the boundaries of language

In the most recent issue of Analysis, Peter Milne has argued that an omniscient being would have to be a dialetheist. The argument proceeds as follows. The assumption that there is an omniscient being entails a contradiction. But rather than merely treating this as a reductio of our assumption, we have the option of holding that the contradiction is true. Since it's true, our omniscient being must know it, so if we want to hold on to an omniscient being, we'll have to recognize that it must know something of the form P & ~P. The argument that the assumption that there is an omniscient being leads to a contradiction runs as follows. Consider the following sentence:

S: No omniscient being knows that which the sentence S expresses.

S is clearly self-referential, but it doesn't trip itself up; it's simply true if there aren't any omniscient beings. Now, suppose that ~S, i.e.,

1. Some omniscient being knows that which S expresses.

Then, given what S expresses,

2. Some omniscient being knows that no omniscient being knows that which the sentence S expresses.

But then by the factivity of knowledge,

3. No omniscient being knows that which the sentence S expresses.

But that's just S. So S is true.

Now suppose that X is an omniscient being. Then X knows all, including what we've just shown, namely (3). So,

4. X knows that no omniscient being knows that which the sentence S expresses,

which, given what S expresses, yields,

5. X knows that which S expresses.

But X is omniscient. So,

6. Some omniscient being knows that which S expresses.

But (6) is just ~S. So, on the assumption that an omniscient being exists, S & ~S.

Milne leaves the reader to draw their own conclusions from his proof. Jesper Kallestrup has picked up the baton, and adapted Milne's proof to show that the knowability principle (KP) entails a contradiction. Again, rather than treating this as a reductio, we might accept that the contradiction is true. A final application of (KP) yields that this contradiction is knowable. Anti-realists are committed to dialetheism.

It can hardly have escaped the attention of readers of this blog that I'm more sympathetic than most to anti-realism, and (unlike Jon Cogburn), I'm not in the market for the conclusion that it entails committment to dialetheism. There's a lot that might be said in order to dodge this conclusion. Jesper notes that so-called restriction strategies that have been offered by Dummett and Tennant in response to the knowability paradox also work here, but he points out they are associated with a whole bunch of problems. Crispin Wright's endorsement of KP as a local thesis also blocks Jesper's reasoning, which requires an assumption that KP holds globally.

Here I just want to suggest that there's a further restriction to KP the anti-realist can make, one that's independently motivated and yet which blocks the proof. Let's begin by examining Jesper's argument that KP entails a contradiction.

Consider the following variant of S:

S*: Nobody possibly knows that which the sentence S expresses.

Again, this is self-referential but does not simply implode. But now suppose that we adopt the following:

KP: P expresses a truth only if it is possible that somebody knows that P expresses a truth.

Actuality Assumption (AA): It is possible that somebody knows that P expresses a truth only if there is some actual person who, perhaps which finite extensions to her capacities, possibly knows that P expresses a truth.

Factivity Assumption (FA): It is possible that somebody knows that P expresses a truth only if P expresses a truth.

AA and FA are usually taken as being part of a package with KP (though, always the exception, Jon Cogburn seems to reject FA here, p238).

Now, suppose the following:

1. That which S* expresses is true.

Then by KP we have,

2. It is possible that somebody knows that which S* expresses.

By AA, we get,

3. Somebody possibly knows that which S* expresses.

But (3) is simply ~S*.

Now suppose,

4. That which S* expresses is false.

Then, by the contrapositive of FA, we arrive at,

5. It is not possible that somebody knows that which S* expresses.

Hence by AA,

6. Nobody possibly knows that which S* expresses.

But this is just S*! So, given KP, AA, and FA (the standard anti-realist package), we get ((S* -> ~S*) & (~S* -> S*)), which intuitionistically yields (~S* & ~~S*), and classically (~S* & S*). Hence the standard package entails a contradiction.

I think the main issue here is how to understand the constraint on KP that FA imposes. We don't want KP to be trivially true or false; surely every party in this debate wants to be affirming a doctrine with some substance. But there's some trickiness here: adopting FA is meant to block trivializing KP by allowing it to be satisfied in cases where P is false in @ and yet it is possible to know that P since there is a world w such that w is not @, and P expresses a truth and thus something knowable in w. The motivating idea behind FA is that KP is supposed to be constrained by the distribution of truth-values at the actual world, since otherwise even falsehoods are knowable - not the interpretation of knowability the anti-realist wants.

