Caicedo Standard Godel Modal

Studia Logica (2010) 94: 189214DOI: 10.1007/s11225-010-9230-1 Springer 2010

Xavier Caicedo

Ricardo O. Rodriguez

Standard Godel Modal

Logics

Abstract. We prove strong completeness of the -version and the -version of a Godel

modal logic based on Kripke models where propositions at each world and the accessibility

relation are both innitely valued in the standard Godel algebra [0,1]. Some asymmetries

are revealed: validity in the rst logic is reducible to the class of frames having two-valued

accessibility relation and this logic does not enjoy the nite model property, while validity

in the second logic requires truly fuzzy accessibility relations and this logic 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.

Keywords: many-valued modal logics, Godel-Dummett logic, fuzzy Kripke semantics,

strong completeness.

1. Introduction

Sometimes it is needed in approximate reasoning to deal simultaneously withboth fuzziness of propositions and modalities, for instance one may try toassign a degree of truth to propositions like John is possibly tall or Johnis 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 alsothese and other kinds of information features like certainty, belief or simi-larity, which have a natural interpretation in terms of modalities. In thiscontext, it is natural to interpret fuzzy modal operators by means of Kripkemodels over fuzzy frames.

We address in this paper the case of the pure modal operators (necessi-tation and possibility) for standard Godel logic, one of the main systems offuzzy logic arising from Hajeks classication in [14]. For this purpose weconsider a many-valued version of Kripke semantics for modal logic whereboth propositions at each world and the accessibility relation are innitelyvalued in the standard Godel algebra [0,1].

We provide strongly complete axiomatizations for the -fragment andthe -fragment of the resulting minimal logic. These fragments are shownto behave quite asymmetrically. Validity in the rst one is determined by

Special Issue: Algebra and Probability in Many-Valued ReasoningEdited by Ioana Leustean and Vincenzo Marra

190 X. Caicedo and R.O. Rodriguez

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 thenumber of worlds or the number of truth values while the -fragment does.

We consider also the Godel analogues of the classical modal systems D,T, S4 and S5 for each modal operator and show that the rst three arecharacterized by the many-valued versions of the frame properties whichcharacterize their classical counterparts. Finally, we extend the strong com-pleteness results to Pavelka-style languages enriched with a set of explicittruth constants denoting a discrete well ordered set of truth values.

Our approach is related to Fittings [10] who considers Kripke modelstaking values in a xed nite Heyting algebra; however, Fittings proof sys-tems and completeness proofs depend essentially on niteness of the algebraand the fact that his languages contain constants for all the truth values ofthe algebra. We must relay on completely dierent methods.

Modal logics with an (intermediate) intuitionistic basis and Kripke stylesemantics have been investigated in a number of relevant papers (see Ono[18], Fischer Servi [8], Boszic and Dosen [5], Font [11], Wolter [20], from anextensive literature), but in all cases the models carry two or more crispaccessibility relations satisfying some commuting properties: a pre-order toaccount for the intuitionistic connectives and one or more binary relations toaccount for the modal operators. Our semantics has, instead, a single fuzzyaccessibility relation and does not seem reducible to those multi-relational se-mantics since the latter enjoy the nite model property for (cf. Grefe [13]).

Recently, Metcalfe and Olivetti [17] have given a proof of weak complete-ness 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-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 dierent choices of the t-norm. However,our methods and results do not generalize to the corresponding modal ver-sions of Lukasiewicz or product logics because they relay on the richnessof endomorphisms of the Godel algebra [0,1]. In fact, we do not know anycompleteness 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 Godel-Kripke semantics for the modal operators is similarly translatableinto fuzzy (Godel) description logic (cf. [15]), and thus our results throwlight on various fragments of this logic and certain Pavelka-style expansionsof them.

Standard Godel Modal Logics 191

We assume the reader is acquainted with modal and Godel logics andthe basic laws of Heyting algebras (cf. Chagrov and Zakharyaschev [6]).

2. Godel-Kripke models

The language L of propositional Godel modal logic is built from a setV ar of propositional variables, logical connectives symbols ,,, and themodal operator symbols and . Other connectives are dened:

:=

:= := (( ) ) (( ) )

:= ( ) ( ).

L and L will denote, respectively, the -fragment and the -fragment ofthe language.

As stated before, the semantics of Godel modal logic will be based onfuzzy Kripke models where the valuations at each world and also the acces-sibility relation between worlds are [0, 1]-valued. The symbols and willdenote the Godel t-norm in [0, 1] and its residuum, respectively:

a b = min{a, b}, a b =

{1, if a bb, otherwise

The maximum is denable, max{a, b} := ((a b) b) ((b a) a), andthe pseudo-complement is denoted a := a 0. This yields the standardGodel algebra; that is, the unique Heyting algebra structure in the linearlyordered interval.

