## A Counterexample to Modus Ponens

• 4.9k
Still valid.

Depends on the symbol interpretation. I think it's more of a problem about how the informal argument codifies into the formal logic.

In the same manner that from "You're fine" you wouldn't be able to conclude "You're pretty" or "You're okay" without knowing the context, you also wouldn't be able to assert "You're fine" is true if and only if you're fine if the quoted thing and the "disquoted" thing were from different speech events. The strings in the statement are like that, "then" in line 1 begins considering only republicans, you don't get that same subsetting effect if you assert it without "then".
• 1.2k
But the puzzle includes an intensional operator "believe'.

Please not. You're inviting the enthusiasts for modal logic to show off, and end up perpetuating the silly libel of a logical error subtlety.

Assume, assert, affirm, hold, "believe"... whatever.

an instance of modus ponens.

Indeed. And perfectly valid.

If you can't stand by all 3 lines at once, don't. They can't be a good expression of what you're trying to say. Don't necessarily involve any logic in expressing yourself, but don't think you need a better one. (I don't mean you.)
• 873
Make explicit 'Republican' and 'Democrat':

'P' stands for 'Republican wins'.
'D' stands for 'Democrat wins'.

P <-> (R v A)
D <-> C
R <-> ~(A v C)
A <-> ~(R v C)
C <-> ~(R v A)
R

So, these follow:

P -> (~R -> A)
P

and conclusion:

~R -> A

/

Or spell it out with constants and predicates.

'c' stands for Carter
'r' stands for Reagan
'a' stands for Anderson

'R' stands for 'is a Republican candidate'
'D' stands for 'is a Democratic candidate'
'W' stands for 'wins the election:

Ax(Rx <-> (x = r v x = a))
Ax(Dx <-> x = c) [but not needed for the argument]
Rr
Ra
Dc [but not needed for the argument]
Wa <-> ~(Wr v Wc)
Wc <-> ~(Wr v Wa)
Wr <-> ~(Wa v Wc)
Wr [but not needed for the argument]

So, these follow:

1. Ex(Rx & Wx) -> (~Wr -> Wa) from background premises
2. Ex(Rx & Wx) from background premises
3. (Rr & Wr) v (Ra & Wa) from 2 and background premises
4. ((Rr & Wr) v (Ra & Wa)) -> (~Wr -.> Wa)
5. ~Wr -> Wa from 3,4 MP

So even with the finer analysis with constants and predicates, we still arrive at MP captured more easily anyway with just sentence letters.

So my point in response to you is that In a context of classical logic, if an argument is valid then it doesn't become invalid by adding premises of finer analysis (such as predicate logic is finer than propositional logic). This is the monotonic property of classical logic.
• 873

I essentially agree. My point is that:

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

is an instance of modus ponens, but

(R v A) -> (~R -> A)
R v A
therefore we have reason to believe ~R -> A

is not.
• 873
But the author might argue this:

If modus ponens is valid, then if we believe the premises, then we believe the conclusion (not always in fact - people err - but in principle). And the premises are believed but the conclusion is not. So, still, there's a puzzle.
• 873
So the argument above is:

1. (A ∧ B) → C
2. A
3. B → C

No, it's:

(R v A) -> (~R -> A)
(R v A)
therefore ~R -> A
• 873
if it is not Reagan who wins, it will be Jimmy Carter

C <-> ~(R v A) is a given

~ R -> (C v A) is a given

C <-> ~A is a given

Lets' say:

~R -> C is a given

Then:

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

• 4.9k
If modus ponens is valid, then if we believe the premises, then we believe the conclusion (not always in fact - people err - but in principle). And the premises are believed but the conclusion is not. So, still, there's a puzzle.

When you believe the premises, you interpret to the string which is used to express them.

(1) If you're an apple, then you're sour or sweet or juicy.
(2) If you're not juicy, then you're sour or sweet.
(3) You're not juicy.
(4) You're either sour or sweet.

Exactly the same thing. You end up transitioning to a space of interpretations that excludes juiciness.
• 873

I take it that you intend (4) as a conclusion from the premises above it.

You end up transitioning to a space of interpretations that excludes juiciness.

I don't know what you mean by "space of interpretations that excludes". It would help if you said it in ordinary terminology for logic.