Unfortunately, adopting FA threatens to make KP come out trivially false. KP is supposed to be a necessary truth by the anti-realist's lights. So it's supposed to be impossible that P express a truth and yet it be impossible for some suitably pimped-out subject to come to know that P expresses a truth. But now suppose that it happens not to be known that Q expresses a truth, even though it in fact does. Given the requirement that we've attempted to capture in FA, that KP be sensitive to the actual distribution of truth-values, the question is whether it's possible that someone might come to know that Q expresses a truth, given the actual distribution of truth-values. But this obviously isn't possible, since the actual distribution of truth-values determines that nobody knows that Q expresses a truth. So, contrary to KP taken as a necessary truth, it is possible that Q express a truth and yet it be impossible that a suitably pimped-out subject to come to know that Q expresses a truth. That's a possibility just because of the simple fact that sometimes truths aren't in fact known by any subject in the actual world. (This line of thought is Sven Rosenkranz's, from p354 of his 2003).

So the anti-realist has a balancing act to do. (S)he wants to hold on to the thought that P expresses a truth if and only if it is possible that somebody knows that P expresses a truth. The right-to-left direction of this is AA, and it constrains the notion of knowability so that what is knowable is determined by the actual distribution of truth-values. But if we allow that what is knowable is determined by the actual distribution of truth-values over sentences expressing our current state of ignorance (or knowledge), we're in trouble, for then KP is clearly false. That was the upshot of Sven's argument. A principled way of avoiding this difficulty is needed, and Sven offers a proposal in his paper.

This has all been pretty torturous, but I've still to show how this all might bear on Kallestrup's proof. We need to look again at the move from line (4) to line (5):

4. That which S* expresses is false.

Then, by the contrapositive of FA, we arrive at,

5. It is not possible that somebody knows that which S* expresses.

But S* is clearly a sentence expressing something about our current state of ignorance, and we resolved, in the light of Sven's point, to understand KP and FA in such a way that the truth-values of such sentences did not go towards determining what is knowable (for example, Sven's suggestion may be put (I hope!) as follows: KP and FA only hold when the truth-value of P isn't alterable by any (perhaps idealized) subject). My suggestion, then, is that once we take Sven's point seriously, and change our understanding of KP and FA accordingly, the move from (4) to (5) will be invalidated. So the upshot, if I'm right, is that an already necessary restriction on the standard anti-realist package serves to invalid Kallestrup's proof of a contradiction.

Homework: What's the bearing of this, if any, on the knowability paradox?

References:

Kallestrup, J. 2007. If omniscient beings are dialetheists, then so are anti-realists. Analysis 67: 252-4.
Milne, P. 2007. Omniscient beings are dialetheists. Analysis 67: 250-1.
Rosenkranz, S. 2003. Realism and Understanding. Erkenntnis 58: 353-78.

Labels: Anti-Realism, Dialetheism, Dummett, Knowability, Philosophy of Logic, Religion

I haven't spent as long as I'd like on the details here, but I don't really have the time just now, so I thought I'd post and let the more logically competent point out in what way I'm misguided (on this, or some other issue of their choosing).

A number of philosophers (Brueckner, Kvanvig, etc.) have argued against Crispin Wright's suggestion that superassertibility is a truth predicate. I'm wondering if Church-Fitch reasoning doesn't give us a very quick route to the conclusion that they're right to challenge Crispin on this point.

A sentence 'P' is superassertible (SA 'P') iff it is warrantly assertible (WA 'P') and that warrant i. will 'survive arbitrarily close and extensive investigation' and ii. is not defeasible (that is, there is no extension, i', to our current information state i such that i' fails to warrant the assertion of 'P').

To set up the problem, we need the following three principles:

1. SA 'P' iff P
2. If SA 'P&Q', then SA 'P' and SA 'Q'
3. If WA 'P' and WA 'Q', then WA 'P&Q'

These are all pretty straightforward. (1) is just the disquotation principle for SA, which Crispin thinks has to hold if SA is genuinely to count as a truth-predicate. (2) says that SA is closed under &-elimination, which follows from the closure of SA under logical implication (which again seems pretty compelling if SA is a genuine truth-predicate). Lastly, (3) says that WA is closed under &-introduction, which again seems compelling (at least when we're restricting attention to sentences with only two conjuncts; perhaps the preface and lottery paradoxes should make us doubt this principle in its full generality).

