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

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."

• If you do not use "probability" operators, what is "likely" ? The possibility modality ? – Mauro ALLEGRANZA Feb 17 '20 at 13:08
• @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 Feb 17 '20 at 13:09
• But I just did? There isn't. Isn't yes/no supposed to be accompanied by an explanation why, a.k.a. "fumbling around"? – Conifold Feb 19 '20 at 11:54
• "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 Feb 19 '20 at 16:00
• My 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 Feb 20 '20 at 9:28

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 Feb 26 '20 at 18:12
• 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 Feb 26 '20 at 21:36
• 1. 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 Feb 27 '20 at 8:32

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.

• 1. 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 Feb 27 '20 at 8:54
• 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 Feb 27 '20 at 14:46
• (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 Feb 27 '20 at 15:02
• 1. 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 Feb 27 '20 at 16:36
• 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 Feb 27 '20 at 16:45