# 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 (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 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). 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. 

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. )