Fuzzy logic interpretations in categories of fuzzy objects

Available online at www.ispacs.com/jfsva

Volume 2016, Issue 1, Year 2016 Article ID jfsva-00269, 12 Pages doi:10.5899/2016/jfsva-00269

Research Article

Fuzzy logic interpretations in categories of fuzzy objects

Jiˇr´ı Moˇckoˇr∗

Centre of Excellence IT4Innovations division of the University of Ostrava,

Institute for Research and Applications of Fuzzy Modeling 30. dubna 22, 701 03 Ostrava 1, Czech Republic

Copyright 2016 c⃝Jiˇr´ı Moˇckoˇr. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Interpretations of first order fuzzy logic formulas in models based on two categories of sets with similarity relations as objects are introduced. Formulas are interpreted either as special morphisms in these categories, or f-cut systems, which generalizeα-cuts of classical fuzzy sets. Relationships between these two interpretations of formulas in corre-sponding categories are investigated.

Keywords:Many-Valued and Fuzzy Logics.

1 introduction

When L.A. Zadeh [13] introduced fuzzy sets, he regarded these new objects as crisp sets with membership degree in interval[0,1]. These membership degrees were soon considered to be truth-values from logical point of view and

this became a starting point for fuzzy logic with truth values in unit interval. The basic idea of fuzzy logic were further elaborated by Zadeh in his publications [14],[15]. In general, fuzzy logic can be characterize as the many-values logic with special properties aiming at modeling of the vagueness phenomenon and some parts of the meaning of natural language via graded approach([11]). The graded approach is mostly expressed by means of a special scale being an ordered set, which is endowed by additional operations. Zadeh in his papers used operations which were originally presented by Łukasiewicz [5] in his infinitely valued logic and then frequently used in many other applications. For technical reasons, truth-values must form a complete lattice and some additional operations are also frequently required. A well known example of a structure standardly used in fuzzy logic is the Łukasiewicz algebra Ł= ([0,1],∨,∧,⊗,→Ł,0,1), where

a∧b=min(a,b),

a∨b=max(a,b),

a⊗b=0∨(a+b−1),

a→_Łb=1∧(1−a+b).

Since the original Zadeh’s paper was published, the notion of ”fuzzy set” has been changed significantly and it is now more general. The first important modification concerns the truth-values set: instead of the unit interval

I= [0,1]mostly with classical Łukasiewicz connectives, more general lattice structuresQare considered. Among

∗_{Corresponding author. Email address: Jiri.Mockor@osu.cz}

these lattice structures, complete residuated lattices play important role, (see e.g. [11]), in modern terminologyunital and commutative quantale, (see [12]), i.e. a structureQ= (L,∧,∨,⊗,→,0,1)such that(L,∧,∨)is a complete lattice,

(L,⊗,1)is a commutative monoid with operation⊗isotone in both arguments and→is a binary operation which is adjoint with respect to⊗, i.e.

α⊗β≤γiffα≤β→γ.

A fuzzy logic, based on these ordered structures as a mathematical object, has a structure, which is analogical to the structure of classical first order predicate logic. A fuzzy logic consists of a first order languageJwhich consists of a set of predicate symbolsP∈P_{, a set of functional symbols f∈}F _{and a set of logical connectives{∧,∨,⇒,¬,⊗}.} MoreoverJcontains also a setQof logical constants. In that language, terms and formulas can be defined by using of inductive principle in the same way as for a classical first order predicate logic.

Fuzzy logic is closely connected with fuzzy sets. Connecting bridge, in this respect, is the concept oftruth valuation

of formulas in fuzzy logic. Ifϕ(x)is a formula in fuzzy logic with a free variablex, which can be substituted by values from a setA, then a truth valuation ofϕ(x)in a basic setAcan defined as a fuzzy set∥ϕ∥:A→[0,1](instead

of[0,1], any other truth-values setQcan be used). The value∥ϕ∥(a)then expresses a degree in which a formulaϕ

is true, if a free variablexis substituted by a constant valuea. A relationship between fuzzy logic and fuzzy sets is further realized by rules which hold for fuzzy sets∥ϕ∥on one hand and logical structure of formulas on the other. For example, basic rules can be written in the following form

∥¬ϕ∥(x) =R¬(∥ϕ∥(x)), ∥ϕ∧ψ∥(x) =R∧(∥ϕ∥(x),∥ψ∥(x)), ∥ϕ∨ψ∥(x) =R∨(∥ϕ∥(x),∥ψ∥(x)),

for some fix mapsR¬:Q→QandR∨,R∧:Q

2_{→Q. IfQ= [0}

,1], thenR¬,R∧,R∨can be defined by, e.g., Łukasiewicz

operations. Then aninterpretationorsemanticof fuzzy logic (based on a setA) is a map∥ − ∥:F _→∪

nQA n

, such that ifϕ is a formula with free variablesx= (x1, . . . ,xn), then∥ϕ∥:A

n_{→Q, whereF is the set of all formulas of}

fuzzy logic.