The predicates are 'is sour', 'is sweet' and 'is juicy'. I guess you mean that your intended interpretation has as its domain the set of apples?

Let 'S' stand for 'is sour', 'W' for 'is sweet' and 'J' for 'is juicy'. Let 'c' be a constant.

If the domain is intended to be the set of apples, then we don't need to symbolize 'is an apple'.

So perhaps this captures your argument:

1. Ax(Sx v Wx v Jx) premise
2. Ax(~Jx -> (Sx v Wx)) from 1
3. ~Jc premise
4 Sc v Wc from 2,3

I don't see a problem.

Or, if the intended domain is not specified, and we have 'P' for 'is an apple':

1. Ax(Px -> (Sx v Wx v Jx)) premise
2. Ax(Px -> (~Jx -> (Sx v Wx))) from 1
3. Pc
4. ~Jc premise
5. Sc v Wc from 2,3,4

I don't see a problem.

/

In my predicate argument about the election, let the intended domain be {Carter, Reagan, Anderson}.

That would make some of the background premises unneeded, but logically I don't see a problem.
• 4.9k
5 Sc v Wc from 2,3,4

I don't see a problem.

I don't think the problem is that ~Jc entails (Sc v Wc) given (Jc v Sc v Wc), it's that it reads as if when one asserts ~Jc, one has established (Sc v Wc) assuming the argument is valid. Whereas in fact all that can be established is (Jc v Sc v Wc).

It couldn't've been a juicy apple!
• 873
it reads as if when one asserts ~Jc, one has established (Sc v Wc) assuming the argument is valid.

Of course, that argument establishes its conclusion only if the premises themselves are established. I don't see a problem.
• 4.9k

Let's see if I can make you see (how I see) the problem. How I see the problem isn't that the op is a counter example to modus ponens (I think modus ponens is valid), I see the problem as that the argument as stated can't be interpreted as a modus ponens, even though it looks like one in terms of the letters that constitute it - how it's written.

I have a box, it contains an apple, an orange, or a banana, you don't know which. But you do know it can only be one of those three. It can't be more than one of those three either. It contains exactly one fruit item.

One of {apple, orange, banana} will be picked out.
Analogously:
One of {Reagan, Carter, Anderson} will win.

Let's say you believe that (assume that, posit, assume as a premise) you will pick out round-ish fruit (apple or orange). In that space of assumptions, (apple, orange), if it's not an orange it must be an apple. not(apple) implies orange holds in that domain, because it consists only of an apple and an orange.

Similarly, Republican consists only of Anderson and Reagan.

If you wanted to conclude that you will receive an apple if you don't receive an orange, you would need to eliminate the possibility of receiving a banana. You can't do that.

What you can do is eliminate the possibility of receiving a banana if you have already assumed, or it is true that you will have received, a roundish fruit. That follows from the assumption. But they can't exclude the banana, so they have no reason to believe (in the OP's terms) that they wouldn't receive a banana (analogously, a democrat, Carter, would win).

So when the words go into your eyeballs, despite the literal characters tracing out a clear instance of modus ponens, as Banno wrote:

Modus ponens:
If p then q
p
therefore, q

p: A Republican wins the election,
q: If it's not Reagan who wins, it will be Anderson

So:

If A Republican wins the election, then If it's not Reagan who wins, it will be Anderson

and

A Republican wins the election

which, by MP, gives

If it's not Reagan who wins, it will be Anderson.

But if it is not Reagan who wins, it will be Jimmy Carter. So there is a prima facie case that MP reaches a false conclusion from true premises.

The overall interpretation is different from what you would expect - writing it like:

If A Republican wins the election, then If it's not Reagan who wins, it will be Anderson

and

A Republican wins the election

which, by MP, gives

(C) If it's not Reagan who wins, it will be Anderson.

Has (C) talking about all the candidates, it's evaluated over the candidates - it could be Carter.

If A Republican wins the election, then If it's not Reagan who wins, it will be Anderson

The context of "it's" in the "then" part of the if-then references only Republicans. A context, current Republican candidates, in which the disjunction between Reagan and Anderson (Reagan or Anderson will win) holds.
• 14.3k
et al...

