So, finally, I’m getting around to three-valued logic!
To start, why are we even looking at three-valued logic? We want to get to fuzzy reasoning, and three valued logic really doesn’t do the job. But it’s a useful starting point, because in general, most of us have only worked with standard logics where statements are either true or false. In Fuzzy logic, we’re don’t have that – and that can lead to a lot of complexity. So looking at something simple that shows us how something either than “true” or “false” can actually make sense in logic.
So, as I said in the intro to the new Scientopical Good Math/Bad Math, I’m
going to writing some articles about fuzzy logic.
Before diving into the fuzzy stuff, let’s start at the beginning. What
is fuzzy logic? More fundamentally, what is logic? And what problem does fuzziness solve?
So, first, what’s logic?
Logic isn’t really a single thing. There are many different logics, for different purposes. But at the core, all logics have the same basic underlying concept: a logic is a symbolic reasoning system. A logic is a system for expressing statements in a form where you reason about them in a completely mechanical way, without even knowing what they mean. If you know that some collection of statements is true, using the logic, you can infer – meaning derive from a sequence of purely mechanical steps – whether or not other statements are true. If the logic is formulated properly, and you started with an initial set of true statements, then any conclusions drawn using the inference process from those statements will also be true.
When we say logic, we tend to automatically think of a particular logic: the first order predicate logic, which is the most common, fundamental logic used in both mathematics, and in rhetoric and debate. But there are an astonishing number of different logics for different purposes. Fuzzy logic is one particular
variation, which tries to provide a way of reasoning about vagueness.