On the other hand, analogously as for classical predicate logic, asyntactic structurecan be defined for fuzzy logic. It means that for any formulaψof a classical predicate logic we can derive if that formula is provable (i.e. truth) in that logic (in symbol⊢ψ) or not. Principal tools for calculations arededuction ruleswhich are used in the logic. In a fuzzy logic, graded versions of deduction rules are used and it means that we receive also a graded notion of a provability of a formula, i.e. ⊢α ψ means thatψ istruein the logic in a degreeα, whereα∈Q. For a relationships between semantic and syntactic structures of a fuzzy logic see, e.g., [11], for some other information about interpretation of fuzzy logic see, e.g., [1],[2].

Fuzzy sets were originally defined on sets. But any setA can be considered to be a couple(A,=), where=is a standard equality relation defined onA. It is then natural, instead of the crisp equality relation=, to consider some more ”fuzzy” equality relationδ defined as a mapA×A→Q, which is called asimilarity relation. Then instead of a classical setAas a basic set and a fuzzy sets:A→Q, we can use a set with similarity relation(A,δ)(called aQ-set) and a maps:(A,δ)→Q. Such a map then represents some new ”fuzzy object” in(A,δ).

In the paper, we use two categories Set(Q)and SetR(Q), withQ-sets as objects. A morphism f:(A,δ)→(B,γ)in the category Set(Q)is a mapf :A→Bsuch thatγ(f(x),f(y))≥δ(x,y)for allx,y∈Aand a morphism(A,δ)→(B,γ)in

the category SetR(Q)is a special relationA×B→Q. It is then natural to speak about afuzzy object(A,δ)→(Q,↔) in the category Set(Q)or SetR(Q), instead of a ”fuzzy set”, where↔is the biresiduation operation inQdefined by

α↔β= (α→β)∧(β→α). These fuzzy objects generalize classical fuzzy setsA→Q.

In the categories Set(Q)and SetR(Q), we can even define fuzzy objects which are not maps or morphisms. The motivation for these fuzzy objects comes fromα-cuts of classical fuzzy sets. Any classically defined fuzzy setXin a setAwith values inQcan be defined equivalently by a system of level setsXα,α∈Q, whereXα={a∈A:X(a)≥α}. Conversely, any (nested) system(Yα)αof subsets ofAsuch that for anya∈Athe set{α∈Q:a∈Yα}has the greatest element defines a fuzzy setY such thatY(a) =∨{β:a∈Y_β}β. In our previous papers [7], [8], we proved that any

morphism(A,δ)→(Q,↔)in the categories Set(Q)and SetR(Q)can be equivalently defined by a system of some special subsets ofAorA×Q, respectively, calledf-cuts. Hence, f-cuts then represent anotherfuzzy objectsin(A,δ) in our categories, which generalize classical fuzzy sets.

Using these two categories Set(Q) and SetR(Q), in the paper we obtain the following two different types of an interpretation of fuzzy logic.

(i) Interpretation of a formula ψ as a fuzzy object ∥ψ∥, i.e. a special morphism∥ψ∥:(A,δ)→(Q,↔) in a corresponding category, or

(ii) Interpretation of a formulaψas afuzzy object(|ψ|α)α∈Q, where(|ψ|α)αis an f-cut in(A,δ)in a corresponding category.

These two interpretations can be described intuitively as follows:

1. If∥ψ∥is a fuzzy object in(A,δ), it means that for any elementa∈(A,δ),∥ψ∥(a)∈Qis adegreein which a formulaψis true, if the value of a free variablexinψis substituted bya∈A.

2. If(|ψ|α)α is an f-cut in(A,δ), then|ψ|α is a”subset”of all substitutions in(A,δ)of free variables ofψ, for which a formulaψis true in a degree at leastα.

As principal results of the paper, we present two types of interpretations of a fuzzy logic based on the categories Set(Q) and SetR(Q), respectively, and for each category we define two additional different types of formulas interpretations. We also prove some relationships between these interpretations, i.e., we show that interpretations∥ψ∥are isomorphic to the interpretations(|ψ|α)α. Since both new interpretations generalize standard interpretation of a fuzzy login in classical fuzzy sets, in that way we are expanding opportunities for obtaining examples of formulas interpretations with the desired (or expected) degree of truth. Some proofs of presented results will be fully published in [10].

