Monthly Archives: February 2008

Metaphysics Conference

Announcing: Perspectives on Ontology

A major international conference on metaphysics to be held at the University of Leeds, Sep 5th-7th 2008.

Speakers:
Karen Bennett (Cornell)
John Hawthorne (Oxford)
Daniel Nolan (Nottingham)
Jill North (Yale)
Helen Steward (Leeds)
Jessica Wilson (Toronto)

Commentators:
Benj Hellie (Toronto)
Kris McDaniel (Syracuse)
Ted Sider (NYU)
Jason Turner (Leeds)
Robbie Williams (Leeds)

This will be a great conference: so keep your diaries free, and spread the word!

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“Supervaluationism”: the word

I’ve got progressively more confused over the years about the word “supervaluations”. It seems lots of people use it in slightly different ways. I’m going to set out my understanding of some of the issues, but I’m very happy to be contradicted—I’m really in search of information here.

The first occurrence I know of is van Fraassen’s treatment of empty names in a 1960’s JP article. IIRC, the view there is that language comes with a partial intended interpretation function, specifying the references of non-empty names. When figuring out what is true in the language, we
look at what is true on all the full interpretations that extend the intended partial interpretation. And the result is that “Zeus is blue” will come out neither true nor false, because on some completions of the intended interpretation the empty name”Zeus” will designate a blue object, and others he won’t.

So that gives us one meaning of a “supervaluation”: a certain technique for defining truth simpliciter out of the model-theoretic notions of truth-relative-to-an-index. It also, so far as I can see, closes off the question of how truth and “supertruth” (=truth on all completions) relate. Supervaluationism, in this original sense, just is the thesis that truth simpliciter should be defined as truth-on-all-interpretations. (Of course, one could argue against supervaluationism in this sense by arguing against the identification; and one could also consistently with this position argue for the ambiguity view that “truth” is ambiguous between supertruth and some other notion—but what’s not open is to be a supervaluationist and deny that supertruth is truth in any sense.)

Notice that there’s nothing in the use of supervaluations in this sense that enforces any connection to “semantic theories of vagueness”. But the technique is obviously suggestive of applications to indeterminacy. So, for example, Thomason in 1970 uses the technique within an “open future” semantics. The idea there is that the future is open between a number of currently-possible histories. And what is true about is what happens on all these histories.

In 1975, Kit Fine published a big and technically sophisticated article mapping out a view of vagueness arising from partially assigned meanings, that used among other things supervaluational techniques. Roughly, the basic move was to assign each predicate with an extension (the set of things to which it definitely applies) and an anti-extension (the set of things to which it definitely doesn’t apply). An interpretation is “admissible” only if it assigns an set of objects to a predicate that is a superset of the extension, and which doesn’t overlap the anti-extension. There are other constraints on admissibility too: so-called “penumbral connections” have to be respected.

Now, Fine does lots of clever stuff with this basic setup, and explores many options (particularly in dealing with “higher-order” vagueness). But one thing that’s been very influential in the folklore is the idea that based on the sort of factors just given, we can get our hands on a set of “admissible” fully precise classical interpretations of the language.

Now the supervaluationist way of working with this would tell you that truth=truth on each admissible interpretation, and falsity=falsity on all such interpretations. But one needn’t be a supervaluationist in this sense to be interested in all the interesting technologies that Fine introduces, or the distinctive way of thinking about semantic indecision he introduces. The supervaluational bit of all this refers only to one stage of the whole process—the step from identifying a set of admissible interpretations to the definition of truth simpliciter.

However, “supervaluationism” has often, I think, been identified with the whole Finean programme. In the context of theories of vagueness, for example, it is often used to refer to the idea that vagueness or indeterminacy arises as a matter of some kind of unsettledness as to what precise extensions are expressions pick out (“semantic indecision”). But even if the topic is indeterminacy, the association with *semantic indecision* wasn’t part of the original conception of supervaluations—Thomason’s use of them in his account of indeterminacy about future contingents illustrates that.

