William of Soissons


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 fundamental logical problem and solution

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 and it is not raining you may infer that there are trees on the moon . In symbolic language: P & ¬P → E.
If a contradiction makes anything true then it makes it impossible to say anything meaningful: whatever you say, its contradiction is also true.

C. I. Lewis's reconstruction of his proof

William's contemporaries compared his proof with a siege engine. Clarence Irving Lewis formalized this proof as follows:
Proof
  • V : or
  • & : and
  • → : inference
  • P : proposition
  • ¬ P : denial of P
  • P &¬ P : contradiction.
  • E : any possible assertion.
P &¬ P → P
P → P∨E
P &¬ P → P∨E P &¬ P → ¬P
P &¬ P → &¬P
&¬P → E
P &¬ P → E and one after the other follows )

Acceptance and criticism in later ages

In the 15th century this proof was rejected by a school in Cologne. They didn't accept step. 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.