2 Preliminary notions and results

In this section we present some preliminary notions and definitions which could be helpful for better understanding of results concerning sets with similarity relations. Most of these notions can be found e.g. in [8, 7]. A principal structure used in the paper is acomplete residuated lattice(see e.g. [11]), i.e., a structureQ= (Q,∧Q,∨Q,⊗Q,→Q

,0Q,1Q)such that(Q,∧Q,∨Q)is a complete lattice,(Q,⊗Q,1Q)is a commutative monoid with operation⊗Qisotone

in both arguments and→Qis a binary operation which is residuated with respect to⊗Q, i.e.,

α⊗Qβ≤γ iff α≤β→Qγ.

For simplicity the indexQwill be sometimes omitted.

Any classical setAcan be considered as a pair(A,=), where=is the equality relation. It is then natural to consider a generalization of that pair, i.e., a pair(A,δ), whereδ is a similarity relation. Recall that a similarity relation inAis a mapδ :A×A→Ωsuch that

(c) (∀x,y,z∈A) δ(x,y)⊗δ(y,z)≤δ(x,z)(generalized transitivity).

A pair(A,δ)will be calledQ-set. In the paper, we will be working not only withQ-sets, but also with ”mappings” betweenQ-sets, i.e., it seems to be useful to use a category theory tools for further investigation of such structures. We basically use two categories withQ-sets as objects and with differently defined morphisms. A morphism f : (A,δ)→(B,γ)in the first category Set(Q)is a map f :A→Bsuch thatγ(f(x),f(y))≥δ(x,y)for allx,y∈A. The

other category SetR(Q)is an analogy of the category of sets with relations between sets as morphisms. Objects of the category SetR(Q)are the same as in the category Set(Q)and morphisms f:(A,δ)→(B,γ)are maps f :A×B→Ω

(i.e.,Q-valued relations) such that

(a) (∀x,z∈A)(∀y∈B) δ(z,x)⊗f(x,y)≤ f(z,y), (b) (∀x∈A)(∀y,z∈B) f(x,y)⊗γ(y,z)≤f(x,z).

(c) (∀a∈A) 1=∨{f(x,y):y∈B}.

If f :(A,δ)→(B,γ)andg:(B,γ)→(C,ω)are two morphisms in SetR(Q), then their composition is a relation

g◦f :A×C→Ωsuch that

g◦f(x,z) =∨

y∈B

(f(x,y)⊗g(y,z)).

It should be mentioned that there is another category of sets with similarity relations as objects, which was intensively investigated. Namely the category SetS(Q)with morphisms satisfying previous conditions (a) and (b) only. Unfor-tunately, that category is not appropriate for logic interpretation, because of a lack of categorical products, which are important for models constructions.

Lemma 2.1. There exists a functor F:Set(Q)→SetR(Q).

As we mentioned in Introduction, a fuzzy object f in anQ-set(A,δ)in a categoryKofQ-sets (in symbol: f ⊂K (A,δ)) is a morphism f :(A,δ)→(Q,↔)in a categoryK, where↔is the biresiduation operation inQdefined by

α↔β= (α→β)∧(β →α). Hence, f ⊂_Set(Q)(A,δ), if f :A→Ωis a map such thatδ(x,y)≤f(x)↔ f(y), or equivalently, f(x)⊗δ(x,y)≤f(y)for allx,y∈A. Such maps are sometimes calledextensional maps. Analogously, a fuzzy object in the category SetR(Q)is a mapf :A×Q→Qwhich has to satisfy the following conditions:

1. f(x,α)⊗δ(x,y)≤f(y,α), for allx,y∈A,α∈Q,

2. f(x,α)⊗(α↔β)≤f(x,β), for allx∈A,α,β∈Q,

3. 1=∨_α∈Qf(x,α), for anyx∈A.

ForK=Set(Q)or SetR(Q), the set of all fuzzy objects f⊂K(A,δ)in aQ-set(A,δ)is an object function of a functor. In fact, there exists a functorF_K:K→Set that is defined byF_K(A,δ) ={s:s⊂K(A,δ)}. If f :(A,δ)→(B,γ)is a morphism inK, then the mapF_K(f):F_K(A,δ)→F_{K(B,γ)is defined differently for categoriesK=Set(Q)and}

K=SetR(Q). We have,

F_Set(

Q)(f)(s)(b) =

∨

x∈A

s(x)⊗γ(b,f(x)),

for allb∈Band anys⊂_Set(Q)(A,δ)(see [8]) and for the category SetR(Q)we have,

(∀b∈B)(∀α∈Q)F_SetR(Q)(f)(s)(b,α) =∨

a∈A

f(a,b)⊗s(a,α),

for allb∈B,α∈Q(see [7]).

