**Fuzzy Logic**

Not everything is so simple as true or false. Even declarative statements may evaluate outside **{0,1}**. So let’s introduce the **kind-of**: truth ∈ **[0,1]**.

*Examples of non-binary declarative statements:*

- Shooting trap, my bullet nicked the
**clay pigeon**but didn’t smash it. I**30%-hit**the mark. - I’m not exactly a
**vegetarian**. I purposely eat**⅔**of my meals without meat, but — like yogini Sadie Nardini — I feel weak if I go**100%**vegetarian. So I’m**⅔**contributing to the social cause of non-animal-eating, and I’m a**⅔ vegetarian**. - I’m
**sixteen years old**. Am I a child, or an adult? Well, I don’t have a career or a mortgage, but I do have a serious boyfriend. This one is going to be hard to assign a single number as a percentage.

So that’s the **motivation** for Fuzzy Logic. It sounds compelling. But the academic field of fuzzy logic seems to have achieved not-very-much, although there are practical applications. Hopefully it’s just not-very-much-yet (Steven Vickers and Ulrich Höhle have two interesting-looking papers I want to read).

I see three problems which a Sensible Fuzzy Logic must overcome:

**Implication.**Classical logic (“the propositional calculus”) uses a screwed up version of “If A, then B”. It equates “if” to “Either not A, or else B is true, or else both.”Fuzzy logic

**inherits**this problem — but also lacks one clear, convincing “t-norm”, which is the fuzzy logic word for fuzzy implication.*Can you come up with a sensible rule for how this should work?*:- A implies B, and A is
**70% true**.*How true is B?* - Furthermore, should there be different numbers attached to “implies” ? Should we have “strongly implies” and “weakly implies” or “strongly implies if Antecedent is above
**70%**and does not imply at all otherwise” ?

You can see where I’m going here. There is an ℵ

_{2}of choices for the number of possible curves / distributions which could be used to define “A implies B”.- A implies B, and A is
**Too specific.**Fuzzy logic uses real numbers, which include transcendental numbers, which are crazy. Bart Kosko’s book explains FL with familiar two-digit percentages, which are for the most part intuitive. So I can accept that something might be**79%**true — but what does it mean for something to be**π/4 %**true? Or**e^e^π^e / 22222222222 %**true?We’re

**encumbering**the theory with all of these unneeded, unintuitive numbers.**One-dimensional.**For all of the**space, breadth, depth, and spaceship adventures**contained in the interval [0,1], it’s still quite limited in terms of the directions it can go. That is [0,1] comprises a total order with an implied norm. Again, why assume distance exists and why assume unidimensionality, if you don’t actually mean to. There are alternatives.**Unidimensionality excludes**survey answers like- N/A
- I don’t know
- Sort of
- Yes and no
- It’s hard to say
- I’m in a delicate superposition

, — or rather it maps effectively different answers onto the same number.

- Sometimes things are
good and bad;*both* - sometimes they are neither good nor bad;
- sometimes things are not up for evaluation;
- sometimes a generalised function (
**distribution**) expresses the membership better than a single number; - sometimes the ideas are
**topologically**related or order related but not necessarily distance related; - sometimes an incomplete
**lattice**might be best.

So those are my **gripes** with fuzzy logic. At the same time, Kosko’s book was my introduction to an interesting, **new way of thinking**. It definitely set my mind spinning. For the logical mind that wants a **rigorous framework** for understanding ambiguity, vagueness, and gray areas, fuzzy logic is a good start.