Fuzzy Logic
How do you handle the ambiguities in your world? What logic do you apply, if any? Traditional logic? Fuzzy logic? Can you tell which you are using? ...
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Logic deals with matters of truth. Science does, too. So what's the difference?
Let's be more logical -- or should that be 'more scientific'? Logic deals with truth -- that's clear from the fact that 'truth' appears in most logic books. And truth involves the use of statements, or what we understand to be propositions, things we normally presume to express information and so to be true or false. (See my efforts here to understand what constitutes information.) On that basis you'd think that it's logic's job to establish the truth (or falsity) of propositions. But that's not quite the case.
Rather, the aim of logic is to establish the truth of statements relative to other statements. In other words, it is concerned with the validity of arguments, the truths that can be deduced from other truths -- not the truth of statements taken independently.
Science, too, deals with truth, but its aim is to establish the truth of statements relative to the methods of measurement, as facts. It's concerned with the validity of observation. And interpretation of the observations depends on the theoretical structure that defines it. One theory -- like Newton's -- leads to one set of observations and information, and another -- like Einstein's -- leads to another system of measurements and information, and still another -- say Quantum Mechanics -- leads to yet another system. Similar relationships occur in most other fields. I refer to the structures as repositories of information. Dependence on theory is crucial.
In this respect, both logic and science use injunction to derive specific results. By following the rules of injunction, you can obtain the same results: The results are repeatable. Do this and that, take this sequence of steps, follow these rules and directives, and you will see what happens!
Deductions
Consider what happens when we deduce consequences from axioms. We first construct axioms (statements we presume true). Then we use rules of deduction (inference) to prove theorems from the axioms. When a theorem is deduced from the axioms in accordance with the deduction rules, we say the information contained in the theorem is true. The idea is that true statements imply certain other true statements. You see the connection when you "read" the underlying pattern. It requires understanding.
In the deduction, information contained "implicitly" in the axioms is "drawn out," or made explicit by the theorem. Information present but unexpressed is given form in the theorem. (In the same way, information contained in experience, as fact, is unexpressed until a statement is formed. In the same way, too, you can know something implicitly without verbalization.) The information is drawn out by the rules by rearranging the symbols or substituting equivalent symbols. (This is called calculation, so we speak of a calculus.) The axioms are shorthand for all the information they contain (repositories).
The Syllogism
To see what deduction means, consider a familiar syllogism. Take as a basic premise the statement:
All men are mortal.
Now let's add the statement:
John is a man.
What can we deduce? Well, because we've asserted that John is a member of the class of men, and that everyone in the class is mortal, it follows that:
John is mortal.
So this is a "new" piece of information that comes out of the premises. Grasping the idea is the understanding.
Information Formatting
Information can be expressed in different ways, using different formats. This is common with math functions. For example, the simple linear function:
y = 2x + 3
holds the special class of information that relates the value of y to the value of x in a special (linear) way. Substituting a specific value of x (numerical data) into the function yields a specific value of y. For instance, if x is 2, then y is 7. We plug in the value for x, and compute the value of y. In English we get the sentence, or proposition:
The sum of three plus twice two equals seven.
The proposition informs you about a specific numerical connection between 3, 2, and 7.
That same linear function can also be expressed in a tabular format:
|
X |
... |
-3 |
-2 |
-1 |
0 |
1 |
2 |
3 |
... |
|
Y |
... |
-3 |
-1 |
1 |
3 |
5 |
7 |
9 |
... |
To draw out the information from this data table, you need to understand how tables are used and apply the underlying rule for interpreting its format -- call it the rule of tabular grammar, if you like. It says that each column in the table relates a particular value of x to a particular value of y. Given the value of x, you read down to get the value of y. For example, you can get information if x equals 1. In that case, y equals 5. (For more details of this view, see my book.)
The Law of the Excluded Middle
This kind of structural formatting, or modeling, has certain fundamental characteristics that determine the way information is defined. Just as with the method of deduction, it presumes the law of the excluded middle, which says that a (meaningful) statement is either true or false, but not both. This is characteristic of traditional logic. It renders the logic two-valued.
Consider the following simple logical function, for example:
If P then Q.
This function is false -- i.e., has the truth-value F -- whenever P is true and Q is false. Otherwise it is true -- i.e., has the truth-value T. The only possible values are T and F. Expressing this in tabular form, we get the truth table:
|
P |
Q |
If P then Q |
|
T |
T |
T |
|
T |
F |
F |
|
F |
T |
T |
|
F |
F |
T |
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The new logic turns the tables on the old logic and considers the fuzziness in things foremost. As Bart Kosko says, in Fuzzy Thinking: The New Science of Fuzzy Logic:
Some things are not fuzzy no matter how closely you look at them. These things tend to come from the world of math. Here by design man or God has kept fuzziness out of the picture. We agree that "Two plus two equals four" is 100% true. But when we move out of the artificial world of math, fuzziness reigns. It blurs borders and deadlines as if our words cut the universe into pieces with a blunt knife.
By contrast with traditional logic, fuzzy logic does not assume the law of the excluded middle. This is to say that propositions can now be both true and false. A new understanding -- a new form -- is now required.
The reason that propositions in this view can be both true and false is that truth-value is considered to be a matter of degree. That is, propositions are true to a certain degree, and false to a certain degree. In the extreme case, if a proposition happens to be completely true -- i.e., to a maximum degree -- then it can not be false in any amount -- i.e., it is false to a minimum degree. One value is the opposite of the other.
It may be helpful to compare the notion of degree with terms in other theories. In statistics, for example, we use the term 'probability.' The corresponding term in neural networks is 'weight.' In fractal theory, the word 'fractal' is used. And multi-valued logic identifies a corresponding 'range' of truth-values.
For convenience, the maximum degree is specified as 1, and the minimum degree is set at 0. This means that propositions can have truth-values anywhere between 1 and 0, so we are now dealing with so-called multi-valued logic.
As a modeling device, this logic changes the way environments of skills are perceived.
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In the real world, problems often arise because of the ambiguity of language -- because of the way we model or form our experience. Many statements that may seem perfectly clear, lose their clarity when examined closely. Propositions may be ambiguous, or have a multiplicity of meaning. These propositions are natural candidates for fuzzy treatment.
See examples discussed by Murray A Ruggiero, Rao & Rao, and Terano, et. al.
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