Proposition 2.1. There exists a natural transformation

Not every mapsA→QorA×Q→Q, respectively, are morphisms in category Set(Q)or SetR(Q), respectively. On the other hand, any such maps can be extended to morphisms, according the following methods.

Lemma 2.2. Let(A,δ),(B,γ)be Q-sets and let g:A×B→Q be a map, such that1=

∨

b∈Bg(a,b), for any a∈A. Letg˜:A×B→Q be defined by the formula

˜

g(a,b) =∨

x∈A

∨

y∈B

g(x,y)⊗δ(a,x)⊗γ(b,y).

Then

1. g is a morphism in SetR˜ (Q),

2. g˜=∧{h:h is a morphism(A,δ)→(B,γ)in SetR(Q),h≥g}, 3. If g is a morphism in SetR(Q), theng˜=g.

Lemma 2.3. Let(A,δ)be an Q-set and let s:A→Q be a map. Then we define a maps_b:A→Q such that_bs(a) =

∨

x∈Aδ(a,x)⊗s(x)for all a∈A. Then

1. _bs:(A,δ)→(Q,↔)is a morphism in Set(Q),

2. If s:(A,δ)→(Q,↔)is a morphism in Set(Q)then_bs=s, 3. _bs=∧{t:t is a morphism(A,δ)→(Q,↔),t≥s in Set(Q)}.

It is well known that any classical fuzzy set (with values in a residuated latticeQ) in a setAcan be alternatively expressed as a system ofα-cutsC= (Cα)α, whereCis a nested system of subsets ofA. In our previous papers ([8, 7]) we proved that analogical representations of fuzzy sets by special cuts, named f-cuts, exists in categories Set(Q),SetR(Q). The definitions of these f-cuts is as follows.

Definition 2.1. Let(A,δ)be a Q-set. Then a systemC= (Cα)α∈Qof subsets in A is called anf-cutin(A,δ)in the

category Set(Q)if

1. ∀a,b∈A, a∈Cα⇒b∈Cα⊗δ(a,b),

2. ∀a∈A,∀α∈Q, ∨_{β:a∈Cβ}β≥α⇒a∈Cα.

Definition 2.2. Let(A,δ)be a Q-set. Then a system(Cα)α∈Qis called anf-cutin(A,δ)in the category SetR(Q), if

1. Cα⊆A×Q, for anyα∈Q,

2. ∀a,b∈A, (a,β)∈Cα⇒(b,β)∈Cα⊗δ(a,b),

3. ∀a∈A,∀γ∈Q, ∨_{β:(a

,γ)∈Cβ}β ≥α⇒(a,γ)∈Cα,

4. ∀a∈A,∀α,γ∈Q, (a,α)∈Cβ ⇒(a,γ)∈Cβ⊗(α↔γ), 5. ∀a∈A, 1=∨{(α,β):(a,α)∈C_β}β

It is cleat that not every system of subsets(Cα)α fromAorA×Qis an f-cut. On the other hand, analogously as for fuzzy set, such systems can be extended to f-cuts, as the following lemmas show.

Proposition 2.2. Let(A,δ)be a Q-set and let(Cα)α be a system of subsets in a set A. For anyα∈Q we set

Cα={a∈A:

∨

{(x,β):x∈Cβ}

β⊗δ(a,x)≥α}.

1. Cα⊆Cα for allα∈Q,

2. (Cα)αis an f-cut system in(A,δ)in the category Set(Q),

3. If(Cα)αis an f-cut system in(A,δ)in the category Set(Q), then Cα=Cαfor allα∈Q.

4. If(Dα)αis a system of subsets in A such that Cα⊆Dαfor allα∈Q, then Cα⊆Dα for allα∈Q. An analogical result holds for the category SetR(Q).

Proposition 2.3.Let(A,δ)be a Q-set and let(Cα)αbe a system of subsets in a set A×Q, such that1=∨{(α,β):(a,α)∈Cβ}β,

for all a∈A. For anyα∈Q we set

Cα={(a,β)∈A×Q:

∨

{(x,τ,ρ):(x,τ)∈Cρ}

ρ⊗δ(a,x)⊗(τ↔β)≥α}.

Then

1. Cα⊆Cα for allα∈Q,

2. (Cα)αis an f-cut in(A,δ)in the category SetR(Q),

3. If(Cα)αis an f-cut in(A,δ)in the category SetR(Q), then Cα=Cαfor allα∈Q.

4. If(Dα)αis a system of subsets in A such that Cα⊆Dαfor allα∈Q, then Cα⊆Dα for allα∈Q.

Analogously as for fuzzy sets, there exists a functorC_{K:K→Set, such that}C_{K(A,δ)is the set of all f-cuts in(A,δ)} in a categoryK=Set(Q),SetR(Q)(for details see [10, 7]). In the same papers the following theorem is proved.

