-Models of a fuzzy logic in two categories of sets with similarity relations are introduced. Interpretations of formulas in these models are defined and some relations between different interpretations are investigated.
k k : A ! Q in a corresponding category, or
(nested) subobjects in A in a corresponding category.
any ”element” a 2 A, k k(a) 2 Q is a degree in which a formula is true in model E , if the value of free variables x in is substituted by elements a 2 A.
of all interpretations in A of free variables, for which a formula is true in a model E in a degree at least .
some fuzzy logic (based on some special deduction rules)
a completeness theorem is true, i.e. `1Q if and only if
Wa2A k k(a) = 1Q for any model E . For more details
concerning fuzzy logic and its models see e.g. [
In the paper we will be interested in constructions of fuzzy
logic models in general settings - in some categories. That
approach enables us to extend significantly a variety of possible
models of fuzzy logic and to create tools for calculating values
k kE depending on models E . An idea to construct a model
of a logic in categories is not new. A comprehensive study
has been done in [
logic based on two different categories Set(Q) and SetS(Q) and for each category we define two different types on formula interpretations and we show some relationships between these interpretations.
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 [
Q iff !Q :
Any classical set A can 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; ), where is a similarity relation. Recall that a similarity relation in A is a map : A A ! such that (a) (b) (c) (8x 2 A) (8x; y 2 A) (8x; y; z 2 A) ized transitivity).
(x; x) = 1, (x; y) = (y; x), (x; y) (y; z) (x; z) (generalA pair (A; ) will be called Q-set. In the paper, we will be working not only with Q-sets, but also with ”mappings” between Q-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 with Q-sets as objects and with differently defined morphisms. A morphism f : (A; ) ! (B; ) in the first category Set(Q) is a map f : A ! B such that (f (x); f (y)) (x; y) for all x; y 2 A. The other category SetS(Q) is an analogy of the category of sets with relations between sets as morphisms. Objects of the category SetS(Q) are the same as in the category Set(Q) and morphisms f : (A; ) ! (B; ) are maps f : A B ! (i.e., Q-valued relations) such that (a) (b) (c) (8x; z 2 A)(8y 2 B) (z; x) f (x; y) (8x 2 A)(8y; z 2 B) f (x; y) (y; z) (8a 2 A) 1 = Wff (x; y) : y 2 Bg. f (z; y), f (x; z).
If f : (A; ) ! (B; ) and g : (B; ) ! (C; !) are two morphisms in SetS(Q), then their composition is a relation g f : A C ! such that _ (f (x; y) y2B 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. Unfortunately, that category is not appropriate for logic interpretation, because of a lack of categorical products, which are important for models constructions.
Lemma II.1 There exists a functor F : Set(Q) ! SetS(Q).
K is a morphism f : (A; ) ! (Q; $) in a category K, where $ is the biresiduation operation in Q defined by $ = ( ! ) ^ ( ! ). Hence, f Set(Q) (A; ), if f : A ! is a map such that (x; y) f (x) $ f (y), or equivalently, f (x) (x; y) f (y) for all x; y 2 A. Analogously, a fuzzy set in the category SetS(Q) is a map f : A Q ! Q which has to satisfy the following conditions: 1) 2) 3) f (x; ) (x; y) f (x; ) ( $ ) Q, 1 = W 2Q f (x; ), for any x 2 A.
f (y; ), for all x; y 2 A; f (x; ), for all x 2 A; ; 2 Q, 2 g f (x; z) = g(y; z)):
_ f (a; b) s(a; ); a2A
K an object function of a functor, for K = Set(Q) or SetS(Q). In fact, there exists a functor FK : K ! Set that is defined by FK(A; ) = fs : s (A; )g. If f : (A; ) ! (B; )
K is a morphism in K, then the map FK(f ) : FK(A; ) ! FK(B; ) is defined differently for categories K = Set(Q) and K = SetS(Q). We have,
FSet(Q)(f )(s)(b) =
(b; f (x));
_ s(x)
x2A
(A; ) (see [
Set(Q) (8b 2 B)(8
2 Q)FSetS(Q)(f )(s)(b; ) = for all b 2 B;
: FSet(Q) ! FSetS(Q) F:
morphisms in category Set(Q) or SetS(Q), respectively. On the other hand, any such maps can be extended to morphisms, according the following methods.
