1

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?

2

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.)

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.