## A more devastating version of the Raven Paradox

C.G. Hempel’s “Raven Paradox” involves derivation of the intuitively unpalatable conclusion that observation of things like a white shoe or a rainbow confirms the raven hypothesis, “All ravens are black.” Here’s how it goes. An earlier author Jean Nicod had put forward the following criteria for confirmation of hypotheses of the form “All A’s are B’s”:

Observation of an object which has the property of being an A and also the property of being a B *confirms* “All A’s are B’s.”

Observation of an object which has the property of being an A but not the property of being a B *disconfirms* “All A’s are B’s.”

Observation of an object which does not have the property of being an A neither *confirms nor disconfirms* “All A’s are B’s.”

Add to these criteria the following highly plausible claim, which Hempel called “the equivalence condition”:

If an hypothesis H1 is logically equivalent to another hypothesis H2, then, if an observation O confirms H1, then O also confirms H2.

The equivalence condition sounds perfectly true, because to say that H1 and H2 are logically equivalent is to say that H1 and H2 make exactly the same claims about the world. Thus if a piece of evidence confirms one of the hypotheses, it must equally confirm the other one.

Now, let H1 be the hypothesis “All nonblack things are nonravens” and H2 be the hypothesis “All ravens are black.” Since H1 is the contrapositive of H2, it is clear that H1 and H2 are logically equivalent hypotheses. Suppose now that our observation O is the observation of a white shoe. A white shoe being a nonblack nonraven, O confirms H1 by Nicod’s first criterion above. And by the equivalence condition, the observation of a white shoe, O, also confirms H2, the hypothesis “All ravens are black.” This result, of course, infuriates our intuitions!

I just gave you a classical, standard exposition of the Raven Paradox. Actually matters look worse. Not only does observation of things like white shoes or red tomatoes or rainbows turn out to confirm the raven hypothesis “All ravens are black,” I will show that, given Nicod’s criteria and the equivalence condition, observation of good old black ravens fails to confirm “All ravens are black.”

Here goes. Consider first the following condition which is logically equivalent to the equivalence condition:

If an hypothesis H1 is logically equivalent to another hypothesis H2, then, if an observation O *fails to confirm* H1, then O also *fails to confirm* H2.

This condition is obtained by an operation of contraposition on a portion of the equivalence condition given above. Hence let us call it “contrap equivalence condition” (for lack of a better term).

Next, let H1 be “All nonblack things are nonravens” and H2 be “All ravens are black.” H1 and H2 are, once again, logically equivalent hypotheses. By Nicod’s third condition above, a black raven does not confirm H1 (nor does it disconfirm it, for that matter), since a black raven does not have the property of being a nonblack thing. Therefore, by the contrap equivalence condition, a black raven fails to confirm “All ravens are black.”

Combining this result with the standard, classical one: Black ravens don’t confirm the raven hypothesis, but white shoes do!

This is a point where our confirmatory intuitions will just go berserk…

This is another argument supporting the claim that the third criterion (“Observation of an object which does not have the property of being an A neither confirms nor disconfirms ‘All A’s are B’s'”) is false. The “more devastating” version of the paradox is stronger than the original one, since in this version, in order to reject that an observation of a white shoe confirms “All ravens are black,” you need to accept that the apparently true proposition “An observation of a black raven confirms ‘All ravens are black'” is false.

Hasan ÇağatayFebruary 27, 2012 at 9:15 pm

I didn’t know that the third criterion was in Nicod. It may be an intuitive principle, but certainly a logical slip. Wason’s card experiments show that indeed many people are inclined to this fallacy of forgetting modus tollens. I find the best explanation of the first unintuitive consequence of Nicod’s first criterion in combination with the ‘logical equivalence condition’ in the Bayesian analyses of the degree of confirmation provided by the observation of different objects in the world. This of course depends on the distribution of ravens and black objects in the world. Assuming there are immensely more black objects in the world than ravens, and that our hypothesis does not change the way we consider these distributions (Vranas has an article on the latter), then the support provided by a white shoe for H1 or for H2 is tiny, almost null. So, the symmetry between H1 and H2 breaks down when one considers the relative numbers of A’s and B’s (or perhaps our ways of encountering them in the world through randomized trials). I wonder if any non-Bayesian analysis would give us some insight on this too. Having recently heard of H. Field on restricted quantification, I wonder if the original puzzle could be removed by modeling the reasoning here using some other kind of logic.

