Category Archives: Logic

Paracompleteness and credences in contradictions.

The last few posts have discussed non-classical approaches to indeterminacy.

One of the big stumbling blocks about “folklore” non-classicism, for me, is the suggestion that contradictions (A&~A) be “half true” where A is indeterminate.

Here’s a way of putting a constraint that appeals to me: I’m inclined to think that an ideal agent ought to fully reject such contradictions.

(Actually, I’m not quite as unsympathetic to contradictions as this makes it sound. I’m interested in the dialethic/paraconsistent package. But in that setting, the right thing to say isn’t that A&~A is half-true, but that it’s true (and probably also false). Attitudinally, the ideal agent ought to fully accept it.)

Now the no-interpretation non-classicist has the resources to satisfy this constraint. She can maintain that the ideal degree of belief in A&~A is always 0. Given that:

p(A)+p(B)=p(AvB)+p(A&B)

we have:

p(A)+p(~A)=p(Av~A)

And now, whenever we fail to fully accept Av~A, it will follow that our credences in A and ~A don’t sum to one. That’s the price we pay for continuing to utterly reject contradictions.

The *natural* view in this setting, it seems to me, is that accepting indeterminacy of A corresponds to rejecting Av~A. So someone fully aware that A is indeterminate should fully reject Av~A. (Here and in the above I’m following Field’s “No fact of the matter” presentation of the nonclassicist).

But now consider the folklore nonclassicist, who does take talk of indeterminate propositions being “half true” (or more generally, degree-of-truth talk) seriously. This is the sort of position that the Smith paper cited in the last post explores. The idea there is that indeterminacy corresponds to half-truth, and fully informed ideal agents should set their partial beliefs to match the degree-of-truth of a proposition (e.g. in a 3-valued setting, an indeterminate A should be believed to degree 0.5). [NB: obviously partial beliefs aren’t going to behave like a probability function if truth-functional degrees of truth are taken as an “expert function” for them.]

Given the usual min/max take on how these multiple truth values get settled over conjunction and negation, for the fullyinformed agent we’ll get p(Av~A) set equal to the degree of truth of Av~A, i.e. 0.5. And exactly the same value will be given to A&~A. So contradictions, far from being rejected, are appropriately given the same doxastic attitude as I assign to “this fair coin will land heads”

Another way of putting this: the difference between our overall attitude to “the coin will land heads” and “Jim is bald and not bald” only comes out when we consider attitudes to contents in which these are embedded. For example, I fully disbelieve B&~B when B=the coin lands heads; but I half-accept it for B=A&~A . That doesn’t at all ameliorate the implausibility of the initial identification, for me, but it’s something to work with.

In short, the Field-like nonclassicist sets A&~A to 0; and that seems exactly right. Given this and one or two other principles, we get a picture where our confidence in Av~A can take any value—right down to 0; and as flagged before, the probabilities of A and ~A carve up this credence between them, so in the limit where Av~A has probability 0, they take probability 0 too.

But the folklore nonclassicist I’ve been considering, for whom degrees-of-truth are an expert function for degrees-of-belief, has 0.5 as a pivot. For the fully informed, Av~A always exceeds this by exactly the amount that A&~A falls below it—and where A is indeterminate, we assign them all probability 0.5.

As will be clear, I’m very much on the Fieldian side here (if I were to be a nonclassicist in the first place). It’d be interesting to know whether folklore nonclassicists do in general have a picture about partial beliefs that works as Smith describes. Consistently with taking semantics seriously, they might think of the probability of A as equal to the measure of the set of possibilities where A is perfectly true. And that will always make the probability of A&~A 0 (since it’s never perfectly true); and meet various other of the Fieldian descriptions of the case. What it does put pressure on is the assumption (more common in degree theorists than 3-value theorists perhaps) that we should describe degree-of-truth-0.5 as a way of being “half true”—why in a situation where we know A is halftrue, would we be compelled to fully reject it? So it does seem to me that the rhetoric of folklore degree theorists fits a lot better with Smith’s suggestions about how partial beliefs work. And I think it’s objectionable on that account.