Definition 2.1. A Godel-Kripke model (GK-model) will be a structureM = W,S, e 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) Sxy. e : W V ar [0, 1] is an arbitrary function (x, p) e(x, p).

The evaluations e(x,) : V ar [0, 1] are extended simultaneously toall formula in L by dening inductively at each world x:

e(x,) := 0

e(x, ) := e(x, ) e(x, )

e(x, ) := e(x, ) e(x, )e(x,) := infyW{Sxy e(y, )}


e(x,) := supyW {Sxy e(y, )}.

It follows that e(x, ) = max{e(x, ), e(x, )} and e(x,) = e(x, ).

The notions of a formula being true at a world x, valid in a modelM = W,S, e, or universally valid, are the usual ones:

is true in M at x, written M |=x , i e(x, ) = 1. is valid in M , written M |= , i M |=x at any world x of M. is GK-valid, written |=GK , if it is valid in all the GK-models.

Clearly, all valid schemas of Godel logic are GK-valid. In addition,

Proposition 2.1. The following modal schemas are GK-valid:

K ( ) ( )Z K ( ) ( ) (in fact, an equivalence)Z F

Proof. Let M = W,S, e be an arbitrary GK-model and x W.

(K) By Denition 2.1 and properties of the residuum we have for anyy 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,); thus, e(x,( )) e(x, ).

(Z) Utilizing the Heyting algebra identity: (x y) = (x y), wehave: e(x,) = e(x,) (Sxy e(y, )) = (Sxy e(y,)).Taking the meet over y, e(x,) e(x,).(K) By properties of suprema and distributivity of over max, e(())= supy{Sxy max{e(y, ), e(y, )}} = max{supy{Sxy e(y, )}, supy{Sxy e(y, )}}.(Z) Sxy e(y,) Sxy e(y, ) = (Sxy e(y, )) e(x,) = e(x,).

(F) e(x,) = supy{Sxy 0} = 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 :

RN :

.

do not preserve truth at a xed world. However,


Proposition 2.2. RN and RN preserve validity in any given model,thus they preserve GK-validity.

Proof. Let W,S, e be a GK-model. (RN) If e(x, ) = 1 for all x Wthen e(x,) = infy{Sxy e(y, )} = inf{1} = 1 for all x. (RN) Ife(x, ) = 1 for all x W then Sxy e(y, ) Sxy e(y, ) e(x,)for any y W Taking the join over y in the left hand side of the lastinequality, e(x,) e(x,).

Semantic consequence is dened for any theory T L, as follows:

Definition 2.2. T |=GK if and only if for any GK-model M and anyworld x of M, M |=x T implies M |=x .

Note that Modus Ponens preserves consequence but this is not the caseof the inference rules RN and RN.

An alternative notion of logical consequence arises naturally. Set e(x, T )= {e(x, ) : T} then:

Definition 2.3. T |=GK if and only if for any GK-model M and anyworld x in M, inf e(x, T ) e(x, ).

Clearly, |=GK implies |=GK , and it will follow from our completenesstheorems that both notions are equivalent for countable theories. This facthas been already observed for pure Godel logic by Baaz and Zach in [3].

3. On strong completeness of Godel logic

To prove strong completeness of the modal fragments L and L we willreduce the problem to pure Godel propositional logic.

In the rest of this paper L(X) will denote the language built from aset X of propositional variables and the connectives ,,. Let G be axed axiomatic calculus for Godel logic (also called Dummetts LC) on thislanguage, say the following one given by Hajek ([14], Def. 4.2.3.):

( ) (( ) ( ))

( )

( ) ( )

( ( )) (( ) )

(( ) ) ( ( ))

( )

(( ) ) ((( ) ) )


MP: From and , infer

The symbol will denote deductive inference in this calculus. It is wellknown that G is deductively equivalent to the intermediate logic obtainedby 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 Godel interpretation of the connectives. We will need thefollowing strong form of standard completeness for Godel logic:

Proposition 3.1. Let T be a countable theory and U a countable set offormulas of L(X) such that for every nite S U we have T

S then

there is a valuation v : X [0, 1] such that v() = 1 for all T andv() < 1 for each U.

Proof. Extend T to a prime theory T (that is, T implies T or T ) satisfying the same hypothesis with respect to U (this isstandard). The Lindenbaum algebra L(X)/T of T

is linearly ordered sinceby primality and the pre-linearity schema T or T .Moreover, the valuation v : X L(X)/T , v(x) = x/T is such that v(T ) =1, v() < 1 for all U. As T is countable we may assume X is countableand thus, being also countable, L(X)/T is embeddable in the Godel algebra[0, 1], therefore, we may assume v : X [0, 1].

Taking S = {} we obtain the usual formulation of completeness. Wecan not expect strong standard completeness of G for uncountable theories,as the following example illustrates.