I'm thinking that your notion of context is along the right track.

The Op was a request for further on the article. There's a trail of academic articles, various detailed accounts.

But I'm going to go with the MP being valid, and hence the conclusion true; Given that a republican won, it is true that If it's not Reagan who wins, it will be Anderson. To get to "it is true that If it's not Reagan who wins, it will be Carter", an additional premiss is needed.
• 1.2k
(R v A) -> (~R -> A)
R v A
therefore we have reason to believe ~R -> A

is not [modus ponens].

Not if you hear it, for no reasons that are obvious to me, as talking about psychology. I hear it, for reasons of charity and extensionalism, as dialect for "therefore we have deductive reason to assert"... i.e. "therefore".
• 873
"therefore we have deductive reason to assert"

That still breaks the form of modus ponens.

'we have deductive reasons to assert is' is intensional.

If it were merely a flourishing touch, then we could delete it, but if we delete it then then the puzzle fizzles in the form it's given, as its form is not modus ponens if it injects a modal operator.

If P then Q
P
Therefore Q

is modus ponens.

If P then Q
P
Therefore [modal operator] Q

is not modus ponens.

extensionalism

On the contrary, it introduces intensionality.
• 4.6k
If it's not Reagan who wins, it will be Anderson.

That is logically equivalent to “Reagan or Anderson wins the election”.

Given that a Republican wins the election, it is valid to conclude that Reagan or Anderson wins the election.

If Reagan wins the election, it’s true that Reagan or Anderson wins the election.

So if they have good reason to think that a Republican will win the election because they have good reason to think Reagan will win the election, then they also have good reason to think that Reagan or Anderson will win, or equivalently, if not Reagan then Anderson. Because in believing that a Republican will win, they’ve ruled out Carter already.
• 873

Of course, but that doesn't address the puzzle.
• 873
the argument as stated can't be interpreted as a modus ponens

The argument as stated is not modus ponens. It injects a modal operator in front of the conclusion.

But there is still a puzzle:

If modus ponens is valid, then if we believe the premises, then we believe the conclusion (not always in fact - people err - but in principle). And the premises are believed but the conclusion is not. So, still, there's a puzzle.

Maybe I'll get time to analyze the rest of your post.
• 4.6k
It absolutely does. The supposedly invalid conclusion is valid after all.
• 873

The conclusion is not valid. The conclusion is contingent.

The modus ponens argument

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

is valid.

But the argument is not of that form. It's of this form:

(R v A) -> (~R -> A)
R v A
therefore [modal operator] ~R -> A

But there is still a puzzle, as I mentioned.
• 4.6k
therefore [modal operator] ~R -> A

I think you're erroneously reading in a modal operator (and which one are you reading in?)

The claim is that the proposition

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

and the proposition that

[2] A Republican will win the election.

do not together give good reason to believe that

[3] If it's not Reagan who wins, it will be Anderson.

That "do not together give good reason to believe" is just the writer denying the validity of the argument; modals about belief are not part of the structure of the argument itself.

Since if it is given that a Republican will win the election, and that the only Republican alternative to Reagan is Anderson, it is valid to conclude that if not Reagan then Anderson will win the election, either there mustn't actually be good reason to believe at least one of those premises, or there is in fact good reason to believe the conclusion, contrary to the writer's claim.
• 873

I see your point. But I haven't been in disagreement.

I don't dispute the author's argument about the modus ponens argument.

My point is to be careful not to take his example in the form he literally gave it.

P -> Q
P
Therefore [modal]Q.

Rather that his analysis can be stated along the lines of:

If modus ponens is valid, then if we believe the premises, then we believe the conclusion (not always in fact - people err - but in principle). And the premises are believed but the conclusion is not.

This does not prove the invalidity of modus ponens. Rather it shows that modus ponens may fail our expectations of belief. And it does seem to me to be a genuine puzzle.
• 14.3k
This does not prove the invalidity of modus ponens. Rather it shows that modus ponens may fail our expectations of belief. And it does seem to me to be a genuine puzzle.

I think we are in agreement.

I think you're erroneously reading in a modal operator

Expressing it as a different propositional syllogism, introducing predicates, and introducing modality are all unjustified and overly complicated.

