William of Soissons

William of Soissons; French: Guillaume de Soissons; was a French logician who lived in Paris in the 12th century. He belonged to a school of logicians, called the Parvipontians.^[1]

William of Soissons fundamental logical problem and solution

William of Soissons^[2] 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.^[1] 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).^[3] Clarence Irving Lewis^[4] formalized this proof as follows:^[5]

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. They didn't accept step (6).^[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.

References

^ ^a ^b 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.
^ His writings are lost, see: The Metalogicon of John Salisbury. A Twelfth-Century Defense of the Verbal and Logical Arts of the Trivium, translated with an Introduction and Notes by Daniel D. McGarry, Gloucester (Mass.), Peter Smith, 1971, Book II, Chapter 10, pp. 98-99.
^ William Kneale and Martha Kneale, The Development of Logic, Clarendon Press Oxford, 1962, p. 201.
^ C. I. Lewis and C. H. Langford, Symbolic Logic, New York, The Century Co, 1932.
^ Christopher J. Martin, William’s Machine, Journal of Philosophy, 83, 1986, pp. 564 – 572. In particular p. 565
^ "Paraconsistent Logic (Stanford Encyclopedia of Philosophy)". Plato.stanford.edu. Retrieved 2017-12-18.

[:1-1] 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.

[2] His writings are lost, see: The Metalogicon of John Salisbury. A Twelfth-Century Defense of the Verbal and Logical Arts of the Trivium, translated with an Introduction and Notes by Daniel D. McGarry, Gloucester (Mass.), Peter Smith, 1971, Book II, Chapter 10, pp. 98-99.

[3] William Kneale and Martha Kneale, The Development of Logic, Clarendon Press Oxford, 1962, p. 201.

[4] C. I. Lewis and C. H. Langford, Symbolic Logic, New York, The Century Co, 1932.

[5] Christopher J. Martin, William’s Machine, Journal of Philosophy, 83, 1986, pp. 564 – 572. In particular p. 565

[6] "Paraconsistent Logic (Stanford Encyclopedia of Philosophy)". Plato.stanford.edu. Retrieved 2017-12-18.

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