Quantification in A-intensional semantics
Once upon a time, two quite different roles were assigned to truth-conditions: 1) they are what you know when you understand a sentence and what people communicate with utterances of the sentence; 2) they determine the truth value of the sentence when prefixed with modal operators. Unfortunately, there are sentences where these two roles come apart, namely context-dependent sentences, like "it's raining" and "I am late", and sentences containing rigid designators, like "London is overcrowded" and "Hesperus = Phosphorus". Since virtually all sentences ever uttered belong to one of these two classes (or both), the idea that we can assign to sentences truth-conditions that serve both (1) and (2) must be given up. The common strategy to deal with this at least among philosophers is to regard truth-conditions in the sense of (2) as the proper topic of compositional semantics and to assume that some other ("pragmatic") story will deliver truth-conditions in the sense of (1) out of the truth-conditions in the sense of (2) and various contextual features. I find that cumbersome and unmotivated. In my view, truth-conditions in the sense of (1) should be the primary topic of semantics, and I don't see any reason for the roundabout two-step procedure via truth-conditions in the sense of (2). I wouldn't complain if that procedure turned out to work sufficiently well, but for all I can tell, it doesn't work well at all. So I think it would be better to do compositional semantics directly for truth-conditions in the sense of (1). Since Frank Jackson calls such truth conditions "A-propositions" or "A-intensions", I use "A-intensional semantics" for that project.
Sets of possible worlds are not suitable as truth-conditions in the sense of (1). We rather need something like sets of possible situations or contexts. For instance, what you communicate in German by uttering "es regnet" is that the present context belongs to the set of possible contexts in which it is raining. (This is obviously not the same as the set of contexts where the sounds "es regnet" are true in whatever language is spoken at that context.) Possible contexts can be modeled as triples of a world, a time and an individual, or, more simply, as a sufficiently small temporal part of a world-bound individual. (Perhaps we should also add a spatial component, but I'll ignore that here.) Hence truth-conditions in this sense closely correspond to properties: a set of triples of a world, a time and an individual is set theoretically equivalent to a function from world-time pairs to sets of individuals, which is a quite intuitive model of a property. For simplicity, I'll assume that the members of truth-conditions are (segments of world-bound) individuals rather than more elaborate tuples in what follows.
One interesting aspect of A-intensional semantics is that there is no obvious default restriction for quantifiers. In modal semantics, where the topic is truth at a possible world, it is very natural to assume that "there are Fs" is true at a world w iff among the individuals existing at w there are Fs. But if our topic is truth at a possible individual, the choice of a domain is less obvious. Of course we can still say that "there are Fs" is true at a possible individual i iff there are Fs among the individuals in the world of i (i, recall, exists at only one world). But we could just as well say that it is true iff there are Fs among the individuals existing at the same time and world as i, or iff there are Fs among the individuals on the same planet as i, or, in the other direction, iff there are Fs among the individuals inhabiting a world with the same laws of nature as that of i. So what should we say?
Well, none of that. When we say "there are Fs", we sometimes mean that there are Fs in our world; more often, we mean that there are Fs presently in our world; sometimes we mean that there are Fs on our planet, or in the room we are currently in, or among the individuals we previously talked about; and sometimes we mean that there are Fs in some nomologically or metaphysically possible world. So the rule should be: "there are Fs" is true at a possible individual i iff there are Fs in the domain of individuals salient at i, where the salient domain at i is ..., followed by rules to figure out the salient domain in a given context.
One of these rules, I suppose, is that if something is mentioned by name, the domain extends so as to include that thing (if it exists). This is why the inference from "a is F" to "something is F" looks so compelling.
The rule must be stated carefully: "London is overcroweded" is not true at an individual i iff the actual city of London is overcrowded in i's world. Rather, it is true at i iff the city that occupies the London role at i is overcrowded, where the London role might be: standing at the origin of a certain causal chain to i's use of the name "London". Or, better I think, we could say that "London is overcroweded" is true at i iff there is something that occupies the London role at i and everything that occupies the London role at i is overcroweded, for arguably there are many overlapping cities that occupy the London role and the sentence is only true if all of them are overcroweded. The role itself will usually see to it that it cannot be occupied by several distinct things.
If nothing occupies the London role at i, the domain cannot extend to include that thing. In this case, and only in this case, "Ex(x=London)" is false at i. That is, what we express by "London exists", at least in normal cases, is that something occupies the London role.
For some years now, I've been very uncertain whether we should teach our students free logic instead of classical predicate logic: on the one hand, I don't want sentences like "London exists" to be logical truths, as they are in classical logic. A logical truth should at least be necessary and a priori, but "London exists" is neither. On the other hand, some inferences invalid in free logic do strike me as valid: surely to prove that ExFx it suffices to find an example of something that is F! You don't need to show that this something, in addition to being F, also has the property of existence: existence is not a property that some things have and others lack so that one can reasonably ask whether this F-thing has it. So I'm stuck with what appear to be convincing reasons against both logics.
By the above considerations, the logic I'm looking for would have to be a third kind of logic. I want "Fa" to entail "ExFx", but "~Fa" to not entail "Ex~Fx": if nothing occupies the London role, "London is self-identical" is false, hence "it is not the case that London is self-identical" is true, yet this doesn't entail that something is not self-identical.
So like in free logic, "AxFx -> Fa" is invalid. This is great, because "AxFx -> Fa" together with the necessitation rule (that all logical truths are necessary) entails the Converse Barcan Formula ("LAxP -> AxLP") and thereby that all things necessarily exist.
Gotta figure out how that logic I want looks like in more detail.