## A Counterexample to Modus Ponens

• 10
the quote in the OP is claiming that having good reason to believe the premises doesn't constitute having good reason to believe the conclusion.

He said there is good reason to believe the premises, but not a reason to believe the conclusion. And that is true*. The part about "constituting" or anything like it, is not in the quote. We might think it is fair to think he intended that (I don't know; it's a fine point); but he didn't actually say it.

* It's true given the background premise that the polls showed Reagan far ahead. But that premise is false, since Reagan was not far ahead in the polls. No matter for the analysis though, as we may take it hypothetically that Reagan was far ahead or that we had good reason to believe he would win on any other grounds.
• 2
MP can be defined generally and abstractly as the composition of arrows in a category. In problems such as the above, the arrows denote conditional probabilities of the form P(B | A) between two propositions A and B , and premises denote arrows of the form 1 -> A, where 1 is a terminal object representing an "empty" premise.

The example also highlights a general problem: given a state of knowledge, is it consistent? and if so, how do you determine what the underlying arrows are?

In the previous example of the OP, the beliefs given are consistent. The arrows are the conditional probabilities of candidates winning given knowledge of the failure of one or more of the remaining candidates, and there is only one premise, namely that a republican wins.
• 10
I don't take logical inference to be "ALL about" [all-caps added] good reasons for belief. Logical inference can take place in a machine that doesn't even have beliefs or reasons for belief. Modus ponens and other deductive forms have settings other than grounds for belief.

To emphasize that point. The validity of modus ponens bears upon grounds for belief, but the validity of modus ponens can be (and often is) understood irrespective of grounds for belief. The validity of modus ponens is that if the premises are true then the conclusion is true, And that "the premises are true then the conclusion is true" is true of modus ponens no matter whether we even wish to raise the subject of grounds for belief.

Inferences may be drawn irrespective of belief. One could draw inferences from modus ponens all day long without even giving a thought as to what one thought are grounds for belief of anything. Indeed, in a formal sense, an argument is an ordered pair <P c> where P is a set of sentences (or formulas but that complicates this with a technicality) and c is a sentence (or formula). A valid argument is an argument such that in all models in which all the members of P are true are models in which c is true. Then sound systems of logic are ones such that proofs only result in valid inferences. And a valid inference is one such that, again, in all models in which all the members of P are true are models in which c is true. There is no requirement that we mention "reason to believe" or anything like that. So inference isn't "ALL about" reasons for belief.
• 13
He said there is good reason to believe the premises, but not a reason to believe the conclusion. And that is true*.

But it’s not, which is my point. If there is good reason to believe those premises, then there is reason (even good reason) to believe the conclusion.

As I elaborated, whatever reason there is to believe the second premise (R leads in polls, A leads in polls, C trails in polls, etc) is also a reason to believe the conclusion (which is true if either R or A wins), provided the first premise is also supported.
• 10
If there is good reason to believe those premises, then there is reason (even good reason) to believe the conclusion

That seems right, of course. But from a different view, there is not a good reason to believe the conclusion, since there is an overwhelming better reason to believe that if Reagan does not win, then Carter wins, so that Anderson does not win. That there is both good reason to believe the conclusion and not good reason to believe the conclusion is the paradox.
• 13
That sounds like it’s just reason to doubt premise 2. Which of course is the actual case in the real world prior to the election: it’s not certain that a Republican will win, a Democrat might win, in which case “if not-R then A” is false too.
• 2

Probability theory, which is currently the most fashionable calculus for representing and reasoning about beliefs and uncertainty, is only defined up to a measure over a sigma-algebra of sets denoting a collection of propositions. Unfortunately, practitioners of the theory don't normally consider this collection to be a model of any specific set of logical axioms, but rather as representing classes of observables, which means that modus ponens is formally absent from probability theory. Whenever an underlying logical system isn't specified in an application of probability theory (which is nearly all of the time), it is undetermined as to whether conditional probabilities or joint probabilities are the more fundamental epistemic principle.

Nevertheless, it is natural for Bayesian practitioners to assume some implicit underlying logic in an ad hoc fashion and to interpret modus ponens in terms of set intersections, in Venn diagram fashion. But as the example demonstrates, probabilities can behave non-intuitively with respect to modus ponens. Formally, Modus ponens speaks only of logical possibilities and not probabilities which are property of a model of a logic.
• 10

I'm not inclined to quibble with the givens of the problem or appeal to lack of certainty. That seems not to face the structure of the problem head on.

I guess we could say that there is good reason to believe the conclusion and that there is good reason not to believe the conclusion. Which in its form is not a contradiction.

