J Robert G Williams

So this was the biggest selection-problem I faced: there are so many many-valued systems out there, and so many ways to think about them, which to choose?

I would have liked to talk a bunch on the interpretation of “third truth values”. This seems often glossed over badly to me. In the vagueness literature, it’s often assumed that once we’ve got a third truth value, we might as well be degree theorists. But it seems to me that “gap” interpretations of the third truth value are super-different from the “half-true” interpretation. But to make the case that this is more than a verbal dispute, I think we have to say a whole lot more about the cognitive role of indeterminacy, the role of logic, etc etc. All good stuff (in fact, very close to what I’m working on right now). But I chose not to go that way.

Another thing I could have done is talk directly on degree theories. Nick Smith has a new book length treatment of them, which makes some really nice moves both in motivating and defending the account. And of course they’re historically popular —and “fuzzy logic” is what you always hear talked about in non-philosophical treatments. In Williamson’s big vagueness book, degree theories are really the focus of the chapter corresponding to this section.

On the other hand, I felt is was really important to get a representative of the “logic first” view into the picture—someone who really treated semantics kind of instrumentally, and who saw the point of talking about vagueness in a rather different way to the way it’s often presented in the extant survey books. And the two that sprang to mind here were Hartry Field and Crispin Wright. Of these, Crispin’s intuitionism is harder to set up, and has less connections to other many valued theories. And his theory of a quandary cognitive role, while really interesting, just takes longer to explain than Hartry’s rejectionist suggestion. Wright’s agnosticism is a bit hard to explain too—I take it the view is supposed to be that we’re poised between the Williamsonian style picture, and an option where you assert the negation of bivalence—and the first seems unbelievable and the second incoherent, so we remain agnostic. But if something is incoherent, how can we remain agnostic? (So, actually, I think the better way to present the view is as agnosticism between bivalence endorsing views and Field-style rejectionist views, albeit carried out in an intuitionist rather than Kleene-based system. But if that’s the way to work things out, rejectionism is conceptually prior to agnosticism).

So in the end I started with a minimal intro to the many-valued truth tables, a brief pointer in the direction of extensions and interpretations, and then concentrate on the elements of the Field view—the Quinean translate-and-deflate theory of language, the logical revisionism (and instrumentalism about model theory) and the cognitive role that flows from it.

Just as with the epistemicism section, there are famous objections I just didn’t have room for. The whole issue over the methodological ok-ness of revising logic, burdens that entails… nothing will remain of that but a few references.

Vagueness survey paper: section V

REVISIONARY LOGIC: MANY-VALUED SETTINGS

A distinctive feature of supervaluationism was that while it threw out bivalence (“Harry is bald” is either true or false) it preserved the corresponding instance of excluded middle (“Harry is either bald or not bald”). Revising the logic in a more thorough-going way would allow for a coherent picture where we can finally reject the claim “there is a  single hair that makes the difference between bald and non-bald” without falling into paradox.

“Many valued” logics can be characterized by increasing the number of truth-values we work with—perhaps to three, perhaps infinitely many—and offering generalizations of the familiar stories of how logical constants behave to accommodate this tweak. There are all sorts of ways in which this can be developed, and even more choice points in extracting notions of “consequence” out of the mass of relations that then become available.

Here is a sample many-valued logic, for a propositional language with conjunctions, disjunctions and negations. To characterize the logic, we postulate three values (let’s call them, neutrally, “1” “1/2” and “0”). For the propositional case, the idea will be that each atomic sentence will be assigned some one of the truth values; and then the truth values get assigned to complex sentences recursively. Thus, a conjunction will get that truth value which is the minimum of the truth values of its conjuncts; a disjunction will get that truth value which is the maximum of the truth values of its disjuncts; and a negation will get assigned 1 minus the truth value of the claim negated (you can easily check that, ignoring the value ½ altogether, we get back exactly classical truth-tables.)

