To: Pearl K.

From: Geoffrey Klempner

Subject: What would it take to justify the law of excluded middle?

Date: 19 April 2007 09:23

Dear Sachiko,

Thank you for your email of 10 April with your University of London essay in response to the question, 'What kind of justification can be given for the Law of Excluded Middle? Is it convincing?'

Your essay is not too bad. With luck, it would fall into the lower end of the 2/i bracket but it could have been a lot better. I think you are taking risks by just relying on the Stanford articles. There must be some book on logic that you can find in your book shop or library that covers at least some of these issues. You might find that taking the time to browse might pay off. Also, there is also a lot more stuff on the internet if you take time to look.

You start off making a good point about the difference between 'internal' and 'external' negation. However, this can be generalised, in a way which makes the Russell point more problematic. What is the fully 'external' negation of a proposition? It is not enough just to move the 'not' sign to the outside, because the proposition contains terms - names and general terms - which we assume to be meaningful. If I say, 'It is not the case that Susan is a witch', I presuppose that there is someone called 'Susan' and also that there is such a thing as being a witch. Someone who wanted to give the fully external negation of 'Susan is a witch' would have to say, 'Either it is not the case that Susan is a witch, or the name 'Susan' lacks a reference, or the concept of 'witch' is incoherent.'

Russell meets the point about names in his theory of descriptions. The possibility of a lack of reference for 'Susan' is taken care of by equating the meaning of 'Susan' with some definite description or descriptions.

I don't think it is correct to say that Godel's incompleteness theorem 'reflects the a priori assumption that all mathematical problems have solutions'. Prior to Godel's theorem, it was believed that arithmetic is finitely axiomatisable, in other words that all the truths of arithmetic could be derived from a finite set of axioms. Godel showed that whatever axiom set one started with, there would be arithmetical truths which are not provable my means of that axiom set. But suppose (prior to Godel) you were confident that your list of axioms was complete. That confidence would not be sufficient to warrant the belief that all mathematical problems have solutions, because one would still lack a mechanical decision procedure for proving all arithmetical theorems on the basis of those axioms.

I have a particular interest in the law of excluded middle going back to the time when I was doing my D.Phil thesis. My main 'target' was Michael Dummett who in numerous writings has put the case for rejecting a truth-conditional theory of meaning in favour of one modelled on the justification of intuitionist logic.

The centrepiece of Dummett's account is the claim that classical logic depends on the law of bivalence, while the law of bivalence is justified by a truth conditions theory of meaning, which he equates with a 'realist' metaphysic according to which propositions have determinate truth values independently of our ability to discover those truth values. If we reject a truth conditions theory of meaning then we lose the justification for the law of bivalence. All that one is permitted to claim is the weaker double negation of the LEM, not-not-(P or not-P).

In short, rejection of the LEM is equated with what Dummett terms 'anti-realism'. If you want to see whether someone is a realist or not -- or a realist with respect to a specific subject matter -- then just see whether they are prepared to use the LEM.

In my thesis, I argued that someone with the strongest anti-realist scruples can still accept the LEM as well as the principle of bivalence, because merely the LEM, or stating, 'Every proposition is either true or false' is not enough to convey what the realist 'means'. The case against realism is simply that there is no question-begging way to convey the realist's 'meaning'.

Here is one illustration of how an anti-realist with respect to a particular subject matter can nevertheless hold the law of excluded middle. You will notice that in spirit it is similar to the account of supervaluation in justifying LEM for vague statements. Consider the statement, 'Either a tree stood here 1000 years ago or not'. The anti-realist would say that the statement, 'A tree stood here 1000 years ago' does not have a determinate truth value. However, it does have a determinate truth value in each possible history, where a 'possible history' is any history consistent with what we know, or with what evidential traces are available (there are various definitions of different degrees of stringency). In that case, 'Either a tree stood here 1000 years ago or not' is true because it is true in each possible history.

What is the significance of the equiconsistency of classical and predicate logic? It leaves the question of the justification of LEM completely open. Logic is consistent with or without LEM. Adding LEM does not make logic inconsistent. So what is the point of having LEM in the first place?

The LEM is needed to prove theorems in classical mathematics, specifically those which depend on deriving a contradiction from the hypothesis that a particular property F does not apply to all numbers and then converting the double negation ('it is not the case that F does not apply...') into the proposition that F does apply to all numbers.

The only time LEM appears to do real work outside mathematics, is when we have a purported statement 'P or not-P' where the negation is not 'fully external' in the sense discussed above. Dummett in his paper 'Truth' gives the example, 'Jones either was brave or not' uttered after his death following a life where Jones was never put to the test. From the proposition 'Either Jones was brave or Jones was not brave' we can derive, via or-elimination, the claim that bravery is a state which determinately exists or fails to exist independently of its being displayed or not. However, I would argue, to challenge this claim one does not need to take an 'anti-realist' stance as Dummett claims. One merely points out that the concept of bravery does not function in that way. Not all terms which purport to refer to dispositions are assumed to have a reference independently of the actualisation of those dispositions.

All the best,

Geoffrey