[Just a quick update. First observation. To get a fix on the “pivot” view, think of the constraint being that P(A)+P(~A)=1. Then we can see that P(Av~A)=1-P(A&~A), which summarizes the result. Second observation. I mentioned above that something that treated the degrees of truth as an expert function “won’t behave like a probability function”. One reflection of that is that the logic-probability link will be violated, given certain choices for the logic. E.g. suppose we require valid arguments to preserve perfect truth (e.g. we’re working with the K3 logic). Then A&~A will be inconsistent. And, for example, P(A&~A) can be 0.5, while for some unrelated B, P(B) is 0. But in the logic A&~A|-B, so probability has decreased over a valid argument. Likewise if we’re preserving non-perfect-falsity (e.g. we’re working with the LP system). Av~A will then be a validity, but P(Av~A) can be 0.5, yet P(B) be 1. These are for the 3-valued case, but clearly that point generalizes to the analogous definitions of validity in a degree valued setting. One of the tricky things about thinking about the area is that there are lots of choice-points around, and one is the definition of validity. So, for example, one might demand that valid arguments preserve both perfect truth and non-perfect falsity; and then the two arguments above drop away since neither |-Av~A nor A&~A|- on this logic. The generalization to this in the many-valued setting is to demand e-truth preservation for every e. Clearly these logics are far more constrained than the K3 or LP logics, and so there’s a better chance of avoiding violations of the logic-probability link. Whether one gets away with it is another matter.]

Non-classical logics: the no interpretation account

In the previous post, I set out what I took to be one folklore conception of a non-classicist treatment of indeterminacy. Essential elements were (a) the postulation of not two, but several truth statuses; (b) the treatment of “it is indeterminate whether” (or degreed variants thereof) as an extensional operator; (c) the generalization to this setting of a classicist picture, where logic is defined as truth preservation over a range of reinterpretations, one amongst which is the interpretation that gets things right.

I said in that post that I thought that folklore non-classicism was a defensible position, though there’s some fairly common maneuvers which I think the folklore non-classicist would be better off ditching. One of these is the idea that the intended interpretation is describable “only non-classically”.

However, there’s a powerful alternative way of being a non-classicist. The last couple of weeks I’ve had a sort of road to Damascus moment about this, through thinking about non-classicist approaches to the Liar paradox—and in particular, by reading Hartry Field’s articles and new book where he defends a “paracomplete” (excluded-middle rejecting) approach to the semantic paradoxes and work by JC Beall on a “paraconsistent” (contradiction-allowing) approach.

One interpretative issue with the non-classical approaches to the Liar and the like is that a crucial element is a truth-predicate that works in a way very unlike the notion of “truth” or “perfect truth” (“semantic value 1”, if you want neutral terminology) that feature in the many-valued semantics. But that’s not necessarily a reason by itself to start questioning the folklore picture. For it might be that “truth” is ambiguous—sometimes picking up on a disquotational notion, sometimes tracking the perfect truth notion featuring in the nonclassicists semantics. But in fact there are tensions here, and they run deep.

Let’s warm up with a picky point. I was loosely throwing around terms like “3-valued logic” in the last post, and mentioned the (strong) Kleene system. But then I said that we could treat “indeterminate whether p” as an extensional operator (the “tertium operator” that makes “indet p” true when p is third-valued, and otherwise false). But that operator doesn’t exist in the Kleene system—the Kleene system isn’t expressively complete with respect to the truth functions definable over three values, and this operator is one of the truth-functions that isn’t there. (Actually, I believe if you add this operator, you do get something that is expressively complete with respect to the three valued truth-functions).

One might take this to be just an expressive limitation of the Kleene system. After all, one might think, in the intended interpretation there is a truth-function behaving in the way just described lying around, and we can introduce an expression that picks up on it if we like.

But it’s absolutely crucial to the nonclassical treatments of the Liar that we can’t do this. The problem is that if we have this operator in the language, then “exclusion negation” is definable—an operator “neg” such that “neg p” is true when p is false or indeterminate, and otherwise false (this will correspond to “not determinately p”—i.e. ~p&~indeterminate p, where ~ is so-called “choice” negation, i.e. |~p|=1-|p|). “p v neg p” will be a tautology; and arbitrary q will follow from the pair {p, neg p}. But this is exactly the sort of device that leads to so-called “revenge” puzzles—Liar paradoxes that are paradoxical even in the 3-valued system. Very roughly, it looks as if on reasonable assumptions a system with exclusion negation can’t have a transparent truth predicate in it (something where p and T(p) are intersubstitutable in all extensional contexts). It’s the whole point of Field and Beall’s approaches to retain something with this property. So they can’t allow that there is such a notion around (so for example, Beall calls such notions “incoherent”).

What’s going on? Aren’t these approaches just denying us the resources to express the real Liar paradox? The key, I think, is a part of the nonclassicist picture that Beall and Field are quite explicit about and which totally runs against the folklore conception. They do not buy into the idea that model theory is ranging over a class of “interpretations” of the language among which we might hope to find the “intended” interpretation. The core role of the model theory is to give an extensionally adequate characterization of the consequence relation. But the significance of this consequence relation is not to be explained in model-theoretic terms (in particular, in terms of one among the models being intended, so that truth-preservation on every model automatically gives us truth-preservation simpliciter).

