Research Article A Modern Syllogistic Method in Intuitionistic Fuzzy...

Research ArticleA Modern Syllogistic Method in Intuitionistic FuzzyLogic with Realistic Tautology

Ali Muhammad Rushdi1 Mohamed Zarouan1

Taleb Mansour Alshehri1 and Muhammad Ali Rushdi2

1Department of Electrical and Computer Engineering Faculty of Engineering King Abdulaziz UniversityPO Box 80204 Jeddah 21589 Saudi Arabia2Department of Biomedical and Systems Engineering Cairo University Giza 12613 Egypt

Correspondence should be addressed to Mohamed Zarouan mzarouankauedusa

Received 9 May 2015 Accepted 27 July 2015

Academic Editor Guilong Liu

Copyright copy 2015 Ali Muhammad Rushdi et alThis is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

The Modern Syllogistic Method (MSM) of propositional logic ferrets out from a set of premises all that can be concluded from itin the most compact form The MSM combines the premises into a single function equated to 1 and then produces the completeproduct of this function Two fuzzy versions ofMSMare developed inOrdinary 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 05The paper formally proves each of the steps needed in the conversion of the ordinary MSM into a fuzzy oneThe proofs rely mainlyon the successful replacement of logic 1 (or ordinary tautology) by an RFT An improved version of Blake-Tison algorithm forgenerating the complete product of a logical function is also presented and shown to be applicable to both crisp and fuzzy versionsof theMSMThe fuzzyMSMmethodology is illustrated by three specific examples which delineate differences with the crispMSMaddress the question of validity values of consequences tackle the problem of inconsistency when it arises and demonstrate theutility of the concept of Realistic Fuzzy Tautology

1 Introduction

Fuzzy deductive reasoning has typically relied on a fuzzifi-cation of the Resolution Principle of Robinson [1] in first-order predicate calculus This principle uses a set of premisesto prove the validity of a single clause or consequent ata time via the refutation (REDUCTIO AD ABSURDUM)method Lee [2] proved that a set of clauses is unsatisfiablein fuzzy logic if and only if it is unsatisfiable in two-valuedlogic He also proved that if the least truthful clause of a setof clauses has a truth value 119886 gt 05 then all the logicalconsequents obtained by repeatedly applying the resolutionprinciple have truth values that are never less than 119886 Later theso-called Mukaidono Fuzzy Resolution Principle developedby a group of Japanese researchers [3ndash6] was used to establisha powerful fuzzy Prolog system The introduction of thisprinciple involved several new concepts including that ofthe contradictory degree cd(119883

119894) of a contradiction (119883

119894and 119883119894)

whose truth value119879(cd(119883119894)) equals the truth value119879(119883

119894and119883119894)

of the contradiction itself Recently a new fuzzy resolutionprinciple was introduced in [7ndash9] wherein refutation isachieved by the antonym not by negation and reasoning ismade more flexible thanks to the existence of a meaninglessrange which is a special set that is not true and also notfalse Other notable work on various aspects and techniquesof fuzzy reasoning and inference is available in [10ndash23]

The purpose of this paper is to implement fuzzy deductivereasoning via fuzzification of a powerful deductive techniqueof propositional logic called the Modern Syllogistic Method(MSM) This method was originally formulated by Blake[24] expounded by Brown [25] and further described orenhanced in [26ndash33] and has a striking similarity with theresolution-based techniques of predicate logic [1 34 35]

The MSM has the distinct advantage that it ferrets outfrom a set of premises all that can be concluded from it withthe resulting conclusions cast in the simplest ormost compact

Hindawi Publishing Corporatione Scientific World JournalVolume 2015 Article ID 327390 12 pageshttpdxdoiorg1011552015327390

2 The Scientific World Journal

form The MSM uses just a single rule of inference ratherthan the many rules of inference conventionally employedin propositional-logic deduction (see eg [36 37]) In factthe MSM includes all such rules of inference as special cases[30 31] The MSM strategy is to convert the set of premisesinto a single equation of the form 119891 = 0 or 119892 = 1 and obtainCS(119891) = the complete sum of 119891 (or CP(119892) = the completeproduct of 119892) The set of all possible prime consequents ofthe original premises are obtained from the final equationCS(119891) = 0 (or CP(119892) = 1)

We describe herein a fuzzy version of the MSM thatutilizes concepts of the Intuitionistic Fuzzy Logic (IFL) [38ndash48] developed mainly by Atanassov [38 40 41 43ndash45] Thisfuzzy MSM reduces to a restricted version in the OrdinaryFuzzy Logic (OFL) of Zadeh [34 49ndash55] The IFL versionof the MSM is more flexible while the OFL version issimpler and computationally fasterWemanaged to adapt theMSM to fuzzy reasoning without any dramatic changes of itsmain steps In particular our algorithm for constructing thecomplete product (or complete sum) of a logic function viaconsensus generation and absorption remains essentially thesame This algorithm was first developed by Blake [24] andlater by Tison [56ndash59] It is usually referred to as the Tisonmethod butwewill name it herein as the Blake-Tisonmethodor algorithm

The organization of the rest of this paper is as followsSection 2 briefly reviews the concept of Intuitionistic FuzzyLogic (IFL) and asserts why it adds necessary flexibility toOrdinary Fuzzy Logic (OFL) Section 3 combines ideas fromLee [2] and Atanassov [38 40 41 43ndash45] to produce a novelsimple concept of a Realistic Fuzzy Tautology (RFT) andexplainswhy such a new concept is needed Section 4 outlinesthe steps of MSM in two-valued Boolean logic and thenadapts it to realistic fuzzy logic which is an IFL in whichthe new RFT concept is embedded Formal proofs of thecorrectness of this adaptation are provided Three examplesare given in Section 5 to demonstrate the computationalsteps and to demonstrate how similar to the result of Lee[2] the validity of the least truthful premise sets an upperlimit on the validity of every logical consequent Section 6concludes the paper The Appendix provides a descriptionof an improved version of the Blake-Tison algorithm forproducing the complete product of a logical function

2 Review of Intuitionistic Fuzzy Logic

In Intuitionistic Fuzzy Logic (IFL) a variable 119883119894is repre-

sented by its validity which is the ordered couple

119881 (119883119894) = ⟨119886

119894 119887119894⟩ (1)

where 119886119894and 119887

119894are degrees of truth and falsity of 119883

119894

respectively such that each of the real numbers 119886119894 119887119894 119886119894+ 119887119894isin

[0 1]Note that when 119886

119894+ 119887119894= 1 then IFL reduces to

Ordinary Fuzzy Logic (OFL) in which 119886119894alone suffices as a

representation for119883119894 since 119887

119894is automatically determined by

119887119894= 1minus 119886

119894 The necessity of allowing the condition (119886

119894+ 119887119894) le

1 is established on the grounds that it allows a degree of

hesitancy ignorance or uncertainty when one can neitherdesignate a variable as true nor label it as false

Since IFL includesOFL as a special case operations in IFLshould be defined such that they serve as extensions to theirOFL counterparts However this allows the existence ofmanydefinitions for pertinent operations such as the negationoperation [45] or the implication operation [43]Wewill stickherein to themost familiar definitionsWehave a single unaryoperation namely the negation operation which producesthe complement119883

119894of a variable119883

119894 We define this operation

as one that interchanges the truth and falsity of the variablethat is

119881(119883119894) = ⟨119887

119894 119886119894⟩ (2)

The most important binary operations are

(i) the intuitionistic conjunction ormeet operation (1198831and

1198832) defined by

119881 (1198831and 1198832) = ⟨min (119886

1 1198862) max (119887

1 1198872)⟩ (3)

(ii) the intuitionistic disjunction or join operation (1198831or

1198832) defined by

119881 (1198831or 1198832) = ⟨max (119886

1 1198862) min (119887

1 1198872)⟩ (4)

(iii) the intuitionistic implication operation (1198831rarr 1198832) equiv

(1198831or 1198832) defined herein by

119881 (1198831997888rarr119883

2) = ⟨max (119887

1 1198862) min (119886

1 1198872)⟩ (5)

With any three intuitionistic fuzzy variables 1198831 1198832 and

1198833 the following pairs of dual theorems are satisfied

(1) idempotency

1198831or 1198831= 1198831

1198831and 1198831= 1198831

(6)

(2) commutativity

1198831or 1198832= 1198832or 1198831

1198831and 1198832= 1198832and 1198831

(7)

(3) associativity

(1198831or 1198832) or 1198833= 1198831or (1198832or 1198833)

(1198831and 1198832) and 1198833= 1198831and (1198832and 1198833)

(8)

(4) absorption

1198831or (1198831and 1198832) = 119883

1

1198831and (1198831or 1198832) = 119883

1

(9)

The Scientific World Journal 3

(5) distributivity

1198831or (1198832and 1198833) = (119883

1or 1198832) and (119883

1or 1198833)

1198831and (1198832or 1198833) = (119883

1and 1198832) or (119883

1and 1198833)

(10)

(6) identities

1198831or 0 = 119883

1

1198831and 1 = 119883

1

(11)

Atanassov [38 41] defined the notion of IntuitionisticFuzzy Tautology (IFT) by the following 119883 is an IFT if andonly if 119886 ge 119887 For comparison 119883 will be a tautology in crispBoolean algebra if and only if 119886 = 1 and 119887 = 0

A variable 1198831is said to be less valid (less truthful) than

another variable 1198832(written 119881(119883

1) le 119881(119883

2)) if and only if

1198861le 1198862and 1198871ge 1198872 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 ofIntuitionistic Fuzzy Tautology (IFT) were not successful wewere obliged to introduce a new concept of tautology that wecall Realistic Fuzzy Tautology (RFT) A variable 119883

119894in IFT

is an RFT if and only if (119886119894gt 05) Note that an RFT is a

more strict particular case of an IFT If 119887119894= 1 minus 119886

119894 then

the concept of an RFT reduces to the representation of FuzzyTautology given by Lee [2] A variable 119883

119894in IFT is a non-

RFT (denoted by nRFT) if and only if (119886119894le 05) Hence

two complementary variables 119883119894and 119883

119894cannot be RFTs

at the same time The conjunction of two complementaryvariables is nRFT If the disjunction of a variable with annRFT is anRFT then this variable is anRFT For conveniencewe will call the Intuitionistic Fuzzy Logic (IFL) with theconcept of RFT embedding in it a Realistic Fuzzy Logic (RFL)The introduction of the RFT concept is utilized herein infuzzifying the MSM but it might have other far-reachingconsequences in fuzzifying other topics

4 The Modern Syllogistic Method

In this section we describe the steps of a powerful techniquefor deductive inference which is called ldquothe Modern Syllo-gistic Methodrdquo (MSM)The great advantage of the method isthat it ferrets out from a given set of premises all that can beconcluded from this set and it casts the resulting conclusionsin the simplest or most compact form [24ndash33]

First we describe the steps of the MSM in conventionalBoolean logic Then we adapt these steps to realistic fuzzylogic Since the MSM has two dual versions one dealingwith propositions equated to zero and the other dealing withpropositions equated to one we are going herein to representthe latter version which corresponds to tautologies

41 The MSM in Conventional Boolean Logic The MSM hasthe following steps

Step 1 Each of the premises is converted into the form of aformula equated to 1 (which we call an equational form) andthen the resulting equational forms are combined togetherinto a single equation of the form 119892 = 1 If we have 119899 logicalequivalence relations of the form

119879119894equiv 119876119894 1 le 119894 le 119899 (12)

they are set in the equational form

119875119894= (119879119894or 119876119894) and (119879

119894or 119876119894) 1 le 119894 le 119899 (13)

We may also have (119898 minus 119899) logical implication (logicalinclusion) relations of the form

119879119894997888rarr 119876

119894 (119899 + 1) le 119894 le 119898 (14)

These relations symbolize the statements ldquoIf 119879119894then 119876

119894rdquo

or equivalently ldquo119879119894if only119876

119894rdquo Conditions (14) can be set into

the equational form

119875119894= 119879119894or 119876119894= 1 (119899 + 1) le 119894 le 119898 (15)

Step 2 The totality of premises in (13) and (15) finally reducesto the single equation 119892 = 1 where 119892 is given by

119892 =

119898

⋀

119894=1

119875119894

=

119899

⋀

119894=1

((119879119894or 119876119894) and (119879

119894or 119876119894)) and

119898

⋀

119894=119899+1

(119879119894or 119876119894)

(16)

Equations (13) and (15) represent the dominant forms thatpremises can take Other less important forms are discussedby Klir and Marin [60] and can be added to (16) whennecessary

Step 3 The function 119892 in (16) is rewritten as a complete prod-uct (a dual Blake canonical form) that is as a conjunctionof all the prime implicates of 119892 There are many manual andcomputer algorithms for developing the complete product ofa switching function [25] Most of these algorithms dependon two logical operations (a) consensus generation and (b)absorption

Step 4 Suppose the complete product of 119892 takes the form

CP (119892) =119897

⋀

119894=1

119862119894= 1 (17)

where 119862119894is the 119894th prime implicate of 119892 Equation (17) is

equivalent to the set of equations

119862119894= 1 1 le 119894 le 119897 (18)

Equations (18) are called prime consequents of 119892 = 1and state in the simplest equational form all that can beconcluded from the original premisesThe conclusions in (18)can also be cast into implication form Suppose 119862

119894is given


by a disjunction of complemented literals119883119894119895and uncomple-

mented literals 119884119894119895 that is

119862119894=

119903

⋁

119895=1

119883119894119895or

119904

⋁

119895=1

119884119894119895 1 le 119894 le 119897 (19)

then (18) can be rewritten as119903

⋀

119895=1

119883119894119895997888rarr

119904

⋁

119895=1

119884119894119895 1 le 119894 le 119897 (20)

42 The MSM in Realistic Fuzzy Logic A crucial prominentfeature of realistic fuzzy logic is that it can be used toimplement the MSM without spoiling any of its essentialfeatures We just need to replace the concept of a crisp logicalldquo1rdquo by that of the realistic fuzzy tautology (RFT) introducedin Section 3 Now a realistic fuzzy version of the MSM hasthe following steps

Step 1 Assume the problem at hand is governed by a setof RFTs 119875

119894 1 le 119894 le 119899 Each of these RFTs might be

assumed from the outset or be constructed from equivalenceor implication relations Let 119875

119894be described by

119881 (119875119894) = ⟨120583

119894 120574119894⟩ (21)

Step 2 The given set of RFT premises are equivalent to thesingle function

119892 =

119898

⋀

119894

119875119894

119881 (119892) = ⟨min119894

120583119894max119894

120574119894⟩

(22)

The function 119892 is also an RFT This equivalence is proved inTheorem 1

Step 3 Replace the function119892 by its complete product CP(119892)The resulting CP(119892) is also an RFT since the operations usedin going from 119892 to CP(119892) preserve the RFT nature Theseoperations are as follows

