Fuzzy Propositions :-

Fuzzy propositions are assigned to fuzzy sets. Suppose a fuzzy proposition ‘P’ is assigned to a fuzzy set ‘A’, then the truth value of the proposition is proposed by T (P) = μ_A(x) where 0 ≤ μ_A(x) ≤ 1

Therefore truthness of a proposition P is membership value of x in fuzzy set A.

The logical connectives like disjunction, conjunction, negation and implication are also defined on fuzzy propositions.

Let, a fuzzy proposition ‘P’ is defined on a fuzzy set A
Q is defined on fuzzy set B

Conjunction

P / Q : x is A and B

T( P / Q) = Min [ T(P), T(Q)]

Negation

T(P^c) = 1 – T(P)

Disjunction

P V Q : x in A or B

T (P V Q) = Max [ T(P), T(Q) ]

Implication

P → Q : x is A then x is B

T( P → Q ) = T (P^c V Q) = Max [ T(P^c, T(Q)]

If P is a proposition defined on set A on universe of discourse X and
Q is another proposition defined on set B on universe of discourse Y,
then the implication P → Q can be represented by the relation R

R = ( A X B) U (A^c X Y) = If A then B

If x ∈ A, where x ∈ X and A ⊂ X
then y ∈ B, where y ∈ Y and B ⊂ Y

Implication of Classical Logic:-

Properties P and Q are given by
P : x ∈ A, where A is defined on x.
Q : y ∈ B, where B is defined on y.

Then the implication P → Q is represented in set theoretic form by a relation R as

R = (A X B) U (A^c X Y)

The implication is equivalent to linguistic rule form, if x ∈ A then y ∈ B.

For the classical predicate logical rule, (P → Q) V (P^c → S) the linguistic rule form is,
if x is A then y is B, else y is C.
Where C is defined as
S : y is C , C ⊂ Y.

The above linguistic rule form is decomposed as if (x is A) then (y is B) or if (x is A) then (y is not B)

In set theoretic form it can be represented by the relation R = (A X B) U (A^c X C)

The characteristic function for above compound proposition is given by

For example, suppose we have two universe of discourse X and Y
X = {1, 2, 3, 4}
Y = {1, 2, 3, 4, 5, 6}

X is the universe of normalized temperatures.
Y is the universe of normalized pressures.

Two crisp sets A and B defined on universe of discourses X and Y are A={2, 3} ; B={3,4}

for the deductive inference if A and B, find the relational matrix R.

sets A and B in Zedah’s notations are given by
A = {0/1 + 1/2 + 1/3 + 0/4}
B = {0/1 + 0/2 + 1/3 + 1/4 + 0/5 + 0/6}

for the deductive inference if A then B, the set theoretic form is given by the relation
R = (A X B) U (A^c X Y)