Contracting Logics

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Contracting Logics

M´arcio M. Ribeiro and Marcelo E. Coniglio Centre for Logic, Epistemology and the History of Science

State University of Campinas, Brazil [email protected] [email protected]

Abstract. In this paper, inspired in the field of belief revision, it is presented a novel operation for defining a new logic given a known logic.

The operation consists in removing some (maybe undesirable) derived rule from a logic. Besides removing the ‘undesirable’ rule, this operation (called contraction) should change the logic in a minimal way. This paper presents formal definitions for contraction operations over logics, both as sets of rationality postulates and by means of concrete constructions. This allowed us to generalize several notions of maximality of logics presented in the literature. Furthermore, the proposed constructions are applied to the study of some paraconsistent and intermediate logics.

1 Introduction

Belief revision is a subfield of knowledge representation that studies the dynamics of propositional theories [AGM85]. The dynamics of the theories is given by a set of operations (contraction, revision, expansion etc.) which are defined via sets of rationality postulates.

The first papers in the field restricted themselves to the study of the dynamics of theories within supra-classical logics, but recently it was showed how this can be generalized to several other non-classical logics [Rib12]. In this paper it is presented a way to generalize belief revision techniques even more.

Instead of considering operations for changing a theory within a given logic, it is presented operations that change the logic itself. This paper focuses on contraction operations i.e. operations that given a logic returns another logic where certain rule doesn’t hold.

Firstly a set of rationality postulates that the operation should satisfy is presented. This includes some kind of minimality criterion concerning the change needed to perform the operation. The techniques involved lead us naturally to the notion of maximality of logics. The constructions proposed here generalize several notions of maximality considered in the literature. After this, representation theorems relating the constructions defined here with sets of postulates are obtained. Finally, several examples involving paraconsistent, paracomplete and super-intuitionistic logics are shown.

2 Preliminaries

In this section we recall the basic notions to be used along the paper.

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Definition 1 A (Tarskian) consequence system is a pair hF or, Cni such that F or is a set (whose elements are calledformulas) and Cn:℘F or→℘F or is a map satisfying, for everyΓ, ∆⊆F or:

– Γ ⊆Cn(Γ) (extensiveness)

– ifΓ ⊆∆ thenCn(Γ)⊆Cn(∆) (monotonicity) – Cn(Cn(Γ))⊆Cn(Γ) (idempotence)

The map Cn : ℘F or → ℘F or is called a consequence operator. We say that hF or, Cniiscompact ifCn(Γ) =S{Cn(Γ0) : Γ0 is a finite subset ofΓ}.

The usual consequence systems are defined over formal languages generated by signatures.

Definition 2 A (propositional) signatureis a family of setsC={Cn}n∈Nsuch that Ci∩Cj=∅ ifi6=j, andC06=∅.

The elements of Cn are called n-ary connectives. The elements of C0, in particular, are called propositional variables. Let Ξ be a given set of symbols calledschema variables.

Definition 3 Let C ={Cn}_n∈N be a signature. The propositionalschema language generated byCis the algebraL(C, Ξ)of typeCfreely generated byΞ∪C0. The propositionallanguage generated by C is the algebra L(C)of type C freely generated by C0.

The elements ofL(C, Ξ) (ofL(C), resp.) are calledschema formulas(formulas, resp.) overC. Note thatC0⊆L(C)⊆L(C, Ξ).

Definition 4 A Hilbert calculus is a pairH =hC, Risuch thatCis a signature, andR is a set ofinference rules, that is, a set of pairs h∆, ϕiwhere∆∪ {ϕ} is a finite subset of L(C, Ξ).

When ∆ = ∅ we say that the inference rule is an axiom. The notion of derivation in a Hilbert calculus is the usual one. Before that it is necessary to introduce the notion of substitution.

A substitution over C is a map σ : Ξ → L(C, Ξ). A substitution can be extended to a unique C-homomorphism ˆσ : L(C, Ξ) → L(C, Ξ) as usual. We denote by ˆσ(∆) the set of schema formulas{σ(ψ) :ˆ ψ∈∆}, for∆⊆L(C, Ξ).

Finally, letSubs(C) ={σ : σ is a substitution overC}.

Definition 5 A derivation of ϕ∈L(C, Ξ) in H from a setΓ ⊆L(C, Ξ) is a sequence ϕ1. . . ϕn such thatϕn =ϕand, fori= 1, . . . , n, eachϕi is either an element ofΓ or there is a substitutionσand an inference ruleh∆, ϕiinH such that σ(ϕ)ˆ isϕi and, for everyψ∈∆,ˆσ(ψ)isϕj for somej < i.

We say that ϕ is derived from Γ in H, denoted by Γ `H ϕ, if there is a derivation of it in H fromΓ.