(i) absorption which is known to be tautology-preserving in general fuzzy logic and intuitionisticfuzzy logic and hence in the current realistic fuzzylogic

(ii) consensus generation which preserves RFTs in thesense that when the conjunction of two clauses is anRFT then it remains so when conjuncted with theconsensus of these two clauses This is proved in theform of Theorem 2

Step 4 Since CP(119892) is an RFT then when it is given by theconjunction in (17) each clause 119862

119894 1 le 119894 le 119897 in (17)

will be an RFT (again thanks to Theorem 1) The fact thateach of the clauses 119862

119894is an RFT is all that can be con-

cluded from the original premises The procedure does notnecessarily provide specific information about the validity of

each consequent 119862119894 However as we show in the examples

below it is possible to obtain such information in specificcases

Theorem 1 Each of the realistic fuzzy variables 119875119894 1 le 119894 le 119898

is an RFT if and only if their conjunction⋀119898119894=1119875119894is an RFT

Proof Consider the following

119875119894is an RFT 1 le 119894 le 119898

lArrrArr 120583119894gt 05 1 le 119894 le 119898

lArrrArr min119894

120583119894gt 05

lArrrArr

119898

⋀

119894=1

119875119894is an RFT

(23)

Theorem 2 The conjunction of two clauses with a singleopposition retains the RFT property when augmented by a thirdclause representing the consensus of the two original clausesSpecifically if (119883

1or 1198832) and (119883

1or 1198833) is an RFT then (119883

1or

1198832) and (119883

1or 1198833) and (119883

2or 1198833) is also an RFT

Proof Let 119881(119883119894) = ⟨119886

119894 119887119894⟩ 119894 = 1 2 3 By virtue ofTheorem 1

the fact that (1198831or 1198832) and (119883

1or 1198833) is an RFT implies that

(1198831or1198832) is an RFT (ie max(119886

1 1198862) gt 05) and that (119883

1or1198833)

is an RFT (ie max(1198871 1198863) gt 05)

Now consider two cases

Case 1 One has 1198861ge 05 rArr 119887

1le 05 and hence

max (1198871 1198863) gt 05 997904rArr 119886

3gt 05

997904rArr max (1198862 1198863) gt 05

997904rArr (1198832or 1198833) is an RFT

(24)

Case 2 One has 1198861le 05

1198861le 05 and max (119886

1 1198862) gt 05 997904rArr 119886

2gt 05

997904rArr max (1198862 1198863) gt 05 997904rArr (119883

2or 1198833) is an RFT

(25)

Now each of (1198831or1198832) (1198831or1198833) and (119883

2or1198833) is an RFT

Hence thanks to Theorem 1 their conjunction (1198831or 1198832) and

(1198831or 1198833) and (119883


One prominent difference between fuzzy MSM and ordi-nary MSM is that the complementary laws

119883119894or 119883119894= 1

119883119894and 119883119894= 0

(26)

in ordinary logic do not hold in any fuzzy logic includingOFL IFL or RFL This means that in implementing our


algorithm for generating the complete product of a switchingfunction a conjunction of the form (119883

119894and 119883119894) might appear

and then it is left as it is and not replaced by 0This point willbe clarified further in Example 2 of Section 5

Table 1 employs the MSM to derive fuzzy versions ofmany famous rules of inference including in particularthe celebrated rules of MODUS PONENS and MODUSTOLLENS The derivation shows that some of the rules havesome intermediate consequences as well as a final particularconsequence

5 Examples

Example 1 A typical example of MSM presented by Brown[25] pp 124ndash127 and taken from Kalish and Montague [61]has the following statements

(1) if Alfred studies then he receives good grades (119878 rarr119866)

(2) if Alfred does not study then he enjoys college (119878 rarr119864)

(3) if Alfred does not receive good grades then he doesnot enjoy college (119866 rarr 119864)

The MSM solution combines the above premises into asingle equation

1198921= (119878 or 119866) and (119878 or 119864) and (119866 or 119864) = 1 (27)

and obtains the complete product of 1198921by adding consensus

alterms or clauses [56] with respect to the biform variables119878 and 119864 and absorbing subsuming alterms (see Appendix)Gradually the formula for 119892

1changes to end up as the

complete product form

1198921= (119878 or 119866) and (119878 or 119864) and (119866 or 119864) and (119866 or 119864)

= (119878 or 119866) and (119878 or 119864) and (119866 or 119864) and (119866 or 119864)

and (119878 or 119866) and 119866

= (119878 or 119864) and 119866

(28)

The last expression for 1198921is CP(119892

1) and is still equated to

1 Hence it asserts the not so-obvious conclusion of (119866 = 1)Alfred receives good grades beside the conclusion (119878 or119864) = 1 which is just a reecho of one of the premisesThese two conclusions are all that can be concluded from thepremises in the simplest form Any other valid conclusionmust subsume one of these two conclusions Now supposethat our knowledge about the premises is fuzzy or uncertainso that each of the premises is no longer a crisp tautology butis weakened to the status of a realistic fuzzy tautology (RFT)To be specific let us assign the following values for the validityof each premise

119881 (119878 997888rarr 119866) = 119881 (119878 or 119866) = ⟨06 03⟩

119881 (119878 997888rarr 119864) = 119881 (119878 or 119864) = ⟨09 01⟩

119881 (119866 997888rarr 119864) = 119881 (119866 or 119864) = ⟨08 01⟩

(29)

The function 1198921in (27) is no longer a crisp tautology (=1)

but rather an RFT with validity

119881 (1198921) = 119881 ((119878 or 119866) and (119878 or 119864) and (119866 or 119864))

= ⟨min (06 09 08) max (03 01 01)⟩

= ⟨06 03⟩

(30)

so 1198921inherits the validity of the first premise which is

the least-truthful premise This validity is also inheritedby CP(119892

1) in the last line of (28) and also by the novel

consequent (119866 = 1) that is

119881 (119866) = ⟨06 03⟩ (31)

This means that the consequent Alfred gets good gradeshas a truth value of 06 and a falsity value of 03 The factthat (06 + 03) = 09 lt 1 leaves room for our uncertaintyor ignorance about this fuzzy proposition

Example 2 The MSM has a built-in capability of detectinginconsistency in a set of premises since this produces CP(119892)as 0 and leads to 0 = 1 which is unacceptable in two-valued logic [30 31] This feature is still enjoyed by the fuzzyMSM since an inconsistency will be revealed in the formof a variable and its complement being both RFT which isa contradiction For a specific example consider the set ofpremises (119860 harr 119861) (119861 harr 119862) and (119862 harr 119860) In equationalform these reduce to

(119860 or 119861) and (119860 or 119861) = 1

(119861 or 119862) and (119861 or 119862) = 1

(119862 or 119860) and (119862 or 119860) = 1

(32)

or equivalently to the single equation


and (119862 or 119860) and (119862 and 119860) = 1

(33)

In two-valued logic the complete product of 1198922is

obtained via the Improved Blake-Tison Method (seeAppendix) as

CP (1198922) = 119860 and 119860 and 119861 and 119861 and 119862 and 119862 = 0 (34)

which leads to the contradiction (0 = 1) However in realisticfuzzy logic we have

CP (1198922) = 119860 and 119860 and 119861 and 119861 and 119862 and 119862 (35)

being an RFTThis means that both119860 and119860 (and also both 119861and 119861 and both 119862 and 119862) are RFTs which is a contradictionHence the original set of premises are inconsistent

Example 3 Consider the set of premises [30 37]

(1) Pollution will increase if government restrictions arerelaxed (119877 rarr 119875)


Table1MSM

deriv

ationof

fuzzyversions

offamou

srules

ofinference

thep

artic

ular

conclusio

nof

aruleish

ighlighted

inbo

ld

Rulename

FuzzyRF

Tantecedents

(premise

s)

Prem

isesa

sseparate

fuzzyequatio

ns119875119894=RF

T119894=1119898

Prem

isesa

sasin

glefuzzy

equatio

n119892=RF

TCon

clusio

nsas

asinglefuzzy

equatio

nCP(119892)=RF

TCon

clusio

nsas

separatefuzzy

equatio

ns119862119894=RF

T119894=1119897

FuzzyRF

Tconsequence

(con

clusio

n)

MODUSPO

NEN

S119860rarr119861

119860

119860or119861=RF

T119860=RF

T(119860or119861)and119860=RF

T119861and119860=RF

TB=RF

T119860=RF

T119861

MODUSTO

LLEN

S119860rarr119861

119861

119860or119861=RF

T119861=RF

T(119860or119861)and119861=RF

T119860and119861=RF

TA=RF

T119861=RF

T119860

HYP

OTH

ETICAL

SYLL

OGISM

119860rarr119861

119861rarr119862

119860or119861=RF

T119861or119862=RF

T(119860or119861)and(119861or119862)=RF

T(119860or119861)and(119861or119862)and(119860or119862)=RF

T119860or119861=RF

T119861or119862=RF

TAorC=RF

T119860rarr119862

SIMPL

IFICAT

ION

119860and119861

119860and119861=RF

T119860and119861=RF

T119860and119861=RF

TA=RF

T119861=RF

T119860

CONJU

NCT

ION

119860 119861

119860=RF

T119861=RF

T119860and119861=RF

T119860and119861=RF

TAandB=RF

T119860and119861

CONST

RUCT

IVE

DILEM

MA

119860rarr119861

119862rarr119863

119860or119862

119860or119861=RF

T119862or119863=RF

T119860or119862=RF

T

(119860or119861)and(119862or119863)

and(119860or119862)=RF

T(119860or119861)and(119862or119863)and(119860or119862)and(119861or119862)

and(119860or119863)and(119861or119863)=RF

T

(119860or119861)=RF

T(119862or119863)=RF

T(119860or119862)=RF

T(119861or119862)=RF

T(119860or119863)=RF

T(BorD)=RF

T

119861or119863

DISJU

NCT

IVESY

LLOGISM119860or119861

119860

119860or119861=RF

T119860=RF

T(119860or119861)and119860=RF

T119861and119860=RF

TB=RF

T119860=RF

T119861

ADDITIO

N119860

119860=RF

T119860=RF

T119860=RF

TAorB=RF

TAnalterm

subsum

ingan

RFT

alterm

isalso

RFT

119860or119861

ABS

ORP

TION

119860rarr119861

119860or119861=RF

T119860or119861=RF

T119860or119861=RF

TAorAB=RF

T119860or119860119861=119860or119861by

reflection

law

119860rarr119860119861

CASE

S

119860 119860rarr(119862or119863)

119862rarr119861

119863rarr119861

119860=RF

T119860or119862or119863=RF

T119862or119861=RF

T119863or119861=RF

T

119860and(119860or119862or119863)and(119862or119861)

and(119863or119861)=RF

T119860and119861and(119862or119863)=RF

T119860=RF

TB=RF

T119862or119863=RF

T119861

CASE

ELIM

INAT

ION

119860or119861

119860rarr(119862and119862)

119860or119861=RF

T119860or(119862and119862)=RF

T(119860or119861)and(119860or(119862and119862))=RF

T(119860or119861)and(119860or(119862and119862))

and(119861or(119862or119862))=RF

T

119860or119861=RF

T119860or(119862and119862)=RF

T119861or(119862and119862)=RF

TB=RF

Tand119862and119862=nR

FT

119861

REDUCT

IOAD

ABS

URD

UM

(CONTR

ADICTION)

119860rarr(119861and119861)

119860or(119861and119861)=RF

T119860or(119861and119861)=RF

T119860or(119861and119861)=RF

T119860or(119861and119861)=RF

TA=RF

T119861and119861=nR

FT

119860


Table 2 Validities of consequences obtained in Example 3

New clause Nature Validity(119877 rarr 119863) equiv (119877 or 119863) Consensus of (119877 or 119875) and (119875 or 119863) ⟨06 03⟩

(119863 rarr 119864) equiv (119863 or 119864) Consensus of (119863 or 119865) and (119864 or 119865) ⟨08 01⟩





(2) If pollution increases there will be a decline in thegeneral health of the population (119875 rarr 119863)

(3) If there is a decline in health in the populationproductivity will fall (119863 rarr 119865)

(4) The economy will remain healthy only if productivitydoes not fall (119864 rarr 119865)

These premises are equivalent to the propositional equa-tion (119892

3= 1) where

1198923= (119877 or 119875) and (119875 or 119863) and (119863 or 119865) and (119864 or 119865) (36)

The complete product of 1198923is obtained via the Improved

Blake-Tison Method (see Appendix) as

CP (1198923) = (119877 or 119875) and (119875 or 119863) and (119863 or 119865) and (119864 or 119865)

and (119877 or 119863) and (119863 or 119864) and (119875 or 119865)


(37)

The fact that CP(1198923) = 1 means that there are six new

consequents (that are not just a reecho of premises) The lastof these consequents is

119877 or 119864 = 1 (38)

or equivalently

119877 997888rarr 119864 (39)

whichmeans that if government restrictions are relaxed thenthe economy will not remain healthy an argument in favor ofa stronger governmental regulatory role

Now suppose that the given premises are not crisptautologies but are just RFTs with respective validities

119881 (119877 997888rarr 119875) = 119881 (119877 or 119875) = ⟨06 03⟩

119881 (119875 997888rarr119863) = 119881 (119875 or 119863) = ⟨07 02⟩

119881 (119863 997888rarr 119865) = 119881 (119863 or 119865) = ⟨08 01⟩

119881 (119864 997888rarr 119865) = 119881 (119864 or 119865) = ⟨09 01⟩

(40)

Hence each of the new clauses in (37) is an RFT of avalidity dependent on the validities of the clauses generating

it Table 2 lists these new clauses identifies their generatorsand hence assigns a validity to each of them The issue of astronger regulatory role for the government nowhas a validityof ⟨06 03⟩ rather than ⟨10 00⟩ This validity is realistic inthe sense that this issue can be viewed as supported by 60of the voters and opposed by 30 of them with 10 of themabstaining or undecided

6 Conclusion

The Modern Syllogistic Method (MSM) is a sound andcomplete single rule of inference that encompasses all rulesof inference It extracts from a given set of premises all thatcan be concluded from it in the simplest possible form Ithas a striking similarity with resolution-based techniques inpredicate logic but while these techniques chain backwardlyfrom a given assertion seeking to refute it the MSM chainsforwardly from the set of premises seeking to prove allpossible consequences [25]

