Standard GÑdel Modal Logics "
Standard Gödel Modal Logics Xavier Caicedo
Ricardo O. Rodríguez
October 23, 2009
Abstract We prove strong completeness of the -version and the 3-version of a Gödel modal logic based on Kripke models where propositions at each world and the accessibility relation are both in…nitely valued in the standard Gödel algebra [0,1]. Some asymmetries are revealed: validity in the …rst logic is reducible to the class of frames having twovalued accessibility relation and this logic does not enjoy the …nite model property, while validity in the second logic requires truly fuzzy accessibility relations and it has the …nite model property. Analogues of the classical modal systems D, T, S4 and S5 are considered also, and the completeness results are extended to languages enriched with a discrete well ordered set of truth constants.
1
Introduction
Sometimes it is needed in approximate reasoning to deal simultaneously with both fuzziness of propositions and modalities, for instance one may try to assign a degree of truth to propositions like “John is possibly tall”or “John is necessarily tall”, where “John is tall”is presented as a fuzzy proposition. Fuzzy logic should be a suitable tool to model not only vagueness but also these and other kinds of information features like certainty, belief or similarity, which have a natural interpretation in terms of modalities. To achieve this end, it is natural to interpret fuzzy modal operators by means of Kripke models over fuzzy frames. We address in this paper the case of the pure modal operators (necessitation and possibility) for standard Gödel logic, one of the main systems of fuzzy logic arising from Hájek´s classi…cation in [16]. For this purpose we consider a many-valued version of Kripke semantics for modal logic where Preprint
1
both propositions at each world and the accessibility relation are in…nitely valued in the standard Gödel algebra [0,1]. We provide strongly complete axiomatizations for the -fragment and the 3-fragment of the resulting minimal logic. These fragments are shown to behave quite asymmetrically. Validity in the …rst one is univocally determined by the class of frames having a crisp (that is, two-valued) accessibility relation, while validity in the second requires truly fuzzy frames. In addition, the -fragment does not enjoy the …nite model property with respect to the number of worlds or the number of truth values while the 3-fragment does. We consider also the Gödel analogues of the classical modal systems D, T, S4 and S5 for each modal operator and show that the …rst three are characterized by the many-valued versions of the frame properties which characterize their classical counterparts. Finally, we extend the strong completeness results to Pavelka-style languages enriched with a set of explicit truth constants denoting a discrete well ordered set of truth values. Our approach is related to Fitting´s [10] who considers Kripke models taking values in a …xed …nite Heyting algebra; however, Fitting´s proof systems and completeness proofs depend essentially on …niteness of the algebra and the fact that his languages contain constants for all the truth values of the algebra. We must relay on completely di¤erent methods. Modal logics with an (intermediate) intuitionistic basis and Kripke style semantics have been investigated in a number of relevant papers (see Ono [18], Fischer Servi [8], Bo¼ szic and Do¼ sen [6], Font [11], Wolter [20], from an extensive literature), but in all cases the models carry two or more crisp accessibility relations satisfying some commuting properties: a pre-order to account for the intuitionistic connectives and one or more binary relations to account for the modal operators. Our semantics has, instead, a single fuzzy accessibility relation and does not seem reducible to those multi-relational semantics since the latter enjoy the …nite model property for (cf. Grefe [15]). Recently, Metcalfe and Olivetti [13] have given a proof of weak completeness of a calculus of sequents of relations for the -fragment of our logic, showing that it is decidable and PSPACE complete. The decidability of the 3-fragment follows from the …nite model property we prove later. Bou, Esteva, and Godo survey in [4] modal logics with analogue [0,1]valued Kripke semantics under di¤erent choices of the t-norm. However, our methods and results do not generalize to the corresponding modal versions of ×ukasiewicz or product logics because they relay on the richness of endomorphisms of the Gödel algebra [0,1]. In fact, we do not know any 2
completeness result for these logics without extra conditions on the frames, strong completeness being known to be untenable. Classical modal logics are inter-translatable with description logics [1]. Our Gödel Kripke semantics for the modal operators is similarly translatable into fuzzy (Gödel) description logic (cf. [17]), and thus our results throw light on various fragments of this logic and certain Pavelka-style expansions of them. We assume the reader is acquainted with modal and Gödel logics and the basic laws of Heyting algebras (cf. Chagrov and Zakharyaschev [5]).
2
Gödel-Kripke models
The language L 3 of propositional Gödel modal logic is built from a set V ar of propositional variables, logical connectives symbols ^; !; ?; and the modal operator symbols and 3. Other connectives are de…ned: > := ' ! ' :' := ' ! ? ' _ := ((' ! ) ! ) ^ (( ! ') ! ') ' $ := (' ! ) ^ ( ! '): L and L3 will denote, respectively, the -fragment and the 3-fragment of the language. As stated before, the semantics of Gödel modal logic will be based on fuzzy Kripke models where the valuations at each world and also the accessibility relation between worlds are [0; 1]-valued. The symbols and ) will denote the Gödel t-norm in [0; 1] and its residuum, respectively: a b = minfa; bg;
a)b=
1; if a b b; otherwise
The maximum is de…nable: maxfa; bg = ((a ) b) ) b) ((b ) a) ) a); and the pseudo-complement is denoted a := a ) 0. This yields the standard Gödel algebra; that is, the unique Heyting algebra structure in the linearly ordered interval. De…nition 2.1 A Gödel-Kripke model (GK-model) will be a structure M = hW; S; ei where: W is a non-empty set of objects that we call worlds of M: S : W W ! [0; 1] is an arbitrary function (x; y) 7 ! Sxy. 3
e:W
V ar ! [0; 1] is an arbitrary function (x; p) 7 ! e(x; p).
The evaluations e(x; ) : V ar ! [0; 1] are extended simultaneously to all formula in L 3 by de…ning inductively at each world x: e(x; ?) := 0 e(x; ' ^ ) := e(x; ') e(x; ) e(x; ' ! ) := e(x; ') ) e(x; ) e(x; 2') := inf y2W fSxy ) e(y; ')g e(x; 3') := supy2W fSxy e(y; ')g. It follows that e(x; ' _ ) = maxfe(x; '); e(x; )g and e(x; :') =
e(x; '):
The notions of a formula ' being true at a world x, valid in a model M = hW; S; ei; or universally valid, are the usual ones: ' is true in M at x, written M j=x '; i¤ e(x; ') = 1. ' is valid in M , written M j= '; i¤ M j=x ' at any world x of M: ' is GK-valid, written j=GK ', if it is valid in all the GK-models. Clearly, all valid schemas of Gödel logic are GK-valid. In addition, Proposition 2.1 K Z K3 Z3 F3
The following modal schemas are GK-valid:
(' ! ) ! ( ' ! ) :: ! :: 3(' _ ) ! (3' _ 3 ) (in fact, an equivalence) 3::' ! ::3' :3?
Proof: Let M = hW; S; ei be an arbitrary GK-model and x 2 W: (K ) By De…nition 2.1 and properties of the residuum we have for any y 2 W : e(x; (' ! )) e(x; ') (Sxy ) (e(y; ') ) e(y; )) (Sxy ) e(y; ')) (Sxy ) e(y; )): Taking the meet over y in the last expression: e(x; (' ! )) e(x; ') e(x; ); hence, e(x; (' ! )) e(x; ' ! ). (Z ) Utilizing the Heyting algebra identity: (x ) y) = (x ) y); we have: e(x; ::2 ) = e(x; 2 ) (Sxy ) e(y; )) = (Sxy ) e(y; )) = (Sxy ) e(y; :: )): Taking the meet over y, e(x; ::2 ) e(x; 2:: ): (K3 ) By properties of suprema and distributivity of over max; e(3(' _ )) = supy fSxy maxfe(y; '); e(y; )gg = maxfsupy fSxy e(y; ')g; supy fSxy e(y; )gg: 4
(Z3 ) Sxy e(y; ::') e(x; 3') = e(x; ::3'):
Sxy
e(y; ') =
(Sxy e(y; '))
(F3 ) e(x; 3?) = supy fSxy 0g = 0: The Modus Ponens rule preserves truth at every world of any GK-model. On the other hand, the classical introduction rules for the modal operators RN :
' '
RN3 :
'! 3' ! 3
:
do not preserve truth at a …xed world. However, Proposition 2.2 RN and RN3 preserve validity in any given model, thus they preserve GK-validity. Proof: Let hW; S; ei be a GK-model: (RN ) If e(x; ') = 1 for all x 2 W then e(x; ') = inf y fSxy ) e(y; ')g = inff1g = 1 for all x. (RN3 ) If e(x; ' ! ) = 1 for all x 2 W then Sxy e(y; ') Sxy e(y; ) e(x; 3 ) for any y 2 W Taking the join over y in the left hand side of the last inequality, e(x; 3') e(x; 3 ): Semantic consequence is de…ned for any theory T
L
3;
as follows:
De…nition 2.2 T j=GK ' if and only if for any GK-model M and any world x of M; M j=x T implies M j=x ': Note that Modus Ponens preserves consequence but this is not the case of the inference rules RN and RN3 : An alternative notion of logical consequence arises naturally. Set e(x; T ) = fe(x; ') : ' 2 T g then: De…nition 2.3 T j= GK ' if and only if for any GK-model M and any world x in M; inf e(x; T ) e(x; '): Clearly, j= GK implies j=GK , and it will follow from our completeness theorems that both notions are equivalent for countable theories. This fact has been already observed for pure Gödel logic by Baaz and Zach in [2].
