William of Soissons; French: Guillaume de Soissons; was a French
logician who lived in
Paris
Paris () is the capital and most populous city of France, with an estimated population of 2,165,423 residents in 2019 in an area of more than 105 km² (41 sq mi), making it the 30th most densely populated city in the world in 2020. S ...
in the 12th century. He belonged to a school of logicians, called the
Parvipontians.
[Graham Priest, 'What's so bad about contradictions?' in Priest, Beall and Armour-Garb, ''The Law of Non-Contradiction'', p. 25, Clarendon Press, Oxford, 2011.]
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
Plato ( ; grc-gre, Πλάτων ; 428/427 or 424/423 – 348/347 BC) was a Greek philosopher born in Athens during the Classical period in Ancient Greece. He founded the Platonist school of thought and the Academy, the first institution ...
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 (¬P)'' you may infer ''that there are trees on the moon (or whatever else)(E)''. 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 (12th century).
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 (Explosion).
(1) P &¬ P → P (If P and ¬ P are both true then P is true)
(2) P → P∨E (If P is true then P or E is true)
(3) P &¬ P → P∨E (If P and ¬ P are both true then P or E are true (from (2))
(4) P &¬ P → ¬P (If P and ¬ P are both true then ¬P is true)
(5) P &¬ P → (P∨E) &¬P (If P and ¬ P are both true then (P∨E) is true (from (3)) and ¬P is true (from (4)))
(6) (P∨E) &¬P → E (If (P∨E) is true and ¬P is true then E is true)
(7) P &¬ P → E (From (5) and (6) one after the other follows (7))
Acceptance and criticism in later ages
In the 15th century this proof was rejected by a school in
Cologne
Cologne ( ; german: Köln ; ksh, Kölle ) is the largest city of the German western States of Germany, state of North Rhine-Westphalia (NRW) and the List of cities in Germany by population, fourth-most populous city of Germany with 1.1 m ...
. They didn't accept step (6).
In 19th-century
classical logic
Classical logic (or standard logic or Frege-Russell logic) is the intensively studied and most widely used class of deductive logic. Classical logic has had much influence on analytic philosophy.
Characteristics
Each logical system in this class ...
, the Principle of Explosion was widely accepted as self-evident, e.g. by logicians like
George Boole and
Gottlob Frege
Friedrich Ludwig Gottlob Frege (; ; 8 November 1848 – 26 July 1925) was a German philosopher, logician, and mathematician. He was a mathematics professor at the University of Jena, and is understood by many to be the father of analytic phil ...
, though the formalization of the Soissons proof by Lewis provided additional grounding for the Principle of Explosion.
References
{{DEFAULTSORT:Soissons, William of
Logicians
Theorems in propositional logic