Tolerance and Continuity

Continuity is an important aspect of what is going on with phenomenal sorites; I think tolerance might be too.

"a necessary condition on a change looking continuous is that narrow enough regions/close enough points appear to be homogenous [the same color]"? (Graff 924)

This is rejected in favor of a more mathematical definition in terms of epsilons and deltas. Here is the criterion for continuous color-change:

"for change in color across a spectrum to look continuous, it is require only that given any positive amount of change in color, there is a narrow enough width such that in any region on the spectrum narrower than that width, the color looks to change less than that amount in that region. Crucially, it is not required that any region, however narrow, look either homogeneous in color, or the same as its immediate neighbors." (925)

Is this too demanding a notion of continuity for continuous change to be represented in our experience at all?

"It may seem that these conditions would be pretty difficult, if not impossible, to satisfy. If so, so much the better for my case, since it is my opponent's first premise that such conditions can be satisfied." (925)

The dialectic here is that the opponent argues from the continuity of the sorites series to the intransitivity of "looks the same as" via the "homogeneity thesis", which has just been rejected.

Unkind Cuts, and different Sorites desiderata*

Is accepting the existence of a cut in the sorites series, into an extension and an antiextension, just like accepting the arbitrariness of other things in nature--the exact value of the gravitational constant, the maximum amount of money the government would spend to save an endangered child, the exact amount of poison that would kill you dead? Fine (1975) suggests that it is:

"I suspect that the temptation to say that [a sorites sentence] is true may have two causes. The first is that the value of a falsifying n appears to be arbitrary. This arbitrariness has nothing to do with vagueness as such. A similar case, but not involving vagueness, is: if n straws do not break a camel's back, neither do (n+1) straws."

Fara cites this as an attempt to answer "the Psychological Question":

"If the universally generalized sorites sentence is not true, why were we so inclined to accept it in the first place? In other words, what is it about vague predicates that makes them seem tolerant, and hence boundaryless to us?" (50)

She contrasts this with two other questions, "the Semantic Question" and "the Epistemological Question":

Semantic Question. If the universal generalization AxAy(Fx & Rxy -> Fy) is not true, then must this classical equivalent of its negation be true?: ExEy(Fx & Rxy & ~Fy) [called "The Sharp Boundaries Claim"].

(i) if Sharp Boundaries is true, how is its truth compatible with the fact that vague predicates have borderline cases?

(ii) If Sharp Boundaries is NOT true, what revision of classical logic/semantics must be made to accommodate this fact?

Epistemological Question. If AxAy(Fx & Rxy -> Fy) is not true, why are we unable to say which instance is untrue, even in the best epistemic situation?

Graff's response is that active consideration of a case (a particular value of x and y which bear the relation R to each other) raises the similarity of x and y to salience, so the boundary cannot fall between them.

The reason this seems odd to me is that active consideration of two shades seems to make one's ability to discriminate them sharper! This observation is compatible with the claim that one would never draw the F/~F line between them, but since active consideration brings their differences to the fore, it's counterintuitive.

***

Delia Graff Fara, "Shifting Sands." Phil. Topics, 2000.

Kit Fine, "Vagueness, Truth and Logic" Synthese. 1975.

2-dimensionalism: WDTM?

``I...outline a two-dimensional intensional framework for handling a posteriori necessity. Every concept has two intensions, a primary intension and a secondary intension. The primary intension delivers a concept's referent in a centered world when the world is considered as actual (i.e., considered as an epistemic possibility); the secondary intension delivers a concept's referent in a world when it is considered as counterfactual. (The primary intension of ``water" picks out roughly the ``watery stuff" in a world; the secondary intension picks out H2O.) A statement is a priori when it has a necessary primary intension; a statement is necessary when it has a necessary secondary intension but a contingent primary intension. A priori necessities (such as ``water is H2O" have a necessary secondary intension but a contingent primary intension. A statement is conceivable (or logically possible) when its primary intension is true in some world; a statement is possible (metaphysically possible) when its secondary intension is true in some world. So the Kripkean gap between conceivability and possibility is explained at the level of statements, without appealing to a distinction between conceivable and possible worlds. The class of worlds in question is always that of the ideally conceivable (or logically possible) worlds. "

--From Chalmers's precis in Philosophy and Phenomenological Research

Broome and Me*

I want to understand the wide-scoping maneuver and why it fails (if indeed it does). In order to understand this I think I'll need (more) modal logic, but I might also need to understand more about wide-scoping itself.

