Three Formats of Prioritized Adaptive Logics: a ... · tion of models. For Minimal Abnormality, this semantics is defined as follows: ... In Appendix C, we show that (iii) fails

Three Formats of Prioritized Adaptive Logics:

a Comparative Study

Frederik Van De Putte & Christian StraßerCentre for Logic and Philosophy of Science

Ghent UniversityBlandijnberg 2, 9000 Gent, Belgium

frvdeput.vandeputte,[email protected]

October 19, 2011

Abstract

A broad range of defeasible reasoning forms has been explicated byprioritized adaptive logics. However, the relative lack in meta-theory ofmany of these logics stands in sharp contrast to the frequency of theirapplication. This paper presents the first comparative study of a largegroup of prioritized adaptive logics. Three formats of such logics are dis-cussed: superpositions of adaptive logics, hierarchic adaptive logics from[20] and logics in the AL⊏-format from [24]. We restrict the scope tologics that use the strategy Minimal Abnormality. It is shown that thesemantic characterizations of these systems are equivalent and that theyare all sound with respect to either of these characterizations. Further-more, sufficient conditions for the completeness and equivalence of theconsequence relations of the three formats are established. Some attrac-tive properties, including Fixed Point and the Deduction Theorem, areshown to hold whenever these conditions are obeyed.

1 Introduction

Prioritized Adaptive Logics. Adaptive logics (henceforth ALs) are formalsystems that model and explicate various forms of human reasoning: reasoningwith inconsistent premises [1], inductive generalization [6, 4], abduction [13,11], reasoning on the basis of conflicting norms [10, 14], argumentation [19],belief revision [25], etc.1 Many consequence relations from the literature havebeen reformulated as ALs, see e.g. [7, 9, 16, 27]. These achievements underlinethe strength of ALs as formal modeling tools and the unificatory power of theadaptive logic program.

ALs are first and foremost developed to capture defeasible reasoning forms(DRFs), i.e. reasoning forms in which certain inferences may be retracted inview of later insights. One of the most important developments within the ALprogram is the definition of a canonical format, the so-called standard format

1Unpublished papers in the reference section are available at the internet addresshttp://logica.UGent.be/centrum/writings/.

1

for ALs. This format encompasses a dynamic proof theory and semantics. Arich and attractive meta-theory has been shown to hold generically for all ALsformulated in the standard format (see [3]): they are sound and complete, theirconsequence relation is idempotent, cautiously monotonic, etc. Most ALs havebeen successfully expressed within this format, whence it provides a good basisfor a unifying study of DRFs.

Every AL in standard format is characterized by a triple: (i) a lower limitlogic (henceforth LLL), (ii) a set of abnormalities Ω and (iii) a strategy. The LLLis a monotonic logic, the rules of which are unconditionally valid in the AL. TheAL strengthens its LLL by considering a certain set of formulas (the elementsof Ω) as abnormal, and by interpreting premises “as normally as possible”.Semantically, this is realized by means of a selection on the set of LLL-models,in the vein of Shoham [15]. How this selection proceeds, depends on the strategyof the AL, which can be either Reliability or Minimal Abnormality.

In this paper, we will confine ourselves to the latter strategy. In this case,the AL selects the LLL-models that verify a minimal set of abnormalities – thedetails will be spelled out in Section 2.2

Notwithstanding its successes, the standard format does not incorporateprioritized ALs. These are elegant tools to model specific kinds of prioritizeddefeasible reasoning, i.e. the kind of defeasible reasoning in which defeasibleassumptions with different degrees of priority play a role. Examples of suchDRFs are: reasoning with prioritized belief bases [8] or imperatives [23]; induc-tive generalization with a preference for the strongest hypotheses [4]; ampliativereasoning supported by background knowledge [6]; inductive generalization ofabduced hypotheses [22].

The most common way to deal with prioritized DRFs within the adaptivelogic program is by the superposition of several ALs in standard format. Roughlyspeaking, this is done as follows: where AL1,AL2, . . . are ALs in standard for-mat that have the same LLL, and where CnALi

(Γ) denotes theALi-consequenceset of Γ, we characterize the superposition of logics, SAL, by

CnSAL(Γ) = “. . . CnAL3(CnAL2

(CnAL1(Γ))) . . .”