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IfH is a Hilbert calculus then its set of rules will be frequently denoted by R_H. Given two Hilbert calculiH_i=hC, R_iioverC(fori= 1,2) we say thatH₁ is a subcalculus ofH₂, denoted byH₁⊆H₂, ifR₁⊆R₂. GivenH =hC, Riover C then theclosure of the set of rules R is the set of rules Cl(R) = {h∆, ϕi :

∆ `H ϕ}. The closure of H is the Hilbert calculus Cl(H) = hC, Cl(R)i. If r ∈Cl(R) thenr is a derived ruleof H. LetCnH : ℘(L(C, Ξ))→℘(L(C, Ξ)) be the consequence operator generated by H as expected: CnH(Γ) = {ϕ : Γ `Hϕ}. This consequence operator is Tarskian andstructural, that is: for every substitution σand every set of schema formulas Γ, ˆσ(CnH(Γ))⊆CnH(ˆσ(Γ)).

We say thathL(C, Ξ), CnHiis the consequence system, associated to the Hilbert calculusH. IfR(C) =def ℘f(L(C, Ξ))×L(C, Ξ) denotes the set of all rules over C, it is easy to see that hR(C), Cli is a compact and structural consequence system, where℘f(X) denotes the set of finite subsets of a setX, and where ˆσ(r) is defined in the obvious way for every r∈R(C) and everyσ∈Subs(C).

3 Rationality Postulates

Given a Hilbert calculus H over C we want to define an operation − that for each ruler=h∆, ϕioverCreturns a Hilbert calculusH−roverCthat satisfies certain properties:

(inclusion)H−r⊆H

Inclusionstates that the result of a contraction inH is a subcalculus ofH. (success)Ifϕ6∈∆(i.e.r6∈Cl(∅)) then∆0H−rϕ.

Successstates that the resulting calculus should not derive the ruler, unless this is impossible. The only case when it is impossible to remove r from H is when ϕis a substitution of some element of∆ i.e. if r is a derived rule in any Hilbert calculus, by extensiveness.

(failure)Ifϕ∈∆(i.e.r∈Cl(∅)) thenH ⊆H−r.

Failure states that if r is a derived rule in any Hilbert calculus, then the contraction − over H should not remove any element from H. Together with inclusion, failure postulate states thatH−r=H ifr∈Cl(∅). In other words, in this case the contraction should fail.

Besides these, we need some postulate that guaranties the minimality of change i.e. that guarantees that only derivations relevant to prove r should be removed. Next section investigate possible minimality postulates.

So far postulates for contraction over an arbitrary Hilbert calculus were presented. A criticism one can make to this approach is that it compromises itself with one specific axiomatization for a logic.

One way to abstract away the specificity of certain choice of a set of rulesRis to consider the closure ofH =hC, Ri, that is, the calculusCl(H) =hC, Cl(R)i.

Observe that hL(C, Ξ), Cl(R)i is a Tarskian, compact and structural consequence system which represents the logic generated by H. For this reason it

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is more robust to consider the contraction over the closure of H instead ofH itself.

Let us then define a contraction−over a closed Hilbert calculusH(i.e. where H =Cl(H)). In this case we want the result of a contraction −over H to be also a closed Hilbert calculus:

(closure)H−r=Cl(H−r)

The other postulates for contraction over closed Hilbert calculi are the same we already stated.

3.1 Minimality Criteria

In belief revision literature we can find several minimality postulates: recovery, relevance, core-retainment, fullness etc. (see [Han99]). We will focus here in two of these postulates, namely, relevance and fullness.

(fullness)Ifr⁰ ∈RH and r⁰∈/R_H−r thenr∈Cl(R_H−r∪ {r⁰}).

Fullnessstates that if some rule r⁰ was removed from H after contraction then re-insertingr⁰ should recover the ruler.

It will be sometimes useful to weaken this postulate to guarantee that only axiomsr⁰=h∅, ϕ⁰imay be removed from H.

(weak fullness)Ifr⁰ ∈RH is an axiom andr⁰∈/R_H−rthenr∈Cl(R_H−r∪ {r⁰}).

However, it is pointed out in the literature that these postulates may be too strong for certain purposes. Relevance is a weaker version of this postulate (see [Han91]):

(relevance)Ifr⁰∈RH andr⁰ ∈/ R_H−r then there isH⁰ such thatH −r⊆ H⁰⊆H,r /∈Cl(H⁰) andr∈Cl(RH⁰∪ {r⁰}).

Relevancestates that ifr⁰is removed then there is some intermediary calculus H⁰ such that ris not a derived rule inH⁰ butr is a derived rule ifr⁰ is added.

Again we can define a weak version of the postulate which is concerned only with derived axioms, not derived rules in general:

(weak relevance)Ifr⁰ ∈RH is an axiom and r⁰ ∈/ RH−r then there isH⁰ such thatH−r⊆H⁰⊆H,r /∈Cl(H⁰) andr∈Cl(RH⁰∪ {r⁰}).

