## A Counterexample to Modus Ponens

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• 10
You regard 1 as the background assumption, whereas I regard (1 v 2 v 3) as the background assumption

I was mistaken to couch it the way I did.
• 10
As I interpret the situation, ~R -> A is not counterintuitive when derived in the appropriate context. Given the polls, a person has good reason to believe a Republican has won (or will win). But Carter might still have won, despite their good reason, since their good reason is not sufficient for truth.

Now I don't think ~R -> A is counterintuitive. Because it has probability of 65%
• 10
given their newly acquired knowledge, if Reagan didn't win, then Anderson did.

Exactly.

I realized that this has an even simpler explanation.
• 10
SMOKE AND MIRRORS desmoked and demirrored

There really isn't a puzzle.

It has not been shown by McGee that modus ponens does not preserve strength for reason to believe.

And the point I mentioned about best strength is not relevant either.

And it has nothing more to do with modus ponens than with tautology itself.

This is so simple that I can't believe I didn't see it.

The only non-logical premise is R v A.

And the conclusion ~R -> A is equivalent with R v A.

So it's just a tautological inference.

The strength of R v A is 65%. And that strength is preserved from the non-logical premise to the non-logical conclusion.

Suppose the only non-logical premise is R v C.

The conclusion ~R -> C is equivalent with R v C.

So it's just a tautological inference.

The strength of R v C is 95%. And that strength is preserved from the non-logical premise to the non-logical conclusion.

That's all. Two different non-logical premises in two different arguments, and their strength preserved in the conclusion in both arguments.

/

For reference, here's the setup of the problem:

Suppose there are three candidates in an election: R, A, and C.

Suppose, the day before the election, the polls show 60% for R, 5% for A, and 35% for C.

Let Jack be a person who knows those poll numbers but he slept through the election so he doesn't know who won.

(1) (R v A) -> (~R -> A)

(2) R v A

Therefore, (3) ~R -> A
• 35
Is this really a puzzle? In my view the following is happening, where p is "republican wins" and q "if it's not Reagan who wins, it will be Anderson." Then you find that Reagan doesn't win:

If p then q,
not q, (because neither Reagan nor Anderson won)
therefore not p

That's modus tollens though.

If you want to stay in the MP, it should be:

If p then q,
not p, (because no Republican won)
therefore not q.

The MP is perfectly valid.

It seems more like slight of hand to play with the implied meaning of Reagan being a Republican and leaving out part of q because that's "either Reagan or Anderson wins" and not only "Reagan wins".
• 10
MODUS PONENS HOCUS POCUS

I think fdrake and Andrew M had the right idea, but it needed a follow-through. I think sime had the solution in a general form.

I found this problem extremely interesting because in human inference making modus ponens is about as basic and ubiquitous an argument form there is, so it would be astounding to find that modus ponens is not reliable.

I can't find McGee's article on the Internet. If there is more in the article that materially qualifies the clip in the first post of this thread, then my remarks might need to be modified. But at this time I'm taking the clip at face value along with the quote "is not strictly valid".

If McGee meant this as a joke or magic trick, then I would say it is a very clever and entertaining joke or magic trick. But I take it that it was meant seriously, so I am curious why he didn't himself see the fallacies in his argument. It turns out that, when you unpack his argument, the solution to his challenge is trivial. So it is fun to see a baffling problem turn out to have a trivial solution.
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I use a hypothetical person named 'Jack' instead of a group of people referred to as 'they.

I take McGee's argument to be fairly couched this way :

We start with premises that Jack has good reason to believe, then we arrive at a conclusion that (a) Jack does not believe and (b) doesn't have good reason to believe. Therefore, modus ponens fails to preserve strength of reason for belief. Therefore, modus ponens is not strictly valid.

That depends on the assumption:

For modus ponens to be strictly valid, modus ponens must preserve strength of reason for belief.

We should accept that assumption, at least for sake of argument.

But (a) is irrelevant. Argument forms don't ensure that people believe the conclusions. People err in their beliefs; that's not the fault of argument forms.

As for (b), argument forms pertain to what is the case, or (granting McGee's assumption) what should be believed to be the case, only relative to the premises. And relative to the premises, it is not the case that there is not good reason to believe the conclusion ~R -> A.

I assigned specific probabilities. But they could be any probabilities, as long as they entail that there is good reason to believe R v A. So here's the example with unspecified probabilities.

prob(R & A) = 0
prob (R & C) = 0

Assume prob(R v A) is great enough that we have good reason to believe R v A.

Assume prob(R v C) > prob(R v A).