The reason for believing that the conclusion is false is a good reason. So maybe its a better reason than the reason for believing the conclusion is true. So maybe its such a better reason that it makes the reason for believing the conclusion is true really not a good reason. But the reason for believing the conclusion is true is that it follows from a sound argument (true premises and modus ponens), and you can't get a better reason than that! Thus, still a puzzle.
• 10

There might be something lurking in the notion of 'good reason' that has to do with degrees of good reason, which also relates to degrees of confidence in beliefs. And Pfhorrest broaches the matter of lack of certainty. I'm not inclined to it, but maybe a solution does lie in that direction.
• 2
There might be something lurking in the notion of 'good reason' that has to do with degrees of good reason, which also relates to degrees of confidence in beliefs. And Pfhorrest broaches the matter of lack of certainty. I'm not inclined to it, but maybe a solution does lie in that direction.

In logic, either an arrow A -> B exists, or it does not. And so for logic there exists only possibility or non-possibility. On the other hand, probability measures over a set of propositions in a model of logic are chosen freely in accordance with external beliefs or experiments.

On the left side below are the axioms of OP's problem that specify every possible election outcome. On the right side is an example of a consistent set of degrees of confidence assigned to each possibility that coheres with every premise of the OP.

Andy or Carter --> Andy 0.25
Andy or Carter --> Carter 0.75

Reagan 0.80
Carter 0.15
Andy 0.05

As usual, Modus Ponens holds while saying nothing about the relative likelihood of possible winners.
• 43
I'm wondering if the English sentence has a different meaning when it nests conditionals than the surface logical syntax would indicate.

After all, it's not like we have parentheses designating which conditional to evaluate first.

It could just be a matter of bad translation.

I don't think I quite grasped the argument before, but I think I get it now. Am wondering if there are other nested conditionals that have a (on the surface) false conditional as its consequent, with a true premise...
• 65
I'm wondering if the English sentence has a different meaning when it nests conditionals than the surface logical syntax would indicate.

Yes, I'd say this.
• 12
Anderson ran as an Independent. If a Republican wins then it had to be Reagan. A single party only runs one candidate in a US presidential election. 1 is false.
• 43
I don't find that very convincing, at least, on the grounds that it can just be translated back -- it's logically equivalent.

I'm saying that the nested conditional in logic does not behave like a string of two if-then statements in English -- so it's not a matter of applying rules of inference to the way premise 1 is set out, but trying to find a different, reasonable interpretation of the English sentence into a logical syntax that keeps MP intact.
• 10

Anderson ran as Independent, but he was a Republican. It doesn't matter anyway, since we don't need to mention 'Republican', as we could just say 'Reagan or Anderson'. Moreover, we could say 'Reagan or x' for any x whatsoever. We could say:

If either Reagan wins or Donald Duck wins, then if Reagan doesn't win then Donald Duck wins.
Either Reagan wins or Donald Duck wins.
Therefore, If Reagan doesn't win then Donald Duck wins.

or

If either Reagan wins or Carter wins, then if Reagan doesn't win then Carter wins.
Either Reagan wins or Carter wins.
Therefore, If Reagan doesn't win, then Carter wins.

But with that argument, there's no puzzle.

/

The actual factual error in the problem is the claim that Reagan was way in the polls. Actually the polls were close between Reagan and Carter.
• 65
I don't find that very convincing, at least, on the grounds that it can just be translated back -- it's logically equivalent.

Yes, and neither is an example of modus ponens. The syntax of the problematic version simply gives the false appearance of one. Much like a sentence such as "this sentence is false" gives the false appearance of a truth-apt proposition.

As explained here "logical forms are semantic, not syntactic constructs", and the semantics of "A → (B → C)" isn't the same as the semantics of "A → D". You can't just substitute D for B → C and have the same logical form.
• 65
I'm saying that the nested conditional in logic does not behave like a string of two if-then statements in English -- so it's not a matter of applying rules of inference to the way premise 1 is set out, but trying to find a different, reasonable interpretation of the English sentence into a logical syntax that keeps MP intact.

That's what I did?
• 10
.I'm looking at this again with a fresh start.

First, we should put aside quibbles about (a) Anderson running as Independent and (b) the mistaken claim that Reagan was far ahead in the polls. We should just take the problem at face value and take as stipulated the hypotheses that 'Republican' includes both Reagan and Anderson and that Reagan was far ahead in the polls.

/

The validity of modus ponens is:

(a) When the premises are true then the conclusion is true.

