Let us recall the setting of McGee’s counterexample. There are three candidates for the 1980 election: the two republicans Reagan and Anderson, and the democrat Carter. The polls see Carter far behind Reagan, with Anderson a distant third. Prima facie, McGee’s counterexample can go counterfactual. Suppose I know about the polls but do not receive any relevant information afterwards, perhaps because I go on a safari trip or because I just don’t care. After the time of the election I consider the following argument:

(1) If a republican had won, then if it had not been Reagan, it would have been Anderson.

(2) A republican won.

(3) Therefore, if Reagan had not won, it would have been Anderson.

Given the polls, I will find the premises highly probable although I will dissent from the conclusion. This comes as a surprise: if an inference is classically valid, the uncertainty of the conclusion cannot exceed the sum of the uncertainties of the premises. This puts pressure on the validity of modus ponens for right-nested counterfactuals.

Posted by Moritz.

Yet intuitions are quite shaky when it comes to the evaluation of counterfactuals with a true antecedent. Since modus ponens is such a valuable tool, we may happily pay the prize of blaming some intuitions which are not stable anyway to be misleading. This strategy would be appealing if the putative counterexamples were simply an idiosyncrasy of our language. But there seems to be a systematic principle in the background which serves to generate the counterexamples, namely the infamous Import/Export principle:

(IE) Counterfactuals of the form (A > (B > C)) are logically equivalent to the corresponding counterfactuals of the form ((A & B) > C).

The Import/Export principle sounds fine for many counterfactuals. Consider for instance its application to (1):

(4) If a republican had won and it had not been Reagan, then it would have been Anderson.

It seems natural to take (1) and (4) to be equivalent. Now, hardly anyone thinks that the inference from ((A & B) > C) and A to (B > C) is valid for counterfactuals. But by the Import/Export principle, any invalidating sequence for this pattern of inference generates a counterexample to modus ponens. Thus, the Import/Export principle may provide a systematic reason of why counterexamples to modus ponens occur. Modus ponens can only be saved if Import/Export fails in all the relevant cases.

In a recent talk, David Etlin considers the following counterexamples to the Import/Export principle for counterfactuals (you can find the corresponding paper here).

(5) Even if this match had lit at noon today, it would not have done so if it had been soaked in water last night.

(6) If this match had lit at noon today and had been soaked in water last night, then it would not have lit at noon today.

(7) If this match had lit at noon today, then even if it had been soaked in water last night it would have lit at noon today.

(8 ) If this match had lit at noon today and had been soaked in water last night, it would have lit at noon today.

(5) seems to be true whereas (6) is false, and (7) seems to be false whereas (8 ) is true. If this is so, then we have counterexamples to Import/Export for both of its directions. But the counterexamples are not as clear as one would like them to be. Both (5) and (7) contain an extra “even” (note also that in (5) the counterfactual in the consequent is presented in reversed order). Let us remove these additional features. We would then arrive at the following sentences:

(5′) If this match had lit at noon today, then if it had been soaked in water last night, it would not have lit at noon today.

(7′) If this match had lit at noon today, then if it had been soaked in water last night, it would have lit at noon today.

(5′) sounds quite bad, and even if (7′) is somewhat pointless, it does not seem to be false. This might be because we would probably assent to

(9) If this match had lit at noon today, then if it had been soaked in water last night, it would have had to have dried rather quickly.

Thus, it does not seem that Etlin has given us clear counterexamples to Import/Export, which is not to deny that there isn’t a challenge here.

Now, “even if”-conditionals may very well differ semantically from the corresponding conditionals in which “even” is dropped. Typically, we take an “even if”-conditional to imply its consequent. But I do not see how this feature could explain why inserting “even” in the counterfactuals under consideration effects our judgement. As you can check, the relevant implications seem to be o.k. What else can be said?

For a start, consider the following conditional which comes from (5) by turning the past past in the antecedent into a simple past tense:

(10) If this match lit at noon today, it would not have done so if it had been soaked in water last night.

This seems indeed to be true. But this is because the outer conditional is now an indicative one which only has a counterfactual in its consequent. Now, a past past in the antecedent does not necessarily command a subjunctive consequent:

(11) I had a look at this match in the afternoon. If it had lit at noon, it had not been soaked in water last night.

