Mathematical intuitionists champion the idea that the Law of Excluded Middle is not, strictly speaking, true. Indeed, because of the problems which arise when using ‘infinities’ in calculations one finds that there are mathematical problems where contradictions necessarily result. Since

Because of such examples some have proposed that our intuition in this matter is wrong. It comes, so it is suggested, from a Platonic view of things; a view of propositions which requires propositions to be either true or false. They simply reject this view, and suggest that such contradictions demonstrate a problem with our logical intuition.

Most philosophers agree with St. Thomas, however, that

“it will be impossible to carry on a dispute with anyone who maintains this… because he equally affirms and denies anything at all. And if this man takes nothing to be definitely true, and similarly thinks and does not think, just as he similarly affirms and denies something in speech, he seems to differ in no way from plants”

~St. Thomas Aquinas, Commentary on Aristotle’s Metaphysics, Book 4 (p.240)

The conviction, then, is that nothing can be both true and false in exactly the same sense at exactly the same time. Technically the Law of Excluded middle and the law of non-contradiction need to be distinguished, though they are often related. Dr. James Anderson defines the terms as follows:

• Law of Non-Contradiction: that no statement can be both true and false

• Law of Excluded Middle: that every statement must be either true or false

In any case, we can imagine for the sake of argument that one goes hand in hand with the other.

However, these Mathematical Intuitionists have suggested that truth is being misconstrued. They recommend that we define truth as simply demonstrability. That is to say, that whatever can be demonstrated ‘is true’. So that when we have in mind a proposition for which we are wondering about it’s correct ‘truth-value assignment’ there is perhaps no such correct ‘truth-value assignment’ as of yet, but that there will be when it is demonstrated. Thus, to demonstrate something is not to ‘show’ it to be true, but it is that in virtue of which something can be said intelligibly to be ‘true’.

Most people intuitively reject this suggestion because they take seriously the rational conviction that there is a ‘truth-of-the-matter’ metaphysically.

Interestingly, Naturalized Epistemology seems to push inevitably towards the same kind of conclusion, and it disregards both this rational intuition, and perhaps metaphysics in general, at least as it is classically understood. Is there really a ‘fact of the matter’ out there in some world which is mind-independent? Even, perhaps, ‘mind-external’? What would that really mean – for something to be ‘true’ without reference to a cognitive language game?

Moreover, one of the interesting features of Quine’s Naturalized Epistemology is the suggestion that there is no rational intuition, no proposition, which cannot in principle be abandoned; no necessary *a priori.* There are only propositions which we hold on to more violently than others because they are more ‘basic’ for us. Mathematical propositions, for instance, can be abandoned, but not as easily as, say, psychological propositions. Thus all ‘*a priori*‘ truths, such as “A v ~A”, can be in principle abandoned. Therefore it seems that the suggestion of the Mathematical Intuitionists is not unacceptable to Quine and those of his persuasion.

Interestingly, I was reading excerpts from an Anthology of Epistemology recently and I read one philosopher, Hilary Putnam offer a criticism. He said that there is at least one proposition which must be *a priori* even for a Naturalized Epistemologist. A proposition which cannot be abandoned in principle. The proposition is:

“Not every statement is both true and false”

~Hilary Putnam, Epistemology: an Anthology, p.585-594

This, he suggests, cannot ever be rationally abandoned. This is what we might call a Weak Law of Excluded Middle (WLEM). I think the WLEM is, as Putnam suggests, an example of at least one proposition without the assumption of which man can absolutely not think. However, if this is true, then it demonstrates that there is at least one *a priori* principle which cannot ever be rejected, and which no Mathematical Intuitionist or Quine-ian philosopher can dismiss.

It also helps those who wish to reject the LEM along with the Law of non-contradiction escape from the retort: “well, if it is true that the LEM and LNC are false, then couldn’t it be (indeed, * isn’t* it) also simultaneously false that the LEM and LNC are false?”

This is a very excellent article. I also have some ideas regarding this precise problem and a very novel solution to it which you should find most useful. It involves applying the concept of equivalence. You have Law of Excluded Middle. You then construct Law of Exclusive Middle. You let these be equivalent. Quantitatively this makes perfect sense because as you know .999… = 1, and there are many other relationships to explore. I have several videos on this, and much more material. Please contact me for more information if interested. Thank you.