# Monthly Archives: April 2019

## 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.

## What’s functionalism anyway?

In reading up for my new project on Group Thinking, I’ve found that people attaching a certain label to a view of the metaphysics of group belief and desire that I find quite attractive. That label is “functionalism”. I’ve found myself very confused about what that common label means, so what follows is where I’ve got to in sorting that out.

Now, at a really rough level, I expect anything deserving the name “functionalism” to have at least two theoretical categories: roles and realizers. For example, if you’re going to be a functionalist about the property being in pain, you’ll be committed to (i) the idea that there is a functional role associated with pain; (ii) if anything is to be in pain, then it needs to have a realizor property i.e. to instantiate a property that plays the functional role.

That allows us a lot of flexibility on how we flesh out the details beyond this. We might have various accounts of what sort of theories of functional roles to give. We might have various accounts of what the realization relation is—and whether we need to allow for multiple realisors, imperfect realizers, etc etc. We might differ in whether we identify the original property of being in pain with the role, the realizor, or something else. But unless we have an account that has the two part structure, it isn’t functionalism as I was taught it or as I teach it.

Okay, with that as the setup, let me say something about the kind of functionalism that I understand best. This starts with Lewis’s story about how to find explicit definitions of theoretical terms. We start with a theory that neologizes—that introduces a set of terms for the first time. That theory will also reuse some old vocabulary. Lewis assumed that the theory is regimented so that all the new terms are names. The old vocabulary will include predicates like “…has the property…” or “…stands in relation …. to …”, if necessary, so that we can do the work of new predicates by means of new names for the relevant properties. If we start with a theory $T(t_1,...,t_n)$, where $t_i$ are the old terms, then the following is the unique-realization sentence for T:

$\exists y_1\ldots \exists y_n \forall x_1\ldots x_n(T(x_1,...,x_n)\leftrightarrow (x_1=y_1\wedge \ldots \wedge x_n=y_n))$

The following one-place predicate is then what we’ll mean by “the theoretical role of $t_1$“, or the “$t_1$“-role:

$\exists y_n\ldots \exists y_n \forall x_1\ldots x_n(T(x_1,...,x_n)\leftrightarrow (x_1=y_1\wedge \ldots \wedge x_n=y_n))$

The explicit definition of the new terms in old vocabulary that Lewis offered was just as the property that played the relevant theoretical role. Using an iota for the definite description operator, for $t_1$ the definition is:

$t_1:=\iota y_1\exists y_2 \ldots \exists y_n \forall x_1\ldots x_n(T(x_1,...,x_n)\leftrightarrow (x_1=y_1\wedge \ldots \wedge x_n=y_n))$

Informally, the definition says that $t_1$ is the property that plays the $t_1$-role.

Now, Lewis proves several nice results about these definitions and their relation to the original theory $T$, using a certain understanding of the definite description operator. I won’t get into that here.

One last thing that will be important: the definite description on the right hand side of the definition sentences is, in general, a non-rigid designator. Since $T$ may be uniquely realized by definite tuples of properties in different worlds, the definite description will in general pick out different properties at different worlds. And sometimes—with empirical investigation—we will be able to say something informative about the property that happens to be picked out at the actual world. For some name $N$ in our old vocabulary, rigidly designating a property, we may discover:

$\exists y_2 \ldots \exists y_n \forall x_1\ldots x_n(T(N,x_n,...,x_n)\leftrightarrow (x_1=y_1\wedge \ldots \wedge x_n=y_n))$

From this and the definition sentence, it will follow that:

$t_1=N$

So here we have a model for how the identification of new theoretical terms with old, familiar terms could go. In these circumstances we would call $N$ the realizer of the $t_1$-role at the actual world. In general, $N_w$ will be the realizer of this role at world w iff the following holds at w: $\exists y_2 \ldots \exists y_n \forall x_1\ldots x_n(T(N_w,x_n,...,x_n)\leftrightarrow (x_1=y_1\wedge \ldots \wedge x_n=y_n))$

It’s up for debate whether $t_1$ is a rigid or non-rigid designator. If it’s a rigid designator, then $t_1=N$ will be necessary if true, but the definition sentence will be contingent (presumably, an example of the contingent a priori). $t_1$ could equally be taken to be non-rigid, allowing the definition sentence to be necessarily true (as well as apriori). In that case, $t_1=N$ will be non-rigid (as well as a posteriori). It seems we could go either way on this, consistent with the rest of the framework.