We deal with it by assuming MP is correct. So we need to look at what the consequence is for our beliefs. The syllogism says nothing about Carter, so no conclusions about Carter can be reached. The only other contender is Anderson, so the conclusion has to be about Anderson or Reagan. Given that, it is true.

The discussion fo domains is close.
• 873
I don't have a solution, but below is one way to lay out the problem by "brute force".

In case it matters, we note that the text mentions both 'good reason to believe' and 'reason to believe'.

1. we have good reason to believe (R v A) -> (~R -> A)
2. we have good reason to believe R v A
therefore 3. we have reason to believe ~R -> A

That does not prove the invalidity of modus ponens. But it is a puzzle.

Mentioning both 'reason to believe' and 'good reason to believe' suggests degrees of reasons to believe. Or perhaps the author didn't mean to imply degrees. In that case we have one of these two:

1. we have good reason to believe (R v A) -> (~R -> A)
2. we have good reason to believe R v A
therefore 3. we have good reason to believe ~R -> A

That seems to preserve the puzzle.

1. we have reason to believe (R v A) -> (~R -> A)
2. we have reason to believe R v A
therefore 3. we have reason to believe ~R -> A

That doesn't seem as strong a puzzle, but still a puzzle.

Or take the modal operator outside the scope of the argument itself and we have three versions. Of the three, the first is closest to the author's text:

1. (R v A) -> (~R -> A)
2. R v A
therefore 3. ~R -> A

we have good reason to believe 1.
we have good reason to believe 2.
we believe that modus ponens is valid, so we have reason to believe 3.

or

1. (R v A) -> (~R -> A)
2. R v A
therefore 3. ~R -> A

we have good reason to believe 1.
we have good reason to believe 2.
we believe that modus ponens is valid, so we have good reason to believe 3.

1. (R v A) -> (~R -> A)
2. R v A
therefore 3. ~R -> A

we have reason to believe 1.
we have reason to believe 2.
we believe that modus ponens is valid, so we have reason to believe 3.

But if it's just 'believes' then there is a chink in the armor:

1. (R v A) -> (~R -> A)
2. R v A
therefore 3. ~R -> A

we believe 1.
we believe 2.
we believe that modus ponens is valid, so we believe 3.

If someone claimed that they believe 1 and 2, but not 3, then I might say, "I don't think you really do believe 2."

Change to 'knows', and the puzzle is even weaker:

1. (R v A) -> (~R -> A)
2. R v A
therefore 3. ~R -> A

we know 1.
we know 2.
so we know 3.

If someone said they know 1 and 2 but not 3, then I might say, "Then wake up and smell the coffee: you're just not following through to accept knowledge implied by what you do know."

So I think the specific nature of the intensionality does have something to do with this puzzle.
• 873
In that space of assumptions, (apple, orange)

The domain is {apple, orange, banana}.

{apple, orange} is a subset of the domain. {apple orange"} is not a "space of assumptions". It is not a set of assumptions. It's a set whose members are two different pieces of fruit.

if it's not an orange it must be an apple. not(apple) implies orange holds in that domain

If x is in {apple orange} and x is not the orange, then x is the apple. And if x is in {apple orange} and x is not the apple then x is the orange.

if you don't receive an orange, you would need to eliminate the possibility of receiving a banana. You can't do that.

No, we can reason from an assumption that you won't receive the banana. That the banana is in the domain doesn't entail that we can't assume that you won't receive the banana. Let 'W' stand for 'we receive x'.

The domain is {apple orange banana} but that doesn't stop us from reasoning from the premise:

W(apple) or W(orange)

Let the domain = {0 1 2}. Let 'W' stand for 'x is the number of pens' in my pocket.

Suppose I know I have a pen in my pocket, so I have the premise:

W(1) or W(2)

Or the author's example:

Let the domain be {Reagan, Anderson, Carter}. (By the way, Anderson ran as an Independent, not as a Republican, though he was a Republican.)

Let 'W' stand for 'x wins'.

Then we have the premise:

W(Reagan) or W(Anderson)

What you can do is eliminate the possibility of receiving a banana if you have already assumed, or it is true that you will have received, a roundish fruit.

Yes, just as we assume a Republican will win.