Now, the objection I have in mind starts from the observation that SA is strictly logically stronger than WA; that is, SA 'P' -> WA 'P' but not vice versa. This doesn't imply, but seems to leave open the possibility of a sentence 'X' and state of information such that:

i. WA 'X' & WA '~SA 'X''

By (3), it immediately follows that,

ii. WA 'X & ~SA 'X''

But we have,

iii. ~WA 'X & ~SA 'X''

Contradiction.

The sub-argument for iii. is just standard Church-Fitch reasoning, with (1) in place of factivity and the knowability principle. Suppose,

iv. X & ~SA 'X'

Then by (1),

v. SA 'X & ~SA 'X''

Whence by (2),

vi. SA 'X' and SA '~SA 'X''

But then, by (1) again,

vii. SA 'X' and ~SA 'X'

So we can conclude,

viii. ~(X & ~SA 'X'),

which presumably suffices for iii.

I'm not spotting any gapping holes in this, though I'm sure others will be quick to point them out. (1) isn't a legitimate supposition if we endorse the following variant of the reflection principle for WA:

4. If WA '~SA 'P'', then ~WA 'P'

But I'm not seeing any good reason to adopt (4). My supposition in i. does seem to be a legitimate possibility; it might be unusual to be in a state of information which both warrants the assertion of a particular sentence and warrants the assertion that some expansion of that information state fails to warrant that sentence, but (in empirical discourse) the idea does not seem incoherent. Yet if I haven't messed up, it is incoherent if superassertibility is a truth-predicate (given the closure of WA under &-introduction).

Labels: Anti-Realism, Crispin Wright, Philosophy of Logic

I was struck by this sentence in the SEP entry on Mill:

'This principle that must implies ought is the converse of the well-known Kantian principle that ought implies can.'

My first thought was 'No it isn't!'. Clearly it's the converse of ought implies must, which is pretty preposterous on the relevant reading: ought implies cannot but; ought implies unable not to (i.e. you ought to help the needy only if you cannot but help the needy, or you ought to help the needy only if you are unable not to help the needy). Not even Kant would want his fingerprints on this train-wreck of a principle.

Is there a charitable interpretation of the claim in the text that I'm missing?

Labels: Philosophy of Logic

This one's for Roy Cook:



(www.xkcd.com)

Labels: Philosophy of Logic, xkcd

I'm currently reading R. Jay Wallace's 'Normativity, Commitment, and Instrumental Reason', and he discusses some issues which reminded me of some tricky issues Bryan Frances raised a while ago in this interesting post. In a discussion of disanalogies between practical and theoretical reason, Wallace raises the question of whether there can be 'strong akrasia of belief' (12); that is, cases in which 'one judges that a given conclusion--say, p--is true, and yet one consciously and without self-deception believes that not-p'. As Wallace notes, Scanlon argues in What We Owe to Each Other that this is a pretty common phenomena, but he's not convinced by Scanlon's purported examples. In fact, Wallace takes strong akrasia of belief to be ruled out (13).

Whether or not akrasia of belief is commonplace, given the way Wallace has described the phenomena, it looks like he has to say that Graham Priest doesn't have the range of beliefs concerning the Liar sentence (etc.) that he claims to have. I don't how on Earth we might begin to settle the question of what beliefs to ascribe to dialetheists, but let me make a couple of observations. Firstly, it's hard not to feel Wallace hasn't quite managed to put his finger on the phenomena in question; intuitively, even if we do attribute clear-eyed inconsistent beliefs to Graham, this isn't a case of strong akrasia of belief (update: what seems to be missing is something like the belief that these are incompatible attitudes). Yet it counts as such given the above characterization. So let us assume for the moment that akrasia of belief is a somewhat different phenomena to mere clear-eyed judging that p is true while believing not-p. My second observation is this; even under this assumption, it seems like it's going to be very hard to find arguments which rule out the possibility of akrasia of belief but which leave open the possibility that one might while clear-eyed judge that p is true while believing not-p. That is, although these seems like different phenomena, one has to wonder if they aren't sufficiently similar to make it extremely difficult to offer considerations which discriminately tell against the possibility of one but not the other.