In [2], a semantics for these systems was proposed, in terms of a sequential selec-tion of models. For Minimal Abnormality, this semantics is defined as follows:first we select the LLL-models of Γ that are minimally abnormal with respectto AL1; from the resulting set of models, we select those that are minimallyabnormal with respect to AL2, etc (see Section 3 for the precise definitions).

Although for some concrete applications, superpositions of ALs were equippedwith a proof theory (see e.g. [19, 6, 18]), there has been no attempt so far toformulate a generic proof theory for SAL, and to prove that its derivabilityrelation coincides with the consequence relation described above, or is soundand complete with respect to the intended semantics of SAL.3

More generally, there has been a substantial lack of meta-theory on thesecombinations of ALs. The results that have been obtained so far are fairly neg-ative for Minimal Abnormality: superpositions of ALs that have the MinimalAbnormality Strategy lack many of the aforementioned properties which render

2We refer to [3] for the definition of Reliability.3In [5], a first proposal is made for such a generic proof theory. However, for superpositions

of ALs that use the Minimal Abnormality Strategy, the derivability relation characterized bythis proof theory does not coincide with the SAL-consequence relation as defined above.

2

the standard format so attractive. To mention but a few: their consequence rela-tion lacks both soundness and completeness with respect to the above semanticcharacterization; it is neither idempotent, nor cautiously monotonic.4

Another way to capture prioritized DRFs by means of a combination of ALsin standard format was put forward in [20] under the name hierarchic adaptivelogics. There the central idea was to take the union of the consequence setsof several ALs. Where each logic ALi has LLL as its lower limit logic, thiscombination is defined as follows:5

CnHAL(Γ) = CnLLL(CnAL1(Γ) ∪ CnAL2

(Γ) ∪ . . .)

Semantically, the set of HAL-models of Γ is the intersection of all the sets ofALi-models. A generic proof theory was defined for HAL, and proven to beadequate with respect to CnHAL(Γ) in [20]. Nevertheless, the hierarchic formatsuffers from the same meta-theoretic problems as superpositions of ALs: nofixed point, no completeness with respect to a non-redundant semantics, etc(see [20] where these results are documented).

In [24], the format AL⊏ for prioritized adaptive logics is presented, and it isproven that the meta-theoretic properties of the standard format easily general-ize to this new format.6 We refer to Section 2 for an overview of some technicalresults from [24]. As for flat ALs, every logic in AL⊏-format is characterized bya triple: an LLL, a sequence of sets of abnormalities 〈Ω1,Ω2, . . .〉 and a strategy:⊏-Reliability or ⊏-Minimal Abnormality. The semantics of AL⊏-logics that use⊏-Minimal Abnormality, is obtained by a lexicographic selection procedure – seeSection 2.1 for more details.

In the remainder, we use the superscriptm to refer to the Minimal Abnormality-variants of the three aformentioned formats: ALm

⊏, SALm and HALm.

Content and Outline of This Paper. We will first provide a general char-acterization of flat adaptive logics and logics in the ALm

⊏-format, and introduce

some notational conventions for the rest of the paper (Section 2). Next, wewill show that three classes of prioritized ALs (logics in ALm

⊏-format, those

in HALm-format and logics in a specific subclass SALmc of superposition-ALs)

stand in a one-to-one relation, i.e. that for every triple of associated logics ALm⊏,

HALm and SALmc each of the following holds:

(i) The semantic consequence relations of all three systems are equivalent(Section 3, Corollary 1)

(ii) The consequence relation of ALm⊏

is always at least as strong as that ofSALm

c and HALm (Section 4, Corollaries 6 and 7)

(iii) SALmc and HALm are complete and equivalent to ALm

⊏, given certain

(weak) restrictions on the premise sets (Section 5, Corollaries 11 and 12)

4See [5, Chapter 6] for more details.5As we will see in Section 4.2, the logics AL1,AL2, . . . have to fulfill certain specific

restrictions in order to get a well-behaved hierarchic logic – we refer to [20] for more details.6In a sense, all formats for prioritized ALs are generalizations of the standard format: if

there is only one priority level, the “prioritized” AL reduces to an AL in standard format.However, the generalization which leads to AL⊏ is more fundamental, in that the crucialconcepts of the standard format are generalized, in order to give the defeasibility a prioritizedflavor – see [24] for the details.