5
3
On strong completeness of Gödel logic
To prove strong completeness of the modal fragments L and L3 we will reduce the problem to pure Gödel propositional logic. In the rest of this paper L(X) will denote the Gödel language built from a set of propositional variables X and the connectives ^; !; ?: Let G be a …xed axiomatic calculus for Gödel logic (also called Dummett’s LC), say the following one given by Hájek ([16], Def. 4.2.3.): (' ! ) ! (( ! ) ! (' ! )) (' ^ ) ! ' (' ^ ) ! ( ^ ') (' ! ( ! )) ! ((' ^ ) ! ) ((' ^ ) ! ) ! (' ! ( ! )) ' ! (' ^ ') ((' ! ) ! ) ! ((( ! ') ! ) ! ) ?!' MP: From ' and ' ! , infer
` will denote deductive inference in this calculus. It is well known that G is deductively equivalent to the intermediate logic obtained by adding to Heyting calculus the pre-linearity schema: (' ! ) _ ( ! '). Given a valuation v : X ! [0; 1]; let v denote the extension of v to L(X) according to the Gödel interpretation of the connectives. We will need the following strong form of standard completeness for Gödel logic: Proposition 3.1 Let T be a countable theory and U a countable W set of formulas of L(X) such that for every …nite S U we have T 0 S then there is a valuation v : X ! [0; 1] such that v( ) = 1 for all 2 T and v( ) < 1 for each 2 U:
Proof: Extend T to a prime theory T 0 (that is, T 0 ` _ implies T 0 ` or T 0 ` ) satisfying the same hypothesis with respect to U (this is standard): The Lindenbaum algebra L(X)= T 0 of T 0 is linearly ordered since by primality and the pre-linearity schema T 0 ` ! or T 0 ` ! . Moreover, the valuation v : X ! L(X)=T 0 ; v(x) = x= T 0 is such that v(T ) = 1; v( ) < 1 for all 2 U: As T 0 is countable we may assume X is countable and thus, being also countable, L(X)= T 0 is embeddable in the Gödel algebra [0; 1]; therefore, we may assume v : X ! [0; 1]. Taking S = f'g we obtain the usual formulation completeness. We can not expect strong standard completeness of G for uncountable theories, as the following example illustrates. 6
Example. Set T = f(p ! p ) ! q : < < ! 1 g where ! 1 is the …rst uncountable cardinal, then T 0 q: Otherwise we would have ` q; for some …nite = f(p i+1 ! p i ) ! q : 1 i < ng; but this is not possible by 1 soundness of G; because the valuation v(q) = 21 , v(p i ) = 12 (1 i+1 ) for 1 1 i n; makes v(p i ) < v(p i+1 ) < 2 and thus v((p i+1 ! p i ) ! q) = 1 for 1 i < n; while v(q) < 1: However, there is no valuation v such that v(T ) = 1 and v(q) < 1, because that would imply v(p ! p ) < 1 for all < < ! 1 ; and thus the set fv(p ) : < ! 1 g would be ordered in type ! 1 ; which is impossible because any well ordered subset of ([0; 1]; 0 implies v(::2 ) = 1; and thus v(2:: ) = 1 since v satis…es axiom Z : This implies that T';v 0G ': Otherwise, 1 ; :::; k `G ' for some i 2 T';v and thus 2 1 ; :::; 2 k `G 2' by Lemma 4.1. Hence, by Lemma 4.2 and the previous observations, < minf2
1 ; :::; 2 k g
v(2');
a contradiction. By Lemma 4.3 we have T';v [ T G 0 ' and by the countability of T';v [ T G we may use the completeness theorem of Gödel logic (Proposition 3.1) to get a Gödel valuation u : V ar [ L ! [0; 1] such that u(T';v ) = 1 and u(') < 1: Then u 2 M and (i) holds by construction. Moreover, (ii) is satis…ed because u(:: ) = 1 and thus u( ) > 0 if v(2 ) > 0: This ends the proof of the claim. 9
Pick now an strictly increasing function g : [0; 1] ! [0; 1] such that g(1) = 1; g(0) = 0; and g[(0; 1)] = ( ;
+ ):
As g is a homomorphism of Heyting algebras, the valuation w = g u preserves the value 1 of the formulas in T G and thus it belongs to M . Moreover, v(2 ) w( ) for all : - if v(2 ) > because w( ) = g(u( )) = g(1) = 1 by (i) above. - if 0 < v(2 ) because then 0 < u( ) 1 by (ii) above, and thus w( ) = g(u( )) 2 ( ; + ) [ f1g: This means S vw = 1, and since u(') < 1 we have, w(') = g(u(')) < + ; which shows (1). De…nition 4.1 Call a GK-model accessibility crisp (a-crisp in short) if S : W W ! f0; 1g; and write T j=Crisp ' if the consequence relation holds at each node of any a-crisp GK-model. Theorem 4.1 For any countable theory T and formula ' in L the following are equivalent: (i) T `G ' (ii) T j= GK ' (iii) T j=GK ' (iv) T j=Crisp ': Proof: By Lemma 4.2, it is enough to show (iv) ) (i). If T 0G ' then T [ T G 0 ' by Lemma 4.3, and by strong completeness of Gödel logic there is a valuation v : V ar [ L ! [0; 1] such that v(T ) = v(T G ) = 1 and v(') < 1: Hence, v 2 W by de…nition, e (v; T ) = v(T ) = 1; and e (v; ') = v(') < 1 by Lemma 4.4, showing that M j=v T but M 2v ': That is, T 2Crisp ' because the canonical model is a-crisp. After the example in Section 3 we can not expect completeness with respect to uncountable theories.
5
G does not have the …nite model property
The following example shows that G does not have the …nite model property with respect to GK-models. The reciprocal of axiom Z : :: ! :: 10
;
fails in the (a-crisp) model M = (N; S; e); where Smn = 1 for all m; n;
e(n; p) =
1 n+1
for all n:
1 Indeed, e(n; ::p) = n+1 = 1 for all n and thus, e(0; 2::p) = inff1g = 1: 1 g = 0; and thus e(0; ::2p) = 0: However, But e(0; 2p) = inf n2N f1 ) n+1
Proposition 5.1 2:: ! ::2 is valid in any GK-model hW; S; ei with …nite W . Proof: Assume e(x; 2:: ) > e(x; ::2 ) then e(x; 2 ) < 1 and thus e(x; 2 ) = 0. This implies the existence of a sequence fyn gn W such that Sxyn > e(yn ; ) for all n 2 N and fe(yn ; )gn converges to 0: If W is …nite so is the range of the latter sequence and there must exist n such that e(yn ; ) = 0: Then (Sxyn ) e(yn ; :: )) = (Sxyn ) 0) = 0 and thus e(x; 2:: ) = 0; a contradiction. Remark. The proof of the proposition shows that 2:: ! ::2 is valid in all 0-witnessed GK-models, those where e(x; 2 ) = 0 implies the existence of y such that e(y; ) = 0 < Sxy: In fact, G + f2:: ! ::2 g is strongly complete for 0-witnessed (a-crisp) models. To see this, notice that if any world v of the canonical model is asked to satisfy the new schema, M becomes 0-witnessed because then v( ) = 0 implies v( :: ) = 0, and thus by the last line in the proof of Lemma 4.4 there is w such that S vw = 1 and w(:: ) < "; hence, w( ) = 0: We do not know if this system has the …nite model property.
6
Completeness of the 3-fragment
The system G3 results by adding to G the following axiom schemas and rule in the language L3 : K3 : 3(' _ ) ! (3' _ 3 ) Z3 : 3::' ! ::3' F3 : :3? RN3 : From ' ! infer 3' ! 3 As in the case of the -fragment, in proofs with assumptions the rule RN3 is to be used in theorems only, and thus we have the deduction theorem DT, hence the rule: Lemma 6.1 If ' `G3
then 3' `G3 3 : 11
Also the soundness theorem: Lemma 6.2 T `G3 ' implies T j=
GK
', hence, T j=GK ':
Moreover, if T G3 is the set of theorems of G3 then: Lemma 6.3 T `G3 ' if and only if T [ T G3 ` ' in Gödel logic. Let 3L3 = f3 : 2 L3 g, then the canonical model M3 = (W ; S ; e ) is de…ned as follows: W is the set of valuations v : V ar [ 3L3 ! [0; 1] such that v(T G3 ) = 1 and its positive values have a positive lower bound: inf fv( ) : v( ) > 0g = 2L3
>0
(2)
when v is extended to L3 = L(V ar [ 3L3 ) as a Gödel valuation. The fuzzy relation between worlds in M3 is given by S vw := inf fw( ) ) v(3 )g: '2L3
e (v; p) := v(p) for any p 2 V ar. Lemma 6.4 For any world v in the canonical model M3 and any ' 2 L3 we have e (v; ') = v('): Proof: The only non trivial step in a proof by induction on complexity of formulas of L3 is that of 3: By induction hypothesis, e (v; 3') = supw fS vw e (w; ')g = supw fS vw w(')g; then we must show supw fS vw w(')g = v(3'): By de…nition S vw
w(') ) v(3');
for any ' 2 L3 and w 2 W ; then S vw w(') v(3'); which yields taking the join over w: e (v; 3') v(3'): The other inequality is trivial if v(3') = 0: For the case v(3') > 0, let w be given as in the following claim then v(3') = = S vw w(') e (v; 3'); concluding the proof of the lemma. Claim. If v is a world of M3 such that v(3') = world w of M3 such that w(') = 1 and S vw = : 12
> 0; there exists a
Proof: Assume v(3') = ';v
> 0 and de…ne
= f 2 L3 : v(3 ) < g [ f:: :
2 L3 , v(3 ) = 0g:
This set is not empty because v(3?) = 0 by axiom F3 : Moreover, for any …nite subset of ';v ; say f 1 ; :::; n g [ f:: 1 ; :::; :: m g; we have ' 0G3
1
_ ::: _
n
_ ::
1
_ ::: _ ::
m:
Otherwise, we would have (Cf. Lemma 6.1) 3' `G3
3( 1 _ ::: _ n _ :: 1 _ ::: _ :: m ) 3 1 _ ::: _ 3 n _ 3:: 1 _ ::: _ 3:: 3 1 _ ::: _ 3 n _ ::3 1 _ ::: _ ::3
RN3 K3 Z3 ;
m m
which would imply by Lemma 6.2 v(3')
max(fv(3 i ) : 1
i
ng [ fv(::3 i ) : 1
i
mg) < ;
a contradiction. Therefore, we have by Lemma 6.3 T G3 ; ' 0
1
_ ::: _
n
_ ::
1
_ ::: _ ::
m;
By Proposition 3.1 there is a Heyting algebra valuation u : V ar [ 3L3 ! [0; 1] such that u(') = u(T G3 ) = 1 and u( ) < 1 for all satis…es the further conditions: (i) u(') = 1 (ii) u( ) < 1 if v(3 ) < ; because then 2 ';v (iii) u( ) = 0 if v(3 ) = 0, because then :: 2 which implies u( ) = 0:
';v
2
';v .
Thus, u
and so u(:: ) < 1
Let g : [0; 1] ! [0; 1] be the strictly increasing function: 8 1 if x = 1 < (x + 1)=2 if 0 < x < 1 g(x) = : 0 if x = 0
where is given by (2). Clearly the valuation w = g u inherits the properties (i), (ii) (iii) of u, with (ii) in the stronger form: (ii0 ) w( )
=2; by construction, and w(T G3 ) = 1 because g is a homomorphism of Heyting algebras, hence, w belongs to M3 : To see that S vw = ; note that w( ) v(3 ) whenever v(3 ) < . If 13
0 < v(3 ) because then w( ) < v(3 ) by (ii0 ) and de…nition of . If v(3 ) = 0 because then w( ) = 0 by (iii). Since (w( ) ) v(3 )) for v(3 ) , and (w(') ) v(3')) = (1 ) ) = , we have S vw = inf '2L3 fw(') ) v(3')g = : Theorem 6.1 For any countable theory T and formula ' in L3 , the following are equivalent (i) T `G3 ' (ii) T j= GK ' (iii) T j=GK ': Proof: Assume T 0G3 ', then T [ T G3 0 ': By the strong completeness of Gödel logic, there is a Heyting algebra valuation v such that v(T [ T G3 ) = 1 and v(') < 1: Since v might not be a world in M3 compose it with the Heyting algebra homomorphism: g(x) = (x + 1)=2 for x > 0; g(0) = 0: Then v 0 = g v belongs to M3 and we still have v 0 (T ) = 1, v 0 (') < 1: Applying Lemma 6.4 to v 0 we have e (v 0 ; T ) = 1; e (v 0 ; ') < 1: That is, M3 j=v0 T and M3 2v0 ': Hence, T 2GK ': By Lemma 6.2 this is enough. j=GK no longer coincides with j=Crisp for the language L3 as the following example illustrates. Example. The schema ::3' ! 3::' is not a theorem of G3 because it fails in the two worlds model hfa; bg; S; ei where Sab = 21 ; e(a; p) = e(b; p) = 1: since e(x; ::3p) = 1; and e(y; 3::') = 12 : However, it holds in all a-crisp models since in any one of them e(x; ::3') > 0 implies the existence of y such that Sxy = 1 and e(y; ') > 0: Thus, e(y; 3::') Sxy ( e(y; ')) = 1. Remark. An interesting question raised by one referee is whether the logic of 3 in a-crisp GK-models is axiomatizable. This is granted in the abstract sense (recursive enumerability of valid formulas) because the logic may be interpreted faithfully in Gödel predicate logic which is axiomatizable (cf. [16], [12]). We do not know an explicit axiomatization but the system G3 [f::3' ! 3::'g is a candidate because the new schema characterizes crisp frames: if (W; S) is not crisp pick x, y 2 S such that 0 < Sxy < 1, then the valuation e(y; p) = 1 and e(z; p) = 0 for every z 6= y, provides a counterexample to the schema since e(x; ::3p) = Sxy = 1 but e(x; 3::p) = Sxy < 1. 14
7
G3 has the …nite model property
For any sentence ' such that 0G3 ' we may construct a …nite counter-model inside M3 : Theorem 7.1 If 0G3 ' then there is a GK-model M with …nitely many worlds such that M 2GK ': Proof: It follows from the Claim in the proof of Lemma 6.4 that for all and v 2 M3 there is w 2 M3 such that v(3 ) = S vw w( ): (if v(3 ) = 0 any w works). Given , let f (v) be a function choosing one such w for each v. For any formula let r( ) (rank of ) be the nesting degree of 3 in , that is, the length of a longest chain of occurrences of 3 in the tree of . Given ' and a world (valuation) v0 in M3 ; let Sj be the set of subformulas of ' of rank j; for each j n = r('), and de…ne inductively the following sets of valuations: M0 = fv0 g
Mi+1 = ff (v) : v 2 Mi ; 3 2 Sn i g Clearly, M = [i n Mi has …nitely many worlds. Consider the model M';v0 induced in M by restricting e and S of M3 to M V ar and M M respectively, that we will call M for simplicity. Then for any formula 3 2 Sj and v 2 Mn j there is w 2 Mn (j 1) such that v(3 ) = S vw w( ); and thus v(3 ) supfS vw w( ) : w is a world in M g: This allows us to show by induction on j n that for all 2 Sj ; v 2 Mn j we have v( ) = eM (v; ): In particular, if 0`G3 '; and v0 is a world in M3 such that v0 (') < 1 then eM (v0 ; ') < 1; which shows M 2 ': The proof of the previous theorem still works if we assume the worlds of M are de…ned in the variables of ' only and the accessibility relation of M is de…ned by using subformulas of ': SM vw := min fw( ) ) v(3 )g: 2[Si
This means that M takes values in a …xed …nite subalgebra of [0; 1] depending only on '. Thus there are only …nitely many models to consider and the decidability of the fragment G3 follows.
