Wolfgang Schwarz

Truth at a world, truth at a banana

Everyone who has taught Kripke and Putnam to undergraduates knows that philosophers nowadays use "truth at a world" in a special, technical sense that requires a lot of explaining. The most straightforward way to assign a sentence a truth value at another world w is to consider an utterance of the same words in w and ask whether or not that utterance is true. But this is not what we mean. Nor do we ask what truth value the sentence has conditional on the assumption that our world is w. (Lewis uses "truth at a world" in roughly this sense in "How to define theoretical terms"; the current convention appears to be really quite new.) What, then, do we mean? I find most introductions of the concept utterly obscure: I'm told to identify the 'proposition expressed' by a sentence in the actual world, and then to 'evaluate' this entity at another possible world. What on earth does that mean?

In my experience, the best way to introduce "truth at a world" is via modal embeddings: first, present the standard argument for the necessity of identity, and infer that water is necessarily H2O. (Ignore the fact that both the argument and the inference from it are fallacious.) Then translate necessity into truth at all worlds: at all worlds, "water is H2O" is true. Optionally, one can then add that with counterfactual conditionals or constructions like "on Twin Earth", we can talk about individual worlds and that the rigidity phenomena observed for the standard modal operators occur here as well.

Behaviour under modal embeddings is a good way to introduce the concept of truth at a world. I think it moreover exhausts that concept: there is nothing else to a sentence S being true at a world w than the truth of "at w, S" (and of related modal constructions).

If this is true, then it makes little sense to speak of a proposition or a belief being true or false at other worlds. But don't we speak that way? Unfortunately yes.

What is going on here is best illustrated with another example. Let truth at a banana be defined as follows:

S is true at a banana b iff the sentence S' that results from S by replacing all singular terms in it by a rigid designator of the banana b is true.

Thus "Hesperus is Phosphorus" is true at all bananas, "Lewis Carroll wrote Alice's Adventures in Wonderland" is true at no banana, and "the Hamas flag is green" is true at all and only the bananas that are green.

Now suppose Fred says "the Hamas flag is green", and I ask you, pointing at a yellow banana, whether what Fred said is true at this banana. Or suppose I tell you that Fred believes that the Hamas flag is yellow, and then ask you whether what he believes is true at this banana. I'm afraid you could answer those questions, even though truth at a banana was introduced only for sentences, not for propositions or beliefs.

The misuse creates trouble when we consider sentential representations of a single belief or proposition that differ with respect to the bananas at which they are true. Suppose Fred believes that some birds can fly backwards. It seems to me that he could express this very belief both by uttering "some birds can fly backwards" and by "there exist birds in this world that can fly backwards". But while the former sentence is true at all bananas inhabiting worlds where some birds can fly backwards, the latter is true only at bananas inside of which there are birds that can fly backwards. So what should we say about Fred's belief and the proposition expressed: are they true at all actual bananas or false at all actual bananas?

The problem gets worse when we proceed to use truth at bananas to individuate beliefs or propositions, stating that propositions A and B are identical iff they are true at the same bananas. This will lead us to say, counter-intuitively, that "London is pretty" and "Canberra is pretty" express the same proposition.

All this, I believe, also holds for truth at other worlds. Since it is derivative from something that applies to sentences only, we get problems with beliefs or propositions that can be expressed with sentences that differ in the worlds at which they are true. We can use the same example as before: "some birds can fly backwards" and "there exist birds in this world that can fly backwards" intuitively express the same belief; but the former is true at all worlds where some birds can fly backwards whereas the latter is true at all worlds whatsoever.

Likewise, when truth at worlds is used to individuate propositions or beliefs, we get absurdly counter-intuitive classifications: "some birds can fly backwards" ends up expressing the same proposition as the tautologous "if some birds can actually fly backwards, then some birds can fly backwards", as do "the evening star is visible in the evening sky" and "the morning star is visible in the evening sky". On the other hand, "some birds can fly backwards" and the analytically equivalent "some birds can actually fly backwards" turn out to express completely different propositions.

The classification of sentences by the worlds or bananas at which they are true results in partitions that are, as far as I can see, wildly out of touch with any pre-theoretic notion of content, and that are moreover completely useless for any theoretical purpose -- except of course for the semantics of modal and bananal embeddings, for interpreting sentences of the form "at world w, S" or "at banana b, S".