William of Soissons
William of Soissons fundamental logical problem and solutionEdit
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 (¬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 proofEdit
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 the 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 agesEdit
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 the Principle of Explosion.
Appendix: A way of rejecting the proofEdit
The above proof can be rejected as is shown below. 
Take the proof above and unpack the justification for line (6), taking it to rely on (6*):
(1) ................. (2) ................. (3) ................. (4) ................. (5) P &¬ P → (P∨E) &¬P (6) (P∨E) &¬P → (P &¬ P) V(¬P&E)
Now only if (P &¬ P) is rejected as invalid E can be inferred:
(7*) (P∨E) &¬P → (¬P&E) (8*) (P∨E) &¬P → E
From (5) and (8*) follows:
(9*) P &¬ P → E
On this reconstruction, only by rejecting (P &¬ P) can E be concluded. So, if (P &¬ P) is not rejected E cannot be concluded. But (P &¬ P) can in this proof only be rejected if E is valid. So this proof is a vicious circle.
(Rejecting this Soissons/Lewis proof does not reject the Principle of Explosion. Therefore a counterexample, in which is shown a contradiction which is not invalid, would do. )
- 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.
- A more a less similar rejection of the Soissons/Lewis proof can be found on the Talkpage of Principle of Explosion. (chapter 3)
- Graham Priest shows counterexamples in Graham Priest, Logic A very short introduction, Chapter 5, Oxford University Press, 2017.