Fog

fog

and the television’s gone
go to the grocery store, buy some new friends
and find out the beginning, the end and the best of it
well, do you need a lot of what you you’ve got to survive?

modest mouse, “teeth like god’s shoeshine” (the lonesome crowded west)

phil 305 ongoing presentation-paper-project-thingy for bill lycan
xxxx words
4 January 2002

tony smith
303b mason farm road
chapel hill, nc 27514
(919) 914 6840
t_rex@unc.edu

making things less transparent

Eventually I’ll get to talking about Fred Dretske. But let’s begin with a valid argument.

    Tegucigalpa is in Nicaragua
    Tegucigalpa is the capital of Honduras
    \ The capital of Honduras is in Nicaragua

Not sound, of course, given that the first premise is false, but valid. Perhaps that premise is not blatantly false, though. The capital of the United States is not in a state. Perhaps “recognised governments in exile” have “capitals in exile,” outside the mother country.
At any rate, if you’ll pardon the sins of using names to symbolize definite descriptions, and using names in identity statements, and ignoring crucial issues to do with necessary and contingent identity, we could symbolise this argument as follows (“a” = “Tegucigalpa,” “b” = “the capital of Honduras,” “c” = “Nicaragua,” “I”= “in”):

    Iac
    a=b
    \ Ibc

Ignoring the sins again, we could transform this argument into a true statement about entailment (whatever entailment actually means):

((Iac & a = b) & ((Iac & a = b) => Ibc)) => Ibc

Which is just a special case of the more general principle of entailment:

(0) (p & (p => q)) => q

Which looks fairly plausible.

Things don’t work out so well when we throw belief operators into the mix. Consider the following argument:

    Dubya believes Tegucigalpa is in Nicaragua
    Tegucigalpa is the capital of Honduras
    \ Dubya believes the capital of Honduras is in Nicaragua

Suppose that premise one this time is actually true, Dubya being a geographically-challenged sort of chap. But Dubya isn’t a terribly subtle thinker all around, and just doesn’t even entertain the possibility that the capital of Honduras is in Nicaragua. So the argument is invalid.
So, doing the same sort of rough-and-ready transformation as before, we can say that the following entailment statement isn’t true (B being the belief operator):

(BIac & a = b & ((Iac & a = b) => Ibc)) => BIbc

Indeed, given that Dubya isn’t a terribly cerebral chap, it’s plausible that he could believe both premises and not believe the conclusion. So the following entailment statement isn’t true either.

(B(Iac & a = b) & ((Iac & a = b) => Ibc)) => BIbc

Which is a counterexample to the following general principle of entailment:

(1) (Bp & (p => q)) => Bq

Note that the following is also not a general principle of entailment:

(2) (Bp & B(p => q)) => Bq

I think we’ve all had days like this in our philosophical careers. We believe something, say “p,” and then some utter bastard points out some entailment to you, say “p => q.” And you believe them. Then you realise that given (0) it must follow that “q,” but you don’t happen to believe “q.” And then, unless you’re reasonably calm and collected about this sort of thing, an unpleasant time is had by all trying to sort out which belief you need to modify, during which time Bp, B(p => q), and ~Bq are all true. This state can persist for some time. It might persist for years.
Let’s go to the knowledge operator now rather than the belief operator. The knowledge analogue of (1) is not a general principle of entailment:

(3) (Kp & (p => q)) => Kq

Switching examples, Dubya (who is slightly better at classics than geography) may know that Cicero denounced Catiline. Given that Cicero is Tully, the truth of Cicero denouncing Tully entails the truth of Tully denouncing Catiline. But it just so happens that Dubya doesn’t even believe that Tully denounced Catiline (he isn’t that good a classicist).
On the other hand, the knowledge analogue of (2) does plausibly look like a general principle of entailment:

(4) (Kp & K(p => q)) => Kq

Perhaps (4) is especially compelling if you treat belief in p as asserting the truth of p with some degree of confidence, and knowledge of p as a justified true belief in p. Analogues of the counterexample to (2) don’t seem to arise. If I know p, it follows that p. If I know p => q, then it follows that q. By (0), it follows that q is true. So it seems that I can’t not know q — awareness that q is entailed by true premises surely counts as adequate justification, q is true, and I can’t assert that I’m not confident at all about the truth of q without denying the truth of the premises (in which case I didn’t know them to begin with), or without denying the truth of (0), which seems heroic, or without claiming to be able to know true contradictions, which is at least as heroic as denying (0), and might just count as a form of denial of (0).<1>
But perhaps you think JTB has been Gettiered to death. You suspect we need some other kind of theory of knowledge. In that case, it seems plausible that the knowledge operator behaves differently from as outlined immediately above. In that case, the symmetry between (0) and (4) and the asymmetry between (0) and (2), and (4) and (2), might start to look suspicious. Adding one kind of epistemic operator to (0) results in a false principle of entailment. Adding a different kind of epistemic operator, in the same places, to (0) does result in a true principle of entailment? That’s not something we might think obviously follows.
This has all been by way of a preemptive strike against certain criticisms of Fred Dretske’s paper “Epistemic operators.” Principles (3) and (4) are closure principles — closure under entailment and closure under known entailment respectively. Keith DeRose points out:

