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Saturday, February 17, 2007

On McGee's Counterexample to Modus Ponens

McGee's counterexample to modus ponens came up in our reading group today at WashU during our discussion of Chapter 4 of Timothy Williamson's forthcoming book The Philosophy of Philosophy. The counterexample runs as follows. It's just before the election in 1980. The polls suggest that Republican Reagan is a clear winner. Democrat Carter is second, and Republican Anderson is third. Someone reasons as follows:

(1) If a Republican wins, then if it's not Reagan who wins, it will be Anderson
(2) A Republican will win
(3) Therefore, if it's not Reagan who wins, it will be Anderson.

Impeccable reasoning. Or maybe not. Intuitively, (1) and (2) are true, while (3) is false. Of course, if we force a logic 101 reading, then the reasoning is impeccable.

The counterexample is not a counterexample to modus ponens, given a reading of 'if ... then ...' as a material conditional. It is a counterexample to modus ponens, given an intuitive assignment of truth-values. Moreover, if we take the intuitive assignment of truth-values seriously, then it is a counterexample to the logic 101 reading.

But if 'if ... then ...' is not to be read as the material conditional, how is it to be read? A possible worlds account (ala the one for subjunctive conditionals) does not seem give us the intuitive result either. Without any further constraints on the closeness relation, a possible worlds analysis of (1) would tell us to go to the closest world in which a Republican wins. That is the actual world (as Reagan won). Then we go to the closest world in which it's not Reagan who wins. The closest world in which he doesn't win is one in which Anderson wins, if we keep the "Republican wins" feature fixed. So, (1) is true. (2), of course, is true. And so is (3) if we keep the "Republican wins" feature fixed. But this is not the intuitive result.

Why do we get a different result on an intuitive reading? Well, because there are two ways of determining closeness. If we keep the "Republican wins" feature fixed, we get one ordering of the worlds, and if we keep the "most likely to win if Reagan doesn't" feature fixed, we get another ordering of the worlds. If we keep the "Republican wins" feature fixed, Anderson wins if Reagan loses. If we keep the "most likely to win if Reagan doesn't" feature fixed, Carter wins if Reagan loses. The "most likely to win" feature would normally be more important, as it would normally be more salient. In the case of (3), for example, we keep the "most likely to win if Reagan doesn't" feature fixed, hence the false reading. In the case of (1) the "Republican wins" feature was just mentioned in the antecedent. That makes it more salient. So when we evaluate the embedded conditional, we keep it fixed, hence the true reading.

Now, there are of course numerous reasons to be suspicious of a possible worlds analysis of indicative conditionals. But at least the possible worlds analysis fares better than the logic 101 analysis in this particular case, as it is able to explain why modus ponens seems to fail.

24 comments:

Mike said...

Hi Brit,

Strange. Here's maybe a way to the intuitive result. Read the wider conditional in (1) as subjunctive and the embedded conditional in the consequent as material. In the closest world in which a Republican is elected, if it's not Reagan, it is Anderson. That seems to be what is intended and it's true, since we hold fixed that a Republican wins. The initial conditional reads like Lewis's subjunctive interpretation of ( / ).

1. (If not Reagan then Anderson/Some Republican wins)

But in the conclusion we do not hold fixed that a Republican wins, it's false that if Reagan does not win, Anderson does.

Brit Brogaard said...

Hi Mike,
So, is the conclusion also a subjunctive conditional, on your proposal? If it is, then your suggestion seems to yield the right result. However, one might worry about the multiple readings of 'if ... then...' introduced by your proposal. It would be strange of 'if' is ambiguous. Or are you suggesting that 'if' is not ambiguous, and that we read (1) the wrong way?

Mike said...

Brit,

That's abundantly fair; my suggestion does seem ad hoc. I plea that my reading of the conditionals vindicates our intuitions. In the conclusion it is material and the wide scope conditional in the premises is subjunctive.

Brit Brogaard said...

Hi Mike,
Shouldn't the conclusion be a "subjunctive"? Otherwise, it's true (as 'it's not Reagan' is false).

Your suggestion does seem to yield the correct evaluation but a possible world account does this too, as long as we keep the context fixed. So, why opt for the "ambiguity thesis"?

Mike said...

Hey Brit,

I'm not sure why you urge that the conclusion be subjunctive. I'm trying to formulate the argument so that it instantiates a valid form. Maybe I'm missing something.

I don't see where the possible worlds analysis gets the right result. The conclusion, that 'were Reagan not to win, Anderson would' is false unless (as you suggest) we "keep fixed the feature that a Republican wins". But that amounts to assuming that 'were Reagan not to win, then a Republican would'. Is the intuition that premise (2) ensures this?
I don't think it does. Suppose Smith is lefthanded and wins the boxing match; Jones is righthanded and loses. Since a lefthanded boxer won, should we say that in the closest world in which Smith loses, he fought another lefthander (instead of Jones) who won?

Brit Brogaard said...

