February 24, 2007

How is Modal Knowledge Possible?

If They Had No Tails...
In Chapter 5 of Williamson's manu The Philosophy of Philosophy we find a proposed answer. To know that it is metaphysically necessarily that A, one needs no more than whatever it takes to know that not-A counterfactually implies a contradiction. And this requires nothing over and above the cognitive faculties employed to acquire ordinary knowledge about the empirical world. Therefore, if (metaphysical) modal knowledge---indeed, if philosophy---is possible at all (and we suppose it is), then it requires no special cognitive faculties over and above what is required for ordinary knowledge of the world. Let's focus on the purported connection between modal and counterfactual knowledge.

Williamson demonstrates that modal claims are logically equivalent to some counterfactual claims. A necessary proposition is one whose negation counterfactual implies a contradiction, and a possible proposition is one that does not counterfactual imply a contradiction. Respectively,


(1) □A (¬A □→ ⊥)


(2) A ¬(A □→ ⊥)



The lessons, according to Williamson, are these: (i) Having what it takes to understand □→ and ¬, implies having what it takes to understand and □, and (ii) modal thinking is a special case of counterfactual thinking.

I take it that (ii) is meant to entail (i)---or is equivalent to it. So there is a natural objection here. Understanding A (or grasping its meaning) is not closed under logical consequence. If A is logically equivalent to B, s may grasp A without grasping B, because B embeds concepts that s doesn't understand---e.g., 'A' is equivalent to 'A or A', and though s understands the former, he doesn't understand the latter because he doesn't grasp the meaning of 'or'. Hence, (i), and so (ii), is false, and we lose the argument for thinking that an epistemology of counterfactuals will give us everything needed for an epistemology of modality. Carrie Jenkins expresses a version of the concern over at Long Words.

I don't think the objection hits Williamson. We need to read him as offering an account of how modal knowledge is possible for beings like us, and not as defending the empirical thesis that our modal and counterfactual knowledge covary. Whether or not our knowledge of necessity and possibility does (or must) take a detour through counterfactual knowledge, is not the question. The epistemological claim is that beings like us, who are in a position to know the right-hand side of the equivalence, are also in a position to know the modality on the left. After all, we might infer the left from the right.

Here's a different objection. It is one that I think is more serious than Williamson suggests. Williamson's proof of the first equivalence requires a certain theory about how to handle counterpossibles (i.e., counterfactuals with impossible antecedents). He sides with Lewis that we should treat them as vacuously true. If this is right, then necessary implications entail counterfactual implications:

(3) □(A → B) (A □→ B)

(3) underwrites Williamson's proof of (1). If this choice about how to handle counterpossibles is a mistake, then things unravel for Williamson's epistemology of modality. For a genuine counterexample to (3) can be turned into a counterexample to (1).

Here's one way to do it:

Let "LEM" be the universal formulation of the law of excluded middle, "∀A(A v ¬A)". And let "DNE" be the classical theorem "∀A(¬¬ A → A)". Suppose classical logic is unrestrictedly valid, so that LEM is metaphysically necessary (since logically necessary). Then the strict conditional, "□(¬ LEM → DNE)", is true. It's vacuously true. However, the corresponding counterfactual appears to be false: "¬LEM □→ DNE". This is false because if LEM were false then intuitionistic logic (and not classical logic) would be unrestrictedly valid. And in intuitionistic logic, ¬LEM does not imply DNE (but rather entails ¬DNE).

If this is right, then we also have a counterexample to (1). Although LEM is in fact metaphysically necessary, it is not equivalent to "¬LEM □→ ⊥". For in a counterfactual circumstance where ¬LEM is true, intuitionistic logic (but not classical logic) obtains. And, ¬LEM does not enjoin an intuitionistic contradiction.

Williamson evaluates other purported counterexamples to (3). I don't see that his replies touch the above example. Williamson attributes a confusion akin to the confusion made by philosophers for centuries prior to the realization that a generalization, "All S are P", is vacuously true when nothing is an S. But I'm not guilty of this confusion. If our counterfactual supposition is the denial of LEM, then not all of our knowledge of classical logical truth is available in the description of the counterfactual circumstances from which we are to develop a path to the consequent. In particular, any logical truth intuitionistically equivalent to LEM is not available in the description of the counterfactual circumstances. But then in such counterfactual (intuitionistic) circumstances, ¬LEM does not materially imply DNE. The reason is that, in those circumstances, ¬LEM is not a logical falsehood. Vacuity vanishes!

