Friday, March 18, 2011

Fuzzy Logical Proofs

Many decent EE Professors will tell you that Fuzzy Logic is used quite a bit in fuzzy control. For those of you who don't know what that is, congratulations; you're a member of normal society. We will not dwell on that here. Instead, we will focus these Fuzzy Logic energies into logical proofs.

For those logicians out there familiar with normal two-column proofs, such as fitch-style natural deduction proofs or similiar, you will find this somewhat familiar. The difference between fuzzy proofs and conventional proofs is that fuzzy proofs provides us a mechanism to deal with uncertainty.

In conventional First Order Logic, we may for any given statement assign a truth value out of the set of two truth values, True and False. In Fuzzy Logic, there is a "spectrum of truth" wherein a statement can be "20% True" and "80% False". Another (more intuitionistic) interpretation of such a value in terms of using it to form a proof is that we are 20% certain that value A is true. Thus fuzzy proofs allow us to use premises with varying degrees of certainty, and in a chosen interpretation allow us to prove results, perhaps even with higher degrees of certainty than our premises. (Side note: those of you familiar with Zadeh's work may cringe at this, but I will explain what I mean when we get there.)

This post starts a series of blog posts which I will publish on the subject of Fuzzy Proofs as I understand them. Please feel free to make comments and give criticisms for my arguments.

In the first post we will define what "fuzzy logic" herein means, and explore the possibilities of the different logical systems at our disposal, and why we may or may not choose them for our proof or to solve a specific problem. Feedback will be requested here. This is a domain which I am trying to understand better even now.

In the second post we will define what an "interpretation" is, or in other words, we will define how to translate our mathy proof back into english so that it may be used to make decisions, as all useful proofs must ultimately provide use for this end.

In the last post, we will walk through a sample fuzzy logic proof. I will comment on its usefulness, its form, and its weaknesses.

From what I can understand, this field has only been explored by few scholars, most deeming it of little use. But I find it relevant here to examine G. H. Hardy, who said "I have never done anything useful," thinking number theory to be an unimportant part of mathematics. Without him, RSA would not be where it is today. (re-quoted from Algorithms, by S. Dasgupta, C.H. Papadimitriou, and U.V. Vazirani, p. 41.)