Example. Set T = {(p p) q : < < 1} where 1 is the rstuncountable cardinal, then T q. Otherwise we would have q, for somenite = {(pi+1 pi) q : 1 i < n}, but this is not possible bysoundness of G, because the valuation v(q) = 12 , v(pi) =

12(1

1i+1) for

1 i n, makes v(pi) < v(pi+1) 0 there is w such that Svw = 1 and w() < + . Toachieve this we prove rst:

Claim. Let v be a world of M and be such that v() = < 1, thenthere exists a world u of M such that u() < 1 and(i) u() = 1 if v() > (ii) u() > 0 if v() > 0.


Proof. Assume v() = < 1 and set

T,v = { : v() > } { : v() > 0}

Notice that v() > for any T,v because v() > 0 impliesv() = 1, and thus v() = 1 since v satises axiom Z. Thisimplies that T,v G . Otherwise, 1, ..., k G for some i T,v andthus

1, ...,k G

by Lemma 4.1. Hence, by Lemma 4.2 and the previous observations,

< min{1, ...,k} v(),

a contradiction. By Lemma 4.3 we have T,v TG and by the count-ability of T,v TG we may use the completeness theorem of Godel logic(Proposition 3.1) to get a Godel valuation u : V ar L [0, 1] suchthat u(T,v) = 1 and u() < 1. Then u M and (i) holds by construc-tion. Moreover, (ii) is satised because u() = 1 and thus u() > 0 ifv() > 0. This ends the proof of the claim.

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 upreserves the value 1 of the formulas in TG and thus it belongs to M.Moreover, v() w() for all :

- if v() > because w() = g(u()) = g(1) = 1 by (i) above.- if 0 < v() because then 0 < u() 1 by (ii) above, and thus

w() = g(u()) (, + ) {1}.This means Svw = 1, and since u() < 1 we have, w() = g(u()) < + ,which shows (1).

Definition 4.1. Call a GK-model accessibility crisp (a-crisp in short) ifS : W W {0, 1}, and write T |=Crisp if the consequence relation holdsat each node of any a-crisp GK-model.

Theorem 4.2. For any countable theory T and formula in L the follow-ing are equivalent:(i) T G (ii) T |=GK (iii) T |=GK (iv) T |=Crisp .


Proof. By Lemma 4.2, it is enough to show (iv) (i). If T G thenT TG by Lemma 4.3, and by strong completeness of Godel logicthere is a valuation v : V ar L [0, 1] such that v(T ) = v(TG) = 1and v() < 1. Hence, v W by denition, e(v, T ) = v(T ) = 1, ande(v, ) = v() < 1 by Lemma 4.4, showing that M |=v T but M v .That is, T Crisp because the canonical model is a-crisp.

After the example in Section 3 we can not expect completeness withrespect to uncountable theories.

5. G does not have the nite model property

The following example shows that G does not have the nite model propertywith respect to GK-models. The reciprocal of axiom Z :

,

fails in the (a-crisp) model M = (N, S, e), where

Smn = 1 for all m,n, e(n, p) = 1n+1 for all n.

Indeed, e(n,p) = 1n+1 = 1 for all n and thus, e(0,p) = inf{1} = 1.

But e(0,p) = infnN{1 1

n+1} = 0, and thus e(0,p) = 0. However,

Proposition 5.1. is valid in any GK-model W,S, e withnite W .

Proof. Assume e(x,) > e(x,) then e(x,) < 1 and thuse(x,) = 0. This implies the existence of a sequence {yn}n W suchthat Sxyn > e(yn, ) for all n N and {e(yn, )}n converges to 0. If W isnite so is the range of the latter sequence and there must exist n such thate(yn, ) = 0. Then (Sxyn e(yn,)) = (Sxyn 0) = 0 and thuse(x,) = 0, a contradiction.

Remark. The proof of the previous proposition shows that is valid in all 0-witnessed GK-models, those where e(x,) = 0 implies theexistence of y such that e(y, ) = 0 < Sxy. In fact, G+{ } isstrongly complete for 0-witnessed (a-crisp) models. To see this, notice thatif 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 thatSvw = 1 and w() < ; hence, w() = 0. We do not know if this systemhas the nite model property.


6. Completeness of the -fragment

The system G results by adding to G the following axiom schemas and rulein the language L :

K: ( ) ( )

Z:

F:

RN: From infer

As in the case of the -fragment, in proofs with assumptions the rule RNis to be used in theorems only, and thus we have the deduction theorem DT,hence the rule:

Lemma 6.1. If G then G .

Also the soundness theorem:

Lemma 6.2. T G implies T |=GK , hence, T |=GK .

Moreover, if TG is the set of theorems of G then:

Lemma 6.3. T G if and only if T TG in Godel logic.

