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Based on the fact that a deductive argument is said to be valid if and only if it takes a form that makes it impossible for the premises to be true and the conclusion nevertheless to be false, I am having trouble understanding why the statement: "If the premises of an argument CANNOT all be true, then said argument is valid" is true.

  • This statement can be simplified to NOT(A) implies NOT(BOTH(A, B)) where A is "all premises are true" and B is "the conclusion is false" – Ben Voigt Feb 28 '18 at 4:16
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    You can read the def of valid argument alternatively as: "there is no circumstance where the premises are true and the conclusion is false". If the premises are contradictory (i.e. always false), there is no circumstance where the premises are true, and thus (a fortiori) no circumstance where the premises are true and the conclusion is false. – Mauro ALLEGRANZA Feb 28 '18 at 7:43
  • Alternatively, look at the definition of INvalid: an argument is invalid iff it is not valid iff there is an interpretation I that makes (a) all the premises true and (b) the conclusion false. In turn, a set of sentences is inconsistent iff there is no interpretation I that makes all sentences in the set true. Thus, if a set of sentences (premises) is inconsistent, there is no I that satisfies (a), and hence no I that satisfies (a) and (b). Thus, if your premises are inconsistent, the argument 'has no chance' of being invalid, whence it must be valid. – MarkOxford Feb 28 '18 at 22:04
  • Are you looking for the intuition behind the tautology ~A => [A =>B]? – Dan Christensen Mar 7 '18 at 3:48
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The rules of logic lead to many counterintuitive results, and this is one of the most fundamental such results: VALID expresses a structural condition, such that it can never happen that all the premises are true and the conclusion is false.

If the premises cannot all be true at at the same time, then the argument is trivially VALID because it can never happen that all the premises are true... (regardless of the truth value of the conclusion). This holds only when the premises are logically contradictory, however, and not in the case where they are incidentally contradictory.

The usefulness of VALID is that it is what is called "truth preserving." If all your arguments are valid, the truth of your conclusions can never be less secure than that of your premises, considered collectively.

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There is a distinction between a sound argument and a valid one. A sound argument actually proves something. A valid argument may not. Instead, a valid argument preserves the truth of its premises.

The idea behind focussing on valid arguments in most logic is that any valid argument could be applied to a wholly different set of premises similar to the actual ones (in both form and truth value). And if those other premises were true, each set would produce a sound argument. So a valid argument can produce a number of different sound proofs. It is, therefore, more useful.

But if there is no truth in the premises, then absolutely any argument preserves 'all' of that nonexistent truth. So if your premises are false, your argument is always valid.

If your premises contradict, so that they cannot all be true, because if some of them are true, others would not be, then, taken together they are false. So your argument is valid.

  • I'm sure you didn't mean this, but, "if your premises are false, your argument is always valid" together with "any valid argument could be applied ... if those other premises were true, each set would produce a sound argument" implies that if your premises are true (and are similar to false premises), then any argument is a sound argument. – H Walters Feb 28 '18 at 17:40
  • ...e.g., P1: Apples are blue. P2: Roses are blue. Argument: P1, P2, therefore apples are roses. Both premises are false; therefore this argument's valid (by "if premises are false your argument is always valid"). P1. Apples are red. P2: Roses are red. Argument: P1, P2, therefore apples are roses. This is now sound (by "any valid argument could be applied... if those other premises were true, each set would produce a sound argument"). So apples are roses? – H Walters Feb 28 '18 at 17:45
  • @HWalters 'Similar', requiring that they have the same truth values... Everything is similar to everything if you don't care about the context. So I added that explicitly, it is not obvious enough. – user9166 Feb 28 '18 at 18:39
  • @hwalters your exa.ple argu.ent is not valid at all. Your example argument commits an elementary fallacy. – Logikal Feb 28 '18 at 19:03
  • @Logikal The fallacy is intentional: "if your premises are false, your argument is always valid" ...suggests that it's valid anyway (so long as the premises are false, as in the first argument). jobermark: Still confused what you're trying to get at. If it's a sound argument, aren't all of its premises true? How then can it be similar to any valid unsound argument? – H Walters Mar 1 '18 at 3:40
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The term valid should be noted to have two contexts in logic: one in which there is no possible way for an argument with all true premises yield a false conclusion; the second is the context that expresses an argument form where a counter example exists of the same format using different terms that yields a blatant false conclusion. When you know there exists a counter example with a false conclusion from true premises the form of that argument is invalid.

There is also the notion that anything can follow from a contradiction. The technical reason is that you will never arrive at two true premises with a false conclusion. You will have either one false premise or both false premises in the argument. That is anything can follow when you start out with falsehood. You cannot guarantee the conclusion.

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