The validity of modus ponens is not:

(b) When there is good reason to believe the premises are true then there is good reason to believe the conclusion is true.

So I don't think the example belies the validity of modus ponens.

But we might claim that if (b) fails then modus ponens is not reliable for informing our belief, but we do expect that modus pones is reliable for informing our belief, as indeed we have not just good reason, but irrefragable reason, to believe modus ponens is reliable for informing our belief. So it is a puzzle.

/

McGee actually wrote not simply about good reason for belief, but about was in fact believed. His argument can be fairly paraphrased:

(1) People believed and had good reason to believe: If either Reagan wins or Anderson wins, then if Reagan does not win then Anderson wins.

(2) People believed and had good reason to believe: Either Reagan wins or Anderson wins.

(3) People did not have reason to believe: If Reagan doesn't win then Anderson wins.

We can add (a) if people did not have reason to believe, then, a fortiori, they did not have good reason to believe, and (b) other than unjustifiably optimistic Anderson supporters, people did not believe that if Reagan doesn't win then Anderson wins.

But I don't know whether the particular wording changes the puzzle.

/

'has reason to believe' and 'has good reason to believe' are intensional:

Suppose there is a spy who stole documents from Interpol and that Smith is that spy. And Jones knows about the caper but little of its details. Then:

Jones has good reason to believe "the spy is the spy". But Jones does not have good reason to believe "Smith is the spy".

If we take out 'has good reason to believe' and leave only 'believed' then we have:

(4) People believed: If either Reagan wins or Anderson wins, then if Reagan does not win then Anderson wins.

(5) People believed: Either Reagan wins or Anderson wins.

(6) People did not believe: Reagan doesn't win then Anderson wins.

That mentions belief, but intensionality is not present. It is just three statements about what people believed.

(4) and (5) are quite unlikely true if by 'people' we mean typical people, even typical people well informed about the campaign, even just journalists and political scientists. Such people never had such thoughts as "If either Reagan wins or Anderson wins, then if Reagan does not win then Anderson wins" and "Either Reagan wins or Anderson wins"*.

* There it does matter that we say "A Republican wins" rather than "Reagan wins or Anderson wins", since the former was believed.

But we should be generous to McGee by revising to (a) "People who were informed about the campaigns and understood formal logic and were presented with such a proposition would have believed". Then there is no harm in taking "People believed" to stand for (a). And then (4) and (5) are true.

All that is shown in this version is that people believed certain premises but not a conclusion that follows from those premises. That's just a factual matter. It doesn't belie modus ponens.

/

Does temporality bear on the puzzle?

McGee's version uses future tense. 'wins' stands for 'will win'. Keeping consistent tense:

(7) People believed and had good reason to believe: If either Reagan will win or Anderson will win, then if Reagan will not win then Anderson will win.

(8) People believed and had good reason to believe: Either Reagan will or Anderson will win.

(9) People did not believe* and did not have reason to believe: If Reagan will not win then Anderson will win.

* I added 'did not believe' because it is true and fits the the pattern.

Recast in past tense, and hypothesize that the people mentioned lost access to information about the election starting with news about the voting results:

(10) People believed and had good reason to believe: If either Reagan won or Anderson won, then if Reagan didn't win then Anderson won.

(11) People believed and had good reason to believe: Either Reagan won or Anderson won.

(12) People did not believe and did not have reason to believe: If Reagan didn't win then Anderson won.

Still obtains as a puzzle. So I don't see temporality as bearing on the puzzle.

/

I mentioned "R -> (R v A))". Most people don't believe it, since they don't even know about, but they would believe it if they knew about formal logic, so they do have good reason to believe it.

Another poster mentioned material implication with its clause "'False antecedent then false consequent' is true". I bet most people don't believe that, since they've never heard of it, and they wouldn't believe it even if they knew about it. And a lot of people who have heard of it don't buy it. But there are people who do believe it, especially if they accept the first chapter in a logic book, so we can limit to those people.
• 3
I only have this example. Does anyone have more?

Here's one (which reflects the election example). Suppose I have a 3-sided die with the following roll statistics:

[odd]  1: 80%
[even] 2: 19%
[odd]  3: 1%


Before the die is rolled, I would have reason to believe 1 would be rolled (since it is rolled 80% of the time). If not 1, then I would not have reason to believe 3 would be rolled (since, when 1 is not rolled, 2 is rolled 95% of the time while 3 is only rolled 5% of the time).

Note: "It" below refers to the upward face of the rolled die.