bernakilincMarch 3, 2012 at 3:30 pm

Thanks Berna. Yes, Nicod has the 3rd criterion also. To my knowledge, at least some of the Bayesian responses to the paradox tend to bite the bullet and affirm that a white shoe does confirm the raven hypothesis, although only to a miniscule degree, and to a degree much smaller than a black raven confirms that hypothesis. Thus their solution to the paradox is that our intuitive judgment that white shoes don’t confirm the raven hypothesis need to be given up. But I am not sure if the Bayesian approaches can be generally accepted to have solved the paradox or not, since Bayesians have their own problems that need to be tackled first. (I am planning to make another post in the near future raising a problem with the Bayesian account of confirmation.) My purpose in my post about the Raven Paradox was not to consider any possible solutions, but to formulate it in another fashion, making the paradox look more devastating than the original formulation of Hempel.

By the way, the non-Bayesian “solution” of Popper’s falsificationism easily dispels the Raven Paradox, but it too has its own troubles to deal with first.

Erdinç SayanMarch 3, 2012 at 5:26 pm

Hi Berna,

Interesting thought. What do you think that a non-classical logic could be used to block? The intuitive plausibility of the original “equivalence condition” or the “operation of contraposition” used to get ““contrap equivalence condition”. Is the thought that the plausibility of the operation of contraposition might rest on an assumption of, say, the law of the excluded middle?

I wonder if Erdinc could say a bit more about the assumptions behind the operation of contrapositon. (What exactly is the logical principle at work here?).

Berna: Do you think there is any intuitive reason to not use classical logic here? (If not, then even if it works to block the paradox it could seem an ad hoc solution).

Lucas ThorpeMarch 3, 2012 at 3:49 pm

Berna: Here’s a thought. Confirmation seems to be an either or thing – but perhaps we are better off thinking in terms of some sort of ‘evidence for’ relationship here and I guess that the ‘evidence for’ relationship might be an essentially vague predicate. It might look like a two place predicate with the first place being the proposition stating the evidence and the second place being the hypothesis. But I guess that is is probably better to think of it as a three place predicate, with the third place having to do with degrees of belief. If this is the case, then vagueness might be built into the evidence for relation. And if so we might wan’t to reject the law of excluded middle for reasoning about evidence. (at least when the evidence suggests that we should have a confidence of less than 1).

Lucas ThorpeMarch 3, 2012 at 5:38 pm

To Lucas: The contraposition operation of classical logic on a conditional statement yields a logically equivalent statement to the statement it is applied to. It can work on a material conditional, but with other kinds of conditional statement contraposition may not work. For instance, a contraposition on a counterfactual conditional doesn’t in general produce a logically equivalent statement to that counterfactual. Thus one may construe “All ravens are black” as some kind of a non-material conditional (though not necessarily as a counterfactual conditional) in a way that invalidates a contraposition on it.

Erdinç SayanMarch 3, 2012 at 6:20 pm

Hi Erdinc,

I think that was Berna’s original suggestion. So the thought is that perhaps the appropriate logic for evidence will be a non-classical logic that denies contrapositon, which would block your new version of the paradox. From talking to Ali about intuitionistic logic, such a logic blocks some contrapositon (although not, I think, the sort of contraposition we need to block here). So Intuitionistic Logic admits if A then B implies if not-B then not-A, but not the converse.

So I guess that if we bite the bullet on the original paradox – but then want to block this new paradox, then we want a logic that blocks the move from if A then B to if not-B then not-A. I wonder if the logic that Hartry Field showed us last week allows for this inference?

Does anyone know a logic that might have some plausibility as a logic for reasoning about evidence that blocks this from of contraposition?

