Secure and Optimal Base Contraction inGraded !ukasiewicz Logics

Pere PARDO a,1, Pilar DELLUNDE a,b and Lluís GODO a

a Institut d’Investigació en Intel·ligència Artificial (IIIA - CSIC)b Universitat Autònoma de Barcelona (UAB)

Abstract. The operation of base contraction was successfully characterized for avery general class of logics using the notion of remainder sets. Nevertheless, in thegeneral case, this notion is inadequate for revision, where it is replaced by maximalconsistent subsets. A natural question is whether this latter notion allows for a defi-nition of contraction-like operators and, in case it does, what differences there existw.r.t. standard contraction. We make some steps towards this direction for the caseof graded expansions of one of the most prominent fuzzy logics, !ukasiewicz logic.We characterize contraction operators with a fixed security-threshold ! > 0; weprove soundness of (an optimal) "-contraction operation, and a collapse theoremfrom "- to some !-contraction for finite theories.

Keywords. Base contraction, deduction theorem, fuzzy logics, !ukasiewicz logic

Introduction

The field of Belief Change studies how can a set of sentences -in some fixed logic S- (i)adapt to make room for some new input in a consistent way (revision), or (ii) stop makingsome old claim provable anymore (contraction). This field originated in the work ofAlchourrón, Gärdenfors and Makinson on theory change [1], i.e. for deductively closedsets of sentences. In this paper we focus rather on base belief change. A base is a simplya set of sentences T in the language of S, that is, it is not required to be closed withrespect to the consequence relation CnS associated to the logic S. (For convenience,in this paper we will also use theory to refer to bases; this convention does not applyto theory change, though.) Base belief change is an important area of research that hasattained a high level of generality in its objects of study. It is known that for any logicsatisfying rather weak conditions, base belief change operators (revision and contraction)can be characterized by the method of partial meet, also introduced in [1] in their studyof theory change. There exist in the literature many other variants of change operations.We refer the reader to [9] for an overview.

Formal results for logics satisfying the deduction theorem (e.g. classical proposi-tional logic) are extremely simple and elegant. This simplicity is lost when we move to amore general classes of logics. For instance, identity between change operators defined

1Corresponding author: Pere Pardo, IIIA - CSIC, Campus UAB s/n, E-08193 Bellaterra (Catalonia), Spain;e-mail: pardo@iiia.csic.es

by (i) remainders, or (ii) maximal consistent subsets (coherers, henceforth), holds forlogics with the deduction theorem but fails in general. In the general case, remainders arenatural candidates to understand contraction [6], while coherers are the natural choicefor the case of revision [6,2,7]. Nonetheless, researching (slightly different) change op-erators from the non-natural tool is interesting in itself.

In this paper we consider the problem of belief base contraction for one of the mostprominent fuzzy logics, !ukasiewicz logic, which lacks the classical form of deduc-tion theorem. Indeed, we make some steps towards a consistency-based definition ofcontraction-like operators in !ukasiewicz t-norm based fuzzy logic !(C) expanded withtruth-constants r for each r ! C " [0, 1], for a suitable (countable) set C, see [3]. RationalPavelka logic RPL, introduced in [5] (simplifying previous work in [8] for C = [0, 1]),corresponds to the particular case of taking C = [0, 1] #Q. In this kind of graded logics,it makes sense to consider contraction-like operators with a security-threshold ! > 0.We characterize first this class of !-contraction operators, for ! > 0. This is a first steptowards an optimal contraction operation, "-contraction, which is proved to be soundwith respect to a set of axioms which are slightly different to the standard, remainder-based contraction from Hansson and Wasserman [6]. Finally, a finite-collapse result isproved showing that for each finite theory, any "-contraction operator collapses to some!-contraction operator.

1. Preliminaries: the (local) deduction property

One of the main differences between the original results and the current, more general,point of view originates in the deduction properties assumed to hold in the class of logics2

where a characterization is possible. In more detail, the logics considered originally wereassumed to have the (classical) deduction property, in other words, were assumed to sat-isfy the (classical) deduction theorem. However, many non-classical logics S do not havea contractive conjunction. This is the case in most t-norm based fuzzy logics [5], wherethe conjunction & is interpreted by a (left-continuous) t-norm $ %= min, e.g. !ukasiewiczor Product logics3 This means that !S # & #n, where #n stands for #& · · ·&# (ntimes). These logics satisfy a weaker property, the so-called local deduction theorem,defined next.

