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In the mid 1980's Vann McGee proposed a counterexample to Modus Ponens:

(a) If a Republicans will win the election, then if Reagan will not win, Anderson will win. (b) A Republican will win the election. (c) So, if Reagan will not win, Anderson will win.

Christian Piller describes it here: "[McGee's] attempt to show that modus ponens is not a valid form of inference - and to show this by the help of a counterexample and not by envisaging an evil demon confusing us - is proof of the ingenuity of a philosopher's ability to doubt."

John MacFarlane here lists two additional statements of the same type:

(a) If that creature is a fish, then if it has lungs, it is a lungfish. (b) That creature is a fish. (c) So, if it has lungs, it is a lungfish.

(a) If Uncle Otto doesn’t find gold, then if he strikes it rich, he will strike it rich by finding silver. (b) Uncle Otto won’t find gold. (c) So, if Uncle Otto strikes it rich, he will strike it rich by finding silver.

Modus Ponens permeates all of mathematics yet the counterexample seems primarily discussed in the philosophical literature. Is it accepted in the mathematical community? Is there a precise, mathematical restatement (eg, in terms of set theory or categorical) - free of subject-matter - that everyone can agree on? Or does it lead to a no-mans land of disputed interpretations?

Recall, proof of the conditional A --> B doesn't require A to be true. But the detachment of B as a true consequence the only follows via Modus Ponens, which requires the antecedent of a conditional to be true.

Lawvere & Rosebrugh write in Sets for Mathematics that substitution, correctly objectified, is composition.

If McGee's counterexample is valid, it would seem that substitutions of the form A --> (B --> C) are a "transitivity trap" so to speak.