I’ve introduced both role and realizer terminology in connection to the Lewis account of the definitions of theoretical terms. It is the model for how I understand role and realizor terminology in the context of functionalism. However, discussion of theoretical neologisms is one thing, and discussion of “functional” vocabulary is another. Lewis’s topic in “how to define theoretical terms” is the former, and comes, and that gives us a particular take on the way that theory and definition sentences relate. For Lewis, the definitions are “implicitly asserted” when we put forward $T$ as a term-introducing theory—presumably we’re doing something that’s equivalent to stipulating that they are to be (a priori) true. This is not an account that can be directly applied to terms—theoretical or otherwise—that are already in common currency. It is not an account, for example, of “pain”. In the case of pain, if “definitions” are to be offered, they have to be offered as a product of analysis, not as the product of stipulation.

Let’s turn, therefore, to a context where we are working only with terms that are already common currency. And let’s suppose that we have found a theory $T$, such that for a suitable set of target vocabulary $t_1,\ldots,t_n$, both $T(t_1,\ldots,t_n)$ and the unique realization sentence is true. The following will be true:

$t_1:=\iota y_1\exists y_2 \ldots \exists y_n \forall x_1\ldots x_n(T(x_1,...,x_n)\leftrightarrow (x_1=y_1\wedge \ldots \wedge x_n=y_n))$

We shouldn’t call these “definition sentences” since it’s not clear in what sense if any they are “definitions”. To highlight this, note that as a limiting case, our “theory” could simply consist in saying “Red is Arnold’s favourite colour”, with Red as the target vocabulary . The unique realization sentence is then that there is an y such that for all x, x is Arnold’s favourite colour, iff x=y—which is true enough. And the putative “definition sentence” would say: Red is the y such that for all x, x is Arnold’s favourite colour iff y=x. But though this is is a true identity, this is quite clearly not a “definition” of the term Red, and is obviously contingent and a posteriori.

Not any old uniquely-realized theory of target old vocabulary will do, therefore. I take it that the step to an “analytic” functionalism of a Lewisian sort imposes the following constraint: we take an analytic/apriori $T(t_1,\ldots,t_n)$. Now if, in addition, the unique realization sentence for this vocabulary is analytic/apriori, then the “definition sentences” will be analytic/apriori. Even if the unique realization sentence is not analytic/apriori, then the conditional whose antecedent is the unique realization sentence and whose consequent is a definition sentence will be analytic/apriori. So we could plausibly claim the definition sentences as “an analysis” of the relevant target vocabulary–perhaps an analysis modulo the assumption of unique realization.  The conjecture, for the special case of analytic functionalism about pain, etc, will be that we could pull off this trick by letting T be systematization of a set of a priori “platitudes” that uniquely characterize the typical causal role of the property of being in pain in causing distinctive kinds of behaviour, and being caused by distinctive kinds of stimuli, and which interacts with other (targeted) mental states in typical kinds of ways.

The assumption that we can find an (a priori) theory T that does the job just described is a major one. But if we can do it, then we can import all the distinctions and terminology from the theoretical terms case. We will have a one-place predicate that is a “theoretical role” for the target term “pain”—which given the nature of the T we’re envisaging we could aptly call a causal-functional role of “pain”. We would be up for discovering that the role is satisfied by a property rigidly designated by some N—say, C-fibres firing. And we could reason, in the fashion Lewis and Armstrong taught us, from the “definition sentence” for pain, plus the putative empirical fact that C-fibres play the pain role, to an identification of the property of being in pain with having one’s C-fibres fire.

