It seems like this is just a case of semantic ambiguity in English--in the first statement Seth Yalcin seems to have implicitly thought of "if ... then" as expressing a *conditional probability*, i.e. the claim that a randomly chosen marble is "likely" red (where likely can be defined in terms of any desired probability threshold, say >50%) *given* that we already know it was observed to be big. Whereas when an "if ... then" construction is used in the verbal description of modus tollens, it's supposed to refer only to material implication.

Suppose instead we try to interpret the "if ... then" only as material implication, i.e. for some marble m we are asserting that "big(m) -> likelyred(m)", where the "big" predicate refers to what's found after checking its size, and the "likelyred" predicate refers to the fact that a rational observer would assign a >50% *unconditional* probability to the event that the marble will be found to be red, prior to actually observing any of its actual features including its size. Here the problem arises that for any marble m that happens to be big, big(m) would be true, but likelyred(m) would be false since the unconditional probability that a marble is red is 40/100. And according to the truth table for material implication, P -> Q is false when statement P is true but statement Q is false. So if we assume the "if ... then" in P1) is supposed to refer to material implication, *and* we use the above translation of the "likelyred" predicate in terms of unconditional probabilities, then P1) would simply be false for any marble m that happens to be big. The fact that you can then use modus tollens to get a false conclusion is hardly an argument against modus tollens if you're starting from a false premise.

On the other hand, suppose we stick with the above translation of "likelyred", but the marble m we have chosen *not* actually big. In that case "big(m) -> likelyred(m)" would be true, since the truth table for material implication says that P -> Q is true when statements P and Q are both individually false. However, in that case it is in fact guaranteed to be true that P2) "likelyred(m) is false" and P3) "big(m) is false", so in this case modus tollens would lead you from true premises to a true conclusion.

If we wanted to capture some idea of conditional probability, we could invent a new predicate "conditionallylikelyredgivenbig" that could be conceptually described as "the marble is big, and upon learning that information, a rational observer who had not yet observed its color would assign a >50% conditional probability to the event of it being found to be red". In that case, if we have a marble m for which big(m) is true, then conditionallylikelyredgivenbig(m) is also true. On the other hand, if we have a marble m for which big(m) is false, then conditionallylikelyredgivenbig(m) is also false. These are the only two combinations that can happen for any of the marbles, and since the truth table for material implication says that P -> Q is true if both P and Q are true *and* if both P and Q are false, P1) big(m) -> conditionallylikelyredgivenbig(m) would be true for any choice of m.

But if we use this translation scheme, then P2) should be translated as "conditionallylikelyredgivenbig(m) is false", and since conditionallylikelyredgivenbig(m) was defined above to mean that the marble is big, conditionallylikelyredgivenbig(m) is false whenever the marble is *not* big, i.e. "conditionallylikelyredgivenbig(m) is false" is *true* when the marble is not big. And in that case, then with P3) translated as "big(m) is false", P3 is guaranteed to be true as well, so modus tollens operating on two true premises has given us a true conclusion. On the other hand, if the marble *is* big, that means P2) is false, and again it's no strike against modus tollens if one of your two starting conclusions is false and you use modus tollens to get a false conclusion.

4If you do not use "probability" operators, what is "likely" ? The

possibilitymodality ? – Mauro ALLEGRANZA – 2020-02-17T13:08:09.1101@MauroALLEGRANZA But we have a proposition "

The marble is likely red", which is either true or false, and we have its negation, "The marble is not likely red". Seems good to me so far. -- We do have a probability operator, "likely", or "probable", but not embedded into φ and ψ. – Speakpigeon – 2020-02-17T13:09:01.937We may read "likely ψ" as implying "not always not-ψ". This is compatible with a case where we have φ and "one ψ". But also "not-likely ψ" is compatible with "one ψ" and thus we have that "φ and one ψ" is compatible with both premises. – Mauro ALLEGRANZA – 2020-02-17T13:27:24.390

@MauroALLEGRANZA But in the first premise, "

φ → probably ψ", the consequent is "probably ψ", and in the second premise "¬ probably ψ", we have exactly the same wording, "probably ψ". So it is irrelevant whether "probably ψ" implies or not "not always not-ψ". – Speakpigeon – 2020-02-17T13:50:44.290It seems that you are treating "likely ψ" as a single entity: a sentence. If we "hide" the probabilistic aspect that way, the conclusion holds: if we pick at random a marble, 60% case is Small, i.e. not-Big. – Mauro ALLEGRANZA – 2020-02-17T15:48:03.383

The argument seems a "variant" of the well-known article Vann McGee, A Counterexample to Modus Ponens (1985) (also discussed in this site).

– Mauro ALLEGRANZA – 2020-02-17T16:02:44.133@MauroALLEGRANZA I don't see how it would be a variant of McGee's argument. So, where is the fallacy? – Speakpigeon – 2020-02-17T17:20:04.633

If we want to get rid of the probabilistic modality we have to treat "likely red" as a single predicate. As such, it is vague and subject to the same issue as the "is a heap" predicate of the sorites paradox, where even

modus ponensfails. Yalcin is explicit that the issue with MT arises only in contexts with modalized indicative conditionals. In other words, not only is modality essential, but also that the conditionals are not the material conditionals of classical logic. – Conifold – 2020-02-17T21:00:06.383@Conifold Sorry, I don't see where you are saying that there is or not a fallacy. – Speakpigeon – 2020-02-18T20:57:35.343

A fallacy is a mistake in argument, there is no mistake here. What makes it "paradoxical" is that a commonly used (elsewhere) rule (MT) does not apply. Same as MP in sorites. – Conifold – 2020-02-18T21:10:49.353

@Conifold This is the question asked:

Where is the fallacy?If you cannot answer that, please don't fudge. If you think there is no fallacy, just state so without fumbling around. – Speakpigeon – 2020-02-19T10:52:57.6271But I just did? There isn't. Isn't yes/no supposed to be accompanied by an explanation why, a.k.a. "fumbling around"? – Conifold – 2020-02-19T11:54:40.810

@Conifold There was no yes/no in your reply that could have been accompanied. And this last comment here still isn't a proper answer in this respect. Never mind. – Speakpigeon – 2020-02-19T14:07:03.863

3"There isn't" a fallacy seems like a "proper" no to me. Do you have a substantive objection to why or is this just a roundabout way of conveying general displeasure? – Conifold – 2020-02-19T16:00:39.293

McGee's paper is included into the ref of Yalcin's paper whose first statement is: "This paper defends a counterexample to Modus Tollens, and uses it to draw some conclusions about the logic and semantics of

indicative conditionalsand probability operators in natural language." – Mauro ALLEGRANZA – 2020-02-20T09:25:57.9233My personal point of view (see my very first comemnt above) is (see para

2 Objections and Replies): "The first reply is that I have misrepresented the logical form of (P1). The probability operator in the sentence is really taking scope over, not under, the conditional operator; and as a result the pattern is a non-instance of MT." – Mauro ALLEGRANZA – 2020-02-20T09:28:06.643Not sure if this worth an answer here given that it's somewhat obvious, but these kinds of "chancy logic" [dis]proofs actually rely on the fact that a conditional probability being high (really) entails nothing about the unconditional probability; see https://philarchive.org/archive/NETCMP In this 2012 example P1 is a bit even more obvious that it's a conditional probability.

– Fizz – 2021-02-25T13:42:56.573@Fizz Sorry, I don't understand how that applies here. – Speakpigeon – 2021-02-25T18:04:19.157