On transitivity of translation

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 L₁ into L₂, that translation itself can be stated in either L₁ or L₂, and consists in an accurate mapping of sentences from L₁ onto sentences of L₂. If the translation is stated in L₁ then, by assumption, it can also be stated in L₂ (after all, there is a mapping from all of L₁ into L₂, i.e., also of the L₁-sentences describing the mapping). So we can safely assume that if there is a translation from L₁ into L₂, it can be formulated in L₂. Now suppose there is a translation from L₂ into L₃: that, again is a mapping, in this case of L₂-sentences onto L₃-sentences. That mapping includes those L₂-sentences that describe how to translate L₁-sentences into L₂-sentences. So what we have then is (among other things) a set of L₃-sentences that tell us which L₂-sentences are translations of which L₁-sentences, and L₃-translations of the L₂-sentences. But that means we have a translation of L₁ into L₃. (More neatly: if a translation is a homomorphism from L to L’, then we know that if there is a homomorphism from L₁ to L₂ and a homomorphism from L₂ to L₃, then there is a homomorphism from L₁ to L₃, 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 L₁ onto a proper subset of sentences of L₂, i.e., there are parts of L₂ that have no counterpart in L₁. 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 L₂, but not in L₁, which is absurd.) Now assume we have a translation from L₂ into L₃: the only way in which that would not give us a translation of L₁ into L₃ (by the reasoning above), is by being restricted to exactly that part of L₂ that does not contain the L₁-to-L₂ translation. But that would mean that it really isn’t a translation of L₂ as such: it must leave out a proper part of L₂. 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