Notice that relevance implies failure:

Lemma 6 Let H be a Hilbert calculus over C. If −over H satisfies relevance thenH satisfies failure.

As will be shown in section 4, these minimality postulates are related with a construction calledremainder set.

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4 Maximality

In previous sections we presented a set of postulates for a contraction operation over a Hilbert calculus which may or may not be closed. We argued that minimality is a desirable property of the operation, i.e. contraction should change the logic as little as possible. This desiderata is closely related to the notion of maximal logics. In this section we will investigate different definitions for maximal logic presented in the literature and we will present a definition which generalizes them.

Usually, a logicL₁is said to bemaximalw.r.t. another logicL₂when both are defined over the same language, the consequence relation`_L₁ofL₁ is contained in `_L₂ and, if ϕ is a schema formula such that `_L₂ ϕ but 0L1 ϕ then the extension of L1 obtained by adding ϕ as a valid schema coincides with L2. In our framework if L1 is given by the closure of a Hilbert calculus H1 then Cl(RL₁∪ {h∅, ϕi}) =RL₂ wheneverh∅, ϕi ∈RL₂\RL₁.

For example, a logic over a signatureCisPost completeif it is maximal w.r.t.

the trivial logicTrivC=hC, R(C)ioverC.

Another definition of maximality found in the literature comes from [AAZ10].

In this article the authors defined a logic L as being maximal w.r.t. a rule (in their case the principle of explosion h{¬ξ₁, ξ₁}, ξ₂i). They define two types of maximality: strong and weak. The logic L1 seeing as a closed Hilbert calculus is strongly maximal w.r.t. a ruler /∈RL₁ if for every logic L2 such thatRL₁⊂ RL₂ we have that r ∈ Cl(L2). Using this definition the authors proved that several three-valued logics, such as P¹ andJ3, are maximal w.r.t. the principle of explosion.

We will borrow a concept from belief revision literature to generalize the notions of maximality.

Definition 7 (Remainder set) Let H be a Hilbert calculus over C and R be a set of rules over C. A remainder setH⊥R is a set such thatX ∈H⊥R iff:

1 X⊆H (X is a subcalculus of H).

2 R*Cl(R_X)(there is some r∈R that is not a derived rule inX).

3 If X ⊂X⁰ ⊆H then R⊆Cl(R_X⁰)(X is maximal).

The remainder setH⊥Ris the set of all maximal subcalculus X ofH such that some rule inRis not a derived rule inX. We can also define a weak version of remainder set, denoted byH⊥_wR, by changing item 3 in the above definition to:

3’ For any axiomr∈RH\RX we have thatR⊆Cl(RX∪ {r}).

Example 8 Consider the following Hilbert calculusH_{CP L}=hC, Ri for Classi- cal Propositional Logic (CPL). LetC={P,{¬},{∧,∨,→}}wherePdenote the set of propositional variables, and letR be the following set of rules¹:

1 In this example we will identify an axiomh∅, ϕiwithϕ.

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(Ax1)ξ1→(ξ2→ξ1) (Ax2)ξ1→(ξ2→(ξ1∧ξ2)) (Ax3)(ξ1∧ξ2)→ξ1 (Ax4)(ξ1∧ξ2)→ξ2

(Ax5)ξ1→(ξ1∨ξ2) (Ax6)ξ2→(ξ1∨ξ2) (Ax7)ξ1→(¬ξ1→ξ2)(Ax8)ξ1∨ ¬ξ1

(Ax9)(ξ1→(ξ2→ξ3))→((ξ1→ξ2)→(ξ1→ξ3)) (Ax10)(ξ1→ξ2)→((ξ3→ξ2)→(ξ1∨ξ3→ξ2)) (Ax11)(ξ1→ξ2)→((ξ1→ ¬ξ2)→ ¬ξ1)

(M P)h{ξ1, ξ1→ξ2}, ξ2i

Let r=h{¬¬ξ}, ξi. It is well known that r∈Cl(HCPL). Consider now the Hilbert calculus HInt = hC, R\ {Ax8}i for Intuitionistic logic. It is also well known thatr /∈Cl(HInt). Hence, it is trivial to show that HInt ∈HCPL⊥{r}.

Notice that this is strongly dependent on the choice of the rules. This is why it is important to consider the contraction operation not over arbitrary Hilbert calculi, but over closed Hilbert calculi.²

The above example shows the limitation of using an arbitrary Hilbert calculus H in the definition of remainder set. We will be more interested in applications whereH is closed. In this case, we can prove that the elements ofH⊥Rare also closed:

Lemma 9 If H =Cl(H) andX∈H⊥Rthen X=Cl(X).