I think I now understand the intuitive distinction: the "undetachable" O(p->q) is a "normative requirement" which tells you that if you believe p, you ought to believe q. It doesn't mean that p's truth obligates you to believe q.

Intuitively, this does seem to capture what is going on in the miners case: O(A->blA) means that, if we believe that they are in A, we ought to believe we should block A. The truth of their being in A isn't enough to rationally require us to believe we ought to block A, since we might not know where they are, in which case blocking A would be foolhardy.

Here is the paradox in the version which is NOT supposed to be helped by wide-scoping. Part of the key here is that deontically ideal worlds (rel. to a pt. of eval.) are a subset of the epistemically possible worlds (rel. to that pt. of eval.). So MUST(p->q) entails OUGHT(p->q). [Another way of putting this is OUGHT implies CAN. MUST(p) means CANNOT(~p), which entails NOT OUGHT(~p), which entails OUGHT(p).] [Note that this is only true in an unsatisfying sense, though--the unsatisfying sense in which Lincoln ought to have been assassinated, since it is now epistemically necessary that this was so.]

1. O(A -> BlA) Premise (in wide-scoped form)

2. O(B -> BlB) ''

3. M(A v B) Premise: either miners are in A or they are in B

4. M(BlA -> BlO) Blocking A entails blocking one shaft

5. M(BlB -> BlO) ''

6. O(A v B) Must implies ought

7. O(BlA -> BlO) ''

8. O(BlB -> BlO) ''

9. O(BlO) Dilemma, MP, MP [not sure how to write out all the steps that go on inside the scope of the "ought" operator in this kind of notation.]

While turning "ought implies can" into "must implies ought" is a bit fishy, surely the problem here still lies with the fact that step 6 generates only two cases for dilemma. M(A v B), from which (6) is descended, is intended to be taken trivalently: it doesn't decompose into M(A) v M(B). A disjunction may be necessary (or obligatory) without having a necessary (or obligatory) disjunct.

Surely this is the situation that putting "A v B" inside the scope of the O was supposed to prevent!

Could this help to explain why restricting MP is better than using wide-scoping?

***

John Broome, "Normative Requirements" Ratio 1999, XII, 4.

MacFarlane and Kolodny, "Ifs and Oughts," manuscript.

Restricting MP vs. restricting discharging*

Does this difference make a difference?

The inference in question is: p -> D(p)

Contraposition: ~D(p) -> ~p [looks bad!]

Imp: ~p v D(p) [looks bad!]

R-Imp: D(p) v ~p

Evans's revised proof:

~D(a=b) Assume for Red.

F(b) Pred. Abst. (F = property of being indef. = to a)

D(a=a) ? [``identity does not admit of borderline cases"]

~F(a) Pred. Abst.

~(a=b) Leibniz' Law

D~(a=b) ? [many strikes: this is a hyp. context, not clear what justifies the rule anyway, esp. in presence of higher-order vagueness.]

[we have no assumptions, so the conclusion of the proof should be an axiom when the assumption is discharged.]

Heck's way of putting the equivocal result:

``Whether Evans's arguent shows that there can be no vague objects may now seem to be but a terminological question...[It] does show that `~D(a=b)' is unsatisfiable; it does not show that `D(a=b)' is valid. If we identify the view that there are vague objects with the view that there are (or might be) true sentences of the form ~D(a=b), Evans has shown there are no vague objects. If, on the other hand, we identify the view that there a re vague objects with the view that `D(a=b)' is not valid, then he has not."

A much simpler way of making the same point appears to be suggested by McF & K. If MP is not valid, then there's no way to even get close to a contradiction. (By ``close", I mean: to get a contradiction even inside the scope of an assumption that, as it turns out, can't be discharged.)