15
8
Modal extensions
The modal systems we have considered so far correspond to minimal modal logic K, the logic of Gödel-Kripke models with an arbitrary accessibility fuzzy relation. We may consider also the fuzzy analogues of the classical modal systems D; T; S4 and S5 for each modal operator, usually presented syntactically as combinations of the following axioms: D T 4 B
: : : :
: ? 2' ! ' 2' ! 22' ' ! 2:2:'
D3 : T3 : 43 : B3 :
3> ' ! 3' 33' ! 3' ' ! :3:3'
and semantically by asking the frames of the Kripke models to satisfy corresponding structural properties. Notice that D follows from T and D3 from T3 : Call a fuzzy frame hW; Si serial if 8x 2 W 9y 2 W : Sxy = 1; re‡exive if Sxx = 1 for all x 2 W , (min)transitive if Sxy Syz Sxz for all x; y; z; and symmetric if Sxy = Syx for all x; y 2 W: Let Ser; Ref l, T rans; and Symm denote the classes of GK-models over frames satisfying, respectively, each one of the above properties, and let j=C denote validity and consequence with respect to models in the class C. Proposition 8.1 j=Ser D ,D3 ; j=Ref l T ,T3 ; j=T rans 4 ,43 ; and j=Symm B ,B3 . Proof: The validity of D ,D3 in serial models is immediate because e(x; 2?) = inf y f Sxyg and e(x; 3>) = supy Sxy: Assume Sxx = 1 for all x: (T ): e(x; 2') (Sxx ) e(x; ')) = e(x; '): (T ): e(x; 3') Sxx e(x; ') = e(x; '): Assume Sxy Syz Sxz for all x; y; z: (4 ): e(x; 2') Sxy Syz (Sxz ) e(z; ')) Sxz e(z; '): Hence, e(x; 2') Sxy (Syz ) e(z; ')): Taking the meet over z in the right hand side: e(x; 2') Sxy e(y; 2'); hence, e(x; 2') (Sxy ) e(y; 2')) for all y and thus e(x; 2') e(x; 22'): (43 ): for any x; y; z; Sxy Syz e(z; ') Sxz e(z; ') e(x; 3'); hence, Syz e(z; ') (Sxy ) e(x; 3')): Taking the join over z in the left; e(x; 3') (Sxy ) e(x; 3')); thus Sxy e(x; 3') e(x; 3')): Taking the join again in the left, e(x; 33') e(x; 3'): Assume Sxy = Syx for all x; y: (B ): we prove the validity of the stronger schema :' ! 2:2'. Assume e(x; :') > 0 then e(x; ') = 0: Take any y such that Sxy > 0; then e(y; 2') (Syx ) e(x; ')) = (Sxy ) 16
e(x; ')) = 0: Therefore, e(y; :2') = 1; and (Sxy ) e(y; :2')) = 1: This shows that x(2:2') = 1: (B3 ): suppose e(x; ') > e(x; :3:3') then e(x; :3:3') = 0 and e(x; 3:3') = 1: This means that there is y such that Sxy e(x; :3') > 0 thus Sxy > 0 and e(x; :3') = 1; hence e(y; 3') = 0; therefore, Syx e(x; ') = 0 which is absurd because Syx = Sxy > 0 and e(x; ') > 0 by construction. Theorem 8.1 (i) G +D and G3 +D3 are strongly complete for j=Ser . (ii) G +T and G3 +T3 are strongly complete for j=Ref l . (iii) G +4 and G3 +43 are strongly complete for j=T rans . (iv) GS4 := G +T +4 and GS43 := G3 +T3 +43 are strongly complete for j=Ref l\T rans : Proof: Soundness follows from Proposition 8.1. Completeness follows, in each case, by asking the worlds of the canonical models M and M3 introduced in the completeness proofs of G and G3 to satisfy the corresponding schemas. The key fact is that the schemas force the accessibility relations S vw and S3 vw to satisfy the respective properties. (i) If v(D ) = 1 in M then then e (v; ?) = v( ?) = 0 for any world v of M and necessarily there is w such that S vw > 0; but the model is a-crisp thus S3 vw = 1. If v(D3 ) = v(3>) = 1 in M3 then by the Claim in the proof of Lemma 6.4 there is w such that S vw = 1: (ii) If v(T ) = 1 then S vv = inf '2L fv( ' ! ')g = 1: If v(T3 ) = 1 then S3 vv = inf '2L fv(' ! 3')g = 1. (iii) If v(4 ) = 1 then v( ') v( ') and so S vv 0 S v 0 v 00
[(v( (v(
') ) v 0 ( ')) (v 0 ( ') ) v 00 ('))]
') ) v 00 ('))
(v( ') ) v 00 ('))
Taking the meet over ' in the last formula we get: S vv 0 S v 0 v 00 (iv) If v(43 ) = 1 then v(33') v(3') and thus S3 vv 0 S3 v 0 v 00
S vv 00 :
[(v 0 (3') ) v(33')) (v 00 (') ) v 0 (3'))] (v 00 (') ) v(33'))
(v 00 (') ) v(3'))
Taking the meet over ' in the last formula we get S3 vv 0 S3 v 0 v 00
S3 vv 00 :
It follows from the proof of Theorem 8.1 that for the given extensions of G we get completeness also with respect to the a-crisp models of the respective class. 17
Call a fuzzy frame hW; Si weakly serial if it satis…es 8x9y Sxy > 0, and let W Serial be the class of GK-models over weakly serial frames. Then it is easily seen that G +D is sound (and thus strongly complete) for j=W Serial but G3 +D3 is not. However, j=W Serial is axiomatized by G3 +f::3>g. Remark. An original motivations of the second author to study fuzzy modal logics was to interpret the possibility operator 3 in the class of Gödel frames Ref l \ T rans \ Symm as a notion of similarity in the sense of Godo and Rodríguez [14], and a reasonable conjecture was that GS53 = GS43 +B3 would axiomatize validity in models over these frames. Unfortunately, the axioms B , B3 do not seem to force symmetry in the canonical models and we have not been able to show completeness of G +B or G3 +B3 for j=Symm , nor completeness of GS53 or the analogue GS5 with respect to j=Ref l\T rans\Symm . Perhaps stronger symmetry axioms such as (' ! ) ! 2(2' ! ) 3(3' ! ) ! (' ! 3 ); which characterize symmetric frames, would do.
9
Adding truth constants
The previous results on strong completeness may be generalized to Pavelkastyle languages [19] with a set Q [0; 1] of truth values added as logical constants, provided Q is well-ordered under the usual order of [0; 1] and discrete in the usual topology of [0; 1]. These conditions include …nite sets and force Q to be at most countable. Without loss of generality, we assume Q contains 0 and 1; to be identi…ed with ? and >, respectively. The logical constant corresponding to r 2 Q will be denoted by r itself. Let L Q be L enriched with elements of Q as atomic constituents, and let G (Q) be the system in this language obtained by adding to the axioms and rule of G the axiom schemas R1 - R4 below. For all r; s 2 Q : R1. (book-keeping axioms) 0 ! ?; > ! 1 r ! s; if r s (r ! s) ! s; if s < r R2. r ! r R3. (r ! ) ! (r ! ) R4. (( ! r) ! r) ! (( ! r) ! r) 18
The system G3 (Q) in the analogue language L3Q is de…ned similarly by adding R1 and R5 - R7 below to G3 . R5. 3r ! r R6. 3(r ! ') ! (r ! 3') R7. 3((' ! r) ! r) ! ((3' ! r) ! r): The double negation shift axioms Z and Z3 become super‡uous in the extended systems due to R4 and R7, respectively; also F3 is super‡uous due to R5. GK-models are extended by de…ning e(x; r) = r at each world x, and validity j=GK ' is de…ned as before in terms of 1-satisfaction. Then R1 to R7 are easily seen to be valid. However, the consequence notion T j=GK ' given in De…nition 2.2 is too rough if there are more than two truth constants (consider 12 j=GK 0). We will utilize the …ner relation T j= GK ' for which it may be shown that G (Q) and G3 (Q) are strongly complete for countable theories if Q is well ordered and discrete. The same holds for the logics mentioned in Theorem 8.1. No conditions on Q are required for weak completeness. This extends substantially a result of Esteva, Godo and Nogera [7] on weak completeness of Gödel logic with rational truth constants. Discreteness of Q is necessary for strong completeness: if r is a limit point of Q then there is a strictly increasing or decreasing sequence of Q converging to r, say frn g increases to sup rn = r; then fr1 ! ; r2 ! ; r3 ! ::::g j=
GK
r!
but no …nite subset of premises can grants this, thus no formal proof is possible. Moreover, discreteness alone is not enough, since Q = fr1 < r2 < ::: ::: < q2 < q1 g with sup ri = inf qi is discrete and fr1 ! ; r2 ! ; :::
,
! q1 ;
! q2 ; :::g j=
GK
!
but no …nite subset of the premises yields the same consequence. Thus well order or a related conditions is needed. We give next the proof of strong completeness for G (Q), a re…nement of that given for G . The deduction theorem and lemmas 4.1 and 4.2 extend readily to the system G (Q): Moreover, any formula of L Q may be seen as a formula of Gödel logic over the vocabulary V ar [ Q [ L Q , and after de…ning T G (Q) = f :
is an axiom of G (Q)g [ f 19
: `G
(Q)
g
it may be shown that T `G
(Q)
' if and only if T [ T G (Q) ` ' in Gödel logic
Call a Gödel valuation v : V ar [ Q [ L Q ! [0; 1] normal if v(r) = r for all r 2 Q: Having v(T G (Q)) = 1 does not make v normal. However, the next lemmas show how to transform such a valuation to a normal one still satisfying T G (Q) and some other useful properties. As before, we will write v for the Gödel extension v: Lemma 9.1 Let v : V ar [ Q [ L Q ! [0; 1] be extended to all formulas of L Q according to the Gödel operations in [0; 1]: If v(R1) = 1; then v(0) = 0; v(1) = 1, and = minfr 2 Q : v(r) = 1g > 0: Moreover, v is weakly increasing in Q and strictly increasing in Q \ [0; ]. Proof: v(0) = 0 and v(1) = 1 by R1; exists and is positive due to the well ordering of Q. If r s in Q then v(r) v(s) by R1. If r < s in Q then v(r) < 1 and so v(s ! r) v(r) < 1 by R1; thus v(s) > v(r): Call a formula of L Q shy if any occurrence of r 2 Q r f0; 1g in the formula is under the scope of an occurrence of : For positive r 2 Q, let r be the supremum (in [0; 1]) of its predecessors in Q. Necessarily r < r because r is isolated, but r may not belong to Q. Lemma 9.2 (Normalization). Let v : V ar [ Q [ L Q ! [0; 1] be Gödel valuation satisfying R1 and let be de…ned as in the previous lemma, then there is a normal w : V ar [ Q [ L Q ! [0; 1] such that for any ' : 1. v(') = 1 ) w(') 2. v(') < 1 ) w(') < 3. v(') v( ) < 1 ) w(') w( ) 4. v(') < v( ) ) w(') < w( ) 5. For any shy formula '; v(') = 1 implies w(') = 1; and v( ! ') = 1 implies w( ! ') = 1: 6. v(T G (Q)) = 1 implies w(T G (Q)) = 1: 7. If r ; r are given so that r < r < r S < r for each positive r 2 Q; then = Q w may be chosen so that w(L Q ) Q [ r2Q [ r ; r ): Hence, no r 2 belongs to the image of w:
20
Proof: Given 0 < r 2 Q; let v(r) = sup[0;1] fv(s) : s < r; s 2 Qg: Clearly, v(r) v(r); moreover, v(r) = v(s); s 2 Q; if and only if s = r or s = r : It should be clear also that S [0; 1] = fv(r) ; v(r) : r 2 Qg [ (v(r) ; v(r)) r2Q\(0; ]
is a partition because v Q \ [0; ] is strictly increasing by the previous lemma and v( ) = 1. Given r ; r as in 7 choose an strictly increasing function g : [0; 1] ! [0; ) [ f1g satisfying. g(1) = 1 g(v(r)) = r for r 2 Q, v(r) < 1 (hence r < ): g((v(r) ; v(r))) = ( r ; r ) if v(r) < v(r) < 1. g(v(r) ) = r if v(r) 62 v(Q) (hence v(r) < v(r)) De…ne w : V ar [ Q [ L
Q
! [0; 1] as follows:
w( ) = g(v( )) if 2 V ar [ L w(r) = r for r 2 Q.