(A) If it's odd then if it's not 1 then it's 3 [per the die characteristics]
(B) It's odd [premise]
(C) If it's not 1 then it's 3 [from (B),(A)]

Note that (B) eliminates face 2 as a possibility. Given that context, (C) can be interpreted as applying to just the odd die faces, in which case the inference is valid. In the broader context of all the die faces, the inference would be invalid (since 2 is a possibility and, furthermore, the possibility I would have reason to believe if 1 has been eliminated).

I gather that is the general point that @fdrake has been making about the domain change and the problem of how to formally capture the informal argument.

My question is, how would the argument be formally written to express that context change, and would that argument involve modus ponens or not?
• 11
I gather that is the general point that fdrake has been making about the domain change and the problem of how to formally capture the informal argument.

:up:
• 2
Modus Ponens is a logical rule for the composition of possibilities but not probabilities, since all logical statements are relative to the truth of premises that are non-logical axioms. So it is perfectly acceptable to disbelieve the actuality of a conclusion of Modus Ponens, for non-logical reasons.

Logic specifies what can happen, but not what will happen. After all, if that weren't the case, then an axiomatic system such as Peano arithmetic wouldn't be a forest of proofs, but merely a single proof of one result consisting of a single chain of reasoning.

Needless to say, there is an (unfortunate) temptation among philosophers and mathematicians to mix the concepts of logic/possibility with statistics/probability by considering conditional-probabilities to be a generalised form of logical implication. This is generally disastrous, because possibilities are easier to state and justify than probabilities which are usually ill-defined and whose use is generally controversial.
• 10

That's really good. It puts the puzzle in stark formal terms and takes out the background noise about the historical election facts. Thanks.

In the broader context of all the die faces, the inference would be invalid

I don't get that. The logic is monotonic. So how can adding premises make the argument invalid? And how would we formalize the inclusion of a broader context? I surely see the point that not mentioning (2) relates to the problem, but I don't know how we would formulate that other than just mentioning it, and how it would overturn an argument in a monotonic logic.

Meanwhile, I'm inclined to think that a solution would center around problems with the notion of "good reason to believe".
• 10
I think I might have a solution.

The solution is that it is not puzzling, let alone paradoxical, to have good reason to believe a statement and also good reason to believe the negation of that statement. Happens all the time in life when we are confronted with a tough decision.

Restating 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

Jack has good reason to believe (1).

Jack has good reason to believe (2).

Jack has good reason to believe (3).

Jack has good reason to believe (4) ~R -> C.

There is not a contradiction there.

To get a contradiction, we need to derive:

(5) Jack does not have good reason to believe (3)

But it wouldn't be by logic alone, since

~R -> A
~R -> C
~(C & A)

is a consistent set as seen by this model (which happens to be the real world):

R is true
C is false
A is false

Yes, we do get ~(~R -> A).

But we haven't yet derived a contradiction about Jack's good reason for belief. So the burden is on McGhee to show a contradiction, especially since it is not puzzling, let alone paradoxical, that one has good reason to believe a statement and also good reason to believe the negation of that statement.

But still, it does stick in the craw to say "Jack has good reason to believe that if R didn't win then A won".
• 10
I think McGhee begs the question when he asserts that Jack does not have good reason to believe (3). He does have good reason to believe it. But he also has good reason to believe (4).

/

But what if we changed the argument to this:

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

(2) R v A

Therefore, (3) ~R -> A

Jack has good reason to believe (1).

Jack has good reason to believe (2).

Jack has good reason to believe (1) and (2) imply (3).

So Jack has good reason to believe (3).

Jack has good reason to believe (4) ~R -> C.

I don't see that it changes anything materially.

/

Also, if 'good' is not being used in the analysis, then we can drop it, and just say 'reason to believe'.
• 2

Precisely. the proposition ~R --> A isn't in contradiction with the proposition ~R -->C because both denote possibilities, as opposed to probabilities or propensities. To get the latter, a non-logical probability measure must be added.

Or alternatively, since precise probabilities are usually difficult and controversial to assign, one simply ranks ~R --> C above ~R --> A to indicate which they believe is the most likely.
• 3
That's really good. It puts the puzzle in stark formal terms and takes out the background noise about the historical election facts. Thanks.

:up:

In the broader context of all the die faces, the inference would be invalid
— Andrew M

I don't get that. The logic is monotonic. So how can adding premises make the argument invalid? And how would we formalize the inclusion of a broader context? I surely see the point that not mentioning (2) relates to the problem, but I don't know how we would formulate that other than just mentioning it, and how it would overturn an argument in a monotonic logic.

