common to exclude individual constants from quantified modal logic in favour of Russellian descriptions. I can see how this works if we have either fixed domains (the same individuals populating all worlds) or possibilist quantifiers. But in such systems individual constants don't cause much trouble anyway. Can one also make the description move in more liberal systems? I don't see how, but I guess I'm just missing something obvious.
Consider a formula "possibly, a is F". We want to replace the name "a" by a description "the A". Does the description get narrow scope ("possibly, the A is F") or wide scope ("the A is possibly F")? Either way, we seem to get the wrong result.
For the wide scope reading, the problem is that "possibly, a is F" can be true at a world w even if the individual denoted by "a" (call her Anne) is not in the domain of w. But then "the A is possibly F" is false at w -- unless "A" applies to an individual other then Anne at w, in which case we can let that individual be non-F at all worlds so that "the A is possibly F" still has a different truth value than "possibly, a is F".
For the narrow scope reading, the problem is that "possibly, a is F" can be true at a world even if Anne (the individual denoted by "a") is not in the domain of any accessible world. (I assume we don't have truth-value gaps, so that at any world either "Fa" or "~Fa" is true.) But with actualist quantifiers, "possibly, the A is F" requires there to be As in at least one accessible world, so again we can construct models where the truth-values come apart.
Hughes and Cresswell say on p.327 of their New Introduction that "it is not difficult to see that individual constants are theoretically dispensable" in favour of definite descriptions, and that with a properly chosen predicate A (picking out the singleton of Anne at every world), scope distinctions won't matter. But it seems to me that this presupposes either fixed domains or possibilist quantifiers. Doesn't it?
Hi Wo,
I wonder what it means to be dispensable? It seems that your argument against wide scoping every occurrence of a name shows that the two sentences can't in general be equivalent-at-a-world. But presumably we're interested in just being equivalent (equivalent at the actual world of the model.)
Doesn't the translation which replaces each name in a sentence with a wide scoped description achieve this? (Of course this translation has some funny properties: e.g. the translation of []phi won't be the same as [] concatenated to the translation of phi. So it's possible that <>Fa is true at every world while its translation is not, while both []<>Fa and its translation are true at the actual world.)