Definition 1.1. A logic S satisfies the deduction theorem iff for all T " Fm, and #, $ !Fm, we have: T ' {#} (S $ ) T (S # & $. Logic S satisfies the local deductiontheorem iff for all T, #, $ as above, T ' {#} (S $ ) *n ! " s.t. T (S #n & $.

Under the deduction theorem, the original tool used to define partial meet, the re-mainder sets, becomes dual to that of maximal consistent subsets of T (introduced in[10]; see also [2])

2Following [4], we define a logic as a finitary and structural consequence relation !S" P(Fm) # Fm,for some algebra of formulas Fm. That is, S satisfies (1) If # $ ! then ! !S #, (2) If ! !S # and ! " "then " !S #, (3) If ! !S # and for every $ $ !, " !S $ then " !S # (consequence relation); (4) If! !S # then for some finite !0 " ! we have !0 !S # (finitarity); (5) If ! !S # then e[!] !S e(#) for allsubstitutions e $ Hom(Fm,Fm) (structurality). In addition, we assume that logics contain the followingsymbols in its language: implication %, conjunction & and falsum 0. In this setting, the negation connective¬ is assumed to be defined as ¬# := # % 0.

3Determined respectively by the t-norms x &! y = max(x + y ' 1, 0) and x &! y = x · y.

Definition 1.2. Let S be some finitary monotonic logic, and T " FmS be some theoryand # ! Fm some input. A remainder set X for T and # is a subset of T "-maximalw.r.t. the property: X !S #. A coherer set X for T and # is a subset T "-maximalw.r.t. the property: X ' {#} !S 0. We denote by Rem(T, #) and Con(T, #) the set ofremainders and coherers of T and #.

Lemma 1.3. Let S be some logic having the deduction property. Then Rem(T,¬#) =Con(T, #).

Definition 1.4. For a given set X of families of sets, a selection function % is a functionselecting a non-empty subset of each family in X.

% : X +&!

X!XP(X), with , %= %(X) " X

A Rem-selection function for a theory T takes X = {Rem(T, #) | # ! Fm}. Analo-gously, a Con-selection function for some theory T takes X = {Con(T, #) | # ! Fm}.

Each Rem-selection function % induces a contraction operator by means of the fol-lowing definition

T -! # ="

%(Rem(T, #))

and each Con-selection function % gives rise to a revision operator of the form

T !! # = {#} '"

%(Con(T, #)) .

The operation of expansion is simply defined as T.# = T'{#}. Observe this operationdoes not depend on any selection function; but in contrast to the other operations, theexpansion operation cannot be guaranteed to output consistent theories, even if input #is consistent.

Now, assume remainders (unprovability) to be the natural tool for partial meet con-traction, while coherers naturally define partial meet revision. The following identitiesremain (provably) true under this assumption.

Theorem 1.5. (Levi and Harper identities) If S has the deduction property, then:

(Levy) T !! # = (T -!! ¬#). # for some Rem-s.f. %"

(Harper) T -! ¬# = (T !!! ¬#) # T for some Con-s.f. %"

The following example shows this coincidence between Rem-based contractionsand Con-based revisions need not hold for logics satisfying only the local deductiontheorem, as e.g. in !ukasiewicz logic.

Exemple 1.6. (In Lukasiewicz logic) Let T = {p2 &! 0}. We have Rem(T, p &! 0) ={T}, but Con(T, p) = {,}. Since there is only one remainder for T , namely T itself,applying Levy’s RHS leads to (T - ¬p). p = T ' {p} which is an inconsistent base4.

4The reason is simple: since p $ T ( {p} we have T ( {p} !! p2; but since T ( {p} !! p2 %! 0, bymodus ponens with we derive that T ( {p} !! 0.

The next result, whose proof depends upon Zorn’s Lemma (i.e. the Axiom of Choicein ZFC set theory), is extensively used later on.

Lemma 1.7. (From [7]) Let S be some finitary logic and T0 be a theory. For any X "T0, if X ' T1 is consistent, then X can be extended to some Y with Y ! Con(T0, T1).