By broader context, I meant a context where we consider only the characteristics of the die where face 1, 2 and 3 are all possibilities. So we might say, "If it's not 1 then it's 2". That's not a valid inference (since 3 is also remotely possible), but it's a reasonable belief based on the stated probabilities.

Whereas the more specific context includes (B) which eliminates face 2 as a possibility. So in that context we might say "If it's not 1 then it's 3" which is a valid inference and also a reasonable belief (since there are no other possibilities).

Meanwhile, I'm inclined to think that a solution would center around problems with the notion of "good reason to believe".

I think so as well. Initially (based on the polls), there's good reason to believe that if Reagan doesn't win then Carter will. But it's not a valid inference, since there is a remote possibility that Anderson will win.

When we subsequently learn that a Republican has won (or will win), then there is no longer good reason to believe that if Reagan doesn't win then Carter will, since Carter has been eliminated as a possibility. So the remote possibility of Anderson winning becomes the only possible alternative to Reagan winning. So there is now good reason to believe that if Reagan doesn't win then Anderson will. It's a valid inference, even though Anderson winning remains only a remote possibility.
• 10
I meant a context where we consider only the characteristics of the die where face 1, 2 and 3 are all possibilities

Right, I understood that.

"If it's not 1 then it's 2". That's not a valid inference (since 3 is also remotely possible)

I don't understand that.

Right, ~1 > 2 is not entailed when there is not a premise 1 v 2. But the reason it is not entailed is just logic. I don't see what the possibility of 3 has to do with.

Maybe you meant that the possibility of 2 should allow ~1 -> 2 as a possibility?

But 'possibility' is bringing a modal operator.

The premises are purely sentential:

[original argument:]
(background assumption) 1
(from background assumption) 1 -> (1 v 3)
(A) (1 v 3) -> (~1 -> 3)
(B) 1 v 3
therefore (C) ~1 -> 3

I do see this:

[revised argument:]
(background assumption) 1
(from background assumption) 1 -> (1 v 2 v 3)
(A') (1 v 2 v 3) -> (~1 -> (2 v3))
(B') 1 v 2 v 3
therefore (C) ~1 -> 3 WRONG

But that doesn't make the original argument incorrect.

if Reagan doesn't win then Carter will. But it's not a valid inference

It is valid from the background assumption that Reason wins.

"Reagan wins" is how we got "a Republican wins", which means "Reagan wins or Anderson wins".

Both ~R -> A and ~R -> C are entailed from the background assumption that Reagan wins.

But ~R -> C is not entailed from just "a Republican wins" which is R v A.

And of course, that is consistent.

So my solution is that there is good reason to believe both ~R -> A and ~R -> C.

Though it is counterintuitive to believe ~R -> A.

So there is good reason to believe something that is counterintuitive. And that is counterintuitive. (Is it paradoxical?) And modus ponens ponens is not invalid. And I think the problem has more to do with disjunction than with modus ponens. That aligns with you and fdrake in the sense that the puzzle results from leaving off Carter in the disjunction.
• 10

A logic form may not be comprehensive. A simple example:

Let P = AxRx
Let Q = Ra

P
therefore Q
INVALID

AxRx
Ra
VALID
• 10
I deleted this post, because I realized the solution is even simpler:

https://thephilosophyforum.com/discussion/comment/567916
• 3
(background assumption) 1

I think we're interpreting the problem differently. You regard 1 as the background assumption, whereas I regard (1 v 2 v 3) as the background assumption (i.e., the die can roll 1, 2 or 3).

What a person has good reason to believe (if not 1 then 2) is distinct from the logic of the situation (1 v 2 v 3).

When the person learns that an odd number has been rolled, then the logic becomes (1 v 3). That knowledge update is a change of context, and the person's reasoning changes. They now have good reason to believe (if not 1 then 3), since 3 is now the only possible alternative to 1, albeit remote.

So my solution is that there is good reason to believe both ~R -> A and ~R -> C.

Though it is counterintuitive to believe ~R -> A.

So there is good reason to believe something that is counterintuitive. And that is counterintuitive. (Is it paradoxical?) And modus ponens ponens is not invalid. And I think the problem has more to do with disjunction than with modus ponens. That aligns with you and fdrake in the sense that the puzzle results from leaving off Carter in the disjunction.

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.

On the other hand, a person could learn that a Republican has won. Given their updated knowledge (a change of context), Carter cannot have won since Carter is not a Republican. So, given their newly acquired knowledge, if Reagan didn't win, then Anderson did.

So I think that interpretation leaves modus ponens as valid and also shows how ~R -> A can be intuitive when derived in the appropriate context.
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