Lemma II.2 Let (A; ); (B; ) be Q-sets and let g : A B ! Q be a map, such that 1 = W g~ : A B ! Q be defined by bt2hBe gfo(ram; bu)la, for any a 2 A. Let g~(a; b) = _ _ g(x; y)
Lemma II.3 Let (A; ) be an Q-set and let s : A ! Q be a map. Then we define a map s : A ! Q such that s(a) = Wx2A (a; x) s(x) for all a 2bA. Then b s : (A; ) ! (Q; $) is a morphism in Set(Q), b If s : (A; ) ! (Q; $) is a morphism in Set(Q) then s = s, b sb = Vft : t is a morphism (A; ) ! (Q; $); t s in Set(Q)g:
It is well known that any classical fuzzy set (with values in
a residuated lattice Q) in a set A can be alternatively expressed
as a system of -cuts C = (C ) , where C is a nested system
of subsets of A. In our previous papers ([
Definition II.1 Let (A; ) be a Q-set. Then a system C = (C ) 2Q of subsets in A is called an f-cut in (A; ) in the category Set(Q) if 1) 2) 1) 2) 3) 4) 5) 1) 2) 3) 4)
1) 2) 3) 4)
Proposition II.3 Let (A; ) be a Q-set and let (C ) be a system of subsets in a set A Q, such that 1 = Wf( ; ):(a; )2C g , for all a 2 A. For any 2 Q we set f(a; ) 2 A
Q :
A Q is 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 II.2 Let (A; ) be a Q-set and let (C ) be a system of subsets in a set A. For any 2 Q we set C = fa 2 A :
_ f(x; ):x2C g
C C for all 2 Q, (C ) is an f-cut system in (A; ) in the category Set(Q),
Set(Q), then C = C for all 2 Q.
D for all 2 Q, then C D for all 2 Q. C C for all 2 Q, (C ) is an f-cut in (A; ) in the category SetS(Q),
then C = C for all 2 Q.
D for all 2 Q, then C D for all 2 Q.
K ! Set, such that CK(A; ) is the set of all f-cuts in (A; )
in a category K = Set(Q); SetS(Q) (for details see [
Theorem II.1 For a category K = Set(Q); SetS(Q), there exists a natural isomorphism
It means, especially, that for any Q-set (A; ) and any category K = Set(Q); SetS(Q), there exists a bijection K;(A; ) : CK(A; ) ! FK(A; ).
: CSet(Q) ! CSetS(Q)
F: We show only how that transformation is defined. Let (A; ) be a Q-set and let C = (C ) 2 CSet(Q)(A; ) be an f-cut in (A; ) in the category Set(Q). Then (A; )(C) = (D ) , where D = f(a; ) : Wa2C $ g.
III.
pretation of a fuzzy logic in models based on Q-sets (see [
Let K be a category with Q-sets as objects and such that for any set of objects f(Ai; i) : i 2 Ig there exists a product (A; ) = Q i 2 Ig in a ic2aIt(eAgoir;yi)K. Risecaanll otbhjaetcta (pAr;od)ucwt iothf mf(oArpi;hisim)s: pri : (A; ) ! (Ai; i) such that for any other object (B; ) with morphisms qi : (B; ) ! (Ai; i) there exists the unique morphism q = Qi qi such that the diagram commutes: (A; )
? pri?
y (Ai; i) qi (A; ) (B; ): x ?q=Qi qi ? Recall, how a product is constructed in our categories Set(Q) and SetS(Q). Let (Ai; i) be Q-sets, i 2 I. Let us consider the category Set(Q), firstly. Then Q ) = (AI ; I ), : wVhere AI is the cartesian product of is2eIts(AAii; aind I (a; b) = i2I i(ai; bi), for any a; b 2 A. The projection morphisms pri : (AI ; I ) ! (Ai; i) are classical projection maps and if qi : (B; ) ! (Ai; i) are morphisms, for i 2 I, the unique morphism Qi qi is such that Qi qi(b) = (qi(b))i 2 AI , for any b 2 B.
