research article conservative intensional extension of...

Hindawi Publishing CorporationAdvances in Artificial IntelligenceVolume 2013 Article ID 920157 10 pageshttpdxdoiorg1011552013920157

Research ArticleConservative Intensional Extension of Tarskirsquos Semantics

Zoran MajkiT

International Society for Research in Science and Technology PO Box 2464 Tallahassee FL 32316-2464 USA

Correspondence should be addressed to Zoran Majkic majk1234yahoocom

Received 30 May 2012 Revised 12 October 2012 Accepted 23 October 2012

Academic Editor Konstantinos Lefkimmiatis

Copyright copy 2013 Zoran MajkicThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We considered an extension of the first-order logic (FOL) by Bealerrsquos intensional abstraction operator Contemporary use of theterm ldquointensionrdquo derives from the traditional logical Frege-Russell doctrine that an idea (logic formula) has both an extension andan intension Although there is divergence in formulation it is accepted that the ldquoextensionrdquo of an idea consists of the subjects towhich the idea applies and the ldquointensionrdquo consists of the attributes implied by the idea From the Montaguersquos point of view themeaning of an idea can be considered as particular extensions in different possible worlds In the case of standard FOL we obtaina commutative homomorphic diagram which is valid in each given possible world of an intensional FOL from a free algebra ofthe FOL syntax into its intensional algebra of concepts and successively into an extensional relational algebra (different fromCylindric algebras)Then we show that this composition corresponds to the Tarskirsquos interpretation of the standard extensional FOLin this possible world

1 Introduction

In ldquoUber Sinn und edeutungrdquo Frege concentrated mostly onthe senses of names holding that all names have a sense(meaning) It is natural to hold that the same considerationsapply to any expression that has an extension But two generalterms can have the same extension and different cognitivesignificance two predicates can have the same extensionand different cognitive significance two sentences can havethe same extension and different cognitive significance Sogeneral terms predicates and sentences all have senses aswellas extensions The same goes for any expression that has anextension or is a candidate for extension

The significant aspect of an expressionrsquos meaning is itsextension We can stipulate that the extension of a sentenceis its truth-value and that the extension of a singular term isits referent The extension of other expressions can be seenas associated entities that contribute to the truth-value of asentence in a manner broadly analogous to the way in whichthe referent of a singular term contributes to the truth-valueof a sentence In many cases the extension of an expressionwill be what we intuitively think of as its referent althoughthis need not hold in all cases While Frege himself is ofteninterpreted as holding that a sentencersquos referent is its truth-value this claim is counterintuitive and widely disputed We

can avoid that issue in the present framework by using thetechnical term ldquoextensionrdquo In this context the claim that theextension of a sentence is its truth-value is a stipulation

ldquoExtensionalrdquo is most definitely a technical term Say thatthe extension of a name is its denotation the extension of apredicate is the set of things it applies to and the extensionof a sentence is its truth value A logic is extensional ifcoextensional expressions can be substituted one for anotherin any sentence of the logic ldquosalva veritaterdquo that is withouta change in truth value The intuitive idea behind thisprinciple is that in an extensional logic the only logicallysignificant notion of meaning that attaches to an expressionis its extension An intensional logics is exactly one in whichsubstitutivity salva veritate fails for some of the sentences ofthe logic

The first conception of intensional entities (or concepts)is built into the possible-worlds treatment of Properties Rela-tions and Propositions (PRPs)This conception is commonlyattributed to Leibniz and underlies Alonzo Churchrsquos alterna-tive formulation of Fregersquos theory of senses (ldquoA formulationof the logic of sense and denotationrdquo in Henle Kallen andLanger 3ndash24 and ldquoOutline of a revised formulation of thelogic of sense and denotationrdquo in two parts Nous VII (1973)24ndash33 and VIII (1974) 135ndash156) This conception of PRPs isideally suited for treating themodalities (necessity possibility

2 Advances in Artificial Intelligence

etc) and to Montaguersquos definition of intension of a givenvirtual predicate 120601(119909

1 119909

119896) (a FOL open-sentence with the

tuple of free variables (1199091 119909

119896)) as amapping frompossible

worlds into extensions of this virtual predicate Among thepossible worlds we distinguish the actual possible worldFor example if we consider a set of predicates of a givenDatabase and their extensions in different time-instancesthen the actual possible world is identified by the currentinstance of the time

The second conception of intensional entities is to befound in Russellrsquos doctrine of logical atomism In this doc-trine it is required that all complete definitions of intensionalentities be finite as well as unique and noncircular it offersan algebraicway for definition of complex intensional entitiesfrom simple (atomic) entities (ie algebra of concepts)conception also evident in Leibnizrsquos remarks In a predicatelogics predicates and open-sentences (with free variables)express classes (properties and relations) and sentencesexpress propositions Note that classes (intensional entities)are reified that is they belong to the same domain asindividual objects (particulars) This endows the intensionallogics with a great deal of uniformity making it possibleto manipulate classes and individual objects in the samelanguage In particular when viewed as an individual objecta class can be a member of another class

The distinction between intensions and extensions isimportant (as in lexicography [1]) considering that exten-sions can be notoriously difficult to handle in an efficientmanner The extensional equality theory of predicates andfunctions under higher-order semantics (eg for two pred-icates with the same set of attributes 119901 = 119902 is true iffthese symbols are interpreted by the same relation) thatis the strong equational theory of intensions is not decid-able in general For example the second-order predicatecalculus and Churchrsquos simple theory of types both underthe standard semantics are not even semi-decidable Thusseparating intensions from extensions makes it possible tohave an equational theory over predicate and function names(intensions) that is separate from the extensional equality ofrelations and functions

Relevant recentwork about the intension and its relation-ship with FOL has been presented in [2] in the considerationof rigid and nonrigid objects with respect to the possibleworlds where the rigid objects like ldquoGeorge Washingtonrdquoare the same things from possible world to possible worldNonrigid objects like ldquothe Secretary-General of UnitedNationsrdquo are varying from circumstance to circumstanceand can be modeled semantically by functions from possibleworlds to domain of rigid objects like intensional entitiesBut in his approach differently from that one fitting changesalso the syntax of the FOL by introducing an ldquoextensionof rdquo operator darr in order to distinguish the intensional entityldquogross-domestic-product-of-Denmarkrdquo and its use in ldquothegross domestic product of Denmark is currently greater thangross domestic product of Finlandrdquo In his approach if 119909is an intensional variable darr 119909 is extensional while darr is notapplicable to extensional variables differently fromourwhereeach variable (concept) has both intensional and extensionMoreover in his approach the problem arises because the

action of letting 119909 designate that is evaluating darr 119909 andthe action of passing to an alternative possible world thatis of interpreting the existential modal operator are notactions that commute To disambiguate this one more pieceof machinery is needed as well which substantially and ad-hock changes the syntax and semantics of FOL introducesthe Higher-order Modal logics and is not a conservativeextension of Tarskirsquos semantics

In most recent work in [3 4] it is given an intensionalversion of first-order hybrid logic which is also a hybridizedversion of Fittingrsquos intensional FOL by a kind of generalizedmodels thus different from our approaches to conservativeextension of Tarskirsquos semantics to intensional FOL

Another recent relevant work is presented by I-logicin [5] which combines both approach to semantics ofintensional objects of Montague and Fitting

We recall that Intensional Logic Programming is a newform of logic programming based on intensional logic andpossible worlds semantics and is a well-defined practice inusing the intensional semantics [6] Intensional logic allowsus to use logic programming to specify nonterminating com-putations and to capture the dynamic aspects of certain prob-lems in a natural and problem-oriented style The meaningsof formulas of an intensional first-order language are givenaccording to intensional interpretations and to elements of aset of possible worlds Neighborhood semantics is employedas an abstract formulation of the denotations of intensionaloperators The model-theoretic and fixpoint semantics ofintensional logic programs are developed in terms of least(minimum) intensional Herbrand models Intensional logicprograms with intensional operator definitions are regardedas metatheories

In what follows we denote by 119861119860 the set of all functionsfrom 119860 to 119861 and by 119860119899 an 119899-folded cartesian product 119860 timessdot sdot sdot times 119860 for 119899 ge 1 By 119891 119905 we denote empty set 0 and singletonset ⟨ ⟩ respectively (with the empty tuple ⟨ ⟩ ie the uniquetuple of 0-ary relation) which may be thought of as falsity 119891and truth 119905 as those used in the relational algebra For a givendomain D we define that D0 is a singleton set ⟨ ⟩ so that119891 119905 = P(D0) whereP is the powerset operator

2 Intensional FOL Language withIntensional Abstraction

Intensional entities are such concepts as propositions andproperties The term ldquointensionalrdquo means that they violatethe principle of extensionality the principle that extensionalequivalence implies identity All (ormost) of these intensionalentities have been classified at one time or another as kinds ofUniversals [7]

We consider a nonempty domainD = 119863minus1⋃119863

119868 where a

subdomain 119863minus1

is made of particulars (extensional entities)and the rest119863

119868= 119863

0⋃119863

1sdot sdot sdot ⋃119863

119899sdot sdot sdot is made of universals (

1198630for propositions (the 0-ary concepts)) and119863

119899 119899 ge 1 for

119899-ary conceptsThe fundamental entities are intensional abstracts or

so-called that-clauses We assume that they are singularterms Intensional expressions like ldquobelieverdquo ldquomeanrdquo ldquoassertrdquo

Advances in Artificial Intelligence 3

ldquoknowrdquo are standard two-place predicates that take ldquothatrdquo-clauses as arguments Expressions like ldquois necessaryrdquo ldquoistruerdquo and ldquois possiblerdquo are one-place predicates that takeldquothatrdquo-clauses as arguments For example in the intensionalsentence ldquoit is necessary that 120601rdquo where 120601 is a proposition theldquothat 120601rdquo is denoted by the ⋖ 120601 ⋗ where ⋖⋗ is the intensionalabstraction operator which transforms a logic formula into aterm Or for example ldquo119909 believes that 120601rdquo is given by formula1199012119894(119909 ⋖ 120601 ⋗) (1199012

119894is binary ldquobelieverdquo predicate)

Here we will present an intensional FOL with slightly dif-ferent intensional abstraction than that originally presentedin [8]

Definition 1 The syntax of the first-order logic language withintensional abstraction ⋖⋗ denoted byL is as follows

logic operators (and not exist) predicate letters in 119875 (functionalletters is considered as particular case of predicate letters)variables 119909 119910 119911 inV abstraction ⋖ ⋗ and punctuationsymbols (comma parenthesis) With the following simulta-neous inductive definition of term and formula

(1) all variables and constants (0-ary functional letters in119875) are terms

(2) if 1199051 119905

119896are terms then 119901119896

119894(1199051 119905

119896) is a formula

(119901119896119894isin 119875 is a 119896-ary predicate letter)

(3) if 120601 and 120595 are formulae then (120601 and 120595) not120601 and (exist119909)120601are formulae

(4) if120601(x) is a formula (virtual predicate) with a list of freevariables in x = (119909

1 119909

119899) (with ordering from-left-

to-right of their appearance in120601) and120572 is its sublist ofdistinct variables then⋖ 120601 ⋗120573

120572is a term where 120573 is the

remaining list of free variables preserving ordering inx as well The externally quantifiable variables are thefree variables not in 120572 When 119899 = 0 ⋖ 120601 ⋗ is a termthat denotes a proposition for 119899 ge 1 it denotes an 119899-ary concept

An occurrence of a variable 119909119894in a formula (or a term) is

bound (free) if and only if it lies (does not lie) within a formulaof the form (exist119909

119894)120601 (or a term of the form ⋖ 120601 ⋗120573

120572with 119909

119894isin 120572)

A variable is free (bound) in a formula (or term) if and onlyif it has (does not have) a free occurrence in that formula (orterm)

A sentence is a formula having no free variables Thebinary predicate letter 1199012

1for identity is singled out as a

distinguished logical predicate and formulae of the form11990121(1199051 1199052) are to be rewritten in the form 119905

1≐ 119905

2 We

denote by119877=the binary relation obtained by standard Tarskirsquos

interpretation of this predicate11990121The logic operatorsforall orrArr

are defined in terms of (and not exist) in the usual way

Remark 2 The 119896-ary functional symbols for 119896 ge 1 in stand-ard (extensional) FOL are considered as (119896 + 1)-ary predicatesymbols 119901119896+1 the function 119891 D119896 rarr D is considered as arelation obtained from its graph 119877 = (119889

1 119889

119896 119891(119889

1

119889119896)) | 119889

119894isin D represented by a predicate symbol 119901119896+1

The universal quantifier is defined by forall = notexistnot Disjunc-tion and implication are expressed by 120601 or 120595 = not(not120601 and not120595)

and 120601 rArr 120595 = not120601or120595 In FOL with the identity ≐ the formula(exist1119909)120601(119909) denotes the formula (exist119909)120601(119909) and (forall119909)(forall119910)(120601(119909) and

120601(119910) rArr (119909 ≐ 119910)) We denote by 119877=the Tarskirsquos interpretation

of ≐In what follows any open-sentence a formula 120601 with

nonempty tuple of free variables (1199091 119909

119898) will be called

a119898-ary virtual predicate denoted also by 120601(1199091 119909

119898) This

definition contains the precise method of establishing theordering of variables in this tuple such a method that will beadopted here is the ordering of appearance from left to rightof free variables in 120601 This method of composing the tuple offree variables is the unique and canonical way of definition ofthe virtual predicate from a given formula

An intensional interpretation of this intensional FOL is amapping between the setL of formulae of the logic languageand intensional entities in D 119868 L rarr D which is akind of ldquoconceptualizationrdquo such that an open-sentence(virtual predicate) 120601(119909

1 119909

119896) with a tuple of all free var-

iables (1199091 119909

119896) is mapped into a 119896-ary concept that is

an intensional entity 119906 = 119868(120601(1199091 119909

119896)) isin 119863

119896 and

(closed) sentence120595 into a proposition (ie logic concept) 119907 =119868(120595) isin 119863

0with 119868(⊤) = 119879119903119906119905ℎ isin 119863

0for a FOL tautology

⊤ A language constant 119888 is mapped into a particular (anextensional entity) 119886 = 119868(119888) isin 119863

minus1if it is a proper name

otherwise in a correspondent concept inDAn assignment 119892 V rarr D for variables inV is applied

only to free variables in terms and formulae Such an assign-ment 119892 isin DV can be recursively uniquely extended intothe assignment 119892lowast T rarr D where T denotes the set ofall terms (here 119868 is an intensional interpretation of this FOLas explained in what follows) by

(1) 119892lowast(119905) = 119892(119909) isin D if the term 119905 is a variable 119909 isin V(2) 119892lowast(119905) = 119868(119888) isin D if the term 119905 is a constant 119888 isin 119875(3) if 119905 is an abstracted term ⋖ 120601 ⋗120573

120572 then 119892lowast(⋖ 120601 ⋗120573

120572) =

119868(120601[120573119892(120573)]) isin 119863119896 119896 = |120572| (ie the number of vari-

ables in 120572) where 119892(120573) = 119892(1199101 119910

119898) = (119892(119910

1)

119892(119910119898)) and [120573119892(120573)] is a uniform replacement of each

119894th variable in the list 120573with the 119894th constant in the list119892(120573) Notice that 120572 is the list of all free variables in theformula 120601[120573119892(120573)]

We denote by 119905119892 (or 120601119892) the ground term (or formula)without free variables obtained by assignment 119892 from a term119905 (or a formula 120601) and by 120601[119909119905] the formula obtained byuniformly replacing 119909 by a term 119905 in 120601

The distinction between intensions and extensions isimportant especially because we are now able to have andequational theory over intensional entities (as ⋖ 120601 ⋗) that ispredicate and function ldquonamesrdquo which is separate from theextensional equality of relations and functions An extension-alization function ℎ assigns to the intensional elements ofD an appropriate extension as follows for each proposition119906 isin 119863

0 ℎ(119906) isin 119891 119905 sube P(119863

minus1) is its extension (true or

false value) for each 119899-ary concept 119906 isin 119863119899 ℎ(119906) is a subset

ofD119899 (119899th Cartesian product ofD) in the case of particulars119906 isin 119863

minus1 ℎ(119906) = 119906

The sets 119891 119905 are empty set and set ⟨ ⟩ (with the emptytuple ⟨ ⟩ isin 119863

minus1 ie the unique tuple of 0-ary relation)


which may be thought of as falsity and truth as those used inthe Coddrsquos relational-database algebra [9] respectively while119879119903119906119905ℎ isin 119863

0is the concept (intension) of the tautology

We define thatD0 = ⟨ ⟩ so that 119891 119905 = P(D0) whereP is the powerset operator Thus we have (we denote thedisjoint union by ldquo+rdquo)

ℎ = (ℎminus1+sum119894ge0

ℎ119894) sum

119894geminus1

119863119894997888rarr 119863

minus1+sum119894ge0

P (119863119894) (1)

where ℎminus1= 119894119889 119863

minus1rarr 119863

minus1is identity mapping the map-

ping ℎ0 119863

0rarr 119891 119905 assigns the truth values in 119891 119905 to all

propositions and the mappings ℎ119894 119863

119894rarr P(119863119894) 119894 ge 1

assign an extension to all concepts Thus the intensions canbe seen as names of abstract or concrete entities while theextensions correspond to various rules that these entities playin different worlds