The only non-logical premise in the modus ponens is R v A. And trivially prob(R v A) = prob(~R -> A).

So modus ponens does preserve the strength of reason to believe from the premises to the conclusion.

We do have greater reason to believe ~R -> C than we have reason to believe ~R -> A. But that does not contradict that, with either R v A or R v C as the non-logical premise, modus ponens did preserve strength of reason to believe.

Modus ponens is a red herring anyway. It is used by McGee as a needless phony baloney armature for a more simple fact: R v A is equivalent with ~R -> A. That's all we need to know.

So this is the slight of hand that McGee uses to pull off his trick:

(1) He puts the argument into an armature of modus ponens. He makes it seem that the supposedly incorrect inference is the fault of modus ponens. But there is no incorrect inference (strength of reason for belief is preserved from premises to conclusion, trivially as the inference is merely tautological), and the inference doesn't require modus ponens.

(2) He distracts by conflating two different arguments. Yes, ~R - C has greater strength of reason for belief than ~R -> A does, but what is at stake is preservation of strength for belief, not an apples and oranges comparison of the conclusions standing alone.
• 3

Here's another interesting example to test your solution on:

1. Either Shakespeare or Hobbes wrote Hamlet.
2. If either Shakespeare or Hobbes wrote Hamlet, then if Shakespeare didn't do it, Hobbes did.
3. Therefore, if Shakespeare didn't write Hamlet, Hobbes did it.

Since Shakespeare did write Hamlet, the first premise is true. The second premise is also true, since starting with a set of possible authors limited to just Shakespeare and Hobbes and eliminating one of them leaves only the other. However, the conclusion may seem false since ruling out Shakespeare as the author of Hamlet would leave numerous possible candidates, many of them more plausible alternatives than Hobbes.

I think the conclusion is true, and MP is valid here.

For a further twist, consider replacing "Hobbes wrote Hamlet" with "1 = 2".
• 10

I agree.

"false -> P" is true.

And again, it's not even about MP:

S v H is equivalent to ~S -> H.
• 27

Seems we have agreement that modus ponens is not invalidated by the argument in the OP; that the premises are true, the argument valid and the conclusion true, but incomplete.
• 10

'incomplete' is not part of my analysis.

MP is valid.

McGee claims MP is not "strictly valid" which I can only take to mean that MP does not preserve strength of reason to believe. But, contrary to his sleight of hand, his example shows MP preserving strength of reason to believe.
• 27
'incomplete' is not part of my analysis.

• 10

My analysis doesn't need to say anything about Carter.

All I need to point out is that the conclusion of the example has strength of reason to belief not less than the strength of reason to believe the premises.

But I did mention that the strength of reason to believe ~R -> C is greater than the strength of reason to believe ~R -> A, though that is not needed to refute McGee.
• 10
Here's the new puzzle for me:

Based on the level of McGee's research in logic and his associations, he must be extremely intelligent and knowledgeable. Not a nano-mote of doubt that I could not possible fathom all the logic he knows. So how could he have made such a rank mistake?

I should have approached the problem more systematically from the start. I thought that the explanation would have to be at a high level involving intensionality and then probability. But then I saw that it is bare bones trivial. That's the magician's trick. He distracts you with a bunch of razzle dazzle hiding the explanation that is the one right in front of your nose.

Then I went on to win the Tour de France, the Indy 500, the Pulitzer Prize, and the Best Pecan Pie Award at the Polk County Fair, all in one week. But everybody already knows all about that, so enough about me.
• 27
In the absence of the article, you can't know that he did.

My analysis doesn't need to say anything about Carter.
But if Reagan did not win, it would have been Carter. And it is this that is in contrast wth the conclusion of the MP. So isn't your explanation incomplete?
• 10
if Reagan did not win, it would have been Carter

Not by the premises of the argument.

The point is not to challenge the premises, but rather to show that the conclusion has as great a reason for belief as the premises. That's all that's needed to refute McGee. And it's trivial. It only looks hard because he razzle dazzles us with a distraction.
• 10
n the absence of the article

I said that I only have the clip from the article to reference plus the locution 'strictly valid'.

Anything I say is in that context alone. If there is more in his article that qualifies the context, then that would be another story.
• 10
I just realized I made a really rookie mistake in some of my attempts several posts back.

prob(x) is not presumably the poll rating of x.

For example, if my memory is in the ballpark, the day before the election, Biden was given about an 85% chance of winning but he only had about 52% in the polls.

But my latest posts don't assign prob(x) but only rank the probabilities as given.

prob(R) > prob(C) > prob(A)

And we don't even need that!