So that’s the way I understand analytic functionalism. And I can understand other forms of functionalism as variations on the theme. For example, we could start with a metaphysically necessary (but not analytic or a priori) theory which necessarily uniquely characterizes a set of target vocabulary, and extract definition sentences from it, obtaining necessarily true (but not analytic or a priori) “definition sentences” that we might go on to present as counting as “metaphysical analyses”. We could take a scientific theory—a theory which uniquely characterizes a set of target terms with nomic necessity, and then extract “nomic analyses”, and so forth. In each case, distinctive functionalist structure of role and realizer, and the relation between them, will be well understood. If functionalism is to be amended (e.g. to allow for imperfect realization, or non-unique realization) then I will want to figure out how to adjust the above theory to make the necessary changes.

It’s one thing to say that functionalisms can be represented as an instance of the how-to-define-theoretical-terms model of extracting definitions from theories. It’s quite another to say that every successful application of that model to common currency terms would be a functionalism. That further claim seems false to me.

For example, suppose we applied this kind of account to a term that for which we already have an analysis ready-to-hand: the property of being a bachelor. An a priori uniquely characterizing theory  says (let’s suppose): bachelorhood is the property of being male and being umarried. So the “definition sentence” here is: bachelorhood is the y such that for all x, x is the intersection of being male and being unmarried iff y=x. What of the role and realizor properties here? The role property is being a y such that for all x, x is the intersection of being male and being unmarried iff y=x. What’s the realizer property?

Well, here’s a way of specifying a property that realizes the role in the minimal sense in which I introduced the terminology earlier: being a bachelor. Here’s another: the property that is the intersection of being unmarried and being male. But this seems dreadfully fishy. It doesn’t seem illuminating in the way typical identifications of realizors of functional roles would and should be. It might be true to say that pain realizes the pain role, and that the property of actually playing the pain role realizes the pain role. But in that paradigm case of functionalism what we are really interested in, and trust to be available, is some more illuminating characterization: e.g. that C-fibres firing plays the pain-role. And what we see from the bachelorhood case, I think, is that it’s entirely possible to apply all this analysis and for there to be no such illuminating identification of the realizor to be given at the end of the day.

To sum this up. In the paradigm cases of functionalism, we expect a two-step methodology. There’s first the step of identifying a relevant uniquely characterizing theory, from which by turning a crank we can extract “functional roles”. And then, we expect a second stage, where we or others do further non-trivial work (in the paradigm cases, empirical work) that gives us an illuminating way of identifying the realizors of those roles, using a vocabulary that differs from that used in characterizing the role itself. The realizors will be some relatively natural “kind” or natural enough property, relative to a somehow-privileged vocabulary. In the paradigm functionalisms, there’s also a suitable distance between the vocabulary used to specify the role, and the vocabulary used in the illuminating identification of the realizor.

Here’s a way of thinking about all this. There’s a genus-level notion of role and realizor here, which we find in functionalism, in understanding theoretical neologisms, and so forth. But in order to have a functionalism worthy of the name, we need more than such minimal roles and realizors—we need roles that are genuinely “functional” and which contrast sufficiently with their natural-enough “realizors”. That vague characterization is probably enough for us to get on with the hard work of finding examples that fit this bill.

But if this is the right way to think of things, then we should resist the thought that whenever we extract definitions  from a theory in the Lewis-style, that we’re engaged in functionalist analysis. And I definitely want to resist the thought that in undertaking that first kind of project, we are committed to there being “realizors” of the theoretical roles used in those definitions in a more-than-minimal sense. Sometimes, perhaps, it will follow from the content of the characterizing theory that realizors of the roles will be more-than-minimal—e.g. perhaps that role is a causal one, and we are independently committed to thinking that only sufficiently natural properties can stand in causal relations. Perhaps part of the characterizing theory itself is the claim that the relevant property is natural enough. That might guarantee that if successful, the analysis will turn out to be a functionalist one. But this needs to be argued out on a case by case basis.

To go back to the beginning: when people talk about functionalist analyses of believing that p and desiring that q, whether in application to groups or individuals, I think that often what they’re picking out are definitions of belief and desire that are extracted from an overall theory of belief and desire in the “theoretical role” way. But it’s a huge step from that to assume that one is committed to full-blown functionalism about belief and desire, with its more-than-minimal realizors of the roles so-characterized. I think it’s misleading to label accounts that aren’t committed to more-than-minimal realizors as kinds of functionalism, and I think that’s one reason that I got myself puzzled at the way the terminology is (sometimes) used in this area.