A many-valued logic (the strong Kleene logic) is defined by looking at the class of those arguments that are “1-preserving”, i.e. such that when all the premises are value 1, the conclusion is value 1 too. It has some distinctive features; e.g. excluded middle “Av~A” is no longer a tautology, since it can be value ½ when A is value ½. “A&~A” is still treated as a logical contradiction, on the other hand (every sentence whatsoever follows from it), since it will never attain value 1, no matter what value A is assigned.

One option at this point is to take this model theory seriously—much as the classicist and supervaluationist do, and hypothesise that natural language has (or is modelled by?) some many-valued interpretation (or set of interpretations?). This view is a major player in the literature (cite cite cite).

For the remainder of this section, I focus on a different framework in which to view the proposal to revise logic. This begins with the rejection of the very idea of an “intended interpretation” of a language, or a semantic treatment of truth. Rather, one treats truth as a “device of disquotation”—perhaps introduced by means of the kind of disquotational principles mentioned earlier (such a device proves useful, argue its fans, in increasing our expressive power—allowing us to endorse or deny such claims as “everything the pope says is true”). The disquotation principles capture all that there is to be said about truth, and the notion doesn’t need any “model theoretic foundation” in an “intended interpretation” to be in good-standing.

In the first instance, such a truth-predicate is “local”—only carving the right distinctions in the very language for which it was introduced (via disquotation). To allow us to speak sensibly of true French sentences (for example), Field (cite) following Quine (cite) analyzes

“la neige est blanc” is true

as:

there’s a good translation of “la neige est blanc” into English, such that the translated version is disquotationally true.

Alongside this disquotationism is a distinctive attitude to logic. On Field’s view, logical consequence does not need to be “analyzed” in model-theoretic terms. Consequence is taken as primitive, and model theory seen as a useful instrument for characterizing the extension of this relation.

Such disquotationism elegantly avoids the sort of worries plaguing the classical, supervaluational, and traditional many-valued approaches – in particular, since there’s no explanatory role for an “intended interpretation”, we simply avoid worries about how such an intended interpretation might be settled on. Moreover, if model-theory is mere instrument, there’s no pressure to say anything about the nature of the “truth values” it uses.

So far, no appeal to revisionary logic has been made. The utility of hypothesizing a non-classical (Kleene-style) logic rather than a classical one comes in explaining the puzzles of vagueness. For Field, when indeterminacy surfaces, we should reject the relevant instance of excluded middle. Thus we should reject “either Harry is bald or he isn’t”—and consequently, also reject the claim that Harry is bald (from which the former follows, in any sensible logic). We then must reject “Harry is bald and the man with one hair less isn’t” (again because something we reject follows from it). So, from our rejection of excluded middle, we derive the core data behind the “little by little” worry—rejection of the horrible conjunctions (N).

So, like one form of supervaluationism, Field sees borderline cases of vague predicates as characterized by forced rejection. No wonder further inquiry into whether Harry is bald seems pointless. Again in parallel, Field accommodates rejection of (N) without accepting (N*)—and this is at least a start on explaining where our perplexity over the sorites comes from. Unlike the supervaluationist, he isn’t committed to the generalizations “there is some bald guy next to a non-bald guy”—the Kleene logic (extended to handle quantifiers) enforces rejection of this claim.

One central concern about this account of vagueness (setting aside very general worries about the disquotational setting) is whether in weakening our logic we have thrown the baby out with the bathwater. Some argue that it is methodologically objectionable to revise logic without overwhelming reason to do so, given the way that classical assumptions are built into successful, progressive science even when vagueness is clearly in play (think of applications of physics, or the classical assumptions in probability and decision theory, for example). This is an important issue: but let’s set it aside for now.

