logicandlanguage.net

Interesting thing number one: Humberstone distinguishes the sense of "two-dimensional" with which his paper is concerned from a thousand other senses with which that expression is used. Did you know that Max gives a logic of two dimensional formulas? (As in "formulas constructed out of columns of ordinary formulas joined by connectives") or that Pratt refers to ordinary modal logics as two dimensional?

The kind of two dimensional logic we (Professor Humberstone, me, and you, gentle reader) are interested in is a semantics which treats the truth of a formula as relativised to elements of the set U₀ × U₁ (i.e. as relative to an ordered pair of say, a time and a place, or a possible world and...another possible world.) Humberstone goes on to distinguish two kinds of 2D semantics in this sense. A 2D semantics is homogenous if U₀= U₁, heterogenous where U₀≠ U₁. A semantics within which the truth of a formula is relativised to say a time and a possible world would be 2D in our sense, only heterogenously so, whereas Humberstone intends to restrict his study to homogenous cases and observes that "the clauses we offer for various operators in the definition of truth (w.r.t. a pair of points in a model) would not make sense without this assumption of homogeneity, since they involve things like moving the occupant of sone of the two coordinates into the position occupied by the other."

I thought it was interesting that Humberstone's conception of a two dimensional modal logic comes apart from David Kaplan's even at this early stage. Kaplan's formulas are true with respect to pairs of circumstances of evaluation (possible worlds) and contexts of utterance (ordered quadruples of agent, time, location and world) and the conceptual difference between the two is an important part of his theory.

Humberstone ends the section with a joke:

This distinction between homogeneous and hetergeneous cases of two-dimensionality arises at the level of informal motivation, it should be added, since the ordered pairs could themselves be considered as single indices in their own right. The fact that we are privately thinking of these indices as ordered pairs of points need not be mentioned as long as the models...are subjected to conditions guaranteeing that they are either isomorphic to or at least indistinguishable for the logical purposes at hand from models in which truth is relative to pairs of points. Kuhn calls this procedure ... "flattening" a two-dimensional modal logic, though the term is not quite apt, since a two-dimensional object is already about as flat as things can get.

I'm not sure how this 2 dimensional semantics business does not reduce to a special case of combining modal logics?

Well, depending on how tightly you want to interpret "modal", it does in a way - but that isn't to dismiss it. The interest is driven by three things i) hope that it can help us to model the semantics of natural languages, (nearly everyone thinks it can help us to model the semantics of indexical expressions and some philosophers think it can be used to rescue descriptivism in the philosophy of language) ii) interest in understanding concepts like validity, necessity and a priority in contexts where we have context sensitive expressions and iii) hope that it might do substantial work in metaphysics and epistemology.

I think that Scott Soames' new book Reference and Description: The Case against Two Dimensionalism is a good introduction, but then, he was my advisor and it's very controversial; no doubt many would think it too opinionated.

Examples of people trying to put 2D frameworks to work can be found in Kaplan's monograph "Demonstratives", Jackson's From Metaphyics to Ethics and in the work of David Chalmers.