McGee's Counterexample to Modus Ponens

Question

I'd like to start off by saying that I have read the other posts in the Math StackExchange and here about this paper, but I think my question is a bit different from those although it does stem from my being unable to understand the argument.

In Van McGee's paper, Counterexample to Modus Ponens (1985), he starts off by giving an inference which he says is not a valid belief. The inference is in the context of the 1980 presidential election in which the polls showed Reagan (Rep.) leading, then Carter (Dem.), and in third, Anderson (Rep.).

The deduction goes as follows,

1. (Premise) If a Republican wins, then if it's not Reagan, then it will be Anderson. (Rep-->(notReg-->And)
2. (Premise) A Republican will win.
3. (Inference via 2,1 Modus Ponens) If it's not Reagan, then it will be Anderson.

Now here McGee says one does not have reason to believe the conclusion. Now I agree that taking the conclusion on it's own, I would not be right to believe the conclusion because Carter is more likely to win than Anderson, BUT in the light of the premises I do hold it to be true. Since a Republican won, it cannot be the case that Carter won.

Now I have attempted to resolve my confusion by speaking with two professors of mine. What I believe (emphasis on believe) they were trying to tell me is that I somehow do not know the premises or something akin to that, and that I had a misunderstanding of the way classical logic is done (which I am completely open to).

The way I thought classical logic was done is that in making my inferences, I hold the premises to be true. That, along with my rules of inference, is what allows my to make conclusions. It seems like the explanations are is that I am supposed to take the conclusion stand-alone which makes absolutely no sense to me because I thought that was the point of the premises. I know them to be true, and believe them to be true throughout my inferences, that knowledge is always there.

I am arguing that it is intuitively valid because there is no case where the premises are true, and the conclusion is false.

I would very very very much appreciate if anyone can try and explain why it's an invalid inference (intuitively) to me. I know it's a valid inference in classical logic via the deduction above. I believe my confusion is in this idea that I always have knowledge of my premises which is not how classical logic is done, maybe a difference in truth and belief? In classical logic, we take the premises to be absolute truths, not probabilistic.

To reiterate, the main question is, why is this an intuitively invalid inference? In what case is it possible for the premises to be true, and the conclusion to be false?

Answer 1

Classical logic, and the material implication conditional, is concerned with propositions that are true or false, categorically, certainly and without exception. Most conditionals in ordinary English do not behave like that. We allow ourselves to say "if A then B" when we are fairly sure but not certain, or when there might be exceptions that we do not ordinarily wish to worry about, or where some background assumptions apply. The result is that some of the classical rules that work for material implication can fail for ordinary conditionals. This is true, for example, of contraposition, hypothetical syllogism and strengthening of the antecedent.

If we try to represent McGee's example using classical logic, with material implication for the conditional, then the argument is valid. Suppose we use the following symbols: A = Anderson wins, R = Reagan wins, E = a republican wins (E for elephant). Then we have

1. E → (¬R → A)
2. E
3. ¬R → A

This is valid, because given the truth of E, it follows that R or A, which implies ¬R → A.

But it is natural to understand the second premise not as a true statement, but as a prediction of which we are fairly, but not absolutely, certain. Indeed, we might reasonably take all the sentences to be statements about what we have good reason to believe. Statements about what it is reasonable to believe typically allow for uncertainty, exceptions and background assumptions. Given this, we need a representation of the argument that is not a simple classical one.

One common approach to representing uncertain conditionals is based on the observation that many conditionals in English behave like judgements of conditional probability. Often, "if A then B" can be glossed as "it is highly probable that B supposing A". If we adopt this approach, we can exhibit how the argument is invalid. In addition to our existing symbols, let's add K = the background information about the election, the candidates and their standing in the opinion polls. We now have:

1A. P( A | E, ¬R, K ) ≈ 1

2A. P( E | K ) ≈ 1

3A. P( A | ¬R, K ) ≈ 1

Here, the conclusion does not follow. Trying to move from 1A and 2A to 3A is an incorrect use of the product rule in probability. What you would need to get to 3A is a different second premise:

2B. P( E | ¬R, K ) ≈ 1

But of course this premise is incorrect: it is not likely that a republican wins if Reagan doesn't.

An alternative to the probabilistic approach would be to appeal to possible world semantics. We might take it that premise 1 means that in the nearest PW in which a republican wins, then in that world, the nearest PW in which Reagan doesn't win is one in which Anderson does. But 3 means that in the actual world, the nearest PW in which Reagan doesn't win is one in which Anderson does. The latter is not true, because the nearest PW in which Reagan doesn't win is one in which Carter does.

The upshot is, I'm inclined to agree that McGee's example is not really a counterexample to MP. It's really just a case of how we need to be careful when representing conditionals.

Answer 2

In modern propositional logic "If p then q" is the same as "~p V q".

In ordinary language, "If p then q" can mean something more like "Q will happen because of p, if p happens."

If you want to fill in "_____ will happen because Reagan lost, if Reagan loses," the answer would be "Carter winning". But if you want to fill in "Either a Democrat won, or: either Reagan won, or ______," then the answer would be "Anderson won".

I believe your final point is that the former question is irrelevant after the election because we know Reagan, a Republican, won. True; but by logical explosion then anything would follow from "Reagan lost". (For all q, if (p&~p) then (if p then q).)

This may all seem a bit trivial but propositional logic is a means to an end, namely clarifying / evaluating lines of reasoning, and if people make arguments and then refine/check them against rules that they mistakenly believe map onto natural-language concepts of explanation, then they become counterproductive. These sorts of considerations are the justification for research into logics with non-classical properties. (Paraconsistent, probabilistic, etc)

Answer 3

(Premise) If a Republican wins, then if it's not Reagan, then it will be Anderson. (Rep-->(notReg-->And) (Premise) A Republican will win. (Inference via 2,1 Modus Ponens) If it's not Reagan, then it will be Anderson.

Now here McGee says one does not have reason to believe the conclusion.

In classical logic, we deal with one or more logical propositions that are unambiguously either true or false all at the same instant in time. The use here of the future tense "will win" is vague as to what instant in time we are talking about. If we tweak the argument a bit so that we are using the present tense, the problem seems to go away:

Premise 1: If a Republican is the winner, then if Reagan is not the winner, then Anderson is the winner.

Premise 2: A Republican is the winner.

Then the inevitable conclusion must be: If Reagan is not the winner, then Anderson is the winner. (Or equivalently: Reagan is the winner OR Anderson is the winner.)