What We Talk About When We Talk About Points

Earlier this decade, I resolved not to kill myself until Blackburn, de Rijke, and Venema’s Modal Logic came out in paperback. It was a good decision, since it’s a terrific book; I still haven’t absorbed everything in it, but I’ve had five years or so to think about the “high points” of their exposition. One of these is their introductory treatment of hybrid logic, which was invented by Arthur Prior in the ’60s and which has gradually become a topic of widespread interest: today there is an entry in the Stanford Encylopedia of Philosophy, but I’ll provide a slightly different exposition for very different purposes here. I’ve already talked quite a bit about Kripke semantics for modal logic, which hinges on “points” or “possible worlds”, at which points modal formulas are evaluated to determine their validity or invalidity. The basic idea of hybrid logic is to stop comparing formulas across possible worlds and instead to consider them at a particular point.

The concept came to Prior in tense-logical form, as a formalization of sentences describing events occurring at a particular point in time; but it can be generalized to semantics for alethic, deontic, and epistemic modalities as well. The main hybrid operator is usually written in the form “@nq”, which like the email symbol indicates that “at” the possible world n under consideration (which is designated by a “nominal”, a name for the state) proposition q holds. Now, the proof- and model theory of hybrid logic is well understood by this point, but the practical functions it might serve in analysis are less clear: I’d like to suggest one that’s been kicking around my head for a while. Ordinary modal logic is often used to formalize phenomena that are “modally robust”, like counterfactual causal dependency: “if the proper conditions obtained, q would happen”. Such things are, in the language of the neo-Kantian Wilhelm Windelband, “nomothetic”: they illustrate a world of laws (such as Wilfrid Sellars claimed concepts were unthinkable without).

Now, to my mind hybrid logic tackles the other area outlined by Windelband, the “idiographic”. In history and other Geisteswissenschaften, attention must be paid to individual persons and events using Verstehen, powers of understanding that do not operate according to strict, exceptionless laws. Hybrid modalities give us the expressive resources to talk about individual “states of affairs”: in fact, it’s possible to translate the whole of first-order logic into hybrid logic (provided we introduce an additional hybrid operatorœ ∀ œœquantifying over points). It seems to me this is also a serviceable representation of the non-metaphorical description of human events, that which does not strictly fall under the ambit of natural laws or those of “poetic politics” without being meaningless: obviously our access to such truths is a bit more fallible than what we can merely deduce, but even those who scoff at “unscientific” social theory can hardly do without such observations.