More locally, the logic we’ve looked at so far seems excessively weak in expressive power. It’s not clear, for example, how one should capture platitudes like “if someone is balder than a bald person, that person too is bald” (translating the “if” here as a kind of disjunction or negated conjunction, as is standard in the classical case, we get something entailing instances of excluded middle we want to reject – we do not seem to yet have in the system a suitable material conditional). For another thing, we haven’t yet said anything about how the central notion “it is determinate whether” fits in. It seems to have interesting logical behaviour—for example, the key connection between excluded middle and indeterminacy would be nicely captured if from Av~A one could infer “it is determinate whether A”. Much of Field’s positive project involves extending the basic Kleene logic to accommodate a suitable conditional and determinacy operators, in particular to capture thoroughly “higher-order” kinds of vagueness (borderline cases of borderline cases, and so on).

Ok, part IV of the survey article. This includes the second of three sample theories: and I’ve chosen to talk about what’s often called “the most popular account of vagueness”: supervaluationism.

I’m personally pretty convinced there are (at least) two different *types* of theses/theories that get the label “supervaluationism” (roughly, the formal logic-semantics stuff, and the semantic indecision story about the source of indeterminacy). And even once you have both on board, I reckon there are at least three *very different* ways of understanding what the theory is saying (that’s separate from, but related to, the various subtle issues that Varzi picks out in the recent Mind paper).

But what I want to present is one way of working out this stuff, so I’m keeping fairly close to the Fine-Keefe (with a bit of Lewis) axis that I think is most people’s reference point around here. I try to strip out the more specialist bits of Fine—which comes at a cost, since I don’t mention “penumbral connection” here. But again, I wanted to keep the focus on the basics of the theory, and the application to the central puzzles, so stuff had to come out.

One thing I’m fairly unapologetic about is to present supervaluationism as a theory where truth=supertruth. It seems terminologically bizarre to do otherwise—given that “supervaluations” have their origins in a semantic-logico technique for defining truth, when you have multiple truth-on-i to play with. I’m happy to think that “semantic indecision” views can be paired with a classical semantics, and the semantics for definiteness given in terms of delineations/sharpenings (as indeed, can the epistemicist D-operator). But, as a matter of terminology, I don’t see these as “supervaluational”. “Ambiguity” views like McGee and McLaughlin are a mixed case, but I don’t have room to fit them in. In any case: bivalence failure is such a common thought that I thought it was worth giving it a run for its money, straightforwardly construed, and I do think that it does lead to interesting and distinctive positions on what’s going on in the sorites.

Actually, the stuff about confusing rejection with accepting-negation is something that’s again a bit of a choice-point. I could have talked some about the “confusion hypothesis” way of explaining the badness of the sorites (and work in particular by Fine, Keefe and Greenough on this—and a fabulous paper by Brian Weatherson that he’s never published “Vagueness and pragmatics”).  But when I tried with this it’s a bit tricky to explain, takes quite a bit of thinking about—and I felt the rejection stuff (also a kind of “confusion hypothesis”) was more organically related to the way I was presenting things. I need to figure out some references for this. I’m sure there must be lots of people making the “since it’s untrue, we’re tempted to say ~p” move, which is essentially what’s involved. I’m wondering whether Greg Restall has some explicit discussion of the temptation of the sorites in his paper on denial and multiple conclusions and supervaluations…

One place below where a big chunk of stuff had to come out to get me near the word limit, was material about the inconsistency of the T-scheme and denials of bivalence. It fitted in very naturally, and I’d always planned that stuff to go in (just because it’s such a neat and simple little argument). But it just didn’t fit, and of all the things I was looking at, it seemed the most like a digression. So, sadly, all that remains is “cite cite cite” where I’m going to give the references.

One thing that does put in a brief appearance at the end of the section is the old bugbear: higher order vagueness. I don’t discuss this very much in the piece, which again is a bit weird, but then it’s very hard to state simply what the issue is there (especially as there seems to be at least three different things going by that name in the literature, the relations between them not being very obvious).

Another issue that occurred to me here is whether I should be strict in separating/distinguishing semantics and model theory. I do think the distinction (between set-theoretic interpretations, and axiomatic specifications of those interpretations) is important, and in the first instance what we get from supervaluations is a model theory, not a semantics. But in the end it seemed not to earn its place. Actually: does anyone know of somewhere where a supervaluational *axiomatic semantics* is given (as opposed to supervaluational models?) I’m guessing it’ll look something like: [“p”]=T on d iff At d, p.—i.e. we’ll carry the relativization through the axioms just as we do in the modal case.