If that hunch is correct, the upshot is that denying the possibility of akrasia of belief might be very difficult indeed, since it might involve getting mired in a debate concerning what beliefs we can/should attribute a self-confessed dialetheist. At any rate, the going will have to be slower than Wallace seems to envisage.

Labels: Dialetheism, Epistemology, Philosophy of Logic

In Vagueness and Contradiction, Roy Sorensen discusses the No-No paradox; abstracting away from details we'll review shortly, this is the puzzle that stands to the truth-teller sentence as the Yes-No paradox stands to the liar. So the (strengthened) Liar sentence is:

S1: S1 is not true,

while the following open-pair generates the Yes-No paradox:

S2: S3 is true
S3: S2 is not true.

What groups these together as a family is that there is no consistent assignment of truth-values to these sentences. In contrast there are consistent assignments of truth-values to the truth-teller sentence:

S4: S4 is true,

and to the pair of sentences that give rise to the No-No paradox:

S5: S6 is not true
S6: S5 is not true.

S4 can consistently receive T or F, we're just at a loss to assign one rather than the other. We can't assign S5 and S6 TT or FF, but TF or FT are fine. Problem is, it seems like symmetry considerations seem to compel us to assign matching truth-values to the two sentences. (We can increase the symmetry between the two sentences by presenting the paradox the way Russell does; we have a sheet of white card, with just "The sentence on the other side of this card is not true" printed on each side. The idea is that there's no grounds on which to conjecture that there has been a context-shift or anything that would justify assigning different true-values to the members of the pair.) So although consistency allows TF or FT, symmetry considerations seem to rule those options out.

More recently, in 'A Definite No-No' (In Beall's Liars and Heaps, 2003) Sorensen presents the following pair of sentences:

S7: S8 is not definitely true
S8: S7 is not definitely true.

(Sorensen's presentation attempts to maximize their 'equal-billing' a little more than mine, again to increase the force of symmetry considerations, but that won't matter here; anyone who feels I've loaded the dice against Sorensen can simply look at his presentation on p225 of his paper.)

The assignment FF (or more carefully, assigning not-true to both) looks like it's immediately ruled out by the logic of definiteness; as Sorensen points out, the untruth of one of the pair implies the definite truth of the other. What about the other options? Sorensen writes:

'. . .the symmetry of the two sentences is destabilizing. We have the following feelings: As twins, [S7] and [S8] should have matching truth-values (or should be equally bereft of truth-values). Since we have already ruled out the possibility that both are untruths, we conclude that [S7] and [S8] are both true.

The truth of the pair is no problem in itself. But there is a ironical difficulty in us possessing such a tidy argument [S7] and [S8] are true. The discovery of a proof for a proposition is generally regarded as a sufficient condition for that proposition being definitely true. Since [S7] and [S8] each deny that the other is definitely true, they would each be false if proven true. Strangely, any cogent argument to the effect that both sentences are true must backfire.' (226)

I just don't find this all that plausible. The crucial move in all of this is the equation of 'cogent argument' with 'proof'. The move is particularly clear in the second and third sentences of the quote; Sorensen passes without a ripple from our possession of a 'tidy argument' to the possession of a proof. Now, this might just be written off as a terminological gripe if Sorensen's argument that the only assignment we can make to S7 and S8 is TT really did constitute a proof. But surely that issue turns on the status of the symmetry principle. After all, it's the symmetry principle that rules out the true-untrue and untrue-true assignments.

So what is the status of the symmetry principle? All Sorensen says is that we have a 'feeling' it should hold. I agree; the principle is pretty intuitively compelling, and likewise I find the argument it underwrites that TT is the only available assignment is pretty compelling. But does such an argument have the credentials to constitute a proof, so that it's uncontroversial that possession of this argument suffices to bestow definiteness on its conclusion? I don't see that Sorensen has motivated that at all. So Sorensen needs to provide an argument that the symmetry principle has some more privileged status, or that there's some intimate but hitherto unrecognized link between intuitively cogent philosophical argument and definiteness. Otherwise all he's done is given us really good reason to think that both S7 and S8 are true.