This paper contributes a fuzzy version of MSM using avariant of Intuitionistic Fuzzy Logic (IFL) called RealisticFuzzy Logic (RFL) Here a propositional variable is char-acterized by 2-tuple validity expressing its truth and falsityAutomatically a third dependent attribute for the variableemerges namely hesitancy or ignorance about the variablewhich complements the sum of truth and falsity to 1 IfIgnorance is 0 then IFL reduces to Ordinary Fuzzy Logic(OFL) and the RFL version of MSM reduces to a simplerbut weaker OFL version The slight restriction of IFL to RFLinvolves the replacement of the concept of an IntuitionisticFuzzy Tautology (IFT) in which truth is greater than orequal to falsity by a restricted concept of Realistic FuzzyTautology (RFT) in which truth is strictly greater than 05The introduction of the RFT enabled us to fuzzify the MSMwithout making any significant changes in it and to formallyprove the correctness of all the steps of the emergent fuzzyMSM As an offshoot the paper contributes an improvementof the main algorithm that constitutes the heart of the MSMwhether it is crisp ordinary fuzzy or realistic fuzzy Theimprovement involves a matrix formulation of the typicalstep of consensus generation thatminimizes the comparisonsamong pairs of alterms that might have consensus altermsThe following task of absorbing subsuming alterms is alsoreduced considerably via a set of novel observations that wereformally proved The concept of consensus used herein isexactly the one used in crisp two-valued propositional logicThere was no need herein to introduce a specific concept of


fuzzy consensus The only significant change is that relations(26) no longer hold

The fuzzy MSM methodology is illustrated by threespecific examples which delineate differences with the crispMSM address the question of validity values of conse-quences tackle the problem of inconsistency when it arisesand demonstrate the utility of RFL compared to ordinaryfuzzy logic

The current paper is one of several new papers by theauthors which are intended to extend the utility and sharpenthe mathematics of the MSM One of these papers [62]presents an incremental version of the MSM in which thecore work of the MSM is not completely repeated but isslightly incremented when additional premises are addedAnother paper [63] utilizes the MSM in the exploration ofhidden aspects in engineering ethical dilemmas by investigat-ing different scenarios describing the situation from variousperspectives

In future work we hope to combine the contributionsof the current paper with those of [62 63] We also hopeto utilize the new RFT concept introduced herein in novelapplications

Appendix

The Improved Blake-Tison Method (ITM)

Thecomplete sum of a switching function119891 to be denoted byCS(119891) is the all-prime-implicant disjunction that expresses119891 that is it is a sum-of-products (SOP) formula whoseproducts are all the prime implicants of 119891 The complete sumis called the ldquoBlake canonical formrdquo by Brown [25] in honorof 119860 Blake who was the first person to study this form inhis thesis [24] Since CS(119891) is a disjunction of all the primeimplicants of 119891 and nothing else it is obviously unique andhence stands for a canonical representation of the switchingfunction [25] The dual quantity of the complete sum is thecomplete product of a switching function 119892 denoted CP(119892)which is the all-prime-implicate conjunction that expresses119892that is it is a product-of-sum (POS) formula whose altermsor sums are all the prime implicates of 119891 [56]

The concept of the complete product of a switchingfunction 119892 is closely related to that of a dual syllogisticformula for119892 However while CP(119892) is unique and canonicalthere are infinitely many dual syllogistic formulas for 119892 Adual syllogistic formula of 119892 can be defined as a POS formulawhose alterms include but are not necessarily confined toall the prime implicates of 119892 that is it is the completeproduct of 119892 conjuncted (possibly) with alterms each ofwhich subsumes some prime implicates of 119892 The complete-product formula CP(119892) is minimal within the class of dualsyllogistic formulas for 119892 that is the set of alterms in anydual syllogistic formula for 119892 is a superset of the set of altermsin CP(119892) Hence CP(119892) can be denoted by ABS(119866) where119866 is any dual syllogistic formula for 119892 and ABS(119866) denotesan equivalent absorptive formula of 119866 that is a formulaobtained from119866 by successive deletion of alterms absorbed inother alterms of119866The complete-product formulaCP(119892)may

be generated by the following two-step iterative-consensusprocedure (a) Find a dual syllogistic formula 119866 for 119892 bycontinually comparing alterms and adding their consensusalterms to the current formula of 119892 and (b) delete absorbedalterms to obtain ABS(119866) Note that two alterms have aconsensus if and only if they have exactly one oppositionthat is exactly one variable that appears complemented in onealterm and appears uncomplemented in the other In such acase the consensus is the ORing of the remaining literals ofthe two alterms with idempotency of the OR operation beingtaken into consideration The concept of a consensus of twoalterms is illustrated in Figure 1

Tison method (see eg [56ndash59 64ndash67]) is a systematicstreamlined version of the iterative-consensus technique forobtaining the CS of a switching function 119891 or dually theCP of a switching function 119892 The original study of Tisonappeared in [57] but amore readable exposition can be foundin [58] and further proofs are available in [58 59] Relatedwork and techniques are also available in [68ndash77] Since Tisonmethod is actually due to Blake [24] we will present it hereunder the name Blake-Tison Method Its essence when usedfor obtaining the complete product is summarized as follows

Blake-Tison Algorithm Start with a set of 1198990alterms or sums

of literals 1199040= 119860

(0)

1 119860(0)

2 119860

(0)

1198990 with biform variables

1198831 1198832 119883

119872and a Boolean function 119892 that is expressed by

conjunction of the alterms in 1199040 Assume that any absorbable

alterms in 1199040have been deleted so that the conjunction of

alterms in 1199040is an absorptive formula For 1 le 119898 le 119872

repeat the following 2-part step that replaces an absorptiveset of alterms 119904

119898minus1by another 119904

119898

(1) For 1 le 119895 le 119896 le 119899(119898minus1)

if 119883119898appears complemented

in one of the two alterms 119860(119898minus1)119895

and 119860(119898minus1)119896

andappears uncomplemented in the other such that thetwo alterms have no other opposition then they havea consensus with respect to119883

119898 Form that consensus

and add it to 119904119898minus1

Finally 119904119898minus1

is replaced by asuperset 119904

119898minus1of 119869(119898minus1)

elements where 119869(119898minus1)

ge

119899(119898minus1)

(2) Consider every pair 119860(119898minus1)119895

119860(119898minus1)119896

119895 = 119896 of (sofar remaining) products in 119904

119898minus1 If 119860(119898minus1)119895

subsumes119860(119898minus1)

119896 then delete 119860(119898minus1)

119895 Otherwise if 119860(119898minus1)

119895is

subsumed by 119860(119898minus1)119896

then delete 119860(119898minus1)119896

Wheneverall subsumptions (and subsequent deletions) areexhausted let the remaining absorptive set be 119904

119898=

119860(119898)

1 119860(119898)

2 119860(119898)

119899119898

Blake [24] and later Cutler et al [58] formally provedTheorem 3 asserting the success of the Blake-Tison algo-rithm in obtaining CP(119892) by merely applying the iterative-consensus procedure to each biform variable one by one

Theorem 3 In the Blake-Tison algorithm above(a) the conjunction of alterms in any of the sets 119904

119898 where

1 le 119898 le 119872 is an expression of 119892(b) the final set 119904

119872consists of all prime implicates of 119892


A or CB or C

A or B

B

C

0

000

A

(a)

B or C

A

B

C

A

00

00 0

(b)

A or BA or BB

C

0

0

0

0

A

(c)

Figure 1 (a) The alterms (119860 or 119861) and (119861 or 119862) have a single opposition (disjoint loops sharing a border) and hence their conjunction can beaugmented by their consensus (119860 or 119862) (b) The alterms 119860 and (119861 or 119862) have zero opposition (nondisjoint or overlapping) loops and henceno consensus (or a consensus of 1) (c) The alterms (119860 or 119861) and (119860 or 119861) have more than one opposition (disjoint faraway loops) and have noconsensus (or a consensus of 1)

Rushdi and Al-Yahya [64] proposed an improvementof Blake-Tisonrsquos Method in which the typical step starts byarranging a given expression for 119892 with respect to a biformvariable119883

119898 1 le 119898 le 119872 in the form

119892 = (119903 or 119883119898) and (119904 or 119883

119898) and 119905 (A1)

where 119903 = ⋀119899119903119894=1119903119894 119904 = ⋀119899119904

119895=1119904119895 and 119905 = ⋀119899119905

119896=1119905119896are POS

formulas that are independent of 119883119898 and the symbols 119903

119894 119904119895

and 119905119896denote alterms or sums of single literals Thanks to

intelligent multiplication [25 64] the function 119892 takes thePOS form

119892 =

119899119903

⋀

119894=1

(119903119894or 119883119898) and

119899119904

⋀

119895=1

(119904119895and 119883119898) and

119899119905

⋀

119896=1

119905119896 (A2)

Next119892 is augmented by all consensus altermswith respectto119883119898 which turn out to be the alterms (119903

119894or 119904119895)which do not

add to 1 in the expression

119899119903

⋀

119894=1

119899119904

⋀

119895=1

(119903119894or 119904119895) (A3)

This is followed by absorbing or deleting alterms thatsubsume others The method repeats this typical step for allbiform variables ending with CP(119892) after the last step

Table 3 suggests an economic layout [64] for implement-ing the typical step in the Improved Blake-Tison Method(IBTM)with a restricted number for the comparisons neededfor implementing absorptions This typical step which per-forms consensus generation with respect to a specific biformvariable 119883

119898 involves a rearrangement of the alterms whose

Table 3The general layout of the consensus generation table of theImproved Blake-Tison Method when producing consensus altermswith respect to 119883

119898 The vertical keys of this table are the alterms

containing119883119898and its horizontal keys are the alterms containing119883

119898

while alterms containing neither119883119898nor119883

119898are set aside

sdot sdot sdot (119860119895or 119883119898) sdot sdot sdot (119860

119896or 119883119898) sdot sdot sdot

(119878119894or 119883119898) sdot sdot sdot 119878

119894or 119860119895 sdot sdot sdot 119878

119894or 119860119896 sdot sdot sdot

(119878119903or 119883119898) sdot sdot sdot 119878

119903or 119860119895 sdot sdot sdot 119878

119903or 119860119896 sdot sdot sdot

Set-aside alterms(alterms containing neither 119883

119898nor119883

119898)

conjunction constitutes the current formula of 119892 at this stepWe construct a consensus-generation table with respect to119883119898

that resembles a multiplication table or matrix Thevertical keys of this table are the alterms containing theuncomplemented literal 119883

119898and its horizontal keys are the

alterms containing the complemented literal 119883119898 while its

entries are the consensus alterms generated by these keyswith respect to 119883

119898 Alterms containing neither the uncom-

plemented literal119883119898nor the complemented literal119883

119898are set

aside and naturally not included in the consensus generationof the table butmight absorb or be absorbed by the consensusalterms produced by the table Table 3 shows typical keys andentries of the consensus-generation table where we use thesymbol 119878

119894or 119860119895 to denote the consensus of the vertical key


(119860119895or 119883119898) with the horizontal key (119878

119894or 119883119898) which is the

ORing of the two alterms 119878119894and119860

119895after deleting any repeated

literals (thanks to the idempotency of the logical operationldquoORrdquo) Of course if the alterms 119878

119894and 119860

119895have at least one

opposition that is one literal that appears complemented inone of them and uncomplemented in the other then 119878

119894or

119860119895 is 1 and hence it is ignored since it does not affect a

POS formula when multiplied with it Now further benefitgained from the above construction is made apparent via thefollowing novel theorem

Theorem 4 In the consensus-generation table of Table 3

(1) there are no absorptions among vertical keys horizon-tal keys and set-aside alterms

(2) a table entry cannot be absorbed by a table key but itcould be absorbed by another table entry or a set-asidealterm A set-aside alterm could be absorbed by a tableentry

(3) if a table entry 119878119903or 119860119896 is to be ever absorbed by

another table entry then it has an absorbing productfor it in the same row 119903 or in the same column 119896

(4) if a table vertical key (119860119896or 119883119898) is to be ever absorbed

by a table entry then it has an absorbing product for itin the same column 119896

(5) if a table horizontal key (119878119903or119883119898) is to be ever absorbed

by a table entry then it has an absorbing product for itin the same row 119903

In the following we outline a proof and reflect on theramifications of Theorem 4

(1) Each of the conjunctions of vertical keys that ofhorizontal keys and that of set-aside alterms consti-tutes an absorptive formula Therefore there are noabsorptions among alterms of such a formula

(2) A table entry cannot be absorbed by a table keybecause the former cannot subsume the latter sincethe former lacks the literal119883

119898or the literal119883

119898

(3) Suppose that the table entry 119878119903or119860119896 subsumes (and

hence is absorbed by) another table entry 119878119894or 119860119895

which lies in a different row (119894 = 119903) and a differentcolumn (119895 = 119896) This means that the set of literals of119878119903or119860119896 is a superset of the set of literals of 119878

119894or119860119895

and hence it is a superset of each of the set of literalsof 119878119894and that of 119860

119895 and hence 119878

119903or 119860119896 subsumes

both 119878119894and 119860

119895 By construction 119878

119903or 119860119896 subsumes

both 119878119903and 119860

119896 Now since 119878

119903or 119860119896 subsumes the

four alterms 119878119894 119860119895 119878119903 and119860

119896 it subsumes each of the

two alterms 119878119894or 119860119896 (which lies in the same column

as 119878119903or 119860119896) and 119878

119903or 119860119895 (which shares the same

row as 119878119903or 119860119896) In conclusion if a general alterm

119878119903or 119860119896 is to be ever absorbed by another alterm in

the table then we can find an absorbing alterm for iteither in the same row 119903 or in the same column 119896

(4) Now suppose that the vertical table key (119860119896or 119883119898)

subsumes (and hence is absorbed by) a table entry

119878119894or 119860119895 which lies in a different column (119895 = 119896)

This means that the set of literals of (119860119896or 119883119898) is a

superset of the set of literals of 119878119894or 119860119895 and hence

it is a superset of each of the set of literals of 119878119894and

that of 119860119895 and hence (119860

119896or 119883119898) subsumes both 119878

119894

and 119860119895 By construction (119860

119896or 119883119898) subsumes 119860

119896

Now since (119860119896or 119883119898) subsumes the two alterms 119878

119894

and 119860119896 it subsumes the alterm 119878

119894or 119860119896 which lies

in the same column as (119860119896or 119883119898) In conclusion if a

table vertical key (119860119896or119883119898) is to be ever absorbed by

a table entry then it has an absorbing alterm for it inthe same column 119896

(5) Likewise it can be shown that if a table horizontal key(119878119903or119883119898) is to be ever absorbed by a table entry then

it has an absorbing alterm for it in the same row 119903

To change the conjunction of alterms in the wholetable (including keys entries and set-aside alterms) intoan absorptive formula there is no need to compare everyaltermwith all other alterms in the whole table Instead everyremaining table entry not equal to 1 is either absorbed inanother in the same row or column of the table or in one ofthe set-aside alterms or it stays unabsorbed A vertical tablekey is either absorbed in a table entry in the same columnof the table or it stays unabsorbed A horizontal table key iseither absorbed in a table entry in the same row of the tableor it stays unabsorbed A set-aside alterm is either absorbedin one of the remaining (not equal to 1) table entries or it staysunabsorbed