3

To arrive at (ii), we will prove that every logic in the format of SALmc is

sound with respect to its semantics (see Section 4, Theorem 15.2).It is worthwhile to stress that all the above results are proven generically, i.e.

the meta-proofs only rely on the properties of the formats, not on particularitiesof the logics defined in these formats. Property (i) contributes to the argumentthat the semantic consequence relation defined by these logics and, in view ofthe soundness and completeness of ALm

⊏, also the ALm

⊏-consequence relation is

a robust concept in the context of prioritized consequence relations.7 (ii) is ofparticular interest in view of the fact that ALm

⊏, SALm

c and HALm are eachcharacterized by their own peculiar semantics and proof theory. This meansthat we obtain a great variety of methods to prove that a formula is an ALm

⊏-

consequence of a set of premises.In Appendix C, we show that (iii) fails in cases where the restrictions re-

ferred to in (iii) are not obeyed. However, as shown in Section 6, whenever therestrictions hold, properties such as Fixed Point and the Deduction Theoremcan be easily transferred from ALm

⊏to HALm and SALm

c , relying on (i)-(iii).The current paper focuses mainly on the semantic characterizations, resp.

the properties of the consequence relations of ALm⊏, SALm

c and HALm. Sinceconsequence relation of flat ALs and logics in ALm

⊏-format is defined in terms

of their respective proof theory, we define the latter in the appendix to makethis paper self-contained. Note however that the consequence relation of SALm

c

and HALm are defined without any reference to proof theories. Hence, we willnot spell out the generic proof theory of HALm in this paper (see [20]), andwe consider that of SALm

c as a topic for future research. Also, we refer to[4, 5, 23, 17, 20, 24] for concrete applications of each of the aforementionedformats.

2 Flat and Prioritized Adaptive Logics

In this section, we introduce and define the format ALm⊂ of flat ALs and the

format ALm⊏

of prioritized ALs from [24]. We restrict ourselves to the semanticsof both formats (the proof theories are defined in Appendix A.1). We will firstprovide the official semantics of the systems (Section 2.1), and next define analternative characterization of their sets of models (Section 2.2). The latter willturn out very useful for certain meta-proofs in the remainder of this paper. Afterthat, we will briefly discuss normal premise sets, i.e. premise sets that do notgive rise to any abnormalities (Section 2.3). Finally, we introduce some notionswhich will facilitate the meta-proofs in the remainder of this paper (Section2.4). Unless specified differently, all results from this section are establishedand illustrated in detail in [3], [5] and [24] – all metaproofs can be found there.Before we start, let us introduce some conventions.

Throughout this paper, all formulas are assumed to be finite strings in a givenformal language. We will use A,B,C, . . . as metavariables for formulas, andΓ,∆,Θ, . . . as metavariables for sets of formulas. Where N is the set of naturalsnumbers without 0, we will use i, j, k, . . . as metavariables for members of N, andI, J,K, . . . as metavariables for initial sequences of N. Where I = 1, . . . , n,

let ~I =def n; where I = N, let ~I =def ∞. Let L be a logic with a characteristic

7By the “robustness” of a concept, we mean that it can be found in many different contexts,under various names and/or characterizations.

4

semantics. Where M is a L-model, we write M A to denote that A is verifiedby M . M is a L-model of Γ iff M A for all A ∈ Γ. The set of L-models of Γ isdenoted by ML(Γ). We say that A is a semantic L-consequence of Γ, Γ |=L A

iff A is verified by all L-models of Γ.

2.1 The Official Semantics

Every (flat or prioritized) adaptive logic is based on a lower limit logic LLL+.LLL+ is required to be compact, transitive, reflexive and monotonic. The su-perscript “+” refers to the fact that the lower limit logic is obtained by enrichinga monotonic logic LLL with classical connectives that are indicated by a checkmark: ¬, ∨, ∧, ⊃, and for the predicative case also ∃, ∀. This enrichment is mo-tivated by technical, meta-theoretic and philosophical reasons – we refer to [24,Section 2.1] where these are spelled out.

