To: Bogdan P.

From: Geoffrey Klempner

Subject: Logic and the principle of excluded middle

Date: 26th February 2010 12:25

Dear Bogdan,

Thank you for your email of 16 February, with your essay for the University of London BA Logic module, in response to the question, 'Are we right to regard the principle of excluded middle as a universal truth, a law of logic?'

You start off by giving two versions of the law of excluded middle (LEM). In the predicate version, a given object x is either P or not-P. In the propositional version, either P or not-P for any given proposition P. You correctly distinguish the LEM from the principle of bivalence, according to which any given proposition P is either true or false.

You say, 'With the principle of excluded middle we are not so much interested in the truth or falsity of a statement, but whether a given object satisfies a predicate.' I think this is misleading. The predicate version of the LEM is the traditional (Aristotelian) version. In modern logic, the contrast would be between the law, P or not-P, and the principle that every proposition P is either true or false. Now, if we consider alternatives to two-valued logic, a case can be made for the LEM where in addition to truth or falsity, P can have a third value. This would be sufficient to show that the two principles are not identical.

However, there is more to say about this: as Dummett has argued with regard to anti-realist theories of meaning, the principle of bivalence, as a semantic principle, requires justification through the appropriate theory of meaning, such as a theory of meaning in terms of truth conditions. This is an important point, which you do not cover in your essay, although you do mention Dummett's example of bravery from his paper, 'Truth'.

Why are laws of logic important, anyway? Because you can use them to prove things. The most spectacular example of this is the use of LEM in classical mathematics, which enables the proof of many theorems which cannot be proved in intuitionist mathematics. Crucially, intuitionists reject the LEM but accept the double negation of the LEM, not-not-(P or not-P). As you note at the end of your essay, if we grant the law of double negation, not-not-P |- P, then the LEM becomes equivalent to the law of non-contradiction.

In your essay, you consider three cases which are considered to pose problems for the law of excluded middle (LEM): vagueness, dispositional properties and future contingents. I shall look at each of these in turn.

Probably the most discussed account of vagueness in recent times is Timothy Williamson's theory that vagueness is a 'form of ignorance'. Take Fred, who is losing his hair. We would say, that at some indeterminate point, it is no longer possible to say whether Fred is bald or not. But Williamson would say, for the sake of preserving classical logic, there *is* an answer, a definite answer as each hair is pulled, only it is an answer we don't know.

This is a classic case of a philosopher 'saying what you've got to say' in order to protect a principle. There's no logical objection. The problem is, if this is 'ignorance' then I no longer know what it means to be 'ignorant'.

You take the case of colour predicates and offer a possible way to deal with the LEM. Colours, you say, are vague only because there are many shades of colour. If we had a name for each shade, then 'The apple is green or the apple is not-green' would always be true. Here, you say something rather strange: you claim that when we say, 'The apple is green' what we mean is that the apple is a precise shade of green, call this Gn. So what we mean is that The apple is Gn or the apple is not Gn. But that is a different claim from the claim that the apple is green or the apple is not green.

What you should have said is that, if there is a precise number of shades of green, then when we say, 'The apple is green or the apple is not green,' what we mean is that Either the apple is G1, or the apple is G2.....or the apple is Gz (the very last shade of green) OR the apple is none of the above shades. This approach can be found in theories of vagueness which involve the idea of successive 'sharpenings'. The problem is that you have made an unjustified assumption right at the start: that the shades of green can be enumerated. This is false if, as seems intuitively correct, there is always a shade of green in between any two given shades of green.

The case of dispositional properties is a good example of how the LEM can be used to prove statements whose truth is dubitable. Take Fred, who has never encountered danger. If we say, 'Fred is brave or Fred is not-brave', meaning, 'If Fred encountered danger he would act bravely or he would not', then we have to consider what it would take to make each of the conditional statements, 'If Fred encountered danger he would act bravely' or 'If Fred encountered danger he would not act bravely' TRUE. As Dummett argues in 'Truth', conditional statements cannot be 'barely true'. In conjunction with the LEM, this enables us to prove that there is in fact a state of Fred, either physical or mental, which his bravery (or the lack of it) consists in. We just don't know what that state is, but it IS there and we can prove it by the LEM!

One response would be to say that LEM is OK, but we should reject the claim that subjunctive conditionals have truth conditions. The attribution of dispositional properties is equivalent to the acceptance of certain subjunctive conditionals, but accepting a subjunctive conditional is not the same as asserting its truth. In which case, the LEM doesn't apply.

There is a way to save the LEM if one accepts Aristotle's claim about future contingents. This is in some ways analogous to what we did with the range of shades of green. The future may not be determinate, but there is nevertheless a determinate range of possible futures, those in which (e.g.) a city is built here, and those in which a city is not built here. Then the statement, 'A city will be built here or a city will not be built' here is true in every possible future. A statement which is true in every possible future is true, period.

In his article, 'The reality of the past', Dummett contrasts this approach to statements about the future, to one which rejects the idea of truth conditions, along the lines of a semantics for intuitionist logic/ mathematics. If Dummett is right, then the LEM cannot be applied to future contingents. We have to make do, as in intuitionist mathematics, with the double negation of the LEM.

I recommend that you read a bit more about the realism/ anti-realism debate, as this is an important issue bearing on the question of whether the LEM should be considered a law of logic.

All the best,

Geoffrey