Opiniatrety

Here are the comments.

Whether Matt presents a persuasive argument against deductive closure depends heavily on one’s prior commitments. We should consider four positions. Matt’s argument might well persuade many proponents of three of those positions of something, although just what varies from case to case. A proponent of the fourth will be unmoved by what Matt says.

The first position is the primary target of Matt’s paper: any form of contextualism about knowledge (or ‘knowledge ascriptions’, if we want to be picky) that endorses deductive closure in any context of knowledge ascription. The second position is that of the contextualist who already embraces at least some failures of deductive closure. The third position is that of the invariantist who embraces at least some failures of closure (for the record, this position is my own). The fourth is that of the invariantist who endorses deductive closure.

Matt primarily argues that a contextualist should accept that knowledge, whatever the context of ascription, whatever knowledge amounts to a given context, is not closed under the logical rule of Conjunction Introduction. It is not without exception true that if one knows A and one knows B, then one knows A and B. If we assume Conjunction Introduction holds in full generality, then we face something like a Sorites paradox which concludes that we know something that we clearly do not know—a giant conjunction which entails that all of the holders of lottery tickets but for the actual winner will never be rich. (If the closure of CI fails at all, then it will presumably fail for mere pairs of propositions provided that each member of the pair is itself a “small enough” conjunction that it can be known, but whose conjunction is “too big” to know.)

If the contextualist gives up on deductive closure for CI, there is little reason for him to resist giving up deductive closure for Modus Ponens. He should accept that if one knows P, and one knows that if P, then Q, it does not follow that one knows Q, however aware one is that Q is entailed by what one does know. In particular, he should accept that in a single context of knowledge ascription, one can know that one’s friend will never be rich and fail to know that he will lose the lottery for which he has a ticket, although that he will lose the lottery follows from the proposition that he will never be rich. Whatever reasons there are for thinking knowledge closed under MP in all contexts are equally reasons for thinking knowledge closed under CI. Since those reasons cannot be decisive in the case of CI (if Matt’s argument is persuasive), they cannot be decisive for MP. Since knowledge appears not to be closed under MP, and only contextualist fancy footwork saves those appearances in the case of MP but not CI, the contextualist should surrender and accept that deductive closure fails in full generality—for at least those two logical rules, in at least the cases in which it intuitively does so fail.

So I reconstruct Matt’s argument, anyway. And I have no objection to make to it. It is easily extended to an argument that says something to a contextualist who already accepts that knowledge is not closed under MP (there are such—I work with one). He should simply say “Interesting! I guess knowledge is not closed under CI either.”

What should an invariantist’s reaction be to Matt’s considerations, however? An invariantist who already holds that deductive closure for MP fails will, I suggest, have the same reaction as the contextualist who holds that closure fails for MP. “Interesting! I guess knowledge is not closed under CI either.” That’s my reaction, anyway. But an invariantist who endorses deductive closure in full generality, and in particular for MP, will find Matt’s argument utterly unpersuasive. Such an invariantist will not claim that I know that my lottery ticket will lose. He will claim that I do not know that I will never be rich, precisely because I do not know that my ticket will lose. He will not claim that I know that my car has not been stolen. He will claim that I do not know where it is parked—although I do, perhaps, know where it is probably parked, and that I probably will never be rich. Consequently, such an invariantist will deny that I know that my feckless friend Bill will never be rich, and likewise for all similar propositions about my friends as discussed by Matt. Matt’s argument will not get off the ground since its initial premises will be flatly denied by an invariantist committed to DC. Matt has not presented a persuasive argument against deductive closure in general, only one that is at best persuasive to a contextualist who antecedently denied closure.

OK, so how might those of us persuaded by Matt’s considerations that knowledge is not closed under CI proceed? Matt disavows offering any solution to Sorites paradoxes. However, I suggest that we can draw some conclusions about what anyone who holds any popular theory of vagueness should say.

Firstly, let us note that there is an important respect in which the case Matt presents is not analogous to a standard Sorites case. There is a natural continuum from (or to) definite heaps to (or from) definite non-heaps. The same natural continua obtain for heads of hair with respect to baldness and hirsuteness, color patches with respect to being red or being orange, and so on in indefinitely many instances where vagueness arises. What is to be said about the Sorites continuum (and all counterpart continua) is widely agreed upon at the most general level by many views about vagueness: in between those entities which are definitely heaps and definitely non-heaps are the borderline cases which are neither definitely heaps nor definitely non-heaps. What differs radically from view to view is the proper interpretation of these claims. Epistemicists will claim that there is a fact of the matter about whether anything is a heap or not; it is just that we cannot know whether the borderline cases are or are not heaps. Supervaluationists and others disagree; ‘definitely’ has metaphysical bite on non-epistemicist views. There is some sense in which there is no fact of the matter about whether borderline cases of heaps are heaps or are not heaps.