Where W is the set of formulas in the language L of LLL, W+ is obtainedby closing W under the checked connectives.8 Intuitively, every AL models areasoning process based on formulas in W, but uses formulas in W+ to expli-cate this reasoning process. Hence, although an AL is defined as a function℘(W+) → ℘(W+), in concrete applications, premise sets are usually subsets ofW. Nevertheless, for metatheoretic purposes (e.g. when considering the super-position of consequence relations), it is easier to let Γ refer to any subset of W+.Unless specified differently, we will do this in the remainder.

The AL strengthens LLL+, by considering formulas of a certain form false“as much as possible”. These formulas are called abnormalities. So a necessaryingredient of any AL is a set of abnormalities Ω, which is a set of formulasspecified by a logical form in L+. A flat AL considers each abnormality to be“equally bad”. For instance, if Γ = A ∨ B, where A,B ∈ Ω and where ∨behaves classically, then neither ¬A, nor ¬B will be an adaptive consequenceof Γ.

Prioritized ALs are also defined in terms of LLL+ and a set of abnormalitiesΩ, but this time Ω is further specified in terms of a sequence of sets, eachassociated with a certain priority level i ∈ I. In other words, we consider notone set of abnormalities, but a (possibly infinite) sequence of such sets: 〈Ωi〉i∈I ,which yields a prioritized set of abnormalities Ω =

⋃i∈I Ωi. Abnormalities of

level 1 are treated as the “worst” abnormalities by the prioritized AL, those oflevel 2 as the “second worst”, and so on. Hence the logic first tries to avoidthe abnormalities from Ω1, next those from Ω2, etc. Suppose that in the aboveexample, A ∈ Ω1 and B ∈ Ω2 −Ω1. In that case, ¬A, and hence also B will bea consequence of Γ.

At the semantic level, every AL selects a subset of the set of LLL+-models, inview of the abnormalities they verify. This requires some notational conventions.We first define the abnormal part of an LLL+-model M : Ab(M) = B ∈ Ω |M B. In ALm

⊂ , the abnormal parts of models are compared in terms of thesubset-relation ⊂. In ALm

⊏, they are compared in terms of a lexicographic order

⊏, which is defined as follows:9

8This means that in W+, checked connectives cannot occur in the scope of the connectivesfrom L.

9Lexicographic orders are well-known in mathematics (see e.g. [12, p. 1170]).

5

Definition 1 Where ∆,∆′ ⊆ Ω: 〈∆ ∩ Ωi〉i∈I ⊏lex 〈∆′ ∩ Ωi〉i∈I iff (1) there isan i ∈ I such that for all j < i, ∆ ∩ Ωj = ∆′ ∩ Ωj, and (2) ∆ ∩ Ωi ⊂ ∆′ ∩ Ωi.We write ∆ ⊏ ∆′ iff 〈∆ ∩ Ωi〉i∈I ⊏lex 〈∆

′ ∩ Ωi〉i∈I .

Let ≺ be a metavariable for ⊏ and ⊂. Then the logic ALm≺ selects the

LLL+-models of Γ whose abnormal part is ≺-minimal:

Definition 2 M ∈ MALm≺(Γ) iff M ∈ MLLL+(Γ) and there is no M ′ ∈

MLLL+(Γ) such that Ab(M ′) ≺ Ab(M).

With the above definitions and the regular definition of semantic consequence(Γ |=L A iff A is true in everyM ∈ ML(Γ)), we obtain the semantic consequencerelations |=ALm

⊂and |=ALm

⊏. As pointed out in Section 1, the standard format

and the AL⊏-format also encompass a proof theory which yields a syntacticconsequence relation ⊢ALm

≺, that is sound and complete with respect to |=ALm

≺.

We refer to Appendix A.1 for this proof theory. We define A ∈ CnALm≺(Γ) iff

Γ ⊢ALm≺A.

We continue this section with some theorems and a corollary, each of whichwill be referred to in the remainder of this paper. Recall that, unless stateddifferently, Γ ⊆ W+:

Theorem 1 Each of the following holds:

1. If Γ ⊢ALm≺A, then Γ |=ALm

≺A.

2. Where Γ ⊆ W: if Γ |=ALm≺A, then Γ ⊢ALm

≺A.