In the remaining of the paper we make some steps towards obtaining consistency-based contraction operators for well particularly suited fuzzy logics, namely, the logics!(C), expansions of !ukasiewicz logic with truth constants from C, Rational Pavelkalogic RPL being the particular case when C = [0, 1] # Q. This is motivated by thepossibility of defining an a Con-based operation of contraction which differs from theRem-based operation as characterized by Hansson and Wasserman.

The idea, roughly, is as follows: we first characterize !-contraction, which is a sub-optimal operation of contraction, in the sense that a security-threshold ! > 0 must befixed beforehand. Then we try to recover optimality in contraction by defining anotheroperation, "-contraction, as the limit of an infinite series of !-contraction operators.

2. Success-based contraction: !-contraction

In what follows we consider a logic !(C) for a some given suitable algebra C of truthconstants. We will refer to graded sentences to !(C)-formulas of the kind r &! #, andwe will denote them as a pair (#, r). Due to !ukasiewicz semantics for the implication&!, a formula r &! # is 1-true (i.e. takes truth-value 1) in a given interpretation when-ever the truth-value of # in that interpretation is greater or equal than r. Therefore, thenotation (#, r) is a convenient form of intuitively expressing that the truth-value of # isat least r.

Next we introduce a graded negation ¬" for graded sentences defined as follows.

Definition 2.1. For a given graded sentence (#, r) and ! ! [0, r] # C, we define

¬"(#, r) = (¬#, 1+ r + !)

According to the above interpretation of pairs, ¬"(#, r) expresses that the value of# is smaller or equal than r + !.

In the remaining of the paper, we will simply denote by selection function for atheory T a Con-selection function % for T , taking X = {Con(T, #) | # ! Fm} anddefining a base !-contraction operator for T :

T -"! (#, r) =

"%(Con(T,¬"(#, r)))

Definition 2.2. Let T be a base. We define a (partial meet) base !-contraction operatorfor T as a function -" : Fm& P(Fm) such that there is some selection function % forT with

T -" (#, r) = T -"! (#, r), for any (#, r) ! Fm.

In general, it is important that the !-parameter is strictly positive, i.e. ! > 0. Thisenforces the property of Success to hold (T - # %( #). For the case ! = 0, consider thenext example.

Example 2.3. Let ! = 0 and T = {#} with # = (p, r). Then Con(T, (¬#, 1 + r)) ={T}. Hence, % is unique and contraction becomes

"{T} = T . But then, the axiom of

Success is not fulfilled, since T -0 (p, r) (!(C) (p, r).

Consider the next set of axioms for a contraction-like operator- for a base T , where! ! [0, r] # C :

(G1") T - (#, r) ' {¬"(#, r)} is consistent, if ¬"(#, r) is so (!-Consistency)(G2") T - (#, r) " T (Inclusion)(G3") For all ($, s) ! Fm, if ($, s) ! T + T - (#, r),

then there exists T " such that T - (#, r) " T " " T ,with T " ' {¬"(#, r)} consistent andT " ' {¬"(#, r), ($, s)} inconsistent (!-Relevance)

(G4") If for all T " " T we haveT " ' {¬"(#, r)} %(!(C) 0 iff T " ' ¬"(#", r") %(!(C) 0

then T - (#, r) = T - (#", r") (!-Uniformity)

Theorem 2.4. For any operator (for T ) - : Fm & P(Fm) we have the following: -satisfies (G1")+ (G4") iff - is a base !-contraction operator for T , i.e. iff - = -"

! forsome selection function % for T .

Proof. See Appendix.

Let (Success) be the (G1)-like axiom T - (#, r) %(!(C) (#, r).

Lemma 2.5. For each logic S = !(C), any ! > 0 and any contraction operator - thefollowing hold:

(i) if - satisfies (G1"), then it also satisfies (Success)(ii) if - satisfies (Success), then it also satisfies (G10)(iii) if - satisfies (G1"), then it also satisfies (G1"!) for any !" < !.