The section itself:

Survey paper on vagueness: part IV

REVISIONARY SEMANTICS: SUPERVALUATIONS

A very common thought about borderline cases is that they’re neither true nor false. Given that one can only know what is true, this would explain our inevitable lack of knowledge in borderline cases. It’s often thought to be a rather plausible suggestion in itself.

Classical semantics builds in the principle that each meaningful claim is either true or false (bivalence). So if we’re to pursue the thought that borderline claims are truth value gaps, we must revise our semantic framework to some extent. Indeed, we can know in advance that any semantic theory with truth-value gaps will diverge from classical semantics even on some of the most intuitively plausible (platitudinous seeming) consequences: for it can be shown under very weak assumptions that truth value gaps are incompatible with accepting disquotational principles such as: “Harry is bald” is true if and only if Harry is bald (see cite cite cite).

How will the alternation of the classical framework go? One suggestion goes under the heading “supervaluationism” (though as we’ll see, the term is somewhat ambiguous).

As an account of the nature of vagueness, supervaluationism is a view on which borderlineness arises from what we might call “semantic indecision”. Think of the sort of things that might fix the meanings of words: conventions to apply the word “bald” to clear cases; conventions to apply “not bald” to clear non-cases; various conventions of a more complex sort—for example, that anyone with less hair than a bald person should count as bald. The idea is that when we list these and other principles constraining the correct interpretation of language, we’ll be able to narrow down the space of acceptable (and entirely classical) interpretations of English down a lot—but not to the single intended interpretation hypothesized by classical semantics. At best, what we’ll get is a cluster of candidates. Let’s call these the sharpenings for English. Each will assign to each vague predicate a sharp boundary. But very plausibly the location of such a boundary is something the different sharpenings will disagree about. A sentence is indeterminate (and if it involves a vague predicate, is a borderline case) just in case there’s a sharpening on which it comes out true, and another on which it comes out false.

As an account of the semantics of vague language, the core of the supervaluationist proposal is a generalization of the idea found in classical semantics, that for something to be true is for it to be true at the intended interpretation. Supervaluationism offers a replacement: it works with a set of “co-intended interpretations”, and says that for a sentence to be true, it must be true at all the co-intended interpretations (this is sometimes called “supertruth”). This dovetails nicely with the semantic indecision picture, since we can take the “co-intended interpretations” to be what we called above the sharpenings—and hence when a sentence is indeterminate (true on one sharpening and false on another) neither it nor its negation will be true: and hence we have a truth value gap. (The core proposal for defining truth finds application in settings where “semantic indecision” idea seems inappropriate: see for example Thomason’s treatment of the semantics of branching time in his (cite)).

The slight tweak to the classical picture leaves a lot unchanged. Consider the tautologies of classical logic, for example. Every classical interpretation will make them true; and so each sharpening is guaranteed to make them true. Any classical tautology will be supertrue, therefore. So – at least at this level – classical logic is retained (It’s a matter of dispute whether more subtle departures from classical logic are involved, and whether this matters: see (cite cite cite)).

So long as (super)truth is a constraint on knowledge, supervaluationists can explain why we can’t know whether borderline bald Harry is bald. On some developments of the position, they can go interestingly beyond this explanation of ignorance. One might argue that insofar as one should only invest credence in a claim to the extent one believes it true, obvious truth-value-gaps are cases where we should utterly reject (invest no credence in) both the claim and its negation. This goes beyond mere lack of knowledge, for it means the information that such-and-such is borderline gives us a direct fix on what our degree of belief should be in such-and-such (by contrast, on the epistemicist picture, though we can’t gain knowledge, we’ve as yet no reason to think that inquiry could raise or lower the probability we assign to Harry being bald, making investigation of the point perfectly sensible, despite initial appearances).