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    I honestly don't see the problem with any of these statements. What is the contradiction/paradox presented here?2012-05-09
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    I think the question could be interesting. I would have voted it up if you had made clear the point outlined by Thomas and if you had added this link http://en.wikipedia.org/wiki/Modus_ponens to the question. Looking forward to vote up your next question! :)2012-05-09
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    Yes, all these examples are perfectly logically sound, and are consistent Modus Ponens as far as I can tell. In Piller's discussion of the first example he seems to be suggesting that people don't fully believe (b). In other words, while it's true that people believed that if Reagan didn't win, then Carter would (rather than Anderson), if they also believe that a Republican will inevitably win then (c) is fine. It just happens that the case that Reagan doesn't win is (considered) impossible, so the statement is vacuous.2012-05-09
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    Ah, after some search, the point is that the statement "if Reagan will not win, Anderson will win" is not obviously true. But modus ponens exists in a universe of facts, and it isn't false in a universe where you know (a) and (b) already. If you don't have the additional facts (a) and (b), then (c) isn't true, but that's the nature of deduction. (c) isn't true absent context, it is dedicble from (a) and (b).2012-05-09
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    Reading John MacFarlane's paper left me with some doubt he has said anything interesting at all. Basically it seems to boil down to the argument that if Regan didn't win, it was more likely that Carter won, but I don't see the problem there because if Carter did win then the original statement is vacuously true...2012-05-09
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    @GiovanniDeGaetano: no need to be stingy with votes!2012-05-09
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    @TheChaz, I hope you (and everyone else) will look at the following not as a polemic but as a friendly expression of my opinion. I try to use my votes to influence the quality of the questions (and answers) of this site. I didn't mean to be stingy, but it seems to me that the quality of this specific question could have been really improved with a very small effort. I tried, instead of simply moving away, to point out what could have been done in this direction. I apologize if my previous comment was interpreted as rude, and I hope that my position is now clear! All the best!2012-05-09
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    I am thoroughly baffled by how on earth that is supposed to be even mildly convincing 'counterexample'.2012-05-09
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    @GiovanniDeGaetano: I actually agree with you. I should have added earlier the possibility of an *eventual* (up)vote, after the OP has edited to include their effort/research. My standards for rudeness are a bit lax, but I didn't find anything rude!2012-05-09
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    @GiovanniDeGaetano, after reading your comment, I edited my Q to include the meaning of MP (rather than a link to wikipedia, since I'm not using any specific result or comment from it), as well as the remark from L&R's most excellent book.2012-05-09
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    I wanted to double-downvote this -- firstly for the ridiculous claim that any of these examples is a counterexample, and secondly for assuming that your readers have any idea who Anderson was. But I couldn't.2012-05-09
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    I remember Anderson. Is everyone else a young whipper-snapper?2012-05-09
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    @GEdgar: I'm not a young whipper-snapper. But neither am I American. (There's plenty of us about, you know.)2012-05-09
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    It's bad enough when mathematical ideas are misused to prove philosophical implications. It's **a lot** worse, though, when philosophical ideas are misused to "prove" mathematical statements.2012-05-09
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    @AsafKaragila To be fair, I don't think that is what MacFarlane (and, by inference, McGee) were doing. They were showing that attempting to carry out modus ponens (as understood mathematically) on certain natural seeming statements leads to counterintuitive results. The lesson is not that philosophers cannot do logic; it's that philosophers care about questions where naive attempts to formalize the reasoning seem problematic.2012-05-09
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    @Yemon: To be fair, I did not point any specific fingers (mostly because I haven't read any of the references) to begin with. I was pointing out that once again non-mathematicians make the mistake of thinking that mathematics has something to do with reality. Indeed, I was exactly saying that philosophers cannot do logic - and applying a philosophical "conundrum" as a pseudo-valid reason to reject mathematically sound rules is even worse than using incompleteness to prove/disprove the existence of god.2012-05-09
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    @AsafKaragila I don't see it as "a pseudo-valid reason to reject mathematically sound rules", I see it as "an argument for not applying mathematically sound rules in contexts where the situation is not entirely amenable to mathematical formalization". Saying philosophers cannot do logic is like saying mathematicians can't do physics properly.2012-05-09
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    @AsafKaragila Why not read the references? Maybe, to quote Signor Montoya, they do not say what the OP thinks they say (or what people on this thread think the OP thinks they say)2012-05-09
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    At the risk of being infra dig by quoting from a primary source, here is an excerpt from that Piller article: "True, McGee does not attack propositional logic. But modus ponens is not used in logic alone. Everyday reasoning, in as well as outside one's particular profession, relies on modus ponens and similar rules. Philosophical, scientific, and everyday arguments proceed in a natural language. What we write, talk, and think is or can be expressed in a natural language. Losing modus ponens for all these purposes seems to be a severe loss, much more severe than a change in propositional logic"2012-05-09
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    @Yemon: Because it's 1:30am and tomorrow is a long day; because someone has to write my thesis down and apparently that someone has to be me; because I just don't want to do that right now; because I am not in the mood for actual reading and would at best skim through a text; because I did not use that word so much and I think it means what I think it means. I can go on with excuses and pseudo-reasons for not reading it for a long time. I also wish to remind you that you said that philosophers cannot do logic before me. I merely agreed that it is consistent, not that it is a provable claim. :)2012-05-09
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    In fairness, Piller goes on to mention some counter-arguments to McGee's own argument. He thinks the counter-arguments do not fully rebut McGee; I'm not sure I agree with him, but would have to think this over more carefully.2012-05-09
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    @Yemon: So what you are saying is that the references in the article discuss science in general? This would make this question quite off topic here, in my taste anyway, and I would then recommend it to be re-asked/migrated to philosophy.SE instead.2012-05-09
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    I did not understand the McGee example, because @alancalvitti omitted the context. Here it is: “Opinion polls taken just before the 1980 election showed the Republican Ronald Reagan decisively ahead of the Democrat Jimmy Carter, with the other Republican in the race, John Anderson, a distant third. Those apprised of the poll results believed, with good reason: [(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.] Yet they did not have reason to believe [(3) If it's not Reagan who wins, it will be Anderson.]”2012-05-10
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    @MarkDominus, I wasn't aware of the context.2012-05-10
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    I read "[On Some Counterexamples to MP and MT](http://web.me.com/sethyalcin/web/work_files/davishandout.pdf)" by Seth Yalcin today; it discusses McGee's argument and some related arguments in some detail. I found it quite interesting, and I hope you do too.2012-05-10
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    @MarkDominus, thanks for the link. The olume of comments here and research on the issue (Yalcin is at Berkeley) show it's not an entirely trivial matter to map logic to real world propositions. But Thomas Andrews and Zhen Lin understood from the start, quantifiers matter.2012-05-11
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    If it were an entirely trivial matter to map logic onto real world propositions, it would have been solved in the time of Aristotle! You might enjoy reading the discussion of implication in Graham Priest's *An Introduction to Non-Classical Logic: From If to Is*. Much of the book—starting on page 12, in fact—is devoted to analyzing the ways in which various formalizations of implication fail to reflect our intuition about how implication should work.2012-05-11
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    @MarkDominus, thanks for the ref to Priest's book, will check it out if my library has it.2012-05-13
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    I downvoted this question. Let me just make this perfectly clear: MCGEE IS NOT MAKING A MATHEMATICAL CLAIM. MacFarlane isn't talking about mathematics either. It is unfortunate that so many of the people on this thread have jumped on the bandwagon into thinking that these philosophers are ignorant of mathematics, when in reality they are doing a completely different kind of work. It is clearly not the place of people on math.SE to make judgments about philosophical work, and yet people have used this question as an opportunity to do so.2016-01-26

