Do Counterfactuals Violate Modus Ponens?

9 08 2008

There has been an intensive debate about whether modus ponens fails for indicative conditionals. Less attention has been paid to the question of whether similar examples can be constructed for counterfactuals as well. This is insofar surprising as McGee claimed that the Import/Export principle (which leads to the counterexamples for indicatives) holds also for counterfactuals. So, are there counterexamples to modus ponens for counterfactuals?

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?




14 responses

9 08 2008
Kenny Easwaran

There’s a paper by Lars Bo Gunderson arguing for a probabilistic understanding of the closeness relation among worlds, which ends up with the consequence that counterfactuals violate modus ponens. There was an interesting response to that paper by Charles Hermes that I was commenting on at the Pacific APA a year or two ago, but I’m not sure what he’s done with that paper since. I think I was agreeing with Charles when I suggested that the probabilistic semantics really can’t be thought of as a closeness relation, but can still give an interesting theory. However, the theory sounds much more plausible as a direct analysis of disposition talk, rather than as an analysis of the counterfactuals that are supposed to analyze dispositions. But maybe thinking of the theory in light of the McGee sorts of counterexamples would make it seem more plausible.

10 08 2008

Hi Kenny,

there may indeed be interesting connections between probabilistic approaches to counterfactuals and the failure of modus ponens, although I am not quite sure how they would go.

In a recent talk at the Konstanz conference on conditionals and ranking functions, Hannes Leitgeb also suggested a probabilistic semantics for counterfactuals. Basically, his idea is to say that a counteractual is true if the objective conditional probability of the consequent given the antecedent is 1. This predicts failures of modus ponens even for unembedded counterfactuals, for if the actual world is super-unlikely, it may well be that it is part of a zero-set such that the antecedent is true at it and the consequent false, even though the conditional probability is 1.

11 08 2008

I’m curious about the notion of a ‘counterfactual with a true antecedent’ which you seem to take as unproblematic. I assumed that counterfactuals characteristically have false antecedents; isn’t that what the etymology suggests? The examples don’t do it for me either; they just seem ungrammatical. In a situation where I know the election has occurred, but not what the result is, wouldn’t the relevant reasoning be better expressed as follows?

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

(2a) A republican has won.

(3a) Therefore, if Reagan has not won, it must have been Anderson.

This just looks like a straightforward indicative conditional.

I think the problem is that your (1) only looks assertible by a speaker who knows that a Republican has not won (just as (3) only looks by a speaker who knows that Reagan has won.) Then the presuppositions of (1) are inconsistent with (2), so it’s not surprising that together they lead to an unacceptable solution. I’m wondering whether all such examples can be treated this way, in which case we might not have to contend with failure of MP for counterfactuals.

I’ve just looked briefly at the Etlin paper. He makes this point in the final section…

12 08 2008

Hi Al,

the taxonomy I used is not ideal. I don’t take “counterfactual with a true antecedent” to be a contradictio in adjecto. Perhaps “subjunctive” would be a better term.

Even though we tend to use a subjunctive when we know that the antecedent is false, this is not always the case. Consider, for instance,
(1) If he had been the murderer, we would have found his fingerprints on the pistol. And look, we indeed found his fingerprints on the pistol.
In this case, we actually want to transfer evidence on the truth of the antecedent by showing that if it had been true, things would have been as they actually are. Once you start looking for examples like this one, you actually find quite a lot of them.

Hence, subjunctives do not even carry the presupposition that the antecedent is false. Perhaps it is true that they trigger the presupposition that it is at least epistemically possible that the antecedent is false. But that presupposition would be satisfied in the subjunctive version of McGee’s counterexample.

Now, even though I can’t really point to it, I share the vague feeling that there is perhaps something odd about the counterexample. However, the challenge would be to argue that it is indeed a semantic oddity we are confronted with. If it is only pragmatic, that would not be enough to get around the failure of modus ponens, since this is a purely semantic claim.