There is no natural continuum of lottery tickets; that they might have numerals assigned to them (hey, they could be imprinted with convoluted shapes) is neither here nor there. Consequently, there is no natural continuum of Alice’s friends who hold lottery tickets and end up losing. She (definitely) knows of any one of her friends, pick one at random and let him be A1, that he will never be rich. And she knows the same of any other, pick one and call him A2; she definitely knows that A2 will never be rich. She also definitely knows that A1 will never be rich and A2 will never be rich. She definitely does not know that A1 will never be rich and . . . A5000 will never be rich. There are numerous conjunctions concerning not too few and not too many of her friends which Alice neither definitely knows nor definitely does not know. Those are borderline cases for extending knowledge by Conjunction Introduction. And that they are borderline cases has nothing to do with which of Alice’s friends they concern and nothing to do with which lottery tickets they happen to hold. It has only to do with the number of conjuncts. The natural continuum here is to be found in the structure of the propositions knowledge of which Matt’s argument concerns, not in the objects or properties the argument’s premises concern, as is the case with standard Sorites arguments.

How does this bear on one of Matt’s concluding paragraphs?

"This picture of how deduction yields knowledge is more realistic than the picture that any known premise may be put in the bank for further deduction. On this picture, we must retain enough information about how knowledge was attained to make sure that we are not compounding the fallibilities of our individual premises. We must not engage in lengthy chains of deduction where each link is fallible and the failure of one link destroys the whole chain. Still, so long as our arguments have the proper redundancy and no new pertinent doubt has been raised, we need not constantly recheck the foundations of our knowledge."

I am not entirely sure what Matt means by these statements concerning what we must and must not do, and so I am not sure what I have to say will conflict with his views—what I wish to do is forestall any reading of them that is too internalist. Small enough conjunctions, n the relevant kinds of case, can be known if we arrive at them by a small enough number of applications of CI. This is so whether or not the knower realizes that too many applications of CI will lead to the knower believing conjunctions that are neither definitely known nor definitely not known, and even more applications of CI will lead to belief in conjunctions that are definitely not known. If one arrives at belief in a small enough conjunction through a small enough number of applications of CI, there is no need to “retain information about how [one’s] knowledge was obtained”; one simply (and definitely) knows the conjunction. If one arrives at belief in a borderline case conjunction through CI, one will neither definitely know nor definitely fail to know that conjunction whether or not one retains such information.

Perhaps we should say that known premises can most certainly be put in the bank for further deduction—but one can still get overdrawn. Keeping a meticulous eye on one’s balance will be of little use here, too, because of higher-order vagueness considerations. There is no size at which a conjunction becomes definitely too big to be definitely known; the size at which that occurs is itself an indeterminate matter. One’s balance has to be big enough— the conjunction believed small enough—for one definitely to know that conjunction. What one believes about that balance—what information one retains about it—is irrelevant to whether one has the money or not. Luckily, as with more familar (alleged) cases of the failure of deductive closure, actual knowers spend within their means. Just as they do not come to believe categorically that their car has not been stolen simply because they believe categorically that it is parked at such-and-such a location, they do not come to believe that all but one (or, more generally, too many) of the lottery ticket holders will never be rich simply because they believe of each one that he will never be rich. The failure of deductive closure is not in conflict with how believers actually extend their knowledge. In order to remain knowers in good standing, there are inferences they should not make—but they are not remotely tempted to do so, and hence do not need explicitly to check whether they have entered the realm of borderline cases for them to remain knowers in good standing.

And here's a draft of my response--it's too long, I may decide to just say "excellent comments" and sit back down.

This response may be a bit dull--I agree with almost everything Jonathan has said. Before moving on to some small areas of disagreement, I would like to express especial agreement with the second half of his comments, in particular his closing paragraphs. What determines whether Deductive Closure is in danger of failing--whether a believer is in danger of losing knowledge by deducing from known premises--is not a matter of whether she is aware that there is a danger. It is a question of the structure of the argument itself. An argument that has built-in self-reinforcement will not lead to a known conclusion if the premises are not known; large enough plain conjunctions will not be known even if the premises are known.

Furthermore, as Jonathan points out, actual knowers will not generally be tempted into the arguments that lead to violations of Deductive Closure. This is the paradox of the Preface: Even if we wholeheartedly believe every single statement in a book, we will not claim, "Every sentence in this book is true." These are important points, and I thank Jonathan for making them.

