Possible World Semantics: What is it good for?

Recently, I’ve been thinking about some issues regarding actualist accounts of possible worlds semantics. Kripkean semantics, at least construed as an applied semantics, poses a number of problems for the actualist realist, since it seems to entail the existence of merely possible individuals. Plantinga offered a solution to this problem: think of the members of “the set of all possible individuals” are individual essences, rather than individuals. We then say things like: “Possibly There is an F” is true at w iff there is some w-accessible world v such that at v, there is an essence that is coexemplified with Fness” and “Necessarily, something is G” is true at w iff every w-accessible world v is such that at v, some essence is coexemplified with Gness”. If we set things up right, we can then invalidate various nasty principles, like the Barcan Formula and its converse.

Question: what on earth has this got to do with English sentences like “there could have been talking donkeys”?

Now, part of the folklore conception of model-theoretic semantics is that the model-theory ranges over a class of interpretations of a language, amongst which we hope to find the “intended” interpretation: the one that “gets the interpretation right”. The role of the model-theory, on this conception, is to provide a constitutive account of both logical consequence (truth-preservation under every interpretation) and truth simpliciter (true on the intended interpretation). If you were thinking of Plantinga’s semantics in this way, I guess I can see what’s odd about it – on the intended interpretation of our language, I’d be quantifying my essence rather myself if I said “I exist”). But I’m also coming to think that the folklore conception is questionable.

One philosopher who explicitly rejects it is Hartry Field. In his recent work on the semantic paradoxes, Field offers a “paracomplete” solution to the paradoxes which secures the full-intersubsitutivity of “True (<A>)” and “A” (in extensional contexts) at the expense of rejecting certain instances of excluded middle. Field’s model-theoretic semantic theory is based upon Kripke’s fixed-point construction and Field explicitly denies that the value of this construction can be found in its provision of a constitutive account of either logical consequence or truth. It’s value, Field contends, is to be found in its provision of an extensionally correct characterization of logical consequence. In other words, the construction provides an answers the question “what follows from what?” and can thereby be used by classical logicians to figure out what inferences can be made in Field’s non-classical account.

If Plantinga’s model-theory is understood as serving a similar role to the role that Field’s model-theory plays, then worries about how Plantinga’s semantics connects up to our ordinary modal talk appear a bit misguided. For such worries assume that Plantinga is deploying his semantics in order to provide a constitutive account of the content of our ordinary modal talk and though many possible worlds theorists might intend their Kripkean semantic theories to provide just that, there is an alternative, Fieldian, understanding of the job that Kripkean semantics does. On this understanding, the semantics is there to answer questions like “does ∀x◇Fx follow from ◇∀xFx?” and “does ◇∀xGx follow from ◇◇∀xGx?” and the value of the semantics is that it provides extensionally correct answers to these questions. Of course, it is a legitimate to demand that Plantinga’s model-theory must deploy only those ontological and conceptual resources that are acceptable by his own lights. But in this regard, it seems to me that Plantinga’s theory is a complete success.

So I’m now thinking that worries about the relationship between Plantinga’s semantics and ordinary English have to be handled pretty carefully. I can see problems if we make certain assumptions about what the semantics is supposed to give us. But if we have a more minimal account of the core role that the model-theory plays, as Field does, then lots of worries appear misconcieved.

FWIW, I’m also thinking that the literature on modal semantics is often totally unclear about what Kripke semantics is supposed to do for us. It seems to me that the “it’s just heuristic” stuff you often here makes alot of sense if you read it as distancing somebody from the folklore conception of model-theoretic semantics sketch earlier.