From any two contradictory premises p and non-p, valid rules of deductive inference can be used to derive any conclusion one likes. Karl Popper, in his essay What is Dialectic? (1940), presented the logical proof of this fact in the course of criticizing Hegel’s “dialectical” account of historical progress. That proof is summarized below, wherein the symbol ∨ is to be read as “and/or”:

Rule of Inference 1: If the premise p is true then the statement p ∨ q must also be true

Rule of Inference 2: If the two premises non-p and p ∨ q are true then the conclusion q must also be true

Contradictory Assumption: There exist two contradictory premises which are both true, such as the following:

(a) Humans did evolve on Earth
(b) Humans did not evolve on Earth

From the two contradictory premises (a) and (b) one can use Rules 1 and 2 above to infer any statement. For example, from (a) and (b) one can validly infer the statement, Genghis Khan fathered one million children.

From the premise (a), one can use Rule 1 to infer the following statement:

(c) Humans did evolve on Earth ∨ Genghis Khan fathered one million children

Now, from the joint premises (b) and (c), one can use Rule 2 to infer the following statement:

(d) Genghis Khan fathered one million children

This proof shows that any theory involving a contradiction entails the truth of any statement, and thus contains no substantive content. Alongside this proof, Popper noted that, contra dialectic, contradictions fuel progress not when they are accepted, but rather if, and only if, they are avoided. For the law of noncontradiction underlies all criticism, and criticism, which is to say the highlighting of contradictions, and the changing of theories that involve contradictions, underlies all progress.