Kaplan and Necessitation

I just came across a weird comment that Kaplan makes in his seminal paper ‘Demonstratives’ that I thought I’d highlight.

It’s familar that Kaplan’s ‘logic of demonstratives’ – his system LD – gives rise to failures of the rule of necessitation, which we can understand as saying that one may infer that “Necessarily, p” is L-valid if one has established that “p” is L-valid. This rule is at the heart of standard modal logics, holding even in very weak systems like K. But Kaplan thinks it fails for LD: we may not infer that “Necessarily p” is LD-valid if one has established that “p” is LD-valid. The standard examples are things like “I exist” and “I am here now”. Both of these sentences are LD-valid. But Kaplan tells us that their necessitations – “Necessarily, I exist” and “Necessarily, I am here now” are false, so we have breakdowns of necessitation.

The weird bit is that Kaplan then says that the counterexamples to necessitation are not limited to cases involving “pure indexicals” like ‘I’, ‘here’ and ‘now’; we can construct counterexamples in the “indexical-free” fragment of the language. The example he gives is “something exists”. This is LD-valid, but its necessitation – “necessarily, something exists” – is false. Or so Kaplan claims, anyway. What’s funny about this case is that “something exists” remains valid when we allow Kaplanian character to vary across models. (“something exists” is not only valid in the logic of demonstratives, it’s valid in classical logic.) Admittedly, this is one of the odd consequences of standard classical logic, but it’s interesting because Kaplan’s rejection of necessitation seems to be more full-blooded than it might initially seem.

The reason I mention this is because one of the things I’ve been playing around with is the idea that the following rule – call it “restricted necessitation” – holds in Kaplan’s system:

From ‘”S” is CL-valid’, infer that “box-S” is LD-valid

Where CL-valid is defined in terms of what’s true in all classical models. If we take Kaplan at his word, then he doesn’t even want this inference rule (since “something exists” is CL-valid, but not, Kaplan thinks, LD-valid).

There is of course stuff about empty domains to be take into account here. As Robbie pointed out to me, there is an obvious rationale for Kaplan to ban the empty domain in the context of LD since every context has an agent (speaker, writer, what-have-you), and every model has a designated context, and every element of the context has to be a member of the domain. So perhaps the way to get the right results is to allow empty domains in the case of logical consequence, but ban them when we come to a priori (LD) consequence.