If Squares Were Circles

In the Semantics class I took last semester, we were taught to analyze counterfactuals in terms of possible worlds, so that the counterfactual "If X were true then Y would be" (for false 'X') is true iff the following truth condition is met:

(w) ('X' is true in w and w is otherwise similar to the actual world -> 'Y' is true in w)

Simply put, the counterfactual is true iff 'Y' is true in all those close possible worlds where 'X' is true.

Now, one problem with this analysis is that it implies that any counterfactual with a necessarily false antecedent will be vacuously true. (Consider: there won't be any falsifying cases where 'X' is true but 'Y' isn't, because 'X' is never true.) But it doesn't quite seem right to say it's true that "if there were a largest prime number then Kerry would be president".

Incidentally, a fun way to deal with this could be to extend our ontology to impossible worlds, and proceeding in an analogous way to that described in the linked-to post (see also here). But that's not really what this post is about.

Perhaps counterfactuals involving necessary falsehoods aren't usually meant as counterfactuals at all. (The above example sounds very unnatural, after all. I can only imagine someone saying it for rhetorical effect, with the implicature "stop wishing for the impossible, it's time to move on!") The notion of putting apparent 'counterfactuals' to factual use is discussed at Siris and Mixing Memory. Anyway, I just want to discuss a particular example I find interesting, which was originally brought up in our Semantics tutorials. The question is, which (if either) of the following two sentences is true?

(1) If squares were circles then cubes would be spheres.
(2) If squares were circles then cubes would be cylinders.

Despite appearances, these statements probably shouldn't be interpreted as counterfactuals. For one thing, according to the previously mentioned 'truth conditions', they'd both be vacuously true. We don't want that, as on any sensible reading they are clearly incompatible! (Though we could just reject the earlier analysis as a flawed or incomplete semantic theory of counterfactuals.)

They seem to me to just be a fancy way of expressing a sort of 'ratio' or comparison. We're effectively being asked: "square is to cube as circle is to ____?"

So what's the answer?

I think almost everyone in our class chose 'sphere'. From a mathematical point of view though, 'cylinder' is probably the better option. It all depends, of course, on how you get from square to cube. The simplest construction is to just take the 2-dimensional object and 'raise' it through the third dimension. Layer a whole bunch of squares atop each other (just pretend they have some small amount of 'thickness' to them) and you eventually get a cube. Do the same with circles and you get a cylinder.

So why do most of use choose 'sphere'? I'd guess it's because we weren't really thinking about how to (geometrically) construct a cube out of squares. Instead, we were comparing our geometric concepts, looking for salient similarities. Now, a cube is the most 'regular' 3-d analog of a square (all sides being of equal length), and something similar can be said of spheres in relation to circles: a sphere has constant radius, just like circles do, whereas the length from the centre of a cylinder to its surface will vary depending on which point of the surface you choose.

For another point in favour of 'sphere', consider working backwards, 'compressing' a 3-d object into its 2-d silhouette. Note that a cube is such that from every side it looks like a square (if looked at front-on, that is - I exclude 'diagonal' or rotated views). And of course the silhouette of a sphere is always a circle (no matter how you rotate it - even better!). But a cylinder? From the wrong side it looks more like a rectangle.

So, overall, I would say that "square is to cube as circle is to sphere". Nevertheless, I think (2) above is true, and (1) is false. So I have to abandon my 'ratio' analysis of the counterfactuals (huh, I didn't see that coming until I actually wrote this post!).

I think the difference is that the 'ratio' question invites us to compare concepts, and the concept of a 'sphere' is the circular analog of our 'cube' concept in a way that 'cylinder' is not. However, the original counterfactual formulations seem more objective, somehow. They're less concerned with our own subjective understanding of geometric concepts, and are instead more concerned with geometric reality - and that means constructions. (Despite our mistaken shortcut of comparing our concepts instead.) If you applied the general "build a cube" method using a circle for your base template instead of a square, the end result would be a cylinder, rather than a sphere. Hence, if squares were circles, then cubes would be cylinders.

Maybe it's a genuine counterfactual after all?