Theorem 2.1. For the categoryK=Set(Q),SetR(Q), there exists a natural isomorphismΨK:FK→CK.

It means, especially, that for any Q-set(A,δ)and any categoryK=Set(Q),SetR(Q), there exists a bijectionΨK

,(A,δ):

C_K(A,δ)→F_K(A,δ).

Proposition 2.4. There exists a natural transformation

τ:C_Set(Q)→C_SetR(Q)◦F.

We show only how that transformation is defined. Let(A,δ)be a Q-set and letC= (Cα)α∈CSet(Q)(A,δ)be an f-cut

in(A,δ)in the category Set(Q). Thenτ(A,δ)(C) = (Dα)α, whereDα={(a,γ):

∨

a∈C_ββ↔γ≥α}.

3 Construction of an interpretation of a fuzzy logic in categories Set(Q)and SetR(Q)

Let us recall some definitions and results concerning an interpretation of a fuzzy logic in models based onQ-sets (see [11, 9]). Recall that a first order predicate fuzzy logic is based on a first order languageJ which consists (as classically) of a set of predicate symbolsP∈P_{, a set of functional symbols f ∈}R_{and a set of logical connectives}

{∧,∨,⇒,¬,⊗}. Terms and formulas are defined analogously as for the classical predicate logic by using of the

inductive principle.

LetKbe a category withQ-sets as objects and such that for any set of objects{(Ai,δi):i∈I}there exists a product (A,δ) =∏i∈I(Ai,δi). Recall that a product of{(Ai,δi):i∈I}in a categoryKis an object(A,δ)with morphisms pri:(A,δ)→(Ai,δi)such that for any other object(B,γ)with morphismsqi:(B,γ)→(Ai,δi)there exists the unique

morphismq=∏iqisuch that the diagram commutes:

(A,δ) (A,δ)

pri

 

y xq=∏iqi

(Ai,δi) qi

Recall, how a product is constructed in our categories Set(Q)and SetR(Q). Let(Ai,δi)beQ-sets,i∈I. Let us consider

the category Set(Q), firstly. Then∏i∈I(Ai,δi) = (AI,δI), whereAI is the cartesian product of setsAiandδI(a,b) =

∧

i∈Iδi(ai,bi), for anya,b∈A. The projection morphismspri:(AI,δI)→(Ai,δi)are classical projection maps and if qi:(B,γ)→(Ai,δi)are morphisms, fori∈I, the unique morphism∏iqiis such that∏iqi(b) = (qi(b))i∈AI, for any b∈B.

Now, let us consider the category SetR(Q). A product(AI,δI)is the same object as in the category Set(Q), but

with different projection morphisms pri defined such thatpri:(AI,δI)→(Ai,δi)in the category SetR(Q), i.e. pri: AI×Ai→Q, such that pri(a,b) =δi(ai,b), for a∈AI,b∈Ai. If qi:(B,γ)→(Ai,δi) are morphisms, for i∈I,

the unique morphism∏iqi is such that∏iqi:B×AI→Q,∏iqi(b,a) =∧iqi(b,ai). If(Ai,δi) = (A,δ)for every i=1, . . . ,n, then∏_i(Ai,δi)will be denoted by(A

n ,δn).

We now introduce two types of models of a fuzzy logic in the categoryK=Set(Q)or SetR(Q)based on different fuzzy objects in these categories.

Definition 3.1. Afuzzy set modelof a language J in a categoryKis

E_{K= ((A,δ)},{PE,K:P∈

P_},{fE,K: f∈

R_}),

where

(1) (A,δ)is an Q-set from a categoryK, (2) PE,Kis a fuzzy object in(A,δ)

n_{in a categoryK, i.e., a morphism(A,δ)}n_→(Q,↔),

(3) fE,K:(A,δ)

n_{→(A,δ)is a morphism in a categoryK.}

Definition 3.2. Acut modelof a language J in a categoryKis

D_{K= ((}A,δ),{PD,K:P∈

P_},{fD,K: f∈

R_}),

where

(1) (A,δ)is an Q-set from a categoryK, (2) PD,Kis an f-cut in(A,δ)

n_{in a categoryK, P}

D,K= (Pα)α,

(3) fD,K:(A,δ)

n_{→(A,δ)is a morphism in a categoryK.}

Ifψis a formula in a fuzzy logic with a setX of free variables, then an interpretation∥ψ∥ofψ will be different for different types of a model.

(a) Interpretation∥ψ∥E,Kin a model

E_{Kis a fuzzy object in a categoryK,}

(b) Interpretation∥ψ∥D,Kin a model

D_{Kis an f-cut in a category}K.