(Field sometimes talks about the “heuristic value” of this or that model and explicitly says that there is something more going on than just the use of model theory as an “algebraic device”. But while I don’t pretend to understand exactly what is being invoked here, it’s quite quite clear that the “added value” doesn’t consist on some classical 3-valued model being “intended”.)

Without appeal to the intended interpretation, I just don’t see how the revenge problem could be argued for. The key thought was that there is a truth-function hanging around just waiting to be given a name, “neg”. But without the intended interpretation, what does this even mean? Isn’t the right thought simply that we’re characterizing a consequence relation using rich set-theoretic resources—and in terms of which we can draw differences that correspond to nothing in the phenomenon being modelled.

So it’s absolutely essential to the nonclassicist treatment of the Liar paradox that we drop the “intended interpretation” view of language. Field, for one, has a ready-made alternative approach to suggest—a Quinean combination of deflationism about truth and reference, with perhaps something like translatability being invoked to explain how such predicates can be applied to expressions in a language other than ones own.

I’m therefore inclined to think of the non-classicism—at least about the Liar—as a position that *requires* something like this deflationist package. Whereas the folklore non-classicist I was describing previously is clearly someone who takes semantics seriously, and who buys into a generalization of the powerful connections between truth and consequence that a semantic theory of truth affords.

When we come to the analysis of vagueness and other (non-semantic-paradox related) kinds of indeterminacy, it’s now natural to consider this “no interpretation” non-classicism. (Field does exactly this—he conceives of his project as giving a unified account of the semantic paradoxes and the paradoxes of vagueness. So at least *this* kind of nonclassicism, we can confidently attribute to a leading figure in the field). All the puzzles described previously for the non-classicist position are thrown into a totally new light. Once we make this move.

To begin with, there’s no obvious place for the thought that there are multiple truth statuses. For you get that by looking at a many valued model, and imagining that to be an image of what the intended interpretation of the language must be like. And that is exactly the move that’s now illegitimate. Notice that this undercuts one motivation for going towards a fuzzy logic—the idea that one represents vague predicates as some smoothly varying in truth status. Likewise, the idea that we’re just “iterating a bad idea” in multiplying truth values doesn’t hold water on this conception—since the many-values assigned to sentences in models just don’t correspond to truth statuses.

Connectedly, one shouldn’t say that contradictions can be “half true” (nor that excluded middle is “half true”. It’s true that (on say the Kleene approach) that you won’t have ~(p&~p) as a tautology. Maybe you could object to *that* feature. But that to me doesn’t seem nearly as difficult to swallow as a contradiction having “some truth to it” despite the fact that from a contradiction, everything follows.

One shouldn’t assume that “determinately” should be treated as the tertium operator. Indeed, if you’re shooting for a combined non-classical theory of vagueness and semantic paradoxes, you *really* shouldn’t treat it this way, since as noted above this would give you paradox back.

There is therefore a central and really important question: what is the non-classical treatment of “determinately” to be? Sample answer (lifted from Field’s discussion of the literature): define D(p) as p&~(p–>~p), where –> is a certain fuzzy logic conditional. This, Field argues, has many of the features we’d intuitively want a determinately operator to have; and in particular, it allows for non-trivial iterations. So if something like this treatment of “determinately” were correct, then higher-order indeterminacy wouldn’t be obviously problematic (Field himself thinks this proposal is on the right lines, but that one must use another kind of conditional to make the case).

“No interpretation” nonclassicism is an utterly, completely different position from the folklore nonclassicism I was talking about before. For me, the reasons to think about indeterminacy and the semantic and vagueness-related paradoxes in the first place, is that they shed light on the nature of language, representation, logic and epistemology. And on these sorts of issues, the no interpretation nonclassicism and the folklore version take diametrically opposed positions on such issues, and flowing from this, the appropriate ways to arguing for or against these views are just very very different.

Non-classical logics: some folklore

Having just finished the final revisions to my Phil Compass survey article on Metaphysical indeterminacy and ontic vagueness (penultimate draft available here) I started thinking some more about how those who favour non-classical logics think of their proposal (in particular, people who think that something like the Kleene 3-valued logic or some continuum valued generalization of it is the appropriate setting for analyzing vagueness or indeterminacy).

The way that I’ve thought of non-classical treatments in the past is I think a natural interpretation of one non-classical picture, and I think it’s reasonably widely shared. In this post, I’m going to lay out some of that folklore-y conception of non-classicism (I won’t attribute views to authors, since I’m starting to wonder whether elements of the folklore conception are characterizations offered by opponents, rather than something that the nonclassicists should accept—ultimately I want to go back through the literature and check exactly what people really do say in defence of non-classicism).