Whatever the prob(R v A) is, it's no greater than prob(~R -> A), as indeed prob(R v A) = prob(~R -> A).

And we don't even need that!

All we have to do is see that whatever the strength of reason for believe for R v A, it's no greater than the strength of reason for belief of ~R -> A. How could that not be? R v A is equivalent with ~R -> A. It's that simple!
• 10

Directly responding to the clip and its one sentence intro:

Vann McGee claims that modus ponens "is not strictly valid" in an article from 1985

MP is valid. McGee does not say MP is not valid. He says it is not "strictly valid". What does "strictly valid" mean? He mentions "good reason to believe". So the only way I can think of regarding "strictly valid" is "If there is good reason to believe the premises then there is good reason to believe the conclusion". He says, of the only non-logical premise, that we have good reason to believe "A Republican will win the election". But "A Republican will win the election" is equivalent to "If it's not Reagan who wins, it will be Anderson". So whatever the good reason we have to believe "A Republican will win the election" is the same good reason we have to believe "If it's not Reagan who wins, it will be Anderson". So this instance of MP that McGee claims shows that strict validity fails, is actually an example in which strict validity succeeds. It's that simple. McGee is refuted.

Opinion polls taken just before the1980 election showed the Republican Ronald Reagan decisively ahead of the Democrat Jimmy Carter,

That's false. But nevermind, we can take it as a hypothetical given. And it doesn't even matter anyway. All we need is the background assumption that people had good reason to believe that a Republican would win. Carter is a red herring.

with the other Republican in the race, John Anderson, a distant third. Those apprised of the poll results believed,

What people believe in premise and conclusion is irrelevant. People can err by believing the premises of a valid argument but not the conclusion. The strict validity of MP couldn't depend on the empirical fact of what erring humans believe.

with good reason:

So it should be "Those apprised of the poll results had good reason to believe".

[1] If a Republican wins the election, then if it's not Reagan who wins it will be Anderson.
[2] A Republican will win the election.

[1] is a logical truth and [2] is a non-logical premise.

In the context, [2] is equivalent to "Either Reagan will win or Anderson will win".

Yet they did not have reason to believe
[3] If it's not Reagan who wins, it will be Anderson

If they had good reason to believe [2] then they had good reason to believe [3].

McGee did not show that strict validity of MP failed.
• 10
From Donkeys and Elephants to Lungfish and Porpoises.

McGee has another supposed impeachment of MP.

https://sites.duke.edu/wsa/papers/files/2011/05/wsa-defenseofmodusponens1986.pdf

"I see what looks like a large fish writhing in a fisherman's net a ways off. I
believe
If that creature is a fish, then if it has lungs, it's a lungfish.
That, after all, is what one means by "lungfish." Yet, even though I
believe the antecedent of this conditional, I do not conclude
If that creature has lungs, it's a lungfish.
Lungfishes are rare, oddly shaped, and, to my knowledge, appear only in
fresh water. It is more likely that, even though it does not look like one,
the animal in the net is a porpoise"

c = that creature
Fx <-> x is a fish
Lx <-> x is a lungfish
Hx <-> x has lungs

(1) Fc -> (Hc -> Lc) [McGee believes]
(2) Fc [McGee believes]
(3) Hc -> Lc [McGee does not believe]

But, again, what people do or do not believe is not relevant. What is relevant is (1) for validity, whether truth is preserved, or I suppose (2) for "strict validity", whether "good reason to believe" is preserved.

Obviously the strength of reason to believe the conclusion Hc -> Lc follows the strength of reason to believe the premise Fc. But we need to show that mathematically. I haven't finished it, but getting close:

prob(Fc -> (Hc -> Lc)) = 100%, since it's true by definition

Let prob(Fc) = x

Let prob(Hc) = y

Let prob(Lc) = z

We also have Lc -> Fc [implicit in the problem; I wish I could dispense with it though]

So prob(Lc) = prob(Fc & Lc) = x*prob(Hc | Fc)

~Hc and Lc are mutually exclusive so:

So prob(Hc -> Lc) = prob(~Hc v Lc) = prob(~Hc)+prob(Lc) = (100-y)+(x*prob(Hc | Fc))

So prob(Hc -> Lc) goes up and down as prob(Fc) goes up and down.

The possibility of the porpoise is not part of this. Just like the possibility of Carter was not part of the earlier problem. The porpoise is Carter! (And, as we know, Paul is the walrus.)
• 10
https://sites.duke.edu/wsa/papers/files/2011/05/wsa-defenseofmodusponens1986.pdf

I just now read that. My argument is basically the same as theirs.
• 3
↪TonesInDeepFreeze ↪Andrew M
Seems we have agreement that modus ponens is not invalidated by the argument in the OP; that the premises are true, the argument valid and the conclusion true, but incomplete.