But they can't exclude the banana, so they have no reason to believe (in the OP's terms) that they wouldn't receive a banana (analogously, a democrat, Carter, would win).

No, they have very good reason: the polls. But it doesn't matter about the factual givens anyway. For sake of argument we accept that we have good reason to believe that a Republican will win and moreover that we assume a Republican will win.

it's evaluated over the candidates

Again, we can take it merely propositionally.

R for 'Reagan wins'
A for 'Anderson wins'
C for 'Carter wins'

(R v A) -> (~R -> A)
R v A
therefore ~R -> A
• 873
Suppose instead of "R v A" our second premise is "R v C". Then there's no puzzle.

But why did we adopt "R v A"? Because Reagan looked bound to win. So we got it from the theorem

R -> (R v A)

So maybe it's not modus ponens that should be in question, but "R -> (R v A)".

I'm not saying we should doubt the validity of "R -> (R v A)". But maybe it's the one not mixing well with intensionality and not so much podus ponens. I think that might be right. Because we can can do it this way, without modus ponens:

1. R
2. R v A
therefore 3. ~R -> A
• 873
When I first read the claim given by the author that Reagan was decisively ahead of Carter in polling, I felt something was wrong, but I let it slide. Then when I took a moment to really think about it, I realized that it is wrong. Indeed it is famous that the polls were close yet Reagan won so decisively.
• 873

I think there is something to what you say. But I don't know whether we need the notion of domains for it.

She has good reason to believe she will receive the apple.
She believes that (A -> (A v O)) is valid.
So she has good reason to believe she will receive the apple or she will receive the orange.
She believes that (A v O) -> (~A -> O) is valid
So she has good reason to believe that if she doesn't receive the apple then she will receive the orange.

But she doesn't have good reason to believe that if she doesn't receive the apple then she will receive the orange.

And that's a puzzle.

But why doesn't she have good reason to believe that if she doesn't receive the apple then she will receive the orange? Because she has good reason to believe that if she doesn't receive the apple then she will receive the banana. (That's where your line of thinking comes in.)

So the banana comes up regarding her beliefs, but it doesn't come up in the argument itself.

So how can that be used to solve the puzzle?

This makes me want to abandon my suggestion that maybe its more about disjunction and intensionality than about modus ponens and intensionality. Maybe it's something about deduction and intensionality in genera (or maybe even more generally about inference and intensionality in general?)
• 4.6k
If modus ponens is valid, then if we believe the premises, then we believe the conclusion (not always in fact - people err - but in principle). And the premises are believed but the conclusion is not.

The quote in the OP is not just about whether someone believes things, but about whether they have good reason to believe them. That's also what logical inferences (like modus ponens) are all about: it's not that believing the premises entails believing the conclusion, but that believing the premises gives reason to believe the premises. As you say, people err, and sometimes don't believe what they have reason to. But 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. That's incorrect: having good reason to believe the premises does constitute having good reason to believe the conclusion.

If you have good reason to believe that:

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

(say, because those are the top two highest-polling candidates in the Republican primary)

and you have good reason to believe that:

[2] A Republican will win the election.

(say, because Reagan is the top-polling candidate in the general election, so you reasonably think that Reagan, a Republican, will win)

then you do have good reason to believe that:

[3] If it's not Reagan who wins, it will be Anderson

(because that's logically equivalent to "Reagan or Anderson will win", and either "Reagan wins" or "Anderson wins" satisfies that, so if you have, say, good reason to believe that Reagan will win, which is your reason for believing [2], then you also have good reason to believe [3]; or if you have good reason to believe that Anderson will win, same thing; or if you don't have good reason to choose between Reagan and Anderson but you have good reason to expect Carter to lose; anything that gives you reason to believe [2] also gives you reason to believe [3], so long as you've also got some reason to believe [1]).
• 873
about whether they have good reason to believe them.

Yes, and I took account of that in followup posts. Actually, he mentions both 'good reason to believe' and 'reason to believe'. I would guess he didn't mean that difference as playing a role, but it might be good to see what happens with the distinction and without the distinction.

That's also what logical inferences (like modus ponens) are all about:

"all about" is sweeping. 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.

Most of the rest of your post is explanation of what I understood when I read the first post in this thread. That's okay though, as other readers may benefit from it.
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