In summary the number of comparisons needed toimplement the absorption operationABS( ) is limited in theworst case to the sum of the following operations

(1) comparing each remaining table entry not equal to1 to the alterms with fewer or the same number ofliterals in (119894) its row and column of the table and (119894119894)the set aside alterms

(2) comparing each vertical table key to the table entriesnot equal to 1 with fewer or the same number ofliterals in its column of the table

(3) comparing each horizontal table key to the tableentries not equal to 1 with fewer or the same numberof literals in its row of the table

(4) comparing each of the set-aside alterms to theremaining table entries not equal to 1 with fewer orthe same number of literals

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This paper was funded by the Deanship of Scientific Research(DSR) King Abdulaziz University Jeddah The authorstherefore acknowledge with thanks DSR technical and finan-cial support


References

[1] J A Robinson ldquoA machine-oriented logic based on the reso-lution principlerdquo Journal of the ACM vol 12 no 1 pp 23ndash411965

[2] R C Lee ldquoFuzzy logic and the resolution principlerdquo Journal ofthe Association for Computing Machinery vol 19 no 1 pp 109ndash119 1972

[3] M Mukaidono Fuzzy Inference of Resolution Style PergamonPress New York NY USA 1982

[4] M Mukaidono Z Shen and L Ding ldquoFuzzy prologrdquo inProceedings of the 2nd IFSAWorldCongress pp 844ndash847 TokyoJapan July 1987

[5] Z Shen L Ding andM Mukaidono ldquoA theoretical frameworkof fuzzy prologmachinerdquo in Fuzzy Computing pp 89ndash100 1988

[6] Z Shen L Ding and M Mukaidono ldquoFuzzy resolution prin-ciplerdquo in Proceedings of the 18th International Symposium onMultiple-Valued Logic pp 210ndash215 IEEE Palma de MallorcaSpain May 1988

[7] C S Kim S J Lee S C Park and D S Kim ldquoFuzzy hyper-resolution a semantic inference rule with fuzzy conceptsrdquoKorea FuzzyMathematics and Systems Society vol 3 no 1 1993

[8] C Kim S Park D Kim and S Lee ldquoA fuzzy hyper-resolutionusing compensatory operatorsrdquo Journal of the Korea Informa-tion Science Society vol 21 no 9 pp 520ndash527 1994

[9] C S Kim D S Kim and J S Park ldquoA new fuzzy resolutionprinciple based on the antonymrdquo Fuzzy Sets and Systems vol113 no 2 pp 299ndash307 2000

[10] B R Gaines ldquoFoundations of fuzzy reasoningrdquo InternationalJournal of Man-Machine Studies vol 8 no 6 pp 623ndash668 1976

[11] Y Tsukamoto ldquoAn approach to fuzzy reasoning methodrdquo inAdvances in Fuzzy Set Theory and Applications vol 137 p 149Elsevier Science 1979

[12] M Mizumoto and H-J Zimmermann ldquoComparison of fuzzyreasoning methodsrdquo Fuzzy Sets and Systems vol 8 no 3 pp253ndash283 1982

[13] D Dubois and H Prade ldquoFuzzy logics and the generalizedmodus ponens revisitedrdquo Cybernetics and Systems vol 15 no3-4 pp 293ndash331 1984

[14] P Magrez and P Smets ldquoFuzzy modus ponens a new modelsuitable for applications in knowledge-based systemsrdquo Interna-tional Journal of Intelligent Systems vol 4 no 2 pp 181ndash2001989

[15] H Takagi and I Hayashi ldquoNN-driven fuzzy reasoningrdquo Inter-national Journal of Approximate Reasoning vol 5 no 3 pp 191ndash212 1991

[16] H Hellendoorn ldquoThe generalized modus ponens considered asa fuzzy relationrdquo Fuzzy Sets and Systems vol 46 no 1 pp 29ndash48 1992

[17] K Demirli and I B Turksen ldquoA review of implications andthe generalized modus ponensrdquo in Proceedings of the 3rdIEEE Conference on Fuzzy Systems IEEE World Congress onComputational Intelligence pp 1440ndash1445 IEEE Orlando FlaUSA June 1994

[18] J C Fodor and T Keresztfalvi ldquoNonstandard conjunctions andimplications in fuzzy logicrdquo International Journal of Approxi-mate Reasoning vol 12 no 2 pp 69ndash84 1995

[19] O Cordon M J Del Jesus and F Herrera ldquoA proposal onreasoning methods in fuzzy rule-based classification systemsrdquoInternational Journal of Approximate Reasoning vol 20 no 1pp 21ndash45 1999

[20] R R Yager ldquoOn global requirements for implication operatorsin fuzzy modus ponensrdquo Fuzzy Sets and Systems vol 106 no 1pp 3ndash10 1999

[21] J Liu D Ruan Y Xu and Z Song ldquoA resolution-like strategybased on a lattice-valued logicrdquo IEEE Transactions on FuzzySystems vol 11 no 4 pp 560ndash567 2003

[22] C Igel and K-H Temme ldquoThe chaining syllogism in fuzzylogicrdquo IEEE Transactions on Fuzzy Systems vol 12 no 6 pp849ndash853 2004

[23] J Tick and J Fodor ldquoFuzzy implications and inference pro-cessesrdquo in Proceedings of the 3rd International Conference onComputational Cybernetics (ICCC rsquo05) pp 105ndash109 IEEE April2005

[24] A BlakeCanonical expressions in boolean algebra [PhD thesis]Department of Mathematics University of Chicago ChicagoIll USA 1937

[25] FM BrownBooleanReasoningTheLogic of Boolean EquationsKluwer Academic Publishers Boston Mass USA 1990

[26] J Gregg Ones and Zeros Understanding Boolean AlgebraDigital Circuits and the Logic of Sets Wiley-IEEE Press 1998

[27] A M Rushdi and A S Al-Shehri ldquoLogical reasoning and itssupporting role in the service of security and justicerdquo Journal ofSecurity Studies vol 11 no 22 pp 115ndash153 2002

[28] A M Rushdi and O M Ba-Rukab ldquoSome engineering applica-tions of the modern syllogistic methodrdquo SEC7 Paper 226 2007

[29] A M Rushdi ldquoThe modern syllogistic method as a tool forengineering problem solvingrdquo Journal of Qassim UniversityEngineering and Computer Sciences vol 1 no 1 pp 57ndash70 2008

[30] A M Rushdi and O M Barukab ldquoAn exposition of themodern syllogistic method of propositional logicrdquo Umm Al-Qura University Journal Engineering and Architecture vol 1 no1 pp 17ndash49 2009

[31] A M Rushdi and O M Ba-Rukab ldquoPowerful features of themodern syllogistic method of propositional logicrdquo Journal ofMathematics and Statistics vol 4 no 3 pp 186ndash193 2008

[32] A M A Rushdi and O M Ba-Rukab ldquoSwitching-algebraicanalysis of relational databasesrdquo Journal of Mathematics andStatistics vol 10 no 2 pp 231ndash243 2014

[33] A M Rushdi and O M BaRukab ldquoMap derivation of theclosures for dependency and attribute sets and all candidatekeys for a relational databaserdquo Journal of King AbdulazizUniversity Engineering Sciences vol 25 no 2 pp 3ndash33 2014

[34] C L Chang and R C Lee Symbolic Logic and MechanicalTheorem Proving Academic Press 1973

[35] M Davis and H Putnam ldquoA computing procedure for quan-tification theoryrdquo Journal of the ACM vol 7 no 3 pp 201ndash2151960

[36] I Copi and C Cohen Introduction to Logic Pearson Prentice-Hall Upper Saddle River NJ USA 14th edition 2010

[37] V Klenk Understanding Symbolic Logic Prentice-Hall Engle-wood Cliffs NJ USA 4th edition 2013

[38] K Atanassov ldquoTwo variants of intuitionistic fuzzy propositionalcalculusrdquo Tech Rep IM-MFAIS-5-88 1988

[39] T Ciftcibasi and D Altunay ldquoFuzzy propositional logic andtwo-sided (intuitionistic) fuzzy propositionsrdquo in Proceedings ofthe 5th IEEE International Conference on Fuzzy Systems vol 1pp 432ndash438 IEEE September 1996

[40] K Atanassov and G Gargov ldquoElements of intuitionistic fuzzylogic Part Irdquo Fuzzy Sets and Systems vol 95 no 1 pp 39ndash521998


[41] K T Atanassov Intuitionistic Fuzzy Sets Springer BerlinGermany 1999

[42] C Cornelis G Deschrijver and E E Kerre ldquoClassificationof intuitionistic fuzzy implicators an algebraic approachrdquo inProceedings of the 6th Joint Conference on Information Sciences(JCIS rsquo02) pp 105ndash108 March 2002

[43] K Atanassov ldquoOn eight new intuitionistic fuzzy implicationsrdquoin Proceedings of the 3rd International IEEE Conference onIntelligent Systems (IS rsquo06) pp 4ndash6 London UK September2006

[44] L Atanassova ldquoA new intuitionistic fuzzy implicationrdquo Cyber-netics and InformationTechnologies vol 9 no 2 pp 21ndash25 2009

[45] K T Atanassov ldquoOn intuitionistic fuzzy negations and lawfor excluded middlerdquo in Proceedings of the IEEE InternationalConference on Intelligent Systems (IS rsquo10) pp 266ndash269 July 2010

[46] S-P Wan and D-F Li ldquoAtanassovrsquos intuitionistic fuzzy pro-gramming method for heterogeneous multiattribute groupdecision making with atanassovrsquos intuitionistic fuzzy truthdegreesrdquo IEEE Transactions on Fuzzy Systems vol 22 no 2 pp300ndash312 2014

[47] J-Q Wang and H-Y Zhang ldquoMulticriteria decision-makingapproach based on atanassovrsquos intuitionistic fuzzy sets withincomplete certain information on weightsrdquo IEEE Transactionson Fuzzy Systems vol 21 no 3 pp 510ndash515 2013

[48] E I Papageorgiou and D K Iakovidis ldquoIntuitionistic fuzzycognitive mapsrdquo IEEE Transactions on Fuzzy Systems vol 21no 2 pp 342ndash354 2013

[49] L Zadeh ldquoFuzzy setrdquo Information and Control vol 8 pp 338ndash353 1965

[50] L A Zadeh ldquoFuzzy algorithmsrdquo Information and Control vol12 no 2 pp 94ndash102 1968

[51] P N Marinos ldquoFuzzy logic and its application to switchingsystemsrdquo IEEE Transactions on Computers vol 18 no 4 pp343ndash348 1969

[52] R C Lee and C-l Chang ldquoSome properties of fuzzy logicrdquoInformation and Computation vol 19 pp 417ndash431 1971

[53] L A Zadeh ldquoFuzzy logicrdquo Computer vol 21 no 4 pp 83ndash931988

[54] G J Klir and T A Folger Fuzzy Sets Uncertainty andInformation Prentice Hall Englewood Cliffs NJ USA 1988

[55] T J Ross Fuzzy Logic with Engineering Applications JohnWileyamp Sons Chichester UK 2010

[56] S Muroga Logic Design and Switching Theory John Wiley ampSons 1979

[57] P Tison ldquoGeneralization of consensus theory and applicationto the minimization of boolean functionsrdquo IEEE Transactionson Electronic Computers vol 16 no 4 pp 446ndash456 1967

[58] R B Cutler K Kinoshita and S Muroga Exposition of TisonrsquosMethod to Derive All Prime Implicants and All Irredundant Dis-junctive Forms for a Given Switching Function Department ofComputer Science University of Illinois atUrbana-ChampaignUrbana Ill USA 1979

[59] M Loui and G Bilardi ldquoThe correctness of Tisonrsquos method forgenerating prime implicantsrdquo Tech Rep DTIC 1982

[60] G J Klir and M A Marin ldquoNew considerations in teachingswitching theoryrdquo IEEE Transactions on Education vol 12 no4 pp 257ndash261 1969

[61] D Kalish and RMontague Logic Techniques of Formal Reason-ing Harcourt Brace Jovanovich New York NY USA 1964

[62] A M Rushdi M Zarouan T M Alshehri and M A RushdildquoThe incremental version of the modern syllogistic methodrdquoJournal of King Abdulaziz University Engineering Sciences vol26 no 2 2015

[63] A M Rushdi T M Alshehri M Zarouan and M A RushdildquoUtilization of themodern syllogisticmethod in the explorationof hidden aspects in engineering ethical dilemmasrdquo Journal ofKing Abdulaziz University Computers and Information Technol-ogy vol 2 no 2 2015

[64] A M Rushdi and H A Al-Yahya ldquoDerivation of the completesum of a switching function with the aid of the variable enteredkarnaugh maprdquo Journal of King Saud University vol 13 no 2pp 239ndash269 2000

[65] AM A Rushdi andHM Albarakati ldquoThe inverse problem forBoolean equationsrdquo Journal of Computer Science vol 8 no 12pp 2098ndash2105 2012

[66] A Kean and G Tsiknis ldquoAn incremental method for generatingprime implicantsimplicatesrdquo Journal of Symbolic Computationvol 9 no 2 pp 185ndash206 1990

[67] AM A Rushdi andHM Albarakati ldquoConstruction of generalsubsumptive solutions of Boolean equations via complete-sumderivationrdquo Journal of Mathematics and Statistics vol 10 no 2pp 155ndash168 2014

[68] J R Slagle C L Chang and R C Lee ldquoA new algorithm forgenerating prime implicantsrdquo IEEE Transactions on Computersvol C-19 no 4 pp 304ndash310 1970

[69] H R Hwa ldquoA method for generating prime implicants of aboolean expressionrdquo IEEE Transactions on Computers vol 23no 6 pp 637ndash641 1974

[70] B Reusch ldquoGeneration of prime implicants from subfunctionsand a unifying approach to the covering problemrdquo IEEETransactions on Computers vol 100 no 9 pp 924ndash930 1975

[71] O Coudert and JMadre ldquoA newmethod to compute prime andessential prime implicants of boolean functionsrdquo in AdvancedResearch in VLSI and Parallel Systems T Knight and J SavageEds pp 113ndash128 MIT Press 1992

[72] A M Rushdi and H A Al-Yahya ldquoA boolean minimizationprocedure using the variable-entered karnaugh map and thegeneralized consensus conceptrdquo International Journal of Elec-tronics vol 87 no 7 pp 769ndash794 2000

[73] A Rushdi ldquoPrime-implicant extraction with the aid of thevariable-entered karnaughmaprdquoUmmAl-QuraUniversity Jour-nal Science Medicine and Engineering vol 13 no 1 pp 53ndash742001

[74] G Alexe S Alexe Y Crama S Foldes P L Hammer andB Simeone ldquoConsensus algorithms for the generation of allmaximal bicliquesrdquo Discrete Applied Mathematics vol 145 no1 pp 11ndash21 2004