Proof. (i) Let - satisfy (G1") for some ! > 0. Then T - (#, r) is consistent with¬"(#, r). This implies (#, r) /! CnS(T-(#, r)). (ii) Assume (#, r) /! CnS(T-(#, r)).We use Lemma 3.3.7 from [5] to obtain T - (#, r) ' {# &! r} is consistent. Sinceinternal negation ¬ is involutive (# &! r) is equivalent to (¬#, 1 + r) = ¬0(#, r).But then T - (#, r) is consistent with ¬0(#, r). Hence, - satisfies(G10). (iii) This isobvious.

Levi and Harper identities hold between !-contraction of (#, r) and revision by¬"(#, r). The former T -"

! (#, r) ' {¬"(#, r)} = T !! ¬"(#, r) follows from thedefinitions. For the latter, we have the following result.

Proposition 2.6. For any theory T , ! > 0 and selection function %, we have that

T -"! (#, r) = (T !! ¬"(#, r)) # T

Proof. We have ($, s) ! ("

%(Con(T,¬"(#, r))) ' {¬"(#, r)}) # T iff ($, s) !("

%(Con(T,¬"(#, r))) # T ) '({¬"(#, r)} # T ). Now consider the next possibilities:Case ¬"(#, r) ! T . Then, by maximality, for each X ! Con(T,¬"(#, r)) we have

¬"(#, r) ! X , so that ¬"(#, r) !"

%(Con(T,¬"(#, r))); and T !! ¬"(#, r) =T -"

! (#, r). Case ¬"(#, r) /! T . Then, trivially the last identity is also the case since{¬"(#, r)} # T = ,. In both cases the identity T -"

! (#, r) = T !! ¬"(#, r) holds.

We saw in Example 2.3 the need for a fixed ! to be > 0. Doing so, though, a positivevalue for ! may result in unnecessary contractions:

Example 2.7. Let T = {(p, 0.9)} and # = (p, 0.93). In case ! > 0.3, we have T -" # =,; otherwise, for ! / 0.3, the output is again T . Intuitively the former case output ,,where ! > 0.3, is wrong due to ! being too high. By setting ! / 0.3 we have T ' {¬"#}is consistent. Observe in this example, one can safely set ! = 0, which looks optimal5.

For comparison purposes between the mentioned Rem-based contraction and ourCon-based proposal, consider the notion of deg-closed bases. This condition requiresbases to be closed in the following sense: if (#, r) ! T and s / r, then (#, s) ! T . Wedenote by CnQ(T ) the deg-closure of T ; then, a base T is deg-closed if T = CnQ(T ).This notion was introduced in [2] as an essential component in the framework adoptedtherein (see [7] for a comparison with pure bases in the case of revision). The condi-tion of being deg-closed makes change operations less drastic, since (possibly) lower butmaximally consistent degrees are kept for formulas being targeted by the operation inquestion. Normal bases, as Example 2.7 shows, suffer major changes. Now, under thecondition of deg-closure, Con-based contraction may be preferred over Rem-based con-traction, since only the former operation preserves the property of being finitely axioma-tizable (for deg-closed bases):

Example 2.8. Let T = {(p, s) : s / 0.9} and # = (p, 0.9). Then, CnQ(T ) -" # =CnQ(T " {(p, s) : s > 0.9+ !}). Observe this is just CnQ({(p, 0.9+ !)}). On the otherside, for Rem-based contraction, we would have T - # = {(p, s) : s < 0.9} whichcannot be finitely axiomatized.

Finally, an example showing the convenience of an alternative contraction whichdoes not commit oneself to any given ! > 0.

Example 2.9. Let T = {(pn, 12n ) | n ! ", n > 0} and #n = (pn, 1

2n + &n). We have

T -" #n =

#T " {(pn, 1

2n )}, for ! > &n, andT otherwise

Now, if we let &n = 12n , then for each ! > 0 there is an n ! " such that for input #n:

T -" #n = T " {(pn, 12n )}, this not being necessary, since we have: T !RPL #n.

5This suggests the possibility of defining, alternatively,

T ) # =

(T if T ( {¬0#} is consistent, (e.g. T !RPL #)

T )! # otherwise

We do not pursue this line here, but note it may also be applied to "-contraction studied below.