As a first application of our framework, let us show that the notions of maximality presented above can be represented using remainder sets:

– A logicL₁seeing as a closed Hilbert calculus is maximal w.r.t. a logicL₂iff L₁∈L₂⊥_wR_L₂.

– A logic L over a signature C is strongly maximal w.r.t. a rule r iff L ∈ TrivC⊥{r}.

– A logicLoverC is weakly maximal w.r.t. a ruler iffL∈Triv_C⊥_w{r}.

– A logicLoverC is Post complete iffL∈TrivC⊥wRTriv_C. Now let us show some useful propositions about remainder sets.

Proposition 10 Let H be an arbitrary Hilbert calculus and let R1 and R2 be sets of rules. IfX ∈H⊥R1,R2⊆R1⊆RHandR2*Cl(RX)thenX ∈H⊥R2. This proposition states that if we know thatX ∈H⊥R1 and we know that a subset R2 of R1 also contains some rule that is not derived from X then X ∈H⊥R2.

Recall thatP¹was introduced in [Set73] as a three valued paraconsistent logic axiomatized by a Hilbert calculus over a signature just containing¬and→. We know from [Set73] thatP¹is maximal w.r.t.CPL, that isP¹∈CPL⊥wRCPL, when both logics are considered as closed Hilbert calculi. Furthermore, we know thatr=h{ξ1,¬ξ1}, ξ2i∈/ R_P1 and thatr∈R_CPL. Hence, we have the following corollary of Proposition 10:

2 Notice that similar problems arise when using belief base contraction instead of belief set contraction.

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Corollary 11 P¹∈CPL⊥_w{h{ξ₁,¬ξ₁}, ξ₂i}.

The logicI¹ was introduced in [SC95] as a three valued paracomplete logic (dual to P¹) axiomatized by a Hilbert calculus over a signature just containing ¬ and →. From [SC95] we know thatI¹ is maximal w.r.t. CPL, i.e. I¹ ∈ CPL⊥wR_CPL. Furthermore, the rule of double negationr=h{¬¬ξ1}, ξ1idoesn’t hold inI¹.

The Smetanich’s logic Sm is another example of logic where the rule of double negation fails. Sm is the greatest super-intuitionistic logic (cf. [CZ97]), i.e.Int⊂Sm∈CPL⊥wR_CPL. From this, clearlyI¹ is not super-intuitionistic.

Sinceh{¬¬ξ1}, ξ1i ∈RCPL, we have the following corollaries:

Corollary 12 I¹∈CPL⊥w{h{¬¬ξ1}, ξ1i}.

Corollary 13 Sm∈CPL⊥w{h{¬¬ξ1}, ξ1i}.

Proposition 10 showed a relation that holds when we fix the Hilbert calculus H and change the set of rules. The following proposition states a relation that holds when we fix the set of rulesR and change the Hilbert calculus.

Proposition 14 Let H₁ andH₂ be arbitrary Hilbert calculi and let R be a set of rules. IfX ∈H₁⊥R andX ⊆H₂⊆H₁ thenX ∈H₂⊥R.

The logic J3, introduced in [DdC70], is a paraconsistent three valued logic that can be defined over the signature C = hP∪ {⊥},{¬},{∧,∨,→}i. From [AAZ10] we know thatJ3is strongly maximal w.r.t. the principle of explosion, that is J₃∈Triv_C⊥{h{¬ξ1, ξ₁}, ξ2i}. The following corollary is a consequence of the fact thatJ₃⊆CPL⊆Triv_C:

Corollary 15 J3∈CPL⊥{h{¬ξ1, ξ1}, ξ2i}.

The following results analyze the connection between the notions of remainder sets and weak remainder sets just introduced. The main purpose is to obtain sufficient conditions in order to guarantee the equivalence between both notions.

Lemma 16 Let H =hC, Ribe a Hilbert calculus over C, r=h{γ1, . . . , γn}, ϕi a rule over C and H^r = hC, R∪ {r}i. Assume that H has (possibly derived) connectives→ (binary) and∼(unary) such that the following holds:

1. ξ1,(ξ1→ξ2)`H ξ2; 2. `_Hξ₁→(ξ₂→ξ₁);

3. ∼ξ1`H(ξ₁→ξ₂);

4. IfΓ, ξ1`H^r ξ2 andΓ,∼ξ1`H^r ξ2 thenΓ `H^r ξ2, for everyΓ. Let ξ be a schema variable not occurring inr. Then

(ξ→γ₁), . . . ,(ξ→γ_n)`_Hr (ξ→ϕ).

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Corollary 17 Let H and H^r as in Lemma 16. Assume that H satisfies the Deduction Meta-Theorem (MTD) with respect to →: Γ, α`_H β implies Γ `_H (α→β), for everyα, β. ThenH^r also satisfies MTD with respect to →.