Remark 3 (Tarskirsquos constraints) This intensional semanticshas to preserve standard Tarskirsquos semantics of the FOL Thatis for any formula 120601 isin L with a tuple of free variables(1199091 119909

119896) and ℎ isin E the following conservative conditions

for all assignments 119892 1198921015840 isin DV have to be satisfied

(T) ℎ(119868(120601119892)) = 119905 if and only if (119892(1199091) 119892(119909

119896)) isin

ℎ(119868(120601)) and if 120601 is a predicate letter 119901119896 119896 ge 2

which represents a (119896minus1)-ary functional symbol 119891119896minus1in standard FOL

(TF) ℎ(119868(120601119892)) = ℎ(119868(1206011198921015840)) = 119905 and forall1le119894le119896minus1

(1198921015840(119909119894) =

119892(119909119894)) implies 1198921015840(119909

119896+1) = 119892(119909

119896+1)

Thus intensional FOL has a simple Tarski first-ordersemantics with a decidable unification problem but we needalso the actual world mapping which maps any intensionalentity to its actual world extension In what follows wewill identify a possible world by a particular mapping whichassigns in such a possible world the extensions to intensionalentities This is a direct bridge between an intensional FOLand a possible worlds representation [10ndash15] where theintension (meaning) of a proposition is a function froma set of possible W worlds into the set of truth valuesConsequentlyEdenotes the set of possible extensionalizationfunctions ℎ satisfying the constraint (T) Each ℎ isin Emay be seen as a possible world (analogously to Montaguersquosintensional semantics for natural language [12 14]) as it hasbeen demonstrated in [16 17] and given by the bijection 119894119904 W ≃ E

Now we are able to define formally this intensionalsemantics [15]

Definition 4 A two-step intensional semanticsLet R = ⋃

119896isinN P(D119896) = sum119896isinN P(119863119896) be the set of all 119896-

ary relations where 119896 isin N = 0 1 2 Notice that 119891 119905 =P(D0) isin R that is the truth values are extensions inR

The intensional semantics of the logic language with theset of formulaeL can be represented by the mapping

L119868

997888997888rarr D 997904rArr119908isinWR (2)

where 119868

997888rarr is a fixed intensional interpretation 119868 L rarr Dand rArr

119908isinW is the set of all extensionalization functions ℎ =119894119904(119908) D rarr R in E where 119894119904 W rarr E is the mappingfrom the set of possible worlds to the set of extensionalizationfunctions

We define the mapping 119868119899 L

119900119901rarr RW where L

119900119901is

the subset of formulae with free variables (virtual predicates)such that for any virtual predicate 120601(119909

1 119909

119896) isin L

119900119901the

mapping 119868119899(120601(119909

1 119909

119896)) W rarr R is the Montaguersquos

meaning (ie intension) of this virtual predicate [10ndash14] thatis the mapping which returns with the extension of this(virtual) predicate in each possible world 119908 isin W

We adopted this two-step intensional semantics insteadof well-known Montaguersquos semantics (which lies in theconstruction of a compositional and recursive semantics thatcovers both intension and extension) because of a number ofweakness of the second semantics

Example 5 Let us consider the following two past participlesldquoboughtrdquo and ldquosoldrdquo (with unary predicates1199011

1(119909) ldquo119909has been

boughtrdquo and 11990112(119909) ldquo119909 has been soldrdquo) These two different

concepts in the Montaguersquos semantics would have not onlythe same extension but also their intension from the fact thattheir extensions are identical in every possible world

Within the two-step formalism we can avoid this prob-lem by assigning two different concepts (meanings) 119906 =

119868(11990111(119909)) and 119907 = 119868(1199011

2(119909)) in isin 119863

1 Note that we have the

same problem in the Montaguersquos semantics for two sentenceswith different meanings which bear the same truth valueacross all possible worlds in Montaguersquos semantics they willbe forced to the samemeaning

Another relevant question with respect to this two-step interpretations of an intensional semantics is how init the extensional identity relation ≐ (binary predicate ofthe identity) of the FOL is managed Here this extensionalidentity relation is mapped into the binary concept 119868119889 = 119868(≐(119909 119910)) isin 119863

2 such that (forall119908 isin W)(119894119904(119908)(119868119889) = 119877

=) where

≐ (119909 119910) (ie 11990121(119909 119910)) denotes an atom of the FOL of the

binary predicate for identity in FOL usually written by FOLformula 119909 ≐ 119910

Note that here we prefer to distinguish this formal symbol≐isin 119875 of the built-in identity binary predicate letter in theFOL from the standard mathematical symbol ldquo=rdquo used in allmathematical definitions in this paper

In what follows we will use the function 119891⟨ ⟩

R rarrR such that for any relation 119877 isin R 119891

⟨ ⟩(119877) = ⟨ ⟩ if

119877 = 0 0 otherwise Let us define the following set of algebraicoperators for relations inR

(1) Binary operator ⋈119878 R timesR rarr R such that for any

two relations 1198771 1198772isin R the 119877

1⋈1198781198772is equal to the

relation obtained by natural join of these two relationsif 119878 is a nonempty set of pairs of joined columns ofrespective relations (where the first argument is thecolumn index of the relation 119877

1while the second

argument is the column index of the joined column of


the relation 1198772) otherwise it is equal to the cartesian

product 1198771times 119877

2

For example the logic formula 120601(119909119894 119909119895 119909119896 119909119897 119909119898) and

120595(119909119897 119910119894 119909119895 119910119895) will be traduced by the algebraic

expression 1198771⋈1198781198772where 119877

1isin P(D5) 119877

2isin P(D4)

are the extensions for a given Tarskirsquos interpretationof the virtual predicate 120601 120595 relatively so that 119878 =(4 1) (2 3) and the resulting relation will havethe following ordering of attributes (119909

119894 119909119895 119909119896 119909119897

119909119898 119910119894 119910119895)

(2) Unary operator sim R rarr R such that for any 119896-ary(with 119896 ge 0) relation 119877 isin P(D119896) sub R we have thatsim (119877) = D119896 119877 isin D119896 where ldquordquo is the substractionof relations For example the logic formula not120601(119909

119894 119909119895

119909119896 119909119897 119909119898)will be traduced by the algebraic expression

D5 119877 where 119877 is the extensions for a given Tarskirsquosinterpretation of the virtual predicate 120601

(3) Unary operator 120587minus119898

R rarr R such that for any 119896-ary (with 119896 ge 0) relation 119877 isin P(D119896) sub R we havethat 120587

minus119898(119877) is equal to the relation obtained by

elimination of the 119898th column of the relation 119877 if1 le 119898 le 119896 and 119896 ge 2 equal to 119891

⟨ ⟩(119877) if 119898 = 119896 = 1

otherwise it is equal to 119877For example the logic formula (exist119909

119896)120601(119909

119894 119909119895 119909119896 119909119897

119909119898) will be traduced by the algebraic expression

120587minus3(119877) where 119877 is the extensions for a given Tarskirsquos

interpretation of the virtual predicate 120601 and theresulting relation will have the following ordering ofattributes (119909

119894 119909119895 119909119897 119909119898)

Notice that the ordering of attributes of resulting relationscorresponds to the method used for generating the orderingof variables in the tuples of free variables adopted for virtualpredicates

Analogously to Boolean algebras which are extensionalmodels of propositional logic we introduce now an inten-sional algebra for this intensional FOL as follows

Definition 6 Intensional algebra for the intensional FOLin Definition 1 is a structure Aint = (D119891119905119868119889119879119903119906119905ℎcon119895

119878119878isinP(N2)negexist119904119899119899isinN) with binary operations

con119895119878 119863

119868times 119863

119868rarr 119863

119868 unary operation neg 119863

119868rarr 119863

119868

unary operations exist119904119899 119863

119868rarr 119863

119868 such that for any exten-

sionalization function ℎ isin E and 119906 isin 119863119896 119907 isin 119863

119895 119896 119895 ge 0

(1) ℎ(119868119889) = 119877=and ℎ(119879119903119906119905ℎ) = ⟨ ⟩

(2) ℎ(con119895119878(119906 119907)) = ℎ(119906) ⋈

119878ℎ(119907) where ⋈

119878is the natural

join operation defined above and con119895119878(119906 119907) isin 119863

119898

where119898 = 119896+119895minus|119878| if for every pair (1198941 1198942) isin 119878 it holds

that 1 le 1198941le 119896 1 le 119894

2le 119895 (otherwise con119895

119878(119906 119907) isin

119863119896+119895

)

(3) ℎ(neg(119906)) =sim (ℎ(119906)) = D119896 (ℎ(119906)) where sim is theoperation defined above and neg(119906) isin 119863

119896

(4) ℎ(exist119904119899(119906)) = 120587

minus119899(ℎ(119906)) where 120587

minus119899is the operation

defined above and exist119904119899(119906) isin 119863

119896minus1if 1 le 119899 le 119896

(otherwise exist119904119899is the identity function)

Notice that for 119906 isin 1198630 ℎ(neg(119906)) =sim(ℎ(119906)) = D0

(ℎ(119906)) = ⟨ ⟩ (ℎ(119906)) isin 119891 119905We define a derived operation union (P(119863

119894)0) rarr 119863

119894

119894 ge 0 such that for any 119861 = 1199061 119906

119899 isin P(119863

119894)

we have that union(1199061 119906

119899)=def1199061 if 119899 = 1 neg(con119895

119878

(neg(1199061) con119895

119878( neg(119906

119899)) ) where 119878 = (119897 119897)|1 le 119897 le 119894

otherwise Than we obtain that for 119899 ge 2

ℎ (union (119861))

= ℎ (neg (con119895119878(neg (119906

1) con119895

119878( neg (119906

119899)) )

= D119894 ((D

119894 ℎ (119906

1)) ⋈

119878sdot sdot sdot ⋈

119878(D

119894 ℎ (119906

119899)))

= D119894 ((D

119894 ℎ (119906

1))⋂ sdot sdot sdot⋂ (D

119894 ℎ (119906

119899)))

= ⋃ℎ (119906119895) | 1 le 119895 le 119899 = ⋃ℎ (119906) | 119906 isin 119861

(3)

Intensional interpretation 119868 L rarr D satisfies the fol-lowing homomorphic extension

(1) The logic formula120601(119909119894 119909119895 119909119896 119909119897 119909119898)and120595(119909

119897 119910119894 119909119895 119910119895)

will be intensionally interpreted by the concept 1199061isin

1198637 obtained by the algebraic expression con119895

119878(119906 119907)

where 119906 = 119868(120601(119909119894 119909119895 119909119896 119909119897 119909119898)) isin 119863

5 119907 = 119868(120595(119909

119897

119910119894 119909119895 119910119895)) isin 119863

4are the concepts of the virtual pre-

dicates 120601 120595 relatively and 119878 = (4 1) (2 3) Conse-quently we have that for any two formulae 120601 120595 isin Land a particular operator con119895

119878uniquely determined

by tuples of free variables in these two formulae 119868(120601and120595) = con119895

119878(119868(120601) 119868(120595))

(2) The logic formula not120601(119909119894 119909119895 119909119896 119909119897 119909119898) will be inten-

sionally interpreted by the concept 1199061isin 119863

5 obtained

by the algebraic expression neg(119906) where 119906 = 119868(120601(119909119894

119909119895 119909119896 119909119897 119909119898)) isin 119863

5is the concept of the virtual pre-

dicate 120601 Consequently we have that for any formula120601 isin L 119868(not120601) = neg(119868(120601))

(3) The logic formula (exist119909119896)120601(119909

119894 119909119895 119909119896 119909119897 119909119898) will be

intensionally interpreted by the concept 1199061isin 119863

4

obtained by the algebraic expression exist1199043(119906) where

119906 = 119868(120601(119909119894 119909119895 119909119896 119909119897 119909119898)) isin 119863

5is the concept of

the virtual predicate 120601 Consequently we have thatfor any formula 120601 isin L and a particular operatorexist119904

119899uniquely determined by the position of the

existentially quantified variable in the tuple of freevariables in 120601 (otherwise 119899 = 0 if this quantifiedvariable is not a free variable in 120601) 119868((exist119909)120601) =exist119904

119899(119868(120601))

Once one has found a method for specifying the inter-pretations of singular terms of L (take in consideration theparticularity of abstracted terms) the Tarski-style definitionsof truth and validity forLmaybe given in the customarywayWhat is proposed specifically is a method for characterizingthe intensional interpretations of singular terms ofL in suchaway that a given singular abstracted term⋖ 120601 ⋗120573

120572will denote

an appropriate property relation or proposition dependingon the value of 119898 = |120572| Thus the mapping of intensionalabstracts (terms) intoD will be defined differently from thatgiven in the version of Bealer [18] as follows


Definition 7 An intensional interpretation 119868 can be extendedto abstracted terms as follows for any abstracted term⋖ 120601 ⋗120573

120572

we define that

119868 (⋖ 120601 ⋗120573

120572) = union(119868(120601[

120573

119892 (120573)]) | 119892 isin D

120573) (4)

where 120573 denotes the set of elements in the list 120573 and theassignments inD120573 are limited only to the variables in 120573

Remark 8 Here we can make the question if there is asense to extend the interpretation also to (abstracted) termsbecause in Tarskirsquos interpretation of FOL we do not haveany interpretation for terms but only the assignments forterms as we defined previously by the mapping 119892lowast T rarrD The answer is positive because the abstraction symbol⋖ ⋗120573

120572can be considered as a kind of the unary built-in

functional symbol of intensional FOL so that we can applythe Tarskirsquos interpretation to this functional symbol into thefixedmapping 119868(⋖ ⋗120573

120572) L rarr D so that for any 120601 isin Lwe

have that 119868(⋖ 120601 ⋗120573120572) is equal to the application of this function

to the value120601 that is to 119868(⋖ ⋗120573120572)(120601) In such an approach we

would introduce also the typed variable119883 for the formulae inL so that the Tarskirsquos assignment for this functional symbolwith variable119883 with 119892(119883) = 120601 isin L can be given by

119892lowast(⋖ ⋗

120573

120572(119883))

= 119868 (⋖ ⋗120573

minus120572) (119892 (119883)) = 119868 (⋖ ⋗

120573

120572) (120601)

=

⟨ ⟩ isin 119863minus1

if 120572⋃120573 is not equal to the set of free variables in 120601

= union(119868(120601[120573

1198921015840 (120573)]) | 1198921015840 isin D120573) isin 119863

|120572|

otherwise(5)

Notice than if 120573 = 0 is the empty list then 119868(⋖ 120601 ⋗120573120572) =

119868(120601) Consequently the denotation of ⋖ 120601 ⋗ is equal to themeaning of a proposition 120601 that is 119868(⋖ 120601 ⋗) = 119868(120601) isin 119863

0

In the case when 120601 is an atom 119901119898119894(1199091 119909

119898) then

119868(⋖ 119901119898119894(1199091 119909

119898) ⋗

1199091119909119898

) = 119868(119901119898119894(1199091 119909

119898)) isin 119863

119898

while 119868(⋖ 119901119898119894(1199091 119909

119898) ⋗

1199091119909119898) = union(119868(119901119898

119894(119892(119909

1)

119892(119909119898))) | 119892 isin D119909

1119909119898) isin 119863

0 with ℎ(119868

(⋖ 119901119898119894(1199091 119909

119898) ⋗

1199091119909119898)) = ℎ(119868((exist119909

1) sdot sdot sdot (exist119909

119898) 119901119898119894(1199091

119909119898))) isin 119891 119905

For example

ℎ (119868 (⋖ 1199011

119894(1199091) and not119901

1

119894(1199091) ⋗

1199091

))

= ℎ (119868 ((exist1199091) (⋖ 119901

1

119894(1199091) and not119901

1

119894(1199091) ⋗

1199091

))) = 119891(6)

The interpretation of a more complex abstract ⋖ 120601 ⋗120573120572

is defined in terms of the interpretations of the relevantsyntactically simpler expressions because the interpretation

of more complex formulae is defined in terms of the interpre-tation of the relevant syntactically simpler formulae based onthe intensional algebra above For example 119868(1199011

119894(119909)and1199011

119896(119909)) =

con119895(11)

(119868(1199011119894(119909)) 119868(1199011

119896(119909))) 119868(not120601) = neg(119868(120601)) 119868(exist119909

119894)120601

(119909119894 119909119895 119909119894 119909119896) = exist119904

3(119868(120601))

Consequently based on the intensional algebra inDefinition 6 and on intensional interpretations of abstractedterms in Definition 7 it holds that the interpretation of anyformula inL (and any abstracted term) will be reduced to analgebraic expression over interpretations of primitive atomsinL This obtained expression is finite for any finite formula(or abstracted term) and represents themeaning of such finiteformula (or abstracted term)

