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The premise is: John is a married bachelor The Conclusion is: Pigs can fly

There is only one premise and one conclusion. Why are they connected and in what way can we say this argument is valid?

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  • The argument above is not valid. Valid applies to an argument form whose possible instances are ALL correct. No argument of form: if P, therefore Q can be valid. May 5, 2023 at 17:36
  • What the example want to show is, maybe, an application of the Explosion principle (aka Ex falso): from a contradiction everything follows. May 5, 2023 at 17:40
  • To do so, you have to add the unstated premise: every bachelor is not married. May 5, 2023 at 17:41
  • There are several questions about the principle of explosion, i.e. the principle in classical logic that anything follows from a contradiction. See for example: philosophy.stackexchange.com/questions/84490/…
    – Bumble
    May 5, 2023 at 17:59
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    Voted to close because the premise of the question is false. The suggested argument is not valid. May 5, 2023 at 18:55

5 Answers 5

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Although the inference A-to-A is typically not suspended, it is still extremely trivial. One question in the theory of deduction is whether deductive inferences are, or at least can be, "ampliative," or carrying more information than the premises. Since converting inference rules themselves into premises (as in Lewis Carroll's poem(?)) is notoriously problematic, the effect of these rules on premises, as delivered in conclusions, might be seen as "transformative" if nothing else: the content of the premises is equipped with structural features like conjunctions, negations, etc. and the inference rules mediate transitions from structures to structures, which give a conclusion that looks different enough from the premises to not seem circular or trivial even though the direct content is presumptively identical.

Or that's the ideal/hope, anyway. Another problematique, here, comes from reflection on the logic of conditionals. Suppose, that is, that some if-A-then-B propositions are not exactly true or false. An argument that took such conditionals as premises could be used to infer some other conditional, in principle. But if we are not starting out with truths, we are not ending up with amplified information (increased knowledge) on the direct-order level, but only more knowledge about what structural transformations are possible on the given rules on a meta-order level. We might say that this makes our knowledge, and its amplification, a procedural more than an ontological manner (we are not learning more objective facts so much as more ways to restructure our concepts of stock facts).


EDITThe trivial/nontrivial distinction can be situated with respect to problem-talk such that we might (uncharitably) say that a trivial solution to a problem "merely restates" the problem, whereas nontrivial solutions actually "fill in the blanks" and expand upon in-play theme(s). However, if restating a problem helps us to understand it more, we might think that there are solutions that are "neither trivial nor nontrivial": not trivial at least because they structurally improve our understanding, not nontrivial at least because they do not extend the content of our understanding (the direct content to which it is applied, not also its content when it is its own content). So in this case, we might avoid intuitionism by saying that, despite appearances, "nontrivial" is not the contradictory but the contrary of "trivial" (and so might better be phrased as "countertrivial"(?)), at least in the problem-talk sense of these things, here.

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It is a feature of classical logic that if you assume two contradictory statements are true, you can end up seeming to 'prove' that patently false statements. Starting with John is a married bachelor, you can 'prove' 2+2=5, the Moon is made of cheese, all politicians are honest, or any other obvious falsehood. You might want to Google 'explosion in logic' to read more about it. I will illustrate the effect for you, as follows...

Let us begin by supposing that two contradictory statements are true, namely:

  1. John is a bachelor

  2. John is married

We can then produce a third true statement by combining either of the above with 'or x' where x is clearly false, for example:

  1. John is a bachelor, or 2+2=5.

Statement 3) is true by virtue of the first part, John being a bachelor, being true.

The so called 'proof' that 2+2=5 is then obtained by holding on to 3) as a true statement, but pointing out that John is married (according to statement 2), so the first part of statement 3) is untrue, so the truth of statement 3) must arise from 2+2 being 5.

Anybody but a manic logician would dismiss the foregoing as utter nonsense*. When, in the argument above, you point out that John is not a bachelor, you should back-out your assertion that statement 3) is true, not retain the truth of it to prove 2+2=5, but apparently that would take all the fun out of it.

However, I jest; the point is not to suggest that it is possible to prove 2+2=5, but instead to show that adopting contradictory assumptions leads to absurdities, lest the inadequate logician fail to click that the adoption is an absurdity in itself.

*And should anyone doubt my assertion, I can prove it logically by assuming John is a married bachelor, etc.

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  • Not very clear... your explanation of Ex falso arguments is quite good. So, what does it mean the statement "lest the inadequate logician..." May 7, 2023 at 12:33
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Somehow John's existence of being a married bachelor has directly and/or indirectly made it a fact of reality that pigs can fly. Presuming that John's existence of being married as such is a fact of reality that has directly and/or indirectly made it a fact of reality that pigs can fly, then the premise and the conclusion are valid.

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    Welcome to SE. I'm afraid you need to explain how this answer addresses the question. It may be obvious to you, but that doesn't mean it is obvious to the person asking the question.
    – Ludwig V
    May 6, 2023 at 14:25
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You seem to be confusing the concepts of a valid argument and a sound one. This is not a sound argument because of the contradiction in "John is a married bachelor".

It is valid because "pigs can fly" is not less true than 'John is a married bachelor". It would be equally valid to state that John is a married bachelor, therefore pigs cannot fly. What the statements mean is irrelevant, only the truth value matters.

This is why we don't usually bother with arguments that begin with false (or contradictory) premises, because can prove both any conclusion we like and its negation, which gets us nowhere.

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The premise is: John is a married bachelor The Conclusion is: Pigs can fly

There is only one premise and one conclusion. Why are they connected and in what way can we say this argument is valid?

There is nothing formally contradictory in being a married bachelor. It is only if you assume the standard semantics of the English vocabulary that there is one.

Still, assuming the standard semantics of the English vocabulary and therefore the self-contradiction in the premise, then nothing follows. As least this is the common sense view.

The idea that from a contradiction everything follows is clearly illogical. Yes, it is the "principle" adopted in mathematical logic and you will find it in all mathematical logic textbooks as the "principle of explosion", PoE for short, but it is just plain wrong. This makes mathematical logic illogical.

Mathematicians adopted this pseudo-logical principle (it is not even a principle) for the simple reason that they didn't know any better. The PoE is a consequence, a logical consequence, of the basic assumptions mathematicians (and philosophers) make about the conditional. This is the formal system they have been able to come up with which is closest to mathematical reasoning. It is wrong and yet nobody has found any better.

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