If one understands “supervaluationism” as tied up with the idea of semantic indecision theories of vagueness, then it does become a live issue whether one should identify truth with truth on all admissible interpretations (Fine himself raises this issue). One might think that the philosophically motivated semantic machinery of partial interpretations, penumbral connections and admissible interpretations is best supplemented by a definition of truth in the way that the original VF-supervaluationists favoured. Or one might think that truth-talk should be handled differently, and that the status of “being true on all admissible assignments” shouldn’t be identified with truth simpliciter (say because the disquotational schemes fail).

If you think that the latter is the way to go, you can be a “supervaluationist” in the sense of favouring a semantic indecision theory of vagueness elaborated along Kit Fine’s lines, without being a supervaluationist in the sense of using Van Fraassen’s techniques.

So we’ve got at least these two disambiguations of “supervaluationism”, potentially cross-cutting:

(A) Formal supervaluationism: the view that truth=truth on each of a range of relevant interpretations (e.g. truth on all admissible interpretations (Fine); on all completions (Van Fraassen); or on all histories (Thomason)).
(B) Semantic indeterminacy supervaluationism: the view that (semantic) indeterminacy is a matter of semantic indecision: there being a range of classical interpretations of the language, which, all-in, have equal claim to be the right one.

A couple of comments on each. (A) of course, needs to be tightened up in each case by saying which are the relevant range of classical interpretations quantified over. Notice that a standard way of defining truth in logic books is actually supervaluationist in this sense. Because if you define what it is for a formula “p” to be true as it being true relative to all variable assignments, then open formulae which vary in truth value from variable-assignment to variable assignment end up exactly analogous to formulae like “Zeus is blue” in Van Fraassen’s setting: they will be neither true nor false.

Even when it’s clear we’re talking about supervaluationism in the sense of (B), there’s continuing ambiguity. Kit Fine’s article is incredibly rich, and as mentioned above, both philosophically and technically he goes far beyond the minimal idea that semantic vagueness has something to do with the meaning-fixing facts not settling on a single classical interpretation.

So there’s room for an understanding of “supervaluationism” in the semantic-indecision sense that is also minimal, and which does not commit itself to Fine’s ideas about partial interpretations, conceptual truths as “penumbral constraints” etc. David Lewis in “Many but also one”, as I read him, has this more minimal understanding of the semantic indecision view—I guess it goes back to Hartry Field’s material on inscrutability and indeterminacy and “partial reference” in the early 1970’s, and Lewis’s own brief comments on related ideas in his (1969).

So even if your understanding of “supervaluationism” is the (B)-sense, and we’re thinking only in terms of semantic indeterminacy, then you still owe elaboration of whether you’re thinking of a minimal “semantic indecision” notion a la Lewis, or the far richer elaboration of that view inspired by Fine. Once you’ve settled this, you can go on to say whether or not you’re a supervaluationist in the formal, (A)-sense—and that’s the debate in the vagueness literature over whether truth should be identified with supertruth.

Finally, there’s the question of whether the “semantic indecision” view (B), should be spelled out in semantic or metasemantic terms. One possible view has the meaning-fixing facts picking out not a single interpretation, but a great range of them, which collectively play the role of “semantic value” of the term. That’s a semantic or “first-level” (in Matti Eklund‘s terminology) view of semantic indeterminacy. Another possible view has the meaning-fixing facts trying to fix on a single interpretation which will give the unique semantic value of each term in the language, but it being unsettled which one they favour. That’s a metasemantic or “second-level” view of the case.

If you want to complain that second view is spelled out quite metaphorically, I’ve some sympathy (I think at least in some settings it can be spelled out a bit more tightly). One might also want to press the case that the distinction between semantic and metasemantic here is somewhat terminological—what we choose to label the facts “semantic” or not. Again, I think there might be something to this. There are also questions about how this relates to the earlier distinctions—it’s quite natural to think of Fine’s elaboration as being a paradigmatically semantic (rather than metasemantic) conception of semantic supervaluationism. It’s also quite natural to take the metasemantic idea to go with a conception that is non-supervaluational in the (A) sense. (Perhaps the Lewis-style “semantic indecision” rhetoric might be taken to suggest a metasemantic reading all along, in which way it is not a good way to cash out what’s the common ground among (B)-theorists is). But there’s room for a lot of debate and negotiation on these and similar points.

