## Where is the fallacy in Seth Yalcin's counterexample to the modus tollens?

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Where is the fallacy, do you think, in Seth Yalcin’s argument (2012) that the Modus Tollens is not a generally valid form of argument?

Seth Yalcin’s counterexample to the Modus Tollens (MT) https://link.springer.com/article/10.1007/s10992-012-9228-4

An urn contains 100 marbles, a mix of blue and red, big and small:

Big & Blue 10

Small & Blue 50

Big & Red 30

small & Red 10

A marble is then drawn at random.

Seth Yalcin's counterexample:

(P1) If the marble is big, then it’s likely red.

(P2) The marble is not likely red.

(C1) The marble is not big.

Seth Yalcin observes that the conclusion does not follow, but that it should follow if the MT was generally valid, and so the MT is not generally valid.

Schematically, the argument is of the following form:

φ → probably ψ

¬ probably ψ

∴ ¬ φ

where φ and ψ are themselves assumed to be free of probability operator.

Seth Yalcin asserts about the schematic form:

"This argument form is invalid. Since it is just a special case of MT, it is a counterexample to the claim that MT is a generally valid pattern."

4If you do not use "probability" operators, what is "likely" ? The possibility modality ? – Mauro ALLEGRANZA – 2020-02-17T13:08:09.110

1@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 – 2020-02-17T13:09:01.937

We 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 – 2020-02-17T13:50:44.290

It 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 ponens fails. 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 – 2020-02-19T10:52:57.627

1But 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 conditionals and probability operators in natural language." – Mauro ALLEGRANZA – 2020-02-20T09:25:57.923

3My 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 – 2020-02-20T09:28:06.643

Not 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

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The use of modus tollens is valid only when used with propositions containing valid logical predicates. And here it is not.

A logical predicate is commonly understood as a boolean function P: X → {true, false} (source).

In other words, "predicate" any kind of a mechanism that, when given an object X, provides you with a yes/no answer to the question "Is this object P?" or "does it possess the quality P?" and does so in a consistent manner i.e. it has to give the same answer every time when presented with the same object.

Therefore, likely red is not a valid predicate. If I show you a marble, can you tell me if it possesses the quality of being likely red? Obviously not, as likely red is not a quality of the marble itself, but depends in the situation where you picked it. On the other hand, you would always be able to tell me if it a given marble is red or not. And that is why red is a valid predicate and likely red is not a valid predicate and thus constitutes an incorrect use of modus tollens.

Another formulation of the same idea is the law of non-contradiction stating that "nothing can both be and not be." To illustrate how the law is broken, imagine that I take out most red marbles from the urm (whatever urm is) and only leave a few of them there - suddenly (and without undergoing any kind of change) those marbles that before a minute were likely red will no longer be likely red.

If you want to make the statement correct, the first thing you have to do is to move the likely at the beginning of the statement (since, as we said, the word "likely" it is clearly meant to be a characteristic of the redness of the marble in question, but rather a characteristic of the whole statement):

(P1) It is likely that if the marble is big, then it’s red.

There are actually a logic that is made to express statements like that - Modal logic. The symbol "◇" is used in modal logic to mean "possibly").

No. 1. A proposition like "the marble likely red" is either true or false. No problem with it in principle. We don't have to verify propositions. We can even reason about absurd propositions that we nonetheless accept as either true or false, for example "All Martians are members of the US senate". - 2. As to the LNC, "is likely red" is not the predicate of any marble in the urn here. - 3. Modal logic is irrelevant here. 4. Your rephrasing of P1 is unnecessary and not appropriate. It is not the whole conditional which is likely. It is its consequent clause. – Speakpigeon – 2020-02-26T18:12:42.050

1 Edited my answer - you may be right about the proposition being valid, but that does not mean automatically that modus tollens can be applied - modus tollens cannot be applied to all propositions. 2 Did not get what you mean here 3. My answer is above the separator, the rest contains some additional remarks, which are not essential for it. 4. Moving the "likely" does not change the meaning of the sentence in any way. If you don't agree, give me an example of a situation where the original sentence applies while the edited version does not. – Boris Marinov – 2020-02-26T21:36:35.593

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• Please, don't make up stuff. I didn't claim the argument valid. I asked why it is fallacious. - 1b. Your explanation for why the MT wouldn't apply in this case is fallacious, and I said why in my point No. 2. - 2. It means what it says. 3. I already replied to your answer - 4. If rephrasing doesn't change the meaning, why do you rephrased? I explained why your rephrasing is wrong.
• < – Speakpigeon – 2020-02-27T08:32:07.547

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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.

