What is so bad about contradictions?
That question is the title of an article published in 1998 by Graham Priest in the Journal of Philosophy. In section 1.2 of my book, I give the traditional answer that contradictions are bad because contradictions imply everything. If even one contradiction is true, then every contradiction is true ‒ and false, and neither true nor false, and so on without restriction. That result is so disastrous that even contradiction-friendly types like Priest recoil from it.
A well-known proof shows how accepting even one contradiction as true leads to disaster. Let p and q be any propositions, let ‘∧’ symbolize ‘and’, let ‘∨’ symbolize ‘at least one of’, and let ‘¬’ before a proposition indicate that the proposition is false:
(1) p ∧ ¬ p
Sample contradiction: p is true and p is false.
(2) p
From (1): If p is true and p is false, then p is true.
(3) p ∨ q
From (2): If p is true, then at least one of p and q is true.
(4) ¬ p
From (1): If p is true and p is false, then p is false.
(5) q
From (3), (4): If at least one of p and q is true, and p is false, then q is true.
Keep in mind that q can be any proposition at all. So if even one contradiction is true, then any proposition at all is true, no matter how outlandish or self-inconsistent.
Priest wants to make it safe for us to believe a limited number of contradictions, so he has spent much of his career attempting to discredit this proof. In section 1.3 of my book, I explain why his attempts fail.
Refusing to accept even one contradiction as true isn’t “a foolish consistency,” to use Ralph Waldo Emerson’s phrase. On the contrary, it’s the only way to avoid committing yourself to the truth of every foolish proposition there is.
