#### 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 *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 Stokhoffrom: Radical Interpretation Discussion Boarddate: 11-2004*