[UPDATE: Berit locates a vacuous assertion of Williamson's that he did not intend to be vacuous, and draws a dark lesson for the prospects of doing metaphysics.]

15 comments:

Carrie Jenkins said...

Hi Joe,

(1) I think your first objection in terms of understanding is different from my worry that pointing out an equivalence does not show that the epistemology of modality is a special case of that of counterfactuals. Even if we accepted that an understanding the RHS invariably goes along with an understanding the LHS, I'd still think there was no reason to think the epistemologies are related as W claims.

(2) If, as you suggest, W merely intends to claim that it is *possible* to obtain modal knowledge via our grasp of counterfactuals, he still owes us more than is provided merely by pointing out the equivalences - he owes us an account of how (on the purportedly possible picture) we get from the one side of the equivalence to the other. (Do we do it by knowing the equivalence facts and deducing? If not, how?)

(3) I don't think W would be convinced by your LEM counterexample. I think he'd say that what would be true if ~LEM were true is settled by what follows from ~LEM in the logic which is in fact correct (i.e. classical logic), not by what follows from ~LEM in intuitionistic logic (even though it is true that if ~LEM were true then intuitionistic logic would be correct).

Aidan said...

"And, ¬LEM does not enjoin an intuitionistic contradiction."

Sorry for being slow again, but isn't ~~LEM a theorem of intuitionist logic?

Joe Salerno said...

¬¬(A v ¬A) is an intuitionistic theorem, but ¬∀A(A v ¬A) is intuitionistically consistent with that.

Aidan said...

Thanks Joe, I guessed it would be something like that. I always get lost in the interplay between the quantifiers in IL.

Mike said...

Suppose classical logic is unrestrictedly valid, so that LEM is metaphysically necessary (since logically necessary). Then the strict conditional, "□(¬ LEM → DNE)", is true. It's vacuously true. However, the corresponding counterfactual appears to be false: "¬LEM □→ DNE". This is false because if LEM were false then intuitionistic logic (and not classical logic) would be unrestrictedly valid.

Hi Joe,

I'm not sure I'm following your second counterexample. It looks to me question-begging. The supposition is that LEM is logically necessary, right? But then you say that ~LEM []-> DNE is false. Of course, by hypothesis, there is no closest ~LEM-world that is not a DNE-world. So why is it false? Are you saying that the condtional is an intuitively false counterpossible?

Joe Salerno said...

Hi mike,
I'm saying that the conditional, ~LEM []-> DNE, *is* false in a discussion about the truth of various logical principles. So, if in fact classical logic is unrestrictedly valid, then the conditional is a false counterpossible.

Julien said...

Hi Carrie,

" I think he'd say that what would be true if ~LEM were true is settled by what follows from ~LEM in the logic which is in fact correct (i.e. classical logic), not by what follows from ~LEM in intuitionistic logic (even though it is true that if ~LEM were true then intuitionistic logic would be correct)."

I still don't get it. After all, Joe could object that, if ~LEM were false, there would be no exclusively classical law to appeal to. For how could the consequences of ~LEM be settled by classical logic, if ~LEM entails that there is no world at which classical logic is the correct logic?

In other words: if true at all, LEM is necessarily so. Hence, the counterfactual assumption that ~LEM really amounts to assuming that there is *no* world at which LEM is true. In such a scenario, however, there would be no 'correct' (i.e. classical) logic to appeal to.

The counterfactual reaction you're attributing to W seems to be that of someone who doesn't really accept the very possibility that LEM might be false. Of course, as an exegetical remark, this is surely correct (we all know W's fervour for classical logic!). But I wonder whether this would be a satisfactory reply after all.

Btw, thanks Joe for the very nice post!

Julien said...

ooops, errata corrige:

"if ~LEM were *true*"

J

Joe Salerno said...

Hi Carrie,
You mentioned that TW wouldn't be convinced by the LEM counterexample:

"I think he'd say that what would be true if ~LEM were true is settled by what follows from ~LEM in the logic which is in fact correct (i.e. classical logic), not by what follows from ~LEM in intuitionistic logic (even though it is true that if ~LEM were true then intuitionistic logic would be correct)"


TW would have to say something like this. But notice that if this position were right, then the final parenthetical in the above quote is vacously true. I don't think Williamson should agree to its vacuousness. He'd also vacuously have to admit: if ~LEM were true then intutionistic logic would *not* be correct.

