Reasoning and Fuzzy Logic - Fuzzy Logic: An Introductory Course for Engineering Students - page 82

Chapter 3
Reasoning and Fuzzy Logic

3.1 What Does It Mean “Logic”?

The question is, in fact, a philosophical one whose discussion does not correspond
to this text, and that received a lot of comments and discussions by philosophers.
Instead of such question, there is the more particular,what is alogic ?, that can be
answered not philosophicaly but in terms of the mathematical definition of what is a
consequence's operator. A definition that corresponds to an abstraction of the term
“deduction”.

3.1.1 Logic and Consequence Operators

A logic is a triplet ( X , A, C ) , where A ↂ P ( X ) is a family of parts of X , and

C : A ₒ A is a mapping verifying

1. For all P ∈ A, P ↂ C ( P ) .

2. If P , Q ∈ A , and P ↂ Q , then C ( P ) ↂ C ( Q ) .

3. For all P ∈ A, C ( C ( P )) ↂ C ( P ) .

It can be said that the pair (A, C ) defines a logic in the set X .

From 1. and 2., it follows C ( P ) ↂ C ( C ( P )) , and from 3. results

3 . For all P or C 2

∈ A, ( ( )) = ( ), =

C .
Hence, a consequence's operator is one that is extensive (1.), monotonic (2.), and a
closure (3 .). These operators were introduced by the logician Alfred Tarski in the
thirties of 20th century, and are called to be compact provided for all P

C C P C P

∈ A ,itexists

a finite set { p 1 ,..., p n }ↂ P such that

4. C ( P ) = C { p 1 ,..., p n } .