The Scientific World Journal

Volume 2015 (2015), Article ID 327390, 12 pages

http://dx.doi.org/10.1155/2015/327390

## A Modern Syllogistic Method in Intuitionistic Fuzzy Logic with Realistic Tautology

^{1}Department of Electrical and Computer Engineering, Faculty of Engineering, King Abdulaziz University, P.O. Box 80204, Jeddah 21589, Saudi Arabia^{2}Department of Biomedical and Systems Engineering, Cairo University, Giza 12613, Egypt

Received 9 May 2015; Accepted 27 July 2015

Academic Editor: Guilong Liu

Copyright © 2015 Ali Muhammad Rushdi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The Modern Syllogistic Method (MSM) of propositional logic ferrets out from a set of premises *all* that can be concluded from it in the most compact form. The MSM combines the premises into a single function equated to 1 and then produces the complete product of this function. Two fuzzy versions of MSM are developed in Ordinary Fuzzy Logic (OFL) and in Intuitionistic Fuzzy Logic (IFL) with these logics augmented by the concept of Realistic Fuzzy Tautology (RFT) which is a variable whose truth exceeds 0.5. The paper formally proves each of the steps needed in the conversion of the ordinary MSM into a fuzzy one. The proofs rely mainly on the successful replacement of logic 1 (or ordinary tautology) by an RFT. An improved version of Blake-Tison algorithm for generating the complete product of a logical function is also presented and shown to be applicable to both crisp and fuzzy versions of the MSM. The fuzzy MSM methodology is illustrated by three specific examples, which delineate differences with the crisp MSM, address the question of validity values of consequences, tackle the problem of inconsistency when it arises, and demonstrate the utility of the concept of Realistic Fuzzy Tautology.

#### 1. Introduction

Fuzzy deductive reasoning has typically relied on a fuzzification of the Resolution Principle of Robinson [1] in first-order predicate calculus. This principle uses a set of premises to prove the validity of a single clause or consequent at a time via the refutation (REDUCTIO AD ABSURDUM) method. Lee [2] proved that a set of clauses is unsatisfiable in fuzzy logic if and only if it is unsatisfiable in two-valued logic. He also proved that if the least truthful clause of a set of clauses has a truth value , then all the logical consequents obtained by repeatedly applying the resolution principle have truth values that are never less than . Later, the so-called Mukaidono Fuzzy Resolution Principle, developed by a group of Japanese researchers [3–6], was used to establish a powerful fuzzy Prolog system. The introduction of this principle involved several new concepts, including that of the contradictory degree of a contradiction whose truth value equals the truth value of the contradiction itself. Recently, a new fuzzy resolution principle was introduced in [7–9], wherein refutation is achieved by the* antonym* not by negation, and reasoning is made more flexible thanks to the existence of a meaningless range, which is a special set that is not true and also not false. Other notable work on various aspects and techniques of fuzzy reasoning and inference is available in [10–23].

The purpose of this paper is to implement fuzzy deductive reasoning via fuzzification of a powerful deductive technique of propositional logic, called the Modern Syllogistic Method (MSM). This method was originally formulated by Blake [24], expounded by Brown [25], and further described or enhanced in [26–33] and has a striking similarity with the resolution-based techniques of predicate logic [1, 34, 35].

The MSM has the distinct advantage that it ferrets out from a set of premises* all* that can be concluded from it, with the resulting conclusions cast in the* simplest* or most compact form. The MSM uses just a single rule of inference, rather than the many rules of inference conventionally employed in propositional-logic deduction (see, e.g., [36, 37]). In fact, the MSM includes all such rules of inference as special cases [30, 31]. The MSM strategy is to convert the set of premises into a single equation of the form or and obtain = the complete sum of (or = the complete product of ). The set of all possible prime consequents of the original premises are obtained from the final equation (or ).

We describe herein a fuzzy version of the MSM that utilizes concepts of the Intuitionistic Fuzzy Logic (IFL) [38–48] developed mainly by Atanassov [38, 40, 41, 43–45]. This fuzzy MSM reduces to a restricted version in the Ordinary Fuzzy Logic (OFL) of Zadeh [34, 49–55]. The IFL version of the MSM is more flexible, while the OFL version is simpler and computationally faster. We managed to adapt the MSM to fuzzy reasoning without any dramatic changes of its main steps. In particular, our algorithm for constructing the complete product (or complete sum) of a logic function via consensus generation and absorption remains essentially the same. This algorithm was first developed by Blake [24] and later by Tison [56–59]. It is usually referred to as the Tison method, but we will name it herein as the Blake-Tison method or algorithm.

