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.
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February 4, 2009 at 6:54 pm
Daniel Elstein
I agree with the point about empty domains in CL. Another way of looking at it is that it’s at least clearer that “Something exists” is 1-necessary than that it’s 2-necessary, and in the context of LD, necessitation would require that various things which are 1-necessary are also 2-necessary (because LD-validity is a subset of 1-necessity). The reason that “Something exists” has to be 1-necessary (and LD-valid) is that it follows from “I exist”, which is 1-necessary. So the failure of LD-necessitation for “Something exists” can be seen as a corollary of the failure of the 1-necessity of “I exist” to imply its 2-necessity.
Also, it’s not obvious that we want to accept necessitation in CL at the same time as holding that “Something exists” is CL-valid. (Obviously we do accept it, but we might not do so whole-heartedly – do we have a purely logical reason for thinking that there couldn’t have been nothing?)
February 4, 2009 at 7:12 pm
richwoodward
Thanks for this – I definitely need to think a bit more about the connections between 1-necessity and LD-validity. But thanks for alternative perspective.
February 7, 2009 at 4:02 pm
Rich Woodward
Here’s a different example of a sentence that is LD-valid but which intuitively has a false necessitation: “there is at least one believer”. There is an interesting discussion of this example in Tim Williamson’s (1986) Analysis paper “The Contingent A Priori: Has It Anything to do with Indexicals”. This seems like a less problematic case than “somthing exists”.