Why not use Ternary Logic?
+1 on
-1 off
0 on and off.

epepke wrote:

cavarka9 wrote:Which brings me to another question, most of our information theory I believe emerges from the zero and one. does multivalued logic fundamentally changes the information theory ?.

Honestly, no, it doesn't, at least not theoretically. To an arbitrary precision, you can do everything with bits.

However, there are also practical considerations. A mathematical argument has to be not only accurate but understandable. You can model all of mathematics with second-order predicate language or a Turing machine, and you can do a hell of a lot with Peano arithmetic. Nobody does, though, because it's just too damn hard to write down all the steps and figure out what they're doing on a large scale.

Oh quite a lot of us do, including Microsoft Research, who have a team currently working towards formalising the classification of all finite simple groups.

You need a computer to figure out the trivial proofs for you, decision procedures for various fragments of logic and arithmetic. It's still laborious, but feasible. And smart engineering guarantees that anything the computer verifies is something that boils down to a small number of inference rules for an appropriate logic.

As Richard Feynman pointed out, mathematics is largely the quest for better notation.

And as we point out, it's all about representation. You depart from standard representations pretty damn quickly once you realise how computationally ineffective they are. For a simple example, consider the fact that the size of numerals in Peano Arithmetic grows linearly with the number represented. Ugh. You move to binary pretty sharpish.

It's not initially obvious, but there's generally no sense asking whether some feature of the natural world or some feature of natural language implies that we should ditch classical logic. Logics quickly become so powerful that they are universal. Each can do the work of the other. As epepke points out, you can get pretty much all of classical mathematics in a relatively simple logic, and once you have classical mathematics, you have every other logic for free.

You never need fuzzy logic. Rather than using propositions which take on real values between 0 and 1, you could talk directly about real values between 0 and 1. Rather than defining conjunction as a maximum function, just talk directly about the maximum function. The definition of fuzzy logic is a definition from classical mathematics. So why not cut out the middle man, stay in classical logic, and just talk about its defining mathematics directly?

It's partly about representation. Some logics may be more succinct, cleaner or more intuitive when it comes to writing stuff down. But then the question of whether to pick one logic over another is about as profound as the question of whether to pick one programming language over another.