Now all this is very confusing to me, and I’m sure I’ve used the terminology confusingly in the past. It kind of seems to me that ideally, we’d go back to using “supervaluationism” in the (A) sense (on which truth=supertruth is analytic of the notion); and that we’d then talk of “semantic indecision” views of vagueness of various forms, with its formal representation stretching from the minimal Lewis version to the rich Fine elaboration, and its semantic/metasemantic status specified. In any case, by depriving ourselves of commonly used terminology, we’d force ourselves to spell out exactly what the subject matter we’re discussing is.

As I say, I’m not sure I’ve got the history straight, so I’d welcome comments and corrections.

Phlox

I just found about about Phlox, a (relatively) new weblog in philosophy of logic, language and metaphysics. It’s attached to a project at Humboldt University in Berlin. As well as following the tradition of philosophy centres with Greek names (this one means “flame”, apparently) “Phlox” is a cunning acronym for the group’s research interests.

There’s several really interesting posts to check out already. Worth heading over!

Aristotelian indeterminacy and partial beliefs

I’ve just finished a first draft of the second paper of my research leave—title the same as this post. There’s a few different ways to think about this material, but since I hadn’t posted for a while I thought I’d write up something about how it connects with/arises from some earlier concerns of mine.

The paper I’m working on ends up with arguments against standard “Aristotelian” accounts of the open future, and standard supervaluational accounts of vague survival. But one starting point was an abstract question in the philosophy of logic: in what sense is standard supervaluationism supposed to be revisionary? So let’s start there.

The basic result—allegedly—is that while all classical tautologies are supervaluational tautologies, certain classical rules of inference (such as reductio, proof by cases, conditional proof, etc) fail in the supervaluational setting.

Now I’ve argued previously that one might plausibly evade even this basic form of revisionism (while sticking to the “global” consequence relation, which preserves traditional connections between logical consequence and truth-preservation). But I don’t think it’s crazy to think that global supervaluational consequence is in this sense revisionary. I just think that it requires an often-unacknowledged premise about what should count as a logical constant (in particular, whether “Definitely” counts as one). So for now let’s suppose that there are genuine counterexamples to conditional proof and the rest.

The standard move at this point is to declare this revisionism a problem for supervaluationists. Conditional proof, argument by cases: all these are theoretical descriptions of widespread, sensible and entrenched modes of reasoning. It is objectionably revisionary to give them up.

Of course some philosophers quite like logical revisionism, and would want to face-down the accusation that there’s anything wrong with such revisionism directly. But there’s a more subtle response available. One can admit that the letter of conditional proof, etc are given up, but the pieces of reasoning we normally call “instances of conditional proof” are all covered by supervaluationally valid inference principles. So there’s no piece of inferential practice that’s thrown into doubt by the revisionism of supervaluational consequence: it seems that all that happens is that the theoretical representation of that practice has to take a slightly more subtle form than one might except (but still quite a neat and elegant one).

One thing I mention in that earlier paper but don’t go into is a different way of drawing out consequences of logical revisionism. Forget inferential practice and the like. Another way in which logic connects with the rest of philosophy is in connection to probability (in the sense of rational credences, or Williamson’s epistemic probabilities, or whatever). As I sketched in a previous post, so long as you accept a basic probability-logic constraint, which says that the probability of a tautology should be 1, and the probability of a contradiction should be 0, then the revisionary supervaluational setting quickly forces you to a non-classical theory of probability: one that allows disjunctions to have probability 1 where each disjunct has probability 0. (Maybe we shouldn’t call such a thing “probability”: I take it that’s terminological).

Folk like Hartry Field have argued completely independently of this connection to Supervaluationism that this is the right and necessary way to handle probabilities in the context of indeterminacy. I’ve heard others say, and argue, that we want something closer to classicism (maybe tweaked to allow sets of probability functions, etc). And there are Dutch Book arguments to consider in favour of the classical setting (though I think the responses to these from the perspective of non-classical probabilities are quite convincing).