The extension of an abstracted term satisfy the followingproperty

Proposition 9 For any abstracted term ⋖ 120601 ⋗120573120572with |120572| ge 1

we have that

ℎ (119868 (⋖ 120601 ⋗120573

120572)) = 120587

minus120573(ℎ (119868 (120601))) (7)

where 120587minus(1199101119910119896)= 120587

minus1199101

∘ sdot sdot sdot ∘ 120587minus1199101

∘ is the sequential composi-tion of functions and 120587

minus0is an identity

Proof Let x be a tuple of all free variables in 120601 so that x =

120572⋃120573 120572 = (1199091 119909

119896) then we have that ℎ(119868(⋖ 120601 ⋗120573

120572)) =

ℎ(union(119868(120601[120573119892(120573)]) | 119892isin D120573)) from Definition 7 =

⋃ℎ(119868(120601[120573119892(120573)])) | 119892isin D120573 = ⋃(1198921(1199091) 119892

1(119909119896)) |

1198921isin D120572 and ℎ(119868(120601[120573119892(120573)][120572119892

1(120572)])) = 119905 | 119892 isin D120573 =

1198921(120572) | 119892

1isin D120572⋃120573 and ℎ(119868(120601119892

1)) = 119905 = 120587

minus120573(119892

1(x) |

1198921isin Dx and ℎ(119868(120601119892

1)) = 119905) = 120587

minus120573(119892

1(x) | 119892

1isin Dx and

1198921(x) isin ℎ(119868(120601))) by (T) = 120587

minus120573(ℎ(119868(120601)))

We can correlateEwith a possible-world semantics Sucha correspondence is a natural identification of intensionallogics with modal Kripke-based logics

Definition 10 (model) A model for intensional FOL withfixed intensional interpretation 119868 which expresses the two-step intensional semantics in Definition 4 is the Kripkestructure Mint = (WD 119881) where W = 119894119904minus1(ℎ)|ℎ isin Ea mapping 119881 W times 119875 rarr ⋃

119899lt120596119905 119891D

119899

with 119875 a set ofpredicate symbols of the language such that for any world119908 = 119894119904minus1(ℎ) isin W 119901119899

119894isin 119875 and (119906

1 119906

119899) isin D119899 it holds that

119881(119908 119901119899119894)(119906

1 119906

119899) = ℎ(119868(119901119899

119894(1199061 119906

119899))) The satisfaction

relation ⊨119908119892

for a given 119908 isin W and assignment 119892 isin DV isdefined as follows

(1) M⊨119908119892119901119896119894(1199091 119909

119896) if and only if 119881(119908 119901119896

119894)

(119892(1199091) 119892(119909

119896)) = 119905

(2) M⊨119908119892120593 and 120601 if and only ifM⊨

119908119892120593 andM⊨

119908119892120601

(3) M⊨119908119892not120593 if and only if notM⊨

119908119892120593

(4) M⊨119908119892(exist119909)120601 if and only if

(41) M⊨119908119892120601 if 119909 is not a free variable in 120601

(42) exists 119906 isin D such that M⊨119908119892120601[119909119906] if 119909 is a

free variable in 120601


It is easy to show that the satisfaction relation ⊨ for thisKripke semantics in aworld119908 = 119894119904minus1(ℎ) is defined byM⊨

119908119892120601

if and only if ℎ(119868(120601119892)) = 119905We can enrich this intensional FOL by another modal

operators as for example the ldquonecessityrdquo universal logicoperator ◻ with accessibility relationR = WtimesW obtainingan S5 Kripke structure Mint = (WRD 119881) In this casewe are able to define the following equivalences betweenthe abstracted terms without free variables ⋖ 120601 ⋗1205731

120572119892 and

⋖ 120595 ⋗1205732120572119892 where all free variables (not in 120572) are instantiated

by 119892 isin DV (here119860 equiv 119861 denotes the formula (119860 rArr 119861)and(119861 rArr119860))

(i) (Strong) Intensional equivalence (or equality) ldquo≍rdquo isdefined by ⋖ 120601 ⋗1205731

120572119892 ≍ ⋖ 120595 ⋗1205732

120572119892 if and only if ◻(120601

[1205731119892(120573

1)] equiv 120595[120573

2119892(120573

2)]) with M ⊨

1199081198921015840◻120593 if

and only if for all 1199081015840 isin W (119908 1199081015840) isin Rimplies M ⊨

11990810158401198921015840120593 From Example 5 we have that

⋖ 11990111(119909) ⋗

119909≍ ⋖ 1199011

2(119909) ⋗

119909 that is ldquo119909 has been

boughtrdquo and ldquo119909 has been soldrdquo are intensionallyequivalent but they have not the same meaning (theconcept 119868(1199011

1(119909)) isin 119863

1is different from 119868(1199011

2(119909)) isin

1198631)

(ii) Weak intensional equivalence ldquoasymprdquo is definedby ⋖ 120601 ⋗1205731

120572119892 asymp ⋖ 120595 ⋗1205732

120572119892 if and only if loz120601[120573

1

119892(1205731)] equiv loz120595[120573

2119892(120573

2)] The symbol loz = not◻not is the

correspondent existential modal operator This weakequivalence is used for P2P database integration in anumber of papers [16 19ndash24]

Note that if we want to use the intensional equalityin our language then we need the correspondent operatorin intensional algebra Aint for the ldquonecessityrdquo modal logicoperator ◻

This semantics is equivalent to the algebraic semanticsfor L in [8] for the case of the conception where inten-sional entities are considered to be equal if and only ifthey are necessarily equivalent Intensional equality is muchstronger that the standard extensional equality in the actualworld just because it requires the extensional equality inall possible worlds in fact if ⋖ 120601 ⋗1205731

120572119892 ≍ ⋖ 120595 ⋗1205731

120572119892 then

ℎ(119868(⋖ 119860 ⋗1205731120572119892)) = ℎ(119868(⋖ 120595 ⋗1205732

120572119892)) for all extensionalization

functions ℎ isin E (ie possible worlds 119894119904minus1(ℎ) isin W)It is easy to verify that the intensional equality means that

in every possible world119908 isin W the intensional entities 1199061and

1199062have the same extensionsLet the logic modal formula ◻120601[120573

1119892(120573

1)] where the

assignment 119892 is applied only to free variables in 1205731of a

formula 120601 not in the list of variables in 120572 = (1199091 119909

119899)

119899 ge 1 represents an 119899-ary intensional concept such that119868(◻120601[120573

1119892(120573

1)]) isin 119863

119899and 119868(120601[120573

1119892(120573

1)]) = 119868(⋖ 120601 ⋗1205731

120572119892) isin

119863119899 Then the extension of this 119899-ary concept is equal to (here

the mapping necess 119863119894rarr 119863

119894for each 119894 ge 0 is a new

operation of the intensional algebraAint in Definition 6)

ℎ(119868(◻120601[1205731

119892 (1205731)]))

= ℎ(necess(119868(120601[1205731

119892 (1205731)])))

= (1198921015840(1199091) 119892

1015840(119909119899)) |

M ⊨1199081198921015840

◻120601[1205731

119892 (1205731)] 119892

1015840isin D

V

= (1198921015840(1199091) 119892

1015840(119909119899)) | 119892

1015840isin D

V forall119908

1

((1199081199081)isinR implies M⊨

11990811198921015840

120601[1205731

119892 (1205731)])

= ⋂ℎ1isinE

ℎ1(119868(120601[

1205731

119892 (1205731)]))

(8)

while

ℎ(119868(loz120601[1205731

119892 (1205731)]))

= ℎ(119868(not◻not120601[1205731

119892 (1205731)]))

= ℎ(neg(necess(119868(not120601[1205731

119892 (1205731)]))))

= D119899 ℎ(necess(119868(not120601[

1205731

119892 (1205731)])))

= D119899 ( ⋂

ℎ1isinE

ℎ1(119868(not120601[

1205731

119892 (1205731)])))

= D119899 ( ⋂

ℎ1isinE

ℎ1(119899119890119892(119868(120601[

1205731

119892 (1205731)]))))

= D119899 ( ⋂

ℎ1isinE

D119899 ℎ

1(119868(120601[

1205731

119892 (1205731)])))

= ⋃ℎ1isinE

ℎ1(119868(120601[

1205731

119892 (1205731)]))

(9)

Consequently the concepts ◻120601[1205731119892(120573

1)] and

loz120601[1205731119892(120573

1)] are the built-in (or rigid) concept as well

whose extensions do not depend on possible worlds


Thus two concepts are intensionally equal that is⋖ 120601 ⋗1205731

120572119892 ≍ ⋖ 120595 ⋗1205732

120572119892 if and only if ℎ(119868(120601[120573

1119892(120573

1)])) =

ℎ(119868(120595[1205732119892(120573

2)])) for every ℎ

Analogously two concepts are weakly equivalent that is⋖ 120601 ⋗1205731

120572119892 asymp ⋖ 120595 ⋗1205732

120572119892 if and only if ℎ(119868(loz120601[120573

1119892(120573

1)])) =

ℎ(119868(loz120595[1205732119892(120573

2)]))

3 Application to the Intensional FOL withoutAbstraction Operator

In the case of the intensional FOL defined in Definition 1without Bealerrsquos intensional abstraction operator ⋖⋗ weobtain the syntax of the standard FOL but with intensionalsemantics as presented in [15]

Such a FOL has a well-known Tarskirsquos interpretationdefined as follows

An interpretation (Tarski) 119868119879consists in a nonempty

domainD and amapping that assigns to any predicateletter 119901119896

119894isin 119875 a relation 119877 = 119868

119879(119901119896119894) sube D119896 to any func-

tional letter 119891119896119894isin 119865 a function 119868

119879(119891119896119894) D119896 rarr D or

equivalently its graph relation 119877 = 119868119879(119891119896119894) sube D119896+1

where the 119896 + 1th column is the resulting functionrsquosvalue and to each individual constant 119888 isin 119865 one givenelement 119868

119879(119888) isin D

Consequently from the intensional point of view aninterpretation of Tarski is a possible world in theMontaguersquos intensional semantics that is 119908 = 119868

119879isin

W The correspondent extensionalization function isℎ = 119894119904(119908) = 119894119904(119868

119879)

We define the satisfaction of a logic formulae in Lfor a given assignment 119892 V rarr D inductively asfollowsIf a formula 120601 is an atomic formula 119901119896

119894(1199051 119905

119896)

then this assignment 119892 satisfies 120601 if and only if(119892lowast(119905

1) 119892lowast(119905

119896)) isin 119868

119879(119901119896119894)119892 satisfiesnot120601 if and only

if it does not satisfy 120601 119892 satisfies 120601 and 120595 iff 119892 satisfies 120601and 119892 satisfies 120595 119892 satisfies (exist119909

119894)120601 if and only if there

exists an assignment 1198921015840 isin DV that may differ from 119892only for the variable 119909

119894isin V and 1198921015840 satisfies 120601

A formula 120601 is true for a given interpretation 119868119879if and

only if 120601 is satisfied by every assignment 119892 isin DVA formula 120601 is valid (ie tautology) if and only if120601 is true for every Tarksirsquos interpretation 119868

119879isin I

119879

An interpretation 119868119879is a model of a set of formulae

Γ if and only if every formula 120601 isin Γ is true inthis interpretation We denote by FOL(Γ) the FOLwith a set of assumptions Γ and by I

119879(Γ) the subset

of Tarskirsquos interpretations that are models of Γ withI119879(0) = I

119879 A formula 120601 is said to be a logical

consequence of Γ denoted by Γ ⊩ 120601 if and only if 120601 istrue in all interpretations in I

119879(Γ) Thus ⊩ 120601 if and

only if 120601 is a tautologyThe basic set of axioms of the FOL are that of thepropositional logic with two additional axioms (A1)(forall119909)(120601 rArr 120595) rArr (120601 rArr (forall119909)120595) (119909 does not occur in 120601

and it is not bound in 120595) and (A2) (forall119909)120601 rArr 120601[119909119905](neither 119909 nor any variable in 119905 occurs bound in 120601)For the FOL with identity we need the proper axiom(A3) 119909

1≐ 119909

2rArr (119909

1≐ 119909

3rArr 119909

2≐ 119909

3)

The inference rules are Modus Ponens and general-ization (G) ldquoif 120601 is a theorem and 119909 is not bound in 120601then (forall119909)120601 is a theoremrdquo

The standard FOL is considered as an extensional logicbecause two open sentences with the same tuple of variables120601(119909

1 119909

119898) and 120595(119909

1 119909

119898) are equal if and only if they

have the same extension in a given interpretation 119868119879 that is if

and only if 119868lowast119879(120601(119909

1 119909

119898)) = 119868lowast

119879(120595(119909

1 119909

119898)) where 119868lowast

119879

is the unique extension of 119868119879to all formulae as follows

(1) For a (closed) sentence 120601119892 we have that 119868lowast119879(120601119892) = 119905

if and only if119892 satisfies120601 as recursively defined above(2) For an open-sentence 120601 with the tuple of free

variables (1199091 119909

119898) we have that 119868lowast

119879(120601(119909

1

119909119898))=def (119892(1199091) 119892(119909119898)) | 119892 isin DV and 119868lowast

119879(120601119892) =

119905

It is easy to verify that for a formula120601with the tuple of freevariables (119909

1 119909

119898) 119868lowast

119879(120601(119909

1 119909

119898)119892) = 119905 if and only if

(119892(1199091) 119892(119909

119898)) isin 119868lowast

119879(120601(119909

1 119909

119898))

This extensional equality of two virtual predicates canbe generalized to the extensional equivalence when bothpredicates 120601 120595 have the same set of free variables but theirordering in the tuples of free variables is not identical suchtwo virtual predicates are equivalent if the extension of thefirst is equal to the proper permutation of columns of theextension of the second virtual predicate It is easy to verifythat such an extensional equivalence corresponds to thelogical equivalence denoted by 120601 equiv 120595

This extensional equivalence between two relations1198771 1198772isin R with the same arity will be denoted by 119877

1cong 119877

2

while the extensional identity will be denoted in the standardway by 119877

1= 119877

2

Let AFOL = (L ≐ ⊤ and not exist) be a free syntax algebrafor ldquofirst-order logic with identity ≐rdquo with the set L offirst-order logic formulae with ⊤ denoting the tautologyformula (the contradiction formula is denoted by not⊤) withthe set of variables in V and the domain of values in DIt is well known that we are able to make the extensionalalgebraization of the FOL by using the cylindric algebras[25] that are the extension of Boolean algebras with a set ofbinary operators for the FOL identity relations and a set ofunary algebraic operators (ldquoprojectionsrdquo) for each case of FOLquantification (exist119909) In what follows we will make an analogextensional algebraization overR but by interpretation of thelogic conjunction and by a set of natural join operators overrelations introduced by Coddrsquos relational algebra [9] and [26]as a kind of a predicate calculus whose interpretations are tiedto the database

Corollary 11 (extensional FOL semantics [15]) Let us definethe extensional relational algebra for the FOL by

AR = (R 119877= ⟨ ⟩ ⋈SSisinP(N2) sim 120587minus119899119899isinN) (10)


where ⟨ ⟩ isin R is the algebraic value correspondent to the logictruth and 119877

=is the binary relation for extensionally equal

elements We will use ldquo=rdquo for the extensional identity forrelations inR

Then for any Tarskirsquos interpretation 119868119879its unique extension

to all formulae 119868lowast119879 L rarr R is also the homomorphism 119868lowast

119879

A119865119874119871

rarr AR from the free syntax FOL algebra into this ext-ensional relational algebra

Proof Directly from definition of the semantics of theoperators in AR defined in precedence let us take thecase of conjunction of logic formulae of the definitionabove where 120593(119909

119894 119909119895 119909119896 119909119897 119909119898 119910119894 119910119895) (its tuple of vari-

ables is obtained by the method defined in the FOLintroduction) is the virtual predicate of the logic formula120601(119909

119894 119909119895 119909119896 119909119897 119909119898) and 120595(119909

119897 119910119894 119909119895 119910119895) 119868lowast

119879(120601 and 120595) = 119868lowast

119879(120593) =

(119892(119909119894) 119892(119909

119895) 119892(119909

119896) 119892(119909

119897) 119892(119909

119898) 119892(119910

119894) 119892(119910

119895)) | 119868lowast

119879(120593119892) =

119905 = (119892(119909119894) 119892(119909

119895) 119892(119909

119896) 119892(119909

119897) 119892(119909

119898)119892(119910

119894) 119892(119910

119895)) | 119868lowast

119879

(120601119892 and 120595119892) = 119905 = (119892(119909119894) 119892(119909

119895) 119892(119909

119896)119892(119909

119897) 119892(119909

119898) 119892(119910

119894)

119892(119910119895)) | 119868lowast

119879(120601119892) = 119905 and 119868lowast

119879(120601119892) = 119905 = (119892(119909

119894) 119892(119909

119895)

119892(119909119896)119892(119909

119897) 119892(119909

119898) 119892(119910

119894) 119892(119910

119895)) | (119892(119909

119894) 119892(119909

119895) 119892(119909

119896)119892(119909

119897)

119892(119909119898)) isin 119868lowast

119879(120601) and (119892(119909

119897) 119892(119910

119894) 119892(119909

119895) 119892(119910

119895)) isin 119868lowast

119879(120601) =

119868lowast119879(120601)⋈

(41)(23)119868lowast119879(120595)