Here’s my take on folklore nonclassicism. While classicists think there are two truth-statuses, non-classicists believe in three, four or continuum many truth-statuses (let’s focus on the 3-valued system for now). They might have various opinions about the structure of these truth-statuses—the most common ones being that they’re linearly ordered (so for any two truth-statuses, one is truer than the other). Some sentences (say, Jimmy is bald) get a status that’s intermediate between perfect truth and perfect falsity. And if we want to understand the operator “it is indeterminate whether” in such settings, we can basically treat it as a one-place extensional connective: “indeterminate(p)” is perfectly true just in case p has the intermediate status; otherwise it is perfectly false.

So interpreted, non-classicism generalizes classicism smoothly. Just as the classicist can think there is an intended interpretation of language (a two valued model which gets the representation properties of words right) the non-classicist can think there’s an intended interpretation (say a three valued model getting the representational features right). And that then dovetails very nicely with a model-theoretic characterization of consequence as truth-preservation under (almost) arbitrary reinterpretations of the language. For if one knows that some pattern is truth-preserving under arbitrary reinterpretations of the language, then that pattern is truth-preserving in particular in the intended interpretation—which is just to say that preserves truth simpliciter. This forges a connection between validity and preserving a status we have all sorts of reason to be interested in—truth. (Of course, one just has to write down this thought to start worrying about the details. Personally, I think this integrated package is tremendously powerful and interesting, deserves detailed scrutiny, and should be given up only as an option of last resort—but maybe others take a different view). All this carries over to the non-classicist position described. So for example, on a Kleene system, validity is a matter of preserving perfect truth under arbitrary reinterpretations—and to the extent we’re interested in reasoning which preserves that status, we’ve got the same reasons as before to be interested in consequence. Of course, one might also think that reasoning that preserves non-perfect-falsity is also an interesting thing to think about. And very nicely, we have a systematic story about that too—this non-perfect falsity sense of validity would be the paraconsistent logic LP (though of course not under an interpretation where contradictions get to be true).

With this much on board, one can put into position various familiar gambits in the literature.

  1. One could say that allowing contradictions to be half-true (i.e. to be indeterminate, to have the middle-status) is just terrible. Or that allowing a parity of truth-status between “Jimmy is bald or he isn’t” and “Jimmy’s both bald and not bald” just gets intuitions wrong (the most powerful way dialectically to deploy this is if the non-classicist backs their position primarily by intuitions about cases—e.g. our reluctance to endorse the first sentence in borderline cases. The accusation is that if our game is taking intuitions about sentences at face value, it’s not at all clear that the non-classicist is doing a good job.)
  2. One could point out that “indeterminacy” for the nonclassicist will trivially iterate. If one defines Determinate(p) as p&~indeterminate(p) (or directly as the one-place connective that is perfectly true if p is, and perfectly false otherwise) then we’ll quickly see that Determinately determinately p will follow from determinately p; and determinately indeterminate whether p will follow from indeterminate whether p. And so on.
  3. In reaction to this, one might abandon the 3-valued setting for a smooth, “fuzzy” setting. It’s not quite so clear what value “indeterminate p” should take (though there are actually some very funky options out there). Perhaps we might just replace such talk with direct talk of “degrees of determinacy” thought of as degrees of truth—with “D(p)=n” being again a one-place extensional operator perfectly true iff p has degree of truth n; and otherwise perfectly false.
  4. One might complain that all this multiplying of truth-values is fundamentally misguiding. Think of people saying that the “third status” view of indeterminacy is all wrong—indeterminacy is not a status that competes with truth and falsity; or the quip (maybe due to Mark Sainsbury?) that one does “not improve a bad idea by iterating it”—i.e. by introducing finer and finer distinctions.

I don’t think these are knock-down worries. (1) I do find persuasive, but I don’t think it’s very dialectically forceful—I wouldn’t know how to argue against someone who claimed their intuitions systematically followed, say, the Kleene tables. (I also think that the nonclassicist can’t really appeal to intuitions against the classicist effectively). Maybe some empirical surveying could break a deadlock. But pursued in this way the debate seems sort of dull to me.

(2) seems pretty interesting. It looks like the non-classicist’s treatment of indeterminacy, if they stick in the 3-valued setting, doesn’t allow for “higher-order” indeterminacy at all. Now, if the nonclassicist is aiming to treat determinacy rather than vagueness *in general* (say if they’re giving an account of the indeterminacy purportedly characteristic of the open future, or of the status of personal identity across fission cases) then it’s not clear one need to posit higher-order indeterminacy.

I should say that there’s one response to the “higher order” issues that I don’t really understand. That’s the move of saying that strictly, the semantics should be done in a non-classical metalanguage, where we can’t assume that “x is true or x is indeterminate or x is false” itself holds. I think Williamson’s complaints here in the chapter of his vagueness book are justified—I just don’t know how what the “non-classical theory” being appealed to here is, or how one would write it down in order to assess its merits (this is of course just a technical challenge: maybe it could be done).