To my ears, the indicative conditional sounds grammatical. Since there is no double subjunctive, nested counterfactuals may be syntactically ambiguous between a counterfactual within a counterfactual and a counterfactual within an indicative. Clearly, the first interpretation is the default one. But context may sometimes favour the outer indicative interpretation. It would be nice if the pragmatics of “even” could be shown to support the indicative interpretation. Compare the following two sequences:

(12) I looked at this match in the afternoon. Probably, it had not lit at noon. Even if it had, it would not have done so if it had been soaked in water last night.

(13) I am certain that the match did not light at noon. Even if it had, it would not have done so if it had been soaked in water last night.

In (12) “probably” invites an “even” within an indicative construction. In (13), however, “certain” forces a counterfactual interpretation. Interestingly, (12) seems to be o.k. whereas (13) appears to be odd. Isn’t this evidence that “even” favours an indicative interpretation with respect to the example under consideration?

Perhaps there is also an alternative explanation. Proponents of Import/Export sometimes hold that the antecedent of the conditional within the consequent of the outer conditional restricts the antecedent of the outer conditional. So, the syntactic structure of an embedded conditional is something like this: [if A: if B] [C]. Perhaps “even” influences the syntax of a sentence by interfering with the default binding structure which then becomes [even if A] [if B: C].

Anyone out there with a better explanation?

Here is the phenomenon. In an initial context, the counterfactual

(1) If she had been at the concert, she would have seen Mick Jagger

may be truly asserted, or so it is assumed. Subsequently, the counterfactual

(2) If she had been at the concert and got stuck behind a group of tall people, she would not have seen Mick Jagger

may be accepted, too. All this is to be expected on Lewis’s account: strengthening the antecedent is not a valid rule of inference. But now suppose that (1) and (2) are uttered in reversed order: it seems that asserting (1) after (2) is not o.k. There is something odd about saying

(3) If she had been at the concert and got stuck behind a group of tall people, she would not have seen Mick Jagger, but if she had been at the concert, she would have seen Mick Jagger.

So, can the order in which these counterfactuals are uttered affect their truth-values?

 Posted by Moritz.

There is a plausible link between “would”-counterfactuals and “might”-counterfactuals. We can capture it with the following principle:

(M) A counterfactual of the form “If A had been the case, B would have been the case” is assertable just in case “If A had been the case, B might not have been the case” is rejectable (if A is taken to be possible).

On Lewis’s account, there is a straightforward explanation for this: the former “would”-counterfactuals are true just in case the latter “might”-counterfactuals are false. To put it another way, “might”-counterfactuals are duals of the corresponding “would”-counterfactuals.

Now, in order for the phenomenon of reversing Sobel sequences to get off the ground, one really needs the fact that (1) is assertable in the initial context. Given (M), this requires that 

(4) If she had been at the concert, she might not have seen Mick Jagger (because she might have been stuck behind a group of tall people)

can be rejected. But in the circumstances most naturally associated with the example, (4) cannot be rejected. This suggests that (1) wasn’t really assertable in the first place. Perhaps it seemed o.k. to assert it because one did not think of the possibility that she might have got stuck behind tall people. Plausibly, if the standards of assertability are low, a counterfactuals passes these standards if most of the close antecedent-worlds are consequent-worlds. But once certain possibilities are mentioned, the standards of assertability are raised, and beeing close to the truth isn’t good enough any longer. 

Note that the picture changes if (4) can be rejected. Suppose it is known that there weren’t any tall people around at the concert. Then (4) can indeed be rejected and (1) is indeed fully assertable in the initial context. But then it doesn’t seem clear to me that (3) would not be o.k. Perhaps the following utterance would lack a decisive point, but from a semantic perspective it seems fine:

(3′) If she had been at the concert and got stuck behind a group of tall people, she would not have seen Mick Jagger, but since there weren’t any tall people around at the concert, if she had been at the concert, she would have seen Mick Jagger.

Posted by Benjamin.