Q
Property 7 is insured for elements of V ar [ Q [ L Q by construction and it extends to all of L Q because the value of a formula under a Gödel valuation is identical to 0; 1; or the value of one of its atomic constituents. For the other properties: 1-2. By simultaneous induction in Gödel connectives, we show 1 and the following strengthening 20 of 2: v(') < 1 ) w(') = g(v(')) < : Atomic. For ' 2 V ar [ L Q by de…nition of w and g: For ' = r 2 Q: v(r) < 1 if and only if w(r) = r < by de…nition of ; and in the later case w(r) = r = g(v(r)) by the way g was chosen. Conjunction. If v(' ^ ) = 1 then v(') = v( ) = 1 and by inductive hypothesis w('^ ) = w(') w( ) : If v('^ ) < 1; say v(') v( ); then v(') < 1 and thus w(') = g(v(')) < by induction hypothesis. Moreover, w( ) w('): if v( ) = 1 because w( ) ; if v( ) < 1 because w( ) = g(v( )) g(v(')): Thus w(' ^ ) = w(') = g(v(' ^ )) < : Implication. Assume v(' ! ) = 1: If v( ) < 1 then v(') v( ) < 1; and by inductive hypothesis and monotonicity of g, w(') = g(v(')) g(v( )) = w( ); thus w(' ! ) = 1: If v( ) = 1 then w(' ! ) w( ) again by the induction hypothesis. Assume now v(' ! ) < 1; then v(') > v( ) < 1 and thus w( ) = g(v( )) < : Moreover, w(') > w( ): if v(') = 1 because w(') ; if v(') < 1 because w(') = g(v(')) > g(v( )): Therefore w(' ! ) = w( ) = g(v(' ! )) < : 3. v(') v( ) < 1 implies by 20 : w(') = g(v(')) g(v( )) = w( ). 21
4. v(') < v( ) implies v( ! ') = v(') < 1; hence w( ! ') < by 2, and thus w(') < w( ): 5. If ' is shy then ' = '0 ( 1 ; :::; n ) where '0 is Gödel formula and i 2 V ar [ L; therefore, w(') = '0 (w( 1 ); :::; w( n )) = '0 (g(v( 1 )); :::; g(v( n ))) = g(v('0 ( 1 ; :::; n )) = g(v(')) because g is an endomorphism of the Heyting algebra [0; 1]: Therefore, v(') = 1 implies w(') = g(1) = 1: Moreover, v( ! ') = 1 implies trivially w( ! ') = 1 when v(') = 1; and w( ) w(') when v(') < 1 by property 4: 6. The axioms of G give 1 under any Gödel valuation. The speci…c axioms of G are shy, w(R1) = 1 because w is the identity in Q; and R2, R3, R4.are of the form ! ' with ' shy. The other elements of T G (Q) are shy by construction. Canonical model M (Q): W : all normal valuations v : V ar [ Q [ L Q ! [0; 1] satisfying v(T G (Q)) = 1 and such S that there are r ; r (r 2 Q) with r < r < < r and Im(v) Q [ r r2Q; r>0 [ r ; r ): S vw = inf
2L
e (v; ) = v
Q
(v(
) ) w( )):
V ar:
Lemma 9.3 e (v; ) = v( ) for any world v of M (Q) and formula : Proof: As in the case of G ; it is enough to check inf w2W (S vw ) w(')) v( ') whenever v( ') < 1: This is done in two stages: Claim 1. If = v( ') < 1 there exists a Gödel valuation u : V ar [ Q [ L Q ! [0; 1] such that u(T G (Q)) = 1; u(') < 1 and for any and r 2 Q 1. u( ) = 1 if v( ) > 2. u(r) u( ) if r v( ) 3. u(r) < u( ) if r < v( ) and r 4. 2 [ ; ) if = minfr 2 Q : u(r) = 1g. Proof: Let T';v be the theory f : v( Then T';v 0G thus
) > g [ fr ! : r 2 Q; r v( )g [ f( ! r) ! r : r 2 Q; r ; r < v( (Q)
': Otherwise, 1 ; :::;
1 ; :::; k
k
`G
`G
(Q)
22
(Q)
':
' for some
)g: i
2 T';v and
But v( ) > for any other 2 T';v : for the …rst group of axioms, by construction; for the second, because v( ) r implies v( (r ! )) v(r ! ) = r ! v( ) = 1 by R3 and normality of v; for the third, because v( ) > r implies by R4 and normality: v( (( ! r) ! r)) (v( ) ! r) ! r = (r ! r) = 1. Hence, we obtain the contradiction < minf
1 ; :::;
kg
v( '):
Therefore, T';v ; T G (Q) 0 ', and we may use the strong completeness theorem of Gödel logic to get a valuation u : V ar [ Q [ L ! [0; 1] such that u(T';v [ T G (Q)) = 1 > u('). Conditions 1, 2 hold by construction, 3 is satis…ed because r = v( ') implies u(r) u(') < 1 by 2 and thus u(( ! r) ! r) = 1 implies u( ) > u(r): To verify 4, notice that r implies u(r) < 1 as just explained, thus > : On the other hand < r < implies v( r) v(r) = r > by R2, and thus u(r) = 1 by 1, contradicting the de…nition of . Therefore, : Claim 2. If = v( ') < 1 then for any M (Q)) such that (Svw ) w(')) < + :
> 0 there exists a world w of
Proof: According to Lemma 9.2, the valuation u of the previous claim may be transformed in a valuation w in M (Q) such that w(') < and the conditions on u become the following conditions on w: 1. w( ) if v( ) > : 2. r w( ) if r v( ) and r < (that is, r 3. r < w( ) if r < v( ) and r . 4. ; w(') 2 [ ; ).
).
Only 4 needs some explanation: if r < then r v( ') = and thus r w(') by 3 above, hence w('): Moreover, we may choose the parameters 0r ; 0r ; of w so that r < for 0 < r < , where
r
;
r
r
e (w; ') in the canonical model, hence T 2 GK(Q) ': Moreover, if the formulas in T have the form or ! ; with shy, then by Lemma 9.2-5 w(T ) = 1 and thus e (w; T ) = 1; hence T 2GK '. Corollary. For any Q and formula ' in L
Q:
`G
(Q)
' , j=GK ':
Proof: The condition in 4 of Theorem 9.1 is trivially satis…ed by empty T , and no condition on Q is required since it is enough to consider the …nitely many truth constants appearing in '. Note that GK-models with crisp accessibility relation are no enough for completeness of G (Q); for example, the formula ( 12 ! 12 ) _ 0 is invalid but is valid in all a-crisp GK-models. However, if Q denotes the topological closure of Q in [0; 1] (still countable or Q itself if Q is …nite) it may be shown that strong completeness of G (Q) holds with respect to GK-models where the accessibility relation takes values in Q only. For weak completeness only Q-valued accessibility has to be considered. 24
The proof of the completeness of G3 (Q) follows similar lines to that of G (Q); and again we have the …nite model property for this logic. Also the results of Section 8 transfer without di¢ culty to the systems with truth constants.
10
Final Comment
The main results of this paper were announced at the meeting on "Logic, Computability and Randomness", Cordoba, Argentina, Sept. 2004. Publication was delayed, aiming to axiomatize the full logic with both modal operators combined, which resulted elusive. It may be seen that the union of the systems G and G3 is not enough for that purpose. However, we have found recently that Fischer Servi [8] "connecting axioms": 3(' ! ) ! (2' ! 3 ) (3' ! 2 ) ! 2(' ! ) together with G [ G3 constitute a strongly complete axiomatization. This will appear elsewhere.
References [1] F. Baader, D. Calvanese, D. L. McGuinness, D. Nardi, P. F. PatelSchneider: The Description Logic Handbook: Theory, Implementation, Applications, Cambridge University Press, Cambridge, 2003. [2] Mathias Baaz and Richard Zach: Compact Propositional Logics. Proc. International Symp on multiple valued logic, IEEE Computer Society Press, pp 108-113, 1998. [3] Matthias Baaz, Norbert Preining, Richard Zach: First-Order Gödel Logics. Annals of Pure and Applied Logic 147 (2007) 23-47 [4] Felix Bou, Francesc Esteva, and Lluis Godo: Exploring a syntactic notion of modal many-valued logics. Mathware and Soft Computing, 15, Number 2 (2008) 175–188. [5] A. Chagrov and M. Zakharyaschev: Modal Logic, Clarendon Press, Oxford, 1997. [6] Milan Bo¼ zi´c and Kosta Do¼ sen: Models for Normal Intuitionistic Modal Logics. Studia Logica 43, Number 3 (1984) 217-245. 25
[7] Francesc Esteva, Lluis Godo, and Carles Noguera: On rational weak nilpotent minimum logics, Journal of Multiple-Valued Logic and Soft Computing 12, Number 1-2 (2006) 9-32. [8] G. Fischer Servi: Axiomatizations for some intutitionistic modal logics, Rend. Sem. Mat. Polit de Torino 42 (1984) 179-194. [9] Melving Fitting: Many valued modal logics. Fundamenta Informaticae 15 (1991) 325-254. [10] Melving Fitting: Many valued modal logics, II. Fundamenta Informaticae 17 (1992) 55-73. [11] Josep M. Font: Modality and Possibility in some intuitionistic modal logics. Notre Dame Journal of Formal Logic 27, 4 (1986) 533-546. [12] Alfred Horn: Logic with truth values in a linearly ordered Heyting algebra. J. Symbolic Logic 34 (1969) 395-408. [13] George Metcalfe and Nicola Olivetti: Proof Systems for a Godel Modal Logic. In M. Giese and A. Waaler (editors), Proceedings of TABLEAUX 2009, volume 5607 of LNAI. Springer (2009) 265-279. [14] Lluis Godo and Ricardo O. Rodríguez: A fuzzy modal logic for similarity reasoning. In Guoqing Chen, Mingsheng Ying, and Kai-Yuan Cai (editors), Fuzzy Logic and Soft Computing, Vol. 6. Kluwer Academic, 1999. [15] C. Grefe: Fischer Servi´s intuitionistic modal logic has the …nite model property, In M. Kracht, M. de Rijke, H. Wansing, M. Zakharyaschev (editors), Advances in Modal Logic Vol.1, CSLI, Stanford, pp 85-98, 1998. [16] Petr Hájek: Metamathematics of fuzzy logic, in: Trends in Logic, Vol. 4, Kluwer Academic Publishers, Dordrecht, 1998. [17] Petr Hájek: Making fuzzy description logic more general Fuzzy Sets and Systems 154 (2005) 1–15 [18] H. Ono: On some intutionistic modal logics. Publications of the Research Institute for Mathematical Sciences, Kyoto University 13 (1977) 13-55. [19] J. Pavelka: On fuzzy logic I, II, III. Zeitschr. f. Math. Logik un Grundl. der Math. 25 (1979) 45-52, 119-134, 447-464. 26
[20] Frank Wolter: Superintuitionistic Companions of Classical Modal Logics. Studia Logica 58, 3 (1997) 229-295.
