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.

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 – 2018-02-28T04:16:25.7301You can read the def of

valid argumentalternatively as: "there is no circumstance where the premises aretrueandthe conclusion isfalse". If the premises arecontradictory(i.e. always false), there is no circumstance where the premises aretrue, and thus (a fortiori) no circumstance where the premises aretrueandthe conclusion isfalse. – Mauro ALLEGRANZA – 2018-02-28T07:43:39.793Alternatively, look at the definition of

INvalid: an argument is invalid iff it is not valid iffthere isan interpretationIthat makes (a) all the premises true and (b) the conclusion false. In turn, a set of sentences isinconsistentiff there is no interpretationIthat makes all sentences in the set true. Thus, if a set of sentences (premises) is inconsistent, there is noIthat satisfies (a), and hence noIthat satisfies (a) and (b). Thus, if your premises are inconsistent, the argument 'has no chance' of being invalid, whence it must be valid. – MarkOxford – 2018-02-28T22:04:59.807Are you looking for the intuition behind the tautology ~A => [A =>B]? – Dan Christensen – 2018-03-07T03:48:24.937