What about the sorites? Every sharpening draws a line between bald and non-bald, so “there is a single hair that makes the difference between baldness and non-baldness” will be supertrue. However, no individual conjunction of form (N) will be true—many of them will instead be truth value gaps, true on some sharpenings and false on others (this highlights one of the distinctive (disturbing?) features of supervaluationist—the ability of disjunctions and existential generalizations to be true, even if no disjunct or instance is.) As truth-value gaps, instances of (N*) will also fail to be true, so some of the needed premises for the sorites paradox are not granted.

(There is one thing that some supervaluationists can point to in attempt to explain the appeal of the paradoxical premises. Suppose that—as I think is plausible—we take as the primary data in the sorites the horribleness of the conjunctions (N). These are untrue, and so (for one kind of supervaluationist) should be utterly rejected. It’s tempting, though mistaken, to try to express that rejection by accepting a negated form of the same claim—that is the move that takes us from the rejection of each of (N) to the acceptance of each of (N*). This temptation is one possible source of the “seductiveness” of sorites reasoning.)

Two points to bear in mind about supervaluationism. First, the supervaluationist endorses the claim that “there is a cut-off” — a pair of men differing by only one hair, with the first bald and the second not. Insofar as one considered that first-order claim to be what was most incredible about (say) epistemicism, one won’t feel much advance has been made. The supervaluationist must try to persuade you that once one understands the sense in which “there’s no fact of the matter” where that cut-off is, the incredulity will dissipate.  Second, many want to press the charge that the supervaluationist makes no progress over the classicist, for reasons of “higher order vagueness”. The thought is the task of explaining how a set of sharpenings gets selected by the meaning fixing facts is no easier or harder than explaining how a single classical interpretation gets picked out. However, (a) the supervaluationist can reasonably argue that if she spells out the notion of “sharpening” in vague terms, she will regard the boundary between the sharpenings and non-sharpenings as vague (see Keefe (cite)); (b) even if both epistemicist and supervaluationist were both in some sense “committed to sharp boundaries”, the account they give of the nature of vagueness is vastly different, and we can evaluate their positive claim on its own merits.

This is the third section of my first draft of a survey paper on vagueness, which I’m distributing in the hope of getting feedback, from the picky to the substantive!

In the first two sections, I introduced some puzzles, and said some general methodology things about accounting for vague language—in effect, some of the issues that come up in giving a theory in a vague metalanguage. The next three sections are the three sample accounts I look at. The first, flowing naturally on from “textbook semantic theories”, is the least revisionary: semantics in a classical setting.

Now, if I lots of time, I’d talk about Delia Graff Fara’s contextualism, the “sneaky classicism” of people like McGee and McLaughlin (and Dave Barnett, and Cian Dorr, and Elizabeth Barnes [joint with me in one place!]). But there’s only so much I can fit in, and Williamson’s epistemicism seems the natural representative theory here.

Then there’s the issue of how to present it. I’m always uncomfortable when people use “epistemicism” as synonym for just the straightforward reading of classical semantics—sharp cut-offs and so on (somebody suggested the name “sharpism” for that view–I can’t remember who though). The point about epistemicism—why people finding it interesting—is surely not that bit of it. It’s that it seems to predict where others retrodict; and it seems principled where others seem ad hoc. Williamson takes formulations of parts of the basic toolkit of epistemology—safety principles; and uses these to explain borderineness (and ultimately the higher-order structure of vagueness). That’s what’s so super-cool about it.

I’m a bit worried in the current version I’ve downplayed the sharpist element so much. After all, that’s where a lot of the discussion has gone on. In part, that betrays my frustration with the debate—there are some fun little arguments around the details, but on the big issue I don’t see that much progress has been made. It feels to me like we’ve got a bit of  a standoff. At minimum I’m going to have to add a bunch of references to this stuff, but I wonder what people think about the balance as it is here.