Labels: Philosophy of Logic

I'm currently engaged in what I think is a very interesting discussion with Colin Caret and others over at What is it like to be a Blog? The discussion started out concerning what the grounds might be to hold that someone who doesn't practice what they preach lacks moral authority, but we've spilled over into a new thread discussing issues that come up when an agent has inconsistent beliefs, but doesn't recognise that this is so. Does this familiar kind of situation motivate a move to paraconsistent logic, or even full blown rational dialetheism, or can the classicist (or less radical logical revisionist) maintain that such a move is unmotivated here? I'm trying to argue for the latter position.

See Colin's posts 'Hypocrisy and Moral Authority' and 'Paraconsistency and Dialetheism'.

(I get rightly criticised here sometimes for not explaining terminology. So here I'm taking paraconsistent logics to be those that lack the explosion rule, A, ~A |- B, and rational dialetheism to be the view that some contradictions are true and that it can be rational to believe some contradictions. The rational dialetheist will adopt a paraconsistent logic, since it clearly isn't rational to have trivial beliefs, but one can adopt a paraconsistent logic without being a dialetheist.)

Labels: Links, Philosophy of Logic

When Jason Stanley was here for the UT Grad Conference in April, I took the chance to talk to him about his draft 'Philosophy of Language in the Twentieth Century'. Now, I haven't managed to finish the paper, but it's an impressive survey of a stupidly large field, and Stanley manages to include nice accounts of the contributions made by Tarski and Carnap and other figures who we all acknowledge as a vital part of the story of our subject, but who often get somewhat glossed over in these kinds of overview pieces.

However, one figure who is really noticeably absent is the later Wittgenstein. Moreover, Stanley's assessment of the contribution of the Tractatus to 20th Century philosophy is surprisingly negative:

'The Tractatus impeded progress in philosophy, because it led philosophers (in particular the Logical Positivists) to expend their energies in the pursuit of developing and honing a criterion of meaningfulness, and using the criterion to argue that traditional philosophical theses failed to satisfy it, and were hence meaningless [...]. ' (p10)

Now, Stanley doesn't deny that the Tractatus did some good; in particular, he notes that the connections the early Wittgenstein saw between modality and meaning turned out to yield insights when studied more systematically by later philosophers (p10). But my own assessment of the Tractatus is overall much more upbeat, and I wanted to say a little bit about why.

Firstly, and most germane to Stanley's topic, the Tractatus contains one of the earliest and clearest statements of the thought that the meaning of a sentence is its truth-conditions:

4.024 'To understand a proposition means to know what is the case, if it is true.

(One can therefore understand it without knowing whether it is true or not.)

One understands it if one understands its constituent parts.'

Stanley takes the idea that the proper form a theory of meaning should be truth-conditional to have been 'extraordinarily fruitful, perhaps the most fruitful insight in the long history of the study of meaning' (p18). As David Wiggins presents the history of that insight (in 'Meaning and Truth Conditions: From Frege's Grand Design to Davidson's' in the Hale and Wright Blackwell Companion), the Tractatus marks the first important development of the idea after Frege. From this perspective, the Tractatus should be a integral part of Stanley's story.

I also have a more positive view of the saying/showing distinction. It would be hard not to agree that it provided inspiration for some philosophical wild-goose chases (though I'm not sure how to assess the claim that such failed projects were a symptom of philosophical progress being impeded). On the other hand, I think the distinction led to some geniune insights in maths and logic. Wittgenstein originally introduced the distinction to offer a response to Lewis Carroll-style worries about the system of logic in Principia Mathematica. Those concerns are evident on the first page of the Notes on Logic (1913), where Wittgenstein writes that '[d]eductions only proceed according to the laws of deduction but these laws cannot justify the deduction.' (Notebooks 1914-18: p108). We find the saying/showing distinction first unveiled as the solution to this problem in the opening sentence of the Notes dictated to Moore in Norway in 1914:

'Logical so-called propositions shew [the] logical properties of language and therefore [the] Universe, but say nothing.' (Notebooks: p108).

That one can infer B from A and A -> B is not to be justified by appeal to a further rule of inference. Rather, anyone who understands the symbols involved can see that B follows from these premises. This difficult thought received its mature statement in 5.132 of the Tractatus:

'If p follows from q, I can conclude from q to p; infer p from q.
The method of inference is to be understood from the two propositions alone.
Only they can justify the inference.
Laws of inference, which - as in Frege and Russell - are to justify the conclusions, are senseless and would be superfluous.'

