Elementary reading about why we might doubt Excluded Middle?

If, by some chance, you were writing a couple of pages of recommendations for “Further reading” for an elementary logic book, and wanted an entry on constructivist doubts about the Law of Excluded Middle, what would you choose?

Ideally something short, accessible (in the sense of being easy to understand for a near beginner), accessible (in the sense of being available online or in most university libraries). Bell/DeVidi/Solomon, Logical Options, pp. 193–195 is, well, a brief option. But any alternatives come to mind?

[Added] Chap 47 of the Open Logic Text is good.

4 thoughts on “Elementary reading about why we might doubt Excluded Middle?”

  1. David Makinson

    To understand and discuss arguments in praise of constructive proofs, students should begin by developing an appreciation of what they are. The best way of doing that is by looking at some striking examples. But as the great majority of everyday proofs are constructive, what we particularly need are good examples of non-constructive proof. I am not thinking merely of instances of intuitionistically inacceptable steps like double negation elimination, wrong-way contraposition, wrong-way reductio ad absurdum, and assertion of excluded middle, but examples of (1) proofs of existential statements (without hidden free variables) that do not supply a witness, and (2) proofs of disjunctive statements (again, without implicit free variables) that do not supply a disjunct. They should be meaningful in ordinary mathematics and accessible to anyone with a basic high-school maths background. Unfortunately, they are in short supply! The standard example in the textbooks is a short proof that there are irrational numbers x, y such that x to the power y is rational. Presumably, one could also articulate some simple applications of the axiom of choice that would make sense to students who have not yet heard of that axiom. And after proving the completeness of an axiomatization of classical propositional logic using the maxiset method, one can profitably discuss the ways in which that proof fails to be fully constructive. But we need more examples, and preferably before the completeness proof is carried out! It would be a great service if someone could build up a compendium of a dozen examples, preferably quite different from each other, for use in teaching.

  2. How about Chapter 8 (‘Whose line is it anyway? The constructivist challenge’) of Stephen Read’s lovely book ‘Thinking about logic’. I’d say this was reasonably accessible to someone who has a done first year logic and philosophy. And of course it is very clear. The example of a non-constructuve proof that he gives is the proof of König’s Lemma.

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