Hi Mike,
Well, I think you were suggesting earlier that the conclusion is false. But if it is a material conditional, it is true. That's why I thought you were committed to the subjunctive reading.

The possible worlds analysis gets the right result, because validity or invalidity obtains only relative to a context. So, if we keep the feature that a republican wins fixed in order to interpret the premise, then we must keep the same feature fixed when we interpret the conclusion (we don't need to but if we don't, then the question of validity does not arise).

I agree with you that we do not normally *read* the conclusion in this way. We allow for a context-shift. Nonetheless, to determine validity, we need to keep the context fixed.

Eric Schwitzgebel said...

But still I'm troubled by the failure of the Logic 101 reading. Shouldn't we be able to treat "a Republican wins" as P and "if it's not Reagan who wins, it will be Anderson" as Q?

If not, then we need rules regarding what propositions can and cannot be P's and Q's, and I'm unaware of a good set of such rules. (But admittedly, my formal logic is not very advanced.)

Brit Brogaard said...

Hi Eric,

(1) If a Republican wins, then if it's not Reagan who wins, it will be Anderson
(2) A Republican will win
(3) Therefore, if it's not Reagan who wins, it will be Anderson.

Given the logic 101 reading, the embedded conidtional in (1) is true, as Reagon won. As the embedded conditional (consequent) is true, (1) is true. The conclusion is then true too, and so is (2). So, the 101 reading doesn't "completely" fail.

But there is a sense in which it fails. It is unable to account for our intuitions that (3) is false. This is one reason some think that logic 101 semantics is not a good semantics for the 'if ... then...' statements of ordinary language.

Eric Schwitzgebel said...

Right, of course. How embarrassingly stupid of me! (3) is of course true in logic 101.

(Thanks for being polite about it.)

Eric Schwitzgebel said...

Wait. I may be cotton-headed, but I wasn't being that cotton-headed.

My thought was that you could treat "A Republican will win" and "If it's not Reagan who wins, it will be Anderson" as atomic (respectively, P and Q). Then their truth value may be evaluated intuitively, not by the logic 101 reading of conditionals, right?

Let's just get rid of "if" in our logic, say, so there's no question of confusion about it; and maybe on this reduced logic it's even less appealing to think we'd be compelled to try to break Q into logical parts.

So then modus ponens is:
(1.) Q (if Reagan will not win, Anderson will win) or not P (a Republican will win).
(2.) P (a Republican will win).
(3.) Therefore, Q (if Reagan will not win, Anderson will win).

Looked at in this way, I guess it becomes clearer to me that (1) is not true. The temptation when (1)is formulated with an "if" is to frame the evaluation of Q under the assumption of not-P. (1) as stated above, however, makes it clearer that that is not legitimate. I reject both disjuncts, Q and not-P.

Mike said...

Well, I think you were suggesting earlier that the conclusion is false.

Right, I see now, in the first post. But as a material c. I'm pretty sure it comes out true.

Brit Brogaard said...

Mike:
Right, but intuitively the conclusion is false.

Eric: That's neat. But then the question becomes: in virtue of what is the conclusion (and embedded conditional) false? Since indicatives are not material conditionals, on your account, we need a different semantics for them. I think a possible world account would do the trick, but possible world accounts of indicatives have independent problems. Anyway, I like your take on it.

Eric Schwitzgebel said...

Right! But now I feel comfortable (again) in seeing the semantics of counterfactual conditionals as a separate (hoary, interesting) problem; and the threat of its dragging down modus ponens no longer seems to me to loom....

Anonymous said...

Au contraire--


indeed looks like a valid tautology (truth functionally), yet as with any claim about future events, is sort of pointless: "modality" has little to nothing to do with logic:

If R wins, then R v A

R will win (or, has won)

thus R v A.

No counterexamples: the premises might not be sound (a last minute write-in conservative GOP could take the election)--so it's not correct to say the premises are true---and that is generally the problem with MP

Anonymous said...

So, I think it's more a problem of verifying premises (and inductive premises are not really "true" as say, the answer to a calculus problem is true) then logical form or validity. The second part of the 1st premise tranlates perfectly well into disjunction: if not Reagan, then Anderson = R v A (so "If Rep. wins, then R v A." OR, If Rep. wins, then if not R, then A--equivalent): that is a valid translation . I'd like to hear to argument of someone who challenges that, who says that is not equivalent (which is to say material implication translation works fine truth functionally).

Anonymous said...

Yet I am not denying the possibility of causal conditionals, as Quine (or perhaps Wittgenstein of the TLP) would.
Yet the causal conditional ---"if Joe drinks 8 shots of whiskey, Joe will be unfit to drive," or something--is not really the same sort of creature as conditional of class membership (if cat, then mammal, if right triangle, then has an angle of 90 degrees, etc.) is it? And the conditional of class membership (set inclusion??) does translate via material implication, to a disjunction--if P, then Q = (not P) v Q. Tho' that seems a bit counterintuitive--" not a cat or mammal,"--better translation perhaps, ~(cat & ~ mammal)--it cannot be the case that there is a cat and not mammal. Conditionals as denied conjunctions...........

