# 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."

Conditional probability

Conditional probability is the probability of one event occurring if another event has already occurred. The conditional probability of event B if the event A has occurred is the conditional probability of "B given A", or the probability of "B under the condition A", usually written as P(B|A).

Here is a slightly more technical definition, but the idea is obviously the same:

The conditional probability of an event relative to another event is a characteristic connecting the two events. If A and B are events and P(B)>0, then the conditional probability P(A∣B) of the event A relative to (or under the condition, or with respect to) B is defined by the equation P(A∣B) = P(A∩B)P(B). The conditional probability P(A∣B) can be regarded as the probability that the event A is realized under the condition that B has taken place. For independent events A and B the conditional probability P(A∣B) coincides with the unconditional probability P(A). -- https://encyclopediaofmath.org/wiki/Conditional_probability

In the example above, the probability to draw a blue marble is 0.6, the conditional probability to draw a marble that is blue if it is big is 0.1, and the conditional probability to draw a marble that is blue if it is small is 0.5.

• If you do not use "probability" operators, what is "likely" ? The possibility modality ? Feb 17, 2020 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. Feb 17, 2020 at 13:09
• 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. Feb 17, 2020 at 15:48
• "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? Feb 19, 2020 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." Feb 20, 2020 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. Feb 26, 2020 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. Feb 26, 2020 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. Feb 27, 2020 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. Feb 27, 2020 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. Feb 27, 2020 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. Feb 27, 2020 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. Feb 27, 2020 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. Feb 27, 2020 at 16:45

The real issue is this: "If the marble is big, then it's likely red" is a false statement. The corresponding true statement is this: "If a rational agent knows the marble is big, then the agent considers it likely that the marble is red."

Probability is fundamentally a question of an individual's lack of knowledge about some proposition. Just because the marble is big does not mean that the individual knows that the marble is big. If the individual does not know the marble is big, then the individual has no reason to suppose that the marble is likely red, even if the marble does happen to be big!

To make this clearer, "If the marble is big, then the marble is likely red" may be translated like this:

A = the information available to the agent

B = the marble is big

R = the marble is red

The sentence is symbolized, "If B, then P(R|A)>0.5." As mentioned, this implication is a false statement. Seth Yalcin's "paradox" is a proof that the implication is a false statement.

As mentioned, the corresponding true statement is, "if the agent knows the marble is big, then the agent considers it likely that the marble is red." We might symbolize it as:

"If P(B|A) = 1, then P(R|A) > 0.5"

And Modus Tollens works perfectly well here. If we consider an agent that has not drawn or looked at any marble, P(R|A) < 0.5, so the antecedent P(B|A) = 1 must be false, as in fact it is. No paradox.

As an aside, the claim presented in some other answers, that:

"If the marble is big, then it’s likely red" is an anaphora for "It's likely that if the marble is big, it's red"

is wrong. There is a difference between the two sentences.

• The first one is an implication whose consequent is the proposition "it's likely that the marble is red" and whose antecedent is the proposition "the marble is big."
• The second one is claim that a certain implication X->Y is likely, where X->Y is "if the marble is big, then it's red."

These two sentences have different constructions and different meanings.

If you want a one word fallacy, I would say it is the modal fallacy, sometimes also called the modal scope fallacy. Usually, the modal fallacy takes the form of saying, "if A then necessarily B" when what is really meant is "necessarily, if A then B". In other words, it is the fallacy of misplacing the scope of the modal operator. In this instance, the relevant modal operator is 'likely'. Yalcin has written, "if the marble is big, then it is likely red" when it should correctly be written as, "it is likely that if the marble is (observed to be) big then it is red". Once we tidy that up, there is no difficulty saying that it is likely that if a marble is drawn at random from the urn and observed to be big then it is red, and also that it is unlikely that a randomly drawn marble is red, from which it follows that a randomly drawn marble is unlikely to be big (but not that it is definitely not big).

• Thank you for an answer which is not rambling. It is also quite interesting that your answer seems to solve the problem quite nicely. However, I disagree on the substance of it. It seems to solve the problem but doesn't. For one, I'm sure you understand that the modal scope fallacy is not meant to apply to conditional probability. In other words, you are reasoning by analogy, a false analogy in this case, and so your reasoning commit the fallacy of false analogy. Also, the probability adverb is properly located with the main clause, even if we often put it in front of the conditional. Mar 1, 2021 at 10:31

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.

Or if you want me to quote Stojnic's exact words on this:

the anaphoricity of modals is captured by requiring that the restriction on the domain of quantification be retrieved in a way similar to how the antecedent of a pronoun is—either provided by the context, or explicitly by the prior discourse. The way anaphora is resolved, in both cases, is determined by discourse structuring mechanisms, in particular, mechanisms of discourse coherence. [...]