[75] D Slęzak ldquoAssociation reducts boolean representationrdquo inRough Sets andKnowledge Technology vol 4062 of LectureNotesin Computer Science pp 305ndash312 Springer Berlin Germany2006

[76] Z Pawlak andA Skowron ldquoRough sets and boolean reasoningrdquoInformation Sciences vol 177 no 1 pp 41ndash73 2007

[77] Y Crama and P L Hammer Boolean Functions Theory Algo-rithms and Applications vol 142 Cambridge University PressCambridge UK 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

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Applied MathematicsJournal of


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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


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Algebra

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Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


form The MSM uses just a single rule of inference ratherthan the many rules of inference conventionally employedin propositional-logic deduction (see eg [36 37]) In factthe MSM includes all such rules of inference as special cases[30 31] The MSM strategy is to convert the set of premisesinto a single equation of the form 119891 = 0 or 119892 = 1 and obtainCS(119891) = the complete sum of 119891 (or CP(119892) = the completeproduct of 119892) The set of all possible prime consequents ofthe original premises are obtained from the final equationCS(119891) = 0 (or CP(119892) = 1)

We describe herein a fuzzy version of the MSM thatutilizes concepts of the Intuitionistic Fuzzy Logic (IFL) [38ndash48] developed mainly by Atanassov [38 40 41 43ndash45] Thisfuzzy MSM reduces to a restricted version in the OrdinaryFuzzy Logic (OFL) of Zadeh [34 49ndash55] The IFL versionof the MSM is more flexible while the OFL version issimpler and computationally fasterWemanaged to adapt theMSM to fuzzy reasoning without any dramatic changes of itsmain steps In particular our algorithm for constructing thecomplete product (or complete sum) of a logic function viaconsensus generation and absorption remains essentially thesame This algorithm was first developed by Blake [24] andlater by Tison [56ndash59] It is usually referred to as the Tisonmethod butwewill name it herein as the Blake-Tisonmethodor algorithm

The organization of the rest of this paper is as followsSection 2 briefly reviews the concept of Intuitionistic FuzzyLogic (IFL) and asserts why it adds necessary flexibility toOrdinary Fuzzy Logic (OFL) Section 3 combines ideas fromLee [2] and Atanassov [38 40 41 43ndash45] to produce a novelsimple concept of a Realistic Fuzzy Tautology (RFT) andexplainswhy such a new concept is needed Section 4 outlinesthe steps of MSM in two-valued Boolean logic and thenadapts it to realistic fuzzy logic which is an IFL in whichthe new RFT concept is embedded Formal proofs of thecorrectness of this adaptation are provided Three examplesare given in Section 5 to demonstrate the computationalsteps and to demonstrate how similar to the result of Lee[2] the validity of the least truthful premise sets an upperlimit on the validity of every logical consequent Section 6concludes the paper The Appendix provides a descriptionof an improved version of the Blake-Tison algorithm forproducing the complete product of a logical function

2 Review of Intuitionistic Fuzzy Logic

In Intuitionistic Fuzzy Logic (IFL) a variable 119883119894is repre-

sented by its validity which is the ordered couple

119881 (119883119894) = ⟨119886

119894 119887119894⟩ (1)

where 119886119894and 119887

119894are degrees of truth and falsity of 119883

119894

respectively such that each of the real numbers 119886119894 119887119894 119886119894+ 119887119894isin

[0 1]Note that when 119886

119894+ 119887119894= 1 then IFL reduces to

Ordinary Fuzzy Logic (OFL) in which 119886119894alone suffices as a

representation for119883119894 since 119887

119894is automatically determined by

119887119894= 1minus 119886

119894 The necessity of allowing the condition (119886

119894+ 119887119894) le

1 is established on the grounds that it allows a degree of

hesitancy ignorance or uncertainty when one can neitherdesignate a variable as true nor label it as false

Since IFL includesOFL as a special case operations in IFLshould be defined such that they serve as extensions to theirOFL counterparts However this allows the existence ofmanydefinitions for pertinent operations such as the negationoperation [45] or the implication operation [43]Wewill stickherein to themost familiar definitionsWehave a single unaryoperation namely the negation operation which producesthe complement119883

119894of a variable119883

119894 We define this operation

as one that interchanges the truth and falsity of the variablethat is

119881(119883119894) = ⟨119887

119894 119886119894⟩ (2)

The most important binary operations are

(i) the intuitionistic conjunction ormeet operation (1198831and

1198832) defined by

119881 (1198831and 1198832) = ⟨min (119886

1 1198862) max (119887

1 1198872)⟩ (3)

(ii) the intuitionistic disjunction or join operation (1198831or

1198832) defined by

119881 (1198831or 1198832) = ⟨max (119886

1 1198862) min (119887

1 1198872)⟩ (4)

(iii) the intuitionistic implication operation (1198831rarr 1198832) equiv

(1198831or 1198832) defined herein by

119881 (1198831997888rarr119883

2) = ⟨max (119887

1 1198862) min (119886

1 1198872)⟩ (5)

With any three intuitionistic fuzzy variables 1198831 1198832 and

1198833 the following pairs of dual theorems are satisfied

(1) idempotency

1198831or 1198831= 1198831

1198831and 1198831= 1198831

(6)

(2) commutativity

1198831or 1198832= 1198832or 1198831

1198831and 1198832= 1198832and 1198831

(7)

(3) associativity

(1198831or 1198832) or 1198833= 1198831or (1198832or 1198833)

(1198831and 1198832) and 1198833= 1198831and (1198832and 1198833)

(8)

(4) absorption

1198831or (1198831and 1198832) = 119883

1

1198831and (1198831or 1198832) = 119883

1

(9)


(5) distributivity

1198831or (1198832and 1198833) = (119883

1or 1198832) and (119883

1or 1198833)

1198831and (1198832or 1198833) = (119883

1and 1198832) or (119883

1and 1198833)

(10)

(6) identities

1198831or 0 = 119883

1

1198831and 1 = 119883

1

(11)




1) le 119881(119883

2)) if and only if


valid than this IFT



119894in IFT



119894 then












119879119894equiv 119876119894 1 le 119894 le 119899 (12)


119875119894= (119879119894or 119876119894) and (119879

119894or 119876119894) 1 le 119894 le 119899 (13)


119879119894997888rarr 119876

119894 (119899 + 1) le 119894 le 119898 (14)


119894rdquo



the equational form

119875119894= 119879119894or 119876119894= 1 (119899 + 1) le 119894 le 119898 (15)


119892 =

119898

⋀

119894=1

119875119894

=

119899

⋀

119894=1

((119879119894or 119876119894) and (119879

119894or 119876119894)) and

119898

⋀

119894=119899+1

(119879119894or 119876119894)

(16)




CP (119892) =119897

⋀

119894=1

119862119894= 1 (17)



119862119894= 1 1 le 119894 le 119897 (18)


119894is given




119862119894=

119903

⋁

119895=1

119883119894119895or

119904

⋁

119895=1

119884119894119895 1 le 119894 le 119897 (19)


⋀

119895=1

119883119894119895997888rarr

119904

⋁

119895=1

119884119894119895 1 le 119894 le 119897 (20)






119881 (119875119894) = ⟨120583

119894 120574119894⟩ (21)


119892 =

119898

⋀

119894

119875119894

119881 (119892) = ⟨min119894

120583119894max119894

120574119894⟩

(22)






119894 1 le 119894 le 119897 in (17)









119875119894is an RFT 1 le 119894 le 119898

lArrrArr 120583119894gt 05 1 le 119894 le 119898

lArrrArr min119894

120583119894gt 05

lArrrArr

119898

⋀

119894=1

119875119894is an RFT

(23)


1or 1198832) and (119883

1or 1198833) is an RFT then (119883

1or

1198832) and (119883

1or 1198833) and (119883


Proof Let 119881(119883119894) = ⟨119886




(1198831or1198832) is an RFT (ie max(119886

1 1198862) gt 05) and that (119883

1or1198833)

is an RFT (ie max(1198871 1198863) gt 05)



1le 05 and hence

max (1198871 1198863) gt 05 997904rArr 119886

3gt 05

997904rArr max (1198862 1198863) gt 05

997904rArr (1198832or 1198833) is an RFT

(24)


1198861le 05 and max (119886

1 1198862) gt 05 997904rArr 119886

2gt 05

997904rArr max (1198862 1198863) gt 05 997904rArr (119883


(25)




(1198831or 1198833) and (119883



119883119894or 119883119894= 1

119883119894and 119883119894= 0

(26)




119894and 119883119894) might appear



5 Examples






1198921= (119878 or 119866) and (119878 or 119864) and (119866 or 119864) = 1 (27)







and (119878 or 119866) and 119866

= (119878 or 119864) and 119866

(28)




119881 (119878 997888rarr 119866) = 119881 (119878 or 119866) = ⟨06 03⟩

119881 (119878 997888rarr 119864) = 119881 (119878 or 119864) = ⟨09 01⟩

119881 (119866 997888rarr 119864) = 119881 (119866 or 119864) = ⟨08 01⟩

(29)



119881 (1198921) = 119881 ((119878 or 119866) and (119878 or 119864) and (119866 or 119864))

= ⟨min (06 09 08) max (03 01 01)⟩

= ⟨06 03⟩

(30)





119881 (119866) = ⟨06 03⟩ (31)



(119860 or 119861) and (119860 or 119861) = 1

(119861 or 119862) and (119861 or 119862) = 1

(119862 or 119860) and (119862 or 119860) = 1

(32)



and (119862 or 119860) and (119862 and 119860) = 1

(33)










Table1MSM

deriv

ationof

fuzzyversions

offamou

srules

ofinference

thep

artic

ular

conclusio

nof

aruleish

ighlighted

inbo

ld

Rulename

FuzzyRF

Tantecedents

(premise

s)

Prem

isesa

sseparate

fuzzyequatio

ns119875119894=RF

T119894=1119898

Prem

isesa

sasin

glefuzzy

equatio

n119892=RF

TCon

clusio

nsas

asinglefuzzy

equatio

nCP(119892)=RF

TCon

clusio

nsas

separatefuzzy

equatio

ns119862119894=RF

T119894=1119897

FuzzyRF

Tconsequence

(con

clusio

n)

MODUSPO

NEN

S119860rarr119861

119860

119860or119861=RF

T119860=RF

T(119860or119861)and119860=RF

T119861and119860=RF

TB=RF

T119860=RF

T119861

MODUSTO

LLEN

S119860rarr119861

119861

119860or119861=RF

T119861=RF

T(119860or119861)and119861=RF

T119860and119861=RF

TA=RF

T119861=RF

T119860

HYP

OTH

ETICAL

SYLL

OGISM

119860rarr119861

119861rarr119862

119860or119861=RF

T119861or119862=RF

T(119860or119861)and(119861or119862)=RF


T119860or119861=RF

T119861or119862=RF

TAorC=RF

T119860rarr119862

SIMPL

IFICAT

ION

119860and119861

119860and119861=RF

T119860and119861=RF

T119860and119861=RF

TA=RF

T119861=RF

T119860

CONJU

NCT

ION

119860 119861

119860=RF

T119861=RF

T119860and119861=RF

T119860and119861=RF

TAandB=RF

T119860and119861

CONST

RUCT

IVE

DILEM

MA

119860rarr119861

119862rarr119863

119860or119862

119860or119861=RF

T119862or119863=RF

T119860or119862=RF

T

(119860or119861)and(119862or119863)

and(119860or119862)=RF


and(119860or119863)and(119861or119863)=RF

T

(119860or119861)=RF

T(119862or119863)=RF

T(119860or119862)=RF

T(119861or119862)=RF

T(119860or119863)=RF

T(BorD)=RF

T

119861or119863

DISJU

NCT

IVESY


119860

119860or119861=RF

T119860=RF

T(119860or119861)and119860=RF

T119861and119860=RF

TB=RF

T119860=RF

T119861

ADDITIO

N119860

119860=RF

T119860=RF

T119860=RF

TAorB=RF

TAnalterm

subsum

ingan

RFT

alterm

isalso

RFT

119860or119861

ABS

ORP

TION

119860rarr119861

119860or119861=RF

T119860or119861=RF

T119860or119861=RF

TAorAB=RF

T119860or119860119861=119860or119861by

reflection

law

119860rarr119860119861

CASE

S

119860 119860rarr(119862or119863)

119862rarr119861

119863rarr119861

119860=RF

T119860or119862or119863=RF

T119862or119861=RF

T119863or119861=RF

T

119860and(119860or119862or119863)and(119862or119861)

and(119863or119861)=RF

T119860and119861and(119862or119863)=RF

T119860=RF

TB=RF

T119862or119863=RF

T119861

CASE

ELIM

INAT

ION

119860or119861

119860rarr(119862and119862)

119860or119861=RF

T119860or(119862and119862)=RF

T(119860or119861)and(119860or(119862and119862))=RF

T(119860or119861)and(119860or(119862and119862))

and(119861or(119862or119862))=RF

T

119860or119861=RF

T119860or(119862and119862)=RF

T119861or(119862and119862)=RF

TB=RF


FT

119861

REDUCT

IOAD

ABS

URD

UM

(CONTR

ADICTION)

119860rarr(119861and119861)

119860or(119861and119861)=RF

T119860or(119861and119861)=RF

T119860or(119861and119861)=RF

T119860or(119861and119861)=RF

TA=RF

T119861and119861=nR

FT

119860













3= 1) where







(37)



119877 or 119864 = 1 (38)

or equivalently

119877 997888rarr 119864 (39)



119881 (119877 997888rarr 119875) = 119881 (119877 or 119875) = ⟨06 03⟩

119881 (119875 997888rarr119863) = 119881 (119875 or 119863) = ⟨07 02⟩

119881 (119863 997888rarr 119865) = 119881 (119863 or 119865) = ⟨08 01⟩

119881 (119864 997888rarr 119865) = 119881 (119864 or 119865) = ⟨09 01⟩

(40)



6 Conclusion








Appendix







of literals 1199040= 119860

(0)

1 119860(0)

2 119860

(0)


1198831 1198832 119883







119898

(1) For 1 le 119895 le 119896 le 119899(119898minus1)



and 119860(119898minus1)119896






119898minus1of 119869(119898minus1)


ge

119899(119898minus1)


119860(119898minus1)119896


119898minus1 If 119860(119898minus1)119895




119895is




119898=

119860(119898)

1 119860(119898)

2 119860(119898)

119899119898



119898 where




A or CB or C

A or B

B

C

0

000

A

(a)

B or C

A

B

C

A

00

00 0

(b)

A or BA or BB

C

0

0

0

0

A

(c)



119898 1 le 119898 le 119872 in the form

119892 = (119903 or 119883119898) and (119904 or 119883

119898) and 119905 (A1)

where 119903 = ⋀119899119903119894=1119903119894 119904 = ⋀119899119904

119895=1119904119895 and 119905 = ⋀119899119905

119896=1119905119896are POS


119894 119904119895



119892 =

119899119903

⋀

119894=1

(119903119894or 119883119898) and

119899119904

⋀

119895=1

(119904119895and 119883119898) and

119899119905

⋀

119896=1

119905119896 (A2)