Lucas ThorpeMarch 3, 2012 at 6:35 pm

Lucas invited me to leave a comment about logical options. I don’t think I have a lot to say about this which is very tightly connected to the issues of confirmation and disconfirmation, but I’ll point to some examples. To answer one question: yes Field’s logic does permit full contraposition.

The question whether or not contraposition is valid largely hinges on our interpretation of the target phenomenon. Are we thinking of the broadest usage of ‘if’ in plain English? That is going to include all kinds of constructions that are not even conditionals. Are we interested in the broadest cross-section of conditionals including subjunctive or counterfactual ones? Are we interested in something more like material implication? Or strict implication? All of this makes a huge difference.

One simple example of a formal system where contraposition fails in a natural way is the Routley-Meyer style ‘situation semantics’ for relevant logics. The basic idea is that we introduce something like an accessibility relation from modal logic, but instead of a two-place relation from one world to another, we are going to a have a three-place relation. We can think of Rxyz as saying something like relative to x, situation y accesses situation z. For al situations x,y,z we have that the conditional A>B is true in x just in case, if Rxyz and A is true in y, then B is true in z. A very basic system of relevant logic has a semantics on which there are no extra structural conditions on the ternary relation.

Imagine that A>B is true in a situation x in this semantics. Does that guarantee that not-B>not-A is also true in situation x? No, because it is compatible with the ternary relational semantics for this conditional that we might have Rxyz and not-B is true y and not-A is true in z. So in the basic relevant logic contraposition fails. We might even find a loose connection between this formal system and issues in philosophy of science because people like Urquart, Mares, and Restall have argued that the ternary relational represents the structure of ‘information flow through channels’.

Colin CaretMarch 3, 2012 at 7:54 pm

Hi Colin,

Thanks for that. So I guess one way to block Erdinc’s new paradox (although I suspect not the original one) would to suggest that when it comes to reasoning about evidential support we should adopt some sort of relevance logic. And there might be independent reasons to adopt such a logic for such reasoning – so this sort of response might not be ad hoc.

Lucas ThorpeMarch 3, 2012 at 8:31 pm

Yeah I wanted to echo your sentiment that this is a definite possibility, at least so far as there are natural and well-understood logical systems we could apply to this problem as you suggest. But there is a lot to think about here. Many people applying weak logics to solve philosophical problems (such as in the paradox literature) are still quite happy to accept contraposition (e.g. Field). On the other hand, there is a separate, loosely related literature on ordinary language conditionals that starts with people like Stalnaker and Lewis. Many of them think contraposition does fail because they say that, e.g. (1) does not entail (2)

1) if it rains, it will not rain heavily.

2) if it rains heavily, it will not rain.

Colin CaretMarch 4, 2012 at 5:14 pm

Yes, one way is to resort to a logic that blocks “All R are B” from being logically equivalent to its contrapositive. But that logic has to pay respect to the meaning of laws like “All R are B”; in other words, we must be able to express “All R are B” in that logic without distorting its meaning one bit.

There are of course “cheaper” ways of blocking my version, by, e.g. denying Nicod’s 3. criterion, which is what the Bayesians too end up doing. But whether we would want to buy the Bayesian solution depends on whether we are willing to buy Bayesianism as a theory of hypothesis confirmation. One may try to find grounds for rejecting the 3. criterion independently of the Bayesian ways, of course.

Erdinç SayanMarch 5, 2012 at 3:27 am

Hi Colin,

So, people like Stalnaker and Lewis reject contraposition in general because they think (2) is not a consequence of (1). Do they will also think that:

(3) if it rains heavily, it will not rain heavily.

is not a consequence of (1)?

If so, I take it they’re also happy to give up transitivity (?).

Best,

Sam

srober21March 5, 2012 at 6:03 pm

A possible way to block the paradox is to construe universal propositions in a more Aristotelian fashion.

http://www.jstor.org/discover/10.2307/186017?uid=3738984&uid=2&uid=4&sid=21100655049176

reineMarch 9, 2012 at 5:05 pm

Thanks very much. I will take a look at it. But certainly there is no lack of “solutions” on the raven market.

Erdinç SayanMarch 11, 2012 at 5:23 pm