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I was reading an answer by Bumble where the topic of modus tollens being valid but contraposition being not valid came up: https://philosophy.stackexchange.com/a/43004/29944

More importantly, there are logics in which contraposition is not valid at all. For example, in David Lewis' logic of counterfactual conditionals, A ◻→ B does not entail ¬B ◻→ ¬A. Also, in Ernest Adams' probability logic it may be highly probable that B given A, but not highly probable that ¬A given ¬B. So in general, when speaking of ordinary English conditionals, one cannot always expect contraposition to be safe. A noteworthy corollary is that in both the Lewis and Adams logics, while contraposition is not valid, modus tollens is valid. Some accounts of logic incorrectly run together contraposition with modus tollens and treat them as the same thing. While both are classically valid, they do not agree across all logics.

I don't doubt this, but I was trying to think of an English language example to illustrate this without success.

Hence the question: Is there an English language example, the simpler the better, where modus tollens is valid but contraposition is not valid?

An example where contraposition is valid but modus tollens is not valid would be nice as well, but I don't want this question to become too broad.

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  • Is there an ambiguity in ‘valid’ at work here, in "an example where modus tollens is valid but contraposition is not valid"? I mean the ambiguity between syntactic and semantic ‘validity’. MT is a deductive rule; so, to say that MT is ‘valid’ means that the step from ~Q and P->Q to ~P counts as a proof, right? Meanwhile, that contraposition is ‘valid’ means that all models that make P->Q true, make ~Q->~P true also, no? I’m not saying that the question can’t be answered, but I think the answer would have to be an English example that distinguishes between syntactic and semantic ‘validity’.
    – MarkOxford
    Aug 29, 2018 at 18:06
  • Maybe useful the example of the fallacy of contraposition in D.Lewis, Counterfactuals, page 35.
    – Mauro ALLEGRANZA
    Aug 29, 2018 at 18:41
  • @MarkOxford I suspect the word "valid" would be in the logic allowing such discrepancy. In classical logic, both modus tollens and contraposition would be valid. So it doesn't work in classical logic. For the English sentence, validity is not so much the issue as some semantic use of language that would illustrate why one should bother with these non-classical logics
    – Frank Hubeny
    Aug 29, 2018 at 18:52
  • @MauroALLEGRANZA Thanks for the link. I think I am mainly having trouble making sense out of this to know when I have an example or not.
    – Frank Hubeny
    Aug 29, 2018 at 18:54
  • You need to be more specific as everyone does not know you are only referring to “Mathematical Logic”. Contraposition in Mathematical Logic does not mean the same thing as Aristotelian logic. There was no mathematical logic before the 19th century. Aristotle did not define contraposition as you would use the term in math today. Philosophy does not use the same inference rule name for a completely different inference rule. What you call contraposition in mathematical logic, philosophers call the rule “Transposition”. Aristotelian logic does not allow contraposition on E propositions.
    – Logikal
    Sep 3, 2018 at 16:42

2 Answers 2

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Here's the example from David Lewis's Counterfactuals (1973):

If Boris had gone to the party, Olga would have gone.

Now suppose that Boris wants to go, but not if Olga goes, because he wants to avoid her. Olga, on the other hand, wants to see Boris, and wants to go if he does. Given this supposition, the contrapositive of the above is false:

If Olga hadn't gone to the party, Boris wouldn't have gone.

As for modus tollens, this is valid:

  1. If Boris had gone to the party, Olga would have gone.
  2. Olga didn't go to the party.
  3. Therefore, Boris didn't go to the party.

The conclusion follows from the premises: we know that Boris didn't go, because Olga didn't -- she would have if he had.

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As well as the counterfactual example given by Eliran, there are an abundance of examples where the conditionals are uncertain. For example, given the poor record of the Norwegian soccer team, I might believe strongly that if Norway reach the final of the next world cup then they won't win. The contrapositive of this is that if Norway win then they won't reach the final, which is impossible. More generally, it may be highly probable that B given A, but not highly probable that not A given not B. Conditional probabilities do not follow the same rules as classical material implications, but many real world uses of conditionals behave like claims of conditional probabilities being high.

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