To: Pat F.

From: Geoffrey Klempner

Subject: Attempting to justify the law of excluded middle

Date: 19 December 2007 12:19

Dear Patrick,

Thank you for your email of 13 December, with your University of London Logic essay in response to the question, 'What kind of justification can be given for the Law of the Excluded Middle? Is the justification convincing?'

This is a much more tightly constructed essay which provides the basis for a good answer to the question.

You have considered three stumbling blocks to the acceptance of LEM: non-denoting singular terms, future contingency and the arguments of the intuitionists in mathematics, as well as rejecting worries based on Godel's Theorem. For the sake of completeness, I would have thought that another major area to consider is vagueness, where there is a strong temptation to reject LEM in order to resist Wang's paradox/ paradox of the heap.

On the face of it the question asks you to examine possible justifications of LEM, rather than justifications for rejecting LEM. However, given the thinness of justifications for LEM, defending against criticisms of LEM may perhaps be seen as a legitimate part of a justification. You need to be careful, though, to show the examiner that you have not forgotten what the question was asking for.

The first hurdle to get over is the seeming emptiness of asserting that LEM 'follows' from bivalence. Why aren't these just two versions of the same claim? What is the significance of a 'semantic' justification?

Dummett has a lot to say about this in 'The Justification of Deduction' and 'The Philosophical Basis of Intuitionist Logic'. I think this is too important to overlook in an essay on this question. Briefly, Dummett's case is that LEM is justified by an account of the form of a theory of meaning which makes truth the central concept, thus yielding a substantial principle of bivalence from which LEM follows. Criticisms of the LEM are based on an alternative theory of meaning, which makes proof rather than truth the central concept. This involves considerable revision of Brouwer's case against classical logic, which is based on the arguably dubious notion that numbers are 'free creations of human thought'.

A point about Aristotle's definition of LEM which you give is that it involves asserting or denying predicates of a subject. This invites the problems addressed by Russell in 'On Denoting' regarding non-denoting singular terms, since it is patently not the case that the present King of France is either bald or not bald.

In your account of Russell's theory of descriptions the text seems to have got a bit garbled. According to Russell, 'It is or it is not the case that the Present King of France is bald' would be equivalent to:

'Either there is a present King of France and the King of France is bald, or there is a present King of France and the King of France is not bald, or it is not the case that there is a present King of France.'

You give the proof of LEM offered in the Principia which seems singularly unilluminating: Given that (p hook q) iff (not-p or q) it follows, substituting p for q that (p hook p) iff (not-p or p), from which not-p or p follows by modus tolens from p hook p.

The point could be made that any system of (classical) propositional logic must either have 'p or not-p' as an axiom or have sufficient axioms and/ or rules of inference to enable the proof of 'p or not-p'. Otherwise, given that 'p or not-p' is shown to be a tautology by its truth table, the system would be incomplete - i.e. lack sufficient resources for proving all tautologies.

What exactly are the consequences of Aristotle's view of future contingency for LEM? Arguably, LEM can still be defended on the basis of a branching model of time, where in every possible course of events either there is a sea battle or there is not a sea battle (Dummett offers this in his essay, 'On the Reality of the Past'). An objector, however, would say that this is not what we *mean* when we assert the LEM about future events. Who is right?

In your account of Brouwer you make an assertion which intuitionists would not accept: not-(p or not-p). Intuitionists do not deny the law of excluded middle, rather, they refuse to allow it. Indeed it is a theorem of intuitionist logic that not-not-(p or not-p). However, your point about the importance of double negation is sound. While it is possible in intuitionist logic to prove p |- not-not-p, it is not possible to prove not-not-p |- p. As double-negation is needed for a considerable part of classical mathematics, its removal leaves mathematics crippled.

You could have mentioned that one possible response is to allow that intuitionist mathematics is a legitimate area within classical mathematics: i.e. it is a worth while project to explore what can be proved by means of constructive proofs. However, perhaps this is too far off the topic of justifications for LEM.

All the best,

Geoffrey