July 30, 2006

Church's Solution to the Knowability Paradox

In Church's "First Anonymous Referee Report on Fitch's 'A Definition of Value'" from 1945 we find not only the first formulation of the proof today known as the knowability paradox, but we find the first proposed solution. Church rejects any principle stating that propositional attitudes necessitate (other) propositional attitudes. For instance, Church rejects plausible closure principles for belief. He explains,

To be sure, one who believes a proposition without believing its more obvious logical consequences is a fool; but it is an empirical fact that there are fools. It is even possible there might be so great a fool as to believe the conjunction of two propositions without believing either of the two propositions; at least, an empirical law to the contrary would seem to be open to doubt. On this ground it is empirically possible that a might believe k' at time t without believing k at time t (although k' is a conjunction one of whose terms is k.


Church here denies that belief is closed under conjunction elimination. The context reveals that he is also denying that knowledge is so closed. More generally the idea is that if a propositional attitude is the result of another, it is so contingently. Notice that we have today departed dramatically from Church's thought. In all of the literature on the knowability paradox, for instance, it is granted that it is at least metaphysically necessary that one knows the conjuncts of known conjunctions. An exeception is found in a position articulated in Kelp and Pritchard's very interesting forthcoming " Anti-realism, Factivity and Fitch".

To pick another example,

...there is no valid law of psychology according to which anything whatsoever about my desires may be inferred from the fact that I know so-and-so.


It should be noted that Church (in the second referee report) allows Russell's theory of types to avoid such obvious counterexamples as "Necessarily, if a knows that she desires that p then a desires that p". Accordingly, the approach blocks the instance of the factivity of knowledge that is employed in the knowability paradox---viz., K(p & ~Kp) => (p & ~Kp). In this regard Church forshadow's Bernie Linsky's independent thoughts on the matter, forthcoming as "Logical Types in Arguments about Knowability and Belief".

In sum, Church's position is that his knowability paradox is indeed an invalid proof. On his view knowledge is neither closed under conjunction elimination nor unrestrictedly factive.

11 comments:

Aidan said...

Sorry, I'm being slow. But how does the proof run if that's the factivity step? We shouldn't get a contradiction from (p & not-Kp), unless realism's in more trouble than we thought.

As it's usually presented the factivity step is from K(not-Kp) to (not-Kp), i.e. an instance of the reflection principle for knowledge in Linsky's '86 terminology. What am I missing?

Joe Salerno said...
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Aidan said...

Ok, thanks, that's cleared things up for me. I wasn't familiar with that way of laying the proof out.

Joe Salerno said...

[I'M RESETTING MY LAST COMMENT WITHOUT THE FORMATTING PROBLEMS. Sorry for the confusion]

Thanks Aidan. In response to your first comment. There are several versions of the result. I was thinking about the relevant reductio in the following way:

1. K(p & ~Kp)
2. p & ~Kp (from 1 by Factivity)
3. ~KP (from 2)
4. Kp (from 1 by K Closure Under &E)
5. Contradiction (3, 4)

Mike said...

"Church here denies that belief is closed under conjunction elimination. The context reveals that he is also denying that knowledge is so closed. More generally the idea is that if a propositional attitude is the result of another, it is so contingently."

I don't think I've ever found the argument against closure convincing. I think I'm just not tracking the argument, but it appears to be an epistemological argument against a proposed logical principle. Let's agree that we are sub-ideal epistemologically (i.e., that we do not believe all of the consequences of our beliefs). How does it follow that the concept of belief is not closed under logical consequence? What does the epistemic point have to do with the logical one? The closure principle--as with all logical principles on my reading--is a normative one, not a descriptive one. It does fail to describe what people in fact believe. But so what? Suppose I argue that MP does not govern deductive inference because it is incompatible with the way we actually draw inferences. Or suppose I argue that it is a contingent matter that people arrive at q, from p -> q and p, so such a rule cannot govern deductive inference. If it did govern deductive inference, why, everyone would draw q from p -> q and p. Silly argument, no? So what am I missing? I'm guessing that there is a better argument available (maybe from the lottery paradox).

Joe Salerno said...

Thanks, Mike.

Whether the "concept of belief" is closed under an operation will of course depend precisely on which concept that is. There is a concept of belief that is clearly closed under entailment---namely, the concept of rational commitment. If one is rationally committed to p then one is rationally committed to all of p's entailments. However, as you admit, "we do not believe all of the consequences of our beliefs". So you grant that there is a concept of belief that is not closed under entailment. It is the latter concept (and corresponding concept(s) of knowledge) that seem to get the attention in mainstream epistemology. I suspect that this is because the questions there tend to be about whether actual subjects deserve credit or have something valuable (in some sense the term) that is deserving of the knowledge attribution. Formal epistemologists have traditionally been more interested in idealized systems outside the realm of actual human situations in which we credit each other and exchange information in humble ways. That I suspect is among the reasons that they tend to embrace strong closure principles without apology.

I don't think that any of us are committed to thinking that the logic of knowledge or belief is descriptive (rather than normative). Affirming the consequent is invalid, even if people never do it. that's because it is not truth preserving. Analogously, affirming belief in a conclusion when there is belief in the premise would be invalid, even if in the actual world were to have the time, resources, and fortuitousness to come to believe all these conclusions. And that is because it is not necessary that we come to believe all these conclusions. Williamson and others make use of the fact that drawing inferences takes time. And so, there are possible worlds where our inferences are thwarted along the way.