Thus it is enough to show that 119868lowast119879(⊤) = ⟨ ⟩ is also valid

and 119868lowast119879(not⊤) = 0 The first property comes from the fact that

⊤ is a tautology thus satisfied by every assignment 119892 thatis it is true that is 119868lowast

119879(⊤) = 119905 (and 119905 is equal to the empty

tuple ⟨ ⟩) The second property comes from the fact that119868lowast119879(not⊤) =sim (119868lowast

119879(⊤)) =sim (⟨ ⟩) = D0 ⟨ ⟩ = ⟨ ⟩ ⟨ ⟩ = 0

That is the tautology and the contradiction have the true andfalse logic value respectively inR

We have also that 119868lowast119879(≐ (119909 119910)) = 119868

119879(≐) = 119877

=for

every interpretation 119868119879

because ≐ is the built-in binarypredicate that is with the same extension in every Tarskirsquosinterpretation

Consequently the mapping 119868lowast119879

(L ≐ ⊤ and not exist) rarrAR is a homomorphism that represents the extensionalTarskian semantics of the FOL

Consequently we obtain the following Intensionalexten-sional FOL semantics [15]

For any Tarskirsquos interpretation 119868119879of the FOL the follow-

ing diagram of homomorphisms commutes

FregeRussellsemantics

Intensional interpret I h (extensionalization)

IlowastT (Tarskirsquos interpretation)119964 (denotation)

119964int (conceptsmeaning)

119964FOL (syntax)

(11)

where ℎ = 119894119904(119908) and 119908 = 119868119879isin W is the explicit possible

world (extensional Tarskirsquos interpretation)This homomorphic diagram formally expresses the fusion

of Fregersquos and Russellrsquos semantics [27ndash29] of meaning anddenotation of the FOL language and renders mathematicallycorrect the definition of what we call an ldquointuitive notion of

intensionalityrdquo in terms of which a language is intensionalif denotation is distinguished from sense that is if bothdenotation and sense are ascribed to its expressions Thisnotion is simply adopted from Fregersquos contribution (withoutits infinite sense-hierarchy avoided by Russellrsquos approachwhere there is only one meaning relation one fundamentalrelation between words and things here represented by onefixed intensional interpretation 119868) where the sense containsmode of presentation (here described algebraically as analgebra of concepts (intensions)Aint) andwhere sense deter-mines denotation for any given extensionalization functionℎ (correspondent to a given Traskirsquos interpretation 119868

119879) More

about the relationships between Fregersquos and Russellrsquos theoriesof meaning may be found in the Chapter 7 ldquoExtensionalityand Meaningrdquo in [18]

As noted by Gottlob Frege and Rudolf Carnap (heuses terms Intensionextension in the place of Fregersquos termssensedenotation [30]) the two logic formulae with the samedenotation (ie the same extension for a given Tarskirsquosinterpretation 119868

119879) need not have the same sense (intension)

thus such codenotational expressions are not substitutable ingeneral

In fact there is exactly one sense (meaning) of a givenlogic formula inL defined by the uniquely fixed intensionalinterpretation 119868 and a set of possible denotations (extensions)each determined by a given Tarskirsquos interpretation of the FOLas follows from Definition 4

L119868

997888rarr D997904rArrℎ=119894119904(119868

119879)amp119868119879isinW=I

119879(Γ)R (12)

Often ldquointensionrdquo has been used exclusively in connectionwith possible worlds semantics however here we use (asmany others as Bealer for example) ldquointensionrdquo in a morewide sense that is as an algebraic expression in the intensionalalgebra of meanings (concepts) Aint which represents thestructural composition of more complex concepts (mean-ings) from the given set of atomic meanings Consequentlynot only the denotation (extension) is compositional but alsothe meaning (intension) is compositional

4 Conclusion

Semantics is a theory concerning the fundamental relationsbetween words and things In Tarskian semantics of the FOLone defines what it takes for a sentence in a language tobe truely relative to a model This puts one in a positionto define what it takes for a sentence in a language to bevalid Tarskian semantics often proves quite useful in logicDespite this Tarskian semantics neglects meaning as if truthin language were autonomous Because of that the Tarskiantheory of truth becomes inessential to the semantics for moreexpressive logics or more ldquonaturalrdquo languages

Both Montaguersquos and Bealerrsquos approaches were useful forthis investigation of the intensional FOL with intensionalabstraction operator but the first is not adequate and explainswhy we adopted two-step intensional semantics (intensionalinterpretation with the set of extensionalization functions)

At the end of this work we defined an extensional algebrafor the FOL (different from standard cylindric algebras) and


the commutative homomorphic diagram that expresses thegeneralization of the Tarskian theory of truth for the FOL intothe FregeRussellrsquos theory of meaning

References

[1] J Pustejovsky and B Boguraev ldquoLexical knowledge representa-tion and natural language processingrdquoArtificial Intelligence vol63 no 1-2 pp 193ndash223 1993

[2] M Fitting ldquoFirst-order intensional logicrdquo Annals of Pure andApplied Logic vol 127 no 1ndash3 pp 171ndash193 2004

[3] T Brauner ldquoAdding intensional machinery to hybrid logicrdquoJournal of Logic and Computation vol 18 no 4 pp 631ndash6482008

[4] T Brauner and S Ghilardi ldquoFirst-order modal logicrdquo in Hand-book of Modal Logic pp 549ndash620 Elsevier 2007

[5] S Bond andM Denecker ldquoI-logic an intensional logic of infor-mationsrdquo in Proceedings of the 19th Belgian-Dutch Conferenceon Artificial Intelligence pp 49ndash56 Utrecht The NetherlandsNovember 2007

[6] M A Orgun and W W Wadge ldquoTowards a unified theory ofintensional logic programmingrdquoThe Journal of Logic Program-ming vol 13 no 4 pp 413ndash440 1992

[7] G Bealer ldquoUniversalsrdquoThe Journal of Philosophy vol 90 no 1pp 5ndash32 1993

[8] G Bealer ldquoTheories of properties relations and propositionsrdquoThe Journal of Philosophy vol 76 no 11 pp 634ndash648 1979

[9] E F Codd ldquoRelational completeness of data base sublanguagesrdquoin Data Base Systems Courant Computer Science SymposiaSeries 6 Prentice Hall Englewood Cliffs NJ USA 1972

[10] D K Lewis On the Plurality of Worlds Blackwell Oxford UK1986

[11] R Stalnaker Inquiry The MIT Press Cambridge Mass USA1984

[12] R Montague ldquoUniversal grammarrdquo Theoria vol 36 no 3 pp373ndash398 1970

[13] R Montague ldquoThe proper treatment of quantification in ordi-nary Englishrdquo in Approaches to Natural Language J HintikkaP Suppes J M E Moravcsik et al Eds pp 221ndash242 ReidelDordrecht The Netherlands 1973

[14] R Montague Formal Philosophy Selected Papers of RichardMontague Yale University Press London UK 1974

[15] Z Majkic ldquoFirst-order logic modality andintensionalityrdquohttparxivorgabs11030680v1

[16] Z Majkic ldquoIntensional first-order logic for P2P database sys-temsrdquo in Journal onData Semantics 12 vol 5480 of LectureNotesin Computer Science pp 131ndash152 Springer Berlin Germany2009

[17] Z Majkic ldquoIntensional semantics for RDF data structuresrdquo inProceedings of the 12th International Symposium on DatabaseEngineering amp Applications Systems (IDEAS rsquo08) pp 69ndash77Coimbra Portugal September 2008

[18] G Bealer Quality and Concept Oxford University Press NewYork NY USA 1982

[19] Z Majkic ldquoWeakly-coupled ontology integration of P2Pdatabase systemsrdquo in Proceedings of the 1st International Work-shop on Peer-to-Peer Knowledge Management (P2PKM rsquo04)Boston Mass USA August 2004

[20] Z Majkic ldquoIntensional P2Pmapping between RDF ontologiesrdquoin Proceedings of the 6th International Conference on Web

Information Systems (WISE rsquo05) M Kitsuregawa Ed vol 3806of Lecture Notes in Computer Science pp 592ndash594 New YorkNY USA November 2005

[21] Z Majkic ldquoIntensional semantics for P2P data integrationrdquoin Journal on Data Semantics 6 vol 4090 of Lecture Notes inComputer Science Special Issue on lsquoEmergent Semanticsrsquo pp47ndash66 2006

[22] Z Majkic ldquoNon omniscient intensional contextual reasoningfor query-agents in P2P systemsrdquo in Proceedings of the 3rdIndian International Conference on Artificial Intelligence (IICAIrsquo07) Pune India December 2007

[23] Z Majkic ldquoCoalgebraic specification of query computation inintensional P2P database systemsrdquo in Proceedings of the Interna-tional Conference onTheoretical and Mathematical Foundationsof Computer Science (TMFCS rsquo08) pp 14ndash23Orlando Fla USAJuly 2008

[24] Z Majkic ldquoRDF view-based interoperability in intensionalFOL for Peer-to-Peer database systemsrdquo in Proceedings of theInternational Conference on Enterprise Information Systems andWeb Technologies (EISWT rsquo08) pp 88ndash96 Orlando Fla USAJuly 2008

[25] L Henkin J DMonk andA TarskiCylindic Algebras I North-Holland Amsterdam The Netherlands 1971

[26] A Pirotte ldquoA precise definition of basic relational notions andof the relational algebrardquo ACM SIGMOD Record vol 13 no 1pp 30ndash45 1982

[27] G Frege ldquoUber Sinn undBedeutungrdquoZeitschrift fur Philosophieund philosophische Kritik vol 100 pp 22ndash50 1892

[28] B Russell ldquoOn Denotingrdquo in Logic and Knowledge vol 14 ofMind Reprinted in Russell pp 479ndash493 1905

[29] A N Whitehead and B Russell Principia Mathematica vol 1Cambridge Mass USA 1910

[30] R CarnapMeaning and Necessity Chicago Ill USA 1947

Submit your manuscripts athttpwwwhindawicom

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ReconfigurableComputing

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Applied Computational Intelligence and Soft Computing

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etc) and to Montaguersquos definition of intension of a givenvirtual predicate 120601(119909

1 119909

119896) (a FOL open-sentence with the

tuple of free variables (1199091 119909

119896)) as amapping frompossible

worlds into extensions of this virtual predicate Among thepossible worlds we distinguish the actual possible worldFor example if we consider a set of predicates of a givenDatabase and their extensions in different time-instancesthen the actual possible world is identified by the currentinstance of the time

The second conception of intensional entities is to befound in Russellrsquos doctrine of logical atomism In this doc-trine it is required that all complete definitions of intensionalentities be finite as well as unique and noncircular it offersan algebraicway for definition of complex intensional entitiesfrom simple (atomic) entities (ie algebra of concepts)conception also evident in Leibnizrsquos remarks In a predicatelogics predicates and open-sentences (with free variables)express classes (properties and relations) and sentencesexpress propositions Note that classes (intensional entities)are reified that is they belong to the same domain asindividual objects (particulars) This endows the intensionallogics with a great deal of uniformity making it possibleto manipulate classes and individual objects in the samelanguage In particular when viewed as an individual objecta class can be a member of another class

The distinction between intensions and extensions isimportant (as in lexicography [1]) considering that exten-sions can be notoriously difficult to handle in an efficientmanner The extensional equality theory of predicates andfunctions under higher-order semantics (eg for two pred-icates with the same set of attributes 119901 = 119902 is true iffthese symbols are interpreted by the same relation) thatis the strong equational theory of intensions is not decid-able in general For example the second-order predicatecalculus and Churchrsquos simple theory of types both underthe standard semantics are not even semi-decidable Thusseparating intensions from extensions makes it possible tohave an equational theory over predicate and function names(intensions) that is separate from the extensional equality ofrelations and functions

Relevant recentwork about the intension and its relation-ship with FOL has been presented in [2] in the considerationof rigid and nonrigid objects with respect to the possibleworlds where the rigid objects like ldquoGeorge Washingtonrdquoare the same things from possible world to possible worldNonrigid objects like ldquothe Secretary-General of UnitedNationsrdquo are varying from circumstance to circumstanceand can be modeled semantically by functions from possibleworlds to domain of rigid objects like intensional entitiesBut in his approach differently from that one fitting changesalso the syntax of the FOL by introducing an ldquoextensionof rdquo operator darr in order to distinguish the intensional entityldquogross-domestic-product-of-Denmarkrdquo and its use in ldquothegross domestic product of Denmark is currently greater thangross domestic product of Finlandrdquo In his approach if 119909is an intensional variable darr 119909 is extensional while darr is notapplicable to extensional variables differently fromourwhereeach variable (concept) has both intensional and extensionMoreover in his approach the problem arises because the

action of letting 119909 designate that is evaluating darr 119909 andthe action of passing to an alternative possible world thatis of interpreting the existential modal operator are notactions that commute To disambiguate this one more pieceof machinery is needed as well which substantially and ad-hock changes the syntax and semantics of FOL introducesthe Higher-order Modal logics and is not a conservativeextension of Tarskirsquos semantics

In most recent work in [3 4] it is given an intensionalversion of first-order hybrid logic which is also a hybridizedversion of Fittingrsquos intensional FOL by a kind of generalizedmodels thus different from our approaches to conservativeextension of Tarskirsquos semantics to intensional FOL

Another recent relevant work is presented by I-logicin [5] which combines both approach to semantics ofintensional objects of Montague and Fitting

We recall that Intensional Logic Programming is a newform of logic programming based on intensional logic andpossible worlds semantics and is a well-defined practice inusing the intensional semantics [6] Intensional logic allowsus to use logic programming to specify nonterminating com-putations and to capture the dynamic aspects of certain prob-lems in a natural and problem-oriented style The meaningsof formulas of an intensional first-order language are givenaccording to intensional interpretations and to elements of aset of possible worlds Neighborhood semantics is employedas an abstract formulation of the denotations of intensionaloperators The model-theoretic and fixpoint semantics ofintensional logic programs are developed in terms of least(minimum) intensional Herbrand models Intensional logicprograms with intensional operator definitions are regardedas metatheories

In what follows we denote by 119861119860 the set of all functionsfrom 119860 to 119861 and by 119860119899 an 119899-folded cartesian product 119860 timessdot sdot sdot times 119860 for 119899 ge 1 By 119891 119905 we denote empty set 0 and singletonset ⟨ ⟩ respectively (with the empty tuple ⟨ ⟩ ie the uniquetuple of 0-ary relation) which may be thought of as falsity 119891and truth 119905 as those used in the relational algebra For a givendomain D we define that D0 is a singleton set ⟨ ⟩ so that119891 119905 = P(D0) whereP is the powerset operator

2 Intensional FOL Language withIntensional Abstraction

Intensional entities are such concepts as propositions andproperties The term ldquointensionalrdquo means that they violatethe principle of extensionality the principle that extensionalequivalence implies identity All (ormost) of these intensionalentities have been classified at one time or another as kinds ofUniversals [7]

We consider a nonempty domainD = 119863minus1⋃119863

119868 where a

subdomain 119863minus1

is made of particulars (extensional entities)and the rest119863

119868= 119863

0⋃119863

1sdot sdot sdot ⋃119863

119899sdot sdot sdot is made of universals (

1198630for propositions (the 0-ary concepts)) and119863

119899 119899 ge 1 for

119899-ary conceptsThe fundamental entities are intensional abstracts or

so-called that-clauses We assume that they are singularterms Intensional expressions like ldquobelieverdquo ldquomeanrdquo ldquoassertrdquo








(2) if 1199051 119905

119896are terms then 119901119896

119894(1199051 119905





1 119909








120572with 119909

119894isin 120572)





1≐ 119905

2 We





1 119889

119896 119891(119889

1

119889119896)) | 119889









119898) This



1 119909


iables (1199091 119909



119896)) isin 119863

119896 and


0with 119868(⊤) = 119879119903119906119905ℎ isin 119863







120572 then 119892lowast(⋖ 120601 ⋗120573

120572) =


ables in 120572) where 119892(120573) = 119892(1199101 119910

119898) = (119892(119910

1)





0 ℎ(119906) isin 119891 119905 sube P(119863




minus1 ℎ(119906) = 119906








ℎ119894) sum

119894geminus1

119863119894997888rarr 119863

minus1+sum119894ge0

P (119863119894) (1)

where ℎminus1= 119894119889 119863

minus1rarr 119863


ping ℎ0 119863



119894rarr P(119863119894) 119894 ge 1





(T) ℎ(119868(120601119892)) = 119905 if and only if (119892(1199091) 119892(119909

119896)) isin




(1198921015840(119909119894) =

119892(119909119894)) implies 1198921015840(119909

119896+1) = 119892(119909

119896+1)







L119868


where 119868





119900119901is


1 119909

119896) isin L

119900119901the

mapping 119868119899(120601(119909

1 119909









119868(11990111(119909)) and 119907 = 119868(1199011

2(119909)) in isin 119863





=) where






⟨ ⟩(119877) = ⟨ ⟩ if






1while the second





2




1isin P(D5) 119877

2isin P(D4)


119894 119909119895 119909119896 119909119897

119909119898 119910119894 119910119895)