I’d like to point out one thing here (probably not new to me!). The “nonclassical metalanguage” move at best evades the challange that by saying that there’s an intended 3-valued interpretation, one is committed to deny higher-order indeterminacy. But we achieve this, supposedly, by saying that the intended interpretation needs to be described non-classically (or perhaps notions like “the intended interpretation” need to be replaced by some more nuanced characterization). The 3-valued logic is standardly defined in terms of what preserves truth over all 3-valued interpretations describable in a classical metalanguage. We might continue with the classical model-theoretic characterization of the logic. But then (a) if the real interpretation is describable only non-classically, it’s not at all clear why truth-preservation in all classical models should entail truth-preservation in the real, non-classical interpretation. And moreover, our object-language “determinacy” operator, treated extensionally, will still trivially iterate—that was a feature of the *logic* itself. This last feature in particular might suggest that we should really be characterizing the logic as truth-preservation under all interpretations including those describable non-classically. But that means we don’t even have a fix on the *logic*, for who knows what will turn out to be truth-preserving on these non-classical models (if only because I just don’t know how to think about them).

To emphasize again—maybe someone could convince me this could all be done. But I’m inclined to think that it’d be much neater for this view to deny higher-order indeterminacy—which as I mentioned above just may not be a cost in some cases. My suggested answer to (4), therefore, is just to take it on directly—to provide independent motivation for wanting however many values that is independent of having higher-order indeterminacy around (I think Nick J.J. Smith’s AJP paper “Vagueness as closeness” pretty explicitly takes this tack for the fuzzy logic folk).

Anyway, I take this to be some of the folklore and dialectical moves that people try out in this setting. Certainly it’s the way I once thought of the debate shaping up. It’s still, I think, something that’s worth thinking about. But in the next post I’m going to say why I think there’s a far far more attractive way of being a non-classicist.

“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.

Logically good inference and the rest

From time to time in my papers, the putative epistemological significance of logically good inference has been cropping up. I’ve been recently trying to think a bit more systematically about the issues involved.

Some terminology. Suppose that the argument “A therefore B” is logically valid. Then I’ll say that reasoning from “A” is true, to “B” is true, is logically good. Two caveats (1) the logical goodness of a piece of reasoning from X to Y doesn’t mean that, all things considered, it’s ok to infer Y from X. At best, the case is pro tanto: if Y were a contradiction, for example, all things considered you should give up X rather than come to believe Y; (2) I think the validity of an argument-type won’t in the end be sufficient for for the logically goodness of a token inference of that type—partly because we probably need to tie it much closer to deductive moves, partially because of worries about the different contexts in play with any given token inference. But let me just ignore those issues for now.

I’m going to blur use-mention a bit by classifying material-mode inferences from A to B (rather than: “A” is true to “B” is true”) as logically good in the same circumstances. I’ll also call a piece of reasoning from A to B “modally good” if A entails B, and “a priori good” if it’s a priori that if A then B (nb: material conditional). If it’s a priori that A entails B, I’ll call it “a priori modally good”.

Suppose now we perform a competent deduction of B from A. What I’m interested in is whether the fact that the inference is logically good is something that we should pay attention to in our epistemological story about what’s going on. You might think this isn’t forced on us. For (arguably: see below) whenever an inference is logically good, it’s also modally and a priori good. So—the thought runs—for all we’ve said we could have an epistemology that just looks at whether inferences are modally/a priori good, and simply sets aside questions of logical goodness. If so, logical goodness may not be epistemically interesting as such.

(That’s obviously a bit quick: it might be that you can’t just stop with declaring something a priori good; rather, any a priori good inference falls into one of a number of subcases, one of which is the class of logically good inferences, and that the real epistemic story proceeds at the level of the “thickly” described subcases. But let’s set that sort of issue aside).

Are there reasons to think competent deduction/logically good inference is an especially interesting epistemological category of inference?

One obvious reason to refuse to subsume logically good inference within modally good inferences (for example) is if you thought that some logically good inferences aren’t necessarily truth-preserving. There’s a precedent for that thought: Kaplan argues in “Demonstratives” that “I am here now” is a logical validity, but isn’t necessarily true. If that’s the case, then logically good inferences won’t be a subclass of the modally good ones, and so the attempt to talk only about the modally good inferences would just miss some of the cases.

I’m not aware of persuasive examples of logically good inferences that aren’t a priori good. And I’m not persuaded that the Kaplanian classification is the right one. So let’s suppose pro tem that the logically good inference are always modally, a priori, and a priori modally, good.

We’re left with the following situation: the logically good inferences are a subclass of inferences that are also fall under other “good” categories. In a particular case where we come to believe B on the basis of A, where is the particular need to talk about its logical “goodness”, rather than simply about its modal, a priori or whatever goodness?