Yes, so why do McGee's examples seem to be counterexamples to modus ponens when they are not? Because the way the counterexamples are expressed suggest an unrestricted set of possibilities for the conclusion when, in fact, the possibilities are restricted by the assumptions. D. E. Over explains (note his use of the pronoun 'he'):

And we can bring out even more clearly what is wrong with these supposed counterexamples by considering the following modification of (1) - (3):

(7) If a Republican wins, then if he is not Reagan he will be Anderson;
(8) A Republican will win;
(9) If he is not Reagan, he will be Anderson.

The antecedent of (7) restricts the possibilities for the interpretation of the pronoun in its consequent. The second assumption (8) does the same job for the conclusion (9), and it would be a transparent mistake to try to interpret 'he' in some other way, in an attempt to show that (7)-(9) is invalid. McGee would make a mistake of this type if he thought of (8) as a relatively long-lasting mental state of justified belief outside of the context of this inference. He would not then see (8) as an assumption in an inference, determining in that context which proposition is expressed by (9).

An inference should be defined in terms of a relationship between assumptions and a conclusion, as is standard in logic. We should remember that the assumptions can restrict the relevant set of possibilities and so affect the propositions expressed under them, just as the antecedents can affect the propositions expressed by the consequents of conditionals. We must therefore be careful about the propositions expressed in inferences, particularly ones containing conditionals, if we wish to question their validity.
— Assumptions and the Supposed Counterexamples to Modus Ponens, D. E. Over, Analysis, 1987
• 10

Of course the premise "A Republican wins" restricts. The impression that there is not good reason to believe "If Reagan doesn't win then Anderson wins" comes from (1) Overlooking that it "If Reagan doesn't win then Anderson wins" is merely a conditional and not a statement about Anderson winning, and (2) overlooking that "If Reagan doesn't win then Anderson wins" is equivalent with "A Republican wins", so whatever the bases are for believing "A Republican wins" are the same bases for believing "If Reagan doesn't win then Anderson wins".

McGee's error is the claim that the conclusion doesn't have as good a reason to believe as the premises. That is the explicit error in his argument about the MP example.

Indeed Carter is left out with the premise "A Republican wins", but stating that Carter is left out and that therefore the conclusion is clouded is not in and of itself a refutation. The follow-through is that Carter being left out is just "A Republican wins" which is equivalent with the conclusion, so whatever basis there is for "A Republican wins" is bases for the conclusion.

The fact that there is more reason to believe "If Reagan does not win then Carter wins" than there is reason to believe "If Reagan does not win then Carter wins" is McGee's red herring.

There is a difference between (1) What is the best way to set up inferences about the election? and (2) Does the conclusion of the particular MP mentioned have as great a reason for believing as the premises have?

The key to refuting McGee is not (1). It's (2).

mental state — Assumptions and the Supposed Counterexamples to Modus Ponens, D. E. Over, Analysis, 1987

McGee also erred by dragging in what people believed. (Probably, nudging in "people believed" is part of the sleight of hand.) What people believed is irrelevant to the example. But we can still couch his argument without concern for what people believed and stick with "reason to believe" only.

/

Another article about the puzzle went into a bunch of stuff about subjunctive mood. For me, that 's a wrong tack: (1) We can rephrase the MP without subjunctive and (2) The puzzle is dissolved much more easily, trivially, anyway.
• 10
I said that my commentary is based only on the clip posted at the top of this thread.

Yesterday I got hold of McGee's paper.

It turns out that his argument does not suppose that the conditionals mentioned are taken in the sense of the material conditional. He says that if the conditionals mentioned are taken in the sense of the material conditional then modus ponens is not impeached by his argument.

Lack of having his paper to know what he actually claimed led to unnecessary disputation about his argument.

This has been a waste of my time and the time of people reading my posts. If I knew from the onset that he's not talking about the material condition, then I wouldn't have unnecessarily bothered.
• 27

This has been a waste of my time and the time of people reading my posts.

I don't agree. Solving crossword puzzles is not time wasted.
• 27
Here is another way to be rid of Modus Ponens...
• 0
(1) is invalid.

With good reason, the pollees believe that a Republican will win the election, not that a Republican might win the election, meaning that (1) is invalid.

IE, it is (1) that is not part of a valid modus ponens, rather than the case that the modus ponens is not valid.
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