The organization of the rest of this paper is as follows. Section 2 briefly reviews the concept of Intuitionistic Fuzzy Logic (IFL) and asserts why it adds necessary flexibility to Ordinary Fuzzy Logic (OFL). Section 3 combines ideas from Lee [2] and Atanassov [38, 40, 41, 43–45] to produce a novel simple concept of a Realistic Fuzzy Tautology (RFT) and explains why such a new concept is needed. Section 4 outlines the steps of MSM in two-valued Boolean logic and then adapts it to realistic fuzzy logic, which is an IFL in which the new RFT concept is embedded. Formal proofs of the correctness of this adaptation are provided. Three examples are given in Section 5 to demonstrate the computational steps and to demonstrate how, similar to the result of Lee [2], the validity of the least truthful premise sets an upper limit on the validity of every logical consequent. Section 6 concludes the paper. The Appendix provides a description of an improved version of the Blake-Tison algorithm for producing the complete product of a logical function.

#### 2. Review of Intuitionistic Fuzzy Logic

In Intuitionistic Fuzzy Logic (IFL), a variable is represented by its validity which is the ordered couple where and are degrees of truth and falsity of , respectively, such that each of the real numbers , , .

Note that when , then IFL reduces to Ordinary Fuzzy Logic (OFL), in which alone suffices as a representation for , since is automatically determined by . The necessity of allowing the condition is established on the grounds that it allows a degree of hesitancy, ignorance, or uncertainty when one can neither designate a variable as true nor label it as false.

Since IFL includes OFL as a special case, operations in IFL should be defined such that they serve as extensions to their OFL counterparts. However, this allows the existence of many definitions for pertinent operations, such as the negation operation [45] or the implication operation [43]. We will stick herein to the most familiar definitions. We have a single unary operation, namely, the negation operation, which produces the complement of a variable . We define this operation as one that interchanges the truth and falsity of the variable, that is,

The most important binary operations are(i)the intuitionistic* conjunction* or* meet* operation defined by (ii)the intuitionistic* disjunction* or* join* operation defined by (iii)the intuitionistic* implication* operation defined herein by

With any three intuitionistic fuzzy variables , , and , the following pairs of dual theorems are satisfied:(1)idempotency: (2)commutativity: (3)associativity: (4)absorption: (5)distributivity: (6)identities:

Atanassov [38, 41] defined the notion of Intuitionistic Fuzzy Tautology (IFT) by the following: is an IFT if and only if . For comparison, will be a tautology in crisp Boolean algebra if and only if and .

A variable is said to be less valid (less truthful) than another variable (written ) if and only if and . Hence, the complement of an IFT is less valid than this IFT.

#### 3. Realistic Fuzzy Tautology

Since our attempts to fuzzify the MSM using the concept of Intuitionistic Fuzzy Tautology (IFT) were not successful, we were obliged to introduce a new concept of tautology that we call Realistic Fuzzy Tautology (RFT). A variable in IFT is an RFT if and only if . Note that an RFT is a more strict particular case of an IFT. If , then the concept of an RFT reduces to the representation of Fuzzy Tautology given by Lee [2]. A variable in IFT is a non-RFT (denoted by nRFT) if and only if . Hence, two complementary variables and cannot be RFTs at the same time. The conjunction of two complementary variables is nRFT. If the disjunction of a variable with an nRFT is an RFT, then this variable is an RFT. For convenience, we will call the Intuitionistic Fuzzy Logic (IFL) with the concept of RFT embedding in it a Realistic Fuzzy Logic (RFL). The introduction of the RFT concept is utilized herein in fuzzifying the MSM, but it might have other far-reaching consequences in fuzzifying other topics.

#### 4. The Modern Syllogistic Method

In this section, we describe the steps of a powerful technique for deductive inference, which is called “the Modern Syllogistic Method” (MSM). The great advantage of the method is that it ferrets out from a given set of premises all that can be concluded from this set, and it casts the resulting conclusions in the simplest or most compact form [24–33].