To make things a little more concrete: suppose that our story about what makes a modally good inference good is that it’s ultra-reliable. Then, since we’re supposing all logically good inferences are modally good, just from their modal goodness, we’re going to get that they’re ultra-reliable. It’s not so clear that epistemologically, we need say any more. (Of course, their logical goodness might explain *why* they’re reliable: but that’s not clearly an *epistemic* explanation, any more than is the biophysical story about perception’s reliability.)

So long as we’re focusing on cases where we deploy reasoning directly, to move from something we believe to something else we believe, I’m not sure how to get traction on this issue (at least, not in such an abstract setting: I’m sure we could fight on the details if they are filled out.). But even in this abstract setting, I do think we can see that the idea just sketched ignores one vital role that logically good reasoning plays: namely, reasoning under a supposition in the course of an indirect proof.

Familiar cases: If reasoning from A to B is logically good, then it’s ok to believe (various) conditional(s) “if A, B”. If reasoning from A1 to B is logically good, and reasoning from A2 to B is logically good, then inferring B from the disjunction A1vA2 is ok. If reasoning from A to a contradiction is logically good, then inferring not-A is good. If reasoning from A to B is logically good, then reasoning from A&C to B is good.

What’s important about these sort of deployments is that if you replace “logically good” by some wider epistemological category of ok reasoning, you’ll be in danger of losing these patterns.

Suppose, for example, that there are “deeply contingent a priori truths”. One schematic example that John Hawthorne offers is the material conditional “My experiences are of kind H > theory T of the world is true”. The idea here is that the experiences specified should be the kind that lead to T via inference to the best explanation. Of course, this’ll be a case where the a priori goodness doesn’t give us modal goodness: it could be that my experiences are H but the world is such that ~T. Nevertheless, I think there’s a pretty strong case that in suitable settings inferring T from H will be (defeasibly but) a priori good.

Now suppose that the correct theory of the world isn’t T, and I don’t undergo experiences H. Consider the counterfactual “were my experiences to be H, theory T would be true”. There’s no reason at all to think this counterfactual would be true in the specified circumstances: it may well be that, given the actual world meets description T*, the closest world where my experiences are H is still an approximately T*-world rather than a T-world. E.g. the nearest world where various tests for general relativity come back negative may well be a world where general relativity is still the right theory, but it’s effects aren’t detectable on the kind of scales initially tested (that’s just a for-instance: I’m sure better cases could be constructed).

Here’s another illustration of the worry. Granted, reasoning from H to T seems a priori. But reasoning from H+X to T seems terrible, for a variety of X. (So: My experiences are of H + my experiences are misleading in way W will plausibly a priori supports some T’ incompatible with T). But if we were allowed to use a priori good reasoning in indirect proofs, then we could simply argue from H+X to H, and thence (a priori) to T, overall getting an a priori route from H+X to T. the moral is that we can’t treat a priori good pieces of reasoning as “lemmas” that we can rely on under the scope of whatever suppositions we like. A priori goodness threatens to be “non-monotonic”: which is fine on its own, but I think does show quite clearly that it can completely crash when we try to make it play a role designed for logical goodness.

This sort of problem isn’t a surprise: the reliability of indirect proofs is going to get *more problematic* the more inclusive the reasoning in play is. Suppose the indirect reasoning says that whenever reasoning of type R is good, one can infer C. The more pieces of reasoning count as “good”, the more potential there is to come into conflict with the rule, because there’s simply more cases of reasoning that are potential counterexamples.

Of course, a priori goodness is just one of the inferential virtues mentioned earlier: modal goodness is another; and a priori modal goodness a third. Modal goodness already looks a bit implausible as an attempt to capture the epistemic status of deduction: it doesn’t seem all that plausible to classify the inferential move from A and B to B as w the same category as the move from this is water to this is H2O. Moreover, we’ll again have trouble with conditional proof: this time for indicative conditionals. Intuitively, and (I’m independently convinced) actually, the indicative conditional “if the watery stuff around here is XYZ, then water is H2O” is false. But the inferential move from the antecedent to the consequent is modally good.

Of the options mentioned, this leaves a priori modal goodness. The hope would be that this’ll cut out the cases of modally good inference that cause trouble (those based around a posteriori necessities). Will this help?

I don’t think so: I think the problems for a priori goodness resurface here. if the move from H to T is a priori good, then it seems that the move from Actually(H) to Actually(T) should equally be a priori good. But in a wide variety of cases, this inference will also be modally good (all cases except H&~T ones). But just as before, thinking that this piece of reasoning preserves its status in indirect proofs gives us very bad results: e.g. that there’s an a priori route from Actually(H) and Actually(X) to Actually (T), which for suitably chosen X looks really bad.