Xavier Caicedo Departamento de Matemáticas Universidad de los Andes A.A. 4976, Bogotá, Colombia [email protected] Ricardo Rodríguez Departamento de Computación, Fac. Ciencias Exactas y Naturales Universidad de Buenos Aires 1428 Buenos Aires, Argentina [email protected]
27
Ricardo O. Rodríguez
October 23, 2009
Abstract We prove strong completeness of the -version and the 3-version of a Gödel modal logic based on Kripke models where propositions at each world and the accessibility relation are both in…nitely valued in the standard Gödel algebra [0,1]. Some asymmetries are revealed: validity in the …rst logic is reducible to the class of frames having twovalued accessibility relation and this logic does not enjoy the …nite model property, while validity in the second logic requires truly fuzzy accessibility relations and it has the …nite model property. Analogues of the classical modal systems D, T, S4 and S5 are considered also, and the completeness results are extended to languages enriched with a discrete well ordered set of truth constants.
1
Introduction
Sometimes it is needed in approximate reasoning to deal simultaneously with both fuzziness of propositions and modalities, for instance one may try to assign a degree of truth to propositions like “John is possibly tall”or “John is necessarily tall”, where “John is tall”is presented as a fuzzy proposition. Fuzzy logic should be a suitable tool to model not only vagueness but also these and other kinds of information features like certainty, belief or similarity, which have a natural interpretation in terms of modalities. To achieve this end, it is natural to interpret fuzzy modal operators by means of Kripke models over fuzzy frames. We address in this paper the case of the pure modal operators (necessitation and possibility) for standard Gödel logic, one of the main systems of fuzzy logic arising from Hájek´s classi…cation in [16]. For this purpose we consider a many-valued version of Kripke semantics for modal logic where Preprint
1
both propositions at each world and the accessibility relation are in…nitely valued in the standard Gödel algebra [0,1]. We provide strongly complete axiomatizations for the -fragment and the 3-fragment of the resulting minimal logic. These fragments are shown to behave quite asymmetrically. Validity in the …rst one is univocally determined by the class of frames having a crisp (that is, two-valued) accessibility relation, while validity in the second requires truly fuzzy frames. In addition, the -fragment does not enjoy the …nite model property with respect to the number of worlds or the number of truth values while the 3-fragment does. We consider also the Gödel analogues of the classical modal systems D, T, S4 and S5 for each modal operator and show that the …rst three are characterized by the many-valued versions of the frame properties which characterize their classical counterparts. Finally, we extend the strong completeness results to Pavelka-style languages enriched with a set of explicit truth constants denoting a discrete well ordered set of truth values. Our approach is related to Fitting´s [10] who considers Kripke models taking values in a …xed …nite Heyting algebra; however, Fitting´s proof systems and completeness proofs depend essentially on …niteness of the algebra and the fact that his languages contain constants for all the truth values of the algebra. We must relay on completely di¤erent methods. Modal logics with an (intermediate) intuitionistic basis and Kripke style semantics have been investigated in a number of relevant papers (see Ono [18], Fischer Servi [8], Bo¼ szic and Do¼ sen [6], Font [11], Wolter [20], from an extensive literature), but in all cases the models carry two or more crisp accessibility relations satisfying some commuting properties: a pre-order to account for the intuitionistic connectives and one or more binary relations to account for the modal operators. Our semantics has, instead, a single fuzzy accessibility relation and does not seem reducible to those multi-relational semantics since the latter enjoy the …nite model property for (cf. Grefe [15]). Recently, Metcalfe and Olivetti [13] have given a proof of weak completeness of a calculus of sequents of relations for the -fragment of our logic, showing that it is decidable and PSPACE complete. The decidability of the 3-fragment follows from the …nite model property we prove later. Bou, Esteva, and Godo survey in [4] modal logics with analogue [0,1]valued Kripke semantics under di¤erent choices of the t-norm. However, our methods and results do not generalize to the corresponding modal versions of ×ukasiewicz or product logics because they relay on the richness of endomorphisms of the Gödel algebra [0,1]. In fact, we do not know any 2
completeness result for these logics without extra conditions on the frames, strong completeness being known to be untenable. Classical modal logics are inter-translatable with description logics [1]. Our Gödel Kripke semantics for the modal operators is similarly translatable into fuzzy (Gödel) description logic (cf. [17]), and thus our results throw light on various fragments of this logic and certain Pavelka-style expansions of them. We assume the reader is acquainted with modal and Gödel logics and the basic laws of Heyting algebras (cf. Chagrov and Zakharyaschev [5]).
2
Gödel-Kripke models
The language L 3 of propositional Gödel modal logic is built from a set V ar of propositional variables, logical connectives symbols ^; !; ?; and the modal operator symbols and 3. Other connectives are de…ned: > := ' ! ' :' := ' ! ? ' _ := ((' ! ) ! ) ^ (( ! ') ! ') ' $ := (' ! ) ^ ( ! '): L and L3 will denote, respectively, the -fragment and the 3-fragment of the language. As stated before, the semantics of Gödel modal logic will be based on fuzzy Kripke models where the valuations at each world and also the accessibility relation between worlds are [0; 1]-valued. The symbols and ) will denote the Gödel t-norm in [0; 1] and its residuum, respectively: a b = minfa; bg;
a)b=
1; if a b b; otherwise
The maximum is de…nable: maxfa; bg = ((a ) b) ) b) ((b ) a) ) a); and the pseudo-complement is denoted a := a ) 0. This yields the standard Gödel algebra; that is, the unique Heyting algebra structure in the linearly ordered interval. De…nition 2.1 A Gödel-Kripke model (GK-model) will be a structure M = hW; S; ei where: W is a non-empty set of objects that we call worlds of M: S : W W ! [0; 1] is an arbitrary function (x; y) 7 ! Sxy. 3
e:W
V ar ! [0; 1] is an arbitrary function (x; p) 7 ! e(x; p).
The evaluations e(x; ) : V ar ! [0; 1] are extended simultaneously to all formula in L 3 by de…ning inductively at each world x: e(x; ?) := 0 e(x; ' ^ ) := e(x; ') e(x; ) e(x; ' ! ) := e(x; ') ) e(x; ) e(x; 2') := inf y2W fSxy ) e(y; ')g e(x; 3') := supy2W fSxy e(y; ')g. It follows that e(x; ' _ ) = maxfe(x; '); e(x; )g and e(x; :') =
e(x; '):
The notions of a formula ' being true at a world x, valid in a model M = hW; S; ei; or universally valid, are the usual ones: ' is true in M at x, written M j=x '; i¤ e(x; ') = 1. ' is valid in M , written M j= '; i¤ M j=x ' at any world x of M: ' is GK-valid, written j=GK ', if it is valid in all the GK-models. Clearly, all valid schemas of Gödel logic are GK-valid. In addition, Proposition 2.1 K Z K3 Z3 F3
The following modal schemas are GK-valid:
(' ! ) ! ( ' ! ) :: ! :: 3(' _ ) ! (3' _ 3 ) (in fact, an equivalence) 3::' ! ::3' :3?
Proof: Let M = hW; S; ei be an arbitrary GK-model and x 2 W: (K ) By De…nition 2.1 and properties of the residuum we have for any y 2 W : e(x; (' ! )) e(x; ') (Sxy ) (e(y; ') ) e(y; )) (Sxy ) e(y; ')) (Sxy ) e(y; )): Taking the meet over y in the last expression: e(x; (' ! )) e(x; ') e(x; ); hence, e(x; (' ! )) e(x; ' ! ). (Z ) Utilizing the Heyting algebra identity: (x ) y) = (x ) y); we have: e(x; ::2 ) = e(x; 2 ) (Sxy ) e(y; )) = (Sxy ) e(y; )) = (Sxy ) e(y; :: )): Taking the meet over y, e(x; ::2 ) e(x; 2:: ): (K3 ) By properties of suprema and distributivity of over max; e(3(' _ )) = supy fSxy maxfe(y; '); e(y; )gg = maxfsupy fSxy e(y; ')g; supy fSxy e(y; )gg: 4
(Z3 ) Sxy e(y; ::') e(x; 3') = e(x; ::3'):
Sxy
e(y; ') =
(Sxy e(y; '))
(F3 ) e(x; 3?) = supy fSxy 0g = 0: The Modus Ponens rule preserves truth at every world of any GK-model. On the other hand, the classical introduction rules for the modal operators RN :
' '
RN3 :
'! 3' ! 3
:
do not preserve truth at a …xed world. However, Proposition 2.2 RN and RN3 preserve validity in any given model, thus they preserve GK-validity. Proof: Let hW; S; ei be a GK-model: (RN ) If e(x; ') = 1 for all x 2 W then e(x; ') = inf y fSxy ) e(y; ')g = inff1g = 1 for all x. (RN3 ) If e(x; ' ! ) = 1 for all x 2 W then Sxy e(y; ') Sxy e(y; ) e(x; 3 ) for any y 2 W Taking the join over y in the left hand side of the last inequality, e(x; 3') e(x; 3 ): Semantic consequence is de…ned for any theory T
L
3;
as follows:
De…nition 2.2 T j=GK ' if and only if for any GK-model M and any world x of M; M j=x T implies M j=x ': Note that Modus Ponens preserves consequence but this is not the case of the inference rules RN and RN3 : An alternative notion of logical consequence arises naturally. Set e(x; T ) = fe(x; ') : ' 2 T g then: De…nition 2.3 T j= GK ' if and only if for any GK-model M and any world x in M; inf e(x; T ) e(x; '): Clearly, j= GK implies j=GK , and it will follow from our completeness theorems that both notions are equivalent for countable theories. This fact has been already observed for pure Gödel logic by Baaz and Zach in [2].