Canonical model M = (W, S, e). Let L = { : L},

then:

W is the set of valuations v : V ar L [0, 1] such that v(TG) = 1and its positive values have a positive lower bound:

infL

{v() : v() > 0} = > 0 (2)

when v is extended to L = L(V ar L) as a Godel valuation.

The fuzzy relation between worlds in M is given by

Svw := infL

{w() v()}.

e(v, p) := v(p) for any p V ar.

Lemma 6.4. For any world v in the canonical model M and any Lwe have e(v, ) = v().

Proof. The only non trivial step in a proof by induction on complex-ity of formulas of L is that of . By induction hypothesis, e

(v,)


= supw{Svw e(w,)} = supw{S

vw w()}, then we must showsupw{S

vw w()} = v(). By denition

Svw w() v(),

for any L and w W, then Svw w() v(), which yields taking

the join over w:e(v,) v().

The other inequality is trivial if v() = 0. For the case v() > 0, let w begiven as in the following claim then v() = = Svw w() e(v,),concluding the proof of the lemma.

Claim. If v is a world of M such that v() = > 0, there exists aworld w of M such that w() = 1 and S

vw = .

Proof. Assume v() = > 0 and dene

,v = { L : v() < } { : L, v() = 0}.

This set is not empty because v() = 0 by axiom F. Moreover, for anynite subset of ,v, say {1, ..., n} {1, ...,m}, we have

G

1 ... n 1 ... m.

Otherwise, we would have (Cf. Lemma 6.1)

G (1 ... n 1 ... m) RN1 ... n 1 ... m K1 ... n 1 ... m Z,

which would imply by Lemma 6.2

v() max({v(i) : 1 i n} {v(i) : 1 i m}) < ,

a contradiction. Therefore, we have by Lemma 6.3

TG, 1 ... n 1 ... m;

By Proposition 3.1 there is a Heyting algebra valuation u : V ar L

[0, 1] such that u() = u(TG) = 1 and u() < 1 for all ,v. Thus, usatises the further conditions:

(i) u() = 1(ii) u() < 1 if v() < , because then ,v


(iii) u() = 0 if v() = 0, because then ,v and so u() < 1which implies u() = 0.

Let g : [0, 1] [0, 1] be the strictly increasing function:

g(x) =

1 if x = 1(x + 1)/2 if 0 < x < 1

0 if x = 0

where is given by (2). Clearly the valuation w = gu inherits the properties(i), (ii) (iii) of u, with (ii) in the stronger form:

(ii) w() < if v() <

Moreover, w() > 0 implies w() > /2, by construction, and w(TG) = 1because g is a homomorphism of Heyting algebras, hence, w belongs to M.To see that Svw = , note that w() v() whenever v() < . If0 < v() because then w() < v() by (ii) and denition of . Ifv() = 0 because then w() = 0 by (iii). Since (w() v()) for v() , and (w() v()) = (1 ) = , we have Svw =infL{w() v()} = .

Theorem 6.1. For any countable theory T and formula in L, the fol-lowing are equivalent(i) T G (ii) T |=GK (iii) T |=GK .

Proof. Assume T G , then T TG . By the strong completeness ofGodel logic, there is a Heyting algebra valuation v such that v(T TG) = 1and v() < 1. Since v might not be a world in M compose it with theHeyting algebra homomorphism: g(x) = (x+1)/2 for x > 0, g(0) = 0. Thenv = g v belongs to M and we still have v

(T ) = 1, v() < 1. ApplyingLemma 6.4 to v we have e(v, T ) = 1, e(v, ) < 1. That is, M |=v Tand M v . Hence, T GK . By Lemma 6.2 this is enough.

|=GK no longer coincides with |=Crisp for the language L as the followingexample illustrates.

Example. The schema is not a theorem of G because itfails in the two worlds model {a, b}, S, e where

Sab = 12 , e(a, p) = e(b, p) = 1.


since e(x,p) = 1, and e(y,) = 12 . However, it holds in all a-crispmodels since then e(x,) > 0 implies the existence of y such that Sxy,e(y, ) > 0.Hence, Sxy = 1 and thus e(y,) Sxy ( e(y, )) = 1.

Remark. An interesting question raised by one referee is whether the logicof in a-crisp GK-models is axiomatizable. This is granted in the abstractsense (recursive enumerability of valid formulas) because the logic may beinterpreted faithfully in Godel predicate logic which is axiomatizable (cf.[14], [16]). We do not know an explicit axiomatization but the system G { } is a candidate because the new schema characterizescrisp frames: if (W,S) is not crisp pick x, y S such that 0 < Sxy < 1,then the valuation e(y, p) = 1 and e(z, p) = 0 for every z = y, providesa counterexample to the schema since e(x,p) = Sxy = 1 bute(x,p) = Sxy < 1.

7. G has the nite model property

For any sentence such that G we may construct a nite counter-modelinside M.