Theorem 2 If M ∈ MLLL+(Γ)−MALm≺(Γ), then there is a M ′ ∈ MALm

≺(Γ)

such that Ab(M ′) ≺ Ab(M). (Strong Reassurance)

Theorem 3 Where Γ ⊆ W: CnALm≺(Γ) = CnLLL+(CnALm

≺(Γ)). (LLL-closure)

Theorem 4 Where ALm⊂ is defined by 〈LLL+,

⋃i∈I Ωi,m〉 and ALm

⊏is defined

by 〈LLL+, 〈Ωi〉i∈I ,m〉:

1. MALm⊏(Γ) ⊆ MALm

⊂(Γ)

2. CnALm⊂(Γ) ⊆ CnALm

⊏(Γ)

The last result which we mention here is proven in Appendix A.2. It statesthat for a very specific kind of premise sets Γ ⊆ W+, the logic ALm

⊂ is soundand complete as well:10

Theorem 5 Where Γ = CnLLL+(Γ): Γ ⊢ALm⊂A iff Γ |=ALm

⊂A.

2.2 The Alternative Characterization of MALm≺(Γ)

The alternative characterization of the set of ALm≺ -models of a given Γ is based

on the set of minimal disjunctions of abnormalities that are LLL+-derivablefrom Γ. For flat ALs, this characterization is well-known from the metatheoryof the standard format – see e.g. [3] where it is established and applied; itwas generalized to the ALm

⊏-format in [24]. Where ∆ is a finite subset of Ω,

let Dab(∆) =def

∨∆. Where ∆ = A, Dab(∆) denotes A; where ∆ = ∅,

10We call upon Theorem 5 in Sections 4 (Lemma 2) and 5 (Lemma 6).

6

∨Dab(∆) denotes the empty string. Dab(∆) is a minimal Dab-consequence ofΓ iff Γ ⊢LLL+ Dab(∆) and there is no ∆′ ⊂ ∆ for which Γ ⊢LLL+ Dab(∆′).Where Dab(∆j) | j ∈ J are the minimal Dab-consequences of Γ, let Σ(Γ) =∆j | j ∈ J.

Let Ψ = ∆k ⊆ Ω | k ∈ K. We say that ϕ ⊆ Ω is a choice set of Ψ iff forevery k ∈ K, ϕ ∩ ∆k 6= ∅. For the border case where Ψ = ∅, this means thatevery set ϕ ⊆ Ω is a choice set of Ψ, including the empty set.

The following two facts give two salient properties of choice sets of Σ(Γ) andtheir relation to LLL+-models of Γ:

Fact 1 Where M ∈ MLLL+(Γ), Ab(M) is a choice set of Σ(Γ).

Fact 2 If Γ has LLL+-models, then for every choice set ϕ of Σ(Γ), there is anLLL+-model M of Γ such that Ab(M) ⊆ ϕ.

Just as we did with the abnormal parts of models, we may rank the choicesets of Σ(Γ) according to the partial orders ⊂ and ⊏. As before, let ≺ be ameta-variable for these partial orders. We say that ϕ is a ≺-minimal choice setof Ψ iff there is no choice set ψ of Ψ such that ψ ≺ ϕ.

Definition 3 Φ≺(Γ) is the set of ≺-minimal choice sets of Σ(Γ).

So, in view of the minimal disjunctions of abnormalities that are LLL+-derivable from Γ, we obtain a set of sets of abnormalities Φ≺(Γ). The followingtheorem provides the link between Φ≺(Γ) and the set of ALm

≺ -models of Γ:

Theorem 6 Each of the following holds:

1. M ∈ MALm≺(Γ) iff (M ∈ MLLL+(Γ) and Ab(M) ∈ Φ≺(Γ))

2. If Γ has LLL+-models, then Φ≺(Γ) = Ab(M) |M ∈ MALm≺(Γ)

In view of this correspondence between MALm≺(Γ) and Φ≺(Γ), Theorem 4.1

has the following counterpart:

Theorem 7 Φ⊏(Γ) ⊆ Φ⊂(Γ).

In view of Theorem 6, we can establish a necessary and sufficient conditionfor the membership of CnALm

≺(Γ), whenever Γ ⊆ W:

Theorem 8 Where Γ ⊆ W: A ∈ CnALm≺(Γ) iff for every ϕ ∈ Φ≺(Γ), there is

a ∆ ⊆ Ω− ϕ such that Γ ⊢LLL+ A ∨Dab(∆).

