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

Question

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."

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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.

Answer 1

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.

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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").

Answer 2

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.

Answer 3

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.

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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.

Answer 4

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).

Answer 5

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.)

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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.

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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.)