The suggestion that the logic of belief and knowledge is *merely descriptive* suggests that their "principles" are contingent facts. But again, even if our inferences were not thwarted in the actual world (and we were to draw all the consequences of what we believe), the relevant closure principles for belief (and knowledge) would still be invalid. That is because the facts about what inferences we do draw are contingent facts.

Mike said...

Thanks Joe, you write a lot that is interesting, but I will focus on this,

"even if our inferences were not thwarted in the actual world (and we were to draw all the consequences of what we believe), the relevant closure principles for belief (and knowledge) would still be invalid. That is because the facts about what inferences we do draw are contingent facts."


I can't see it. Whether the relevant closure principle is invalid or not depends on what is the correct logic of belief. Whether (say, LB) is the correct logic of belief will depend on whether the theorems of LB are validated in the semantic models that are correct for belief. The facts about what inferences we do draw do not seem (non-circularly) relevant to what the correct models are.
So, for instance, whether the right semantics for necessity include models that validate all of the theorems of S5 does not depend on how people happen to use the necessity operator. It's perfectly possible that talk of logical necessity is, in many contexts, just badly mistaken (and indeed it is). Why would we take consistently bad modal inferences employing the necessity operator as indicative of anything about the logic of necessity? There is good reason to doubt that it is relevant (witness the mess in some medieval modal discussions, though they were highly sophisticated in their discussions of necessity). So I can't see why determining the correct logic of belief should take it's cue from the casual use of 'belief' in ordinary contexts. When someone says, "but S believes (p & q) but doubts p" our response should not be "oh, that can't be, given the logic of belief". Our response should be "he logically should believe p, despite the fact that he doubts it."
I guess one way to avoid this problem, and maybe this is what you're getting at, is to urge that the very concept of belief whose logic we are seeking is determined entirely by the ordinary way that 'belief' is used. In that case we try to read the logic off of the use. But that seems to me a strange idea. We don't take so empirical an approach for the logic of necessity or for the logic of probability, and so on. Further we run the risk of locating a parochial logic of belief: why should such a parochial logic give us any worries in paradoxes? Just wait a while: the logic will change and the paradox will be gone.

Brit Brogaard said...

"Whether the relevant closure principle is invalid or not depends on what is the correct logic of belief. Whether (say, LB) is the correct logic of belief will depend on whether the theorems of LB are validated in the semantic models that are correct for belief. The facts about what inferences we do draw do not seem (non-circularly) relevant to what the correct models are".

That's interesting. But what determines whether the theorems of LB are validated in the semantic models? Isn't it ultimately our intuitions about the concept of belief that determines this? Are those intuitions empirically grounded? Probably.

In any event, it could be that Church is employing a different concept of belief, viz. one that denies that belief is closed under conjunction elimination.

Kevin said...

Hi Joe, -- I couldn't be more pleased to have stumbled onto your blog as I browsed other philosophy blogs. I wish I had something to contribute here, but this is all above my head. If you have time you may want to check out my blog at www.civilwarmemory.typepad.com. Hope all is well with you in St. Louis. Best, -- Kevin M. Levin

Mike said...

Brit, you write,
"Are those intuitions empirically grounded? Probably."

I gather that they're a priori. Questions about the logic of belief at least seem to be questions about the concept of belief. So I think I can observe lots of misuse empirically.


"In any event, it could be that Church is employing a different concept of belief, viz. one that denies that belief is closed under conjunction elimination".

Yes, I agree. But this takes lots of the interest from the problem. Suppose we are arguing about whether the S5 axiom governs logical necessity (or broad logical necessity) and I settle the issue this way: "well, the concept of necessity I happen to have in mind is governed by S5". Ok, but I think I've taken all of the interest out of the dispute. I thought the dispute was whether *the* logic of necessity is S5, not whether there is some concept or other of necessity that has the logic of S5.

Mark Jago said...

Hi. Mike said: "The facts about what inferences we do draw do not seem (non-circularly) relevant to what the correct models are." Well, drawing inferences is relevant to belief formation. This is non-circular because *inferences* here can be supra-classical, default or downright invalid inferences. Such inferences produce new beliefs, even if you assume belief is classically closed. On the other hand, acts of inference never produce new necessities. That's why *actual* inference is relevant to belief.

My main worry in this discussion is the idea of a logic of belief. It seems odd to me, for the following reason. Belief cannot be classically closed because it's an empirical fact that we have inconsistent beliefs, but don't believe anything in the language. The way we revise our beliefs is again an empirical matter: it isn't governed *just* by some AGM-style operation or whatever. Actual belief states do not uniquely determine a non-trivial ideal belief state. Because of this, it is hard to claim that doxastic idealisation is harmless.

Against this, we think there's something odd about believing p&q but not p. This oddness goes beyond rational commitment; it sounds downright irrational. I think we have a kind of sorities problem with belief ascriptions: it is very uncomfortable ascribing B(p&q) & -Bp. But it is equally uncomfortable ascribing belief in any tautology whatsoever. Think of the number of inference steps required to get from belief 1 to belief 2: if it's small (ie p from p&q) we're uneasy. And if you believe p, and belief in q is just one inference step away, we want to say you believe q as well. But we don't want full transitivity, such that you also believe whatever follows from p.

I think an argument for non-closure can be given by assuming a conceptual link between belief, desire and action. Take a chess player who makes a stupid move, but who had all the logical machinery to make a better one. Assuming closure prohibits our explaining that action in terms of belief and desire. But if we can't explain belief inthat way (or at least assume some such conceptual link), then belief becomes very obscure.

Mark