I think Schlick or someone pointed that out years ago. Inductive, causal premises (or probable events that have not occurred--likelihood) do not work so well in argument forms, except informally. though its difficult to avoid them. But probability ain't logic, regardless of what some hepcat Kripkeans or possible worlds people say.

Andrew said...

That's very interesting.

If we're comparing theories, it seems like the Lewis/Kratzer theory comes out tops on this one (I *think*). On that view 'if' serves as a device for restricting operators (in this case 'will'.) Presumably the embedded conditional serves like just two restrictors on 'it will be Anderson' (that is we consider only worlds in which a republican who isn't Reagan wins.) Under this analysis 1, 2 but not 3 come out true.

As for why MP fails - this might be to do with the operator in question, namely 'will'. You could run similar counterexamples to MP with subjunctive conditionals - but why is this less surprising given the similarities between subjunctives and 'will' conditionals?

Brit said...

Hi Andrew,
Right, the Lewis/Kratzer theory seems to deliver the intuitive verdict in this case.

We might write the argument as follows:

(1) If a Republican wins and it's not Reagan: it will be Anderson.
(2) A Republican wins
(3) If it's not Reagan who wins: it will be Anderson.

This is not valid.

But how would the restrictor theory handle the following sort of case:

-If Bush has been fired, then if he hadn't been fired, he would still be in office.

Andrew said...

Hi Brit,

I'm having a little difficulty seeing what that sentence actually says. I think that there is a plausible reading in which the first antecedent has a covert epistemic modal (so it will look like 'If Bush has been fired: given what we know p' where p is the subjunctive conditional'.) The Lewis/Kratzer view does postulate covert epistemic Modals, especially in the case of (bare) indicatives. In this case it seems like we are treating Bush being fired as an epistemic possibility.

The other reading, if there is one, in which there is no covert modal, doesn't make too much sense to me. But I'm assuming this reading, if it exists, is captured by 'if Bush has and hasn't been fired: he would still be in office'.

Andrew said...

I guess if you modified your example so it read wholly in the subjunctive mood you have something potentially problematic:

(1) If Bush had been fired, then if he hadn't been fired, he would still be in office.

I'm not sure how the Lewis/Kratzer view deals with subjunctives, but I think its subtly different. For me this sentence reads

(2) If Bush had been fired: would[if he hadn't been fired: he would still be in office]

Similarly this modified sentence seems to me to have a good and a bad reading, the bad one being (2) minus the covert 'would'.

Also note that the analogous sentences in the indicative mood, or in the last combination, are deviant:

-If Bush has been fired, then if he hasn't been fired, he will still be in office.

-If Bush had been fired, then if he hasn't been fired, he will still be in office.

Perezoso said...

No Lewis is wrong, as is most modal theorizing, except when read as probability/likelihood, etc.

We "might" write the argument as follows, but we don't have to. The right translation is with the consequent ass a disjunction.

(1) If a Republican wins, then it will be Reagan OR it will be Anderson.
(2) A Republican DID win
(3) Then it was either Reagan or Anderson. (temporality another issue--which is to say, if you are talking about future events only, then logic really doesn't apply anyways, and it's a matter of probability, not modus ponens)

Now you might say that's not correct, but then you merely attack the soundness of the first premise (ie some unknown GOPers wins in a write-in)
Either way the argument IS valid.

brit brogaard said...

Andrew:
Yes, I think the Lewis/Kratzer view runs into trouble with subjunctives but like you I am not totally sure how they deal with them. Otherwise your suggestion seems to work.

Perezoso: Nice! But where does the "or" come from? (compositionally speaking)

Andrew Bacon said...

Perezoso: I took the puzzle here to be to explain why the original argument seems invalid. That your translation of the argument is valid, prima facie, suggests to me that it is not in fact a translation. At least, arguing that it *is* a translation will probably amount to explaining why the original argument seemed invalid yet isn't. So why do you think your argument is a correct translation?

(Also, I'm not quite sure why you transposed the tenses? What you write afterwards suggests that you do think the tense makes a difference.)

Brit: the thing I do find puzzling about the Lewis/Kratzer view is that it validates all instances of 'if p then it must be the case that p'. You can do a similar trick here (add a covert modal so it's doubly modalised.) But here, unlike before, this seems wildly ad hoc (in that it doesn't correspond to a natural reading of the sentence.)

brit said...

ah, very interesting! I (and my co-author Joe Salerno) was in fact hoping that something like Nolan's extension of Lewis' account would work for subjunctives and indicatives. The fact that the Lewis/Kratzer view validates all instances of 'if p then it must be that p' is indeed problematic. I don't think the Nolan view has that problem. But it may run into other problems (especially with adverbs of quantification)