Corollary 18 Let H andH^r as in Corollary 17 for r=h{γ1, . . . , γn}, ϕi. Let Hr=hC, R∪ {h∅, γ1 →(γ2 →(. . .→(γn →ϕ). . .))i}i. Then the consequence systems generated by H^rand Hr coincide, that is: Cl(H^r) =Cl(Hr).

Theorem 19 LetH andH⁰ be closed Hilbert calculi overC such thatH ⊆H⁰. Assume that H satisfies the conditions of Corollary 17 for every rulerover C, and thatH⁰ satisfiesM T Dwith respect to→. Then,H ∈H⁰⊥wRiffH ∈H⁰⊥R for every R⊆R(C).

With Theorem 19 we can now prove that bothI¹andP¹are not only weakly maximal, but also strongly maximal w.r.t.CPL:

Proposition 20 LetCPL be the Classical Propositional Logic in the signature containing just→and¬, seeing as closed Hilbert set. Then we have the following:

1. I¹∈CPL⊥RCPL and 2. I¹∈CPL⊥{h{¬¬ξ1}, ξ1i}.

Proposition 21 LetCPLbe the Classical Propositional Logic as in Proposition 20. Then the following holds:

1. P¹∈CPL⊥RCPL and 2. P¹∈CPL⊥{h{¬ξ1, ξ1}, ξ2i}.

5 Representation Theorems

In Section 3 we enumerated a set of rationality postulates for a contraction operation. In Section 4 we claimed that this operation is related with a notion of maximality which we explored. This section presents constructions for contraction. Each of them is proved to be characterized by a specific set of rationality postulates.

The following is an important lemma related to remainder sets calledupper- bound lemma. This result was adapted from a similar one in belief revision field found in [AM81].

Lemma 22 (upper-bound) LetH andX be Hilbert Calculi such thatX⊆H and letR be a finite set of rules, all over a signature C. If R*Cl(X)then

1. there is someH⁰ such that X⊆H⁰∈H⊥Rand 2. there is someH⁰⁰ such thatX ⊆H⁰⁰∈H⊥wR.

Consider now a Hilbert calculusH and a rulerboth over the same signature.

A (strong)subset selection functionis any functionΥ that satisfies the following properties:

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1. Υ(H, r)6=∅;

2. Υ(H, r)⊆H⊥{r} ifH⊥{r} 6=∅;

3. Υ(H, r) ={H}ifH⊥{r}=∅.

Aweak subset selection functionis defined analogously using weak remainder set instead of remainder set. Now consider the following construction for some selection functionΥ:

H−Υ r=\

Υ(H, r)

In the belief revision field this construction is calledpartial meet contraction (cf. [AGM85]). Any partial meet contraction satisfiessuccess,inclusionandfail- ure. It also satisfies relevance or weak relevancedepending if Υ is a weak or a strong selection function. Furthermore, ifH is closed, by Lemma 9 and the fact that the intersection of closed Hilbert calculi is closed, partial meet contraction satisfiesclosure.

Theorem 23 For any weak subset selection functionΥ, the contraction−Υ over H defined asH−Υ r=T

Υ(H, r) satisfiessuccess, inclusion,failure and weak relevance. Furthermore:

– IfΥ is a (strong) subset selection function then −Υ also satisfies relevance.

– IfH is closed then −Υ satisfies closure

Besides satisfying the postulates, the following theorem shows that, in fact, the postulates fully characterize the construction.

Theorem 24 Let H be a Hilbert calculus. If a contraction − overH satisfies success, inclusion, failure and weak relevance then there is some weak subset selection function Υ such that:

H−r=H−Υ r=\

Υ(H, r)

If−satisfiesrelevancethen the above equation holds for some (not weak) subset selection function. In this case,failureproperty become redundant by Lemma 6.

Now let us define anelement selection functionas any functionΨ such that:

1. Ψ(H, r)∈H⊥{r}ifH⊥{r} 6=∅.

2. Ψ(H, r) =H if otherwise.

Weak element selection functionis defined analogously using a weak remainder set. A maxi-choice contractionis defined as follows:

H−_Ψ r=Ψ(H, r) Weak maxi-choice contractionis defined analogously.

As for partial meet contraction, maxi-choice contraction is fully characterized by a set of postulates, namely: success, inclusion, failureand fullness. Further- more, weak maxi-choice contraction is characterized by the same postulates with fullness exchanged byweak fullness.

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Theorem 25 Let H be a Hilbert calculus. An operation − over H is a maxi- choice contraction iff it satisfies success,inclusion,failure andfullness. − is a weak maxi-choice contraction iff instead of fullness it satisfiesweak fullness.