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• You seem to understand where the fallacy is. However, your answer is so protracted and confused, I am not going to accept it, not as it is. The fallacy can be identified in just one word, and it can be explained in 15 lines. 2. Material implication is irrelevant here. The question is about a logical argument, couched in plain English, you need to assume only as much. 3. Still, assuming I understand your charabia, congratulation. 4. You really need to train to express yourself clearly. Ce que l'on conçoit bien s'énonce clairement.
• < – Speakpigeon – 2020-02-27T08:54:37.917

I don't think a purely abstract description of the fallacy is convincing, and that to show how the argument rests on verbal vagueness, you need to actually come up with some specific precise definitions of what could be meant by the "likely" predicate (ones involving very clearly defined probability calculations, whether conditional or unconditional) and then methodically show that in all cases (whether the marble in question is big or not) modus tollens either leads to correct conclusions, or one of the first two premises of Yalcin's argument is false. – Hypnosifl – 2020-02-27T14:46:43.437

(cont) Do you think my answer is overly complicated/unclear even granted that this strategy of argument may be a good one? Or is your objection just that this strategy is overly complicated in itself, no matter how clearly one tries to execute it? In the former case I'd be happy to try to edit my answer, maybe starting with more of an overview of my strategy before diving into the nitty gritty details, if you think it might help. But if your objection is more the latter, I'm not convinced that a less specific type of argument would convince people who found Yalcin's initial argument plausible. – Hypnosifl – 2020-02-27T15:02:09.577

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• I didn't suggest an "abstract" description. - 2. I suggested clarity. What is convincing is what is clearly described. - 3. We don't need to define "likely", dictionaries are not for nothing. Further, the presentation of the argument includes the contents of the urn and this is all we need to decide whether "likely" is true or not. - 4. We don't need to show that the MT leads to correct conclusions. We know it does. The question is as to which is the fallacy of Seth Yalcin's counterexample.
• < – Speakpigeon – 2020-02-27T16:36:53.970

Further, the presentation of the argument includes the contents of the urn and this is all we need to decide whether "likely" is true or not. The problem is that you can use "likely" in a sense that is correct in terms of probability theory, but inconsistent bt. statement P1 (where you are calculating a conditional probability) and statement P2 (where you are calculating an unconditional one). So I was trying to show that if you give a precise definition of what probability calculation the "likely red" predicate is actually referring to and make sure it's consistent, the paradox goes away. – Hypnosifl – 2020-02-27T16:45:49.127

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• Your answer is needlessly complicated. It is 675 words long. It should be 115 words or less. - 2. The fallacy is identified by just one word. An explanation as to why this is this fallacy can be 115 words long. - 3. I'm not asking you to edit your answer. Whenever the penny drops, you'll do it yourself.
• < – Speakpigeon – 2020-02-27T16:47:08.343

Your last comment proves you can do it. However, you seem to be ignorant of the conventional vocabulary used to describe fallacies. Is that the case? – Speakpigeon – 2020-02-27T16:50:28.067

I can't think of a specific fallacy this would fall under, other than a sort of equivocation fallacy. The problem is that it's neither a formal logical fallacy nor a fallacy of calculating probability, it's more a semantic inconsistency or lack of clear definition in what implicitly means by "likely red" in P1 and P2. Do you think a person educated in logic & probability who read the whole paper and found it convincing (or Seth Yalcin himself) would be likely to have a lightbulb moment that changed their mind if someone just named a fallacy and gave 115 words of explanation? – Hypnosifl – 2020-02-27T17:09:39.657

In any case, the stack exchange rules do let people answer their own questions, so if no one else posts the answer you're thinking of reasonably soon, I'd recommend giving your answer, people can discuss in the comments if they want. – Hypnosifl – 2020-02-27T17:11:01.157

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As e.g. Una Stojnic argues, a natural-language sentence like

If the marble is big, then it’s likely red.

is actually an anaphora for

It’s likely that if the marble is big, then it is red.

The latter is basically a conditional probability: Pr(red|big) > some "likely" threshold.

As Sven Neth argues these kinds of "chancy" paradoxes generally rely on the fact that (knowing) a conditional probability (really) implies nothing about the unconditional probability.

E.g. in this 2012 Yalcin/example problem, Pr(red) (and Pr(big)) can be vanishingly small if there are very few red balls aside from the big subset, which is actually the case here as there are many more small & blue balls, rendering [the unconditional] P(red) small.

The actual premises in proper form are:

P1: Pr(red|big) > likely-threshold
P2: Pr(red) < likely-threshold


In proper probabilistic reasoning, you can only infer about probabilities, so the conclusion has to translated too in that form

C1: Pr(big) = 0


For the sake of making this a simpler calculation, using equalities rather than inequalities for the probabilities as given the problem setup, those numbers for the premises are

Pr(red|big) = 0.75
Pr(red) = 0.4


These allow you to infer (using the conditional probability formula) that Pr(red⋂big) = 0.3 (i.e. red and big) but you cannot infer anything whatsoever about Pr(big) from those two premises alone. (Which is basically Neth's point about these kinds of paradoxes, generally.)

In general, "probabilistic logic" is a "work in progress", meaning various formalism have proposed, but none (of the modern ones) are as "dumb" as what Yalcin suggests in that example, as far as I know.

To give you a basic insight here as to why this is difficult, a basic ‘probable’ operator (which I'm gonna call is-likely) was e.g. suggested by Hamblin with the meaning of exceeding some set probability value (e.g. 0.5). Alas doing much inference that "logic" way (instead of calculating probabilities) doesn't work too well because that operator is not a normal modal operator, meaning that

P1: x is-likely
P2: y is-likely


does not imply that

C:  (x and y) is-likely.