In general, supervaluationism and relativism seem to fare better as accounts of future contingents than as theories of vagueness, since there is no equivalent of the problem of higher-order vagueness for future contingents. In this note, I will try to challenge this assumption by pointing to a problem which seems to arise for supervaluationism and relativism with respect to future contingents without being as problematic in the case of vagueness. 

Posted by Moritz.

The problem I have in mind concerns our credences in sentences like ‘The coin will come down heads’. In most situations, it will be the right thing to think that it is roughly 50% likely that the coin will come down heads. The question is whether Supervaluationism and Relativism can explain this kind of epistemic attitude. The corresponding problem in the case of vagueness would concern our credences in statements about borderline cases. But it is less clear whether we should have definite credences with respect to borderline cases. It may well be that we should simply suspend jugdment (or stick with a coarse grained ordering of likelihoods). So, the problem of our epistemic attitudes with respect to sentences which are neither supertrue nor superfalse is less pressing in the case of vagueness than it is in the case of future contingents. 

Let’s state the case. Consider a convinced indeterminist who is certain that it is a truely chancy matter whether the coin will come down heads or not. Also, he thinks that there is an objective chance of 50% that the coin will come down heads. Since he has no other relevant information about the future, he assings subjective probability 1/2 to this statement. In this situation, this seems to be the right thing to think. Now given supervaluationist’s and relativist’s views about how to evaluate such statements, it is hard to see how this can be explained. For, our indeterminist has only trivial attitudes towards the sentence’s semantic value as described by supervaluationism and relativism: he can be certain that the sentence is neither supertrue nor superfalse, since he is certain that there is both an objective possibility which would verify the sentence and an objective possibility which would falsify it. Consequently, he can be certain that it is neither supertrue nor superfalse that the coin will come down heads. Hence, his credence in this statement does not aim at supertruth (nor at some weighted average between the three values as is obvious from the fact that supertruth and superfalsity get credence 0). So, how can our non-trivial credences in statements about the future be justified within the framework of supervaluationism and relativism?

In a forthcoming paper entitled ‘Knowledge and Objective Chance’, Hawthorne and Lasonen mention in passing a counterexample to the Principal Principle (a draft of which you can find here). Essentially, they think that instances of the contingent a priori provide a source of potential counterexamples. This idea stands in an interesting relation to a recent paper of Williamson (‘Indicative versus Subjunctive, Congruential versus Non-Hyperintensional Contexts’), in which the modal status of statements involving objective and subjective probabilities is discussed (a draft of which you can find there). It seems to me that Williamson’s considerations may provide a deeper reason to explain why we should not expect something like the Principal Principle to hold. In the following note, I will present a structurally similar counterexample by way of relating it to Williamson’s claims about the modal nature of the two kinds of probability.

Posted by Moritz.

The Principal Principle can partly be motivated by way of examples. Suppose a fair coin is going to be tossed tomorrow. How likely should we think it to be true that it will come down heads? 1/2, of course. Why? Because its present objective chance of coming down heads is 1/2. The Principal Principal generalises this pattern of reasoning. It states that in the absence of evidence which bears more directly on a proposition A, we should adjust our credence in A to what we take to be the objective chance of A. More precisely, the Principal Principal can be stated as follows:

(The Principal Principle)
Let be any reasonable initial credence function, any
proposition within the domain of objective chances, any time,
and any proposition which is admissible at . Now, let be
the proposition that the objective chance of at is . Then
.

A few remarks may be in order. The reference to times is needed, since objective chances obtain relative to times. Today the chances of the coin coming down heads may be 50%. Tomorrow, after the coin was tossed, the chances will be either 1 or 0, depending on whether it comes down heads or not. Now to the idea of admissible evidence. It requires a great deal of work to specify the admissible propositions. As a first approximation, one can start by saying that information is admissible if it is solely concerned with the history up to time t. For instance, every proposition which is solely about the history up to now will be admissible for the proposition that the coin will come down heads tomorrow. Since the question of which propositions are admissible will not play any role in the argument, I will leave it at that.

