many-valued logic

In standard logic there are only two truth values: 0 and 1. Here 0 stands for "False" and 1 stands for "True".

In quantum mechanics, due to the prevailing uncertainty, the truthfulness of most statements are simply unknown. In computer science, not all programs terminate for every input. In other words, viewed as functions, programs are undefined at some points of their domain. Hence, again some questions regarding the behaviour of your program, may have no answer. In other words, some statements will be neither True nor False.

Among many other motivations, these are the two reasons why a need arose for a three-value logic whose truth values consist of: 0, 1/2, 1. Here 1/2 stands for "Undefined".

Now it takes a small leap of imagination to jump into a more generalized logical system which has uncountably many truth values: [0,1] namely all the real numbers between 0 and 1. How should these varying degrees of truth be interpreted?

One interpretation is as follows. If the truth value of a statement P is greater than 1/2, then P is more likely to be true than false. In other words, proximity to 0 or 1 indicate proximity to "being false" or "being true".

What is the problem with this interpretation?

Say you have done an experiment, but the results are somewhat conflicting and therefore do not allow you to conclude whether P is true or false. Nevertheless you claim that P is more likely to be true than false. If someone asks you why, you say that "there is more evidence in favour of the truth of P."

But what does it mean to have "more" evidence for something? How do you measure evidence? There is obviously also some evidence for ~P. (Otherwise you would have concluded that P is true.) What makes this evidence less worthwhile? Of course, the link between "evidence for P" and P, and the link between "evidence for ~P" and ~P have to be tenuous. (Otherwise the totality of all evidences is inconsistent.) Hence one inevitably has to look at the contents of these argumentative links and decide which one has more strength. These measurements and decisions are obviously all subjective. Moreover, with the arrival of new evidences you may change your opinion regarding the truth value of P.

If the state of affairs is that dynamic and vague, why assign a truth value to P at all? The truth value should not be a malleable quantity. (The word itself has static underpinnings.) Until a decisive experiment is made, it should simply not exist (or be undefined so to say). Here is an exemplary attitude by an experimental physicist:

How has the universe’s expansion, and hence the influence of dark energy, changed since the Big Bang?

For cosmologists there’s this interesting moment in the very, very early universe—10–35 seconds or so after the Big Bang—called the inflationary period. Inflation was another period of acceleration, and we don’t know what caused that acceleration, either. It’s possible that there was another kind of dark energy back then. After inflation there was so much mass so close together that gravity dominated and the expansion slowed. That lasted until about halfway through the life of the universe. It was some 7 billion years before the universe expanded to the point where matter was too scattered to keep the expansion slowing. At that point, dark energy’s power started to be felt and the universe started to accelerate again.

What does this discovery mean for the fate of the universe? Will dark energy ever let up?

Well, you can just take the naive approach of saying that the universe is accelerating now, so that means it will accelerate forever and lead to a very dark, empty, cold end, and that’s all we have to look forward to. However, we should remember that we don’t know what’s causing the current acceleration, and we don’t know what caused that acceleration during inflation at the very beginning of the universe. That inflation turned around—it stopped and the universe started to decelerate. Who knows whether we’re seeing something now that might also decay away, and then the universe could collapse. So I would say that the fate of the universe has to remain in the category of unknown until we have any clue as to why it is currently accelerating.

(Discover Interview with Saul Perlmutter)

There is also something disturbing with the plurality of logics. How do you select which one is more appropriate for a given decision process? Intuitionistic logic, classical logic, three-valued logic or infinite-valued logic? (There are many more.) Making such a selection necessitates a higher level decision process. In other words the question becomes "Using which logic do you decide which logic is appropriate for the given decision process?" Of course, making this higher level selection necessitates yet another higher level decision process... Ad infinitum. In order to avoid this ad infinitum, you somehow need to reduce the number of legitimate logics available. You do not need to reduce the number to one. You only have to ensure that, at some point in the chain of decision processes, the choice of logic (from the reduced set) will make no difference. (i.e. It will be unnecessary to go up another level.)