Bonus: A Puzzle about Conceptual Changes

Let us say a bit more about a puzzle that came about in previous posts.

Let p be "the sum of the angles of a triangle is lower than 180 degrees" and C a modal operator "it is conceivable that". Consider the situation before anyone even conceived of non-euclidean geometries as a consistent alternative to euclidean geometry. Then, intuitively speaking, p was not conceivable (¬Cp). But we now know that p is true of some triangles, because the geometry of the universe is non-euclidean (p). On the other hand, it is quite natural to think that conceptual modality is factive, that is, that if something stems from conceptual necessity, then it is also true (¬C¬p→p), and this implies the converse, that if something is true, then it is conceivable (p→Cp). But these three premises are mutually inconsistent, hence our trilemma: one of the following statements must be false.

1. ¬Cp
2. p
3. p→Cp

So, we have exactly three options: (A) either we can claim that p always was conceivable, it was merely unconceived of at the time, or (B) we can deny that p is actually the case in this context, presumably because it should be interpreted relative to a conceptual scheme (we are talking about Euclidean triangles only), or (C) we can give up on factivity for conceptual modality, there are unconceivable truths (the same goes mutadis mutandis if we replace truth with epistemic possibility: then the third option would deny that they are a subset of conceivable possibilities).

I think A and B are both standard ways of responding to the problem. Option A amounts to enlarging the conceptual possibility space as much as we can, so as to include everything that any cognitive agent could ever conceive in any conceptual scheme whatsoever. A potential problem is that it makes it harder to conceive of conceptual progress in a meaningful way, because it abstracts away from particular cognitive agents and conceptual schemes. In what sense could we say that non-euclidean geometries enlarged our conceptual space, if they always were part of it? We could define a notion of accessible conceptual sub-space to solve this, but then the trilemma appears again at this new level.

Option B is also quite standard. The idea is that sentence meanings are relative to conceptual schemes, and moving from one to the other changes meaning, so that the same sentence expresses different propositions in different conceptual schemes. Hence our sentence about triangle "becomes" true only because it now expresses a different proposition (we could also relativise truth, to the same effect). I think it is the most viable option. An objection could come from externalists about meaning, who would assume that the meaning of "triangle" does not change from one geometry to the other, only our conception of triangle does.

The remaining option, C, fits well with externalism, because it disconnects actual truths (and meaning) from conceptual schemes. It could be conceptually necessary, but false, that the sum of the angles of a triangle is 180. But this blurs the distinction between the realm of epistemic and conceptual truths, since everything becomes an expression of beliefs about external entities (and non-factivity just comes from the fact that we can be wrong: we are not associating necessity with knowledge anymore, only with some beliefs!). This sounds too much like plain scepticism about conceptual necessities to me.

One could attempt to justify option C on the ground that conceptual necessity is normative (it is about the right way to think), and that norms can be broken. But remember that according to our taxonomy, conceptual normativity constrains thoughts, not external facts. The point is not that we want physical triangles to be Euclidean, but that if we adopt the Euclidean conceptual scheme, what we call a triangle is something Euclidean (and we have no name for the physical triangles before knowing about non-Euclidean geometry). A conceptual norm can be broken, sure: it is broken if we entertain an unconceivable thought, for example if we come to believe that the angles of an Euclidean triangle make less than 180. But this does not make it true, so factivity is still in place.

I tend to think that conceptual necessity is both normative and factive. It does not really bear on external facts, but only on how we decide to shape them into thoughts, and evaluating such conceptual decisions should not be done in terms of truth versus falsity, but only in terms of utility and fruitfulness. Conceptual necessities are always true by their own light, and conceptual impossibilities always false. This means taking option B (or almost equivalently, option A and introduce a notion of conceptual sub-space that respects factivity).