3. Towards Con-based (optimal) "-contraction

A problem posed by !-contraction operators is their lack of flexibility: if we are to choosethe value of ! beforehand, we may find that a given problem –say, !-contraction of T by(#, r)– is not optimally solved (! being unnecessarily high, hence violating the princi-ple of minimizing information loss. (Observe this problem does not affect Rem-basedcontraction operators from [6].) To solve this inconvenience we will define a contractionoperation which is not committed to any particular !.

Definition 3.1. A selection function % for T is conservative iff for all !" < ! we have

T -"! (#, r) " T -"!

! (#, r)

Notice first that *** condition (iii) from the previous Lemma *** is weaker thanconservativeness: a selection function -defined for both families Con(T,¬"(#, r)) andCon(T,¬"!(#, r))- can satisfy (G3"), hence (G3"!), with T -"

! (#, r) # T -"!! (#, r).

(For example, consider % to be maxichoice in both cases but with selected elements X"

and X "" not forming a "-chain.) Conservativeness can be enforced by imposing a further

condition upon selection functions, which is defined next.

Definition 3.2. For X, Y families of sets, we define:

X0+&Y i! for each X ! X there is some Y ! Y such that X " Y

X01+Y i! for each Y ! Y there is some X ! X such that X " Y

A selection function defined on P(P(Fm))\{,} is family-conservative iff X0+&Y implies%(X)01+%(Y).

We call such arrow-reversing selection functions % family-conservative because theyonly select sets in the greater family Y which extend some set in the lesser familyX. Observe that (1) for any X, Y, X01+Y implies

"X "

"Y; and (2) (by Lemma

1.7) for each theory T , sentence (#, r) and degrees ! > 0 we have: Con(T, (#, r +!))0+&Con(T, (#, r)). Facts (1) and (2) jointly imply that any family-conservative selec-tion function is conservative.

By fixing some ! > 0 in the definition of contraction, one enforces contraction tosatisfy (Success), at the cost of committing ourselves to possibly suboptimal contractionoperations for some problems. To overcome this problem, it would be desirable thatparameter ! in -"-contraction automatically adjusted itself to find the optimal solutionfor the problem at hand –contracting T by (#, r)–, while granting that (Success) stillholds. With this idea in mind, we define partial meet base "-contraction as follows.

Definition 3.3. Let T be a base and let % be a selection function for T . We define the(partial meet) base "-contraction operator -#

! for T induced by % as

T -#! (#, r) =

$0<"#r T -"

! (#, r)

In comparison to !-contraction, we have that Levi identity does not hold for "-contraction; but Harper identity remains valid:

Corollary 3.4. For each theory T , selection function % and (#, r) we have that

T -#! (#, r) = (

$">0 T !! ¬"(#, r)) # T

Proof. By Proposition 2.6, for each ! > 0 we have T -"! (#, r) = T !! ¬"(#, r) # T .

Hence$

">0 T -"! (#, r) =

$">0(T !! ¬"(#, r)#T ) = (

$">0 T !! ¬"(#, r))#T .

Next we show soundness of partial meet base "-contraction operators in some logicS = !(C) with respect to the following axioms, where ! ! [0, 1 + r]. Completeness isleft as an open problem.

(G1#) (#, r) /! CnS(T - (#, r)), if !S (#, r) (Success)(G2#) T - (#, r) " T (Inclusion)(G3#) For all ($, s) ! Fm, if ($, s) ! T + T - (#, r), then

for all ! > 0 there exists T "" with (T ! ¬"(#, r) # T ) "

" T "" " T and T "

" ' {¬"(#, r)} consistent andT "

" ' {¬"(#, r), ($, s)} inconsistent ("-Relevance)(G4#) For any (#, r), (#, r"), if for all T " " T we have that

for all ! exists !" (and viceversa for all !" exists !) suchthat T " ' {¬"(#, r)} !S 0 iff T " ' {¬"(#", r")} !S 0 )then T - (#, r) = T - (#", r") ("-Uniformity)

Notice the difference between (G3#) and (G3) from Hansson and Wassermann: theformer implies the latter only for logics with the deduction property.

Theorem 3.5. Within a given !(C) logic, for any conservative selection function % for atheory T , the operator -#

! satisfies (G1#)+ (G4#).

Proof. See Appendix.