Many, in fact, consider the anticlosure implications of Dretske’s and Nozick’s theories to be reductios of those theories. To their credit, both Dretske and Nozick admit the intuitive power of closure.<2>

I’m just not convinced that we do have such powerful intuitions about closure with respect to epistemic operators. The somewhat perverse construal of Quine’s examples of failure of substitution of logical equivalents into what he called (at the time) “intensional contexts,” as examples of failures of “belief closure” principles is meant to illustrate this.<3> We’ve known for some time that something decidedly funny goes on with epistemic operators in similar circumstances. So our intuitions about closure and knowledge operators are hardly as monolithic and robust as DeRose suggests.
    Indeed, here’s an argument that (3), closure under entailment, just is false. The idea is that so long as we know at least one thing we know an arbitrarily large (perhaps infinite) number of things. I’ll take something that I’m pretty sure I know — I exist (I’ve never been able to come up with a convincing response to the cogito). Let “A” = “I exist.” So KA. Now “A and A” certainly seems to be entailed by A. So K(A and A). But “A and A and A” also certainly seems to be entailed by A. So K(A and A and A). And so on, ad infinitum. Do I really know an arbitrarily large number of such things? I don’t think so. I don’t have an arbitrarily large brain. Where would I put all these pieces of knowledge? I suppose that if I went in for a gonadology, or for a Roger Penrose-style “quantum superposition” model of the mind, then perhaps I could know all these things. But I’m pretty sure that I actually don’t have anywhere to put all these different pieces of knowledge. On that plausible assumption, I can’t know all these things. So we have a counterexample to closure under entailment.
    Indeed, so long as I also know how conjunction works, we can work this example up into a counterexample for (4), closure under known entailment. Once again KA. I happen to know, given the way that conjunction works, that A entails any string of n greater-than-or-equal-to 1 A’s joined together with n-1 conjunctions. So once again, I seem to get KA, K(A and A), K(A and A and A), and so on ad infinitum, but I don’t have enough space for all those pieces of knowledge.
    A piece of skeptical equipment borrowed from Kripke on Wittgenstein might come to my assistance here in blocking closure. Perhaps I don’t know how “and” works. Instead I know how “qunand” works. (Qunand is typically spelt and pronounced “and,” but for the sake of illustration, I’ll use this deviant spelling.) Qunand is a bit like quus, only slightly less precisely defined. The idea is that qunand works much like “and,” for small numbers of iterations of quonjunctions. (Which is typically spelt and pronounced “conjunction.”) A entails A qunand A, and A qunand A qunand A, and so on. But for the nth such addition of a quonjunction (precisely what number n is for me is a little obscure but we know that n greater-than-or-equal-to 7), n+1 A’s joined by n qunands is not entailed. What is entailed is shouting “A, damn it, just A!” So if I know A, I only know a finite number of quonjunctions of A, and I know I exist damn it, I just know I exist!
    I think variants of the quonjunction block are how we actually deny epistemic closure in ordinary life. After a little while of being pressed on matters, we start shouting that we know some stuff, and don’t know any of this other nonsense, even if it is entailed or known to be entailed by what we do know. It does come at a bit of a philosophical cost however. Effectively, the quonjunction suggestion involves denying that we really know how things like the truth functional operators are supposed to work. It would probably be nice if we could keep our knowledge of bog-standard deductive logic, and still deny epistemic closure. Enter Fred Dretske.

degrees of transparency

Begin with some terminology. A sentential operator (or just operator) transforms one sentence into another sentence. Let Q be a necessary consequence of P. An operator O is said to be fully penetrating when O(Q) is a necessary consequence of O(P). Putting things into entailment language:

Definition of full penetration: An operator is fully penetrating if it possesses the quality that if P entails Q, then O(P) entails O(Q).