119894 119909119895







⟨ ⟩(119877) if 119898 = 119896 = 1


119896)120601(119909

119894 119909119895 119909119896 119909119897




119894 119909119895 119909119897 119909119898)





con119895119878 119863

119868times 119863

119868rarr 119863


119868rarr 119863

119868


119868rarr 119863



119895 119896 119895 ge 0

(1) ℎ(119868119889) = 119877=and ℎ(119879119903119906119905ℎ) = ⟨ ⟩

(2) ℎ(con119895119878(119906 119907)) = ℎ(119906) ⋈

119878ℎ(119907) where ⋈



119898


that 1 le 1198941le 119896 1 le 119894


119878(119906 119907) isin

119863119896+119895

)


119896

(4) ℎ(exist119904119899(119906)) = 120587

minus119899(ℎ(119906)) where 120587



119896minus1if 1 le 119899 le 119896




119894)0) rarr 119863

119894


119899 isin P(119863

119894)


119899)=def1199061 if 119899 = 1 neg(con119895

119878

(neg(1199061) con119895

119878( neg(119906

119899)) ) where 119878 = (119897 119897)|1 le 119897 le 119894


ℎ (union (119861))

= ℎ (neg (con119895119878(neg (119906

1) con119895

119878( neg (119906

119899)) )

= D119894 ((D

119894 ℎ (119906

1)) ⋈


119878(D

119894 ℎ (119906

119899)))

= D119894 ((D

119894 ℎ (119906


119894 ℎ (119906

119899)))

= ⋃ℎ (119906119895) | 1 le 119895 le 119899 = ⋃ℎ (119906) | 119906 isin 119861

(3)



119897 119910119894 119909119895 119910119895)



119878(119906 119907)

where 119906 = 119868(120601(119909119894 119909119895 119909119896 119909119897 119909119898)) isin 119863

5 119907 = 119868(120595(119909

119897

119910119894 119909119895 119910119895)) isin 119863





119878(119868(120601) 119868(120595))



5 obtained


119909119895 119909119896 119909119897 119909119898)) isin 119863




119894 119909119895 119909119896 119909119897 119909119898) will be


4


119906 = 119868(120601(119909119894 119909119895 119909119896 119909119897 119909119898)) isin 119863

5is the concept of




119899(119868(120601))


120572will denote




120572

we define that

119868 (⋖ 120601 ⋗120573

120572) = union(119868(120601[

120573

119892 (120573)]) | 119892 isin D

120573) (4)









119892lowast(⋖ ⋗

120573

120572(119883))

= 119868 (⋖ ⋗120573

minus120572) (119892 (119883)) = 119868 (⋖ ⋗

120573

120572) (120601)

=



= union(119868(120601[120573

1198921015840 (120573)]) | 1198921015840 isin D120573) isin 119863

|120572|

otherwise(5)



0


119898) then

119868(⋖ 119901119898119894(1199091 119909

119898) ⋗

1199091119909119898

) = 119868(119901119898119894(1199091 119909

119898)) isin 119863

119898

while 119868(⋖ 119901119898119894(1199091 119909

119898) ⋗

1199091119909119898) = union(119868(119901119898

119894(119892(119909

1)

119892(119909119898))) | 119892 isin D119909

1119909119898) isin 119863

0 with ℎ(119868

(⋖ 119901119898119894(1199091 119909

119898) ⋗

1199091119909119898)) = ℎ(119868((exist119909


119898) 119901119898119894(1199091

119909119898))) isin 119891 119905

For example

ℎ (119868 (⋖ 1199011

119894(1199091) and not119901

1

119894(1199091) ⋗

1199091

))

= ℎ (119868 ((exist1199091) (⋖ 119901

1

119894(1199091) and not119901

1

119894(1199091) ⋗

1199091

))) = 119891(6)




119894(119909)and1199011

119896(119909)) =

con119895(11)

(119868(1199011119894(119909)) 119868(1199011

119896(119909))) 119868(not120601) = neg(119868(120601)) 119868(exist119909

119894)120601

(119909119894 119909119895 119909119894 119909119896) = exist119904

3(119868(120601))




we have that

ℎ (119868 (⋖ 120601 ⋗120573

120572)) = 120587

minus120573(ℎ (119868 (120601))) (7)

where 120587minus(1199101119910119896)= 120587

minus1199101





120572⋃120573 120572 = (1199091 119909


120572)) =


⋃ℎ(119868(120601[120573119892(120573)])) | 119892isin D120573 = ⋃(1198921(1199091) 119892

1(119909119896)) |

1198921isin D120572 and ℎ(119868(120601[120573119892(120573)][120572119892

1(120572)])) = 119905 | 119892 isin D120573 =

1198921(120572) | 119892

1isin D120572⋃120573 and ℎ(119868(120601119892

1)) = 119905 = 120587

minus120573(119892

1(x) |

1198921isin Dx and ℎ(119868(120601119892

1)) = 119905) = 120587

minus120573(119892

1(x) | 119892

1isin Dx and

1198921(x) isin ℎ(119868(120601))) by (T) = 120587

minus120573(ℎ(119868(120601)))



119899lt120596119905 119891D

119899


119894isin 119875 and (119906

1 119906


119881(119908 119901119899119894)(119906

1 119906

119899) = ℎ(119868(119901119899

119894(1199061 119906


relation ⊨119908119892


(1) M⊨119908119892119901119896119894(1199091 119909

119896) if and only if 119881(119908 119901119896

119894)

(119892(1199091) 119892(119909

119896)) = 119905


119908119892120593 andM⊨

119908119892120601


119908119892120593







119908119892120601



120572119892 and




120572119892 ≍ ⋖ 120595 ⋗1205732

120572119892 if and only if ◻(120601

[1205731119892(120573

1)] equiv 120595[120573

2119892(120573

2)]) with M ⊨

1199081198921015840◻120593 if



⋖ 11990111(119909) ⋗

119909≍ ⋖ 1199011

2(119909) ⋗



1(119909)) isin 119863


2(119909)) isin

1198631)


120572119892 asymp ⋖ 120595 ⋗1205732

120572119892 if and only if loz120601[120573

1

119892(1205731)] equiv loz120595[120573

2119892(120573





120572119892 ≍ ⋖ 120595 ⋗1205731

120572119892 then

ℎ(119868(⋖ 119860 ⋗1205731120572119892)) = ℎ(119868(⋖ 120595 ⋗1205732





1119892(120573

1)] where the



119899)


1119892(120573

1)]) isin 119863

119899and 119868(120601[120573

1119892(120573

1)]) = 119868(⋖ 120601 ⋗1205731

120572119892) isin





ℎ(119868(◻120601[1205731

119892 (1205731)]))

= ℎ(necess(119868(120601[1205731

119892 (1205731)])))

= (1198921015840(1199091) 119892

1015840(119909119899)) |

M ⊨1199081198921015840

◻120601[1205731

119892 (1205731)] 119892

1015840isin D

V

= (1198921015840(1199091) 119892

1015840(119909119899)) | 119892

1015840isin D

V forall119908

1


11990811198921015840

120601[1205731

119892 (1205731)])

= ⋂ℎ1isinE

ℎ1(119868(120601[

1205731

119892 (1205731)]))

(8)

while

ℎ(119868(loz120601[1205731

119892 (1205731)]))

= ℎ(119868(not◻not120601[1205731

119892 (1205731)]))

= ℎ(neg(necess(119868(not120601[1205731

119892 (1205731)]))))

= D119899 ℎ(necess(119868(not120601[

1205731

119892 (1205731)])))

= D119899 ( ⋂

ℎ1isinE

ℎ1(119868(not120601[

1205731

119892 (1205731)])))

= D119899 ( ⋂

ℎ1isinE

ℎ1(119899119890119892(119868(120601[

1205731

119892 (1205731)]))))

= D119899 ( ⋂

ℎ1isinE

D119899 ℎ

1(119868(120601[

1205731

119892 (1205731)])))

= ⋃ℎ1isinE

ℎ1(119868(120601[

1205731

119892 (1205731)]))

(9)


1)] and

loz120601[1205731119892(120573





120572119892 ≍ ⋖ 120595 ⋗1205732

120572119892 if and only if ℎ(119868(120601[120573

1119892(120573

1)])) =

ℎ(119868(120595[1205732119892(120573

2)])) for every ℎ


120572119892 asymp ⋖ 120595 ⋗1205732


1119892(120573

1)])) =

ℎ(119868(loz120595[1205732119892(120573

2)]))






119894isin 119875 a relation 119877 = 119868

119879(119901119896119894) sube D119896 to any func-


119879(119891119896119894) D119896 rarr D or



119879(119888) isin D


119879isin


119879)


119894(1199051 119905

119896)


1) 119892lowast(119905

119896)) isin 119868








119879isin I

119879







119879(Γ) Thus ⊩ 120601 if and



1≐ 119909

2rArr (119909

1≐ 119909

3rArr 119909

2≐ 119909

3)



1 119909

119898) and 120595(119909

1 119909




1 119909

119898)) = 119868lowast

119879(120595(119909

1 119909


119879




variables (1199091 119909


119879(120601(119909

1


119879(120601119892) =

119905


1 119909

119898) 119868lowast

119879(120601(119909

1 119909

119898)119892) = 119905 if and only if

(119892(1199091) 119892(119909

119898)) isin 119868lowast

119879(120601(119909

1 119909

119898))



1cong 119877

2


1= 119877

2










119879

A119865119874119871



119894 119909119895 119909119896 119909119897 119909119898 119910119894 119910119895) (its tuple of vari-


119894 119909119895 119909119896 119909119897 119909119898) and 120595(119909

119897 119910119894 119909119895 119910119895) 119868lowast

119879(120601 and 120595) = 119868lowast

119879(120593) =

(119892(119909119894) 119892(119909

119895) 119892(119909

119896) 119892(119909

119897) 119892(119909

119898) 119892(119910

119894) 119892(119910

119895)) | 119868lowast

119879(120593119892) =

119905 = (119892(119909119894) 119892(119909

119895) 119892(119909

119896) 119892(119909

119897) 119892(119909

119898)119892(119910

119894) 119892(119910

119895)) | 119868lowast

119879

(120601119892 and 120595119892) = 119905 = (119892(119909119894) 119892(119909

119895) 119892(119909

119896)119892(119909

119897) 119892(119909

119898) 119892(119910

119894)

119892(119910119895)) | 119868lowast

119879(120601119892) = 119905 and 119868lowast

119879(120601119892) = 119905 = (119892(119909

119894) 119892(119909

119895)

119892(119909119896)119892(119909

119897) 119892(119909

119898) 119892(119910

119894) 119892(119910

119895)) | (119892(119909

119894) 119892(119909

119895) 119892(119909

119896)119892(119909

119897)

119892(119909119898)) isin 119868lowast

119879(120601) and (119892(119909

119897) 119892(119910

119894) 119892(119909

119895) 119892(119910

119895)) isin 119868lowast

119879(120601) =

119868lowast119879(120601)⋈

(41)(23)119868lowast119879(120595)






119879(⊤)) =sim (⟨ ⟩) = D0 ⟨ ⟩ = ⟨ ⟩ ⟨ ⟩ = 0



119879(≐) = 119877

=for












119964FOL (syntax)

(11)





119879) More






L119868

997888rarr D997904rArrℎ=119894119904(119868

119879)amp119868119879isinW=I

119879(Γ)R (12)


4 Conclusion






References







































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in










(2) if 1199051 119905

119896are terms then 119901119896

119894(1199051 119905





1 119909








120572with 119909

119894isin 120572)





1≐ 119905

2 We





1 119889

119896 119891(119889

1

119889119896)) | 119889









119898) This



1 119909


iables (1199091 119909



119896)) isin 119863

119896 and


0with 119868(⊤) = 119879119903119906119905ℎ isin 119863







120572 then 119892lowast(⋖ 120601 ⋗120573

120572) =


ables in 120572) where 119892(120573) = 119892(1199101 119910

119898) = (119892(119910

1)





0 ℎ(119906) isin 119891 119905 sube P(119863




minus1 ℎ(119906) = 119906








ℎ119894) sum

119894geminus1

119863119894997888rarr 119863

minus1+sum119894ge0

P (119863119894) (1)

where ℎminus1= 119894119889 119863

minus1rarr 119863


ping ℎ0 119863



119894rarr P(119863119894) 119894 ge 1





(T) ℎ(119868(120601119892)) = 119905 if and only if (119892(1199091) 119892(119909

119896)) isin




(1198921015840(119909119894) =

119892(119909119894)) implies 1198921015840(119909

119896+1) = 119892(119909

119896+1)







L119868


where 119868





119900119901is


1 119909

119896) isin L

119900119901the

mapping 119868119899(120601(119909

1 119909









119868(11990111(119909)) and 119907 = 119868(1199011

2(119909)) in isin 119863





=) where






⟨ ⟩(119877) = ⟨ ⟩ if






1while the second





2




1isin P(D5) 119877

2isin P(D4)


119894 119909119895 119909119896 119909119897

119909119898 119910119894 119910119895)


119894 119909119895







⟨ ⟩(119877) if 119898 = 119896 = 1


119896)120601(119909

119894 119909119895 119909119896 119909119897




119894 119909119895 119909119897 119909119898)





con119895119878 119863

119868times 119863

119868rarr 119863


119868rarr 119863

119868


119868rarr 119863



119895 119896 119895 ge 0

(1) ℎ(119868119889) = 119877=and ℎ(119879119903119906119905ℎ) = ⟨ ⟩

(2) ℎ(con119895119878(119906 119907)) = ℎ(119906) ⋈

119878ℎ(119907) where ⋈



119898


that 1 le 1198941le 119896 1 le 119894


119878(119906 119907) isin

119863119896+119895

)


119896

(4) ℎ(exist119904119899(119906)) = 120587

minus119899(ℎ(119906)) where 120587



119896minus1if 1 le 119899 le 119896




119894)0) rarr 119863

119894


119899 isin P(119863

119894)


119899)=def1199061 if 119899 = 1 neg(con119895

119878

(neg(1199061) con119895

119878( neg(119906

119899)) ) where 119878 = (119897 119897)|1 le 119897 le 119894


ℎ (union (119861))

= ℎ (neg (con119895119878(neg (119906

1) con119895

119878( neg (119906

119899)) )

= D119894 ((D

119894 ℎ (119906

1)) ⋈


119878(D

119894 ℎ (119906

119899)))

= D119894 ((D

119894 ℎ (119906


119894 ℎ (119906

119899)))

= ⋃ℎ (119906119895) | 1 le 119895 le 119899 = ⋃ℎ (119906) | 119906 isin 119861

(3)



119897 119910119894 119909119895 119910119895)



119878(119906 119907)

where 119906 = 119868(120601(119909119894 119909119895 119909119896 119909119897 119909119898)) isin 119863

5 119907 = 119868(120595(119909

119897

119910119894 119909119895 119910119895)) isin 119863





119878(119868(120601) 119868(120595))



5 obtained


119909119895 119909119896 119909119897 119909119898)) isin 119863




119894 119909119895 119909119896 119909119897 119909119898) will be


4


119906 = 119868(120601(119909119894 119909119895 119909119896 119909119897 119909119898)) isin 119863

5is the concept of




119899(119868(120601))


120572will denote




120572

we define that

119868 (⋖ 120601 ⋗120573

120572) = union(119868(120601[

120573

119892 (120573)]) | 119892 isin D

120573) (4)









119892lowast(⋖ ⋗

120573

120572(119883))

= 119868 (⋖ ⋗120573

minus120572) (119892 (119883)) = 119868 (⋖ ⋗

120573

120572) (120601)

=



= union(119868(120601[120573

1198921015840 (120573)]) | 1198921015840 isin D120573) isin 119863

|120572|

otherwise(5)



0


119898) then

119868(⋖ 119901119898119894(1199091 119909

119898) ⋗

1199091119909119898

) = 119868(119901119898119894(1199091 119909

119898)) isin 119863

119898

while 119868(⋖ 119901119898119894(1199091 119909

119898) ⋗

1199091119909119898) = union(119868(119901119898

119894(119892(119909

1)

119892(119909119898))) | 119892 isin D119909

1119909119898) isin 119863

0 with ℎ(119868

(⋖ 119901119898119894(1199091 119909

119898) ⋗

1199091119909119898)) = ℎ(119868((exist119909


119898) 119901119898119894(1199091

119909119898))) isin 119891 119905

For example

ℎ (119868 (⋖ 1199011

119894(1199091) and not119901

1

119894(1199091) ⋗

1199091

))

= ℎ (119868 ((exist1199091) (⋖ 119901

1

119894(1199091) and not119901

1

119894(1199091) ⋗

1199091

))) = 119891(6)




119894(119909)and1199011

119896(119909)) =

con119895(11)

(119868(1199011119894(119909)) 119868(1199011

119896(119909))) 119868(not120601) = neg(119868(120601)) 119868(exist119909

119894)120601

(119909119894 119909119895 119909119894 119909119896) = exist119904

3(119868(120601))




we have that

ℎ (119868 (⋖ 120601 ⋗120573

120572)) = 120587

minus120573(ℎ (119868 (120601))) (7)

where 120587minus(1199101119910119896)= 120587

minus1199101





120572⋃120573 120572 = (1199091 119909


120572)) =


⋃ℎ(119868(120601[120573119892(120573)])) | 119892isin D120573 = ⋃(1198921(1199091) 119892