Theorem 3.6. (In !(C)) Let -! be an "-contraction operator defined from a conserva-tive selection function % for T . If T is finite, then there is some ! > 0 with

T -#! (#, r) = T -"

! (#, r)

Proof. Assume % is conservative. Since T is finite, so are T -! (#, r) and$

">0 T !!

¬"(#, r) (otherwise, these sets could not be subsets of the former; recall contractionis standard, not C-closed). By being

$">0

"%(Con(T,¬"(#, r))) finite, there must be

some n ! " and !0 > · · · > !n > 0 with$

">0(T !! ¬"(#, r) # T ) =$

i#n(T !!

¬"i(#, r)#T ). Since % is conservative and !n / !i (for i / n), we have T-"!¬"i(#, r) "

T -! ¬"n(#, r). Hence, the join of all such i’s is also a subset of T !! ¬"n(#, r). Inconsequence, T-! (#, r) " (T !!¬"n(#, r)#T ). Since the other direction2 is obviousfrom the definition of -! , we finally obtain T -! (#, r) = T -"

! (#, r).

This result shows that for finite bases in !ukasiewicz logic, "-contraction operator-! , with % conservative, will produce no new outputs which cannot also be obtained bymeans of some !-contraction. Still, this operation is interesting since it grants optimalityof solutions, so we need not choose an !-value from advance.

4. Conclusions and Future Work

We motivated the convenience of obtaining an alternative (consistency-based) operationof base contraction in graded expansions of !ukasiewicz logic and made some stepstowards this result for (optimal) "-contraction, including characterization of (secure)!-contraction, soundness of "-contraction, and a finite-collapse theorem between thesebase change operations. In the future, we hope to completely characterize the latter op-eration, as well as to obtain complexity results, whenever possible, for the operationsstudied in the present contribution.

Acknowledgements

The authors acknowledge partial support of the the Spanish MICCIN Consolider projectAT (CSD2007-022). P. Dellunde and L. Godo also acknowledge partial support of theMICINN project FFI2008-03126-E/FILO related to the Eurocores-LogICCC Project Lo-MoReVI (FP006).

References

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Appendix: proofs of Theorems 2.4 and 3.5

Theorem 2.4. For any operator (for T ) ! : Fm " P(Fm) we have the following: ! satisfies(G1!)# (G4!) iff ! is a base !-contraction operator for T , i.e. iff ! = !!

" for some selectionfunction " for T .

Proof. (Soundness) (G1!) Since for all X $ Con(T,¬!(#, r)), we have X%{¬!(#, r)} is consis-tent, the same holds for X $ "(Con(T,¬!(#, r))). Hence,

T"(Con(T,¬!(#, r)))% {¬!(#, r)}

is consistent. (G2!) Trivial, by definition of !!" . (G3!) Let ($, s) $ T # T !!

" (#, r). Then,($, s) /$ X , for some X $ "(Con(T,¬!(#, r))). In consequence, X % {¬!(#, r)} is consistent.On the other side, since ($, s) $ T , from ($, s) /$ X we infer X % {¬!(#, r), ($, s)} is incon-sistent. Such X is what we are looking for, so we put T ! = X . (G4!) Assume for all T ! & Twe have T ! is consistent with ¬!(#, r) iff it is consistent with ¬!(#

!, r!). Then by [7, Lemma 6],we have Con(T,¬!(#, r)) = Con(T,¬!(#

!, r!)). Since " is a function, applying " to both sets

preserves the identity. Finally, applyingT

to each side of the corresponding equation preservesidentity too, so T !!

" (#, r) = T !!" (#!, r!).

(Completeness) Let ! : Fm " P(Fm) be an operator (for T ) satisfying (G1!)# (G4!). Wedefine a function "

"(Con(T,¬!(#, r))) = {X $ Con(T,¬!(#, r)) : X ' T !! (#, r)}

We have to show that for " for T , " is: (1) well-defined (for ¬!-sentences), (2) a ¬!-selectionfunction for T , and (3) that for all (#, r) $ Fm, we have T ! (#, r) = T !!