[As to Yalcin's 2012 example,] the modal ‘likely’ in the consequent, which is searching for the most prominent epistemically accessible possibility, selects this possibility as the restrictor for its domain of quantification. The consequent is thus understood as further describing the possibility introduced in the antecedent, providing the intuitively correct restricted reading—the marble is likely red, given that it is big.

And this analysis is actually repeated/reasserted in Stojnic's paper [I've changed the premise/clause indices to refer to them as given in the OP's question in the quote below]:

A familiar view is that modals are quantifiers over possible worlds, but just which worlds depends on the context (Kratzer, 1977, 1981). We can exploit this to argue that the problematic counterexample can be explained away by maintaining that the modal ‘likely’ in (P1) contributes a different semantic content than the one in (P2), due to contextual effects on the interpretation of the two occurrences of the modal; and so, (P2) and the consequent of (1) fail to contradict each other. Accordingly, (P1,P2,C1) is not really an instance of MT.

This strategy captures the intuition that the consequent of (P1) talks about a restricted (conditional) probability, while (P2) talks about an unrestricted one. The challenge is to explain exactly why and how the context secures different (and intuitively correct) interpretations for the two occurrences of the modal. To do so in a non-ad hoc way is notoriously difficult.

Properly interpreted, the natural language construct P1 is basically a statement about 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.
``````

Since the fact that there's some anaphora in Yalcin's natural-language P1 ("If the marble is big, then it’s likely red") was challenged in the comments, to say a bit more on that: if look more carefully at the [formal] MT Yalcin sets up here, he makes it a simple propositional matter, but that alone is a bad formalization because the words "marble" and "it" refer to the same thing in the natural language expression, but there's no referent in common between "φ" and "probably ψ" as Yalcin formalized that natural language statement as (P1) "φ → probably ψ".

To push the analogy (without resorting to probabilities) of why that lack of any syntactic commonality between φ and ψ is a bad formalization (for this), you might as well write a simpler rhetorical style argument that somewhat captures that idea:

``````P1: If Biden is the president of the USA, then I'm Biden.
P2: I'm not Biden.
C1: [Therefore,] Biden is not the president of the USA.
``````

(As yourself why P1 might "sounds alright" at some pub, but it's actually not properly formalized merely as some "φ → ψ" where φ and ψ are propositions, in the formal sense of that word.)

And to include the obvious here, if you change P1 to something lacking an overt implication symbol, MT is not directly applicable anymore. You need the whole FOL + ZFC + probability space axioms (which also needs some construction of real numbers) to syntactically reason about equations with probabilities symbols like "Pr". Simpler "alternative" systems (like Hamblin's) turned out not to be very useful in terms of what you can prove in them, relative to "reality" (which really is just a word here for the pre-theoretic, frequentist intuition we might have about a problem setup like that in the first quote you gave.)

• 1. Read again the Wikipedia article on anaphora, because the notion does not apply here. 2. The whole point of conditional probability is precisely that it applies to the main clause of the conditional and not to the conditional itself, contrary to what you say here. 3. It is correct that we cannot infer the probability P(Big) from the premises as they are, but this still does not say where the fallacy.is in Seth Yalcin claiming that this is a counterexample to the modus tollens. Feb 28, 2021 at 18:14
• @Speakpigeon: Stojnic uses the term anaphora/anaphoric in a slightly broader sense if you bother to read her paper (from p.12 onwards to save you some effort). And there's no MT left (as a form) if you replace P1 with a conditional probability as you don't have any arrow/implication in P1 anymore, so that obviously solves the "paradox".
– Fizz
Feb 28, 2021 at 18:17
• @Speakpigeon While I concur that the answer does not explicitly name a fallacy, I think it is a good answer insofar as it points out a (probably deliberately) misleading use of a conditional in the first premise. That answers why it is a fallacious (deliberately misleading via unsound premises but valid structure) argument. And it does so with actual references. I don't see how pointing out semantical equivalence is a redaction of the argument, really. Mar 1, 2021 at 7:49
• @Speakpigeon If it pleases you to maintain that you know better than people who actually publish papers on that subject, you may do so. But that does not make your opinion on the subject more authoritative nor does it lessen the validity of the points mentioned. And it does not justify how you lecture people or disparage their contributions by suggesting they would not understand the topic just because they do not share your understanding of it. That is where you habitually walk the line from discussion of a topic to ad hominem. Mar 1, 2021 at 11:00
• @Speakpigeon See, I had three university courses on statistics and probability theory. That's why I know that conditional probability is a unitary expression and not a logical conditional. Its meaning is "the probability of a single defined event X (first A with probability p1 of its own, then B with probability p2 of its own happens) is p", not "If A already happened, then the probability of B is p". It is the probability of the decision path as a whole, not the second step. Thus, what we have here is misleading formulations coupled with a wrong formalisation and nothing more. Mar 1, 2021 at 13:05