If, in such logical contexts, questions about what would be the case must be settled against the backdrop of the correct logic (ex hypothesi, classical logic), then we can't engage non-vacuously about alternative logics. Or at least, we can't so engage non-vacuously, when we engage counterfactually.

Mike said...

I'm saying that the conditional, ~LEM []-> DNE, *is* false in a discussion about the truth of various logical principles.

Actually, you're saying it is false while conceding for the sake of argument that LEM is necessarily true. Why doesn't that beg the question? What gives traction to the assertion that the conditional is false? Is it that the conditional is intuitively false even under the assumption that LEM is necessarily true? I'm pretty sure I don't have that intuition under that assumption. Oh well.

Joe Salerno said...

Thanks, Mike. It's not an intuition that gives traction to my assertion that
¬∀A(A v ¬A) □→ ∀A(¬¬ A → A)
is false. It's the fact that, intuitionistically, '∀A(A v ¬A)' and '∀A(¬¬ A → A)' stand or fall together. Since a nearby world where one of these is false is a world where intuitionistic logic is correct, it follows that a nearby world where one of these is false is a world where the other is false as well. Therefore, it is false that the nearby worlds where '∀A(A v ¬A)' is false are worlds where '∀A(¬¬ A → A)' is true.

Carrie Jenkins said...

Williamson is of course happy to admit that all counterpossibles are vacuous. The intuition that some are not might be best accommodated by saying that these ones (e.g. 'If ~LEM were true then intuitionistic logic would be correct') have some special epistemic status: e.g., that we can tell they are true just by thinking about what LEM and intuitionistic logic are, without making assumptions as to which logic is correct (or to put it in a more Williamsonian way, by engaging in a certain kind of 'off-line reasoning' about LEM and intuitionism).

Mike said...

. . . intuitionistically, '∀A(A v ¬A)' and '∀A(¬¬ A → A)' stand or fall together. Since a nearby world where one of these is false is a world where intuitionistic logic is correct, it follows that a nearby world where one of these is false is a world where the other is false as well.

I see the argument, thanks Joe. I guess I'm wondering why I couldn't reason this way instead. Any nearby world in which '∀A(A v ¬A)' is false is a world where intuitionist logic is not correct. This is because, by hypothesis, there just are no worlds where '∀A(A v ¬A)' is false.
I guess I'd like to argue that it follows from the falsity of '∀A(A v ¬A)' in any world w that w is not an intuitionistic world in the same way that it follows from the falsity of '~(A & ~A)' that intuitionism is false: viz., trivially by []((A & ~A) -> intuitionism is false)). But then doesn't that upset the inference to your claim that ". . .'∀A(A v ¬A)' and '∀A(¬¬ A → A)' stand or fall together"? Since there is no world in which '∀A(A v ¬A)' false, it sure seems like we can conclude that ∀(~∀A(A v ¬A)-> ~∀A(¬¬ A → A). That seems contrary to your conclusion.

p.s. Maybe the disagreement abut how to argue here is related to your almost-last post. I just glanced at it.

Mike said...

Where I write,

Since there is no world in which '∀A(A v ¬A)' false, it sure seems like we can conclude that ∀(~∀A(A v ¬A)-> ~∀A(¬¬ A → A).

I should have written,

Since there is no world in which '∀A(A v ¬A)' false, it sure seems like we can conclude that ~∀A(A v ¬A)-> ∀A(¬¬ A → A).

The negation of a necessary truth (viz., ~∀A(A v ¬A)) entails anything, including ∀A(¬¬ A → A). So '∀A(A v ¬A)' and '∀A(¬¬ A → A)' do not stand or fall together. Or so goes the argument I've offered. Sorry for the typos, maybe the argument is clearer now.

Perezoso said...

Prologema to modal chat, and really to formal logic:

Decide on realism or nominalism (which itself seems to be matter of inference rather than axiomatic).

Nominalism far more likely--thus supposed a priori givens such as law of excluded middle, and even law of contradiction are inductive. And this can be seen even in ordinary language when one can say quite correctly "she loves joe, and she doesn't love Joe." Love is not discrete. Or "it is raining and it isn't raining." Raining itself an event and process; not discrete. Discreteness must established as parameter, stipulated (not a priori), as must classes, functions, objects, numbers-----one might say the same about UG and EG as well (infinity---nicht)--tho EG can most likely be established........


Modality then applies only to observable events. Or in other words, modality is probability.

How easy was dat!

;)

(if some of the great modal theorists had bothered with Goodman/Quine on constructive nominalism--even refuting it (gut gluck) this wouldn't be an issue)