1(119909119896)) |

1198921isin D120572 and ℎ(119868(120601[120573119892(120573)][120572119892

1(120572)])) = 119905 | 119892 isin D120573 =

1198921(120572) | 119892

1isin D120572⋃120573 and ℎ(119868(120601119892

1)) = 119905 = 120587

minus120573(119892

1(x) |

1198921isin Dx and ℎ(119868(120601119892

1)) = 119905) = 120587

minus120573(119892

1(x) | 119892

1isin Dx and

1198921(x) isin ℎ(119868(120601))) by (T) = 120587

minus120573(ℎ(119868(120601)))



119899lt120596119905 119891D

119899


119894isin 119875 and (119906

1 119906


119881(119908 119901119899119894)(119906

1 119906

119899) = ℎ(119868(119901119899

119894(1199061 119906


relation ⊨119908119892


(1) M⊨119908119892119901119896119894(1199091 119909

119896) if and only if 119881(119908 119901119896

119894)

(119892(1199091) 119892(119909

119896)) = 119905


119908119892120593 andM⊨

119908119892120601


119908119892120593







119908119892120601



120572119892 and




120572119892 ≍ ⋖ 120595 ⋗1205732

120572119892 if and only if ◻(120601

[1205731119892(120573

1)] equiv 120595[120573

2119892(120573

2)]) with M ⊨

1199081198921015840◻120593 if



⋖ 11990111(119909) ⋗

119909≍ ⋖ 1199011

2(119909) ⋗



1(119909)) isin 119863


2(119909)) isin

1198631)


120572119892 asymp ⋖ 120595 ⋗1205732

120572119892 if and only if loz120601[120573

1

119892(1205731)] equiv loz120595[120573

2119892(120573





120572119892 ≍ ⋖ 120595 ⋗1205731

120572119892 then

ℎ(119868(⋖ 119860 ⋗1205731120572119892)) = ℎ(119868(⋖ 120595 ⋗1205732





1119892(120573

1)] where the



119899)


1119892(120573

1)]) isin 119863

119899and 119868(120601[120573

1119892(120573

1)]) = 119868(⋖ 120601 ⋗1205731

120572119892) isin





ℎ(119868(◻120601[1205731

119892 (1205731)]))

= ℎ(necess(119868(120601[1205731

119892 (1205731)])))

= (1198921015840(1199091) 119892

1015840(119909119899)) |

M ⊨1199081198921015840

◻120601[1205731

119892 (1205731)] 119892

1015840isin D

V

= (1198921015840(1199091) 119892

1015840(119909119899)) | 119892

1015840isin D

V forall119908

1


11990811198921015840

120601[1205731

119892 (1205731)])

= ⋂ℎ1isinE

ℎ1(119868(120601[

1205731

119892 (1205731)]))

(8)

while

ℎ(119868(loz120601[1205731

119892 (1205731)]))

= ℎ(119868(not◻not120601[1205731

119892 (1205731)]))

= ℎ(neg(necess(119868(not120601[1205731

119892 (1205731)]))))

= D119899 ℎ(necess(119868(not120601[

1205731

119892 (1205731)])))

= D119899 ( ⋂

ℎ1isinE

ℎ1(119868(not120601[

1205731

119892 (1205731)])))

= D119899 ( ⋂

ℎ1isinE

ℎ1(119899119890119892(119868(120601[

1205731

119892 (1205731)]))))

= D119899 ( ⋂

ℎ1isinE

D119899 ℎ

1(119868(120601[

1205731

119892 (1205731)])))

= ⋃ℎ1isinE

ℎ1(119868(120601[

1205731

119892 (1205731)]))

(9)


1)] and

loz120601[1205731119892(120573





120572119892 ≍ ⋖ 120595 ⋗1205732

120572119892 if and only if ℎ(119868(120601[120573

1119892(120573

1)])) =

ℎ(119868(120595[1205732119892(120573

2)])) for every ℎ


120572119892 asymp ⋖ 120595 ⋗1205732


1119892(120573

1)])) =

ℎ(119868(loz120595[1205732119892(120573

2)]))






119894isin 119875 a relation 119877 = 119868

119879(119901119896119894) sube D119896 to any func-


119879(119891119896119894) D119896 rarr D or



119879(119888) isin D


119879isin


119879)


119894(1199051 119905

119896)


1) 119892lowast(119905

119896)) isin 119868








119879isin I

119879







119879(Γ) Thus ⊩ 120601 if and



1≐ 119909

2rArr (119909

1≐ 119909

3rArr 119909

2≐ 119909

3)



1 119909

119898) and 120595(119909

1 119909




1 119909

119898)) = 119868lowast

119879(120595(119909

1 119909


119879




variables (1199091 119909


119879(120601(119909

1


119879(120601119892) =

119905


1 119909

119898) 119868lowast

119879(120601(119909

1 119909

119898)119892) = 119905 if and only if

(119892(1199091) 119892(119909

119898)) isin 119868lowast

119879(120601(119909

1 119909

119898))



1cong 119877

2


1= 119877

2










119879

A119865119874119871



119894 119909119895 119909119896 119909119897 119909119898 119910119894 119910119895) (its tuple of vari-


119894 119909119895 119909119896 119909119897 119909119898) and 120595(119909

119897 119910119894 119909119895 119910119895) 119868lowast

119879(120601 and 120595) = 119868lowast

119879(120593) =

(119892(119909119894) 119892(119909

119895) 119892(119909

119896) 119892(119909

119897) 119892(119909

119898) 119892(119910

119894) 119892(119910

119895)) | 119868lowast

119879(120593119892) =

119905 = (119892(119909119894) 119892(119909

119895) 119892(119909

119896) 119892(119909

119897) 119892(119909

119898)119892(119910

119894) 119892(119910

119895)) | 119868lowast

119879

(120601119892 and 120595119892) = 119905 = (119892(119909119894) 119892(119909

119895) 119892(119909

119896)119892(119909

119897) 119892(119909

119898) 119892(119910

119894)

119892(119910119895)) | 119868lowast

119879(120601119892) = 119905 and 119868lowast

119879(120601119892) = 119905 = (119892(119909

119894) 119892(119909

119895)

119892(119909119896)119892(119909

119897) 119892(119909

119898) 119892(119910

119894) 119892(119910

119895)) | (119892(119909

119894) 119892(119909

119895) 119892(119909

119896)119892(119909

119897)

119892(119909119898)) isin 119868lowast

119879(120601) and (119892(119909

119897) 119892(119910

119894) 119892(119909

119895) 119892(119910

119895)) isin 119868lowast

119879(120601) =

119868lowast119879(120601)⋈

(41)(23)119868lowast119879(120595)






119879(⊤)) =sim (⟨ ⟩) = D0 ⟨ ⟩ = ⟨ ⟩ ⟨ ⟩ = 0



119879(≐) = 119877

=for












119964FOL (syntax)

(11)





119879) More






L119868

997888rarr D997904rArrℎ=119894119904(119868

119879)amp119868119879isinW=I

119879(Γ)R (12)


4 Conclusion






References







































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in








ℎ119894) sum

119894geminus1

119863119894997888rarr 119863

minus1+sum119894ge0

P (119863119894) (1)

where ℎminus1= 119894119889 119863

minus1rarr 119863


ping ℎ0 119863



119894rarr P(119863119894) 119894 ge 1





(T) ℎ(119868(120601119892)) = 119905 if and only if (119892(1199091) 119892(119909

119896)) isin




(1198921015840(119909119894) =

119892(119909119894)) implies 1198921015840(119909

119896+1) = 119892(119909

119896+1)







L119868


where 119868





119900119901is


1 119909

119896) isin L

119900119901the

mapping 119868119899(120601(119909

1 119909









119868(11990111(119909)) and 119907 = 119868(1199011

2(119909)) in isin 119863





=) where






⟨ ⟩(119877) = ⟨ ⟩ if






1while the second





2




1isin P(D5) 119877

2isin P(D4)


119894 119909119895 119909119896 119909119897

119909119898 119910119894 119910119895)


119894 119909119895







⟨ ⟩(119877) if 119898 = 119896 = 1


119896)120601(119909

119894 119909119895 119909119896 119909119897




119894 119909119895 119909119897 119909119898)





con119895119878 119863

119868times 119863

119868rarr 119863


119868rarr 119863

119868


119868rarr 119863



119895 119896 119895 ge 0

(1) ℎ(119868119889) = 119877=and ℎ(119879119903119906119905ℎ) = ⟨ ⟩

(2) ℎ(con119895119878(119906 119907)) = ℎ(119906) ⋈

119878ℎ(119907) where ⋈



119898


that 1 le 1198941le 119896 1 le 119894


119878(119906 119907) isin

119863119896+119895

)


119896

(4) ℎ(exist119904119899(119906)) = 120587

minus119899(ℎ(119906)) where 120587



119896minus1if 1 le 119899 le 119896




119894)0) rarr 119863

119894


119899 isin P(119863

119894)


119899)=def1199061 if 119899 = 1 neg(con119895

119878

(neg(1199061) con119895

119878( neg(119906

119899)) ) where 119878 = (119897 119897)|1 le 119897 le 119894


ℎ (union (119861))

= ℎ (neg (con119895119878(neg (119906

1) con119895

119878( neg (119906

119899)) )

= D119894 ((D

119894 ℎ (119906

1)) ⋈


119878(D

119894 ℎ (119906

119899)))

= D119894 ((D

119894 ℎ (119906


119894 ℎ (119906

119899)))

= ⋃ℎ (119906119895) | 1 le 119895 le 119899 = ⋃ℎ (119906) | 119906 isin 119861

(3)



119897 119910119894 119909119895 119910119895)



119878(119906 119907)

where 119906 = 119868(120601(119909119894 119909119895 119909119896 119909119897 119909119898)) isin 119863

5 119907 = 119868(120595(119909

119897

119910119894 119909119895 119910119895)) isin 119863





119878(119868(120601) 119868(120595))



5 obtained


119909119895 119909119896 119909119897 119909119898)) isin 119863




119894 119909119895 119909119896 119909119897 119909119898) will be


4


119906 = 119868(120601(119909119894 119909119895 119909119896 119909119897 119909119898)) isin 119863

5is the concept of




119899(119868(120601))


120572will denote




120572

we define that

119868 (⋖ 120601 ⋗120573

120572) = union(119868(120601[

120573

119892 (120573)]) | 119892 isin D

120573) (4)









119892lowast(⋖ ⋗

120573

120572(119883))

= 119868 (⋖ ⋗120573

minus120572) (119892 (119883)) = 119868 (⋖ ⋗

120573

120572) (120601)

=



= union(119868(120601[120573

1198921015840 (120573)]) | 1198921015840 isin D120573) isin 119863

|120572|

otherwise(5)



0


119898) then

119868(⋖ 119901119898119894(1199091 119909

119898) ⋗

1199091119909119898

) = 119868(119901119898119894(1199091 119909

119898)) isin 119863

119898

while 119868(⋖ 119901119898119894(1199091 119909

119898) ⋗

1199091119909119898) = union(119868(119901119898

119894(119892(119909

1)

119892(119909119898))) | 119892 isin D119909

1119909119898) isin 119863

0 with ℎ(119868

(⋖ 119901119898119894(1199091 119909

119898) ⋗

1199091119909119898)) = ℎ(119868((exist119909


119898) 119901119898119894(1199091

119909119898))) isin 119891 119905

For example

ℎ (119868 (⋖ 1199011

119894(1199091) and not119901

1

119894(1199091) ⋗

1199091

))

= ℎ (119868 ((exist1199091) (⋖ 119901

1

119894(1199091) and not119901

1

119894(1199091) ⋗

1199091

))) = 119891(6)




119894(119909)and1199011

119896(119909)) =

con119895(11)

(119868(1199011119894(119909)) 119868(1199011

119896(119909))) 119868(not120601) = neg(119868(120601)) 119868(exist119909

119894)120601

(119909119894 119909119895 119909119894 119909119896) = exist119904

3(119868(120601))




we have that

ℎ (119868 (⋖ 120601 ⋗120573

120572)) = 120587

minus120573(ℎ (119868 (120601))) (7)

where 120587minus(1199101119910119896)= 120587

minus1199101





120572⋃120573 120572 = (1199091 119909


120572)) =


⋃ℎ(119868(120601[120573119892(120573)])) | 119892isin D120573 = ⋃(1198921(1199091) 119892

1(119909119896)) |

1198921isin D120572 and ℎ(119868(120601[120573119892(120573)][120572119892

1(120572)])) = 119905 | 119892 isin D120573 =

1198921(120572) | 119892

1isin D120572⋃120573 and ℎ(119868(120601119892

1)) = 119905 = 120587

minus120573(119892

1(x) |

1198921isin Dx and ℎ(119868(120601119892

1)) = 119905) = 120587

minus120573(119892

1(x) | 119892

1isin Dx and

1198921(x) isin ℎ(119868(120601))) by (T) = 120587

minus120573(ℎ(119868(120601)))



119899lt120596119905 119891D

119899


119894isin 119875 and (119906

1 119906


119881(119908 119901119899119894)(119906

1 119906

119899) = ℎ(119868(119901119899

119894(1199061 119906


relation ⊨119908119892


(1) M⊨119908119892119901119896119894(1199091 119909

119896) if and only if 119881(119908 119901119896

119894)

(119892(1199091) 119892(119909

119896)) = 119905


119908119892120593 andM⊨

119908119892120601


119908119892120593







119908119892120601



120572119892 and




120572119892 ≍ ⋖ 120595 ⋗1205732

120572119892 if and only if ◻(120601

[1205731119892(120573

1)] equiv 120595[120573

2119892(120573

2)]) with M ⊨

1199081198921015840◻120593 if



⋖ 11990111(119909) ⋗

119909≍ ⋖ 1199011

2(119909) ⋗



1(119909)) isin 119863


2(119909)) isin

1198631)


120572119892 asymp ⋖ 120595 ⋗1205732

120572119892 if and only if loz120601[120573

1

119892(1205731)] equiv loz120595[120573

2119892(120573





120572119892 ≍ ⋖ 120595 ⋗1205731

120572119892 then

ℎ(119868(⋖ 119860 ⋗1205731120572119892)) = ℎ(119868(⋖ 120595 ⋗1205732





1119892(120573

1)] where the



119899)


1119892(120573

1)]) isin 119863

119899and 119868(120601[120573

1119892(120573

1)]) = 119868(⋖ 120601 ⋗1205731

120572119892) isin





ℎ(119868(◻120601[1205731

119892 (1205731)]))

= ℎ(necess(119868(120601[1205731

119892 (1205731)])))

= (1198921015840(1199091) 119892

1015840(119909119899)) |

M ⊨1199081198921015840

◻120601[1205731

119892 (1205731)] 119892

1015840isin D

V

= (1198921015840(1199091) 119892

1015840(119909119899)) | 119892

1015840isin D

V forall119908

1


11990811198921015840

120601[1205731

119892 (1205731)])

= ⋂ℎ1isinE

ℎ1(119868(120601[

1205731

119892 (1205731)]))

(8)

while

ℎ(119868(loz120601[1205731

119892 (1205731)]))

= ℎ(119868(not◻not120601[1205731

119892 (1205731)]))

= ℎ(neg(necess(119868(not120601[1205731

119892 (1205731)]))))

= D119899 ℎ(necess(119868(not120601[

1205731

119892 (1205731)])))

= D119899 ( ⋂

ℎ1isinE

ℎ1(119868(not120601[

1205731

119892 (1205731)])))

= D119899 ( ⋂

ℎ1isinE

ℎ1(119899119890119892(119868(120601[

1205731

119892 (1205731)]))))

= D119899 ( ⋂

ℎ1isinE

D119899 ℎ

1(119868(120601[

1205731

119892 (1205731)])))

= ⋃ℎ1isinE

ℎ1(119868(120601[

1205731

119892 (1205731)]))

(9)


1)] and

loz120601[1205731119892(120573





120572119892 ≍ ⋖ 120595 ⋗1205732

120572119892 if and only if ℎ(119868(120601[120573

1119892(120573

1)])) =

ℎ(119868(120595[1205732119892(120573

2)])) for every ℎ


120572119892 asymp ⋖ 120595 ⋗1205732


1119892(120573

1)])) =

ℎ(119868(loz120595[1205732119892(120573

2)]))






119894isin 119875 a relation 119877 = 119868

119879(119901119896119894) sube D119896 to any func-


119879(119891119896119894) D119896 rarr D or



119879(119888) isin D


119879isin


119879)


119894(1199051 119905

119896)


1) 119892lowast(119905

119896)) isin 119868








119879isin I

119879







119879(Γ) Thus ⊩ 120601 if and



1≐ 119909

2rArr (119909

1≐ 119909

3rArr 119909

2≐ 119909

3)