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In the first example, it seems like the problem is with an intuition of truth being "high likelihood." You can't start from

(a') If a Republican wins, then if Ronald Reagan doesn't win, Anderson will win
(b') It is highly likely that a Republican will win

and deduce:

(c') If Reagan doesn't win it is highly likely that Anderson will win

That certainly is not a valid statement, even if (a') and (b') are true. But it also isn't an application of Modus Ponens.

(For those not old enough to remember, in 1980, the US presidential election was between Reagan, a Republican, Carter, a Democrat, and Anderson, a Republican running as an independent. Anderson was not very likely to win - if Reagan did not win, then it was highly likely that Carter would be the winner. But, given that a Republican was going to win, if Reagan did not win, it was most likely that Anderson would have won.)

Vann McGee, then, appears to be unaware of the fact (or just playing qwith it) that the truths used in logic are absolute. Modus Ponens only works if you are careful about your language. If you are lazy about your language, as in all things, logical deduction is useless.

If you want to deal with degrees of likelihood, you want probability. If you want degrees of truth other than pure "true" and pure "false," you want fuzzy logic. Modus Ponens fails in these variants of logic, and it is worth exploring how it fails and what sorts of deductions you can do in these spaces, but it is hardly a failure of modus ponens - it is more a failure of imprecise colloquial language.

The lungfish example is actually a different sort of error, fundamentally related to the difference between Propositional Logic, in which the only types are propositions, and First Order Logic, in which you can make propositions about "all" things. In first order logic, you would write:

(a) For any thing, if the thing is a fish, then if the thing has lungs, then the thing is a lungfish.
(b) This thing is a fish
(c) Therefore, if this thing has lungs, then this thing is a lungfish.

(c) Is not the same as saying, "For any thing, if the thing has lungs, then the thing is a lungfish," but rather, a statement about a specific thing about which we have some (possibly incomplete) information.

If you start with the statements:

(u) For all X, If X won the election, then X is a Republican.
(v) Y won the election

You can conclude:

(w) Y is a Republican

But that doesn't mean that (w) is true for all Y, it only means it is true given the statement (v).

