New Method of Fuzzy Conditional Inference and Reasoning

ABSTRACT We consider fuzzy inference of the form ‘if ··· then ··· ’. ‘if ··· then ··· else ···’ and ‘if ··· and/or ··· and/or ··· then ···’. Muzomoto proposed logical constructs for ‘if. then.’ by applying Godelien and Standard Sequence methods. These method do not satisfy all the intuitions In this paper, We propose method of fuzzy inference and applied on logical constructs. We have shown that our fuzzy inference method satisfy all the intuitions under several criterions.


Introduction
Classical logics unable to provide a methods to reasoning uncertainty, vague, incomplete or imprecise propositions. Fuzzy logic is capable of reasoning such inexact propositions [1,2] Zadeh [3] and Mamdani [4] proposed methods for fuzzy conditional propositions of the form 'if ··· then ··· '. Fukami et al. [5] developed logical constructs for fuzzy implications using Standard Sequence and Godelien Sequence Methods. The Mizumoto shown that the Zadeh [1] and Mamdani [2] methods do not fit Muzimoto logical constructs. Muzimoto logical constructs are satisfy some intuitions using Standard Sequence and Godelien Sequence Methods. We apply our method on these logical constructs and satisfy all the intuition for 'if ··· then ··· ', 'if ··· then ··· else ··· ' and 'if.. and/or.. then ··· ' propositions. We show four method which fit all types of fuzzy conditional inferences.
Fukami et al. [3]. Consider the following types of inference for these propositions.
If x is red or x is ripe and x is big then x is taste x is red or x is ripe and x is very big -----y is ?

Fuzzy Conditional Inference
A fuzzy set P is define by its characteristic function μ P (x)/x, x X, where x is individual and Xis universe of discourse.
The fuzzy conditional propositions are of the form 'if (precedent part) then (consequent part) '.
Consider the proposition of type 'if x is P then y is Q' [2] fuzzy conditional inference of type 'if x is P then y is Q' is given by [2] fuzzy conditional inference of type 'if x is P' then y is R' is given by Logical system of Godelian sequence G method is given by

Fuzzy plausibility
Plausibility theory will perform inconsistent information into consistent.
The inference is given using generalization and specialization Consider fuzzy conditional inference Fuzzy conditional inference for Type-1 is given by Fuzzy conditional inference for Type-2 is given by The fuzzy inference is given by using generalization and specialization

New Methods of Fuzzy Conditional Inference
The fuzzy conditional propositions is of the form 'if (precedent part) then (consequent part)'. Mamdani [2] fuzzy conditional inference given by P → Q == {P × Q}. The consequent part is derived from precedent part for fuzzy conditional inference [8,9]. if x is P then y is Q= P μ Q (y)= μ P (x), i.e. Q ⊆ P and P ⊆ Q Consider fuzzy quantifiers P α and Q α P α ⊆ Q Q α ⊆ P The fuzzy conditional inference is given by i.e. R ⊆ P and P ⊆ R Consider fuzzy quantifiers P α and R α P α ⊆ R R α ⊆ P } From the Type-1, Type-2 and Type-3, we have Criterion-1 and Criterion-2

Verification of Fuzzy Criterion-1
The fuzzy intuitions are give by Table 1 .
Verification of fuzzy intuitions x is very P y is very Q II-2 y is very Q x is very P III-1 x is more or less P yis more or less Q III-2 y is more or less Q xis more or less P IV-1 x is not P y is not Q IV-2 y is not Q x is not P

In the Case of Intuition II-
=y is very Q Intuition II-1 satisfied.

In the Case of Intuition III-2
=x is more or less P Intuition III-2 satisfied.

In the Case of Intuition IV-2
x is more or less P' y is more or less R III'-2 y is More or less R xis more or less P' Intuition IV-2 satisfied. Criteria-1 is satisfies I-1,I-2, II-1, II-2, III-1 and III-2, IV-1, IV-2.

Verification of Fuzzy Criterion-2
The fuzzy intuitions are give by Table 2 .

In the Case of Intuition II'-1
Using specialization = μ veryR (y) =y is very R Intuition II'-1 satisfied.
The fuzzy conditional inferences of types may proved similar lines

Conclusion
We introduced fuzzy natural deduction as an easily proof procedure. The fuzzy intuitions are studied for fuzzy conditional inference with our method for the proposition containing 'if ··· then ··· ', 'if ··· and/or ··· and/or then ··· ', and 'if ··· then ··· else ··· '. All the Fuzzy intuitions are satisfied with our method.

Disclosure statement
No potential conflict of interest was reported by the author(s).