Intuitively,

1. If∥ψ∥ is a fuzzy set in(A,δ)X_{, it means that for anya= (a}

x)x∈X ∈AX,∥ψ∥(a)∈Qis adegreein which a

formulaψis true in modelE_{, if the value of a variable}xis substituted by elementax∈A.

2. If∥ψ∥is an f-cut(|ψ|α)α in(A,δ)X, then|ψ|αis asetof all interpretations (inA) of free variables fromX, for which a formulaψis true in a modelE _{in a degree at leastα.}

Now we present these definitions of∥ψ∥in our two types of models. First of all, we need to define an interpretation of terms in our models. The definition will be the same for all two types of models. LetG ₌E_KorD_{Kbe a model of} a languageJin a categoryK. An interpretation of a termtwith a set of variables contained in a setX is a morphism

∥t∥G,K:(A,δ)

X_{→(A,δ)in a categoryK, defined by:}

1. Lett=x, wherex∈X. Then∥t∥G,K:=prx:(A,δ)

2. Lett=f(t1, . . . ,tn). Then∥t∥G,Kis a composition (inK) of morphisms

(A,δ)X ∏i∥ti∥G,K

−−−−−→ (A,δ)n fG,K

−−−−→ (A,δ).

For example, forK=SetR(Q)we have∥t∥G,SetR(Q)(a,b) =

∨

x∈An∧n_i=1∥ti∥(a,xi)⊗fG(x,b), wherea∈A X

,b∈A.

A definition of∥ψ∥will differs for different types of models and different categoriesK=Set(Q),SetR(Q).

Defini-tions will be done by the induction principle depending on a structure ofψ.

Definition 3.3(Interpretation in modelsE_Set(

Q),ESetR(_Q)). LetKbe the category Set(Q)or SetR(Q), respectively. An interpretation∥ψ∥=∥ψ∥E,K,Xofψ with free variables in a set X in a categoryKis defined as follows.

1. Letψ≡P(t1, . . . ,tn). Then∥ψ∥E,Kis defined as the composition of the following morphisms inK:

(A,δ)X ∏i∥ti∥E,K

−−−−−→ (A,δ)n PE,K

−−−−→ (Q,↔).

2. Letψ≡t1=t2. Then∥ψ∥E,Kis the composition of the following morphisms inK:

(A,δ)X ∥t1∥E,K×∥t2∥E,K

−−−−−−−−−→ (A,δ)2 ∆K,E

−−−−→ (Q,↔),

where∆K,E is a morphism inK, which interprets equality in a model

E_K.

3. Letψ≡σ∇τ, where∇represents logical connectives∧,∨,=⇒,⊗, respectively. Then∥ψ∥E,Kis the

compo-sition of the following morphisms:

(A,δ)X ∥σ∥E,K×∥τ∥E,K

−−−−−−−−−→ (Q,↔)2 ⊙K,E

−−−−→ (Q,↔),

where⊙K,E is a morphism inK, which interprets logical operations∧,∨,→,⊗, respectively, in a model

E_K.

4. Letψ≡ ¬σ. Then∥ψ∥E,Kis the composition of the following morphisms:

(A,δ)X ∥σ∥E,K

−−−−→ (Q,↔) ¬K,E

−−−−→ (Q,↔),

where¬K,E is a morphism inK, which interprets logical negation in a model

E_K.

5. Letψ≡(∀x)σ. Then∥σ∥E,K,X∪{x}is already defined as a morphism(A,δ)

X_{×(A,δ)→(Q,↔)inK. Then we} set

∥ψ∥E,K,X(a) =

∧

x∈A

∥σ∥E,K,X∪{x}(a,x).

Let us now consider some examples of interpretations of logical connectives in categories Set(Q)and SetR(Q), pre-sented in the definition.

Example 3.1.

1) A reasonable example of a morphism∆Set(Q),E could be the morphismδ:(A,δ)

2_→(Q,↔).

2) A morphism∆SetR(Q),E can be also defined as a map A

2_{×Q→Q by using results from Lemmas 2.1 and 2.2, by}

∆SetR(Q),E((a,b),α) =F(δ)((a,b),α),

3) ∧Set(Q),E which interprets logical connective∧can be∧Q. In fact, the following holds for any a,a

′ ,b,b

′_∈A:

(a∧Qa′)↔(b∧Qb′)≥(a↔b)∧Q(a′↔b′),

4) ∧SetR(Q),E can be defined by∧SetR(Q),E((β,γ),α) =α↔(β∧γ). In fact,∧SetR(Q),E :(Q,↔)

2_{→(Q,↔)is a}

morphism in SetR(Q), as it can be proved by a simple calculation.