119894or 119904119895)which do not


119899119903

⋀

119894=1

119899119904

⋀

119895=1

(119903119894or 119904119895) (A3)







119898


119898are set aside


119896or 119883119898) sdot sdot sdot

(119878119894or 119883119898) sdot sdot sdot 119878

119894or 119860119895 sdot sdot sdot 119878

119894or 119860119896 sdot sdot sdot

(119878119903or 119883119898) sdot sdot sdot 119878

119903or 119860119895 sdot sdot sdot 119878

119903or 119860119896 sdot sdot sdot


119898nor119883

119898)








119898are set





119894or 119883119898) which is the




119894and 119860



119894or
















119898




119894or119860119895


119895 and hence 119878

119903or 119860119896 subsumes

both 119878119894and 119860


119903or 119860119896 subsumes

both 119878119903and 119860

119896 Now since 119878

119903or 119860119896 subsumes the

four alterms 119878119894 119860119895 119878119903 and119860



as 119878119903or 119860119896) and 119878











that of 119860119895 and hence (119860

119896or 119883119898) subsumes both 119878

119894


119896or 119883119898) subsumes 119860

119896


119894


119894or 119860119896 which lies














Acknowledgment



References






















































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(5) distributivity

1198831or (1198832and 1198833) = (119883

1or 1198832) and (119883

1or 1198833)

1198831and (1198832or 1198833) = (119883

1and 1198832) or (119883

1and 1198833)

(10)

(6) identities

1198831or 0 = 119883

1

1198831and 1 = 119883

1

(11)




1) le 119881(119883

2)) if and only if


valid than this IFT



119894in IFT



119894 then












119879119894equiv 119876119894 1 le 119894 le 119899 (12)


119875119894= (119879119894or 119876119894) and (119879

119894or 119876119894) 1 le 119894 le 119899 (13)


119879119894997888rarr 119876

119894 (119899 + 1) le 119894 le 119898 (14)


119894rdquo



the equational form

119875119894= 119879119894or 119876119894= 1 (119899 + 1) le 119894 le 119898 (15)


119892 =

119898

⋀

119894=1

119875119894

=

119899

⋀

119894=1

((119879119894or 119876119894) and (119879

119894or 119876119894)) and

119898

⋀

119894=119899+1

(119879119894or 119876119894)

(16)




CP (119892) =119897

⋀

119894=1

119862119894= 1 (17)



119862119894= 1 1 le 119894 le 119897 (18)


119894is given




119862119894=

119903

⋁

119895=1

119883119894119895or

119904

⋁

119895=1

119884119894119895 1 le 119894 le 119897 (19)


⋀

119895=1

119883119894119895997888rarr

119904

⋁

119895=1

119884119894119895 1 le 119894 le 119897 (20)






119881 (119875119894) = ⟨120583

119894 120574119894⟩ (21)


119892 =

119898

⋀

119894

119875119894

119881 (119892) = ⟨min119894

120583119894max119894

120574119894⟩

(22)






119894 1 le 119894 le 119897 in (17)









119875119894is an RFT 1 le 119894 le 119898

lArrrArr 120583119894gt 05 1 le 119894 le 119898

lArrrArr min119894

120583119894gt 05

lArrrArr

119898

⋀

119894=1

119875119894is an RFT

(23)


1or 1198832) and (119883

1or 1198833) is an RFT then (119883

1or

1198832) and (119883

1or 1198833) and (119883


Proof Let 119881(119883119894) = ⟨119886




(1198831or1198832) is an RFT (ie max(119886

1 1198862) gt 05) and that (119883

1or1198833)

is an RFT (ie max(1198871 1198863) gt 05)



1le 05 and hence

max (1198871 1198863) gt 05 997904rArr 119886

3gt 05

997904rArr max (1198862 1198863) gt 05

997904rArr (1198832or 1198833) is an RFT

(24)


1198861le 05 and max (119886

1 1198862) gt 05 997904rArr 119886

2gt 05

997904rArr max (1198862 1198863) gt 05 997904rArr (119883


(25)




(1198831or 1198833) and (119883



119883119894or 119883119894= 1

119883119894and 119883119894= 0

(26)




119894and 119883119894) might appear



5 Examples






1198921= (119878 or 119866) and (119878 or 119864) and (119866 or 119864) = 1 (27)







and (119878 or 119866) and 119866

= (119878 or 119864) and 119866

(28)




119881 (119878 997888rarr 119866) = 119881 (119878 or 119866) = ⟨06 03⟩

119881 (119878 997888rarr 119864) = 119881 (119878 or 119864) = ⟨09 01⟩

119881 (119866 997888rarr 119864) = 119881 (119866 or 119864) = ⟨08 01⟩

(29)



119881 (1198921) = 119881 ((119878 or 119866) and (119878 or 119864) and (119866 or 119864))

= ⟨min (06 09 08) max (03 01 01)⟩

= ⟨06 03⟩

(30)





119881 (119866) = ⟨06 03⟩ (31)



(119860 or 119861) and (119860 or 119861) = 1

(119861 or 119862) and (119861 or 119862) = 1

(119862 or 119860) and (119862 or 119860) = 1

(32)



and (119862 or 119860) and (119862 and 119860) = 1

(33)










Table1MSM

deriv

ationof

fuzzyversions

offamou

srules

ofinference

thep

artic

ular

conclusio

nof

aruleish

ighlighted

inbo

ld

Rulename

FuzzyRF

Tantecedents

(premise

s)

Prem

isesa

sseparate

fuzzyequatio

ns119875119894=RF

T119894=1119898

Prem

isesa

sasin

glefuzzy

equatio

n119892=RF

TCon

clusio

nsas

asinglefuzzy

equatio

nCP(119892)=RF

TCon

clusio

nsas

separatefuzzy

equatio

ns119862119894=RF

T119894=1119897

FuzzyRF

Tconsequence

(con

clusio

n)

MODUSPO

NEN

S119860rarr119861

119860

119860or119861=RF

T119860=RF

T(119860or119861)and119860=RF

T119861and119860=RF

TB=RF

T119860=RF

T119861

MODUSTO

LLEN

S119860rarr119861

119861

119860or119861=RF

T119861=RF

T(119860or119861)and119861=RF

T119860and119861=RF

TA=RF

T119861=RF

T119860

HYP

OTH

ETICAL

SYLL

OGISM

119860rarr119861

119861rarr119862

119860or119861=RF

T119861or119862=RF

T(119860or119861)and(119861or119862)=RF


T119860or119861=RF

T119861or119862=RF

TAorC=RF

T119860rarr119862

SIMPL

IFICAT

ION

119860and119861

119860and119861=RF

T119860and119861=RF

T119860and119861=RF

TA=RF

T119861=RF

T119860

CONJU

NCT

ION

119860 119861

119860=RF

T119861=RF

T119860and119861=RF

T119860and119861=RF

TAandB=RF

T119860and119861

CONST

RUCT

IVE

DILEM

MA

119860rarr119861

119862rarr119863

119860or119862

119860or119861=RF

T119862or119863=RF

T119860or119862=RF

T

(119860or119861)and(119862or119863)

and(119860or119862)=RF


and(119860or119863)and(119861or119863)=RF

T

(119860or119861)=RF

T(119862or119863)=RF

T(119860or119862)=RF

T(119861or119862)=RF

T(119860or119863)=RF

T(BorD)=RF

T

119861or119863

DISJU

NCT

IVESY


119860

119860or119861=RF

T119860=RF

T(119860or119861)and119860=RF

T119861and119860=RF

TB=RF

T119860=RF

T119861

ADDITIO

N119860

119860=RF

T119860=RF

T119860=RF

TAorB=RF

TAnalterm

subsum

ingan

RFT

alterm

isalso

RFT

119860or119861

ABS

ORP

TION

119860rarr119861

119860or119861=RF

T119860or119861=RF

T119860or119861=RF

TAorAB=RF

T119860or119860119861=119860or119861by

reflection

law

119860rarr119860119861

CASE

S

119860 119860rarr(119862or119863)

119862rarr119861

119863rarr119861

119860=RF

T119860or119862or119863=RF

T119862or119861=RF

T119863or119861=RF

T

119860and(119860or119862or119863)and(119862or119861)

and(119863or119861)=RF

T119860and119861and(119862or119863)=RF

T119860=RF

TB=RF

T119862or119863=RF

T119861

CASE

ELIM

INAT

ION

119860or119861

119860rarr(119862and119862)

119860or119861=RF

T119860or(119862and119862)=RF

T(119860or119861)and(119860or(119862and119862))=RF

T(119860or119861)and(119860or(119862and119862))

and(119861or(119862or119862))=RF

T

119860or119861=RF

T119860or(119862and119862)=RF

T119861or(119862and119862)=RF

TB=RF


FT

119861

REDUCT

IOAD

ABS

URD

UM

(CONTR

ADICTION)

119860rarr(119861and119861)

119860or(119861and119861)=RF

T119860or(119861and119861)=RF

T119860or(119861and119861)=RF

T119860or(119861and119861)=RF

TA=RF

T119861and119861=nR

FT

119860













3= 1) where







(37)



119877 or 119864 = 1 (38)

or equivalently

119877 997888rarr 119864 (39)



119881 (119877 997888rarr 119875) = 119881 (119877 or 119875) = ⟨06 03⟩

119881 (119875 997888rarr119863) = 119881 (119875 or 119863) = ⟨07 02⟩

119881 (119863 997888rarr 119865) = 119881 (119863 or 119865) = ⟨08 01⟩

119881 (119864 997888rarr 119865) = 119881 (119864 or 119865) = ⟨09 01⟩

(40)



6 Conclusion








Appendix







of literals 1199040= 119860

(0)

1 119860(0)

2 119860

(0)


1198831 1198832 119883







119898

(1) For 1 le 119895 le 119896 le 119899(119898minus1)



and 119860(119898minus1)119896






119898minus1of 119869(119898minus1)


ge

119899(119898minus1)


119860(119898minus1)119896


119898minus1 If 119860(119898minus1)119895




119895is




119898=

119860(119898)

1 119860(119898)

2 119860(119898)

119899119898



119898 where




A or CB or C

A or B

B

C

0

000

A

(a)

B or C

A

B

C

A

00

00 0

(b)

A or BA or BB

C

0

0

0

0

A

(c)



119898 1 le 119898 le 119872 in the form

119892 = (119903 or 119883119898) and (119904 or 119883

119898) and 119905 (A1)

where 119903 = ⋀119899119903119894=1119903119894 119904 = ⋀119899119904

119895=1119904119895 and 119905 = ⋀119899119905

119896=1119905119896are POS


119894 119904119895



119892 =

119899119903

⋀

119894=1

(119903119894or 119883119898) and

119899119904

⋀

119895=1

(119904119895and 119883119898) and

119899119905

⋀

119896=1

119905119896 (A2)


119894or 119904119895)which do not


119899119903

⋀

119894=1

119899119904

⋀

119895=1

(119903119894or 119904119895) (A3)







119898


119898are set aside


119896or 119883119898) sdot sdot sdot

(119878119894or 119883119898) sdot sdot sdot 119878

119894or 119860119895 sdot sdot sdot 119878

119894or 119860119896 sdot sdot sdot

(119878119903or 119883119898) sdot sdot sdot 119878

119903or 119860119895 sdot sdot sdot 119878

119903or 119860119896 sdot sdot sdot


119898nor119883

119898)








119898are set





119894or 119883119898) which is the




119894and 119860



119894or
















119898




119894or119860119895


119895 and hence 119878

119903or 119860119896 subsumes

both 119878119894and 119860


119903or 119860119896 subsumes

both 119878119903and 119860

119896 Now since 119878

119903or 119860119896 subsumes the

four alterms 119878119894 119860119895 119878119903 and119860



as 119878119903or 119860119896) and 119878











that of 119860119895 and hence (119860

119896or 119883119898) subsumes both 119878

119894


119896or 119883119898) subsumes 119860

119896


119894


119894or 119860119896 which lies














Acknowledgment



References






















































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119862119894=

119903

⋁

119895=1

119883119894119895or

119904

⋁

119895=1

119884119894119895 1 le 119894 le 119897 (19)


⋀

119895=1

119883119894119895997888rarr

119904

⋁

119895=1

119884119894119895 1 le 119894 le 119897 (20)






119881 (119875119894) = ⟨120583

119894 120574119894⟩ (21)


119892 =

119898

⋀

119894

119875119894

119881 (119892) = ⟨min119894

120583119894max119894

120574119894⟩

(22)






119894 1 le 119894 le 119897 in (17)









119875119894is an RFT 1 le 119894 le 119898

lArrrArr 120583119894gt 05 1 le 119894 le 119898

lArrrArr min119894

120583119894gt 05

lArrrArr

119898

⋀

119894=1

119875119894is an RFT

(23)


1or 1198832) and (119883

1or 1198833) is an RFT then (119883

1or

1198832) and (119883

1or 1198833) and (119883


Proof Let 119881(119883119894) = ⟨119886




(1198831or1198832) is an RFT (ie max(119886

1 1198862) gt 05) and that (119883

1or1198833)

is an RFT (ie max(1198871 1198863) gt 05)



1le 05 and hence

max (1198871 1198863) gt 05 997904rArr 119886

3gt 05

997904rArr max (1198862 1198863) gt 05

997904rArr (1198832or 1198833) is an RFT

(24)


1198861le 05 and max (119886

1 1198862) gt 05 997904rArr 119886

2gt 05

997904rArr max (1198862 1198863) gt 05 997904rArr (119883


(25)




(1198831or 1198833) and (119883



119883119894or 119883119894= 1

119883119894and 119883119894= 0

(26)




119894and 119883119894) might appear



5 Examples






1198921= (119878 or 119866) and (119878 or 119864) and (119866 or 119864) = 1 (27)







and (119878 or 119866) and 119866

= (119878 or 119864) and 119866

(28)




119881 (119878 997888rarr 119866) = 119881 (119878 or 119866) = ⟨06 03⟩

119881 (119878 997888rarr 119864) = 119881 (119878 or 119864) = ⟨09 01⟩

119881 (119866 997888rarr 119864) = 119881 (119866 or 119864) = ⟨08 01⟩

(29)



119881 (1198921) = 119881 ((119878 or 119866) and (119878 or 119864) and (119866 or 119864))

= ⟨min (06 09 08) max (03 01 01)⟩

= ⟨06 03⟩

(30)





119881 (119866) = ⟨06 03⟩ (31)



(119860 or 119861) and (119860 or 119861) = 1

(119861 or 119862) and (119861 or 119862) = 1

(119862 or 119860) and (119862 or 119860) = 1

(32)



and (119862 or 119860) and (119862 and 119860) = 1

(33)










