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.[1]

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.[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.

Lewis's reconstruction of his proof

William's contemporairies compared his proof with a siege engine (12th century).[2] In the 1800s, Clarence Irving Lewis formalized this proof as follows:[3]

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 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 ages

In the 15th century this proof was rejected by a school in Cologne. They didn't accept step (6).[4] 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 proof

The above proof can be rejected as is shown below. [5]

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 infered:

(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. [6])

References

  1. 1 2 Graham Priest, 'What's so bad about contradictions?' in Priest, Beal and Armour-Garb, The law of non-contradicton, p. 25, Clarendon Press, Oxford, 2011.
  2. ↑ Kneale and Kneale, The development of logic, Clarendon Press Oxford, 1978, p. 201.
  3. ↑ Christopher J. Martin, William’s Machine, Journal of Philosophy, 83, 1986, pp. 564 – 572. In particular p. 565
  4. ↑ "Paraconsistent Logic (Stanford Encyclopedia of Philosophy)". Plato.stanford.edu. Retrieved 2017-12-18.
  5. ↑ A more a less similar rejection of the Soissons/Lewis proof can be found on the Talkpage of Principle of Explosion. (chapter 3)
  6. ↑ Graham Priest shows counterexamples in Graham Priest, Logic A very short introduction , Chapter 5, Oxford University Press, 2017.
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