I have indulged a little bit in raising one of the features that always puzzles me about epistemicism: I see that Williamson has an elegant explanation about why we can’t ever identify a cut-off. But I just don’t see what the story is about why we find the existential itself so awful. The analogy to lottery cases seems helpful here. Anyway, on with the section:

Vagueness Survey Paper, Part III.

VAGUENESS IN A CLASSICAL SETTING: EPISTEMICISM

One way to try to explain the puzzles of vagueness look to resources outwith the philosophy of language. This is the direction pursued by epistemicists such as Timothy Williamson.

One distinctive feature of the epistemicist package is retaining classical logic and semantics. It’s a big advantage of this view that we can keep textbook semantic clauses described earlier, as well as seemingly obvious truths such as that “Harry is bald” is true iff Harry is bald (revisionary semantic theorists have great trouble saving this apparently platudinous claim). Another part of the package is a robust face-value reading of what’s involved in doing this. There really is a specific set that is the extension of “bald”—a particular cut-off in the sorites series for bald, and so on (some one of the horrible conjunctions given earlier is just true). Some other theorists say these things but try to sweeten the pill—to say that admitting all this is compatible with saying that in a strong sense there’s no fact of the matter where this cut-off is (see McGee McLaughlin; Barnett; Dorr; Barnes). Williamson takes the medicine straight: incredible as it might sound, our words really do carve the world in a sharp, non-fuzzy way.

The hard-nosed endorsement of classical logic and semantics at a face-value reading is just scene-setting: the real task is to explain the puzzles that vagueness poses. If the attempt to make sense of “no fact of the matter” rhetoric is given up, what else can we appeal to?

As the name suggests, Williamson and his ilk appeal to epistemology to defuse the puzzle. Let us consider borderlineness first. Start again from the idea that we are ignorant of whether Harry is bald, when he is a borderline case. The puzzle was to explain why this was so, and why the unknowability was of such a strong and ineliminable sort.

Williamson’s proposal makes use of a general constraint on knowledge: the idea that in order to know that p, it cannot be a matter of luck that one’s belief that p is true. Williamson articulates this as the following “safety principle”:

For “S knows that p” to be true (in such situation s), “p” must be true in any marginally different situation s* (where one forms the same beliefs using the same methods) in which “S believes p” is true.

The idea is that the situations s* represent “easy possibilities”: falsity at an easy possibility makes a true belief too lucky to count as knowledge.

This first element of Williamson’s view is independently motivated epistemology. The second element is that the extensions of vague predicates, though sharp, are unstable. They depend on exact details of the patterns of use of vague predicates, and small shifts in the latter can induce small shifts in the (sharp) boundaries of vague predicates.

Given these two, we can explain our ignorance in borderline cases. A borderline case of “bald”  is just one where the boundary of “bald” is close enough that a marginally different pattern of usage could induce a switch from (say) Harry being a member of the extension of “bald” to not being in that extension. If that’s the case, then even if one truly believed that Harry was bald, there will be an easy possibility where one forms the same beliefs for the same reasons, but that sentence is false. Applying the safety principle,  the belief can’t count as knowledge.

Given that the source of ignorance resides in the sharp but unstable boundaries of vague predicates, one can see why gathering information about hair-distributions won’t overcome the relevant obstacle to knowledge. This is why the ignorance in borderline cases seems ineliminable.

What about the sorites? Williamson, of course, will say that one of the premises if false—there is a sharp boundary, we simply can’t know what that is. It’s unclear whether this is enough to “solve” the sorites paradox however. As well as knowing what premise to reject, we’d like to know why we found the case paradoxical in the first place. Why do we find the idea of a sharp cut off so incredible (especially since there’s a very simple, valid argument from obvious premises to this effect available)? Williamson can give an account of why we’d never feel able to accept any one of the individual conjunctions (Man n is bald and man n+1 is not). But that doesn’t explain why we’re uneasy (to say the least) with the thought that some such conjunction is true—i.e. that there is a sharp cut-off. I’ll never know in advance which ticket will win a lottery; but I’m entirely comfortable with the thought that one will win. Why don’t we feel the same about the sorites?