Table1MSM

deriv

ationof

fuzzyversions

offamou

srules

ofinference

thep

artic

ular

conclusio

nof

aruleish

ighlighted

inbo

ld

Rulename

FuzzyRF

Tantecedents

(premise

s)

Prem

isesa

sseparate

fuzzyequatio

ns119875119894=RF

T119894=1119898

Prem

isesa

sasin

glefuzzy

equatio

n119892=RF

TCon

clusio

nsas

asinglefuzzy

equatio

nCP(119892)=RF

TCon

clusio

nsas

separatefuzzy

equatio

ns119862119894=RF

T119894=1119897

FuzzyRF

Tconsequence

(con

clusio

n)

MODUSPO

NEN

S119860rarr119861

119860

119860or119861=RF

T119860=RF

T(119860or119861)and119860=RF

T119861and119860=RF

TB=RF

T119860=RF

T119861

MODUSTO

LLEN

S119860rarr119861

119861

119860or119861=RF

T119861=RF

T(119860or119861)and119861=RF

T119860and119861=RF

TA=RF

T119861=RF

T119860

HYP

OTH

ETICAL

SYLL

OGISM

119860rarr119861

119861rarr119862

119860or119861=RF

T119861or119862=RF

T(119860or119861)and(119861or119862)=RF


T119860or119861=RF

T119861or119862=RF

TAorC=RF

T119860rarr119862

SIMPL

IFICAT

ION

119860and119861

119860and119861=RF

T119860and119861=RF

T119860and119861=RF

TA=RF

T119861=RF

T119860

CONJU

NCT

ION

119860 119861

119860=RF

T119861=RF

T119860and119861=RF

T119860and119861=RF

TAandB=RF

T119860and119861

CONST

RUCT

IVE

DILEM

MA

119860rarr119861

119862rarr119863

119860or119862

119860or119861=RF

T119862or119863=RF

T119860or119862=RF

T

(119860or119861)and(119862or119863)

and(119860or119862)=RF


and(119860or119863)and(119861or119863)=RF

T

(119860or119861)=RF

T(119862or119863)=RF

T(119860or119862)=RF

T(119861or119862)=RF

T(119860or119863)=RF

T(BorD)=RF

T

119861or119863

DISJU

NCT

IVESY


119860

119860or119861=RF

T119860=RF

T(119860or119861)and119860=RF

T119861and119860=RF

TB=RF

T119860=RF

T119861

ADDITIO

N119860

119860=RF

T119860=RF

T119860=RF

TAorB=RF

TAnalterm

subsum

ingan

RFT

alterm

isalso

RFT

119860or119861

ABS

ORP

TION

119860rarr119861

119860or119861=RF

T119860or119861=RF

T119860or119861=RF

TAorAB=RF

T119860or119860119861=119860or119861by

reflection

law

119860rarr119860119861

CASE

S

119860 119860rarr(119862or119863)

119862rarr119861

119863rarr119861

119860=RF

T119860or119862or119863=RF

T119862or119861=RF

T119863or119861=RF

T

119860and(119860or119862or119863)and(119862or119861)

and(119863or119861)=RF

T119860and119861and(119862or119863)=RF

T119860=RF

TB=RF

T119862or119863=RF

T119861

CASE

ELIM

INAT

ION

119860or119861

119860rarr(119862and119862)

119860or119861=RF

T119860or(119862and119862)=RF

T(119860or119861)and(119860or(119862and119862))=RF

T(119860or119861)and(119860or(119862and119862))

and(119861or(119862or119862))=RF

T

119860or119861=RF

T119860or(119862and119862)=RF

T119861or(119862and119862)=RF

TB=RF


FT

119861

REDUCT

IOAD

ABS

URD

UM

(CONTR

ADICTION)

119860rarr(119861and119861)

119860or(119861and119861)=RF

T119860or(119861and119861)=RF

T119860or(119861and119861)=RF

T119860or(119861and119861)=RF

TA=RF

T119861and119861=nR

FT

119860













3= 1) where







(37)



119877 or 119864 = 1 (38)

or equivalently

119877 997888rarr 119864 (39)



119881 (119877 997888rarr 119875) = 119881 (119877 or 119875) = ⟨06 03⟩

119881 (119875 997888rarr119863) = 119881 (119875 or 119863) = ⟨07 02⟩

119881 (119863 997888rarr 119865) = 119881 (119863 or 119865) = ⟨08 01⟩

119881 (119864 997888rarr 119865) = 119881 (119864 or 119865) = ⟨09 01⟩

(40)



6 Conclusion








Appendix







of literals 1199040= 119860

(0)

1 119860(0)

2 119860

(0)


1198831 1198832 119883







119898

(1) For 1 le 119895 le 119896 le 119899(119898minus1)



and 119860(119898minus1)119896






119898minus1of 119869(119898minus1)


ge

119899(119898minus1)


119860(119898minus1)119896


119898minus1 If 119860(119898minus1)119895




119895is




119898=

119860(119898)

1 119860(119898)

2 119860(119898)

119899119898



119898 where




A or CB or C

A or B

B

C

0

000

A

(a)

B or C

A

B

C

A

00

00 0

(b)

A or BA or BB

C

0

0

0

0

A

(c)



119898 1 le 119898 le 119872 in the form

119892 = (119903 or 119883119898) and (119904 or 119883

119898) and 119905 (A1)

where 119903 = ⋀119899119903119894=1119903119894 119904 = ⋀119899119904

119895=1119904119895 and 119905 = ⋀119899119905

119896=1119905119896are POS


119894 119904119895



119892 =

119899119903

⋀

119894=1

(119903119894or 119883119898) and

119899119904

⋀

119895=1

(119904119895and 119883119898) and

119899119905

⋀

119896=1

119905119896 (A2)


119894or 119904119895)which do not


119899119903

⋀

119894=1

119899119904

⋀

119895=1

(119903119894or 119904119895) (A3)







119898


119898are set aside


119896or 119883119898) sdot sdot sdot

(119878119894or 119883119898) sdot sdot sdot 119878

119894or 119860119895 sdot sdot sdot 119878

119894or 119860119896 sdot sdot sdot

(119878119903or 119883119898) sdot sdot sdot 119878

119903or 119860119895 sdot sdot sdot 119878

119903or 119860119896 sdot sdot sdot


119898nor119883

119898)








119898are set





119894or 119883119898) which is the




119894and 119860



119894or
















119898




119894or119860119895


119895 and hence 119878

119903or 119860119896 subsumes

both 119878119894and 119860


119903or 119860119896 subsumes

both 119878119903and 119860

119896 Now since 119878

119903or 119860119896 subsumes the

four alterms 119878119894 119860119895 119878119903 and119860



as 119878119903or 119860119896) and 119878











that of 119860119895 and hence (119860

119896or 119883119898) subsumes both 119878

119894


119896or 119883119898) subsumes 119860

119896


119894


119894or 119860119896 which lies














Acknowledgment



References






















































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119894and 119883119894) might appear



5 Examples






1198921= (119878 or 119866) and (119878 or 119864) and (119866 or 119864) = 1 (27)







and (119878 or 119866) and 119866

= (119878 or 119864) and 119866

(28)




119881 (119878 997888rarr 119866) = 119881 (119878 or 119866) = ⟨06 03⟩

119881 (119878 997888rarr 119864) = 119881 (119878 or 119864) = ⟨09 01⟩

119881 (119866 997888rarr 119864) = 119881 (119866 or 119864) = ⟨08 01⟩

(29)



119881 (1198921) = 119881 ((119878 or 119866) and (119878 or 119864) and (119866 or 119864))

= ⟨min (06 09 08) max (03 01 01)⟩

= ⟨06 03⟩

(30)





119881 (119866) = ⟨06 03⟩ (31)



(119860 or 119861) and (119860 or 119861) = 1

(119861 or 119862) and (119861 or 119862) = 1

(119862 or 119860) and (119862 or 119860) = 1

(32)



and (119862 or 119860) and (119862 and 119860) = 1

(33)










Table1MSM

deriv

ationof

fuzzyversions

offamou

srules

ofinference

thep

artic

ular

conclusio

nof

aruleish

ighlighted

inbo

ld

Rulename

FuzzyRF

Tantecedents

(premise

s)

Prem

isesa

sseparate

fuzzyequatio

ns119875119894=RF

T119894=1119898

Prem

isesa

sasin

glefuzzy

equatio

n119892=RF

TCon

clusio

nsas

asinglefuzzy

equatio

nCP(119892)=RF

TCon

clusio

nsas

separatefuzzy

equatio

ns119862119894=RF

T119894=1119897

FuzzyRF

Tconsequence

(con

clusio

n)

MODUSPO

NEN

S119860rarr119861

119860

119860or119861=RF

T119860=RF

T(119860or119861)and119860=RF

T119861and119860=RF

TB=RF

T119860=RF

T119861

MODUSTO

LLEN

S119860rarr119861

119861

119860or119861=RF

T119861=RF

T(119860or119861)and119861=RF

T119860and119861=RF

TA=RF

T119861=RF

T119860

HYP

OTH

ETICAL

SYLL

OGISM

119860rarr119861

119861rarr119862

119860or119861=RF

T119861or119862=RF

T(119860or119861)and(119861or119862)=RF


T119860or119861=RF

T119861or119862=RF

TAorC=RF

T119860rarr119862

SIMPL

IFICAT

ION

119860and119861

119860and119861=RF

T119860and119861=RF

T119860and119861=RF

TA=RF

T119861=RF

T119860

CONJU

NCT

ION

119860 119861

119860=RF

T119861=RF

T119860and119861=RF

T119860and119861=RF

TAandB=RF

T119860and119861

CONST

RUCT

IVE

DILEM

MA

119860rarr119861

119862rarr119863

119860or119862

119860or119861=RF

T119862or119863=RF

T119860or119862=RF

T

(119860or119861)and(119862or119863)

and(119860or119862)=RF


and(119860or119863)and(119861or119863)=RF

T

(119860or119861)=RF

T(119862or119863)=RF

T(119860or119862)=RF

T(119861or119862)=RF

T(119860or119863)=RF

T(BorD)=RF

T

119861or119863

DISJU

NCT

IVESY


119860

119860or119861=RF

T119860=RF

T(119860or119861)and119860=RF

T119861and119860=RF

TB=RF

T119860=RF

T119861

ADDITIO

N119860

119860=RF

T119860=RF

T119860=RF

TAorB=RF

TAnalterm

subsum

ingan

RFT

alterm

isalso

RFT

119860or119861

ABS

ORP

TION

119860rarr119861

119860or119861=RF

T119860or119861=RF

T119860or119861=RF

TAorAB=RF

T119860or119860119861=119860or119861by

reflection

law

119860rarr119860119861

CASE

S

119860 119860rarr(119862or119863)

119862rarr119861

119863rarr119861

119860=RF

T119860or119862or119863=RF

T119862or119861=RF

T119863or119861=RF

T

119860and(119860or119862or119863)and(119862or119861)

and(119863or119861)=RF

T119860and119861and(119862or119863)=RF

T119860=RF

TB=RF

T119862or119863=RF

T119861

CASE

ELIM

INAT

ION

119860or119861

119860rarr(119862and119862)

119860or119861=RF

T119860or(119862and119862)=RF

T(119860or119861)and(119860or(119862and119862))=RF

T(119860or119861)and(119860or(119862and119862))

and(119861or(119862or119862))=RF

T

119860or119861=RF

T119860or(119862and119862)=RF

T119861or(119862and119862)=RF

TB=RF


FT

119861

REDUCT

IOAD

ABS

URD

UM

(CONTR

ADICTION)

119860rarr(119861and119861)

119860or(119861and119861)=RF

T119860or(119861and119861)=RF

T119860or(119861and119861)=RF

T119860or(119861and119861)=RF

TA=RF

T119861and119861=nR

FT

119860













3= 1) where







(37)



119877 or 119864 = 1 (38)

or equivalently

119877 997888rarr 119864 (39)



119881 (119877 997888rarr 119875) = 119881 (119877 or 119875) = ⟨06 03⟩

119881 (119875 997888rarr119863) = 119881 (119875 or 119863) = ⟨07 02⟩

119881 (119863 997888rarr 119865) = 119881 (119863 or 119865) = ⟨08 01⟩

119881 (119864 997888rarr 119865) = 119881 (119864 or 119865) = ⟨09 01⟩

(40)



6 Conclusion








Appendix







of literals 1199040= 119860

(0)

1 119860(0)

2 119860

(0)


1198831 1198832 119883







119898

(1) For 1 le 119895 le 119896 le 119899(119898minus1)



and 119860(119898minus1)119896






119898minus1of 119869(119898minus1)


ge

119899(119898minus1)


119860(119898minus1)119896


119898minus1 If 119860(119898minus1)119895




119895is




119898=

119860(119898)

1 119860(119898)

2 119860(119898)

119899119898



119898 where




A or CB or C

A or B

B

C

0

000

A

(a)

B or C

A

B

C

A

00

00 0

(b)

A or BA or BB

C

0

0

0

0

A

(c)



119898 1 le 119898 le 119872 in the form

119892 = (119903 or 119883119898) and (119904 or 119883

119898) and 119905 (A1)

where 119903 = ⋀119899119903119894=1119903119894 119904 = ⋀119899119904

119895=1119904119895 and 119905 = ⋀119899119905

119896=1119905119896are POS


119894 119904119895



119892 =

119899119903

⋀

119894=1

(119903119894or 119883119898) and

119899119904

⋀

119895=1

(119904119895and 119883119898) and

119899119905

⋀

119896=1

119905119896 (A2)


119894or 119904119895)which do not


119899119903

⋀

119894=1

119899119904

⋀

119895=1

(119903119894or 119904119895) (A3)







119898


119898are set aside


119896or 119883119898) sdot sdot sdot

(119878119894or 119883119898) sdot sdot sdot 119878

119894or 119860119895 sdot sdot sdot 119878

119894or 119860119896 sdot sdot sdot

(119878119903or 119883119898) sdot sdot sdot 119878

119903or 119860119895 sdot sdot sdot 119878

119903or 119860119896 sdot sdot sdot


119898nor119883

119898)








119898are set





119894or 119883119898) which is the




119894and 119860



119894or
















119898




119894or119860119895


119895 and hence 119878

119903or 119860119896 subsumes

both 119878119894and 119860


119903or 119860119896 subsumes

both 119878119903and 119860

119896 Now since 119878

119903or 119860119896 subsumes the

four alterms 119878119894 119860119895 119878119903 and119860



as 119878119903or 119860119896) and 119878











that of 119860119895 and hence (119860

119896or 119883119898) subsumes both 119878

119894


119896or 119883119898) subsumes 119860

119896


119894


119894or 119860119896 which lies














Acknowledgment



References






















































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Table1MSM

deriv

ationof

fuzzyversions

offamou

srules

ofinference

thep

artic

ular

conclusio

nof

aruleish

ighlighted

inbo

ld

Rulename

FuzzyRF

Tantecedents

(premise

s)