5
3
On strong completeness of Gödel logic
To prove strong completeness of the modal fragments L and L3 we will reduce the problem to pure Gödel propositional logic. In the rest of this paper L(X) will denote the Gödel language built from a set of propositional variables X and the connectives ^; !; ?: Let G be a …xed axiomatic calculus for Gödel logic (also called Dummett’s LC), say the following one given by Hájek ([16], Def. 4.2.3.): (' ! ) ! (( ! ) ! (' ! )) (' ^ ) ! ' (' ^ ) ! ( ^ ') (' ! ( ! )) ! ((' ^ ) ! ) ((' ^ ) ! ) ! (' ! ( ! )) ' ! (' ^ ') ((' ! ) ! ) ! ((( ! ') ! ) ! ) ?!' MP: From ' and ' ! , infer
` will denote deductive inference in this calculus. It is well known that G is deductively equivalent to the intermediate logic obtained by adding to Heyting calculus the pre-linearity schema: (' ! ) _ ( ! '). Given a valuation v : X ! [0; 1]; let v denote the extension of v to L(X) according to the Gödel interpretation of the connectives. We will need the following strong form of standard completeness for Gödel logic: Proposition 3.1 Let T be a countable theory and U a countable W set of formulas of L(X) such that for every …nite S U we have T 0 S then there is a valuation v : X ! [0; 1] such that v( ) = 1 for all 2 T and v( ) < 1 for each 2 U:
Proof: Extend T to a prime theory T 0 (that is, T 0 ` _ implies T 0 ` or T 0 ` ) satisfying the same hypothesis with respect to U (this is standard): The Lindenbaum algebra L(X)= T 0 of T 0 is linearly ordered since by primality and the pre-linearity schema T 0 ` ! or T 0 ` ! . Moreover, the valuation v : X ! L(X)=T 0 ; v(x) = x= T 0 is such that v(T ) = 1; v( ) < 1 for all 2 U: As T 0 is countable we may assume X is countable and thus, being also countable, L(X)= T 0 is embeddable in the Gödel algebra [0; 1]; therefore, we may assume v : X ! [0; 1]. Taking S = f'g we obtain the usual formulation completeness. We can not expect strong standard completeness of G for uncountable theories, as the following example illustrates. 6
Example. Set T = f(p ! p ) ! q : < < ! 1 g where ! 1 is the …rst uncountable cardinal, then T 0 q: Otherwise we would have ` q; for some …nite = f(p i+1 ! p i ) ! q : 1 i < ng; but this is not possible by 1 soundness of G; because the valuation v(q) = 21 , v(p i ) = 12 (1 i+1 ) for 1 1 i n; makes v(p i ) < v(p i+1 ) < 2 and thus v((p i+1 ! p i ) ! q) = 1 for 1 i < n; while v(q) < 1: However, there is no valuation v such that v(T ) = 1 and v(q) < 1, because that would imply v(p ! p ) < 1 for all < < ! 1 ; and thus the set fv(p ) : < ! 1 g would be ordered in type ! 1 ; which is impossible because any well ordered subset of ([0; 1]; 0 implies v(::2 ) = 1; and thus v(2:: ) = 1 since v satis…es axiom Z : This implies that T';v 0G ': Otherwise, 1 ; :::; k `G ' for some i 2 T';v and thus 2 1 ; :::; 2 k `G 2' by Lemma 4.1. Hence, by Lemma 4.2 and the previous observations, < minf2
1 ; :::; 2 k g
v(2');
a contradiction. By Lemma 4.3 we have T';v [ T G 0 ' and by the countability of T';v [ T G we may use the completeness theorem of Gödel logic (Proposition 3.1) to get a Gödel valuation u : V ar [ L ! [0; 1] such that u(T';v ) = 1 and u(') < 1: Then u 2 M and (i) holds by construction. Moreover, (ii) is satis…ed because u(:: ) = 1 and thus u( ) > 0 if v(2 ) > 0: This ends the proof of the claim. 9
Pick now an strictly increasing function g : [0; 1] ! [0; 1] such that g(1) = 1; g(0) = 0; and g[(0; 1)] = ( ;
+ ):
As g is a homomorphism of Heyting algebras, the valuation w = g u preserves the value 1 of the formulas in T G and thus it belongs to M . Moreover, v(2 ) w( ) for all : - if v(2 ) > because w( ) = g(u( )) = g(1) = 1 by (i) above. - if 0 < v(2 ) because then 0 < u( ) 1 by (ii) above, and thus w( ) = g(u( )) 2 ( ; + ) [ f1g: This means S vw = 1, and since u(') < 1 we have, w(') = g(u(')) < + ; which shows (1). De…nition 4.1 Call a GK-model accessibility crisp (a-crisp in short) if S : W W ! f0; 1g; and write T j=Crisp ' if the consequence relation holds at each node of any a-crisp GK-model. Theorem 4.1 For any countable theory T and formula ' in L the following are equivalent: (i) T `G ' (ii) T j= GK ' (iii) T j=GK ' (iv) T j=Crisp ': Proof: By Lemma 4.2, it is enough to show (iv) ) (i). If T 0G ' then T [ T G 0 ' by Lemma 4.3, and by strong completeness of Gödel logic there is a valuation v : V ar [ L ! [0; 1] such that v(T ) = v(T G ) = 1 and v(') < 1: Hence, v 2 W by de…nition, e (v; T ) = v(T ) = 1; and e (v; ') = v(') < 1 by Lemma 4.4, showing that M j=v T but M 2v ': That is, T 2Crisp ' because the canonical model is a-crisp. After the example in Section 3 we can not expect completeness with respect to uncountable theories.
5
G does not have the …nite model property
The following example shows that G does not have the …nite model property with respect to GK-models. The reciprocal of axiom Z : :: ! :: 10
;
fails in the (a-crisp) model M = (N; S; e); where Smn = 1 for all m; n;
e(n; p) =
1 n+1
for all n:
1 Indeed, e(n; ::p) = n+1 = 1 for all n and thus, e(0; 2::p) = inff1g = 1: 1 g = 0; and thus e(0; ::2p) = 0: However, But e(0; 2p) = inf n2N f1 ) n+1
Proposition 5.1 2:: ! ::2 is valid in any GK-model hW; S; ei with …nite W . Proof: Assume e(x; 2:: ) > e(x; ::2 ) then e(x; 2 ) < 1 and thus e(x; 2 ) = 0. This implies the existence of a sequence fyn gn W such that Sxyn > e(yn ; ) for all n 2 N and fe(yn ; )gn converges to 0: If W is …nite so is the range of the latter sequence and there must exist n such that e(yn ; ) = 0: Then (Sxyn ) e(yn ; :: )) = (Sxyn ) 0) = 0 and thus e(x; 2:: ) = 0; a contradiction. Remark. The proof of the proposition shows that 2:: ! ::2 is valid in all 0-witnessed GK-models, those where e(x; 2 ) = 0 implies the existence of y such that e(y; ) = 0 < Sxy: In fact, G + f2:: ! ::2 g is strongly complete for 0-witnessed (a-crisp) models. To see this, notice that if any world v of the canonical model is asked to satisfy the new schema, M becomes 0-witnessed because then v( ) = 0 implies v( :: ) = 0, and thus by the last line in the proof of Lemma 4.4 there is w such that S vw = 1 and w(:: ) < "; hence, w( ) = 0: We do not know if this system has the …nite model property.
6
Completeness of the 3-fragment
The system G3 results by adding to G the following axiom schemas and rule in the language L3 : K3 : 3(' _ ) ! (3' _ 3 ) Z3 : 3::' ! ::3' F3 : :3? RN3 : From ' ! infer 3' ! 3 As in the case of the -fragment, in proofs with assumptions the rule RN3 is to be used in theorems only, and thus we have the deduction theorem DT, hence the rule: Lemma 6.1 If ' `G3
then 3' `G3 3 : 11
Also the soundness theorem: Lemma 6.2 T `G3 ' implies T j=
GK
', hence, T j=GK ':
Moreover, if T G3 is the set of theorems of G3 then: Lemma 6.3 T `G3 ' if and only if T [ T G3 ` ' in Gödel logic. Let 3L3 = f3 : 2 L3 g, then the canonical model M3 = (W ; S ; e ) is de…ned as follows: W is the set of valuations v : V ar [ 3L3 ! [0; 1] such that v(T G3 ) = 1 and its positive values have a positive lower bound: inf fv( ) : v( ) > 0g = 2L3
>0
(2)
when v is extended to L3 = L(V ar [ 3L3 ) as a Gödel valuation. The fuzzy relation between worlds in M3 is given by S vw := inf fw( ) ) v(3 )g: '2L3
e (v; p) := v(p) for any p 2 V ar. Lemma 6.4 For any world v in the canonical model M3 and any ' 2 L3 we have e (v; ') = v('): Proof: The only non trivial step in a proof by induction on complexity of formulas of L3 is that of 3: By induction hypothesis, e (v; 3') = supw fS vw e (w; ')g = supw fS vw w(')g; then we must show supw fS vw w(')g = v(3'): By de…nition S vw
w(') ) v(3');
for any ' 2 L3 and w 2 W ; then S vw w(') v(3'); which yields taking the join over w: e (v; 3') v(3'): The other inequality is trivial if v(3') = 0: For the case v(3') > 0, let w be given as in the following claim then v(3') = = S vw w(') e (v; 3'); concluding the proof of the lemma. Claim. If v is a world of M3 such that v(3') = world w of M3 such that w(') = 1 and S vw = : 12
> 0; there exists a
Proof: Assume v(3') = ';v
> 0 and de…ne
= f 2 L3 : v(3 ) < g [ f:: :
2 L3 , v(3 ) = 0g:
This set is not empty because v(3?) = 0 by axiom F3 : Moreover, for any …nite subset of ';v ; say f 1 ; :::; n g [ f:: 1 ; :::; :: m g; we have ' 0G3
1
_ ::: _
n
_ ::
1
_ ::: _ ::
m:
Otherwise, we would have (Cf. Lemma 6.1) 3' `G3
3( 1 _ ::: _ n _ :: 1 _ ::: _ :: m ) 3 1 _ ::: _ 3 n _ 3:: 1 _ ::: _ 3:: 3 1 _ ::: _ 3 n _ ::3 1 _ ::: _ ::3
RN3 K3 Z3 ;
m m
which would imply by Lemma 6.2 v(3')
max(fv(3 i ) : 1
i
ng [ fv(::3 i ) : 1
i
mg) < ;
a contradiction. Therefore, we have by Lemma 6.3 T G3 ; ' 0
1
_ ::: _
n
_ ::
1
_ ::: _ ::
m;
By Proposition 3.1 there is a Heyting algebra valuation u : V ar [ 3L3 ! [0; 1] such that u(') = u(T G3 ) = 1 and u( ) < 1 for all satis…es the further conditions: (i) u(') = 1 (ii) u( ) < 1 if v(3 ) < ; because then 2 ';v (iii) u( ) = 0 if v(3 ) = 0, because then :: 2 which implies u( ) = 0:
';v
2
';v .
Thus, u
and so u(:: ) < 1
Let g : [0; 1] ! [0; 1] be the strictly increasing function: 8 1 if x = 1 < (x + 1)=2 if 0 < x < 1 g(x) = : 0 if x = 0
where is given by (2). Clearly the valuation w = g u inherits the properties (i), (ii) (iii) of u, with (ii) in the stronger form: (ii0 ) w( )
=2; by construction, and w(T G3 ) = 1 because g is a homomorphism of Heyting algebras, hence, w belongs to M3 : To see that S vw = ; note that w( ) v(3 ) whenever v(3 ) < . If 13
0 < v(3 ) because then w( ) < v(3 ) by (ii0 ) and de…nition of . If v(3 ) = 0 because then w( ) = 0 by (iii). Since (w( ) ) v(3 )) for v(3 ) , and (w(') ) v(3')) = (1 ) ) = , we have S vw = inf '2L3 fw(') ) v(3')g = : Theorem 6.1 For any countable theory T and formula ' in L3 , the following are equivalent (i) T `G3 ' (ii) T j= GK ' (iii) T j=GK ': Proof: Assume T 0G3 ', then T [ T G3 0 ': By the strong completeness of Gödel logic, there is a Heyting algebra valuation v such that v(T [ T G3 ) = 1 and v(') < 1: Since v might not be a world in M3 compose it with the Heyting algebra homomorphism: g(x) = (x + 1)=2 for x > 0; g(0) = 0: Then v 0 = g v belongs to M3 and we still have v 0 (T ) = 1, v 0 (') < 1: Applying Lemma 6.4 to v 0 we have e (v 0 ; T ) = 1; e (v 0 ; ') < 1: That is, M3 j=v0 T and M3 2v0 ': Hence, T 2GK ': By Lemma 6.2 this is enough. j=GK no longer coincides with j=Crisp for the language L3 as the following example illustrates. Example. The schema ::3' ! 3::' is not a theorem of G3 because it fails in the two worlds model hfa; bg; S; ei where Sab = 21 ; e(a; p) = e(b; p) = 1: since e(x; ::3p) = 1; and e(y; 3::') = 12 : However, it holds in all a-crisp models since in any one of them e(x; ::3') > 0 implies the existence of y such that Sxy = 1 and e(y; ') > 0: Thus, e(y; 3::') Sxy ( e(y; ')) = 1. Remark. An interesting question raised by one referee is whether the logic of 3 in a-crisp GK-models is axiomatizable. This is granted in the abstract sense (recursive enumerability of valid formulas) because the logic may be interpreted faithfully in Gödel predicate logic which is axiomatizable (cf. [16], [12]). We do not know an explicit axiomatization but the system G3 [f::3' ! 3::'g is a candidate because the new schema characterizes crisp frames: if (W; S) is not crisp pick x, y 2 S such that 0 < Sxy < 1, then the valuation e(y; p) = 1 and e(z; p) = 0 for every z 6= y, provides a counterexample to the schema since e(x; ::3p) = Sxy = 1 but e(x; 3::p) = Sxy < 1. 14
7
G3 has the …nite model property
For any sentence ' such that 0G3 ' we may construct a …nite counter-model inside M3 : Theorem 7.1 If 0G3 ' then there is a GK-model M with …nitely many worlds such that M 2GK ': Proof: It follows from the Claim in the proof of Lemma 6.4 that for all and v 2 M3 there is w 2 M3 such that v(3 ) = S vw w( ): (if v(3 ) = 0 any w works). Given , let f (v) be a function choosing one such w for each v. For any formula let r( ) (rank of ) be the nesting degree of 3 in , that is, the length of a longest chain of occurrences of 3 in the tree of . Given ' and a world (valuation) v0 in M3 ; let Sj be the set of subformulas of ' of rank j; for each j n = r('), and de…ne inductively the following sets of valuations: M0 = fv0 g
Mi+1 = ff (v) : v 2 Mi ; 3 2 Sn i g Clearly, M = [i n Mi has …nitely many worlds. Consider the model M';v0 induced in M by restricting e and S of M3 to M V ar and M M respectively, that we will call M for simplicity. Then for any formula 3 2 Sj and v 2 Mn j there is w 2 Mn (j 1) such that v(3 ) = S vw w( ); and thus v(3 ) supfS vw w( ) : w is a world in M g: This allows us to show by induction on j n that for all 2 Sj ; v 2 Mn j we have v( ) = eM (v; ): In particular, if 0`G3 '; and v0 is a world in M3 such that v0 (') < 1 then eM (v0 ; ') < 1; which shows M 2 ': The proof of the previous theorem still works if we assume the worlds of M are de…ned in the variables of ' only and the accessibility relation of M is de…ned by using subformulas of ': SM vw := min fw( ) ) v(3 )g: 2[Si
This means that M takes values in a …xed …nite subalgebra of [0; 1] depending only on '. Thus there are only …nitely many models to consider and the decidability of the fragment G3 follows.
