# Conventions vs. ideal theory

I’m often in the market for a metaphysics of semantic properties of language. And what I’m shopping for is the best instance I can find of “top down” or “interpretationist” accounts. These have a two-step pattern. First: you identify a target set of pairings of sentences with some kind of semantic property. Second: you give a story about how those target pairings select the correct interpretation of all linguistic expressions.

One instance of this is a form of Lewis’s conventionalism. On this story, there is a collection of sentences-in-use, X, and for each of these, there are conventions of truthfulness and trust that link each S in X to a specific proposition p (construed as a set of possible worlds). That gives you the target pairings that we need for Fit. Truthfulness is a conventionally-entrenched regularity of uttering S only if one believes p, and trust is a similarly entrenched regularity of forming the belief p if one hears S uttered.

In the selection step, the correct interpretation of the whole language—not just sentences but individual words, and sentences that are never used—is fixed as the *simplest* interpretation that “fits with” the pairings.

Clearly, there’ll be many difficulties and nuances in spelling this out, but for my purposes here, I’ll assume that to fit with a set of sentence-propositions pairs, the selected interpretation needs to mapping the first element of each such pair to the second element—so that the sentence, according to the interpretation, expresses the proposition it is conventionally associated with. I’ll also assume that considerations of fit take lexical priority over considerations of simplicity, so simplicity’s role is just to select among fitting interpretations.

Here’s another instance of the two-step pattern. On this story, from language-use we find a privileged set of sentences, an “ideal theory”. The pairing of sentences with semantic properties this induces is just the following: every sentence in the ideal theory is paired with The True. The second step is as before: the correct overall interpretation is the simplest theory that fits with these pairings.

The latter is the kind of story that sometimes goes under the label “global descriptivism” and is associated with Lewis’s response to Putnam. There’s controversy about whether it ever was a view that Lewis endorsed. In that context, the appeal to simplicity in the selection story is replaced by an appeal to naturalness or eligibility. I think that these amount to the same thing, given Lewis’s understanding of simplicity. But I won’t argue or further explain that here.

Are these stories compatible? Might they amount to the same thing? (I’m grateful to discussions here with Fraser MacBride that prompted these questions). This all depends on what ideal theory amounts to. Consider the set of sentences-in-use, X. At every world w, the set of sentence-proposition pairs induces a map from X to truth values. Let $D_w$ be a complete world-description of w in some privileged world-making sublanguage L. Let the $I_w$ be a set of sentences of the form “Necessarily, if $D_w$, then S”, for each S which is mapped to the True at w. Let $I$ be the union of the $I_w$ for arbitrary w.

Consider an interpretation that fits with the sentence-proposition pairs. This will make a S sentence in X true at w iff  “Necessarily, if $D_w$, then S” is in $I$—that’s guaranteed by the way we constructed I. Does that interpretation make the sentence true, as truth-maximization would demand? Yes it does, on the condition that the interpretation is correct for the world-making sublanguage and “necessarily” and the connectives “if… then…” and “not”. In those circumstances the antecedent of the conditional is true only at world w,  since S is true at w, the strict conditional is true.  Conversely, suppose that we have an interpretation that makes true all of $I$, and which is again faithful to the world making language, “necessarily” and “if then”. Since the “Necessarily, if $D_w$, then S” is $I$, it is made true, and that requires that the sentence S is true at w. In sum: making this particular “ideal theory” true is equivalent to fitting with the sentence-proposition pairs from which it is built—though only among a restricted range of interpretations that are already guaranteed to be faithful to the worldmaking language, etc.

I’m not sure that the need to be faithful to the worldmaking language is too big a deal, on this repackaging. One way of thinking about this is that we start with a set of expressions to interpret—a certain signature $\sigma$. The sentences-in-use S are included within this set. Then we as theorists consider an expanded signature $\sigma^+$, which we get by adjoining a new set of terms (the necessity operator, the conditional, the worldmaking vocabulary) for which we explicitly stipulate an interpretation. Using the expanded signature, we build the ideal theory, and then indirectly get a fix on the correct interpretation of the original signature by requiring that the ideal theory in the expanded signature be true. Since we have stipulated the interpretation of the added vocabulary, we introduce no new parameters.