Now, let us consider the category SetS(Q). A product (AI ; I ) is the same object as in the category Set(Q), but with different projection morphisms pri defined such that pri : (AI ; I ) ! (Ai; i) in the category SetS(Q), i.e. pri : AI Ai ! Q, such that pri(a; b) = i(ai; b), for a 2 AI ; b 2 Ai. If qi : (B; ) ! (Ai; i) are morphisms, for i 2 I, the unique morphism Qi qi is such that Qi qi : B AI ! Q, Qi qi(b; a) = Vi qi(b; ai). If (Ai; i) = (A; ) for every i = 1; : : : ; n, then Qi(Ai; i) will be denoted by (An; n).
We now introduce two types of models of a fuzzy logic in a category K = Set(Q) or SetS(Q). where (1) (2) (3) where (1) (2) (3) (a) (b) 1) 2) 1) 2)
(A; ) is an Q-set from a category K, PE;K is a fuzzy set in (A; )n in a category K, i.e., a morphism (A; )n ! (Q; $), fE;K : (A; )n ! (A; ) is a morphism in a category K.
Definition III.2 A cut model of a language J in a category K is
DK = ((A; ); fPD;K : P 2 Pg; ffD;K : f 2 Rg); (A; ) is an Q-set from a category K, PD;K is an f-cut in (A; )n in a category K, PD;K = (P ) , fD;K : (A; )n ! (A; ) is a morphism in a category K.
variables, then an interpretation k k of will be different for different types of a model.
in a category K,
a category K.
If k k is a fuzzy set in (A; )X , it means that for any a = (ax)x2X 2 AX , k k(a) 2 Q is a degree in which a formula is true in model E , if the value of a variable x is substituted by element ax 2 A. If k k is an f-cut (j j ) in (A; )X , then j j is a set of all interpretations (in A) of free variables from
degree at least .
Definition III.1 A fuzzy set model of a language J in a category K is
EK = ((A; ); fPE;K : P 2 Pg; ffE;K : f 2 Rg);
models and different categories K = Set(Q); SetS(Q). Definitions will be done by the induction principle depending on a structure of .
Now we present these definitions of k k 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. Let G = EK or DK be a model of a language J in a category K. An interpretation of a term with a set of variables contained in a set X is a morphism ktkG;K : (A; )X ! (A; ) in a category K, defined as follows:
Let t = x, where x 2 X. Then ktkG;K := prx : (A; )X ! (A; ) is the x-projection morphism in the category K.
Let t = f (t1; : : : ; tn). Then ktkG;K is a composition (in K) of morphisms (A; )X
Qi ktikG;!K (A; )n fG;K! (A; ): For example, for K = SetS(Q) we have ktkG;SetS(Q)(a; b) = Wx2An Vin=1 ktik(a; xi) fG (x; b), where a 2 AX ; b 2 A. Definition III.3 (Interpretation in models ESet(Q); ESetS(Q)) Let K be the category Set(Q) or SetS(Q), respectively. An interpretation k k = k kE;K;X of with free variables in a set X in a category K is defined as follows.
1) 2) 3) 4) 5)
Let P (t1; : : : ; tn). Then k kE;K is defined as the composition of the following morphisms in K: (A; )X
Qi ktikE;!K (A; )n
PE;K ! (Q; $): Let t1 = t2. Then k kE;K is the composition of the following morphisms in K: (A; )X
kt1kE;K kt2kE;!K (A; )2 (A; )2
K;E
! (Q; $); where K;E is a morphism in K, which interprets equality in a model EK.