Next, I'd like to explain a bit more who my argument is meant to reach. Jonathan claims that the invariantist who accepts closure will be unmoved by my argument. In this too, he is correct. The invariantist who accepts closure will deny knowledge of the premises of the arguments I have cited. And Jonathan brings out why I find that unappealing--if I don't know that my car hasn't been stolen, I don't know where it's parked, and I don't know that I will be home within an hour of arriving at Salt Lake City airport. More disturbing, for me, it seems to me that if I do not know that the stranger giving me directions is not lying or mistaken, I do not know where the Salt Palace is. And if I can't gain knowledge through testimony, I don't know much. So my arguments won't provide the invariantist reason to deny closure, but such an invariantist may face much worse problems.
What my arguments are meant to do is to soften the sting of denying closure. Deductive closure seems so obvious that denying it seems like a great cost. But many other sorites seem obvious, and are usually reliable, yet cannot be accepted without reservation. My proposal is that denying unqualified deductive closure is no more costly than denying any other unqualified sorites premise. This is a bullet we have already bitten. So those who would like to abandon closure--perhaps out of fear of skepticism--may find my way of denying it congenial.

Finally, I'd like to say a bit about the circumstances in which deductive closure fails. It is true that we can expect failures to occur both with repeated uses of CI and with repeated uses of MP. But that doesn't mean that all applications of CI and of MP are equally impugned. In particular, consider a view on which knowledge is simply justified true belief, where a belief is justified if it is likely enough on the information the believer has, where "likely enough" is a vague standard. Suppose that it is determinate that 99% probability is likely enough, and that any probability over 75% is determinately not determinately likely enough. (I know, this is dreadfully oversimplified, and knowledge isn't justified true belief anyway--but the argument will carry over to at least some other accounts).

Note first that a single application of CI is safe in the following way: It won't take you from two premises that are determinately known to a conclusion that is determinately not known. Hence closure of CI does not fail completely for pairs of propositions, if it makes sense to speak of incomplete failure.

More important, applications of MP will be completely safe, so long as one of the premises is known with complete certainty. This is most relevant for the apparent failures of MP that motivate contextualism.

Take the lottery case. Suppose that I am 99.98% sure that Bill will never be rich. I am 100% sure that, if Bill wins the lottery and collects on his ticket, then he will be rich. So I am just as sure that Bill will not win the lottery and collect on his ticket as that he will never be rich. I could perform as many MPs as I like without my certitude dipping below 99%, so long as every conditional were one that was absolutely certain on my information.

But most of us have the intuition that this will not do--that, even if I know that Bill will never be rich, I do not know that he will not win the lottery and collect on his winnings. Hence the motivation for contextualism. The contextualist can say that, when we are considering the prospect of the lottery, the standards for being likely enough go way up, or can provide another account of how the possibility of Bill's ticket winning is relevant when considering the lottery and not when considering his future riches. Viewing deductive closure as a sorites may not solve this particular problem; it may only solve the problem of how knowledge can leak away when the context clearly does not shift.

In the age of the Internet, one gets the rare opportunity to respond to the response to one's comments a little more frequently than was previously the case! A number of times in the past, I would have relished that opportunity in order to say that the speaker had not understood my comments *at all.* That is most definitely *not* true here. And I am very happy that you have found my comments "excellent," Matt; I found your paper to be excellent, too, and it will influence my own work substantially.

As far as the Paradox of the Preface goes, I am not entirely sure that your model case for denying deductive closure under CI will generalize. That is, if I really know the proposition expressed by each sentence in my book, it is not clear to me that I will not know their conjunction. I have nothing to say about *why* that is not clear to me; I just mean that my intuitions go in neither direction even given that I am convinced that the kind of case you describe is an instance of the failure of DC under CI.

My intuitions are similarly neutral with respect to your claim that:

"a single application of CI is safe in the following way: It won't take you from two premises that are determinately known to a conclusion that is determinately not known. Hence closure of CI does not fail completely for pairs of propositions, if it makes sense to speak of incomplete failure."

if the known premises are both big enough conjunctions. (When I wrote the comments, I simply assumed the negation of the quoted proposition for big enough conjunctions -- but I should have been neutral, I think.)

Well, the above is all little more than psychobiography and incidental comments. So let me try something a little more substantive.

I think that the reason that the denial of DC seems so outrageous to many is a background assumption that deductive logic provides examples of the very best kinds of inference there are -- those that are necessarily truth-preserving. But when it comes to the evaluation of inferences construed as what thinkers actually engage in, rather than abstract arguments, I think that a good inference is one that takes one from premises that one justifiably believes to a conclusion that one justifiably believes in virtue of performing the inference. And one is not justified in believing (1) that one's car has not been stolen -- just that it probably has not (in normal situations) -- although one is justified in believing (2) that it is parked at location X, and one is justified in believing (3) that if it is parked at X, then it has not been stolen (one knows those propositions, after all), and one cannot become justified in believing (1) by an inference from (2) and (3). Luckily, people are not remotely tempted to make that inference -- we do not have to say that reasoning that is actually performed on a regular basis is no good if we take this line on DC and what makes for a "good" inference. As with the instances that you describe in which DC under CI fails.