One of the frequent flaws in elementary logic is that people think "implication" actually implicitly means "for all cases." (Often it also is taken to imply causality.) It doesn't. Implication is always about individual instances. The only way you get a "for all" added to implication is by explicitly adding that phrase to the sentence. In common language, it often doesn't need to be there. But the meaning in hard logic of the "P implies Q" is always about an individual instance, and the only way to make it general is by adding a "for all" explicitly to the sentence and adding a variable to the expression.

Modus ponens is a purely Propositional Logic statement.

The symbol $\forall$ is used to represent "For all" in First Order Logic. What you are trying to do is start with the statements:

(a) $\forall X: P(X)\implies Q(X)$
(b) $P(Y)$

and conclude:

(c) $\forall Y: Q(Y)$

But that is not how modus ponens of First Order Logic works. You cannot add back the $\forall$ part of the sentence. What you can do, from (a), and (b) is conclude:

(a') $P(Y)\implies Q(Y)$ (by the substitution rule for $\forall$)
(d) $Q(Y)$ (By modus ponens)

$Q(Y)$ is not the same statement as $\forall Y: Q(Y)$. $Q(Y)$ is a conclusion given that you've already stated that you know $P(Y)$ is true.

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    Some form of Modus Ponens should still hold even in probabilistic logic, right?2012-05-09
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    Sure, you can find some variants, and most probability is done inside absolute logic, but if your base logical language is dealing with probabilistic truth rather than absolute truth, you've got some work to do. @Neal2012-05-09
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    I see. I guess in some sense, the more complicated the expression, the "fuzzier" its truth is going to be. $(P\Rightarrow[Q\Rightarrow R]\wedge P)\Rightarrow(Q\Rightarrow R)$ is pretty complicated, so if you're working probabilistically, and you only know $Pr[P,Q,R]$, then the truth of the modus ponens is going to be much less certain than the individual truths of $P,Q,R$.2012-05-09
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    @ThomasAndrews, thanks for rephrasing the lungfish counterexample. It is more specific, as you point out. But the statements are all yes/no: doesn't seem to require any gradation in truth values or likelyhood (leaving aside the complexity of biology). It seems that your restatement can be phrased in Boolean or classical logic. Do you concur?2012-05-09
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    The lungfish example is pure language error, having nothing to do with grades of truth. The reason (c) appears false in the first example is that it implicitly seems untrue, but that's only if you treat the premise (b) as a high likelihood, rather than as an actual hard truth. If (a) and (b) are absolutely true, then (c) is clearly absolutely true. @alancalvitti2012-05-09
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    @Neal, I agree with the beginning of your comment. And there are today non-classical logic systems like Heyting algebras and also "relaxed" syllogisms (eg Polya's system). However, probability theory based on Kolmogorov's approach is classical: there are measures, and random variables from sample space to the real line, and functors from Borel algebras back to the sample space and so on (see Rota's definition). But the underlying logic is classical.2012-05-09
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    @ThomasAndrews, the motivation for my Q is whether *natural* language can be factored out of this problem. Can the lungfish be restated in terms of categories, objects, subobjects and part-of relations? (W3C semantic web technologies like RDF and OWL are designed to build knowledge bases based on relations that get ever closer to natural language but in fact are computable data - and underlying computation is Boolean logic correct?)2012-05-09
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    @ThomasAndrews again: in addition to part-of, other typical relations used in semantic web IT include is-a, instance_of, and many more (it's open-ended). But these are all binary-valued. There is no fuzzyness at this level.2012-05-09
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    @alancalvitti You've lost me, and comments aren't a good place to chat. Modus ponens applies to absolutely truths in logic. If your computer system represents individual facts in an absolute sense, you can apply modus ponens to those facts, as long as they mean what you think they mean. If you are not talking about absolute truths, but mere likely truths or fuzzy truths, then you cannot simply apply modus ponens to your set of facts.2012-05-09
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    @alancalvitti: Modus ponens is unequivocally valid in Heyting algebras, and in the intuitionistic logic that they algebraise. Intuitionistic logic is not fuzzy logic either. If you want to formalise the lungfish example, look up "conditional proof".2012-05-09
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    @ThomasAndrews I'll be happy to chat (I checked you weren't online) Your lunglish counterexample seems to be built out of absolute truths (ie, binary-valued logic) yet also seems to fail MP. I'm simply asking if you can help formalize it.2012-05-09
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    @ZhenLin, thanks. [Khami & Kirk](http://onlinelibrary.wiley.com/store/10.1002/9781118033074.fmatter/asset/fmatter.pdf;jsessionid=CEB096823C7C6801C1F6D54AE6AF5F1B.d03t04?v=1&t=h20nj5vj&s=7030aff5495422aa3952391458dd719c568ee79a) list 3 different types of intuitionistic logic, and none are fuzzy logic, so I believe you... I deferenced to wikipedia's entry for [conditional proof](http://en.wikipedia.org/wiki/Conditional_proof) but it doesn't seem to add much that isn't already inherent in my Q. Can you help formalize ThomasAndrew's lungfish counterexample?2012-05-09