Let me start by pointing to the fact, highlighted in Williamson’s paper, that statements about objective chances do not generate hyperintensional contexts. A sentential operator D is said to be hyperintensional if there are necessarily equivalent sentences A and B such that D(A) is true but D(B) is false. The idea that objective chances do not give rise to hyperintensional contexts can be put like this:

(Objective Chances Are Not Hyperintensional)
If and are necessarily equivalent, then the objective chance
of always equals the objective chance of . More formally, for
all times (where denotes the objective chance distribution
at ):
.

The argument for this thesis is straightforward. Objective chances measure objective possibilities. Thus, the objective chance of a proposition or sentence to be true derives from its modal properties. But if two propositions or sentences are necessarily equivalent, they have the same modal profile and therefore the same modal properties. Hence, the expression of objective chances does not constitute a hyperintensional context.

Subjective probabilities, on the other hand, seem to give rise to hyperintensional contexts. Consider the following example. Suppose a proposition is actually true. Then the proposition is necessarily equivalent to (I use as the ‘actually’-operator). For, if we evaluate at a counterfactual world , will be true at just in case is true at , since is actually true. Now, as an obvious instance of the contingent a priori, we should always be certain about . However, we should not always be certain about any proposition which happens to be true. For instance, we should not be certain that the coin will come down heads tomorrow (even if in fact the coin will come down heads tomorrow). But, of course, we should be certain that the coin will come down heads just in case the coin will actually come down heads. Hence, subjective probabilities constitute a hyperintensional context:

(Subjective Probabilities Are Hyperintensional)
There are necessarily equivalent propositions or sentences and
such that the subjective probability of should not always
equal the subjective probability of . More formally ( being a
reasonable credence function),
.

Of course, this does not hold if one identifies propositions with sets of possible worlds as, for instance, Lewis (1980) did. But the foregoing example shows that this way of individuating propositions is too coarse grained to be sensitive to the distinctive features of our epistemic lifes.
The observation is thus that subjective possibilities measure epistemic possibilities which are, as examples of the contingent a priori and the necessary a posteriori show, incomparable in strength with metaphysical possibilities. Since subjective probabilities fall on the epistemic side of this distinction, it takes no wonder that they give rise to hyperintensional contexts.

Now to the counterexample. Let be the proposition that the coin will come down heads tomorrow, and let be today. As above, is the objective chance distribution at time (and, of course, at the actual world). The crucial observation is that the following two statements are a priori equivalent:

(1) ,
(2) .

To see this, note that the logic of ‘actually’ implies the following theorem:

(3) .

We have already argued for this principle. If is true at the actual world, then is true exactly at the -worlds, and if is false at the actual world, then is true exactly at the -worlds.

From (3), the equivalence of (1) and (2) follows by using the non-hyperintensionality of objective chances. For the direction from (1) to (2): By (3) and the non hyperintensionality, we get that the objective chance of is either the one of or the one of ; since both are the same, (2) follows. For the direction from (2) to (1): By (3) and the non hyperintensionality of objective chances, it follows from (2) that either the objective chance of is or the objective chance of is ; since both disjuncts are equivalent, (1) follows.

Now, the Principal Principle makes the following prediction:

(4) .

But this is wrong. We should assign credence to no matter what, since we can always be certain that the coin will land heads just in case it will actually land heads. Hence, there seems to be a counterexample to the Principal Principle.

One may think that conditionalizing on should undermine our certainty in . Even though I do not take this idea to be a live option (we should always be certain about a logical truth such as !), one can demonstrate the coherency of the epistemic state described by

(5) .

relative to the coherency of another state. Clearly, the following epistemic state is rational:

(6) .

My thinking that the objective chance of the coin landing heads is should not undermine my certainty that the coin will land heads just in case it will actually land heads. However, since is a priori equivalent to (as we have seen in arguing for the equivalence of (1) and (2) above), (6) is a rational epistemic state just in case (5) is. So, we have shown the relative coherency of an epistemic state described by (5): it is coherent just in case (6) desribes a coherent state. So, there is a counterexample to the Principle Principal if (6) is correct. And, as I have argued, (6) is correct.

What is the source of the counterexample to the Principal Principal? An answer suggests itself: the counterexample derives from the fact that subjective chances are hyperintensional whereas objective chances are not.

You don’t think so? Then read the New York Times’s review of Coppola’s new movie “Youth Without Youth”.

Posted by Miguel.