Tricky stuff. But the passage forms part of a pretty deep critique of the universalist conception of logic coming from Frege and Russell. The issue here is not just how inference is to be justified, though that's obviously a central and important part of Wittgenstein's concerns. In the background here is the issue of how the logical propositions are to be demarcated. For the universalist the mark of the logical is unrestricted validity and their self-evidence; from his earliest recorded philosophical remarks it is clear that Wittgenstein is starkly opposed to universalism. Logic 'must turn out to be a totally different kind than any other science' (Letter to Russell 1912, Notebooks: p120).

I should get to the point. The saying/showing distinction is the centrepiece of Wittgenstein's critique of the universalist conception of logic, and this critique had a number of really interesting and important consequences. For example, it provided grounds on which Wittgenstein could check Russell's tendency to claim that the axioms he needed to adopt to try to get his logicism to fly were logical; a crucial moment in the history of set theory, and its demarcation from logic.

Of course, as I noted above Stanley doesn't claim that the Tractatus didn't have positive influence. But here I wanted to say a little about why I disagree with his overall assessment; the Tractatus certainly has its flaws, particularly stylistic flaws, but ultimately I think it was an impressive, at times poignant, and overall beneficial contribution to 20th Century Philosophy. In any case, I strongly recommend reading Stanley's paper; I've learned a lot from it, and from thinking through these points of disagreement.

Labels: Jason Stanley, Philosophy of Language, Philosophy of Logic, Wittgenstein

(Cross-posted at Arche)

In a talk I've been giving various places recently, I've argued that Delia Graff's theory of vagueness, like other theories which hold onto bivalence, runs into trouble with the forced-march Sorites. Here's the worry in brief. Graff explains our ignorance of where the boundary in a given series lies with the following similarity constraint:

'..if two things are saliently similar [in the relevant respect], then it cannot be that one is in the extension of a vague predicate, or in its anti-extension, while the other is not.' (2000: 57)

The idea is that whenever one is evaluting a pair of items in a suitably constructed Sorites series, both will be saliently similar, and so either both are in the extension of the predicate in question or both are in the anti-extension; the boundary is never where we are looking for it. This also gives an explanation of why we are liable to find Sorites reasoning compelling despite its invalidity (something critics have complained is entirely missing from Williamson-brand Epistemicism, though it's a central concern in Sorensen's Vagueness and Contradiction).

But now imagine a subject shown the items in a Sorites series in order, and asked of each whether it is in the extension of the relevant predicate or the anti-extension. Given the way the series has been constructed, she can neither competently judge all of the items to be in the extension, nor all of the items to be in the anti-extension. So at some point she must 'jump'; that is, judge some item n to be in the extension, yet judge item n + 1 (differing only marginally in the relevant respects from n) to be in the anti-extension. But by the similarity constraint, the boundary in known not to lie between n and n + 1, so how can her judgements be viewed as competent?

As Derek Ball, Geoff Georgi and Crispin Wright all rightly pointed out to me in various discussions, Graff has an answer to that question ready. It's essentially that of Diana Raffman in 'Vagueness without Paradox' (1994); when the subject judges n to be in the extension, n + 1 is too, but when the subject jumps, the boundary shifts so that now both n and n + 1 are in the anti-extension. So the subject's judgements can be seen as competent, yet the similarity constraint is never violated. As Raffman writes (1994: 57):

'There are shifts - events that occur - but these are best viewed as Gestalt like changes of “perspective” or “anchor,” not as boundary crossings - at least not if by ‘boundary’ you mean something that installs a simultaneous and/or fixed category difference between adjacent patches.'

It's clear that my original objection fails to engage with this kind of response, but I've been wondering how it's actually meant to work, given Graff's committment to bivalence - I've reached the conclusion that it doesn't. In presenting the forced-march above, I assumed that the subject only had two available responses. But we should be able to drop this assumption and allow the subject to respond as she likes (see for example Raffman 1994: 45-6). Suppose the subject judges n to be in the extension; then n + 1 is also in the extension. Now suppose when confronted with n + 1, the subjects responds 'I don't know', or 'There's no fact of the matter', or 'It's neither in the extension nor the anti-extension'. How does a shift in the location of the boundary such that both items are now in the anti-extension help to ensure that the subject says something competent? The problem is this; the boundary is supposed to shift so as to accomodate the speakers judgements - that's how we can see the subject who jumps in the face of the forced-march as doing something competent - but while there are a number of judgements a subject might return, the endorser of bivalence only recognises binary options for accomodating them.

