What would be a good example of explicit deductive reasoning that doesn't seem to be possibly interpreted correctly as a conditional (If A, then B)?

  • Disjunctive syllogism. – Conifold Jun 27 at 8:30
  • @CanBeSaidClearly "Doesn't seem to be possibly" is an English phrase meaning "doesn't seem to be possibly". Don't think it could be expressed differently, I suspect it cannot, but prove me wrong if I am. If you did, you would prove that it is understandable. And if you don't, you confort my belief that it cannot be expressed differently. But no doubt it means something. I don't see how you could prove that this is not the case, since failing to understand something which is perfectly understandable is such a common occurrence. – Speakpigeon Jun 27 at 9:14
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    You can help by articulating yours in more than one sentence. – Conifold Jun 27 at 9:35

It would be worthwhile distinguishing between a conditional sentence in the object language and a conditional in the metalanguage. Some deductive arguments have a conditional in the object language, e.g. those of the form modus ponens or modus tollens. Some arguments do not, e.g. those of the form conjunction elimination, disjunction elimination, etc.

But what is always possible (and I think this is what your question is asking) is that a deductive argument can be expressed as a kind of modal conditional at the meta level. A valid argument is one such that necessarily if all the premises are true then the conclusion is true.

The modal term 'necessarily' can be replaced by some other modality or generality, depending on what account of validity you find most congenial. Some common ones are: under all interpretations, under all substitutions of the non-logical constants, under all permutations of the domain of quantification, in all possible worlds, it is a priori knowable that, it is provable that, it is conceptually certain that, or whatever.

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Perhaps you're looking for objections to characterizing natural language conditionals with the material conditional ("->"). If so, two examples come to mind.

First, from Van McGee:

(1) If that creature is a fish, then if it has lungs, it is a lungfish
(2) That creature is a fish
(3) Hence, if it has lungs, it is a lungfish

If we treat "if...then" in natural language as the material conditional, then we have:

(1*) A -> (B -> C)
(2*) A
(3*) B -> C

Van McGee claimed, however, that while (1*) and (2*) might be stipulated to entail (3*), one can construct scenarios in which (1) and (2) are true, but (3) false.

Second, natural language counterfactuals raise similar problems for material conditionals:

(4) If Hoover had been born in Russia, he would've been a communist
(5) If Hoover were a communist, he would've been a traitor
(6) Hence, if Hoover had been born in Russia, he would've been a traitor

Which - if using the material conditional to model - results in:

(4*) A -> B
(5*) B -> C
(6*) A -> C

But while (4*) and (5*) entail (6*), many believe (4) and (5) can be true while (6) false.

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