12 08 2008

OK – I understand better if what you’re talking about is (in my terms) ‘conditionals in the subjunctive mood’ rather than ‘counterfactuals’. I still think the reasoning in the scenario you discuss (person on a desert trek unaware of the results) would be better expressed by non-subjunctive conditionals (1a and 3a).

But you’re right that some non-counterfactual conditionals do use the subjunctive mood. The question is: can we construct a McGee-style problem case with a a non-counterfactual subjunctive conditional? I can’t myself see a way to do it – but maybe I’m just being thick. If such a counterexample can’t be constructed, then maybe the relevant dividing line is between counterfactuals and non-counterfactuals, and not between subjunctives and indicatives.

14 08 2008

Hi Al,

I guess that McGee-style examples for subjunctives with a true antecedent will all more or less look like the one discussed. So, probably the story one tells about this special case is going to generalize (although one never quite knows). On the semantic level, however, one still needs to say something about Import/Export, i.e. justify that it fails in the majority of cases contra to prima facie evidence.

15 08 2008
Andrew Bacon

Hi Moritz, Al,

I’m not sure I quite got the counterexample:

(1)* If he had been the murderer, we would have found his fingerprints on the pistol. And look, we indeed found his fingerprints on the pistol.

This sounds bad to me. The the use of the past perfect in the consequent strongly suggests that they didn’t find fingerprints on the pistol (and, further, that he’s not the murderer.) What is ok for me is the following mixed conditional, with a simple past in the consequent:

(2) If he had been the murderer, we would find his fingerprints on the pistol. And look, here are his fingerprints on the pistol.

But I wouldn’t count this as a ‘proper’ subjunctive, for it is only partly in the subjunctive mood.

That said, I don’t think there is anything incoherent about the idea of a proper subjunctive with a true antecedent. These sentences only sound bad when it is *believed* between the speaker, and the relevant audience, that the antecedent is true. If I say “If a republican had won, it would have been Reagan”, thinking that the republicans lost, I don’t think anyone could object to my assertion upon later discovering that I’d been mistaken and the republicans lost.

So my conjecture is that every *proper* subjunctive (i.e. excluding mixed conditionals like (2)) has a presupposition that the antecedent is believed to be false. What do you think?

15 08 2008
Andrew Bacon

BTW, the MIT theory of conditionals has a really neat explanation of the failure of MP you refer to: it’s not really an application of MP! On the MIT theory, ‘if’ clauses act as restrictors on modals, so the logical forms of conditionals are very different from the ordinary analysis as connectives. E.g. in premise (1) in the argument the two if clauses are doubly restricting the modal ‘would’. The logical form of the argument does not instantiate a valid inference pattern – in particular – it does not instantiate modus ponens.

Note also that on the MIT theory MP can fail even for simple (non-embedded) conditionals if the accessibility relation of the modal in the consequent fails to be reflexive. (Take, for example, doxastic ‘must’, or ‘should’: “If John did it, he should go to prison” may be true, even though the legal system failed to put John in jail despite his being guilty.)

16 08 2008

Hi Andrew,

what about an example involving `actually’ which is quite often used:

I want to argue that Jack is likely to be the murderer. You are unsure. I say: If Jack had been the murderer, we would have found exactly the traces we actually found.

Seems I use here a proper subjunctive whose antecedent I believe to be true. Even though the hearer does not believe it, she can be assumed not to disbelieve it either. So, my guess is that a subjunctive can be used in a context in which no participant categorically believes the antecedent to be false.

By the way, I have come to like the Kratzer restrictor theory a lot. However, it is not easy to say what it means that modus ponens holds for the conditional. For on her theory, there are no bare conditionals, there are only (sometimes implicitly) modalized conditionals. So, on the level of logical form, there is nothing like (A ARROW B) to be found. Strictly speaking, there is no instance of an application of modus ponens, be it valid or invalid. One could say that the question whether modus ponens is valid arises when we suppose the modal to be an epistemic or perhaps an ontic `must’. But then the validity of modus ponens is not so much a question concerning the conditional, but rather a question about the logic of restricted necessities.

