I found the following passage in Mark Sainsbury's 'Logical Forms'

If a collection of propositions is inconsistent, any argument whose premises consist of all but one of the collection, and whose conclusion is a contradictory of the remaining proposition, is valid (23).

This seems to be true for three propositions, or where only one of the collection is inconsistent with the others, but what about where there are multiple inconsistencies? Please could someone verify in which cases Sainsbury's statement is true.

  • 2
    This reduces to the case of just two premises, simply take take the conjunction of all but one as a single premise.
    – Conifold
    Aug 27, 2015 at 17:56

1 Answer 1


The statement holds in general.

Proof. (A_1 and ... and A_n) false

<=> not (A_1 and ...and A_n) true

<=> (not A_1 or ... or not A_n) true

<=> (not A_1 or ... or not A_n-1) or not A_n true

<=> not (A_1 and ... and A_n-1) or not A_n true

<=> not ( (A_1 and ... and A_n-1) and A_n) true

<=> (A_1 and ... and A_n-1 => not A_n) true, q.e.d.

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