5) On the other hand, the operation→Qis not a morphism in Set(Q)and a reasonable candidate for interpretation →Set(Q),E of a logical implication in Set(Q)can be thenc→.

6) Analogously,→SetR(Q),E can be defined naturally in two different ways:

(i) →SetR(Q),E=

^ F(→Set(Q),E),

(ii) →SetR(Q),E=F(→\Set(Q),E).

7) ¬Set(Q),E can be defined as a morphism(Q,↔)→(Q,↔), such that¬Set(Q),E(α) =b¬(α) =

∨

β∈Q(β→0Q)⊗

(α↔β).

Proposition 3.1. ∥ψ∥E,K,Xis a fuzzy object in(A,δ)

X_{, for any categoryK=Set(Q)}

,SetR(Q).

Definition 3.4(Interpretation in the modelD_Set(Q)). An interpretation∥ψ∥=∥ψ∥D,Set(Q),X ofψ with free variables

in a set X in the category Set(Q)is an f-cut(|ψ|α)αin(A,δ)Xdefined as follows (instead of∥t∥D,Set(Q)we write∥t∥,

for any term t).

1. Letψ≡P(t1, . . . ,tn). Then

|ψ|α={a∈AX:(∥t1∥(a), . . . ,∥tn∥(a))∈Pα}. 2. Letψ≡t1=t2. Then

|ψ|α={a∈AX:∆Set(Q),D(∥t1∥(a),∥t2∥(a))≥α}, where∆Set(Q),Dis a morphism in Set(Q), which interprets equality in a model

D_Set(Q).

3. Letψ ≡σ∇τ, where∇represents logical connectives∧,∨,=⇒,⊗, respectively. Let∥σ∥= (|σ|β)β,∥τ∥=

(|τ|γ)γbe f-cuts in(A,δ)X. Then

|ψ|α={a∈AX:(

∨

a∈|σ|β

β)⊙Set(Q),D(

∨

a∈|τ|γ

γ)≥α},

where⊙Set(Q),D is a morphism(Q,↔)

2_{→(Q,↔)in Set(Q), which interprets logical operations∧,∨,→,⊗,}

respectively, in a modelD_Set(Q).

4. Letψ≡ ¬σ. Let∥σ∥= (|σ|β)β be an f-cut. Then

|ψ|α={a∈AX:¬Set(Q),D(

∨

a∈|σ|β

β)≥α},

where¬Set(Q),Dis a morphism in Set(Q), which interprets logical negation in a model

D

Set(Q).

5. Letψ≡(∀x)σ. Then∥σ∥D,Set(Q),X∪{x}is already defined as an f-cut(|σ|β)βin(A,δ)

X∪{x}_{and we set}

|ψ|α={a∈AX:

∧

{(x,β):x∈A,β∈Q,(a,x)∈|σ|β}

β≥α}.

Proposition 3.2. ∥ψ∥D,Set(Q),X= (|ψ|α)αis an f-cut in(A,δ)

X_{in the category Set(Q).}

Finally, we will describe an interpretation in the modelD_SetR(

Q).

Definition 3.5(Interpretation in the modelD_SetR(

Q)). An interpretation∥ψ∥=∥ψ∥D,SetR(Q),Xofψwith free variables

1. Letψ≡P(t1, . . . ,tn). Then

|ψ|α={(a,β)∈AX×Q:

∨

x∈An n

∧

i=1

∥ti∥(a,xi)⊗

∨

{γ:(x,β)∈Pγ}

γ≥α}.

2. Letψ≡t1=t2. Then

|ψ|α={(a,β)∈A

X_×Q

: ∨

(x,y)∈A

2

(∥t1∥(a,x)∧ ∥t2∥(a,y))⊗∆_SetR(_Q)

,D((x,y),β)≥α},

where∆SetR(Q),Dis a morphism in SetR(Q), which interprets equality in a model

D_SetR(Q).

3. Letψ ≡σ∇τ, where∇represents logical connectives∧,∨,=⇒,⊗, respectively. Let∥σ∥= (|σ|β)β,∥τ∥=

(|τ|γ)γbe f-cuts in(A,δ)Xin SetR(Q). Then

|ψ|α={(a,ϕ)∈A

X_×Q

:∨ ρ,ε

( ∨

{(γ,ω):(a,ρ)∈|σ|γ,(a,ε)∈|τ|ω}

γ∧ω)⊗ ⊙SetR(Q),D((ρ,ε),ϕ)≥α},

where⊙SetR(Q),Dis a morphism(Q,↔)

2_{→(Q,↔)in SetR(Q), which interprets logical operations∧,∨,→,⊗,}

respectively, in a modelD

SetR(Q).