That connects with your final point. I am not convinced that it should be seen as a failure of modus ponens for simple indicative conditionals. After all, we would conclude that John should go to prison. So, since on the surface `should’ has narrow scope, an application of modus ponens is o.k. What is not o.k. rather concerns the logic of deontic necessities: For an deontic `must’, `Must p’ does not imply p.

However, we can also give `should’ wide scope:

(1) The following should be the case: if John did it, he goes to prison.

We can then suppose with Kratzer that the `if’-clause restricts the wide scope `should’. But the failure of the inference from (1) and

(2) John did it


(3) John goes to prison

seems to reveal not much more than the unfortunate truth that not everything what should be the case is the case.

17 08 2008

Hi Moritz,

Thanks for that example. I’ll have to think a bit more about these cases – I can’t really see what generates the presuppositions in my case, but not in yours.

As for modus ponens, syntactically, it’s not so obvious on the Kratzer theory what counts as an instance of MP in English. But I don’t think we should restrict our modus ponens talk just to formal languages. After all, the inference rule no doubt was originally stated for natural language conditionals, and we certainly seem to be able to recognise instances of it in natural language reasoning when we see them.

Since conditionals have the form ‘if p, Oq’ (a three place relation between two propositions and an operator O) a natural way to reconstruct modus ponens is as follows.

1) p
2) if p, Oq
3) q

It is then natural to ask for what operators, O, is the above a valid inference form. And it seems it is valid precisely if O has a reflexive accessibility relation.

You say there are no genuine ‘bare’ conditionals of the form (A ARROW B). One possible substitute is just to take O to be the truth operator (with the identity accessibility relation), then ‘if p, Oq’ is just a material conditional which certainly obeys MP, and seems to behave much like you’d expect a bare conditional to, if I’m understanding the term correctly.

If I’m getting your comments at the end right, your reconstruction of MP is something like the following (at least when O = ‘should’ )

1′ ) p
2′ ) if p, Oq
3′ ) Oq

But this inference rule is not valid. At least, not if you’re following the Kratzer theory. For suppose there are O-accessible ~p worlds, where q is false, but q is true at all the accessible p worlds – then 1′ ) may be true, as may 2′ ), but 3′ ) is false. The only reason the inference sounded so good when O=should above was because any world, w, where John did it, John goes to jail in all the worlds deontically accessible to w.

17 08 2008

Ooops, I didn’t want all those smilies in there!

17 08 2008


I agree that modus ponens may be discussed simply with respect to the surface grammar. The only reason why I mentioned the subtle issue about logical form was the following. One advantage of the Kratzer view may be, as you point out, that iterated conditionals are not really seen as the iteration of a binary connective, but rather as one if-clause restricted by another, which together form a complex if-clause. Given this, one may claim that apparent applications of modus ponens to iterated conditionals are not really applications of modus ponens. But it looks to me that this is a claim about logical form. On the surface, modus ponens fails for iterated conditionals on the Kratzer view. And once one has moved to the level of logical form, it is not easy to state what on the Kratzer view amounts to an application of modus ponens.

I really liked your suggestion to define bare conditionals on the Kratzer view by taking a verum operator as the implicit modal. I had thought about this idea a bit before because it stroke me as quite ad hoc that Kratzer tends to take the implicit modal in an indicative conditional to be an epistemic ‘must’. It seems much more natural to check whether we can get away with simple truth. Of course, one would need to defend that bare conditionals behave like the material conditional…

Concerning your final point: that looks a bit like a misunderstanding. I agree that the inference you look at is not valid and thought I was actually making a similar point.

P.S. I don’t know how to get rid of ths smilies. For the time being I inserted an emty space at the relevant points.

19 08 2008


Sorry for the misunderstanding. But now I can’t see what is wrong with the simple counterexample I had to MP above?

Actually, I’ve written up some of my thoughts here if you’re interested. I think the restrictor is pretty much comitted to failures of MP no matter how she characterises the MP rule of inference for restrictors.

20 08 2008

Hi Anrew,

I am off to the ECAP in Krakow right now, but I hope to get to read your stuff pretty soon…

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