Dretske’s examples of fully penetrating operators are “it is true that,” “it is a fact that,” “it is necessary that” and “it is possible that.”
    Not all operators are fully penetrating. Some hardly penetrate the simplest logical consequences of sentences. For example, “it is strange that” doesn’t penetrate conjunctions. “P and Q” entails “P” and “Q.” But it might be strange that P and Q, but not at all strange that P and not at all strange that Q. Dretske calls “it is strange that” a non-penetrating operator. (Other examples are “it is accidental that” and “it is strange that”.) Note that Dretske doesn’t think that these operators are totally impotent (his pun, not mine), but that they are good examples of operators “on the other end of the spectrum” of penetration from the fully penetrating operators.
    If you have two ends of a spectrum of penetration, it stands to reason that some operators lie in the middle of the spectrum (including perhaps, “it stands to reason that”). Dretske calls these semi-penetrating operators and claims that all epistemic operators are semi-penetrating. His examples are: “S knows that,” “S sees that,” “S has reason to believe that,” “there is evidence to suggest that,” “S can prove that,” “S learned that” and “in relation to our evidence it is probable that.”
    These operators apparently do penetrate elementary logical consequences. Quoting Dretske, p133:

… it seems to me fairly obvious that if someone knows that P and Q, has a reason to believe P and Q, or can prove P and Q, he thereby knows that Q, has a reason to believe that Q, or can prove (in the appropriate epistemic sense of the term) that Q.

We also get to know P or Q if we know that P (or we know that Q). It seems therefore, although Dretske doesn’t explicitly mention the point, that if we know P we get to know P and P. I’ll return to that point later, but for the moment note that Dretske’s attack on closure is going to be proceeding along different lines from my quonjunction solution. We do get to know the basic logical operators.
The tricky job is showing that the epistemic operators aren’t as penetrative as the fully penetrative operators. First point Dretske makes is to note what would count as a failure of full penetration. Quoting again, p134:

When we are dealing with the epistemic operators, it becomes crucial to specify whether the agent in question knows that P entails Q. That is to say, P may entail Q, and S may know that Q, but he may not know that Q because, and perhaps only because, he fails to appreciate the fact that P entails Q.

In other words, (3), or closure under entailment fails. It fails all the time. Dretske thinks it’s a boring sort of failure, and isn’t really interested in it. What he wants to show is that (4) fails, closure under known entailment fails. Establishing this, he thinks, will block a lot of skeptical arguments. Quoting again, p135:

S claims to know that this is a tomato. A necessary consequence of its being a tomato is that it is not only a clever imitation which only looks and feels (and, if you will, tastes) like a tomato. But S does not know that it is not a clever imitation that only looks and feels (and tastes) like a tomato … Therefore, S does not know that this is a tomato.

Let “T” = “tomato,” and “I” = “imitation.” Substituting into (4) we get (KT & K(T => ~I)) => K~I. But ~K~I. By hypothesis, K(T => ~I). So ~KT after all. This all depends on the truth of the principle of closure under known entailment. If Dretske can show that closure under known entailment fails — by virtue of the epistemic operators only being semipenetrating — we do get to know that this is a tomato, even though we do not know it is not a clever imitation.
First an example, then the theory. Quoting p136:

Suppose you have a reason to believe that the church is empty. Must you have a reason to believe it is a church? … Your reason for believing the church to be empty may be that you made a thorough inspection of it without finding anyone. That is a good reason to believe the church empty. Just as clearly, however, it is not a reason, much less a good reason, to believe that what is empty is a church.

Good point, but an odd point. We’re in the game of blocking closure under known entailment. Aren’t we supposed to know the church being empty entails that it is a church? Otherwise we’re not in the game of attacking closure under known entailment. If this is an example of semipenetration of an epistemic operator, it seems to amount to a denial that we do know the relevant entailment.
Perhaps the theory will make things a little clearer. Quoting again, p137:

The general point may be put this way: there are certain presuppositions associated with a statement. These presuppositions, although their truth is entailed by the truth of the statement, are not part of what is operated on when we operate on the statement with one of our epistemic operators. The epistemic operators do not penetrate to these presuppositions.

This doesn’t help. We may well have a failure of penetration here, but it isn’t relevant to closure under known entailment. His point is that the truth of any statement entails certain presuppositions. But “knowing” the statement doesn’t entail knowing the presuppositions. True enough, but this is just an example of failure of (3), closure under entailment, not known entailment.
    Let’s try another tack. Dretske introduces the notion of a contrast consequence. Let x have some predicate A. Let B be some predicate incompatible with A — x cannot both be A and B. So it follows from the truth of x is A that x is not-B. It also follows that, for any predicate Q, that x is not (B and Q). Call all of these consequences of the form “x is not (B and Q)” contrast consequences.
    I’ll diverge from Dretske’s presentation at this point, but I think what I’m going to say preserves the sense of his point. Refer to a contrast consequence of Ax as “~Cx.” There’s an infinite number of them, all entailed by Ax. Grant that we can know this. If you’ll pardon the lapse into the second-order predicate calculus, we can know that K(Ax => (C)~Cx). If we plug this and the knowledge claim Kax into (4), it seems that any given knowledge claim entails that we know something odd-looking: it entails K(C)~Cx. But surely we don’t know that. It suggests we know that every contrast consequence we can think of is false, that every contrast consequence that we haven’t thought of is false, that every contrast consequence that we will never even dream of is false. So it seems to know anyone thing involves a claim that we know an infinite number of other things are false, which seems wildly hubristic. This is all grist to the skeptical mill.
    But this only follows if we think that epistemic operators are fully penetrating. Dretske’s claim at this point is that the operators only penetrate to a subset of the contrast consequences — relevant alternatives. To know Ax is to know that some but not all ~Cx’s are false. Quoting again, pp137–8:

