## Archive for December 31st, 2022

### Fuzzy Logic? Truth or false statements? What about a third option that we need to decide on a proposition?

Posted on: December 31, 2022

In mathematics, we generally use Classical or Standard logic, known as Aristotle’s Logic, which presents us with only two options for defining the truth value of propositions.

This mathematical tool is still beneficial in producing solutions for many problems in math and real life.

In classical logic, to define the correct value for a proposition, we use only ‘1’ and ‘0’, which respectively means that it is either true or false.

For example:

The truth value of “1 + 1 = 2” is true, which we can denote as T.

The truth value of “11 is not an odd number” is false, we can denote it as F.

On the other side, this traditional perspective of classical logic limited our ability to produce solutions to many problems.

Science continued to improve itself.

Eventually, for the first time, L.A.Zadeh, a scientist from Azerbaijan, introduced this new idea (1965). He said that propositions do not have to just have two value sides. Our brains and our decisions do not work like this.

This example may be beneficial to understand this idea further.

Imagine that you and your mathematician friend are in the grocery. You decided to buy three sets of apples. Here we are given three options: green apples, red apples and mixed colour apples, which are mixed with green and red apples.

You bought a sample of all of them. Later you came home, and your mathematician friend offered to arrange these apples in green, red and mixed. You started to play a game. The game is about choosing apples.

You are the first player and you have to select the red coloured ones. You can’t choose one fully coloured red apple as mixed colours apples have a red colour, too. If you don’t have this one, you will fail the game. But if you choose a mixed coloured apple, you will fail again because a mixed coloured apple has an ingredient green colour.

If you had realized some linguistic confusion, you could understand the idea easily.

Players are making two mistakes.

1) If you don’t want to fail the game, you should know what colour is exactly wanted from you. Only red or green?

2) Players must know what is the degree of mixed colours.

What is the degree of red colour in the mixed-coloured apple? What is the degree of green colour in the mixed-coloured apple?

This example takes us to this idea:

If the fully coloured red apple’s correctness is ‘1’ and the fully coloured green apple’s correctness is ‘0’, what is the correctness of the mixed coloured apple?

Maybe it has 0.1 red colours and 0.9 green colours.

Maybe it has 0.26 red colours and 0.74 green colours.

.
We can continue to do this until we find a convenient degree.

And if we want it, we can continue to do this until we find the conclusion below:

It has ‘0’ red colour and ‘1’ green colour.

It has ‘1’ red colour and ‘0’ green colour.

Both of them are true.

Their only difference is that when you work with classical logic, a proposition’s correctness is just true or false. However, when you work with fuzzy logic, you will have a proposition’s correctness degree inside a closed interval from 0 to 1.

But, classical logic and fuzzy logic have different types of perspectives for looking at propositions.

Fuzzy logic gives more options and elasticity when working on real-life’s problems.

If our proposition’s correctness is 0, we say exactly wrong, like in classical logic. If our proposition’s correctness is 1, we say exactly correct, like in classical logic. If our proposition’s correctness is 0.5 and incorrectness 0.5, we say it is neither true nor wrong.

In our example, we can say that it has some red colour pieces and some green colour pieces, but the apple is neither red nor green.

Fuzzy logic is common in many areas of science due to its capability to measure the correctness degree of sensitivity.

In math, for instance, it helps us in solutions to decision-making problems. It is slightly similar to our feelings, decisions, emotions, etc.

Like measuring life.

When I write these notes, I am inspired by this article :

KAVDIR, İSMAİL and GUYER, DANIEL E. (2003) “Apple Grading Using Fuzzy Logic,” Turkish Journal of Agriculture and Forestry: Vol. 27: №6, Article 8.

Note: Most progress in Artificial Intelligence is based on Fuzzy Logic.

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