1 119909

119898) and 120595(119909

1 119909




1 119909

119898)) = 119868lowast

119879(120595(119909

1 119909


119879




variables (1199091 119909


119879(120601(119909

1


119879(120601119892) =

119905


1 119909

119898) 119868lowast

119879(120601(119909

1 119909

119898)119892) = 119905 if and only if

(119892(1199091) 119892(119909

119898)) isin 119868lowast

119879(120601(119909

1 119909

119898))



1cong 119877

2


1= 119877

2










119879

A119865119874119871



119894 119909119895 119909119896 119909119897 119909119898 119910119894 119910119895) (its tuple of vari-


119894 119909119895 119909119896 119909119897 119909119898) and 120595(119909

119897 119910119894 119909119895 119910119895) 119868lowast

119879(120601 and 120595) = 119868lowast

119879(120593) =

(119892(119909119894) 119892(119909

119895) 119892(119909

119896) 119892(119909

119897) 119892(119909

119898) 119892(119910

119894) 119892(119910

119895)) | 119868lowast

119879(120593119892) =

119905 = (119892(119909119894) 119892(119909

119895) 119892(119909

119896) 119892(119909

119897) 119892(119909

119898)119892(119910

119894) 119892(119910

119895)) | 119868lowast

119879

(120601119892 and 120595119892) = 119905 = (119892(119909119894) 119892(119909

119895) 119892(119909

119896)119892(119909

119897) 119892(119909

119898) 119892(119910

119894)

119892(119910119895)) | 119868lowast

119879(120601119892) = 119905 and 119868lowast

119879(120601119892) = 119905 = (119892(119909

119894) 119892(119909

119895)

119892(119909119896)119892(119909

119897) 119892(119909

119898) 119892(119910

119894) 119892(119910

119895)) | (119892(119909

119894) 119892(119909

119895) 119892(119909

119896)119892(119909

119897)

119892(119909119898)) isin 119868lowast

119879(120601) and (119892(119909

119897) 119892(119910

119894) 119892(119909

119895) 119892(119910

119895)) isin 119868lowast

119879(120601) =

119868lowast119879(120601)⋈

(41)(23)119868lowast119879(120595)






119879(⊤)) =sim (⟨ ⟩) = D0 ⟨ ⟩ = ⟨ ⟩ ⟨ ⟩ = 0



119879(≐) = 119877

=for












119964FOL (syntax)

(11)





119879) More






L119868

997888rarr D997904rArrℎ=119894119904(119868

119879)amp119868119879isinW=I

119879(Γ)R (12)


4 Conclusion






References







































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in






2




1isin P(D5) 119877

2isin P(D4)


119894 119909119895 119909119896 119909119897

119909119898 119910119894 119910119895)


119894 119909119895







⟨ ⟩(119877) if 119898 = 119896 = 1


119896)120601(119909

119894 119909119895 119909119896 119909119897




119894 119909119895 119909119897 119909119898)





con119895119878 119863

119868times 119863

119868rarr 119863


119868rarr 119863

119868


119868rarr 119863



119895 119896 119895 ge 0

(1) ℎ(119868119889) = 119877=and ℎ(119879119903119906119905ℎ) = ⟨ ⟩

(2) ℎ(con119895119878(119906 119907)) = ℎ(119906) ⋈

119878ℎ(119907) where ⋈



119898


that 1 le 1198941le 119896 1 le 119894


119878(119906 119907) isin

119863119896+119895

)


119896

(4) ℎ(exist119904119899(119906)) = 120587

minus119899(ℎ(119906)) where 120587



119896minus1if 1 le 119899 le 119896




119894)0) rarr 119863

119894


119899 isin P(119863

119894)


119899)=def1199061 if 119899 = 1 neg(con119895

119878

(neg(1199061) con119895

119878( neg(119906

119899)) ) where 119878 = (119897 119897)|1 le 119897 le 119894


ℎ (union (119861))

= ℎ (neg (con119895119878(neg (119906

1) con119895

119878( neg (119906

119899)) )

= D119894 ((D

119894 ℎ (119906

1)) ⋈


119878(D

119894 ℎ (119906

119899)))

= D119894 ((D

119894 ℎ (119906


119894 ℎ (119906

119899)))

= ⋃ℎ (119906119895) | 1 le 119895 le 119899 = ⋃ℎ (119906) | 119906 isin 119861

(3)



119897 119910119894 119909119895 119910119895)



119878(119906 119907)

where 119906 = 119868(120601(119909119894 119909119895 119909119896 119909119897 119909119898)) isin 119863

5 119907 = 119868(120595(119909

119897

119910119894 119909119895 119910119895)) isin 119863





119878(119868(120601) 119868(120595))



5 obtained


119909119895 119909119896 119909119897 119909119898)) isin 119863




119894 119909119895 119909119896 119909119897 119909119898) will be


4


119906 = 119868(120601(119909119894 119909119895 119909119896 119909119897 119909119898)) isin 119863

5is the concept of




119899(119868(120601))


120572will denote




120572

we define that

119868 (⋖ 120601 ⋗120573

120572) = union(119868(120601[

120573

119892 (120573)]) | 119892 isin D

120573) (4)









119892lowast(⋖ ⋗

120573

120572(119883))

= 119868 (⋖ ⋗120573

minus120572) (119892 (119883)) = 119868 (⋖ ⋗

120573

120572) (120601)

=



= union(119868(120601[120573

1198921015840 (120573)]) | 1198921015840 isin D120573) isin 119863

|120572|

otherwise(5)



0


119898) then

119868(⋖ 119901119898119894(1199091 119909

119898) ⋗

1199091119909119898

) = 119868(119901119898119894(1199091 119909

119898)) isin 119863

119898

while 119868(⋖ 119901119898119894(1199091 119909

119898) ⋗

1199091119909119898) = union(119868(119901119898

119894(119892(119909

1)

119892(119909119898))) | 119892 isin D119909

1119909119898) isin 119863

0 with ℎ(119868

(⋖ 119901119898119894(1199091 119909

119898) ⋗

1199091119909119898)) = ℎ(119868((exist119909


119898) 119901119898119894(1199091

119909119898))) isin 119891 119905

For example

ℎ (119868 (⋖ 1199011

119894(1199091) and not119901

1

119894(1199091) ⋗

1199091

))

= ℎ (119868 ((exist1199091) (⋖ 119901

1

119894(1199091) and not119901

1

119894(1199091) ⋗

1199091

))) = 119891(6)




119894(119909)and1199011

119896(119909)) =

con119895(11)

(119868(1199011119894(119909)) 119868(1199011

119896(119909))) 119868(not120601) = neg(119868(120601)) 119868(exist119909

119894)120601

(119909119894 119909119895 119909119894 119909119896) = exist119904

3(119868(120601))




we have that

ℎ (119868 (⋖ 120601 ⋗120573

120572)) = 120587

minus120573(ℎ (119868 (120601))) (7)

where 120587minus(1199101119910119896)= 120587

minus1199101





120572⋃120573 120572 = (1199091 119909


120572)) =


⋃ℎ(119868(120601[120573119892(120573)])) | 119892isin D120573 = ⋃(1198921(1199091) 119892

1(119909119896)) |

1198921isin D120572 and ℎ(119868(120601[120573119892(120573)][120572119892

1(120572)])) = 119905 | 119892 isin D120573 =

1198921(120572) | 119892

1isin D120572⋃120573 and ℎ(119868(120601119892

1)) = 119905 = 120587

minus120573(119892

1(x) |

1198921isin Dx and ℎ(119868(120601119892

1)) = 119905) = 120587

minus120573(119892

1(x) | 119892

1isin Dx and

1198921(x) isin ℎ(119868(120601))) by (T) = 120587

minus120573(ℎ(119868(120601)))



119899lt120596119905 119891D

119899


119894isin 119875 and (119906

1 119906


119881(119908 119901119899119894)(119906

1 119906

119899) = ℎ(119868(119901119899

119894(1199061 119906


relation ⊨119908119892


(1) M⊨119908119892119901119896119894(1199091 119909

119896) if and only if 119881(119908 119901119896

119894)

(119892(1199091) 119892(119909

119896)) = 119905


119908119892120593 andM⊨

119908119892120601


119908119892120593







119908119892120601



120572119892 and




120572119892 ≍ ⋖ 120595 ⋗1205732

120572119892 if and only if ◻(120601

[1205731119892(120573

1)] equiv 120595[120573

2119892(120573

2)]) with M ⊨

1199081198921015840◻120593 if



⋖ 11990111(119909) ⋗

119909≍ ⋖ 1199011

2(119909) ⋗



1(119909)) isin 119863


2(119909)) isin

1198631)


120572119892 asymp ⋖ 120595 ⋗1205732

120572119892 if and only if loz120601[120573

1

119892(1205731)] equiv loz120595[120573

2119892(120573





120572119892 ≍ ⋖ 120595 ⋗1205731

120572119892 then

ℎ(119868(⋖ 119860 ⋗1205731120572119892)) = ℎ(119868(⋖ 120595 ⋗1205732





1119892(120573

1)] where the



119899)


1119892(120573

1)]) isin 119863

119899and 119868(120601[120573

1119892(120573

1)]) = 119868(⋖ 120601 ⋗1205731

120572119892) isin





ℎ(119868(◻120601[1205731

119892 (1205731)]))

= ℎ(necess(119868(120601[1205731

119892 (1205731)])))

= (1198921015840(1199091) 119892

1015840(119909119899)) |

M ⊨1199081198921015840

◻120601[1205731

119892 (1205731)] 119892

1015840isin D

V

= (1198921015840(1199091) 119892

1015840(119909119899)) | 119892

1015840isin D

V forall119908

1


11990811198921015840

120601[1205731

119892 (1205731)])

= ⋂ℎ1isinE

ℎ1(119868(120601[

1205731

119892 (1205731)]))

(8)

while

ℎ(119868(loz120601[1205731

119892 (1205731)]))

= ℎ(119868(not◻not120601[1205731

119892 (1205731)]))

= ℎ(neg(necess(119868(not120601[1205731

119892 (1205731)]))))

= D119899 ℎ(necess(119868(not120601[

1205731

119892 (1205731)])))

= D119899 ( ⋂

ℎ1isinE

ℎ1(119868(not120601[

1205731

119892 (1205731)])))

= D119899 ( ⋂

ℎ1isinE

ℎ1(119899119890119892(119868(120601[

1205731

119892 (1205731)]))))

= D119899 ( ⋂

ℎ1isinE

D119899 ℎ

1(119868(120601[

1205731

119892 (1205731)])))

= ⋃ℎ1isinE

ℎ1(119868(120601[

1205731

119892 (1205731)]))

(9)


1)] and

loz120601[1205731119892(120573





120572119892 ≍ ⋖ 120595 ⋗1205732

120572119892 if and only if ℎ(119868(120601[120573

1119892(120573

1)])) =

ℎ(119868(120595[1205732119892(120573

2)])) for every ℎ


120572119892 asymp ⋖ 120595 ⋗1205732


1119892(120573

1)])) =

ℎ(119868(loz120595[1205732119892(120573

2)]))






119894isin 119875 a relation 119877 = 119868

119879(119901119896119894) sube D119896 to any func-


119879(119891119896119894) D119896 rarr D or



119879(119888) isin D


119879isin


119879)


119894(1199051 119905

119896)


1) 119892lowast(119905

119896)) isin 119868








119879isin I

119879







119879(Γ) Thus ⊩ 120601 if and



1≐ 119909

2rArr (119909

1≐ 119909

3rArr 119909

2≐ 119909

3)



1 119909

119898) and 120595(119909

1 119909




1 119909

119898)) = 119868lowast

119879(120595(119909

1 119909


119879




variables (1199091 119909


119879(120601(119909

1


119879(120601119892) =

119905


1 119909

119898) 119868lowast

119879(120601(119909

1 119909

119898)119892) = 119905 if and only if

(119892(1199091) 119892(119909

119898)) isin 119868lowast

119879(120601(119909

1 119909

119898))



1cong 119877

2


1= 119877

2










119879

A119865119874119871



119894 119909119895 119909119896 119909119897 119909119898 119910119894 119910119895) (its tuple of vari-


119894 119909119895 119909119896 119909119897 119909119898) and 120595(119909

119897 119910119894 119909119895 119910119895) 119868lowast

119879(120601 and 120595) = 119868lowast

119879(120593) =

(119892(119909119894) 119892(119909

119895) 119892(119909

119896) 119892(119909

119897) 119892(119909

119898) 119892(119910

119894) 119892(119910

119895)) | 119868lowast

119879(120593119892) =

119905 = (119892(119909119894) 119892(119909

119895) 119892(119909

119896) 119892(119909

119897) 119892(119909

119898)119892(119910

119894) 119892(119910

119895)) | 119868lowast

119879

(120601119892 and 120595119892) = 119905 = (119892(119909119894) 119892(119909

119895) 119892(119909

119896)119892(119909

119897) 119892(119909

119898) 119892(119910

119894)

119892(119910119895)) | 119868lowast

119879(120601119892) = 119905 and 119868lowast

119879(120601119892) = 119905 = (119892(119909

119894) 119892(119909

119895)

119892(119909119896)119892(119909

119897) 119892(119909

119898) 119892(119910

119894) 119892(119910

119895)) | (119892(119909

119894) 119892(119909

119895) 119892(119909

119896)119892(119909

119897)

119892(119909119898)) isin 119868lowast

119879(120601) and (119892(119909

119897) 119892(119910

119894) 119892(119909

119895) 119892(119910

119895)) isin 119868lowast

119879(120601) =

119868lowast119879(120601)⋈

(41)(23)119868lowast119879(120595)






119879(⊤)) =sim (⟨ ⟩) = D0 ⟨ ⟩ = ⟨ ⟩ ⟨ ⟩ = 0



119879(≐) = 119877

=for












119964FOL (syntax)

(11)





119879) More






L119868

997888rarr D997904rArrℎ=119894119904(119868

119879)amp119868119879isinW=I

119879(Γ)R (12)


4 Conclusion






References







































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in





120572

we define that

119868 (⋖ 120601 ⋗120573

120572) = union(119868(120601[

120573

119892 (120573)]) | 119892 isin D

120573) (4)









119892lowast(⋖ ⋗

120573

120572(119883))

= 119868 (⋖ ⋗120573

minus120572) (119892 (119883)) = 119868 (⋖ ⋗

120573

120572) (120601)

=



= union(119868(120601[120573

1198921015840 (120573)]) | 1198921015840 isin D120573) isin 119863

|120572|

otherwise(5)



0


119898) then

119868(⋖ 119901119898119894(1199091 119909

119898) ⋗

1199091119909119898

) = 119868(119901119898119894(1199091 119909

119898)) isin 119863

119898

while 119868(⋖ 119901119898119894(1199091 119909

119898) ⋗

1199091119909119898) = union(119868(119901119898

119894(119892(119909

1)

119892(119909119898))) | 119892 isin D119909

1119909119898) isin 119863

0 with ℎ(119868

(⋖ 119901119898119894(1199091 119909

119898) ⋗

1199091119909119898)) = ℎ(119868((exist119909


119898) 119901119898119894(1199091

119909119898))) isin 119891 119905

For example

ℎ (119868 (⋖ 1199011

119894(1199091) and not119901

1

119894(1199091) ⋗

1199091

))

= ℎ (119868 ((exist1199091) (⋖ 119901

1

119894(1199091) and not119901

1

119894(1199091) ⋗

1199091

))) = 119891(6)




119894(119909)and1199011

119896(119909)) =

con119895(11)

(119868(1199011119894(119909)) 119868(1199011

119896(119909))) 119868(not120601) = neg(119868(120601)) 119868(exist119909

119894)120601

(119909119894 119909119895 119909119894 119909119896) = exist119904

3(119868(120601))




we have that

ℎ (119868 (⋖ 120601 ⋗120573

120572)) = 120587

minus120573(ℎ (119868 (120601))) (7)

where 120587minus(1199101119910119896)= 120587

minus1199101





120572⋃120573 120572 = (1199091 119909


120572)) =


⋃ℎ(119868(120601[120573119892(120573)])) | 119892isin D120573 = ⋃(1198921(1199091) 119892

1(119909119896)) |

1198921isin D120572 and ℎ(119868(120601[120573119892(120573)][120572119892

1(120572)])) = 119905 | 119892 isin D120573 =

1198921(120572) | 119892

1isin D120572⋃120573 and ℎ(119868(120601119892

1)) = 119905 = 120587

minus120573(119892

1(x) |

1198921isin Dx and ℎ(119868(120601119892

1)) = 119905) = 120587

minus120573(119892

1(x) | 119892

1isin Dx and

1198921(x) isin ℎ(119868(120601))) by (T) = 120587

minus120573(ℎ(119868(120601)))



119899lt120596119905 119891D

119899


119894isin 119875 and (119906

1 119906


119881(119908 119901119899119894)(119906

1 119906

119899) = ℎ(119868(119901119899

119894(1199061 119906


relation ⊨119908119892


(1) M⊨119908119892119901119896119894(1199091 119909

119896) if and only if 119881(119908 119901119896

119894)