Example 26 Consider P¹ and CPL defined in the same signature of J3 and letr=h{¬ξ1, ξ1}, ξ2i. For certain choice of element selection functions we have that P¹=CPL−Ψ1r andJ3=CPL−Ψ2r. Furthermore, for certain choice of subsetselection functionΥ, we haveP¹∩J₃=CPL−Υr. An analogous result can be obtained by considering weak selection functions,I¹,SmandCPL(over the same signature) and r=h{¬¬ξ1}, ξ1i. Note that I¹ can be choose even for non weak selection functions.

6 Conclusion and Future Work

In this paper it was presented a formal framework for defining new logics by contracting a derived rule from a known logic. The framework consists in defining operations −over possibly closed Hilbert Calculi. These operation are related with contraction operation in belief revision field. Constructions for the operations as well as postulates that characterize them were presented.

This framework is related with the study of maximal logics w.r.t. another logic (cf. [Set73,SC95]) and the study of maximal logics w.r.t. a principle (cf.

[AAZ10]). Furthermore, the relation between the notion of strong remainder set and weak remainder set was analyzed. As an application it was proved that the logicsP¹andI¹are strongly maximal w.r.t. Classical Propositional Logic.

This paper focused in the contraction ofonerule from a logic. It would be interesting to analyze the result of contracting a set of rules. Once again the field of belief revision may help us in this task.

As future work we intend to extend this framework to sequent calculi and also to study the analogous of a revision operation over logics.

Acknowledgements: This research was financed by FAPESP (Brazil), The- matic Project LogCons 2010/51038-0. The first author was also supported by a Post-doctoral grant by FAPESP, process 2011/01384-1. The second author was also financed by an individual research grant from The National Council for Sci- entific and Technological Development (CNPq), Brazil, process 305237/2011-0.

References

[AAZ10] O. Arieli, A. Avron, and A. Zamansky. Maximally paraconsistent three- valued logics. In Proceedings of the 12th International Conference on the Principles of Knowledge Representation and Reasoning (KR’10), Toronto, Canada, 2010.

[AGM85] C. Alchourr´on, P. G¨ardenfors, and D. Makinson. On the logic of theory change. Journal of Symbolic Logic, 50(2):510–530, 1985.

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[AM81] C. Alchourr´on and D. Makinson. Hierarchies of regulation and their logic.

In Hilpinen, editor, New studies in deontic logic, pages 125–148. D. Reidel Publishing Company, 1981.

[CZ97] A. Chagrov and M. Zakharyaschev. Modal Logic. Oxford Logic Guides.

Clarendon Press, 1997.

[DdC70] I. M. L. D’Ottaviano and N. C. A. da Costa. Sur un probl`eme de Ja´skowski.

Comptes Rendus de l’Acad´emie de Sciences de Paris, 270:1349–1353, 1970.

[Han91] S. O. Hansson. Belief contraction without recovery.Studia Logica, 50(2):251–

260, 1991.

[Han99] S. O. Hansson. A Textbook of Belief Dynamics. Kluwer Academic, 1999.

[Rib12] M. M. Ribeiro. Belief Revision in Non-Classical Logics, volume XII of Springer Briefs in Computer Science. Springer, 2012.

[SC95] A. M. Sette and W. A. Carnielli. Maximal weakly-intuitionistic logics.Studia Logica, 55(1):181–203, 1995.

[Set73] A. M. Sette. On the propositional calculus P¹. Mathematica Japonicae, 18:173–180, 1973.

[W´oj88] R. W´ojcicki.Theory of logical calculi: basic theory of consequence operations.

Synthese library. Kluwer Academic Publishers, 1988.

Appendix A: Proofs of the main results

Lemma 6: Proof. Letr be a rule overC such thatr∈Cl(∅). IfH *H−r then there is r⁰ ∈RH\R_H−r. By relevance, there is H⁰ such thatr /∈Cl(H⁰).

But this is an absurd, by monotonicity ofCl.

Lemma 9: Proof. LetH=Cl(H), andX∈H⊥R. By extensiveness it follows that X ⊆ Cl(X). Now, suppose that X ⊂ Cl(X). By monotonicity Cl(X) ⊆ Cl(H) =H, and soX ⊂Cl(X)⊆H. SinceX ∈H⊥Rwe have, by idempotence, that R ⊆Cl(R_X), which is a contradiction. Hence, there is no r∈ Cl(X)\X

i.e.X =Cl(X).

Proposition 10: Proof. To prove that X ∈ H⊥R2 we need to show 1) that X ⊆ H, 2) R2 * Cl(RX) and 3) if X ⊂ X⁰ ⊆ H then R2 ⊆ Cl(RX⁰). 1) follows directly from the fact thatX∈H⊥R1 and 2) follows by hypothesis. To prove 3) notice that if X ⊂X⁰ ⊆H thenR₁ ⊆Cl(R_X0) and, since R₂ ⊆R₁,

R₂⊆Cl(X⁰).