4. Letψ≡ ¬σ. Let∥σ∥= (|σ|β)β be an f-cut. Then

|ψ|α={(a,ρ)∈A

X_×Q: ∨

{(β,γ):(a,β)∈|σ|γ}

γ⊗ ¬SetR(Q),D(β,ρ)≥α},

where¬SetR(Q),D is a morphism (Q,↔)→(Q,↔)in SetR(Q), which interprets logical negation in a model

D

SetR(Q).

5. Letψ≡(∀x)σ. Then∥σ∥D,SetR(Q),X∪{x}is already defined as an f-cut(|σ|β)β in(A,δ)

X∪{x}_{×(Q,↔)and we}

set

|ψ|α={(a,ρ)∈A

X_×Q

: ∧

{(x,β):x∈A,β∈Q,((a,x),ρ)∈|σ|β}

β≥α}.

Proposition 3.3. ∥ψ∥D,SetR(Q),X= (|ψ|α)αis an f-cut in(A,δ)

X_{in the category SetR(Q).}

4 Relations between interpretations

As we know from Section II, there are some relations between categories Set(Q)and SetR(Q)and between fuzzy objects and f-cuts in these categories. Roughly speaking, fuzzy objects (i.e., special morphisms) and f-cuts in one of these categories represent the same objects. It is then natural to ask a question, if some relationships exist also between interpretations of formulas in modelsE_Set(

Q),ESetR(Q),DSet(Q)andDSetR(Q). In that section we show some

principal relations between these interpretations in the case, that corresponding models are derived from one generic model. In the following definition we use a notation from Lemma 2.1 and Theorem 2.1.

Definition 4.1. We say that modelsE_SetR(Q),DSet(Q) orDSetR(Q) are associated with a modelESet(Q), if the following

hold:

1. ModelE_SetR(Q)is associated withE_Set(Q), if

(a) PE,SetR(Q)=F(PE,Set(Q)), for any P∈

P,

(b) fE,SetR(Q)=F(fE,Set(Q)), for any f∈

(c) ∆SetR(Q),E =F(∆Set(Q),E),

(d) ⊙SetR(Q),E =F(⊙Set(Q),E),

(e) ¬SetR(Q),E =F(¬Set(Q),E).

2. ModelD

Set(Q)is associated withESet(Q), if

(a) PD,Set(Q)=Ψ(A,δ)n(PE,Set(Q)), for any P∈

P_,

(b) fD,Set(Q)=fE,Set(Q), for all f∈

R_,

(c) ∆Set(Q),D=∆Set(Q),E,

(d) ⊙Set(Q),D=⊙Set(Q),E,

(e) ¬Set(Q),D=¬Set(Q),E.

3. ModelD_{SetR(Q)is associated with}E_{Set(Q), if}

(a) PD,SetR(Q)=Ψ(A,δ)

n(F(PE,Set(Q)),

(b) fD,SetR(Q)=F(fE,Set(Q)),

(c) ∆SetR(Q),D=F(∆Set(Q),E),

(d) ⊙SetR(Q),D=F(⊙Set(Q),E),

(e) ¬SetR(Q),D=F(¬Set(Q),E).

Proposition 4.1. Let t be a term and letE

SetR(Q),DSet(Q)andDSetR(Q)be associated with modelESet(Q). Then we have

∥t∥E,Set(Q)=∥t∥D,Set(Q),

∥t∥E,SetR(Q)=∥t∥D,SetR(Q)=F(∥t∥E,Set(Q)).

Theorem 4.1. Letψbe a formula and let modelE_SetR(Q)be associated with modelE_Set(Q).

(i) Letψdoes not contain quantifier∀. Then

∥ψ∥E,SetR(Q)=F(∥ψ∥E,Set(Q)).

(ii) Letψ= (∀x)σ. Then

∥ψ∥E,SetR(Q)≤F(∥ψ∥E,Set(Q)).

Theorem 4.2. Let a model D_{Set(Q) be associated with a model}E_{Set(Q). Let for a formulaψ with free variables}

contained in a set X ,∥ψ∥D,Set(Q)be an f-cut(|ψ|α)α∈Qin the category Set(Q). Then

(∀α∈Q) |ψ|α={a∈AX:∥ψ∥E,Set(Q)(a)≥α}.

An analogical theorem holds for interpretations in the category SetR(Q).

Theorem 4.3. Let modelsD

SetR(Q) andESetR(Q) be associated with a modelESet(Q). Let for a formulaψ with free

variables contained in a set X ,∥ψ∥D,SetR(Q)be an f-cut(|ψ|α)α∈Qin the category SetR(Q). Then

|ψ|α={(a,β)∈A

X_×Q

:∥ψ∥E,SetR(Q)(a,β)≥α},

for allα∈Q,a∈AX.

5 Acknowledgement

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