Theorem 7.1. If G then there is a GK-model M with nitely manyworlds such that M .

Proof. It follows from the Claim in the proof of Lemma 6.4 that for all and v M there is w M such that v() = S

vw w(). (if v() = 0any w works). Given , let f(v) be a function choosing one such w foreach v. For any formula let r() (rank of ) be the nesting degree of in ,that is, the length of a longest chain of occurrences of in the tree of .

Given and a world (valuation) v0 in M, let Sj be the set of sub-formulas of of rank j, for each j n = r(), and dene inductively thefollowing sets of valuations:

M0 = {v0}

Mi+1 = {f(v) : v Mi, Sni}

Clearly, M = inMi has nitely many worlds. Consider the modelM,v0 induced in M by restricting e

and S of M to M V ar andMM respectively, that we will call M for simplicity. Then for any formula Sj and v Mnj there is w Mn(j1) such that v() = S

vw w(),and thus

v() sup{Svw w() : w is a world in M}.


This allows us to show by induction on j n that for all Sj, v Mnjwe have v() = eM (v, ). In particular, if G , and v0 is a world in Msuch that v0() < 1 then eM (v0, ) < 1, which shows M .

The proof of the previous theorem still works if we assume the worldsof M are valuations dened in the variables of only and the accessibilityrelation of M is dened by using subformulas of :

SMvw := minSi

{w() v()}.

This means that M takes values in a xed nite subalgebra of [0, 1] dependingonly on . Thus there are only nitely many models to consider and thedecidability of the fragment G follows.

8. Modal extensions

The modal systems we have considered so far correspond to minimal modallogic K, the logic of Godel-Kripke models with an arbitrary accessibilityfuzzy relation. We may consider also the fuzzy analogues of the classicalmodal systems D,T, S4 and S5 for each modal operator, usually presentedsyntactically as combinations of the following axioms:

D: D: T: T: 4: 4: B: B:

and semantically by asking the frames of the Kripke models to satisfy cor-responding structural properties. Notice that D follows from T and Dfrom T.

Call a fuzzy frame W,S serial if x W y W : Sxy = 1, reexiveif Sxx = 1 for all x W , (min)transitive if Sxy Syz Sxz for all x, y, z,and symmetric if Sxy = Syx for all x, y W.

Let Ser, Refl, Trans, and Symm denote the classes of GK-models overframes satisfying, respectively, each one of the above properties, and let |=Cdenote validity and consequence with respect to models in the class C.

Proposition 8.1. |=Ser D,D; |=Refl T,T; |=Trans 4,4; and |=SymmB,B.

Proof. The validity of D,D in serial models is immediate becausee(x,) = infy{Sxy} and e(x,) = supy Sxy.


Assume Sxx = 1 for all x. (T): e(x,) (Sxx e(x, )) = e(x, ).(T): e(x,) Sxx e(x, ) = e(x, ).

Assume Sxy Syz Sxz for all x, y, z. (4): e(x,) Sxy Syz (Sxz e(z, )) Sxz e(z, ). Hence, e(x,) Sxy (Syz e(z, )).Taking the meet over z in the right hand side: e(x,) Sxy e(y,);hence, e(x,) (Sxy e(y,)) for all y and thus e(x,) e(x,).(4): for any x, y, z, Sxy Syz e(z, ) Sxz e(z, ) e(x,), hence, Syz e(z, ) (Sxy e(x,)). Taking the join over z in the left, e(x,) (Sxy e(x,)), thus Sxy e(x,) e(x,)). Taking the join againin the left, e(x,) e(x,).

Assume Sxy = Syx for all x, y. (B): we prove the validity of thestronger schema . Assume e(x,) > 0 then e(x, ) = 0. Takeany y such that Sxy > 0, then e(y,) (Syx e(x, )) = (Sxy e(x, )) = 0. Therefore, e(y,) = 1, and (Sxy e(y,)) = 1. Thisshows that x() = 1. (B): suppose e(x, ) > e(x,) thene(x,) = 0 and e(x,) = 1. This means that there is y such thatSxy e(x,) > 0 thus Sxy > 0 and e(x,) = 1, hence e(y,) = 0,therefore, Syx e(x, ) = 0 which is absurd because Syx = Sxy > 0 ande(x, ) > 0 by construction.

Theorem 8.1.

(i) G+D and G+D are strongly complete for |=Ser.

(ii) G+T and G+T are strongly complete for |=Refl.

(iii) G+4 and G+4 are strongly complete for |=Trans.

(iv) GS4 := G+T+4 and GS4 := G+T+4 are strongly completefor |=ReflTrans .

Proof. Soundness follows from Proposition 8.1. Completeness follows, ineach case, by asking the worlds of the canonical models M and M intro-duced in the completeness proofs of G and G to satisfy the correspondingschemas. The key fact is that the schemas force the accessibility relationsSvw and S

vw to satisfy the respective properties.