You take you son to the zoo, see several zebras, and, when questioned by your son, tell him that they are all zebras. Do you know they are zebras? Well, most of us would have little hesitation in saying that we do know this. We know what zebras look like, and besides, this is the city zoo and the animals are in a pen clearly marked “Zebras.” Yet something’s being a zebra implies that it is not a mule and, in particular, not a mule cleverly disguised by the zoo authorities to look like a zebra. Do you know that these animals are not mules cleverly disguised by the zoo authorities to look like zebras? … I don’t think you do. In this I agree with the skeptic. I part company with the skeptic only when he concludes from this that, therefore, you do not know that the animals in the pens are zebras. I part with him because I reject the principle he uses in reaching this conclusion — the principle that if you do not know that Q is true, when it is known that P entails Q, then you do not know that P is true.

This looks better. This genuinely is an attack on closure under known entailment involving a failure of penetration. The epistemic operators do not penetrate fully to all the contrast consequences, merely the relevant alternatives. So I do know that those are zebras in the pen, even though I don’t know that they are not disguised mules. I know I have two hands, even though I do not know I am not a brain in a vat. And so on.
Pretty. We get to keep the penetration through simple logical implications, but we don’t get an endorsement of closure under known entailment. We might even get to keep a version of JTB, where knowledge is a true belief for which all the relevant alternatives have been ruled out. But I’m not quite convinced the story works, or solves the problem I noted in the previous section.
To begin with, why doesn’t the knowledge operator penetrate through to all the members of the contrast class? Epistemic operators penetrate entailments involving ands and ors. It seems to make sense to suppose that they also penetrate entailments involving negations. Any given contrast consequence can be constructed by an appeal to an “exclusive or” statement and some fairly primitive rules of truth functional logic.

    Either Ax or Bx and not both
    Ax
    ~Bx (exclusive variant on the disjunctive syllogism)
    ~Bx v ~Qx (addition)
    ~(Bx & Qx) (one of de Morgan’s laws, I think)

So if epistemic operators semipenetrate, they ought to be able to penetrate through these rather simple logical operations to any contrast consequence, whether or not it’s a relevant alternative. To not get through, the operators would have to be non-penetrating. So it looks like knowing that Ax should involve knowing that all the contrast consequences are false. So Dretske doesn’t get to part company with the skeptic after all. In order to know something is true, we have to know that an infinite number of things are false. So we don’t know anything.
And, as I hinted at before, if we did happen to know anything, we would also know an infinite number of things. Epistemic operators are sentential operator and semipenetrate. If it is the case that knowing P and Q entails knowing P and knowing Q, and knowing P entails knowing P or Q, then it is surely the case that knowing P entails knowing P and P. If P is a sentence, “P and P” is a perfectly legitimate (if boring) sentence that can be plugged perfectly legitimately into a sentential operator. If Dretske is allowing epistemic operators to penetrate through simple and-eliminations, and or-introductions, surely they penetrate through simple and-introductions. After all, if we know P and know Q, surely we know P and Q? So it looks like we do know P and P. And we know P and P and P. And so on, ad infinitum. Where are we supposed to keep all that knowledge?
So. My line is that Dretske’s enterprise is thoroughly legitimate, and much more intuitive than, say, DeRose, or even Dretske is prepared to grant. I’m just not convinced that this enterprise start-up ends up working.

Footnotes

1. Note however, that I’m going to be talking about Fred Dretske, who doesn’t want to be giving up JTB but rather offers a variant of JTB in which J = “all relevant alternatives have been ruled out.”

2. p15, DeRose, Keith; Warfield, Ted; editors: Skepticism: a contemporary reader (New York: Oxford University Press, 1999)

3. Quine, W V O: “Notes on existence and necessity,” pp77–91, Linsky, Leonard: Semantics and the philosophy of language (Urbana: University of Illinois Press, 1952)