15
8
Modal extensions
The modal systems we have considered so far correspond to minimal modal logic K, the logic of Gödel-Kripke models with an arbitrary accessibility fuzzy relation. We may consider also the fuzzy analogues of the classical modal systems D; T; S4 and S5 for each modal operator, usually presented syntactically as combinations of the following axioms: D T 4 B
: : : :
: ? 2' ! ' 2' ! 22' ' ! 2:2:'
D3 : T3 : 43 : B3 :
3> ' ! 3' 33' ! 3' ' ! :3:3'
and semantically by asking the frames of the Kripke models to satisfy corresponding structural properties. Notice that D follows from T and D3 from T3 : Call a fuzzy frame hW; Si serial if 8x 2 W 9y 2 W : Sxy = 1; re‡exive if Sxx = 1 for all x 2 W , (min)transitive if Sxy Syz Sxz for all x; y; z; and symmetric if Sxy = Syx for all x; y 2 W: Let Ser; Ref l, T rans; and Symm denote the classes of GK-models over frames satisfying, respectively, each one of the above properties, and let j=C denote validity and consequence with respect to models in the class C. Proposition 8.1 j=Ser D ,D3 ; j=Ref l T ,T3 ; j=T rans 4 ,43 ; and j=Symm B ,B3 . Proof: The validity of D ,D3 in serial models is immediate because e(x; 2?) = inf y f Sxyg and e(x; 3>) = supy Sxy: Assume Sxx = 1 for all x: (T ): e(x; 2') (Sxx ) e(x; ')) = e(x; '): (T ): e(x; 3') Sxx e(x; ') = e(x; '): Assume Sxy Syz Sxz for all x; y; z: (4 ): e(x; 2') Sxy Syz (Sxz ) e(z; ')) Sxz e(z; '): Hence, e(x; 2') Sxy (Syz ) e(z; ')): Taking the meet over z in the right hand side: e(x; 2') Sxy e(y; 2'); hence, e(x; 2') (Sxy ) e(y; 2')) for all y and thus e(x; 2') e(x; 22'): (43 ): for any x; y; z; Sxy Syz e(z; ') Sxz e(z; ') e(x; 3'); hence, Syz e(z; ') (Sxy ) e(x; 3')): Taking the join over z in the left; e(x; 3') (Sxy ) e(x; 3')); thus Sxy e(x; 3') e(x; 3')): Taking the join again in the left, e(x; 33') e(x; 3'): Assume Sxy = Syx for all x; y: (B ): we prove the validity of the stronger schema :' ! 2:2'. Assume e(x; :') > 0 then e(x; ') = 0: Take any y such that Sxy > 0; then e(y; 2') (Syx ) e(x; ')) = (Sxy ) 16
e(x; ')) = 0: Therefore, e(y; :2') = 1; and (Sxy ) e(y; :2')) = 1: This shows that x(2:2') = 1: (B3 ): suppose e(x; ') > e(x; :3:3') then e(x; :3:3') = 0 and e(x; 3:3') = 1: This means that there is y such that Sxy e(x; :3') > 0 thus Sxy > 0 and e(x; :3') = 1; hence e(y; 3') = 0; therefore, Syx e(x; ') = 0 which is absurd because Syx = Sxy > 0 and e(x; ') > 0 by construction. Theorem 8.1 (i) G +D and G3 +D3 are strongly complete for j=Ser . (ii) G +T and G3 +T3 are strongly complete for j=Ref l . (iii) G +4 and G3 +43 are strongly complete for j=T rans . (iv) GS4 := G +T +4 and GS43 := G3 +T3 +43 are strongly complete for j=Ref l\T rans : Proof: Soundness follows from Proposition 8.1. Completeness follows, in each case, by asking the worlds of the canonical models M and M3 introduced in the completeness proofs of G and G3 to satisfy the corresponding schemas. The key fact is that the schemas force the accessibility relations S vw and S3 vw to satisfy the respective properties. (i) If v(D ) = 1 in M then then e (v; ?) = v( ?) = 0 for any world v of M and necessarily there is w such that S vw > 0; but the model is a-crisp thus S3 vw = 1. If v(D3 ) = v(3>) = 1 in M3 then by the Claim in the proof of Lemma 6.4 there is w such that S vw = 1: (ii) If v(T ) = 1 then S vv = inf '2L fv( ' ! ')g = 1: If v(T3 ) = 1 then S3 vv = inf '2L fv(' ! 3')g = 1. (iii) If v(4 ) = 1 then v( ') v( ') and so S vv 0 S v 0 v 00
[(v( (v(
') ) v 0 ( ')) (v 0 ( ') ) v 00 ('))]
') ) v 00 ('))
(v( ') ) v 00 ('))
Taking the meet over ' in the last formula we get: S vv 0 S v 0 v 00 (iv) If v(43 ) = 1 then v(33') v(3') and thus S3 vv 0 S3 v 0 v 00
S vv 00 :
[(v 0 (3') ) v(33')) (v 00 (') ) v 0 (3'))] (v 00 (') ) v(33'))
(v 00 (') ) v(3'))
Taking the meet over ' in the last formula we get S3 vv 0 S3 v 0 v 00
S3 vv 00 :
It follows from the proof of Theorem 8.1 that for the given extensions of G we get completeness also with respect to the a-crisp models of the respective class. 17
Call a fuzzy frame hW; Si weakly serial if it satis…es 8x9y Sxy > 0, and let W Serial be the class of GK-models over weakly serial frames. Then it is easily seen that G +D is sound (and thus strongly complete) for j=W Serial but G3 +D3 is not. However, j=W Serial is axiomatized by G3 +f::3>g. Remark. An original motivations of the second author to study fuzzy modal logics was to interpret the possibility operator 3 in the class of Gödel frames Ref l \ T rans \ Symm as a notion of similarity in the sense of Godo and Rodríguez [14], and a reasonable conjecture was that GS53 = GS43 +B3 would axiomatize validity in models over these frames. Unfortunately, the axioms B , B3 do not seem to force symmetry in the canonical models and we have not been able to show completeness of G +B or G3 +B3 for j=Symm , nor completeness of GS53 or the analogue GS5 with respect to j=Ref l\T rans\Symm . Perhaps stronger symmetry axioms such as (' ! ) ! 2(2' ! ) 3(3' ! ) ! (' ! 3 ); which characterize symmetric frames, would do.
9
Adding truth constants
The previous results on strong completeness may be generalized to Pavelkastyle languages [19] with a set Q [0; 1] of truth values added as logical constants, provided Q is well-ordered under the usual order of [0; 1] and discrete in the usual topology of [0; 1]. These conditions include …nite sets and force Q to be at most countable. Without loss of generality, we assume Q contains 0 and 1; to be identi…ed with ? and >, respectively. The logical constant corresponding to r 2 Q will be denoted by r itself. Let L Q be L enriched with elements of Q as atomic constituents, and let G (Q) be the system in this language obtained by adding to the axioms and rule of G the axiom schemas R1 - R4 below. For all r; s 2 Q : R1. (book-keeping axioms) 0 ! ?; > ! 1 r ! s; if r s (r ! s) ! s; if s < r R2. r ! r R3. (r ! ) ! (r ! ) R4. (( ! r) ! r) ! (( ! r) ! r) 18
The system G3 (Q) in the analogue language L3Q is de…ned similarly by adding R1 and R5 - R7 below to G3 . R5. 3r ! r R6. 3(r ! ') ! (r ! 3') R7. 3((' ! r) ! r) ! ((3' ! r) ! r): The double negation shift axioms Z and Z3 become super‡uous in the extended systems due to R4 and R7, respectively; also F3 is super‡uous due to R5. GK-models are extended by de…ning e(x; r) = r at each world x, and validity j=GK ' is de…ned as before in terms of 1-satisfaction. Then R1 to R7 are easily seen to be valid. However, the consequence notion T j=GK ' given in De…nition 2.2 is too rough if there are more than two truth constants (consider 12 j=GK 0). We will utilize the …ner relation T j= GK ' for which it may be shown that G (Q) and G3 (Q) are strongly complete for countable theories if Q is well ordered and discrete. The same holds for the logics mentioned in Theorem 8.1. No conditions on Q are required for weak completeness. This extends substantially a result of Esteva, Godo and Nogera [7] on weak completeness of Gödel logic with rational truth constants. Discreteness of Q is necessary for strong completeness: if r is a limit point of Q then there is a strictly increasing or decreasing sequence of Q converging to r, say frn g increases to sup rn = r; then fr1 ! ; r2 ! ; r3 ! ::::g j=
GK
r!
but no …nite subset of premises can grants this, thus no formal proof is possible. Moreover, discreteness alone is not enough, since Q = fr1 < r2 < ::: ::: < q2 < q1 g with sup ri = inf qi is discrete and fr1 ! ; r2 ! ; :::
,
! q1 ;
! q2 ; :::g j=
GK
!