(i) If v(D) = 1 inM then then e(v,) = v() = 0 for any world

v of M and necessarily there is w such that Svw > 0, but the model is

a-crisp thus Svw = 1. If v(D) = v() = 1 in M then by the Claim

in the proof of Lemma 6.4 there is w such that Svw = 1.

(ii) If v(T) = 1 then Svv = infL{v( )} = 1. If v(T) = 1

then Svv = infL{v( )} = 1.

(iii) If v(4) = 1 then v() v() and so


Svv Sv

v [(v() v()) (v() v())]

(v() v()) (v() v())

Taking the meet over in the last formula we get: Svv S

vv S

vv.

(iv) If v(4) = 1 then v() v() and thus

Svv Sv

v [(v() v()) (v() v())]

(v() v()) (v() v())

Taking themeet over in the last formula we get Svv Sv

v Svv.

It follows from the proof of Theorem 8.1 that for the given extensionsof G we get completeness also with respect to the a-crisp models of therespective class.

Call a fuzzy frame W,S weakly serial if it satises xy Sxy > 0, andlet WSerial be the class of GK-models over weakly serial frames. Then it iseasily seen that G+D is sound (and thus strongly complete) for |=WSerialbut G+D is not. However, |=WSerial is axiomatized by G+{}.

Remark. An original motivations of the second author to study fuzzy modallogics was to interpret the possibility operator in the class of Godel framesRefl Trans Symm as a notion of similarity in the sense of Godo andRodrguez [12], and a reasonable conjecture was that GS5 = GS4+Bwould axiomatize validity in models over these frames. Unfortunately, theaxioms B, B do not seem to force symmetry in the canonical modelsand we have not been able to show completeness of G+B or G+B for|=Symm, nor completeness of GS5 or the analogue GS5 with respect to|=ReflTransSymm. Perhaps stronger symmetry axioms such as

( ) ( )( ) ( ),

which characterize symmetric frames, would do.

9. Adding truth constants

The previous results on strong completeness may be generalized to Pavelka-style languages [19] with a set Q [0, 1] of truth values added as logicalconstants, provided Q is well-ordered under the usual order of [0, 1] anddiscrete in the usual topology of [0, 1]. These conditions are satised bynite sets and force Q to be at most countable.


Without loss of generality, we assume Q contains 0 and 1, to be identiedwith and , respectively. The logical constant corresponding to r Qwill be denoted by r itself.

Let LQ be L enriched with elements of Q as atomic constituents, andlet G(Q) be the system in this language obtained by adding to the axiomsand rule of G the axiom schemas R1 - R4 below.

For all r, s Q :

R1. (book-keeping axioms)0 , 1r s, if r s(r s) s, if s < r

R2. r r

R3. (r ) (r )

R4. (( r) r) (( r) r)

The system G(Q) in the analogue language LQ is dened similarly byadding R1 and R5 - R7 below to G.

R5. r r

R6. (r ) (r )

R7. (( r) r) (( r) r).

The double negation shift axioms Z and Z become superuous in theextended systems due to R4 and R7, respectively; also F is superuousdue to R5.

GK-models are extended by dening e(x, r) = r at each world x, andvalidity |=GK is dened as before in terms of 1-satisfaction. Then R1 toR7 are easily seen to be valid.

However, the consequence notion T |=GK given in Denition 2.2 is toorough if there are two or more truth constants (consider 12 |=GK 0). We willutilize the ner relation T |=GK for which it may be shown that G(Q)and G(Q) are strongly complete for countable theories if Q is well orderedand discrete, and the same holds for the logics mentioned in Theorem 8.1.No conditions are required on Q to obtain weak completeness. Thus theseresults extend substantially a result of Esteva, Godo and Nogera [7] on weakcompleteness of Godel logic with rational truth constants.

Discreteness of Q is necessary for strong completeness: if r is a limitpoint of Q then there is a strictly increasing or decreasing sequence of Qconverging to r, say {rn} increases to sup rn = r, then

{r1 , r2 , r3 ....} |=GK r


but no nite subset of premises can grants this, thus no formal proof ispossible. But discreteness alone is not enough, since Q = {r1 < r2 < ...... < q2 < q1} with sup ri = inf qi is discrete and

{r1 , r2 , ... , q1, q2, ...} |=GK

but no nite subset of the premises yields the same consequence. Thus wellorder or a related conditions is needed.

We give next a proof of strong completeness for G(Q), a renement ofthat given for G. The deduction theorem and lemmas 4.1 and 4.2 extendreadily to the system G(Q). Moreover, any formula of LQ may be seen asa formula of Godel logic over the vocabulary V ar Q LQ, and afterdening

TG(Q) = { : is an axiom of G(Q)} { : G(Q) }

it may be shown that

T G(Q) if and only if T TG(Q) in Godel logic.