Anyway, of course there’s wriggle room here, and I’m sure a suitably ingenious defender of one of these positions could spin a story (and I’d be genuinely interested in hearing it). But my main interest is just to block the dialectical maneuver that says: well, all logically good inferences are X-good ones, so we can get everything we want having a decent epistemology of X-good inferences. The cases of indirect reasoning I think show that the *limitations* on what inferences are logically good can be epistemologically central: and anyone wanting to ignore logic better have a story to tell about how their story plays in these cases.

[NB: one kind of good inference I haven’t talked about is that backed by what 2-dimensionalists might call “1-necessary truth preservation”: I.e. truth preservation at every centred world considered as actual. I’ve got no guarantees to offer that this notion won’t run into problems, but I haven’t as yet constructed cases against it. Happily, for my purposes, logically good inference and this sort of 1-modally good inference give rise to the same issues, so if I had to concede that this was a viable epistemological category for subsuming logically good inference, it wouldn’t substantially effect my wider project.]

CEM journalism

The literature on the linguistics/philosophy interface on conditionals is full of excellent stuff. Here’s just one nice thing we get. (Directly drawn from a paper by von Fintel and Iatridou). Nothing here is due to me: but it’s something I want to put down so I don’t forget it, since it looks like it’ll be useful all over the place. Think of what follows as a bit of journalism.

Here’s a general puzzle for people who like “iffy” analyses of conditionals.

  • No student passes if they goof off.

The obvious first-pass regimentation is:

  • [No x: x is a student](if x goofs off, x passes)

But for a wide variety of accounts, this’ll give you the wrong truth-conditions. E.g. if you read “if” as a material conditional, you’ll get it coming out true if all the students goof and succeed! What is wanted, as Higgenbotham urges, is something with the effect:

  • [No x: x is a student](x goofs off and x passes)

This seems to suggest that under some embeddings “if” expresses conjunction! But that’s hardly what a believer in the iffness of if wants.

What the paper cited above notes is that so long as we’ve got CEM, we won’t go wrong. For [No x:Fx]Gx is equivalent to [All x:Fx]~Gx. And where G is the conditional “if x goofs off, x passes”, the negated conditional “not: if x goofs off, x passes” is equivalent to “if x goofs off, x doesn’t pass” if we have the relevant instance of conditional excluded middle. What we wind up with is an equivalence between the obvious first-pass regimentation and:

  • [All x: x is a student](if x goofs off, x won’t pass).

And this seems to get the right results. What it *doesn’t* automatically get us is an equivalence to the Higgenbotham regimentation in terms of a conjunction (nor with the Kratzer restrictor analysis). And maybe when we look at the data more generally, we’ll can get some traction on which of these theories best fits with usage.

Suppose we’re convinced by this that we need the relevant instances of CEM. There remains a question of *how* to secure these instances. The suggestion in the paper is that rules governing legitimate contexts for conditionals give us the result (paired with a contextually shifty strict conditional account of conditionals). An obvious alternative is to hard-wire in CEM into the semantics, as Stalnaker does. So unless you’re prepared (with von Fintel, Gillies et al) to defend in detail fine-tuned shiftiness of the contexts in which conditionals can be uttered then it looks like you should smile upon the Stalnaker analysis.

[Update: It’s interesting to think how this would look as an argument for (instances of) CEM.

Premise 1: The following are equivalent:
A. No student will pass if she goofs off
B. Every student will fail to pass if she goofs off

Premise 2: A and B can be regimented respectively as follows:
A*. [No x: student x](if x goofs off, x passes)
B*. [Every x: student x](if x goofs off, ~x passes)

Premise 3: [No x: Fx]Gx is equivalent to [Every x: Fx]~Gx

Premise 4: if [Every x: Fx]Hx is equivalent to [Every x: Fx]Ix, then Hx is equivalent to Ix.

We argue as follows. By an instance of premise 3, A* is equivalent to:

C*. [Every x: student x] not(if x goofs off, x passes)

But C* is equivalent to A*, which is equivalent to A (premise 2) which is equivalent to B (premise 1) which is equivalent to B* (premise 2). So C* is equivalent to B*.

But this equivalence is of the form of the antecedent of premise 4, so we get:

(Neg/Cond instances) ~(if x goofs off, x passes) iff if x goofs off, ~x passes.

And we quickly get from the law of excluded middle and a bit of logic:

(CEM instances) (if x goofs off, x passes) or (if x goofs off, ~ x passes). QED.

The present version is phrased in terms of indicative conditionals. But it looks like parallel arguments can be run for CEM for counterfactuals (Thanks to Richard Woodward for asking about this). For one of the controversial cases, for example, the basic premise will be that the following are equivalent:

D. No coin would have landed heads, if it had been flipped.
E. Every coin would have landed tails, if it had been flipped.