I’ve got the feeling the debate is at a stand-off, at least at this level of generality. I’m particularly unmoved by people swapping intuitions about degrees of belief it is appropriate to have in borderline cases of vague predicates, and the like (NB: I don’t think that Field ever argues from intuition like this, but others do). Sometimes introspection suggests intriguing things (for example, Schiffer makes the interesting suggestion that one’s degree of belief in a conjunction of two vague propositions is typically matches one’s degree of belief in the propositions themselves). But I can’t see any real dialectical force here. In my own case, I don’t have robust intuitions about these cases. And if I’m to go on testimonial evidence on others intuitions, it’s just too unclear what people are reporting on for me to feel comfortable taking their word for it. I’m worried, for example, they might just be reporting the phenomenological level of confidence they have in the proposition in question: surely that needn’t coincide with one’s degree of belief in the proposition (thinking of an exam you are highly nervous about, but are fairly certain you will pass… your behaviour may well manifest a high degree of belief, even in the absence of phenomenological trappings of confidence). In paradigm cases of indeterminacy, it’s hard to see how to do better than this.

However, I think in application to particular debates we might be able to make much more progress. Let us suppose that the topic for the day is the open future, construed, minimally, as the claim that while there are definite facts about the past and present, the future is indefinite.

Might we model this indefiniteness supervaluationally? Something like this idea (with possible futures playing the role of precisifications) is pretty widespread, perhaps orthodoxy (among friends of the open future). It’s a feature of MacFarlane’s relativistic take on the open future, for example. Even though he’s not a straightforward supervaluationist, he still has truth-value gaps, and he still treats them in a recognizably supervaluational-style way.

The link between supervaluational consequence and the revisionionary behaviour of partial beliefs should now kick in. For if you know with certainty that some P is neither true nor false, we can argue that you should invest no credence at all in P (or in its negation). Likewise, in a framework of evidential probabilities, P gets no evidential probability at all (nor does its negation).

But think what this says in the context of the open future. It’s open which way this fair coin lands: it could be heads, it could be tails. On the “Aristotelian” truth-value conception of this openness, we can know that “the coin will land heads” is gappy. So we should have credence 0 in it, and none of our evidence supports it.

But that’s just silly. This is pretty much a paradigmatic case where we know what partial belief we have and should have in the coin landing heads: one half. And our evidence gives exactly that too. No amount of fancy footwork and messing around with the technicalities of Dempster-Shafer theory leads to a sensible story here, as far as I can see. It’s just plainly the wrong result. (One doesn’t improve matters very much by relaxing the assumptions, e.g. taking the degree of belief in a failure of bivalence in such cases to fall short of one: you can still argue for a clearly incorrect degree of belief in the heads-proposition).

Where does that leave us? Well, you might reject the logic-probability link (I think that’d be a bad idea). Or you might try to argue that supervaluational consequence isn’t revisionary in any sense (I sketched one line of thought in support of this in the paper cited). You might give up on it being indeterminate which way the coin will land—i.e. deny the open future, a reasonably popular option. My own favoured reaction, in moods when I’m feeling sympathetic to the open future, is to go for a treatment of metaphysical indeterminacy where bivalence can continue to hold—my colleague Elizabeth Barnes has been advocating such a framework for a while, and it’s taken a long time for me to come round.

All of these reactions will concede the broader point—that at least in this case, we’ve got an independent grip on what the probabilities should be, and that gives us traction against the Supervaluationist.

I think there are other cases where we can find similar grounds for rejecting the structure of partial beliefs/evidential probabilities that supervaluational logic forces upon us. One is simply a case where empirical data on folk judgements has been collected—in connection with indicative conditions. I talk about this in some other work in progress here. Another which I talk about in the current paper, and which I’m particularly interested in, concerns cases of indeterminate survival. The considerations here are much more involved than in indeterminacy we find in connection to the open future or conditionals. But I think the case against the sort of partial beliefs supervaluationism induces can be made out.

All these results turn on very local issues. None, so far as see, generalizes to the case of paradigmatic borderline cases of baldness and the rest. I think that makes the arguments even more interesting: potentially, they can serve as a kind of diagnostic: this style of theory of indeterminacy is suitable over here; that theory over there. That’s a useful thing to have in one’s toolkit.