‘Rooks console each other.’ (NRC Handelsblad, 23/01/2007) Question: can a rook console me? Can I console a rook? If in both cases the answer is ‘No’, then we have two isolated subframes, each characterised by a ‘console’-relation that has certain (formal and material) properties within that subframe, but which is limited to that subframe. The analogy with (in)translatability is obvious. The question that arises then is this: What reasons do we have (could we have) to call two such completely isolated relations both an ‘x-relation’? In what sense are these two the same relation?

Martin Stokhof from: Aantekeningen/Notes date: 23/01/2007

Part of the argumentation in Davidson’s ‘On the Very Idea of a Conceptual Scheme’ centers around transitivity of the translation relation. The point Davidson is trying to make here, I think, is that if there exists a translation from L_{1} into L_{2}, that translation itself can be stated in either L_{1} or L_{2}, and consists in an accurate mapping of sentences from L_{1} onto sentences of L_{2}. If the translation is stated in L_{1} then, by assumption, it can also be stated in L_{2} (after all, there is a mapping from all of L_{1} into L_{2}, i.e., also of the L_{1}-sentences describing the mapping). So we can safely assume that if there is a translation from L_{1} into L_{2}, it can be formulated in L_{2}. Now suppose there is a translation from L_{2} into L_{3}: that, again is a mapping, in this case of L_{2}-sentences onto L_{3}-sentences. That mapping includes those L_{2}-sentences that describe how to translate L_{1}-sentences into L_{2}-sentences. So what we have then is (among other things) a set of L_{3}-sentences that tell us which L_{2}-sentences are translations of which L_{1}-sentences, and L_{3}-translations of the L_{2}-sentences. But that means we have a translation of L_{1} into L_{3}. (More neatly: if a translation is a homomorphism from L to L’, then we know that if there is a homomorphism from L_{1} to L_{2} and a homomorphism from L_{2} to L_{3}, then there is a homomorphism from L_{1} to L_{3}, viz.: the composition of the two.)

Of course the above argument only works if we assume that translations are total, i.e., that they map all sentences of L onto sentences of L’ and vice versa. What if we drop that assumption? First of all we have to ask ourselves whether we are still dealing with translation in such a case. But let that pass, and suppose we have a mapping of all the sentences of L_{1} onto a proper subset of sentences of L_{2}, i.e., there are parts of L_{2} that have no counterpart in L_{1}. Notice that the formulation of the translation can not be in the latter set (for that would mean that the translation would be statable in L_{2}, but not in L_{1}, which is absurd.) Now assume we have a translation from L_{2} into L_{3}: the only way in which that would not give us a translation of L_{1} into L_{3} (by the reasoning above), is by being restricted to exactly that part of L_{2} that does not contain the L_{1}-to-L_{2} translation. But that would mean that it really isn’t a translation of L_{2} as such: it must leave out a proper part of L_{2}. Of course, such a mapping could exists, but we would lack any justification for calling it a translation. (And if we would insist that we could call it that, then we actually are assuming what we set out to prove, viz., the failure of transitivity of translation.)

Note that the fact that in actual cases there are bound to be discrepancies between languages, i.e., things in L that don’t have an exact counterpart in L’, or things in L’ that lack a counterpart in L, does not really affect this line of argumentation. The central point is that the translation itself concerns such a substantial part of both languages, that that by itself guarantees transitivity to a sufficient degree.

Martin Stokhof from: Radical Interpretation Discussion Board date: 11-2004