This looks pretty good, so the argument can run just as before.]

Must, Might and Moore.

I’ve just been enjoying reading a paper by Thony Gillies. One thing that’s very striking is the dilemma he poses—quite generally—for “iffy” accounts of “if” (i.e. accounts that see English “if” as expressing a sentential connective, pace Kratzer’s restrictor account).

The dilemma is constructed around finding a story that handles the interaction between modals and conditionals. The prima facie data is that the following pairs are equivalent:

  • If p, must be q
  • If p, q

and

  • If p, might be q
  • Might be (p&q)

The dilemma proceeds by first looking at whether you want to say that the modals scope over the conditional or vice versa, and then (on the view where the modal is wide-scoped) looking into the details of how the “if” is supposed to work and showing that one or other of the pairs come out inequivalent. The suggestion in the paper is if we have the right theory of context-shiftiness, and narrow-scope the modals, then we can be faithful to the data. I don’t want to take issue with that positive proposal. I’m just a bit worried about the alleged data itself.

It’s a really familiar tactic, when presented with a putative equivalence that causes trouble for your favourite theory, to say that the pairs aren’t equivalent at all, but can be “reasonably inferred” from each other (think of various ways of explaining away “or-to-if” inferences). But taken cold such pragmatic explanations can look a bit ad hoc.

So it’d be nice if we could find independent motivation for the inequivalence we need. In a related setting, Bob Stalnaker uses the acceptability of Moorean-patterns to do this job. To me, the Stalnaker point seems to bear directly on the Gillies dilemma above.

Before we even consider conditionals, notice that “p but it might be that not p” sounds terrible. Attractive story: this is because you shouldn’t assert something unless you know it to be true; and to say that p might not be the case is (inter alia) to deny you know it. One way of bringing out the pretty obviously pragmatic nature of the tension in uttering the conjunction here is to note that asserting the following sort of thing looks much much better:

  • it might be that not p; but I believe that p

(“I might miss the train; but I believe I’ll just make it”). The point is that whereas asserting “p” is appropriate only if you know that p, asserting “I believe that p” (arguably) is appropriate even if you know you don’t know it. So looking at these conjunctions and figuring out whether they sound “Moorean” seems like a nice way of filtering out some of the noise generated by knowledge-rules for assertion.

(I can sometimes still hear a little tension in the example: what are you doing believing that you’ll catch the train if you know you might not? But for me this goes away if we replace “I believe that” with “I’m confident that” (which still, in vanilla cases, gives you Moorean phenomena). I think in the examples to be given below, residual tension can be eliminated in the same way. The folks who work on norms of assertion I’m sure have explored this sort of territory lots.)

That’s the prototypical case. Let’s move on to examples where there are more moving parts. David Lewis famously alleged that the following pair are equivalent:

  • it’s not the case that: if were the case that p, it would have been that q
  • if were that p, it might have been that ~q

Stalnaker thinks that this is wrong, since instances of the following sound ok:

  • if it were that p, it might have been that not q; but I believe if it were that p it would have been that q.

Consider for example: “if I’d left only 5 mins to walk down the hill, (of course!) I might have missed the train; but I believe that, even if I’d only left 5 mins, I’d have caught it. ” That sounds totally fine to me. There’s a few decorations to that speech (“even” “of course” “only”). But I think the general pattern here is robust, once we fill in the background context. Stalnaker thinks this cuts against Lewis, since if mights and woulds were obvious contradictories, then the latter speech would be straightforwardly equivalent to something of the form “A and I don’t believe that A”. But things like that sounds terrible, in a way that the speech above doesn’t.

We find pretty much the same cases for “must” and indicative “if”.

  • It’s not true that if p, then it must be that q; but I believe that if p, q.

(“it’s not true that if Gerry is at the party, Jill must be too—Jill sometimes gets called away unexpectedly by her work. But nevertheless I believe that if Gerry’s there, Jill’s there.”). Again, this sounds ok to me; but if the bare conditional and the must-conditional were straightforwardly equivalent, surely this should sound terrible.

These sorts of patterns make me very suspicious of claims that “if p, must q” and “if p, q” are equivalent, just as the analogous patterns make me suspicious of the Lewis idea that “if p, might ~q” and “if p, q” are contradictories when the “if” is subjunctive. So I’m thinking the horns of Gillies’ dilemma aren’t equal: denying the must conditional/bare conditional equivalence is independently motivated.

None of this is meant to undermine the positive theory that Thony Gillies is presenting in the paper: his way of accounting for lots of the data looks super-interesting, and I’ve got no reason to suppose his positive story won’t have a story about everything I’ve said here. I’m just wondering whether the dilemma that frames the debate should suck us in.