The McF & K way is to refer us to Restricted MP, which is (always) valid when the antecedent is either world- or info-invariant and the consequent is info-invariant.

The method of evaluating the conditional (``if phi, then psi") is to contract the original information state until it is (i.a) a unique maximal phi-subset or (i.b) until you have one of n maximal phi subsets, in which case you'll need to do step (ii) for all of them; (ii) check that psi is true throughout the remaining i.

Now: let's say p -> D(p) is a rule of inference, in Heck's sense (as opposed to a rule of proof.) It does seem that it is exactly the kind of conditional which is NOT VALID in hypothetical contexts.

Compare Quine's proof, with Nec() and Pos():

Pos(~(9 = num planets)) Assume for red.

F(num pl.) Pred. Abst. (F = prop of pos. being nonidentical 9)

9 = num planets Astronomy

Nec(9=9) premise [axiom, iron law, etc.]

~Pos(~(9=9)) Modal Shift

~F(9) Pred. Abst. (note: both ok bc 9 is rigid desig.)

~F(num planets) Sub. of identicals

F(num pl.) & ~F(num pl.) & Intro

[Contradiction] ~Intro

Do we have an instance of the inference form p -> D(p) here? We do have Nec(9=9), which COULD be gotten from 9=9 and the suspect inference. But the intuition underlying Nec(9=9) isn't that at all. Also, the proof WOULD go through if we had gone from step 3, ``9 = num pl." to something stronger, namely ``Nec(9 = num pl.)" But this wouldn't have been a truth-preserving step. ``p = num pl." is only a contingent identity (if it is an identity at all; perhaps it's more perspicuous to say that it isn't. Instead, the number-of-planets-hood is predicated of 9 in the actual world. It would be off-the-wall to give ``identity doesn't admit of borderline cases" as a justification for such a step.)

Conclusion: the indefinitist case (Heck's terminology) is really stronger than he makes out. Re-interpreting the restriction on rules of inference as a restriction on Modus Ponens (although the two are equivalent in terms of what can be proved) makes Evans's argument look much more insignificant.

****

Heck, ``That there might be vague objects (so far as concerns logic)"

MacFarlane and Kolodny, ``Ifs and Oughts"

Tolerance and the phenomenal sorites

A tolerant object, in my sense of tolerant (the topological one) isn't vague, because, amongst other things, it has no borderline cases: if I pick a (rational) point, it is either definitely inside or definitely outside of the object.*

The sense in which it is tolerant is a different one: there is no last point in the set.

*Graff rejects "admits of borderline cases" as a definition of vagueness because it includes as vague such predicates as "dommel" and Sainsbury's "child*". This is rejecting "admits of borderline cases" as a SUFFICIENT condition for vagueness. Are there complaints about it as a NECESSARY condition? Perhaps only Weatherson (VAI) has challenged that.

What about a set of shades, or a set of heights? It seems that here we return to issues of discriminability and identity. What Graff's other article suggested to me is that indiscriminability is extremely context-sensitive even when the mode of presentation is held fixed. That's why, for example, you bring paint chips home.

We can challenge ourselves when it comes to discriminability. In fact, we can play a game: show me a red thing, and I'll show you an oranger red thing. We can play this game a long time...maybe even forever, if the circumstances were right! And we will never get into the range of things that are definitely orange (and thereby definitely NOT red.) This is particularly true if we're picking points on a spectrum. The experience of the spectrum tells us that for every two points x and y, if x is e.g. on the left of y, then it is orange-er than y. So in those terms, even if there WAS a ``sharp line" dividing the orange things from the red things, we'd still be able to play the game forever.

Could this be what a visual experience of a spectrum tells us? After all, we aren't really sure what it does tell us; we puzzle over the fact that it seems we can only perceive a finite number of shades while being unable to get a fix on just how different two shades need to be to be discriminable. The answer seems to be, "it varies." And the visual experience of the whole spectrum, it seems, gives us more information than that: it tells us precisely that there are an infinite number of shades arranged according to the rule "left of" = "oranger than." There is no limit to the number of ways we can use this information once our visual experience has imparted it to us; we will know that we can always (or almost always) be able to pick an oranger red shade.