The left-right direction of the above theorem also holds for the more generalcase where Γ ⊆ W+:

Theorem 9 If A ∈ CnALm≺(Γ), then for every ϕ ∈ Φ≺(Γ), there is a ∆ ⊆ Ω−ϕ

such that Γ ⊢LLL+ A ∨Dab(∆).

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2.3 Normal Premise Sets

Every flat adaptive logic in standard format has a unique upper limit logic ULL,which is the logic that trivializes all abnormalities of the AL. Semantically, ULL

is obtained as follows. We say that M ∈ MLLL+(Γ) is normal iff M 6 A forevery A ∈ Ω. We write Γ |=ULL A iff A is true in every normal LLL+-model ofΓ. Similarly, for the prioritized logic ALm

⊏= 〈LLL+, 〈Ωi〉i∈I ,m〉, we obtain the

associated ULL by defining normal models as those LLL+-models that falsifyevery member of Ω =

⋃i∈I Ωi. The following is immediate:

Fact 3 MULL(Γ) ⊆ MALm≺(Γ)

Where Θ ⊆ W+, let Θ¬ = ¬A | A ∈ Θ. Syntactically, we defineCnULL(Γ) =def CnLLL+(Γ ∪ Ω¬). In [3], it is shown that ULL is a mono-tonic logic that is sound and complete. Hence by Fact 3 and Theorem 1.1:

Theorem 10 CnALm≺(Γ) ⊆ CnULL(Γ).

We say that a premise set Γ is normal iff it has normal models; alternatively,iff Γ ∪ Ω¬ is LLL+-satisfiable. The following is proven in [24]:

Theorem 11 If Γ is normal, then CnALm≺(Γ) = CnULL(Γ) = CnLLL+(Γ∪Ω¬).

In other words, if it is possible to avoid all abnormalities, then the adaptivelogic will do so. In Section 4.3, we will see that this result can be extendedto the formats SALm

c and HALm. There we will also formulate a stronger,prioritized variant of this theorem and prove it for each of ALm

⊏, SALm

c andHALm.

2.4 Some Notational Conventions

To close the gap between, on the one hand, combinations in terms of flat adap-tive logics, and on the other hand logics in theALm

⊏-format, it will be convenient

to define logics that only consider abnormalities up to a certain rank i ∈ I. Forevery i ∈ I, let Ω(i) = Ω1 ∪ . . . ∪ Ωi. We define the lexicographic order up tolevel i as follows:

Definition 4 〈∆ ∩ Ωj〉j≤i ⊏(i)lex〈∆′ ∩ Ωj〉j≤i iff (1) there is a j ≤ i such that

for all k < j, ∆ ∩ Ωk = ∆′ ∩ Ωk, and (2) ∆ ∩ Ωj ⊂ ∆′ ∩ Ωj.We write ∆⊏(i) ∆

′ iff 〈∆ ∩ Ωj〉j≤i ⊏(i)lex〈∆′ ∩ Ωj〉j≤i.

We define the prioritized adaptive logic ALm⊏(i)

just as ALm⊏, but replacing

the whole sequence 〈Ωi〉i∈I by 〈Ωj〉j≤i, and replacing ⊏ by ⊏(i). Similarly, wecharacterize the flat adaptive logic ALm

(i) by the triple 〈LLL,Ω(i),m〉. Note thatsince these logics are all defined in ALm

≺ -format, all the theorems mentioned inSection 2 hold for them as well.

In a similar vain as for ALm⊏, we can characterize the ALm

⊏(i)-semantics

in terms of minimal choice sets. We say that Dab(∆) is a minimal Dab(i)-consequence of Γ iff Γ ⊢LLL+ Dab(∆) and there is no ∆′ ⊂ ∆ such that Γ ⊢LLL+

Dab(∆). Where Dab(∆j) | j ∈ J are the minimal Dab(i)-consequences of Γ,

let Σ(i)(Γ) = ∆j | j ∈ J. Let Φ⊏(i)(Γ) be the set of ⊏(i)-minimal choice sets

of Σ(i)(Γ) and let Φ(i)(Γ) be the set of ⊂-minimal choice sets of Σ(i)(Γ).

