Contradictory Beliefs
- Posted by Andrew Bailey on Thursday, April 12, 2007 at 1:10 PM
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12 Comments |
Some doxastic logics endorse the following principle: if S believes p at t, then it's not the case that s believes ~p at t. Call this rule No Contradictions.
No Contradictions gets some traction, I suppose, because of its similarity to the Law of Non-Contradiction. That contradictions aren't possible suggests to some that contradictory beliefs aren't possible either. And abiding by No Contradictions does seem to be a necessary condition for proper function or rational thought. It's natural to think that sane, rational, epistemically responsible agents just don't hold contradictory beliefs.
I'm inclined to think that this is mistaken. But first a distinction. It's one thing to believe p at t while also believing ~p at t. It's another to believe that (p&~p) at t. One can reject No Contradictions while still thinking that it's impossible for someone to believe the conjunction of a proposition and its negation (this view requires that belief not be closed under conjunction introduction).
So why reject No Contradictions? Here are two reasons. First, No Contradictions assumes that actual epistemic agents have a degree of rationality that they may not have. So it’s perhaps open to empirical falsification. This is something for the psychologists to tell us, I suppose; but for all I know we might actually find someone (perhaps someone who's not very epistemically responsible) who believes p and ~p at the same time.
But second, it seems to me that No Contradiction is subject to counterexample. If No Contradiction is going to be an axiom or a theorem of some doxastic logic based on K, then it will have to be broadly logically necessary. Any broadly logically possible case in which No Contradiction fails to hold will suffice to unseat it, then. Here's a counterexample which seems to me, at least, to be broadly logically possible: I believe that Hesperus is a star. I also happen to believe that Phosphorus isn’t (I’m convinced it’s a planet, say). In this case, I believe p and I believe ~p.
The case is interesting in that it doesn't require any acts of wild epistemic irresponsibility. So long as the subject doesn't know that Hesperus is Phosphorus, at least. To elaborate: ‘Hesperus’ and ‘Phosphorus’ are proper names that rigidly designate the same thing. The English sentence S1: “Hesperus is a star” expresses the same proposition (call it p) as the sentence S2: “Phosphorus is a star.” Suppose so. Then I’d think the sentence S3: “It’s not the case that Phosphorus is a star” expresses the negation of p. A subject believing the propositions expressed by both S1 and S3 will then hold a set of beliefs with contradictory propositional content (though not necessarily believing a contradiction proper, given that belief isn't closed under conjunction introduction). One cannot, of course, derive a contradiction from S1 and S3 without some further premise (viz., that Heperus=Phosphorus). But this doesn’t imply that the propositions expressed by S1 and S3 aren’t contradictory. And that's enough to unseat No Contradiction.
(Thanks to John DePoe for bringing these issues to my attention in this post on Liar Paradoxes; in the comments, I reproduce some of these ideas)
12 Comments:
at 3:34 PM said... If you're committed to saying that "Hesperus is a star" and "Phosphorus is a star" are both P, then it seems like you would also be committed to saying that "Hesperus is Hesperus" and "Hesperus is Phosphorus" are both Q. In other words, you seem to be fine with substituting two proper names with the same references for each other in a proposition, and the proposition remaining the same. But "Hesperus is Hesperus" is analytic and a priori, while "Hesperus is Phosphorus" is a posteriori. That's a pretty significant difference. So, my question is why would we call them both Q? Or, in your case, why are you calling the propositions both P?
Andrew Bailey at 3:54 PM said... First, I'd think that analyticity is a property of sentences, not of propositions. Propositions, for example, have no "subject" or "predicate" place, so the account according to which analytic truths are those whose subjects include their predicates can apply only to sentences with this grammatical form. In fact, I'd think that analyticity is a property that can be had only by sentences in certain languages; viz., those with a subject/predicate structure. It doesn't make much sense, then, to say that p or q are analytic, when p and q are variables ranging over propositions.
Second, I wouldn't think that "being a priori" is a property of propositions, strictly speaking. If it's anything at all, it's rather a polyadic relation between subjects, propositions, and times. What's a priori for one subject at one time is not for another subject at another time.
Here's a classic example to motivate the claim. For a skilled mathematician, it's a priori that 783x387=303021. But for the calculator-dependent, that truth is known only a posteriori.
Once we get clear on these distinctions, then, it's not at all clear to me that there's any objection left.
Andrew Bailey at 4:07 PM said... And Eric, can you elaborate that point? The Russelian (as I understand her) says that propositions are structured entities consisting of objects and properties. But why am I committed to *that*?
So far as I can tell, what I say is consistent with the Ludovician account of propositions as sets of possible worlds, or a primitivist account which says next to nothing about what propositions are (aside from specifying their functional role in one's ontology, say).
at 4:22 PM said... Hmmm... I see a bit more where you're coming from. I agree with Eric that you seem to be assuming Frege is wrong on a number of things (most significantly the role of a Sinn), though I'm not so sure he's right in saying that your view is Russellian.
