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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.

The Amplitwist
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alancalvitti
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    I honestly don't see the problem with any of these statements. What is the contradiction/paradox presented here? – Thomas Andrews May 09 '12 at 14:56
  • 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! :) – Giovanni De Gaetano May 09 '12 at 15:00
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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. – mdp May 09 '12 at 15:01
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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). – Thomas Andrews May 09 '12 at 15:02
  • 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... – mboratko May 09 '12 at 15:05
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    @GiovanniDeGaetano: no need to be stingy with votes! – The Chaz 2.0 May 09 '12 at 15:34
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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! – Giovanni De Gaetano May 09 '12 at 15:43
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    I am thoroughly baffled by how on earth that is supposed to be even mildly convincing 'counterexample'. – Tara B May 09 '12 at 16:01
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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! – The Chaz 2.0 May 09 '12 at 16:01
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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. – alancalvitti May 09 '12 at 16:12
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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. – TonyK May 09 '12 at 20:02
  • I remember Anderson. Is everyone else a young whipper-snapper? – GEdgar May 09 '12 at 20:07
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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.) – TonyK May 09 '12 at 20:11
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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. – Asaf Karagila May 09 '12 at 20:48
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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. –  May 09 '12 at 22:09
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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. – Asaf Karagila May 09 '12 at 22:16
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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. –  May 09 '12 at 22:23
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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) –  May 09 '12 at 22:24
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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" –  May 09 '12 at 22:28
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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. :) – Asaf Karagila May 09 '12 at 22:30
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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. –  May 09 '12 at 22:31
  • @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. – Asaf Karagila May 09 '12 at 22:32
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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.]” – MJD May 10 '12 at 15:16
  • @MarkDominus, I wasn't aware of the context. – alancalvitti May 10 '12 at 17:37
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    I read "On Some Counterexamples to MP and MT" 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. – MJD May 10 '12 at 22:18
  • @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. – alancalvitti May 11 '12 at 14:36
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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. – MJD May 11 '12 at 15:39
  • @MarkDominus, thanks for the ref to Priest's book, will check it out if my library has it. – alancalvitti May 13 '12 at 04:24
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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. – Caleb Stanford Jan 26 '16 at 01:12
  • For anyone who might still care, a decade after the original question was posted: all three of those examples--the election, the lungfish, and Uncle Otto--are from McGee's 1985 article. MacFarlane is just summarizing McGee. Doesn't make a huge difference to the discussion, I guess, but it's kind of an indicator of how disconnected the discussion was from what McGee actually was up to. – StumpyLeg Nov 14 '22 at 07:12

7 Answers7

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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.

Thomas Andrews
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  • Some form of Modus Ponens should still hold even in probabilistic logic, right? – Neal May 09 '12 at 15:31
  • 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. @Neal – Thomas Andrews May 09 '12 at 15:34
  • 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 $P,Q,R$. – Neal May 09 '12 at 15:39
  • @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? – alancalvitti May 09 '12 at 16:21
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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. @alancalvitti – Thomas Andrews May 09 '12 at 16:24
  • @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. – alancalvitti May 09 '12 at 16:25
  • @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?) – alancalvitti May 09 '12 at 16:32
  • @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. – alancalvitti May 09 '12 at 16:41
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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. – Thomas Andrews May 09 '12 at 16:46
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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". – Zhen Lin May 09 '12 at 17:10
  • @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. – alancalvitti May 09 '12 at 17:10
  • @ZhenLin, thanks. Khami & Kirk 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 but it doesn't seem to add much that isn't already inherent in my Q. Can you help formalize ThomasAndrew's lungfish counterexample? – alancalvitti May 09 '12 at 17:19
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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. – Zhen Lin May 09 '12 at 17:28
  • @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" – alancalvitti May 09 '12 at 17:33
  • @alancalvitti: Thomas has just done exactly that. It should be clearer now what's happening. – Zhen Lin May 09 '12 at 20:09
  • @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. – Thomas Andrews May 09 '12 at 20:13
  • ah, now I understand the question. thanks! – Ronald May 09 '12 at 21:45
  • @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. – alancalvitti May 09 '12 at 23:28
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    @alancalvitti Have you seen "A defense of modus ponens" by Sinnott-Armstrong et al.? It seems to concur with Thomas Andrews' analysis. – MJD May 10 '12 at 15:24
  • @MarkDominus, I hadn't - thanks for the link – alancalvitti May 13 '12 at 04:23
  • 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. – Caleb Stanford Jan 26 '16 at 00:23
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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. – Thomas Andrews Jan 26 '16 at 00:25
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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!) – Thomas Andrews Jan 26 '16 at 00:27
  • 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. – Caleb Stanford Jan 26 '16 at 00:29
  • A question is never misguided. @6005 – Thomas Andrews Jan 26 '16 at 00:32
  • 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. – Caleb Stanford Jan 26 '16 at 00:35
  • 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. – Caleb Stanford Jan 26 '16 at 00:35
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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? @6005 – Thomas Andrews Jan 26 '16 at 00:38
  • @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. – Caleb Stanford Jan 26 '16 at 00:39
  • I was randomly skimming the comments, noticed Caleb’s “if it is raining” comment, was compelled to respond along the lines of types of “implies” and implicitly-quantified “implies”, then saw that this answer has already said pretty much the same. $\quad$ To save readers’ time, this Question benefits from an explicit ‘NB’ edit indicating that it may just be a strawman. – ryang Mar 17 '23 at 11:07
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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.