Prem

isesa

sseparate

fuzzyequatio

ns119875119894=RF

T119894=1119898

Prem

isesa

sasin

glefuzzy

equatio

n119892=RF

TCon

clusio

nsas

asinglefuzzy

equatio

nCP(119892)=RF

TCon

clusio

nsas

separatefuzzy

equatio

ns119862119894=RF

T119894=1119897

FuzzyRF

Tconsequence

(con

clusio

n)

MODUSPO

NEN

S119860rarr119861

119860

119860or119861=RF

T119860=RF

T(119860or119861)and119860=RF

T119861and119860=RF

TB=RF

T119860=RF

T119861

MODUSTO

LLEN

S119860rarr119861

119861

119860or119861=RF

T119861=RF

T(119860or119861)and119861=RF

T119860and119861=RF

TA=RF

T119861=RF

T119860

HYP

OTH

ETICAL

SYLL

OGISM

119860rarr119861

119861rarr119862

119860or119861=RF

T119861or119862=RF

T(119860or119861)and(119861or119862)=RF


T119860or119861=RF

T119861or119862=RF

TAorC=RF

T119860rarr119862

SIMPL

IFICAT

ION

119860and119861

119860and119861=RF

T119860and119861=RF

T119860and119861=RF

TA=RF

T119861=RF

T119860

CONJU

NCT

ION

119860 119861

119860=RF

T119861=RF

T119860and119861=RF

T119860and119861=RF

TAandB=RF

T119860and119861

CONST

RUCT

IVE

DILEM

MA

119860rarr119861

119862rarr119863

119860or119862

119860or119861=RF

T119862or119863=RF

T119860or119862=RF

T

(119860or119861)and(119862or119863)

and(119860or119862)=RF


and(119860or119863)and(119861or119863)=RF

T

(119860or119861)=RF

T(119862or119863)=RF

T(119860or119862)=RF

T(119861or119862)=RF

T(119860or119863)=RF

T(BorD)=RF

T

119861or119863

DISJU

NCT

IVESY


119860

119860or119861=RF

T119860=RF

T(119860or119861)and119860=RF

T119861and119860=RF

TB=RF

T119860=RF

T119861

ADDITIO

N119860

119860=RF

T119860=RF

T119860=RF

TAorB=RF

TAnalterm

subsum

ingan

RFT

alterm

isalso

RFT

119860or119861

ABS

ORP

TION

119860rarr119861

119860or119861=RF

T119860or119861=RF

T119860or119861=RF

TAorAB=RF

T119860or119860119861=119860or119861by

reflection

law

119860rarr119860119861

CASE

S

119860 119860rarr(119862or119863)

119862rarr119861

119863rarr119861

119860=RF

T119860or119862or119863=RF

T119862or119861=RF

T119863or119861=RF

T

119860and(119860or119862or119863)and(119862or119861)

and(119863or119861)=RF

T119860and119861and(119862or119863)=RF

T119860=RF

TB=RF

T119862or119863=RF

T119861

CASE

ELIM

INAT

ION

119860or119861

119860rarr(119862and119862)

119860or119861=RF

T119860or(119862and119862)=RF

T(119860or119861)and(119860or(119862and119862))=RF

T(119860or119861)and(119860or(119862and119862))

and(119861or(119862or119862))=RF

T

119860or119861=RF

T119860or(119862and119862)=RF

T119861or(119862and119862)=RF

TB=RF


FT

119861

REDUCT

IOAD

ABS

URD

UM

(CONTR

ADICTION)

119860rarr(119861and119861)

119860or(119861and119861)=RF

T119860or(119861and119861)=RF

T119860or(119861and119861)=RF

T119860or(119861and119861)=RF

TA=RF

T119861and119861=nR

FT

119860













3= 1) where







(37)



119877 or 119864 = 1 (38)

or equivalently

119877 997888rarr 119864 (39)



119881 (119877 997888rarr 119875) = 119881 (119877 or 119875) = ⟨06 03⟩

119881 (119875 997888rarr119863) = 119881 (119875 or 119863) = ⟨07 02⟩

119881 (119863 997888rarr 119865) = 119881 (119863 or 119865) = ⟨08 01⟩

119881 (119864 997888rarr 119865) = 119881 (119864 or 119865) = ⟨09 01⟩

(40)



6 Conclusion








Appendix







of literals 1199040= 119860

(0)

1 119860(0)

2 119860

(0)


1198831 1198832 119883







119898

(1) For 1 le 119895 le 119896 le 119899(119898minus1)



and 119860(119898minus1)119896






119898minus1of 119869(119898minus1)


ge

119899(119898minus1)


119860(119898minus1)119896


119898minus1 If 119860(119898minus1)119895




119895is




119898=

119860(119898)

1 119860(119898)

2 119860(119898)

119899119898



119898 where




A or CB or C

A or B

B

C

0

000

A

(a)

B or C

A

B

C

A

00

00 0

(b)

A or BA or BB

C

0

0

0

0

A

(c)



119898 1 le 119898 le 119872 in the form

119892 = (119903 or 119883119898) and (119904 or 119883

119898) and 119905 (A1)

where 119903 = ⋀119899119903119894=1119903119894 119904 = ⋀119899119904

119895=1119904119895 and 119905 = ⋀119899119905

119896=1119905119896are POS


119894 119904119895



119892 =

119899119903

⋀

119894=1

(119903119894or 119883119898) and

119899119904

⋀

119895=1

(119904119895and 119883119898) and

119899119905

⋀

119896=1

119905119896 (A2)


119894or 119904119895)which do not


119899119903

⋀

119894=1

119899119904

⋀

119895=1

(119903119894or 119904119895) (A3)







119898


119898are set aside


119896or 119883119898) sdot sdot sdot

(119878119894or 119883119898) sdot sdot sdot 119878

119894or 119860119895 sdot sdot sdot 119878

119894or 119860119896 sdot sdot sdot

(119878119903or 119883119898) sdot sdot sdot 119878

119903or 119860119895 sdot sdot sdot 119878

119903or 119860119896 sdot sdot sdot


119898nor119883

119898)








119898are set





119894or 119883119898) which is the




119894and 119860



119894or
















119898




119894or119860119895


119895 and hence 119878

119903or 119860119896 subsumes

both 119878119894and 119860


119903or 119860119896 subsumes

both 119878119903and 119860

119896 Now since 119878

119903or 119860119896 subsumes the

four alterms 119878119894 119860119895 119878119903 and119860



as 119878119903or 119860119896) and 119878











that of 119860119895 and hence (119860

119896or 119883119898) subsumes both 119878

119894


119896or 119883119898) subsumes 119860

119896


119894


119894or 119860119896 which lies














Acknowledgment



References






















































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





















3= 1) where







(37)



119877 or 119864 = 1 (38)

or equivalently

119877 997888rarr 119864 (39)



119881 (119877 997888rarr 119875) = 119881 (119877 or 119875) = ⟨06 03⟩

119881 (119875 997888rarr119863) = 119881 (119875 or 119863) = ⟨07 02⟩

119881 (119863 997888rarr 119865) = 119881 (119863 or 119865) = ⟨08 01⟩

119881 (119864 997888rarr 119865) = 119881 (119864 or 119865) = ⟨09 01⟩

(40)



6 Conclusion








Appendix







of literals 1199040= 119860

(0)

1 119860(0)

2 119860

(0)


1198831 1198832 119883







119898

(1) For 1 le 119895 le 119896 le 119899(119898minus1)



and 119860(119898minus1)119896






119898minus1of 119869(119898minus1)


ge

119899(119898minus1)


119860(119898minus1)119896


119898minus1 If 119860(119898minus1)119895




119895is




119898=

119860(119898)

1 119860(119898)

2 119860(119898)

119899119898



119898 where




A or CB or C

A or B

B

C

0

000

A

(a)

B or C

A

B

C

A

00

00 0

(b)

A or BA or BB

C

0

0

0

0

A

(c)



119898 1 le 119898 le 119872 in the form

119892 = (119903 or 119883119898) and (119904 or 119883

119898) and 119905 (A1)

where 119903 = ⋀119899119903119894=1119903119894 119904 = ⋀119899119904

119895=1119904119895 and 119905 = ⋀119899119905

119896=1119905119896are POS


119894 119904119895



119892 =

119899119903

⋀

119894=1

(119903119894or 119883119898) and

119899119904

⋀

119895=1

(119904119895and 119883119898) and

119899119905

⋀

119896=1

119905119896 (A2)


119894or 119904119895)which do not


119899119903

⋀

119894=1

119899119904

⋀

119895=1

(119903119894or 119904119895) (A3)







119898


119898are set aside


119896or 119883119898) sdot sdot sdot

(119878119894or 119883119898) sdot sdot sdot 119878

119894or 119860119895 sdot sdot sdot 119878

119894or 119860119896 sdot sdot sdot

(119878119903or 119883119898) sdot sdot sdot 119878

119903or 119860119895 sdot sdot sdot 119878

119903or 119860119896 sdot sdot sdot


119898nor119883

119898)








119898are set





119894or 119883119898) which is the




119894and 119860



119894or
















119898




119894or119860119895


119895 and hence 119878

119903or 119860119896 subsumes

both 119878119894and 119860


119903or 119860119896 subsumes

both 119878119903and 119860

119896 Now since 119878

119903or 119860119896 subsumes the

four alterms 119878119894 119860119895 119878119903 and119860



as 119878119903or 119860119896) and 119878











that of 119860119895 and hence (119860

119896or 119883119898) subsumes both 119878

119894


119896or 119883119898) subsumes 119860

119896


119894


119894or 119860119896 which lies














Acknowledgment



References






















































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














Appendix







of literals 1199040= 119860

(0)

1 119860(0)

2 119860

(0)


1198831 1198832 119883







119898

(1) For 1 le 119895 le 119896 le 119899(119898minus1)



and 119860(119898minus1)119896






119898minus1of 119869(119898minus1)


ge

119899(119898minus1)


119860(119898minus1)119896


119898minus1 If 119860(119898minus1)119895




119895is




119898=

119860(119898)

1 119860(119898)

2 119860(119898)

119899119898



119898 where




A or CB or C

A or B

B

C

0

000

A

(a)

B or C

A

B

C

A

00

00 0

(b)

A or BA or BB

C

0

0

0

0

A

(c)



119898 1 le 119898 le 119872 in the form

119892 = (119903 or 119883119898) and (119904 or 119883

119898) and 119905 (A1)

where 119903 = ⋀119899119903119894=1119903119894 119904 = ⋀119899119904

119895=1119904119895 and 119905 = ⋀119899119905

119896=1119905119896are POS


119894 119904119895



119892 =

119899119903

⋀

119894=1

(119903119894or 119883119898) and

119899119904

⋀

119895=1

(119904119895and 119883119898) and

119899119905

⋀

119896=1

119905119896 (A2)


119894or 119904119895)which do not


119899119903

⋀

119894=1

119899119904

⋀

119895=1

(119903119894or 119904119895) (A3)







119898


119898are set aside


119896or 119883119898) sdot sdot sdot

(119878119894or 119883119898) sdot sdot sdot 119878

119894or 119860119895 sdot sdot sdot 119878

119894or 119860119896 sdot sdot sdot

(119878119903or 119883119898) sdot sdot sdot 119878

119903or 119860119895 sdot sdot sdot 119878

119903or 119860119896 sdot sdot sdot


119898nor119883

119898)








119898are set





119894or 119883119898) which is the




119894and 119860



119894or
















119898




119894or119860119895


119895 and hence 119878

119903or 119860119896 subsumes

both 119878119894and 119860


119903or 119860119896 subsumes

both 119878119903and 119860

119896 Now since 119878

119903or 119860119896 subsumes the

four alterms 119878119894 119860119895 119878119903 and119860



as 119878119903or 119860119896) and 119878











that of 119860119895 and hence (119860

119896or 119883119898) subsumes both 119878

119894


119896or 119883119898) subsumes 119860

119896


119894


119894or 119860119896 which lies














Acknowledgment



References






















































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










A or CB or C

A or B

B

C

0

000

A

(a)

B or C

A

B

C

A

00

00 0

(b)

A or BA or BB

C

0

0

0

0

A

(c)



119898 1 le 119898 le 119872 in the form

119892 = (119903 or 119883119898) and (119904 or 119883

119898) and 119905 (A1)

where 119903 = ⋀119899119903119894=1119903119894 119904 = ⋀119899119904

119895=1119904119895 and 119905 = ⋀119899119905

119896=1119905119896are POS


119894 119904119895



119892 =

119899119903

⋀

119894=1

(119903119894or 119883119898) and

119899119904

⋀

119895=1

(119904119895and 119883119898) and

119899119905

⋀

119896=1

119905119896 (A2)


119894or 119904119895)which do not


119899119903

⋀

119894=1

119899119904

⋀

119895=1

(119903119894or 119904119895) (A3)







119898


119898are set aside


119896or 119883119898) sdot sdot sdot

(119878119894or 119883119898) sdot sdot sdot 119878

119894or 119860119895 sdot sdot sdot 119878

119894or 119860119896 sdot sdot sdot

(119878119903or 119883119898) sdot sdot sdot 119878

119903or 119860119895 sdot sdot sdot 119878

119903or 119860119896 sdot sdot sdot


119898nor119883

119898)








119898are set





119894or 119883119898) which is the




119894and 119860



119894or
















119898




119894or119860119895


119895 and hence 119878

119903or 119860119896 subsumes

both 119878119894and 119860


119903or 119860119896 subsumes

both 119878119903and 119860

119896 Now since 119878

119903or 119860119896 subsumes the

four alterms 119878119894 119860119895 119878119903 and119860



as 119878119903or 119860119896) and 119878











that of 119860119895 and hence (119860

119896or 119883119898) subsumes both 119878

119894


119896or 119883119898) subsumes 119860

119896


119894


119894or 119860119896 which lies














Acknowledgment



References






















































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119894or 119883119898) which is the




119894and 119860



119894or
















119898




119894or119860119895


119895 and hence 119878

119903or 119860119896 subsumes

both 119878119894and 119860


119903or 119860119896 subsumes

both 119878119903and 119860

119896 Now since 119878

119903or 119860119896 subsumes the

four alterms 119878119894 119860119895 119878119903 and119860



as 119878119903or 119860119896) and 119878











that of 119860119895 and hence (119860

119896or 119883119898) subsumes both 119878

119894


119896or 119883119898) subsumes 119860

119896


119894


119894or 119860119896 which lies














Acknowledgment



References






















































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










References






















































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Research Article A Modern Syllogistic Method in Intuitionistic Fuzzy...

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Transcript of Research Article A Modern Syllogistic Method in Intuitionistic Fuzzy...