First, we describe the steps of the MSM in conventional Boolean logic. Then, we adapt these steps to realistic fuzzy logic. Since the MSM has two dual versions, one dealing with propositions equated to zero and the other dealing with propositions equated to one, we are going herein to represent the latter version which corresponds to tautologies.

##### 4.1. The MSM in Conventional Boolean Logic

The MSM has the following steps.

*Step 1. *Each of the premises is converted into the form of a formula equated to 1 (which we call an equational form), and then the resulting equational forms are combined together into a single equation of the form . If we have logical equivalence relations of the formthey are set in the equational form We may also have logical implication (logical inclusion) relations of the form These relations symbolize the statements “If then ” or equivalently “ if only ”. Conditions (14) can be set into the equational form

*Step 2. *The totality of premises in (13) and (15) finally reduces to the single equation , where is given byEquations (13) and (15) represent the dominant forms that premises can take. Other less important forms are discussed by Klir and Marin [60] and can be added to (16) when necessary.

*Step 3. *The function in (16) is rewritten as a complete product (a dual Blake canonical form), that is, as a conjunction of all the prime implicates of . There are many manual and computer algorithms for developing the complete product of a switching function [25]. Most of these algorithms depend on two logical operations: (a) consensus generation and (b) absorption.

*Step 4. *Suppose the complete product of takes the form where is the th prime implicate of . Equation (17) is equivalent to the set of equationsEquations (18) are called prime consequents of and state in the simplest equational form all that can be concluded from the original premises. The conclusions in (18) can also be cast into implication form. Suppose is given by a disjunction of complemented literals and uncomplemented literals , that is,then (18) can be rewritten as

##### 4.2. The MSM in Realistic Fuzzy Logic

A crucial prominent feature of realistic fuzzy logic is that it can be used to implement the MSM without spoiling any of its essential features. We just need to replace the concept of a crisp logical “1” by that of the realistic fuzzy tautology (RFT) introduced in Section 3. Now, a realistic fuzzy version of the MSM has the following steps.

*Step 1. *Assume the problem at hand is governed by a set of RFTs . Each of these RFTs might be assumed from the outset or be constructed from equivalence or implication relations. Let be described by

*Step 2. *The given set of RFT premises are equivalent to the single functionThe function is also an RFT. This equivalence is proved in Theorem 1.

*Step 3. *Replace the function by its complete product . The resulting is also an RFT since the operations used in going from to preserve the RFT nature. These operations are as follows:(i)absorption, which is known to be tautology-preserving in general fuzzy logic and intuitionistic fuzzy logic and hence in the current realistic fuzzy logic,(ii)consensus generation, which preserves RFTs in the sense that when the conjunction of two clauses is an RFT, then it remains so when conjuncted with the consensus of these two clauses. This is proved in the form of Theorem 2.

*Step 4. *Since is an RFT, then when it is given by the conjunction in (17), each clause , in (17) will be an RFT (again thanks to Theorem 1). The fact that each of the clauses is an RFT is all that can be concluded from the original premises. The procedure does not necessarily provide specific information about the validity of each consequent . However, as we show in the examples below, it is possible to obtain such information in specific cases.

Theorem 1. *Each of the realistic fuzzy variables is an RFT if and only if their conjunction is an RFT.*

*Proof. *Consider the following:

Theorem 2. *The conjunction of two clauses with a single opposition retains the RFT property when augmented by a third clause representing the consensus of the two original clauses. Specifically, if is an RFT, then is also an RFT.*

*Proof. *Let , By virtue of Theorem 1, the fact that is an RFT implies that is an RFT (i.e., ) and that is an RFT (i.e., ).

Now consider two cases.*Case **1*. One has , and hence*Case **2*. One has Now each of , , and is an RFT. Hence, thanks to Theorem 1, their conjunction is an RFT.

One prominent difference between fuzzy MSM and ordinary MSM is that the complementary laws in ordinary logic do not hold in any fuzzy logic including OFL, IFL, or RFL. This means that in implementing our algorithm for generating the complete product of a switching function, a conjunction of the form might appear, and then it is left as it is, and not replaced by . This point will be clarified further in Example 2 of Section 5.

Table 1 employs the MSM to derive fuzzy versions of many famous rules of inference, including, in particular, the celebrated rules of MODUS PONENS and MODUS TOLLENS. The derivation shows that some of the rules have some intermediate consequences as well as a final particular consequence.