# Challenging the law of excluded middle

‘p or not p’

used with a disjunctive syllogism.

Is this begging the question?

And is that statement ‘p or not p’ a true dichotomy?

Jones, sales assistant in an East London grocery store, enlisted for King and Country just in time for the Battle of the Somme in 1916. But he never went ‘over the top’. The day before a sniper’s bullet put an end to his young life.

We will never know whether or not Jones was brave. However, using the law of excluded middle it seems we can make the following deduction:

1. Either Jones was brave or he was not.

2. If Jones was brave, then there was some fact about Jones’ physical or mental state that constituted his disposition to act bravely if and when the occasion arose.

3. If Jones was not brave, then there was some fact about Jones’ physical or mental state that constituted his disposition to fail to act bravely if and when the occasion arose.

4. Either way (the disjunctive syllogism), there was some fact about Jones’ physical or mental state that constituted his bravery or lack thereof.

The argument is a fallacy.

There might, or might not, exist in the yet untested Jones, a physical or mental state that his bravery or the lack thereof consists in. We simply do not know. You can’t use pure logic to prove an existence claim.

The correct response is to point out that in this case, both the statement ‘Jones was brave’ and the statement ‘Jones was not brave’ share a common assumption. All the disjunctive syllogism (‘or-elimination’ in propositional logic) shows is that there is a shared assumption.

However, it would not be correct to draw the conclusion that the law of excluded middle should be rejected. What it shows that, ‘Jones was not brave’ is not the correct way of stating the proper, or full negation of ‘Jones was brave’.

What we should say, instead, is that, ‘Either Jones possessed the disposition to act unbravely, or there does not exist any such disposition, or Jones does not exist, or there is no such thing as bravery (just as there is no such thing as being a ‘witch’), or… (anything else you can think of).’

What is interesting about the Jones example is that the celebrated Oxford philosopher Michael Dummett in his seminal article, ‘Truth’ (reprinted in ‘Truth and Other Enigmas’ 1978), uses it as leverage against realism about meaning and truth. An anti-realist rejects the law of excluded middle.

I think he was wrong about this (as I argued in my doctoral thesis The Metaphysics of Meaning) but that’s a discussion for another occasion.

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