Perhaps another test to run is this: if we were given a bunch of tiles, could we arrange them from left to right according to the rule "left of" = "oranger than"? At a fine enough level, the answer would be "no." But just because we cannot do it ourselves does not mean we don't jump to the conclusion that it is correctly done, when we see a color sorites series.

*****

Delia Graff Fara, "Shifting Sands: An Interest-Relative Theory of Vagueness." Phil. Topics

Brian Weatherson, Vagueness as Indeterminacy, manuscript.

Delia Graff Fara, "Phenomenal Continua and the Sorites." Mind, 2001.

Distributing vagueness: shifting the burden from singular to general terms

Orthodoxy has it that vagueness is first and foremost a property of sentences:

(1) That [pointing at my car] is red

or

(2) Bruce is bald. [where Bruce is Bruce Willis]

The vagueness of these sentences is usually analyzed by giving an account of the general terms red and bald, not the singular terms that and Bruce. So any account of vague objects will have to shift the intuitive source of the vagueness of paradigmatically vague sentences from the predicate to the subject. (Williamson suggests the canonical view's tendency to localize vagueness in predicates as a reflection of the `truthmakers are objects' principle, which is really, he says, an unjustifiable dogma.)

It is worth asking why we think that the vagueness of (1) and (2) should be explained by focusing on properties. I suspect the answer is not any insight into, or conviction regarding, the manifest precision of objects (whatever that would be) but rather our strongly-held beliefs about the modal plasticity of Bruce and my car:

(1') My car might have been red.

(2') Bruce might have had abundant hair all his life.

...while Bruce and my car are borderline cases, it is not essential to them that they are. But is the truth of (1') and (2')---which I take for granted---really material to the question of whether there are vague objects? The significance of (1') and (2') should be left an open question pending an account of the relationship between vagueness and modality.

Note also that while we have robust intuitions about the modal plasticity of persons and everyday objects, we generally lack intuitions about the modal profile of properties. While there is talk of vague predicates' being essentially vague, it is not clear what this means; to the extent that it is clear, it is a statement about the predicates' meaning, not their extension. By contrast the intuitions in (1') and (2') are definitely intuitions about the denotation (or extension) of `That' and `Bruce.' This is all the more easily shown given that the current going theory of proper names and demonstratives is that they do not have meanings (`meanings' understood here as Fregean senses.)

Two questions:

(#1) Are all these intuitions about general and singular terms really consistent? If we think of e.g. the predicate blue as the set of blue things, then if you change the color of my blue shirt, you will change the predicate blue...because you change the set of which the shirt is a member, and the identity conditions of sets are given by their members.

So the (1') intuition about the modal plasticity of my car [my car itself, not just the mode of presentation ``Melissa's car"] is equally an intuition about the modal plasticity of blue [blue itself--if such a distinction between the sense and the reference of a general term is even possible!*]

The identity conditions of properties are usually given by modal extensions (c.f. solving the renate-cordate problem.) But, again, that means that one fewer blue object in logical space will change blue into an entirely different property (because it would denote a different set.) Given our limited knowledge of what goes on in the outer reaches of logical space, this is not a very helpful definition of `blue', nor is it one which informs our intuitions about the term. But the fact remains that we are not sure what blue could be like, if it weren't like it is. (Is 2-dimensionalism helpful here?)

(#2) Can I make the same move I made with Sorensen? Here is my cri du coeur:

I just don't think this approach--starting first and foremost with vague sentences and then looking for a suspect to fix the vagueness on--is very profitable. After all, it does not seem to me that sentences are the primary locus of vagueness. The phenomenology of vagueness is something that strikes us when we look at continua: color continua, height spectra, etc. So it is, for example, a feature of color, and only derivatively a feature of sentences with color predicates. At the level of sentential form, we don't have enough information to determine whether ``Tim is pink" is vague. What we need is to see Tim, and to see his color.

This seems right to me even still. I do not think what is really at issue is an enormous debate over what it would mean for a general term to be a rigid designator, etc.

****

*Is it made possible by distinguishing between analytic truths, as McGee does, and truths constituted by use + world?