(In a recent Analysis note ('How to understand contextualism about vagueness', 2005) Raffman explicitly considers what would happen on her earlier view when a subject switches from judging items to lie in the extension of the relevant predicate to judging them to be borderline. But the issues I raise above don't get into sight, simply because in a parenthetical remark she assumes '[f]or the sake of argument only' that we're working in a three-valued logic (246). With the assumption in place the subject's judgements can be accomodated, but that doesn't help us see how the trick is to be pulled whilst retaining bivalence.)

Labels: Philosophy of Language, Philosophy of Logic, Vagueness

It seems appropriate to open this blog with a post on boundarylessness and language. In 'Concepts Without Boundaries', Mark Sainsbury argues against the common (if not ubiquitous) assumption that vague concepts classify by making set-theoretic divisions. The epistemicist, for example, holds that there is a particular number of grains of sand that marks the boundary between the heaps and the non-heaps. Rival pictures of the semantics of vague concepts try to avoid this committment by positing more divisions; so supervaluationists hold that vague concepts divide objects into an extension, an anti-extension, and those that have borderline status, while supporters of many-valued logics think that they parition the domain into even more sets.

Sainsbury thinks that this whole approach is radically misguided. There are borderline cases of 'red', i.e. things that are neither red nor not red. But sets are sharp, so there is no set of red things. Positing a set of objects that are borderline only helps if we ignore the phenomenon of higher-order vagueness; once we appreciate that there are borderline cases of 'borderline red', we can see that the problem just recurs. Positing yet more boundaries, and so partioning the domain even more finely, is just to repeat the original mistake a bunch more times. Vague concepts are boundaryless.

It's worth noting that the paper was delivered and published several years before people really started taking epistemicism seriously, and the argument that there is no set of red things looks undermined to the extent that characterising vagueness as an epistemic phenomenon is a viable option. More generally, Roy Cook has argued at length that Sainsbury's attack on many-valued approaches rests on bad philosophy of logic ('Vagueness and Mathematical Precision', Mind 2002). To formalise a region of natural langauge is to build a mathematical model, and if our model introduces sharpness that isn't present in the concepts we were trying to capture,we can see that as just useful idealisation - idealisation of the sort that has facilitated the success of formal semantics. We don't have to, and shouldn't, see our mathematical treatment of natural language as mis-describing its workings in the way Sainsbury alleges.

I don't want to evaluate either of these responses to Sainsbury, but rather to draw attention to a point in his paper that I think hasn't been sufficiently discussed in this debate. He observes that:

'Boundaryless concepts tend to come in systems of contraries: opposed pairs like child/adult, hot/cold, weak/strong, true/false, and the more complex systems exemplified by our colour terms.' (p258 in the Keefe and Smith reader).

Further down the same page he writes:

'Not just any clear case of the non-applicability of a concept will serve to help a learner see what the concept excludes. Television sets, mountains and French horns are all absolutely definite cases of non-children; but only the contrast with adult will help the learner grasp what child excludes. So it is no accident that boundaryless concepts come in groups of contraries.'

This observation goes awol on the picture Sainsbury is attacking. If we think of 'red' as classifying by partisioning the domain into an extension and an anti-extension, we don't seem to be able to explain the relationship between contraries. The anti-extension of a vague concept would contain all the junk that Sainsbury observes plays no role in grasping what the concept excludes.

I think Sainsbury has pointed out an important feature of vague concepts, and one that may be difficult to capture on the set-theoretic approach. Of course, Cook's response can still be given; just as our formalisation can introduce useful artefacts, it can abstract away from certain features of the discourse we want to study. But it would be interesting to see how far Sainsbury's observation can be pushed; if the set theoretic framework really does just lack the resources to model the feature he draws attention to, that provides some support for Sainsbury's conclusion that we've a bad picture of how vague concepts classify.

Labels: Mark Sainsbury, Philosophy of Logic, Vagueness