As for the mathematician example, I'm not so sure we can agree enough to argue, since I agree with Kant that such claims are (even for skilled mathematicians) a posteriori. But I'll sheepishly admit that accepting this view requires a robust notion of Kantian intuition.
at 4:25 PM said... Sorry, I meant "I agree with Kant that such claims are (even for skilled mathematicians) synthetic a priori."
at 4:59 PM said... I'm also curious as to what you say about the cases:
1. Lois Lane believes Clark Kent is Clark Kent.
2. Lois Lane believes Clark Kent is Superman.
So are they the same proposition, even though they seem to have different truth values? Or do they have the same truth values?
at 5:30 PM said... To follow up on my last point. If in your example, "Hesperus is a star" expresses the same proposition as "Phosphorus is a star," then shouldn't they be inter-substitutable in a belief context? But this is exactly what you're denying with your example. So, I would take it that "Hesperus is a star" and "Phosphorus is a star" do not express the same proposition.
Andrew Bailey at 6:31 PM said... I'm no historian, so I'll refrain from saying anything about Kant.
The Superman case makes prima facie trouble for the view I've described. While there are a number of escape hatches out there in the literature, I'm not sure which one to take. So I'll simply suggest a few (not necessarily exclusive) options and leave it at that.
1. Bullet biting. 1 and 2 express the same proposition.
2. Nonetheless, Lois doesn't know that the sentence "Clark Kent is Clark Kent" expresses the same proposition as the sentence "Clark Kent is Superman." This explains why she might affirm one sentence and deny the other.
3. The retreat. While proper names in *general* rigidly designate, some don't. "Superman" is a definite description in disguise, so it's one of these exceptions. Thus, "Clark Kent is Clark Kent" doesn't express the same proposition as "Clark Kent is Superman," so 1 and 2 might very well differ in truth value.
4. Other funny phil-language tricks that I don't understand.
Finally: I said that in my case that subject doesn't know that Hesperus is Phosphorus; but this isn't quite right. What I should have said is that the subject doesn't know that 'Hesperus' refers to the same thing that 'Phosphorus' does.
Eric at 8:39 PM said... Andrew,
I'm no expert at the philosophy of language, but as I understand the terms, the Fregean view of propositions claims that propositions are distinguished by Fregean senses, so that "Hesperus is a star" and "Phosphorus is a star" pick out different propositions, due to the senses of the names being different.
I understand Russellianism to be the rejection of the Fregean claim that senses are relevant to propositions in this way.
Again, maybe I'm misusing the terms; my original comment was just that someone who thinks that propositions can be distinguished by senses will most likely deny that S1 and S2 pick out the same proposition, because Hesperus and Phosphorus have different senses.
Noumena at 6:45 AM said... A couple of possibilities occurred to me as I was failing to fall asleep last night.
1) Chisholm, for one, defines `disbelieving p' as `believing not-p'. In that case, No Contradictions says it's impossible to both believe p and disbelieve p at the same time, for any p.
2) Go Intuitionist. First, assert that propositions are beliefs or thoughts, not the object of beliefs or thoughts. Second, define not-p as not holding the belief p. Then No Contradictions says it's impossible to both hold the belief p and not hold the belief p, which is just an instance of ordinary (non-doxastic) non-contradiction.
1) ends up saying something a bit more prima facie plausible, but I still don't see an argument. 2) ends up saying something very plausible, but Intuitionist logic is usually considered highly counter-intuitive (pun intended).
Penultimately, I think the only case to be made here is one of empirical psychology: you're asking whether (or not) actual human beings ever believe two contradictory propositions, so you need to go out and find at least one person who believes both p and not-p at the same time. Armchair psychology isn't going to work unless you discover that you yourself have contradictory beliefs, but then you're really just doing empirical psychology with the most readily available subject.
And ultimately this seems irrelevant to the question of normative epistemology. Cf this line of moral thinking: `No killing except possibly in cases of self-defence' is a pretty plausible moral principle. But look, here are all these cases where people have killed and it wasn't a matter of self-defence. Conclude that the principle is false.
Or this, from non-doxastic logic: `From p->q and p, conclude q' seems like a pretty plausible principle. But look, here are all these cases where people have affirmed p->q and p and denied q. Conclude that the principle is false.
I don't think that empirical psychology is or should be irrelevant to normative epistemology -- I'm a naturalist in the Quinean vein, after all. But the fact that people have failed to live up to the canons of logic or normative epistemology or normative ethics falls far short of an argument that these canons are false.
Noumena at 7:01 AM said... As an aside, I would like to point out that your Hesperus is a star / Phosphorus is not a star counterexample depends critically on two sentences being tied to the same (ie, one numerically identical) proposition. That is, it needs identity conditions for propositions.
steve at 12:48 PM said... No contradictions seems false. My main reason for thinking so is the clear counterexamples. Plenty of perfectly rational, really smart people maintain that some instances of (p & ~p) are true. Graham Priest and other dialetheists are the most prominent examples that come to mind.
