Wolfgang Schwarz

Centering C-propositions

Let's call the class of counterfactual circumstances at which a sentence S is true the C-proposition expressed by S. This is more or less what Kaplan calls the "content" of S. Here are three reasons why the circumstances constituting a C-proposition should be understood as centered possible worlds rather than old-fashioned uncentered worlds.

First reason: centering is needed for modal embeddings. The standard use of C-propositions is the analysis of modal constructions: "it is possible that hummingbirds can fly backwards" is true iff there is at least one relevant circumstance w at which "hummingbirds can fly backwards" is true. Now take a sentence such as "it is early afternoon", or "it is starting to rain". It doesn't make much sense to say of an entire world that it is early afternoon there, or starting to rain. So on the standard view, on which the circumstances in C-propositions are uncentered worlds, we first have to fix a time and place, presumably by drawing on the utterance context: "necessarily, it is early afternoon" is true iff it is early afternoon at every possible world at the time and place of the utterance. So "necessarily, it is early afternoon" is true whenever it is uttered on an early afternoon. That seems wrong.

It gets worse with counterfactuals: when I say, on Sunday, "if it was Monday, there would be lots of people in the courtyard now", the circumstance of evaluation isn't meant to be shifted to the closest worlds where it is Monday on a Sunday. You have to shift the time as well. And for "if we were in China, you would be arrested for saying this", you have to shift the place.

To get these cases right, we should say that "necessarily S" is true iff S is true at all centered worlds; and "if it had been the case that A, it would have been the case that C" is true iff C is true at the closest centered world at which A is true.

Second reason: centering helps with attitude reports. Many philosophers are attracted to the view that attitude reports ("Fred believes that S", "Fred said that S") somehow operate on the C-proposition expressed by the embedded sentence S. Evidence for this is that we can replace indexicals by co-refering names when moving from direct to indirect reports: when Fred said, "she is tired", we can report that "Fred said that Mary is tired". A simple explanation is that the sentence in our report preserves the C-proposition expressed by Fred's sentence: both are true in a world w iff Mary is tired in w, and that is all that is required of a true report (or maybe not all, but let's ignore complications introduced by "actually" and hyperintensionality). However, sometimes centering matters. "Fred said that Mary had been tired two days ago" can be understood as true if Fred said at t that Mary was tired two days before t. (Example stolen from Philippe Schlenker.) If C-propositions are uncentered, this is hard to explain. But suppose they are centered, and the C-proposition expressed by Fred's sentence is true at all and only those centered worlds in which Mary was tired two days before the center. Then the simple account of attitude reports straightforwardly entails that we can also use "two days ago" when reporting his speech act: the C-proposition of the embedded sentence will match the C-proposition of his utterance. (The other reading, on which we report Fred as saying that Mary was sick two days back from now, might be explained as sort of de re, with "two days ago" taking wide scope, like "the man over there" in "Fred claimed that the man over there had left the country".)

Third reason: centering unifies indices. Suppose that, just as modal operators shift the world of evaluation, temporal operators shift the time, and location operators the location of evaluation: "at some time, London was a small town" is true iff "London was a small town" is true at some time. If C-propositions are uncentered worlds, they are of no help in anaysing temporal operators. We need to introduce further 'index coordinates', taking semantic values to be something like functions from triples of a world, a time and a location to a truth-value. But if modal operators shift centered worlds, then tense operators can operate on the very same coordinate: "at some time, S" is true at a circumstance w iff S is true at every circumstance w' that differs from w at most in its temporal location. To evaluate "in the Sherlock Holmes world, in 1900, in London, S", we don't have to shift three independent world, time and place coordinates and then evaluate S at the resulting index; rather, the modal operator shifts a single, centered coordinate to the Holmes world (ignore the indeterminacy), then the tense operator shifts that same coordinate back to 1900, and the location operator shifts it to London.

If a centered world is itself a triple of an uncentered world, a time and a place, the difference between having only one index coordinate: a centered world, and having three: a world, a time, and a place, might appear spurious. The only news is that 1) modal operators are taken to shift all coordinates, not just the world (that is in fact also Montague's interpretation of modal operators), and 2) all indices are automatically proper; that is, sentences don't have interesting truth-value distributions over (w,t,p) triples such that t doesn't exist at w, or p doesn't exist at t at w: nothing interesting can be said about what happens at place and time at a world where this place or time doesn't exist. That's only a small gain in theoretical simplicity. However, we need centered worlds for the modal operators even for languages that lack temporal and location operators. So there's also some potential cross-language unification to be gained. Moreover, it is not clear that centered worlds can be identified with world-time-place triples, and if not, the effect of centering cannot be mimicked with triple-indices. For instance, we arguably need an orientation aspect for things like "necessarily, if A is to left of B, then B is to the right of A".

(A certain degree of theoretical simplicity is also gained with respect to two-dimensionalism: if horizontal propositions are centered (and proper), the 2D matrix becomes a simple square, and the diagonal a real diagonal.)