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    @alancalvitti: It does not fail MP, so it is not a counterexample. _I_ can formalise it, but in order to appreciate it, _you_ have to learn some formal logic first.2012-05-09
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    @ZhenLin, since you're much more comfortable with the formal logic you please interpret Thomas' lungfish, in particular his distinction between (c) Therefore, if this thing has lungs, this thing is a lungfish and (c') "If something has lungs, it is a lungfish"2012-05-09
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    @alancalvitti: Thomas has just done exactly that. It should be clearer now what's happening.2012-05-09
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    @alancalvitti: I've restructured the answer somewhat to reflect the real problem in the lungfish example, namely, a confusion between Propositional Logic and First Order Logic.2012-05-09
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    ah, now I understand the question. thanks!2012-05-09
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    @ThomasAndrews, very nice, thanks for the clarification. The argument is clear now and also a testament to the power of *notation*, as Babbage pointed out some time ago.2012-05-09
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    @alancalvitti Have you seen "[A defense of modus ponens](http://sites.duke.edu/wsa/papers/files/2011/05/wsa-defenseofmodusponens1986.pdf)" by Sinnott-Armstrong et al.? It seems to concur with Thomas Andrews' analysis.2012-05-10
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    @MarkDominus, I hadn't - thanks for the link2012-05-13
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    This answer is the correct understanding from a mathematical view, but does not do justice to the philosophical issues in question. McGee is not, as this answer claims, "unaware of the fact that the truths used in logic are absolute" -- and the many philosophers who take his counterexample seriously are not idiots. The thing you're missing is that McGee is not giving a counterexample to the LOGICAL RULE of Modus Ponens; he's saying that Modus Ponens does not hold in the case of "natural language", or everyday English.2016-01-26
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    Since this is a mathematics site, and not philosophy, I'll take your comment as a compliment. @6005 If philosophers use lazy logical term usage, then they'll be pretty shoddy in their reasoning, too.2016-01-26
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    Specifically, if you actually read the question "Is McGee's counterexample to Modus Ponens accepted by the mathematical community?" The question is about the math of it, not the philosophical question or the limitations of natural language. @6005 That's for another question (on another site!)2016-01-26
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    Yes, the question is quite misguided, and your answer correctly points out why McGee's argument would not hold up in any sense to be a counterexample to the mathematical rule of modus ponens. I did think it was generally a good answer. However, there is another kind of "modus ponens", which many philosophers actually believe to be true, and that is the principle that if someone says "if P then Q" and you later find out that "P", you are allowed to believe that "Q". This is a counterexample to that principle.2016-01-26
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    A question is never misguided. @60052016-01-26
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    By "the question is quite misguided" I mean that the OP seems to think that McGee's philosophical modus ponens and mathematical modus ponens are the same, when they are two entirely different things.2016-01-26
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    There is an entire body of philosophical and linguistic work related to understanding conditionals--in English, "if P then Q" statements. If anything is certain in this field, it is that the mathematical conditional is not the conditional that people actually use in everyday use. An easy example is, you don't believe the statement "if it is raining, the street is dry" is true just because the street happens to be dry. So philosophers have developed several different models of the conditional, and in many of these models, modus ponens holds. McGee is (correctly) criticizing those models.2016-01-26
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    Still, not on topic for this site, and not related to the question at hand. Are you trying to bore me to death? @60052016-01-26
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    @ThomasAndrews Fine, I will not continue to bore you. However, I do request that you modify your offensive and inaccurate statement about McGee in this answer--surely, that is not on-topic for this site either.2016-01-26
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To put it briefly, McGee's "counterexample" is not accepted by the mathematical community because it is not, per se, a statement about mathematics. Modus ponens certainly holds in the context of logic, with its absolute interpretations of "true" and "false", and the references you give acknowledge that. But those authors (who are philosophers, not mathematicians) appear to be considering other possible notions of truth, different from those of logic, which they believe may better describe the way humans routinely think, and noting that modus ponens can fail to hold for those.