Definition 9.1. Call a Godel valuation v : V arQLQ [0, 1] normalif v(r) = r for all r Q.

Having v(TG(Q)) = 1 does not make v normal. However, the next lem-mas show how to transform such a valuation to a normal one still satisfyingTG(Q) and some other useful properties. As before, we will write v for theGodel extension v.

Lemma 9.1. Let v : V ar QLQ [0, 1] be extended to all formulas ofLQ according to the Godel operations in [0, 1]. If v(R1) = 1, then v(0) = 0,v(1) = 1, and

= min{r Q : v(r) = 1} > 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 thewell ordering of Q. If r s in Q then v(r) v(s) by R1. If r < s in Qthen v(r) < 1 and so v(s r) v(r) < 1 by R1, thus v(s) > v(r).

Call a formula of LQ shy if any occurrence of r Q {0, 1} in theformula is under the scope of an occurrence of .

For positive r Q, let r be the supremum (in [0, 1]) of its predecessorsin Q. Necessarily r < r because r is isolated, but r may not belong to Q.


Lemma 9.2. (Normalization). Let v : V ar Q LQ [0, 1] be Godelvaluation satisfying R1 and let be dened as in the previous lemma, thenthere is a normal w : V ar Q LQ [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( ) = 1implies w( ) = 1.6. v(TG(Q)) = 1 implies w(TG(Q)) = 1.7. If r , r are given so that r

< r < r < r for each positive r Q, thenw may be chosen so that w(LQ) Q

rQ[

r , r). Hence, no r

/ Qbelongs to the image of w.

Proof. Given 0 < r Q, let v(r) = sup[0,1]{v(s) : s < r, s Q}. Clearly,v(r) v(r); moreover, v(r) = v(s), s Q, if and only if s = r or s = r.It should be clear also that

[0, 1] = {v(r), v(r) : r Q}

rQ(0,]

(v(r), v(r))

is a partition because v Q [0, ] is strictly increasing by the previouslemma and v() = 1. Given r , r as in 7 choose an strictly increasingfunction g : [0, 1] [0, ) {1} satisfying.

g(1) = 1g(v(r)) = r for r 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)

v(Q) (hence v(r) < v(r))

Dene w : V ar Q LQ [0, 1] as follows:

w() = g(v()) if V ar LQw(r) = r for r Q.

Property 7 is insured for elements of V arQLQ by construction and itextends to all of LQ because the value of a formula under a Godel valuationis identical to 0, 1, or the value of one of its atomic constituents. For theother properties:

1-2. By simultaneous induction in Godel connectives, we show 1 andthe following strengthening 2 of 2: v() < 1 w() = g(v()) < .

Atomic. For V ar LQ by denition of w and g. For = r Q:v(r) < 1 if and only if w(r) = r < by denition of , and in the later casew(r) = r = g(v(r)) by the way g was chosen.


Conjunction. If v( ) = 1 then v() = v() = 1 and by inductivehypothesis w() = w()w() . If v() < 1, say v() v(), thenv() < 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() v() < 1 and thus w() = g(v()) < . Moreover, w() > w(): ifv() = 1 because w() , if v() < 1 because w() = g(v()) > g(v()).Therefore w( ) = w() = g(v( )) < .

3. v() v() < 1 implies by 2: w() = g(v()) g(v()) = w().

4. v() < v() implies v( ) = v() < 1, hence w( ) < by 2, and thus w() < w().

5. If is shy then = (1, ..., n) where is Godel for-

mula and i V ar L; therefore, w() = (w(1), ..., w(n)) =

(g(v(1)), ..., g(v(n))) = g(v((1, ..., n)) = g(v()) because g is an

endomorphism of the Heyting algebra [0, 1]. Therefore, v() = 1 impliesw() = g(1) = 1. Moreover, v( ) = 1 implies trivially w( ) = 1when v() = 1, and w() w() when v() < 1 by property 4.

6. The axioms of G give 1 under any Godel valuation. The specicaxioms 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 TG(Q)are shy by construction.

Canonical model M(Q):

W : all normal valuations v : V ar Q LQ [0, 1] satisfyingv(TG(Q)) = 1 and such that there are

r , r (r Q) with r

< r < r < rand Im(v) Q

rQ, r>0[

r , r).

Svw = infLQ(v() w()).

e(v,) = v 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 infwW (Svw

w()) v() whenever v() < 1. This is done in two stages:

Claim 1. If = v() < 1 there exists a Godel valuation u : V ar Q LQ [0, 1] such that u(TG(Q)) = 1, u() < 1 and for any and r Q


1. u() = 1 if v() > 2. u(r) u() if r v()3. u(r) < u() if r < v() and r 4. [, ) if = min{r Q : u(r) = 1}.