Let r , where r represents logical connectives ^; _; =) ; , respectively. Then k kE;K is the composition of the following morphisms: (A; )X
k kE;K k kE;!K (Q; $)2 (Q; $)2
K;E
! (Q; $); where K;E is a morphism in K, which interprets logical operations ^; _; !; , respectively, in a model EK.
Let : . Then k kE;K is the composition of the following morphisms: (A; )X k kE;!K (Q; $) :K;E ! (Q; $); where :K;E is a morphism in K, which interprets logical negation in a model EK.
Let (8x) . Then k kE;K;X[fxg is already defined as a morphism (A; )X (A; ) ! (Q; $) in K. Then we set k kE;K;X (a) =
k kE;K;X[fxg(a; x): ^ x2A
of logical connectives in categories Set(Q) and SetS(Q), presented in the definition.
Example III.1
_ x;y2A
the morphism b : (A; )2 ! (Q; $) (see Lemma I.3), i.e.
b(a; b) = (x; y) ( (a; x) ^ (b; y)): 2) A morphism SetS(Q);E can be also defined as a map A2 Q ! Q by using results from Lemmas I.1-I.3 in several natural ways, i.e.: (i)
SetS(Q);E ((a; b); ) = F (b)((a; b); ), 3) ^Set(Q);E which interprets logical connective ^ can be ^Q. In fact, the following holds for any a; a0; b; b0 2 A: (a ^Q a0) $ (b ^Q b0) (a $ b) ^Q (a0 $ b0); which represents a fact that ^Q : (Q; $)2 ! (Q; $) is a morphism in Set(Q).
4) ^SetS(Q);E can be defined by ^SetS(Q);E (( ; ); ) = $ ( ^ ). In fact, ^SetS(Q);E : (Q; $)2 ! (Q; $) is a morphism in SetS(Q), as it can be proved by a simple calculation.
5) On the other hand, the operation !Q is not a morphism in Set(Q) and a reasonable candidate for interpretation !Set(Q);E of a logical implication in Set(Q) can be then !. c 6) Analogously, !SetS(Q);E can be defined naturally in two different ways: (i) (ii) !SetS(Q);E = F (!^Set(Q);E ), !SetS(Q);E = F (!\Set(Q);E ).
(Q; $), such that :Set(Q);E ( ) = :b( ) = W 2Q( ! 0Q) ( $ ).
Proposition III.1 k kE;K;X is a fuzzy set in (A; )X in any category K = Set(Q); SetS(Q).
Definition III.4 (Interpretation in model DSet(Q)) An interpretation k k = k kD;Set(Q);X of with free variables in a set X in the category Set(Q) is an f-cut (j j ) in (A; )X defined as follows (instead of ktkD;Set(Q) we write ktk, for any term t).
P (t1; : : : ; tn). Then t1 = t2. Then j j
= fa 2 AX : (kt1k(a); : : : ; ktnk(a)) 2 P g: = fa 2 AX :
Set(Q);D(kt1k(a); kt2k(a)) g; where Set(Q);D is a morphism in Set(Q), which interprets equality in a model DSet(Q).
Let r , where r represents logical connectives ^; _; =) ; , respectively. Let k k = (j j ) ; k k = (j j ) be f-cuts in (A; )X . Then j j = fa 2 AX : ( _ ) Set(Q);D( g; a2j j
_ a2j j where Set(Q);D is a morphism (Q; $)2 ! (Q; $) in Set(Q), which interprets logical operations ^; _; ! ; , respectively, in a model DSet(Q).
Let : . Let k k = (j j ) be an f-cut. Then j j = fa 2 AX : :Set(Q);D( ) g; _ where :Set(Q);D is a morphism in Set(Q), which interprets logical negation in a model DSet(Q). 1) 2) 3) 4) j j
Definition III.5 (Interpretation in model DSetS(Q)) An interpretation k k = k kD;SetS(Q);X of with free variables in a set X in the category SetS(Q) is an f-cut (j j ) in (A; )X in the category SetS(Q), defined as follows (instead of ktkD;SetS(Q) we write ktk for any term t).