Some of those models make sense to describe mathematically, but mathematicians would not confuse those models with plain logical truth, and indeed would probably avoid using the words "true" and "false" to describe anything else.

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    Can natural language can be factored out of McGee's counterexample? Thomas's lungfish in particular seems to consist of binary-valued, "plain logical truth" statements amenable to [conditional proof](http://en.wikipedia.org/wiki/Conditional_proof) (thanks to Zhen Lin for the ref). Can you help formalize the lungfish using categories, objects, subobjects and part-of and is-a relations to see if the conditional proof holds?2012-05-09
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    @alancalvitti: The proof obviously holds, and the statement (c) is obviously true (given the assumptions (a) and (b)). What is the contradiction/paradox/problem here?2012-05-09
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    @ShreevatsaR: I'm confused about what the issue is, too. I think it's that they believe Modus Ponens implies that statement `c` is true in *all* cases, not just the cases where `b` is true. I get the impression that this is the result of philosophers trying to reason about mathematical logic, without first building up the underlying mathematical intuition that we all take for granted.2012-05-09
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    I respectfully disagree that philosophers are somehow deficient in logic. Skimming the link to Macfarlane's note, it seems more that they are discussing rules of inference in settings to which mathematical logic cannot always be applied consistently2012-05-09
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    @BlueRaja-DannyPflughoeft: My opinion is the same as that of Yemon. These philosophers are *not* trying to reason about mathematical logic. They are reasoning about some other system which looks superficially similar to mathematical logic and uses similar terminology, but is actually very different. (Disclaimer: I know little of philosophy and do not claim to actually understand what they are doing, but this is my understanding so far.)2012-05-09
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Mc Gee's counterexample points one problematic application of classical propositionnal logic to natural language. In general mathematicians are not interested with it because they are happy with classical logic and consider that it gives a correct model of their way to reason. As far as I know, it is not possible to produce the same counterexample applied to mathematical objects. Perhaps it is due to the necessary relations between objects in mathematical sentences.

For those who do not see any interest in this kind of counterexample, I just want to point that the problematic application of classical logic to reasoning in natural language does not interest only philosophers but also computer scientists and many other researchers. This question concerns non-classical logics, a field perhaps not useful for mathematics but important to other areas as Artificial Intelligence.

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It looks to me like McGee is making the following error.

Suppose, for a moment, that $A \to (B \to C)$ is a tautology.

Because of this, we know

$$ A \vdash B \to C$$

which means, among other things, that

$$\mathcal{M} \models A \quad \text{implies} \quad \mathcal{M} \models B \to C$$

However, McGee seems to have fallen into a trap of some sort, and is concluding

$$ \vdash B \to C$$

which is incorrect.

(the other answer does say this too, but in a more verbose language)