" (#, r).(1) " is well-defined (for ¬!-sentences). Let Con(T,¬!(#, r)) = Con(T,¬!(#

!, r!)). ByLemma 6 in [7], since ! satisfies (G4!), we infer T ! (#, r) = T ! (#!, r!). From this andthe initial assumption we obtain, using the above definition of ", that "(Con(T,¬!(#, r))) ="(Con(T,¬!(#

!, r!))). (2) " is a ¬!-selection function for T (for ¬!-sentences). LetCon(T,¬!(#, r)) be non-empty. Since, by (G1!), T ! (#, r) % {¬!(#, r)} !S 0, we obtainfrom Lemma 1.7 that T ! (#, r) & X $ Con(T,¬!(#, r)), for some X $ Con(T,¬!(#, r)).By the above definition of ", this implies X $ "(T,¬!(#, r)), so this set is not empty.(3) Wecheck the T !!

" (#, r) & T ! (#, r) first. Let ($, s) $ T !!" (#, r). Hence, ($, s) in T .

Now, assume the contrary of what we want to prove: ($, s) /$ T ! (#, r). Then, by (G3!),there is T ! with T ! (#, r) & T ! & T . Then we have (i) T ! % {¬!(#, r)} consistent while(ii) T ! % {¬!(#, r), ($, s)} is inconsistent. We have that (i) plus T ! (#, r) & T (which holdsby (G2!)) jointly imply T ! (#, r) & X $ Con(T,¬!(#, r)). We also have (ii) impliesX % {¬!(#, r), ($, s)} is inconsistent, so ($, s) /$ X . Since X ' T !!

" (#, r) we have thatX $ "(Con(T,¬!(#, r))). But then ($, s) /$

T"(Con(T,¬!(#, r))), thus contradicting the ini-

tial assumption ($, s) $ T !!" (#, r). In consequence, ($, s) $ T ! (#, r). For the other direction

T ! (#, r) & T !!" (#, r) consider: ($, s) $ T ! (#, r). Then, let X $ "(Con(T,¬!(#, r))).

Since X & T ! (#, r), we have ($, s) $ X . In consequence ($, s) $T

"(Con(T,¬!(#, r))),so that ($, s) $ T !!

" (#, r).

Theorem 3.5. Within a given !(C) logic, for any conservative selection function " for a theory T ,the operator !#

" satisfies (G1#)# (G4#).

Proof. (G1#) Since T !!" (#, r) is consistent with ¬!(#, r) we have that (#, r) /$ CnS(T !!

"

(#, r)). Since " is conservative, {T !!" (#, r)}!>0 is a&-chain. Since S is finitary, these two facts

imply that (#, r) /$ CnS(S

!>0 T!!" (#, r)). (G2#) By Corollary 3.4, T!" (#, r) = (

S!>0 T !"

¬!(#, r))(T , which is a subset of T . (G3#) Let ($, s) $ T #T !" (#, r). This implies ($, s) /$T !"¬!(#, r), for any ! > 0, so that for each ! > 0 there is X $ "(Con(T, (#, r))) with ($, s) /$X . Let T !

! = X . We have that T !" ¬!(#, r) & X % {¬!(#, r)}; since X & T , T !" ¬!(#, r)(T & X % ({¬!(#, r)}(T ). Wether ¬!(#, r) $ T or not, maximality of X implies the latter set isX . Also, it is clear that X & T , that X%{¬!(#, r)} is consistent, and that X%{¬!(#, r), ($, s)}is not consistent, by definition of Con(T, ·). (G4#) Assume for each ! > 0 there exists % > 0such that for all T ! & T , T ! % {¬!(#, r)} is consistent i! T ! % {¬$(#

!, r!)} is consistent, andviceversa. Then, for each ! > 0 there is % > 0 with Con(T,¬!(#, r)) = Con(T,¬$(#

!, r!)),hence with "(Con(T,¬!(#, r))) = "(Con(T,¬$(#

!, r!))). Let ($, s) $ T !" (#, r). Then($, s) $

S!>0(T !"¬!(#, r)(T ), so that for some ! > 0, ($, s) $

T"(Con(T,¬!(#, r)))(T .

By the assumption, there is % > 0 with ($, s) $T

"(Con(T,¬$(#!, r!))). Finally, ($, s) $S

$>0 T !" ¬$(#!, r!) so that ($, s) $ T !" (#!, r!). The other direction is similar.