In the above, I started with sentence-proposition pairs fixed by convention and extracted an “ideal theory”. One could reverse the process, if our ideal theory already consists in a bunch of strict conditionals whose antecedents are world descriptions and whose consequents are sentences in a set X. It’ll help if we assume the ideal theory is X-complete in the sense that for each world-description $D_w$, and sentence S in X, either “Necessarily, if $D_w$, then S”  or “Necessarily, if $D_w$, then not-S” is in the set. Each S in X can now be paired with the proposition consisting of all the worlds w such that  “Necessarily, if $D_w$, then S” occurs in the ideal theory. The same reasoning as before will show that maximizing the truth of ideal theory is equivalent to fitting the sentence-proposition pairs.

If we wanted to give a metasemantics that incorporated this second direction of explanation, then we have some additional challenges. We need an independent fix on what it takes for a conditional “Necessarily, if $D_w$, then S” to be included in ideal theory. Answers could be given: for example, we could say that such a conditional is included in the ideal theory relavant to interpreting agent x iff that agent is disposed to endorse sentence S, conditional on believing the world to satisfy D_w. I think of this as a Chalmers-style approach to these questions, though I haven’t yet done the work of going back to pin down how it relates to the manifold distinctions and projects included in his book “Constructing the World”. Here again, the actual language to be interpreted might not include the world-making vocabulary—that could be reserved to the theorist. But in this case, in giving the story about constructing the ideal theory, the theorist needs to use that vocabulary in specifying part of a psychological state of an individual–a possible belief state. So to apply this, even in principle, we would need some independent fix what it takes for an individual to have propositional attitudes with contents corresponding to elements of the world-making language.

In Chalmers, we find the suggestion that there is a privileged set of concepts whose meaning is fixed by acquaintance, in a thick, Russellian sense. So one option would be to run the above acquaintance-based story as the metasemantics for a basic chunk of the language of thought, and then run (scare-quotes) “global” descriptivism for the rest of that same language.

There is a more Lewisian way to run the story though. Here we will firmly distinguish between interpreting public language and ascribing mental content. The convention-based story notoriously leans heavily on this, anyway. Our starting point for the metasemantics of public language will be a fix on the psychological states of individuals sufficiently rich to make sense of psychological states whose content we theorists describe using the world-making language. Dispositions of subjects to endorse public-language sentences under those conditions then look like legitimate resources for us to use. And using them, we can give a principled characterization of an ideal theory which (as argued previously) will be equivalent to fitting with certain sentence-proposition pairs.

So truth-maximization (of certain strict conditional sentences) and proposition-fit maximization do seem compatible if the targets are related in the right way–even equivalent. And it may even be that what we get from looking to the sentence-proposition pairs fixed by Lewisian conventions is the same as sentence-proposition pairs extracted from an ideal theory constructed by the above method, and vice versa—at least for subjects who were within a community where conventions of truthfulness and trust prevailed in the first place. That would, however, take further argument.

An interpretation of Lewis that I’ve favoured elsewhere was that he really believed in the convention-based metasemantics, and the stuff about global descriptivism and truth-maximization was just something adopted for dialectical purposes in the context of a discussion with Putnam (this is something that Wo Schwarz has pushed). A lot of time in the literature one finds the global descriptivist/truth-maximizing theory being worked with, but with “ideal theory” being handled fairly loosely—when I do this myself, for example, I think of it as something like a global folk theory of the world. But given the above, I guess one interpretative option here is that Lewis had in mind the sort of equivalences described above, and so was happy to discuss the account in either formulation.

And here’s a final thought about this. Though dispositions-to-endorse might line up with conventions where such conventions exist, it’s pretty clear that subjects can have dispositions to endorse sentences even where the conditions for conventionality are violated. So one way of presenting this is that the ideal theory characterization, grounded in dispositions-to-endorse, is a general metasemantics for language that coincides *in the limit where there are conventions* with Lewis’s convention-based story, but which has far wider application. The prospect of that kind of generalization seems to me a good reason to look closer at ways to characterize this kind of ideal theory metasemantics and study its relation to convention.