(119892(1199091) 119892(119909

119896)) = 119905


119908119892120593 andM⊨

119908119892120601


119908119892120593







119908119892120601



120572119892 and




120572119892 ≍ ⋖ 120595 ⋗1205732

120572119892 if and only if ◻(120601

[1205731119892(120573

1)] equiv 120595[120573

2119892(120573

2)]) with M ⊨

1199081198921015840◻120593 if



⋖ 11990111(119909) ⋗

119909≍ ⋖ 1199011

2(119909) ⋗



1(119909)) isin 119863


2(119909)) isin

1198631)


120572119892 asymp ⋖ 120595 ⋗1205732

120572119892 if and only if loz120601[120573

1

119892(1205731)] equiv loz120595[120573

2119892(120573





120572119892 ≍ ⋖ 120595 ⋗1205731

120572119892 then

ℎ(119868(⋖ 119860 ⋗1205731120572119892)) = ℎ(119868(⋖ 120595 ⋗1205732





1119892(120573

1)] where the



119899)


1119892(120573

1)]) isin 119863

119899and 119868(120601[120573

1119892(120573

1)]) = 119868(⋖ 120601 ⋗1205731

120572119892) isin





ℎ(119868(◻120601[1205731

119892 (1205731)]))

= ℎ(necess(119868(120601[1205731

119892 (1205731)])))

= (1198921015840(1199091) 119892

1015840(119909119899)) |

M ⊨1199081198921015840

◻120601[1205731

119892 (1205731)] 119892

1015840isin D

V

= (1198921015840(1199091) 119892

1015840(119909119899)) | 119892

1015840isin D

V forall119908

1


11990811198921015840

120601[1205731

119892 (1205731)])

= ⋂ℎ1isinE

ℎ1(119868(120601[

1205731

119892 (1205731)]))

(8)

while

ℎ(119868(loz120601[1205731

119892 (1205731)]))

= ℎ(119868(not◻not120601[1205731

119892 (1205731)]))

= ℎ(neg(necess(119868(not120601[1205731

119892 (1205731)]))))

= D119899 ℎ(necess(119868(not120601[

1205731

119892 (1205731)])))

= D119899 ( ⋂

ℎ1isinE

ℎ1(119868(not120601[

1205731

119892 (1205731)])))

= D119899 ( ⋂

ℎ1isinE

ℎ1(119899119890119892(119868(120601[

1205731

119892 (1205731)]))))

= D119899 ( ⋂

ℎ1isinE

D119899 ℎ

1(119868(120601[

1205731

119892 (1205731)])))

= ⋃ℎ1isinE

ℎ1(119868(120601[

1205731

119892 (1205731)]))

(9)


1)] and

loz120601[1205731119892(120573





120572119892 ≍ ⋖ 120595 ⋗1205732

120572119892 if and only if ℎ(119868(120601[120573

1119892(120573

1)])) =

ℎ(119868(120595[1205732119892(120573

2)])) for every ℎ


120572119892 asymp ⋖ 120595 ⋗1205732


1119892(120573

1)])) =

ℎ(119868(loz120595[1205732119892(120573

2)]))






119894isin 119875 a relation 119877 = 119868

119879(119901119896119894) sube D119896 to any func-


119879(119891119896119894) D119896 rarr D or



119879(119888) isin D


119879isin


119879)


119894(1199051 119905

119896)


1) 119892lowast(119905

119896)) isin 119868








119879isin I

119879







119879(Γ) Thus ⊩ 120601 if and



1≐ 119909

2rArr (119909

1≐ 119909

3rArr 119909

2≐ 119909

3)



1 119909

119898) and 120595(119909

1 119909




1 119909

119898)) = 119868lowast

119879(120595(119909

1 119909


119879




variables (1199091 119909


119879(120601(119909

1


119879(120601119892) =

119905


1 119909

119898) 119868lowast

119879(120601(119909

1 119909

119898)119892) = 119905 if and only if

(119892(1199091) 119892(119909

119898)) isin 119868lowast

119879(120601(119909

1 119909

119898))



1cong 119877

2


1= 119877

2










119879

A119865119874119871



119894 119909119895 119909119896 119909119897 119909119898 119910119894 119910119895) (its tuple of vari-


119894 119909119895 119909119896 119909119897 119909119898) and 120595(119909

119897 119910119894 119909119895 119910119895) 119868lowast

119879(120601 and 120595) = 119868lowast

119879(120593) =

(119892(119909119894) 119892(119909

119895) 119892(119909

119896) 119892(119909

119897) 119892(119909

119898) 119892(119910

119894) 119892(119910

119895)) | 119868lowast

119879(120593119892) =

119905 = (119892(119909119894) 119892(119909

119895) 119892(119909

119896) 119892(119909

119897) 119892(119909

119898)119892(119910

119894) 119892(119910

119895)) | 119868lowast

119879

(120601119892 and 120595119892) = 119905 = (119892(119909119894) 119892(119909

119895) 119892(119909

119896)119892(119909

119897) 119892(119909

119898) 119892(119910

119894)

119892(119910119895)) | 119868lowast

119879(120601119892) = 119905 and 119868lowast

119879(120601119892) = 119905 = (119892(119909

119894) 119892(119909

119895)

119892(119909119896)119892(119909

119897) 119892(119909

119898) 119892(119910

119894) 119892(119910

119895)) | (119892(119909

119894) 119892(119909

119895) 119892(119909

119896)119892(119909

119897)

119892(119909119898)) isin 119868lowast

119879(120601) and (119892(119909

119897) 119892(119910

119894) 119892(119909

119895) 119892(119910

119895)) isin 119868lowast

119879(120601) =

119868lowast119879(120601)⋈

(41)(23)119868lowast119879(120595)






119879(⊤)) =sim (⟨ ⟩) = D0 ⟨ ⟩ = ⟨ ⟩ ⟨ ⟩ = 0



119879(≐) = 119877

=for












119964FOL (syntax)

(11)





119879) More






L119868

997888rarr D997904rArrℎ=119894119904(119868

119879)amp119868119879isinW=I

119879(Γ)R (12)


4 Conclusion






References







































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in





119908119892120601



120572119892 and




120572119892 ≍ ⋖ 120595 ⋗1205732

120572119892 if and only if ◻(120601

[1205731119892(120573

1)] equiv 120595[120573

2119892(120573

2)]) with M ⊨

1199081198921015840◻120593 if



⋖ 11990111(119909) ⋗

119909≍ ⋖ 1199011

2(119909) ⋗



1(119909)) isin 119863


2(119909)) isin

1198631)


120572119892 asymp ⋖ 120595 ⋗1205732

120572119892 if and only if loz120601[120573

1

119892(1205731)] equiv loz120595[120573

2119892(120573





120572119892 ≍ ⋖ 120595 ⋗1205731

120572119892 then

ℎ(119868(⋖ 119860 ⋗1205731120572119892)) = ℎ(119868(⋖ 120595 ⋗1205732





1119892(120573

1)] where the



119899)


1119892(120573

1)]) isin 119863

119899and 119868(120601[120573

1119892(120573

1)]) = 119868(⋖ 120601 ⋗1205731

120572119892) isin





ℎ(119868(◻120601[1205731

119892 (1205731)]))

= ℎ(necess(119868(120601[1205731

119892 (1205731)])))

= (1198921015840(1199091) 119892

1015840(119909119899)) |

M ⊨1199081198921015840

◻120601[1205731

119892 (1205731)] 119892

1015840isin D

V

= (1198921015840(1199091) 119892

1015840(119909119899)) | 119892

1015840isin D

V forall119908

1


11990811198921015840

120601[1205731

119892 (1205731)])

= ⋂ℎ1isinE

ℎ1(119868(120601[

1205731

119892 (1205731)]))

(8)

while

ℎ(119868(loz120601[1205731

119892 (1205731)]))

= ℎ(119868(not◻not120601[1205731

119892 (1205731)]))

= ℎ(neg(necess(119868(not120601[1205731

119892 (1205731)]))))

= D119899 ℎ(necess(119868(not120601[

1205731

119892 (1205731)])))

= D119899 ( ⋂

ℎ1isinE

ℎ1(119868(not120601[

1205731

119892 (1205731)])))

= D119899 ( ⋂

ℎ1isinE

ℎ1(119899119890119892(119868(120601[

1205731

119892 (1205731)]))))

= D119899 ( ⋂

ℎ1isinE

D119899 ℎ

1(119868(120601[

1205731

119892 (1205731)])))

= ⋃ℎ1isinE

ℎ1(119868(120601[

1205731

119892 (1205731)]))

(9)


1)] and

loz120601[1205731119892(120573





120572119892 ≍ ⋖ 120595 ⋗1205732

120572119892 if and only if ℎ(119868(120601[120573

1119892(120573

1)])) =

ℎ(119868(120595[1205732119892(120573

2)])) for every ℎ


120572119892 asymp ⋖ 120595 ⋗1205732


1119892(120573

1)])) =

ℎ(119868(loz120595[1205732119892(120573

2)]))






119894isin 119875 a relation 119877 = 119868

119879(119901119896119894) sube D119896 to any func-


119879(119891119896119894) D119896 rarr D or



119879(119888) isin D


119879isin


119879)


119894(1199051 119905

119896)


1) 119892lowast(119905

119896)) isin 119868








119879isin I

119879







119879(Γ) Thus ⊩ 120601 if and



1≐ 119909

2rArr (119909

1≐ 119909

3rArr 119909

2≐ 119909

3)



1 119909

119898) and 120595(119909

1 119909




1 119909

119898)) = 119868lowast

119879(120595(119909

1 119909


119879




variables (1199091 119909


119879(120601(119909

1


119879(120601119892) =

119905


1 119909

119898) 119868lowast

119879(120601(119909

1 119909

119898)119892) = 119905 if and only if

(119892(1199091) 119892(119909

119898)) isin 119868lowast

119879(120601(119909

1 119909

119898))



1cong 119877

2


1= 119877

2










119879

A119865119874119871



119894 119909119895 119909119896 119909119897 119909119898 119910119894 119910119895) (its tuple of vari-


119894 119909119895 119909119896 119909119897 119909119898) and 120595(119909

119897 119910119894 119909119895 119910119895) 119868lowast

119879(120601 and 120595) = 119868lowast

119879(120593) =

(119892(119909119894) 119892(119909

119895) 119892(119909

119896) 119892(119909

119897) 119892(119909

119898) 119892(119910

119894) 119892(119910

119895)) | 119868lowast

119879(120593119892) =

119905 = (119892(119909119894) 119892(119909

119895) 119892(119909

119896) 119892(119909

119897) 119892(119909

119898)119892(119910

119894) 119892(119910

119895)) | 119868lowast

119879

(120601119892 and 120595119892) = 119905 = (119892(119909119894) 119892(119909

119895) 119892(119909

119896)119892(119909

119897) 119892(119909

119898) 119892(119910

119894)

119892(119910119895)) | 119868lowast

119879(120601119892) = 119905 and 119868lowast

119879(120601119892) = 119905 = (119892(119909

119894) 119892(119909

119895)

119892(119909119896)119892(119909

119897) 119892(119909

119898) 119892(119910

119894) 119892(119910

119895)) | (119892(119909

119894) 119892(119909

119895) 119892(119909

119896)119892(119909

119897)

119892(119909119898)) isin 119868lowast

119879(120601) and (119892(119909

119897) 119892(119910

119894) 119892(119909

119895) 119892(119910

119895)) isin 119868lowast

119879(120601) =

119868lowast119879(120601)⋈

(41)(23)119868lowast119879(120595)






119879(⊤)) =sim (⟨ ⟩) = D0 ⟨ ⟩ = ⟨ ⟩ ⟨ ⟩ = 0



119879(≐) = 119877

=for












119964FOL (syntax)

(11)





119879) More






L119868

997888rarr D997904rArrℎ=119894119904(119868

119879)amp119868119879isinW=I

119879(Γ)R (12)


4 Conclusion






References







































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in





120572119892 ≍ ⋖ 120595 ⋗1205732

120572119892 if and only if ℎ(119868(120601[120573

1119892(120573

1)])) =

ℎ(119868(120595[1205732119892(120573

2)])) for every ℎ


120572119892 asymp ⋖ 120595 ⋗1205732


1119892(120573

1)])) =

ℎ(119868(loz120595[1205732119892(120573

2)]))






119894isin 119875 a relation 119877 = 119868

119879(119901119896119894) sube D119896 to any func-


119879(119891119896119894) D119896 rarr D or



119879(119888) isin D


119879isin


119879)


119894(1199051 119905

119896)


1) 119892lowast(119905

119896)) isin 119868








119879isin I

119879







119879(Γ) Thus ⊩ 120601 if and



1≐ 119909

2rArr (119909

1≐ 119909

3rArr 119909

2≐ 119909

3)



1 119909

119898) and 120595(119909

1 119909




1 119909

119898)) = 119868lowast

119879(120595(119909

1 119909


119879




variables (1199091 119909


119879(120601(119909

1


119879(120601119892) =

119905


1 119909

119898) 119868lowast

119879(120601(119909

1 119909

119898)119892) = 119905 if and only if

(119892(1199091) 119892(119909

119898)) isin 119868lowast

119879(120601(119909

1 119909

119898))



1cong 119877

2


1= 119877

2










119879

A119865119874119871



119894 119909119895 119909119896 119909119897 119909119898 119910119894 119910119895) (its tuple of vari-


119894 119909119895 119909119896 119909119897 119909119898) and 120595(119909

119897 119910119894 119909119895 119910119895) 119868lowast

119879(120601 and 120595) = 119868lowast

119879(120593) =

(119892(119909119894) 119892(119909

119895) 119892(119909

119896) 119892(119909

119897) 119892(119909

119898) 119892(119910

119894) 119892(119910

119895)) | 119868lowast

119879(120593119892) =

119905 = (119892(119909119894) 119892(119909

119895) 119892(119909

119896) 119892(119909

119897) 119892(119909

119898)119892(119910

119894) 119892(119910

119895)) | 119868lowast

119879

(120601119892 and 120595119892) = 119905 = (119892(119909119894) 119892(119909

119895) 119892(119909

119896)119892(119909

119897) 119892(119909

119898) 119892(119910

119894)

119892(119910119895)) | 119868lowast

119879(120601119892) = 119905 and 119868lowast

119879(120601119892) = 119905 = (119892(119909

119894) 119892(119909

119895)

119892(119909119896)119892(119909

119897) 119892(119909

119898) 119892(119910

119894) 119892(119910

119895)) | (119892(119909

119894) 119892(119909

119895) 119892(119909

119896)119892(119909

119897)

119892(119909119898)) isin 119868lowast

119879(120601) and (119892(119909

119897) 119892(119910

119894) 119892(119909

119895) 119892(119910

119895)) isin 119868lowast

119879(120601) =

119868lowast119879(120601)⋈

(41)(23)119868lowast119879(120595)






119879(⊤)) =sim (⟨ ⟩) = D0 ⟨ ⟩ = ⟨ ⟩ ⟨ ⟩ = 0



119879(≐) = 119877

=for












119964FOL (syntax)

(11)





119879) More






L119868

997888rarr D997904rArrℎ=119894119904(119868

119879)amp119868119879isinW=I

119879(Γ)R (12)


4 Conclusion






References







































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in









119879

A119865119874119871



119894 119909119895 119909119896 119909119897 119909119898 119910119894 119910119895) (its tuple of vari-


119894 119909119895 119909119896 119909119897 119909119898) and 120595(119909

119897 119910119894 119909119895 119910119895) 119868lowast

119879(120601 and 120595) = 119868lowast

119879(120593) =

(119892(119909119894) 119892(119909

119895) 119892(119909

119896) 119892(119909

119897) 119892(119909

119898) 119892(119910

119894) 119892(119910

119895)) | 119868lowast

119879(120593119892) =

119905 = (119892(119909119894) 119892(119909

119895) 119892(119909

119896) 119892(119909

119897) 119892(119909

119898)119892(119910

119894) 119892(119910

119895)) | 119868lowast

119879

(120601119892 and 120595119892) = 119905 = (119892(119909119894) 119892(119909

119895) 119892(119909

119896)119892(119909

119897) 119892(119909

119898) 119892(119910

119894)

119892(119910119895)) | 119868lowast

119879(120601119892) = 119905 and 119868lowast

119879(120601119892) = 119905 = (119892(119909

119894) 119892(119909

119895)

119892(119909119896)119892(119909

119897) 119892(119909

119898) 119892(119910

119894) 119892(119910

119895)) | (119892(119909

119894) 119892(119909

119895) 119892(119909

119896)119892(119909

119897)

119892(119909119898)) isin 119868lowast

119879(120601) and (119892(119909

119897) 119892(119910

119894) 119892(119909

119895) 119892(119910

119895)) isin 119868lowast

119879(120601) =

119868lowast119879(120601)⋈

(41)(23)119868lowast119879(120595)






119879(⊤)) =sim (⟨ ⟩) = D0 ⟨ ⟩ = ⟨ ⟩ ⟨ ⟩ = 0



119879(≐) = 119877

=for












119964FOL (syntax)

(11)





119879) More






L119868

997888rarr D997904rArrℎ=119894119904(119868

119879)amp119868119879isinW=I

119879(Γ)R (12)


4 Conclusion






References







































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in





References







































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in










Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in



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