1)
Let
P (t1; : : : ; tn). Then 2)
Let
t1 = t2. Then _( j j = f(a; ') 2 AX
_ ;" f( ;!):(a; )2j j ;(a;")2j j!g
SetS(Q);D(( ; "); ') j j _ f( ; ):(a; )2j j g where SetS(Q);D is a morphism (Q; $)2 ! (Q; $ ) in SetS(Q), which interprets logical operations ^; _; !; , respectively, in a model DSetS(Q).
Let : . Let k k = (j j ) be an f-cut. Then g; where :SetS(Q);D is a morphism (Q; $) ! (Q; $) in
DSetS(Q).
Let (8x) . Then k kD;SetS(Q);X[fxg is already defined as an f-cut (j j ) in (A; )X[fxg (Q; $) and we set j j = f(a; ) 2 AX ^
Q : g: f(x; ):x2A; 2Q;((a;x); )2j j g j j where SetS(Q);D is a morphism in SetS(Q), which interprets equality in a model DSetS(Q).
Let r , where r represents logical connectives ^; _; =) ; , breespfe-cctuitvselyin. L(eAt ; k)Xk i=n (j j ) ; k k = (j j )
3) 4) 5) Q : Q : g; Q : g; ^ !) Proposition III.3 k kD;SetS(Q);X (A; )X in the category SetS(Q). = (j j ) is an f-cut in
between categories Set(Q) and SetS(Q) and between fuzzy sets and f-cuts in these categories. Roughly speaking, fuzzy sets and f-cuts in one of these categories represent the same objects. It is then natural to ask a question, if some relations exist also between interpretations of formulas in models ESet(Q); ESetS(Q); DSet(Q) and DSetS(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
Definition IV.1 We say that models ESetS(Q); DSet(Q) or DSetS(Q) are associated with a model ESet(Q), if the following hold: 1) 2) 3)
Model ESetS(Q) is associated with ESet(Q), if a) PE;SetS(Q) = F (PE;Set(Q)), for any P 2 P , b) fE;SetS(Q) = F (fE;Set(Q)), for any f 2 R, c) SetS(Q);E = F ( Set(Q);E ), d) SetS(Q);E = F ( Set(Q);E ), e) :SetS(Q);E = F (:Set(Q);E ).
Model DSet(Q) is associated with ESet(Q), if a)
(A; )n (PE;Set(Q)), for any P 2 PD;Set(Q) =
P , b) fD;Set(Q) = fE;Set(Q), for all f 2 R, c) Set(Q);D = Set(Q);E , d) Set(Q);D = Set(Q);E , e) :Set(Q);D = :Set(Q);E .
Model DSetS(Q) is associated with ESet(Q), if a) b) c) d) e)
PD;SetS(Q) = (A; )n (F (PE;Set(Q)), fD;SetS(Q) = F (fE;Set(Q)),
SetS(Q);D = F ( Set(Q);E ),
SetS(Q);D = F ( Set(Q);E ), :SetS(Q);D = F (:Set(Q);E ).
Proposition IV.1 Let t be a term and let ESetS(Q), DSet(Q) and DSetS(Q) be associated with model ESet(Q). Then we have ktkE;Set(Q) = ktkD;Set(Q); ktkE;SetS(Q) = ktkD;SetS(Q) = F (ktkE;Set(Q)): Theorem IV.1 Let be a formula and let model ESetS(Q) be associated with model ESet(Q).
(i) (ii) does not contain quantifier 8. Then
k kE;SetS(Q) = F (k kE;Set(Q)): = (8x) . Then k kE;SetS(Q)
F (k kE;Set(Q)): Theorem IV.2 Let a model DSet(Q) be associated with a model ESet(Q). Let for a formula with free variables contained in a set X , k kD;Set(Q) be an f-cut (j j ) 2Q in the category
category SetS(Q).
Theorem IV.3 Let models DSetS(Q) and ESetS(Q) be associated with a model ESet(Q). Let for a formula with free variables contained in a set X , k kD;SetS(Q) be an f-cut (j j ) 2Q in the category SetS(Q). Then
j j for all