Nate Eldredge
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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 (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? – alancalvitti May 09 '12 at 17:26
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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? – ShreevatsaR May 09 '12 at 17:28
  • @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. – BlueRaja - Danny Pflughoeft May 09 '12 at 19:59
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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 consistently –  May 09 '12 at 20:21
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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.) – Nate Eldredge May 09 '12 at 21:19
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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.

Mathieu Vidal
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McGee isn't very explicit in his article, but I do not believe he was proposing counterexamples to Modus Ponens as a rule of inference (contra the title of his article). If that were what we was proposing, he'd have to show that (c) is false when both (a) and (b) are true. But nowhere in his article is that shown. Rather, I think his article shows that knowledge/justification is not closed under modus ponens. This means that we can be justified in believing (a) and (b) and not be justfied in believing (c). This is certainly an odd feature of Modus Ponens and worth discussing, but it doesn't show it to be formally invalid. In fact, he so much as says this in his article but he uses 'good grounds for believing' instead of 'justifies'. Again, even if Modus Ponens does not preserve knowledge or justification, it does not follow that it does not preserve truth (i.e. is invalid). There is distinction between what is true and what we are justified in believing.

To answer the question explicity, having studied both math and philosophy, I don't know any academics in either field that reject the validity of Modus Ponens as a rule of inference. But it is a point of contention as to whether or not knowledge/justification is closed under implication, and McGee seems to provide reasons for thinking that it is not.

Aside: to the interpretations of McGee that imply philosophers do not know how to use logic, this is definitely mistaken. Logic is as much a branch of philosophy as it is of mathematics.

SIR
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0

The comments here seem to be a bit confused about what McGee was arguing. He's arguing that modus ponens is invalid for the ordinary language conditional (if, then statement). I mean, it's obviously valid for the material conditional (horseshoe statement) in logic and math: that's true by stipulation. (We've stipulated that horseshoe statements have the truth table they have, and modus ponens is obviously truth-preserving for horseshoe statements given those stipulations.)

I think that it's hard to deny that McGee's examples show that modus ponens isn't a universally good "rule of thought" for ordinary language conditionals: that there are cases where we shouldn't reason with ordinary language conditionals using modus ponens. His Republican argument is an instance of ordinary language modus ponens, and it seems very clear that while it was rational to believe the premises, it would have been irrational to accept the conclusion. Hence, ordinary language indicative conditionals are not material conditionals, as is commonly thought, since while modus ponens is a good rule of thought for material conditionals, it is not a good rule of thought for indicative conditionals.

It doesn't immediately follow that modus ponens is invalid (in the sense of being non-truth-preserving), but that does follow given some fairly plausible bridge principles about rationality and truth. Maybe we should give up on those principles rather than the validity of (ordinary language) modus ponens, but that's still an interesting result.

John Keller
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    "I think that it's hard to deny that McGee's examples show that modus ponens isn't a universally good "rule of thought" for ordinary language conditionals" I personally completely disagree with this (and I think so do some of the commenters above); I just don't see how any of the conclusions are particularly counterintuitive. – Noah Schweber Feb 04 '20 at 19:48
  • @John Keller, what does 'true by stipulation' mean? – alancalvitti Feb 04 '20 at 20:13
0

Perhaps an ambiguity is occurring? Here is the original outline of the argument:

Attempted Counterexample

It seems that there is little reason to believe $(3)$, because you might as well respond with "But what about Carter?". But when reading the consequent of $(1)$, we wouldn't naturally respond that way, because the antecedent ("a Republican wins the election") restricts the scope of possibilities to either Reagan or Anderson.

In other words, there seems to be an ambiguity (for all that I can tell): when we say the word it in "If it's not Reagan who wins it will be Anderson", I think we should be careful about whether we are referring to a possible winning candidate out of all Republican candidates vs out of all available candidates. To motivate this idea, think of the following example:

$(1')$ If the analogue clock reads $5$ o-clock and is correct, then if it is not $5$ AM it is $5$ PM.

$(2')$ The analogue clock reads $5$ o-clock and is correct.

So, $(3')$ If it is not $5$ AM, it is $5$ PM.

You might disagree a priori with $(3')$, because if it is not $5$ AM, then, for all you know, it could be any other time $-$ and there are more possible alternative times than just $5$ PM. Put differently, just because it is not $5$ AM does not guarantee that it is $5$ PM. Here, we are reading $(3')$ under the scope of all times on the clock: $1$ AM, $2$ AM, $\ldots$, $1$ PM, $2$ PM, $\ldots$.

But we would not read the consequent of $(1')$ the same way, because therein by "it" we do not refer to a possible alternative among all times on the clock, but rather merely among all times given a correct $5$ o-clock reading $-$ in which case there are only two alternatives available (compared to twenty-four alternatives).

Putting all that together, perhaps $(3')$ does not express the same proposition as that expressed by the consequent of $(1')$ (despite being the same sentence in symbols). Likewise, perhaps the same goes for $(3)$ and the consequent of $(1)$. On this view, then, the "problem" appears to dissolve into a semantic ambiguity with respect to scope. This is probably a reason why we should be careful as to how we decide to formalise natural-language sentences.

Mr Pie
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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)

Community
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