Proposition 14: Proof. We need to prove 1) that X ⊆ H2, 2) R * Cl(RX) and 3) if X ⊂ X⁰ ⊆ H2 then R ⊆ Cl(RX⁰). 1) and 2) follow directly from hypothesis. Now let considerX⁰ such thatX⊂X⁰⊆H2. We have by hypothesis that H2 ⊆ H1 and, hence, X⁰ ⊆ H1. It follows that R ⊆ Cl(R⁰_X0), because

X ∈H1⊥R.

2 Lemmas 6 and 9 were adapted from [Han99].

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Lemma 16: Proof. Fori = 1, . . . , n it holds that (ξ →γ_i), ξ `_Hr γ_i, by (1), and so (ξ→γ₁), . . . ,(ξ→γ_n), ξ`_Hr γ_i, for everyi. Sinceγ₁, . . . , γ_n`_Hr ϕthen (ξ→γ₁), . . . ,(ξ→γ_n), ξ `_Hr ϕ. But then (ξ→γ₁), . . . ,(ξ→γ_n), ξ `_Hr (ξ→ ϕ), since `H^r ϕ →(ξ → ϕ) and by (1). On the other hand ∼ξ `H^r (ξ →ϕ), by (3), and so (ξ→ γ1), . . . ,(ξ→γn),∼ξ `H^r (ξ →ϕ). By (4) it follows that (ξ→γ1), . . . ,(ξ→γn)`H^r (ξ→ϕ) as required.

Corollary 17: Proof. Assume thatΓ, α`_Hr β. By induction on the length of a derivationϕ₁. . . ϕ_kofβfromΓ∪{α}inH^rit will be shown thatΓ `_Hr (α→β).

SinceH satisfies MTD with respect to→, the only case to be analyzed is when β is obtained by the use of the rule r=h{γ1, . . . , γn}, ϕi. Thus, there is some substitution σ such that β = ˆσ(ϕ) and {σ(γˆ 1), . . . ,σ(γˆ n)} ⊆ {ϕ1, . . . , ϕ_k−1}.

By induction hypothesis,Γ `H^r (α→σ(γˆ i)) for every i. By Lemma 16, (α→ ˆ

σ(γ1)), . . . ,(α → σ(γˆ n)) `H^r (α → σ(ϕ)) (by takingˆ σ(ξ) = α). Therefore Γ `H^r (α→σ(ϕ)), that is,ˆ Γ `H^r (α→β).

Corollary 18: Proof. Clearly γ₁, . . . , γ_n `_Hr ϕ, and then `_Hr γ₁ → (γ₂ → (. . .→(γn→ϕ). . .), by MTD. ThusCl(Hr)⊆Cl(H^r).

On the other hand, by (i) of Lemma 16 it holds that γ₁, . . . , γ_n `_H_r ϕ and

soCl(H^r)⊆Cl(Hr). This completes the proof.

Theorem 19: Proof. Assume that H and H⁰ satisfy the hypothesis of the theorem. The ‘if’ part is obviously true. For the ‘only if’ part, assume that H ∈H⁰⊥_wR and let r =h{γ₁, . . . , γ_n}, ϕibe a rule such that r ∈ R_H⁰ \R_H. SinceH⁰satisfies MTD with respect to→if follows that`_H⁰ γ₁→(γ₂→(. . .→ (γ_n→ϕ). . .)). SinceH satisfies (1) of Lemma 16 it follows that0H γ₁→(γ₂→ (. . .→(γn →ϕ). . .)). Given thatH ∈H⁰⊥wRit follows thatR*Cl(RH) but R ⊆Cl(Hr), where Hr =hC, RH∪ {h∅, γ1 →(γ2 → (. . . →(γn → ϕ). . .))i}i.

But Cl(Hr) = Cl(H^r), by Corollary 18, where H^r = hC, RH ∪ {r}i. Then

R⊆Cl(H^r) and soH ∈H⁰⊥R.

Proposition 20: Proof. 1. It is known that the logic I¹ is maximal with respect toCPL in the signature just containing →and ¬(cf.[SC95]). That is, I¹ ∈ CPL⊥wRCPL. Let ∼α =def (α → ¬α) and α∨β =def (¬(β → β) → β) → ((α → α) → α). Then `_I1 (∼α∨α) for every α. On the other hand, it is easy to show the following: Γ, α `_(I1)^r γ and Γ, β `_(I1)^r γ implies that Γ,(α∨β)`_(I1)^r γ, for everyΓ, α, β, γ and for every ruler, where (I¹)^r is as in Lemma 16. Thus,I¹satisfies the conditions (1)-(4) of Lemma 16, for every rule r. On the other hand, bothCPLandI¹ satisfy MTD with respect to→. Then, by Theorem 19 it follows thatI¹is strongly maximal with respect toCPL, that is,I¹∈CPL⊥RCPL.

