This question was inspired by Three statements that contradict each other

I’m wondering if there is any set of 4 statements that contradict each other when taken as a whole, but any combined 3 of those statements do not have a contradiction.

If the above is possible, is there always at least 1 set of n number of statements that when taken as a whole are contradictory, and all groups of n-1 statements in that set are not contradictory for all n values?

1 Answer 1


Sure: take for example A,B,C, ~(A and B and C).

More generally, take A1, A2, A3,... , ~(A1 and A2 and A3 and ...).

Interestingly, this cannot happen with infinitely many sentences: the compactness theorem says that if T is a set of sentences such that every finite subset of T is consistent, then T itself is consistent.

(Strictly speaking we haven't specified what logic we're working in, but the above is true for both propositional and first-order logic. Meanwhile, the examples I gave above were in propositional logic but easily lift to first-order examples.)

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