From: Geoffrey Klempner

Subject: Justifying the law of excluded middle

Date: 9 January 2007 12:19

Dear Catherine,

Thank you for your email of 29 December, with your University of London essay in response to the Logic 2006 paper question 'What kind of justification can be given for the law of excluded middle? Is it convincing?'

Although you clearly felt that your response to the question is inadequate, you have in fact covered the main issues (perhaps guided by the examiners' report!).

The most important point to make is that the law of excluded middle, P or not-P, is NOT the same as the law of bivalence, 'Every proposition has the value true, or the value false.'

The law of excluded middle is a law of logic. The law of bivalence describes a semantic model for a system of logic, in this case classical logic. A seminal essay to read about this is Michael Dummett's British Academy lecture 'The Justification of Deduction' (reproduced in his collection 'Truth and Other Enigmas' Duckworth).

If you haven't already done so, read up in an elementary logic text book (e.g. Guttenplan 'Languages of Logic') about truth tables and the difference between syntax and semantics. A logical system is defined by its syntax, the connectives and the rules governing them. The semantics describes the intended interpretation. Thus the 'truth table' for the connective 'v' (inclusive 'or') is as follows:

P Q P v Q

T T T

T F T

F T T

F F F

In other words 'P v Q' is only false when both P and Q are false.

Applying this to the special case of 'P v -P' (P or not-P) we get:

P P v -P

T T

F T

In this sense, the law of bivalence 'justifies' the law of excluded middle. What the essay asks is whether this justification is convincing.

One obvious objection is that the idea behind the rule, 'P or not-P' just is the idea of bivalence. However, it is not difficult to show that the two notions - excluded middle and bivalence - are in fact distinct.

Consider a statement about the future, 'CH will pass the Logic exam'. On Aristotle's view of 'future contingency' there is no determinate 'fact' about the future, in virtue of which that statement, asserted now, 'has' a determinate truth value. The future doesn't 'exist' yet. However, it remains the case that there is no middle possibility in between 'CH passes the Logic exam' and 'It is not the case that CH passes the logic exam' (as in your example from Russell, note that we are taking care to use 'not' in its primary occurrence). Every possible future history - including a future where the universe is obliterated before the date of the exam - is one in which either 'CH passes the Logic exam' is true or one in which 'It is not the case that CH passes the Logic exam' is true. (But note that we are in fact invoking the law of bivalence for each possible future history.)

The biggest challenges to the law of excluded middle, as you note, are from examples of unverifiable statements - including statements of unrestricted generality - and examples of vague statements. For verifiability, Dummett's essay 'Truth' (also in the above collection) is the place to start. Dummett uses the example of intuitionist mathematics to describe an 'anti-realist' view of the world which sees things 'coming into existence when we probe' rather than existing independently of our knowledge.

In mathematics, classical logic allows one to prove conclusions which are not provable in intuitionist logic. In his essay 'Truth' Dummett gives an example of an attempt to do a similar proof in the case of an empirical proposition. Either Jones, who died never facing a situation which required bravery, was brave or not. Therefore, there is something which existed in Jones by virtue of he possessed bravery or failed to possess bravery. I find this unconvincing. To speculate that 'It is not the case that Jones was brave' (i.e. with 'not' in its primary occurrence) leaves it open whether we are considering that Jones actually possessed an attribute of character inconsistent with bravery, or whether we are merely considering - without any extra assumptions - the absence of the attribute of bravery.

Similarly, in the case of vagueness, Sainsbury's pornography example appears to show that matters of fact can be fallaciously inferred from cases of excluded middle. But this appearance is superficial. Once more, we need to pay heed to whether 'not' is in its primary or secondary occurrence. Either Fred is an adult, or it is not the case that Fred is an adult. However, to derive the unacceptable consequence, one needs the additional premise that if it is not the case that Fred is an adult, then he is a child.

Susan Haack in her book 'Deviant Logics' describes various alternative systems of multi-valued logic which have been put forward for a variety of motives, including the perceived need for a logic which works for vague statements. However, it is not clear that any system of logic, with however many values, solves the problem of vagueness.

All the best,

Geoffrey