The principium tertii exclusi, the principle or law of the excluded middle, what is that? If there is a proposition B then it is either the case that B is true or that B is false. There is no third possibility. Either B = “you, the reader, are an American citizen” or not-B, you are not a citizen. Either B = “you have cancer” or not-B, you do not. Either B = “you are virgin” or not-B, you are not. The possibility of being just a little bit virginal does not exist.
The principle states that this matter of fact is true for all propositions. How do we know the principle holds in all cases? We don’t. We have to accept that it does axiomatically. But what if we rejected the axiom, what then? We enter the realm of intuitionism, a re-thinking of what mathematics is and what it means. According to the Stanford Encyclopedia of Philosophy:
Intuitionism is based on the idea that mathematics is a creation of the mind. The truth of a mathematical statement can only be conceived via a mental construction that proves it to be true, and the communication between mathematicians only serves as a means to create the same mental process in different minds.
The Riemann hypothesis, a non-absurd statement about the distribution of prime numbers, is either true or it is not; at least, that is so if we accept the principium. Regardless whether the principium holds, we can say that the hypothesis is not yet proved. What do we mean by that?
Well, starting from a set of simple axioms about the nature of math and some accepted-as-true rules guiding the manipulation of mathematical objects, all deduced paths—strings of statements, where each farther along on the path is true given the ones that came before—none have so far led to the hypothesis. Stated another way, given all the known streams of argument, none have allowed us to deduce the hypothesis.
Given these paths the probability that the Riemann hypothesis is true is neither 0 nor 1. These paths certainly do not say the probability is 0; i.e. that the RH is false. And neither do they say the probability is 1; i..e that the RH is true. If we accept the principium, then we can say that it is true (there is probability equal to 1) that either the RH is true or that it is false. But isn’t a rather strong statement to make, especially considering we must believe it for all propositions?
Let’s remind ourselves that logic is a matter between propositions, and that the propositions themselves are not part of logic. All knowledge is conditional. You can’t say “It is true that B” without adding the condition on which you base this claim. This is a constructivist position. It isn’t true that B = “George wears a hat” unless we construct evidence that makes this so; such as E = “All Martians wear hats and George is a Martian.” Other E are certainly possible. As are E that make B false, or give it probabilities in between 0 and 1.
We can’t say E is true or false or in between unless we offer a different constructive evidentiary proposition relevant to the question. Whether or not such evidence exists is irrelevant to the question whether B is true given E. We accept E is true. Then B is necessarily true given E. Even though, in this case, we have other evidence that suggests E is in fact false.
When we say, for example, that B = “Fermat’s last theorem is true” we imply that there is a condition E which makes it so—even if we do not know what E is. This is important. The civilian saying “B is true” does not know E; he is relying on the premise that his mathematical betters have said E exists. His argument is that experts have said E is true and that given E therefore B is true. The civilian’s argument is therefore either circular or an appeal to authority. But because there is an E that does indeed let us deduce B, this only proves that there are “forms” of fallacies that give true results (this is another argument which David Stove gave).
This is different when we ask what is the probability that the RH is true. Here, we have a jumble of evidence: the beauty of the hypothesis, that many of the consequences of the RH are themselves useful and wide ranging and that other theorems once unproved (but now proved) shared the similar property that its consequences were useful and wide ranging and so on; not all of this is made articulate. Given this evidence we can say that the probability that the RH is true is high. But we’re hard pressed to deduce a quantification for this probability.
What then is the unconditional (intuitionist) probability that the RH is true? There isn’t one. There is no unconditional validity, invalidity, or probability of any argument, not just this one. If our only evidence is that the principium is true, then we begin our argument with a tautology and prefixing any tautology to any argument does not change the validity, invalidity, or probability of its conclusion. If our only evidence is that the principium is false, then we have nothing and can go nowhere. The RH neither follows nor doesn’t follow from knowledge that the principium is true or false. We have constructed nothing so no probability exists.
Finally, the intuitionist turns the question around and asked what the probability the principium itself is true. Given what evidence? is the question we must ask. Since for any proposition B we do not have constructive evidence that either B or not-B, we cannot claim that the principium is always true.