Topics Logic

Question Re: a third state. Sophists seem to be concerned with two things: being and nonbeing. Mathematics is based on this very same concept (the law of excluded middle): p or Non-p. What about a third state? How could we construct a logical system that would have a third state? I was told, and told again that the Law of excluded middle works fine and we should be content. Why not explore a system with more than 2 states, why not 3 or more than 3 states? I look forward to hearing from you. Ben V.

Accepted:
March 12, 2009

Comments

There is in fact a tradition of 'constructivist' mathematics which does not endorse the law of excluded middle. Very roughly, suppose you think that mathematical truth consists in the possibility of a constructing a proof. Then there is no reason to suppose that, inevitably, either P or not-P -- because there is no reason to suppose that (for each and every P) there is a possibility of proving P or is a possibility of disproving P. To find out more about this, you can read (at least the opening couple of sections of) this article. Note, however, that refusing to endorse the law of excluded middle for mathematical propositions is not to deny the law, nor is it to assert that there is a "third state" between truth and falsehood.

Still, for certain other purposes, it can be useful to explore three-valued logical systems (even multi-valued logics), which allow more "states" that the classical true/false pair. From the same source, here's another article -- note the section on "applications".

I'm afraid both articles become quite technical quite quickly. But that's in the nature of the case: as soon as we move on from very vague motivating thoughts and try to develop a non-classical logical system, we'll need worked out details if we are to assess the attractiveness and utility of the proposals.