Proof. Let T,v be the theory

{ : v() > } {r : r Q, r v()} {( r) r : r Q, r , r < v()}.

Then T,v G(Q) . Otherwise, 1, ..., k G(Q) for some i T,v andthus

1, ...,k G(Q) .

But v() > for any other T,v: for the rst group of axioms, byconstruction; 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

< min{1, ...,k} v().

Therefore, T,v, TG(Q) , and we may use the strong completeness the-orem of Godel logic to get a valuation u : V ar Q L [0, 1] such thatu(T,v TG(Q)) = 1 > u(). Conditions 1, 2 hold by construction, 3 issatised because r = v() implies u(r) u() < 1 by 2 and thusu(( 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, contradictingthe denition of . Therefore, .

Claim 2. If = v() < 1 then for any > 0 there exists a world w ofM(Q)) such that (Svw w()) < + .

Proof. According to Lemma 9.2, the valuation u of the previous claimmay be transformed in a valuation w in M(Q) such that w() < and theconditions 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() [, ).


Only 4 needs some explanation: if r < then r v() = andthus r w() by 3 above, hence w(). Moreover, we may choose theparameters r ,

r, of w so that

r < r < r r <

r < r

for 0 < r < , where r , r are the pairs associated to v, and

< < + , .

Then we have w() < < + . To conclude (Svw w()) = w() this follows from 1 above. If v() < we have the cases:

(i) r < v() < r , then r w() by 2. If r Q we apply 3to conclude that r < w(), and if r Q we may conclude similarly thatr < w() because r is not in the image of w. In any case, v() r r w() and v() w() = 1

(ii) v() = r < , then r w() by 2 above.

Before stating the completeness theorem, we introduce an alternativeconsequence relation:

Definition 9.2. T |=GK(Q) i sup[0,1]{r Q : r e(x, T )} e(x, ) atany world x of any GK-model.

Obviously, T |=GK implies T |=GK(Q) . In fact, both notions coin-cide, and they coincide with |=GK under mild conditions on T.

Theorem 9.3. The following are equivalent for T, in LQ:1. T G(Q) 2. T |=GK 3. T |=GK(Q) In case any formula in T is shy or has the form with shy:4. T |=GK .

Proof. Since T G(Q) T |=GK T |=GK(Q) T |=GK we have to prove only 3 1 and 4 1 (for shy T ). If T G(Q) thenTTG(Q) , and there is a Godel valuation v such that v(T TG(Q)) =1 and v() < 1. By the Normalization Lemma 9.2 we may transform v to win M(Q) so that inf w(T ) > w() for certain Q. By the previouslemma, this means inf e(w, T ) > e(w,) in the canonical model, henceT 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; henceT GK .


Corollary. For any Q and formula in LQ: G(Q) |=GK .

Proof. The condition in 4 of Theorem 9.3 is trivially satised by empty T ,and no condition on Q is required since it is enough to consider the nitelymany truth constants appearing in .

Note that GK-models with crisp accessibility relation are no enough forcompleteness 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 topologicalclosure of Q in [0, 1] (still countable or Q itself if Q is nite) it may be shownthat strong completeness of G(Q) holds with respect to GK-models wherethe accessibility relation takes values in Q only. For weak completeness onlyQ-valued accessibility has to be considered.

The proof of the completeness of G(Q) follows similar lines to that ofG(Q), and again we have the nite model property for this logic. Alsothe results of Section 8 transfer without diculty to the systems with truthconstants.

10. Final comment

The main results of this paper were announced at the meeting on Logic,Computability and Randomness, Cordoba, Argentina, Sept. 2004. Pub-lication was delayed, aiming to axiomatize the full logic with both modaloperators combined, which resulted elusive. It may be seen that the unionof the systems G and G is not enough for that purpose. However, we havefound recently that Fischer Servi [8] connecting axioms:

( ) ( )( ) ( )

together with G G constitute a strongly complete axiomatization. Thiswill appear elsewhere.

Acknowledgements. The authors are indebted to F. Bou, F. Esteva andL. Godo for helpful comments on dierent drafts of this paper. The sec-ond author acknowledges partial support of Argentinean projects: PIP 112-200801-02543 2009-2011 and CyT-UBA X484.


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Xavier Caicedo

Departamento de MatematicasUniversidad de los AndesA.A. 4976, Bogota, [email protected]

Ricardo O. Rodrguez

Departamento de ComputacionUniversidad de Buenos Aires1428 Buenos Aires, [email protected]

Standard Goumldel Modal LogicsAbstract1. Introduction2. Goumldel-Kripke models3. On strong completeness of Goumldel logic4. Completeness of the square-fragment5. gsquare does not have the finite model property6. Completeness of the lozenge-fragment7. glozenge has the finite model property8. Modal extensions9. Adding truth constants10. Final commentAcknowledgementsReferences

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