William of Soissons () was a French logician who lived in Paris in the 12th century. He belonged to a school of logicians called the Parvipontians.
William of Soissons seems to have been the first one to answer the question, "Why is a contradiction not accepted in logic reasoning?" by the principle of explosion. Exposing a contradiction was already in the ancient days of Plato a way of showing that some reasoning was wrong, but there was no explicit argument as to why contradictions were incorrect. William of Soissons gave a proof in which he showed that from a contradiction any assertion can be inferred as true. In example from: It is raining (P) and it is not raining (ìP) you may infer that there are trees on the moon (or whatever else)(E). In symbolic language: P & ìP â E.
If a contradiction makes anything true then it makes it impossible to say anything meaningful: whatever you say, its contradiction is also true.
William's contemporaries compared his proof with a siege engine (12th century). Clarence Irving Lewis formalized this proof as follows:
Proof
(1) P &ì P â P (If P and ì P are both true then P is true) (2) P â Pâ¨E (If P is true then P or E is true) (3) P &ì P â Pâ¨E (If P and ì P are both true then P or E are true (from (2)) (4) P &ì P â ìP (If P and ì P are both true then ìP is true) (5) P &ì P â (Pâ¨E) &ìP (If P and ì P are both true then (Pâ¨E) is true (from (3)) and ìP is true (from (4))) (6) (Pâ¨E) &ìP â E (If (Pâ¨E) is true and ìP is true then E is true) (7) P &ì P â E (From (5) and (6) one after the other follows (7))
In the 15th century this proof was rejected by a school in Cologne. They didn't accept step (6). In 19th-century classical logic, the Principle of Explosion was widely accepted as self-evident, e.g. by logicians like George Boole and Gottlob Frege, though the formalization of the Soissons proof by Lewis provided additional grounding for the Principle of Explosion.