but no …nite subset of the premises yields the same consequence. Thus well order or a related conditions is needed. We give next the proof of strong completeness for G (Q), a re…nement of that given for G . The deduction theorem and lemmas 4.1 and 4.2 extend readily to the system G (Q): Moreover, any formula of L Q may be seen as a formula of Gödel logic over the vocabulary V ar [ Q [ L Q , and after de…ning T G (Q) = f :
is an axiom of G (Q)g [ f 19
: `G
(Q)
g
it may be shown that T `G
(Q)
' if and only if T [ T G (Q) ` ' in Gödel logic
Call a Gödel valuation v : V ar [ Q [ L Q ! [0; 1] normal if v(r) = r for all r 2 Q: Having v(T G (Q)) = 1 does not make v normal. However, the next lemmas show how to transform such a valuation to a normal one still satisfying T G (Q) and some other useful properties. As before, we will write v for the Gödel extension v: Lemma 9.1 Let v : V ar [ Q [ L Q ! [0; 1] be extended to all formulas of L Q according to the Gödel operations in [0; 1]: If v(R1) = 1; then v(0) = 0; v(1) = 1, and = minfr 2 Q : v(r) = 1g > 0: Moreover, v is weakly increasing in Q and strictly increasing in Q \ [0; ]. Proof: v(0) = 0 and v(1) = 1 by R1; exists and is positive due to the well ordering of Q. If r s in Q then v(r) v(s) by R1. If r < s in Q then v(r) < 1 and so v(s ! r) v(r) < 1 by R1; thus v(s) > v(r): Call a formula of L Q shy if any occurrence of r 2 Q r f0; 1g in the formula is under the scope of an occurrence of : For positive r 2 Q, let r be the supremum (in [0; 1]) of its predecessors in Q. Necessarily r < r because r is isolated, but r may not belong to Q. Lemma 9.2 (Normalization). Let v : V ar [ Q [ L Q ! [0; 1] be Gödel valuation satisfying R1 and let be de…ned as in the previous lemma, then there is a normal w : V ar [ Q [ L Q ! [0; 1] such that for any ' : 1. v(') = 1 ) w(') 2. v(') < 1 ) w(') < 3. v(') v( ) < 1 ) w(') w( ) 4. v(') < v( ) ) w(') < w( ) 5. For any shy formula '; v(') = 1 implies w(') = 1; and v( ! ') = 1 implies w( ! ') = 1: 6. v(T G (Q)) = 1 implies w(T G (Q)) = 1: 7. If r ; r are given so that r < r < r S < r for each positive r 2 Q; then = Q w may be chosen so that w(L Q ) Q [ r2Q [ r ; r ): Hence, no r 2 belongs to the image of w:
20
Proof: Given 0 < r 2 Q; let v(r) = sup[0;1] fv(s) : s < r; s 2 Qg: Clearly, v(r) v(r); moreover, v(r) = v(s); s 2 Q; if and only if s = r or s = r : It should be clear also that S [0; 1] = fv(r) ; v(r) : r 2 Qg [ (v(r) ; v(r)) r2Q\(0; ]
is a partition because v Q \ [0; ] is strictly increasing by the previous lemma and v( ) = 1. Given r ; r as in 7 choose an strictly increasing function g : [0; 1] ! [0; ) [ f1g satisfying. g(1) = 1 g(v(r)) = r for r 2 Q, v(r) < 1 (hence r < ): g((v(r) ; v(r))) = ( r ; r ) if v(r) < v(r) < 1. g(v(r) ) = r if v(r) 62 v(Q) (hence v(r) < v(r)) De…ne w : V ar [ Q [ L
Q
! [0; 1] as follows:
w( ) = g(v( )) if 2 V ar [ L w(r) = r for r 2 Q.
Q
Property 7 is insured for elements of V ar [ Q [ L Q by construction and it extends to all of L Q because the value of a formula under a Gödel valuation is identical to 0; 1; or the value of one of its atomic constituents. For the other properties: 1-2. By simultaneous induction in Gödel connectives, we show 1 and the following strengthening 20 of 2: v(') < 1 ) w(') = g(v(')) < : Atomic. For ' 2 V ar [ L Q by de…nition of w and g: For ' = r 2 Q: v(r) < 1 if and only if w(r) = r < by de…nition of ; and in the later case w(r) = r = g(v(r)) by the way g was chosen. Conjunction. If v(' ^ ) = 1 then v(') = v( ) = 1 and by inductive hypothesis w('^ ) = w(') w( ) : If v('^ ) < 1; say v(') v( ); then v(') < 1 and thus w(') = g(v(')) < by induction hypothesis. Moreover, w( ) w('): if v( ) = 1 because w( ) ; if v( ) < 1 because w( ) = g(v( )) g(v(')): Thus w(' ^ ) = w(') = g(v(' ^ )) < : Implication. Assume v(' ! ) = 1: If v( ) < 1 then v(') v( ) < 1; and by inductive hypothesis and monotonicity of g, w(') = g(v(')) g(v( )) = w( ); thus w(' ! ) = 1: If v( ) = 1 then w(' ! ) w( ) again by the induction hypothesis. Assume now v(' ! ) < 1; then v(') > v( ) < 1 and thus w( ) = g(v( )) < : Moreover, w(') > w( ): if v(') = 1 because w(') ; if v(') < 1 because w(') = g(v(')) > g(v( )): Therefore w(' ! ) = w( ) = g(v(' ! )) < : 3. v(') v( ) < 1 implies by 20 : w(') = g(v(')) g(v( )) = w( ). 21
4. v(') < v( ) implies v( ! ') = v(') < 1; hence w( ! ') < by 2, and thus w(') < w( ): 5. If ' is shy then ' = '0 ( 1 ; :::; n ) where '0 is Gödel formula and i 2 V ar [ L; therefore, w(') = '0 (w( 1 ); :::; w( n )) = '0 (g(v( 1 )); :::; g(v( n ))) = g(v('0 ( 1 ; :::; n )) = g(v(')) because g is an endomorphism of the Heyting algebra [0; 1]: Therefore, v(') = 1 implies w(') = g(1) = 1: Moreover, v( ! ') = 1 implies trivially w( ! ') = 1 when v(') = 1; and w( ) w(') when v(') < 1 by property 4: 6. The axioms of G give 1 under any Gödel valuation. The speci…c axioms of G are shy, w(R1) = 1 because w is the identity in Q; and R2, R3, R4.are of the form ! ' with ' shy. The other elements of T G (Q) are shy by construction. Canonical model M (Q): W : all normal valuations v : V ar [ Q [ L Q ! [0; 1] satisfying v(T G (Q)) = 1 and such S that there are r ; r (r 2 Q) with r < r < < r and Im(v) Q [ r r2Q; r>0 [ r ; r ): S vw = inf
2L
e (v; ) = v
Q
(v(
) ) w( )):
V ar:
Lemma 9.3 e (v; ) = v( ) for any world v of M (Q) and formula : Proof: As in the case of G ; it is enough to check inf w2W (S vw ) w(')) v( ') whenever v( ') < 1: This is done in two stages: Claim 1. If = v( ') < 1 there exists a Gödel valuation u : V ar [ Q [ L Q ! [0; 1] such that u(T G (Q)) = 1; u(') < 1 and for any and r 2 Q 1. u( ) = 1 if v( ) > 2. u(r) u( ) if r v( ) 3. u(r) < u( ) if r < v( ) and r 4. 2 [ ; ) if = minfr 2 Q : u(r) = 1g. Proof: Let T';v be the theory f : v( Then T';v 0G thus
) > g [ fr ! : r 2 Q; r v( )g [ f( ! r) ! r : r 2 Q; r ; r < v( (Q)
': Otherwise, 1 ; :::;
1 ; :::; k
k
`G
`G
(Q)
22
(Q)
':
' for some
)g: i
2 T';v and
But v( ) > for any other 2 T';v : for the …rst group of axioms, by construction; for the second, because v( ) r implies v( (r ! )) v(r ! ) = r ! v( ) = 1 by R3 and normality of v; for the third, because v( ) > r implies by R4 and normality: v( (( ! r) ! r)) (v( ) ! r) ! r = (r ! r) = 1. Hence, we obtain the contradiction < minf
1 ; :::;
kg
v( '):
Therefore, T';v ; T G (Q) 0 ', and we may use the strong completeness theorem of Gödel logic to get a valuation u : V ar [ Q [ L ! [0; 1] such that u(T';v [ T G (Q)) = 1 > u('). Conditions 1, 2 hold by construction, 3 is satis…ed because r = v( ') implies u(r) u(') < 1 by 2 and thus u(( ! r) ! r) = 1 implies u( ) > u(r): To verify 4, notice that r implies u(r) < 1 as just explained, thus > : On the other hand < r < implies v( r) v(r) = r > by R2, and thus u(r) = 1 by 1, contradicting the de…nition of . Therefore, : Claim 2. If = v( ') < 1 then for any M (Q)) such that (Svw ) w(')) < + :
> 0 there exists a world w of
Proof: According to Lemma 9.2, the valuation u of the previous claim may be transformed in a valuation w in M (Q) such that w(') < and the conditions on u become the following conditions on w: 1. w( ) if v( ) > : 2. r w( ) if r v( ) and r < (that is, r 3. r < w( ) if r < v( ) and r . 4. ; w(') 2 [ ; ).
).
Only 4 needs some explanation: if r < then r v( ') = and thus r w(') by 3 above, hence w('): Moreover, we may choose the parameters 0r ; 0r ; of w so that r < for 0 < r < , where
r
;
r
r
e (w; ') in the canonical model, hence T 2 GK(Q) ': Moreover, if the formulas in T have the form or ! ; with shy, then by Lemma 9.2-5 w(T ) = 1 and thus e (w; T ) = 1; hence T 2GK '. Corollary. For any Q and formula ' in L
Q:
`G
(Q)
' , j=GK ':
Proof: The condition in 4 of Theorem 9.1 is trivially satis…ed by empty T , and no condition on Q is required since it is enough to consider the …nitely many truth constants appearing in '. Note that GK-models with crisp accessibility relation are no enough for completeness of G (Q); for example, the formula ( 12 ! 12 ) _ 0 is invalid but is valid in all a-crisp GK-models. However, if Q denotes the topological closure of Q in [0; 1] (still countable or Q itself if Q is …nite) it may be shown that strong completeness of G (Q) holds with respect to GK-models where the accessibility relation takes values in Q only. For weak completeness only Q-valued accessibility has to be considered. 24
The proof of the completeness of G3 (Q) follows similar lines to that of G (Q); and again we have the …nite model property for this logic. Also the results of Section 8 transfer without di¢ culty to the systems with truth constants.
10
Final Comment
The main results of this paper were announced at the meeting on "Logic, Computability and Randomness", Cordoba, Argentina, Sept. 2004. Publication was delayed, aiming to axiomatize the full logic with both modal operators combined, which resulted elusive. It may be seen that the union of the systems G and G3 is not enough for that purpose. However, we have found recently that Fischer Servi [8] "connecting axioms": 3(' ! ) ! (2' ! 3 ) (3' ! 2 ) ! 2(' ! ) together with G [ G3 constitute a strongly complete axiomatization. This will appear elsewhere.
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[7] Francesc Esteva, Lluis Godo, and Carles Noguera: On rational weak nilpotent minimum logics, Journal of Multiple-Valued Logic and Soft Computing 12, Number 1-2 (2006) 9-32. [8] G. Fischer Servi: Axiomatizations for some intutitionistic modal logics, Rend. Sem. Mat. Polit de Torino 42 (1984) 179-194. [9] Melving Fitting: Many valued modal logics. Fundamenta Informaticae 15 (1991) 325-254. [10] Melving Fitting: Many valued modal logics, II. Fundamenta Informaticae 17 (1992) 55-73. [11] Josep M. Font: Modality and Possibility in some intuitionistic modal logics. Notre Dame Journal of Formal Logic 27, 4 (1986) 533-546. [12] Alfred Horn: Logic with truth values in a linearly ordered Heyting algebra. J. Symbolic Logic 34 (1969) 395-408. [13] George Metcalfe and Nicola Olivetti: Proof Systems for a Godel Modal Logic. In M. Giese and A. Waaler (editors), Proceedings of TABLEAUX 2009, volume 5607 of LNAI. Springer (2009) 265-279. [14] Lluis Godo and Ricardo O. Rodríguez: A fuzzy modal logic for similarity reasoning. In Guoqing Chen, Mingsheng Ying, and Kai-Yuan Cai (editors), Fuzzy Logic and Soft Computing, Vol. 6. Kluwer Academic, 1999. [15] C. Grefe: Fischer Servi´s intuitionistic modal logic has the …nite model property, In M. Kracht, M. de Rijke, H. Wansing, M. Zakharyaschev (editors), Advances in Modal Logic Vol.1, CSLI, Stanford, pp 85-98, 1998. [16] Petr Hájek: Metamathematics of fuzzy logic, in: Trends in Logic, Vol. 4, Kluwer Academic Publishers, Dordrecht, 1998. [17] Petr Hájek: Making fuzzy description logic more general Fuzzy Sets and Systems 154 (2005) 1–15 [18] H. Ono: On some intutionistic modal logics. Publications of the Research Institute for Mathematical Sciences, Kyoto University 13 (1977) 13-55. [19] J. Pavelka: On fuzzy logic I, II, III. Zeitschr. f. Math. Logik un Grundl. der Math. 25 (1979) 45-52, 119-134, 447-464. 26
[20] Frank Wolter: Superintuitionistic Companions of Classical Modal Logics. Studia Logica 58, 3 (1997) 229-295.
Xavier Caicedo Departamento de Matemáticas Universidad de los Andes A.A. 4976, Bogotá, Colombia [email protected] Ricardo Rodríguez Departamento de Computación, Fac. Ciencias Exactas y Naturales Universidad de Buenos Aires 1428 Buenos Aires, Argentina [email protected]
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