I hope the general idea there is clear -- there is more that can be said (and I have said it in work that is as yet unpublished), but I have gone on here for long enough already.

The view here is kinda neat, and it gives us logic-types something to think about. Instead of thinking of what we know as what is true in all of the relevant alternatives (which will give us full deductive closure), we've got to think of it in another way if we take these purported counterexamples seriously.

So how do we do it?

[Oops! I hit the post button instead of the preview button. Oh well.]

Here's another way to think about it: Think of all of the propositions as ordered by entailment. So A <= B iff A entails B, and this ordering is reflexive and transitive. None of the objections in Matt's paper lead us to think that we could ever have A <= B where we're in a position to know A, but not in a position to know B. So, think of the things we're in a position to know at some point. This collection of propositions must be closed upwards under the ordering <=. But it is not necessarily closed under conjunction, because of the failure of CI to preserve knowledge.

Or if you, like Matt, prefer talking of degrees of justification, then don't think of an epistemic state as a class of propositions, but rather, as a ranking of propositions with degrees, and the ordering on degrees must cohere with the entailment ordering on propositions. If A entails B, if A is justified to degree x, then so is B.

That seems like an interesting way to think about it, and you can then go on and ask further questions: what are these degrees? Are all epistemic rankings (or sets of propositions, if you think about it in the "flat" manner) formally equal, or are there more constraints you can put on these classes?

Thanks for the kind words again, Jonathan. I'm blushing.

I wanted to pick up on this:

But when it comes to the evaluation of inferences construed as what thinkers actually engage in, rather than abstract arguments, I think that a good inference is one that takes one from premises that one justifiably believes to a conclusion that one justifiably believes in virtue of performing the inference.

That seems exactly right to me--and it's helpful to my larger project of replacing knowledge with justification as the property that epistemologists should study. The best inferences aren't those that preserve knowledge but those that preserve justification.

I'm here inching toward the Harman line that implication and inference are completely different things (if I'm quoting that right); that's not somewhere I've liked to be in the past, but maybe I'll have to swallow it.

An interesting question is what is meant exactly by "what thinkers engage in." If we mean the voluntary actions by which thinkers affect their beliefs, those may not look much like arguments at all--I take it most inferences just happen. Or you might take it that the best inferences are as Harman describes in Change in View--briefly, you reconsider your beliefs when the cost of doing so is low enough, and you only carry out low-cost kinds of revision. I hope I don't have to adopt that Harmanian view, but I'll have to think about it.

I like Greg's suggestion a lot, and maybe it provides a way of avoiding the Harman view. My current view is that purely epistemic justification is a matter of how much the available information supports a belief, without regard to whether your cognitive limitations let you figure out whether the available information supports the belief. (You can be blameless in a non-epistemic way for unavoidable cognitive error, but that doesn't mean the belief itself is justified.) That obeys Greg's constraint--if A entails B, you must be at least as justified in believing B as in believing A. But your level of justification can drop when you use CI.

Right, Matt. There's something nice in obeying just a little of the connection between entailment and justification (or degree of justification), because then you're free from the worry that justification doesn't measure the content of the belief, but inheres in something else. (If you reject deductive closure, the issue comes up: just how much deductive closure do you reject?) It seems like distinguishing multiple premise arguments from single premise arguments might give you enough wiggle room. The nice thing then is that CI is all you need to add to single premise arguments to define all multiple premise arguments, so your diagnosis of the problems with CI are really characteristic of all multiple premise arguments.

(Well, those with finitely many premises anyway)

But when it comes to the evaluation of inferences construed as what thinkers actually engage in, rather than abstract arguments, I think that a good inference is one that takes one from premises that one justifiably believes to a conclusion that one justifiably believes in virtue of performing the inference.

That seems exactly right to me--and it's helpful to my larger project of replacing knowledge with justification as the property that epistemologists should study. The best inferences aren't those that preserve knowledge but those that preserve justification.

As I think you know, for me justified belief and knowledge are the same thing, so either characterization of what is preserved in the best inferences is fine with me! But that is a long story...

An interesting question is what is meant exactly by "what thinkers engage in." If we mean the voluntary actions by which thinkers affect their beliefs, those may not look much like arguments at all--I take it most inferences just happen.

I mean a psychological event or process culminating in belief formation as opposed to something purely formal, and, whatever inferences in this sense are, they had better be happening all the time -- and often pretty unreflectively. Beyond that, I have no particular views on the precise (or vague) extension of 'inference', or on whether there are sufficiently similar processes culminating in states other than belief that we might also profitably term 'inferences'.