2. We know from Corollary 12 thatI¹∈CPL⊥w{h{¬¬ξ1}, ξ1i}. The proof that I¹∈CPL⊥{h{¬¬ξ1}, ξ1i}is analogous to that of item 1.

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Proposition 21: Proof. The proof is similar to that of Proposition 20.

1. We begin by observing that the logicP¹is maximal with respect toCPLin the signature just containing→and¬(cf. [Set73]). That is,P¹∈CPL⊥_wR_CPL. It is enough to define in P¹ a classical negation ∼ and a disjunction ∨ which guarantee, as in the case ofI¹, the satisfaction of conditions (1)-(4) of Lemma 16, for every rule r. The derived connectives∼α=def ¬(¬α→α) and α∨β =def

(∼α→β) satisfy the required properties. Since bothP¹andCPLsatisfy MTD with respect to →then, by Theorem 19, it follows that P¹is strongly maximal with respect to CPL. That is,P¹∈CPL⊥RCPL.

2. We know from Corollary 11 thatP¹ ∈CPL⊥_w{h{ξ₁,¬ξ₁}, ξ₂i}. The proof that P¹∈CPL⊥{h{ξ₁,¬ξ₁}, ξ₂i} is analogous to that of item 1.

Lemma 22: Proof. Let C be the signature of H and enumerate the rules in RH:{r1, r2, . . .}. LetR0=RX and for eachi≥1 let:

Ri=

R_i−1∪ {ri}ifR*Cl(R_i−1∪ {ri}) R_i−1 otherwise.

Now considerH⁰ =hC,S

iR_ii. Clearly,X ⊆H⁰. Suppose thatR={r₁⁰, . . . , r⁰_n} is contained inCl(R_H⁰). SinceCl is compact, there exists a finite setR⁰_j⊆R_H⁰ such thatr⁰_j ∈Cl(R⁰_j) for j = 1, . . . , n. But R_i ⊆R_i+1 and so R⊆Cl(R_j) for some j, which is a contradiction. We conclude thatR*Cl(R_H⁰).

The other conditions forH⁰ ∈H⊥Rare easy to verify.³ Theorem 23: Proof. Inclusionfollows trivially andsuccessfollows directly from the upper-bound lemma with X =∅. To proverelevancenote that ifr⁰ ∈RH\ TΥ(H, r) then there is X ∈Υ(H, r) such that r⁰ ∈/ X. Of course T

Υ(H, r)⊆ X ⊆H,r /∈Cl(X) and, sinceX is maximal, thenr∈Cl(RX∪ {r⁰}). This same argument holds ifr⁰is an axiom andΥ is a weak subset selection funcion, hence weak relevance also holds. Finally, if H is closed then closure follows directly

from Lemma 9.

Theorem 24: Proof. We will show only a sketch of a proof for strong subset selection function. The proof for weak subset selection function is completely analogous. Let Υ(H, r) = {X ∈ H⊥{r} : H −r ⊆ X} if H⊥{r} 6= ∅ and Υ(H, r) ={H} otherwise. We need to show that Υ is well defined, that it is a selection function and thatH−r=Tγ(H, r).

Proving thatΥis well defined is trivial, since we definedΥover the generators (that is, the pairshH, ri).

It is also trivial to verify thatΥ(H, r)⊆H⊥{r}. Fromsuccess,inclusionand the upper-bound lemma, we show thatΥ(H, r)6=∅.

Now if H⊥{r} = ∅ then r ∈ Cl(∅). In this case TΥ(H, r) = H and, by failure and inclusion we have that H −r = H. If H⊥{r} 6= ∅ then trivially

3 This proof was adapted from a very similar to the proof of Lindenbaum’s lemma found in [W´oj88].

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H −r ⊆ TΥ(H, r). To prove the converse, suppose by absurdum that r⁰ ∈/ H −r. If r⁰ ∈/ H then r⁰ ∈/ TΥ(H, r) and we are done. Consider then that r⁰ ∈H. Then byrelevancethere isH⁰ such that H−r⊆H⁰ ⊆H,r /∈Cl(H⁰) and r ∈ Cl(RH⁰ ∪ {r⁰}). By the upper bound lemma there is X such that H⁰⊆X ∈H⊥{r}. It follows thatr⁰ ∈/X ∈Υ(H, r) and, hence,r⁰∈/ TΥ(H, r).

Theorem 25: Proof. We will sketch the proof of maxi-choice characterization.

The proof for weak maxi-choice characterization is completely analogous.

LetΨ(H, r) =H −r. We must prove that Ψ(H, r)∈H⊥{r}if H⊥{r} 6=∅ andΨ(H, r) =H otherwise.

The second situation follows directly fromfailure. Now let us assume that r /∈Cl(∅), thenΨ(H, r)∈H⊥{r}bysuccess,inclusion andfullness.