8

Fact 4 Let ∆,∆′ ⊆ Ω. Then ∆∩Ω1 ⊂ ∆′∩Ω1 iff ∆⊏(1) ∆′. Hence CnALm

1(Γ) =

CnALm(1)

(Γ) = CnALm⊏(1)

(Γ), MALm(1)

(Γ) = MALm1(Γ) = MALm

⊏(1)(Γ), and

Φ(1)(Γ) = Φ⊏(1)(Γ).

In view of the definitions of ⊏(i), ALm⊏(i)

and Φ⊏(i)(Γ), we have:11

Fact 5 Each of the following holds for every i ∈ I:

1. ∆ ⊏(i) ∆′ iff ∆ ∩ Ω(i) ⊏(i) ∆

′ ∩ Ω(i)

2. If ∆⊏(i) ∆′, then ∆ ⊏ ∆′

3. If ∆⊏(i) ∆′, then ∆⊏(j) ∆

′ for every j ∈ I, i ≤ j

4. Where Γ ⊆ W: CnALm⊏(i)

(Γ) ⊆ CnALm⊏(Γ).

In the remainder of this paper, we will skip the sub- and superscript ⊂, inorder to facilitate the reading and to stay as close as possible to the notationalconventions of the adaptive logics program. Hence we will write Φ(Γ) insteadof Φ⊂(Γ) and ALm instead of ALm

⊂ .

3 Two equivalent Semantic Characterizations

In this section we will define the semantics for superposed adaptive logics and forhierarchic adaptive logics. As explained in Section 1, both formats are defined interms of a combination of a sequence of flat ALs 〈ALm

1 ,ALm2 , . . .〉. As a central

result we will show that for a specific class of such sequences, the semantics ofSALm and HALm are equivalent.

3.1 Superposition of Selections

In the introduction we have already informally explicated the idea of the se-quential selections that constitute the semantics of SALm. We will now makethis idea formally precise. Where SALm is characterized by the sequence of flatadaptive logics 〈ALm

i〉i∈I , its semantics is defined as follows:

Definition 5 Let MSALm0(Γ) = MLLL+(Γ). For every i ∈ I, let MSALm

i(Γ) =

M ∈ MSALmi−1

(Γ) | there is no M ′ ∈ MSALmi−1

(Γ) : Ab(M ′) ∩ Ωi ⊂ Ab(M) ∩

Ωi. Let MSALm(Γ) = lim infi→~I

MSALmi(Γ) = lim sup

i→~IMSALm

i(Γ) =

⋂i∈I MSALm

i(Γ).12

Fact 6 MSALm1(Γ) = MALm

1(Γ) = MALm

(1)(Γ) = MALm

⊏(1)(Γ).

Fact 7 Each of the following holds for every i ∈ I:

1. MSALmi(Γ) ⊆ MSALm

i−1(Γ).

2. MSALmi(Γ) =

⋂j≤i MSALm

j(Γ)

11For the proofs of Facts 5.1-5.3, it suffices to refer to Definitions 1 and 4. For Fact 5.4, weneed Definition 2, Fact 5.2, resp. Fact 5.3, the soundness of ALm

⊏(i)and the completeness of

ALm⊏.

12Note that the sequence 〈MSALmi(Γ)〉i∈I converges to its limes inferior resp. to its limes

superior due to the fact that the sequence is anti-monotonic (see Fact 7.1).

9

Lemma 1 If M ∈ MSALmi(Γ) and M ′ ∈ MLLL+(Γ) is such that Ab(M ′) ∩

Ω(i) ⊆ Ab(M) ∩ Ω(i), then M′ ∈ MSALm

i(Γ).

Proof. Assume that the antecedent holds, but M ′ 6∈ MSALmi(Γ). Let j ≤ i

be the smallest j ∈ I such that M ′ 6∈ MSALmj(Γ). By Definition 5, there is

an M ′′ ∈ MSALmj−1

(Γ) such that Ab(M ′′) ∩ Ωj ⊂ Ab(M ′) ∩ Ωj . In view of the

supposition, Ab(M ′)∩Ωj ⊆ Ab(M)∩Ωj , whence (†) Ab(M′′)∩Ωj ⊂ Ab(M)∩Ωj .

By the supposition and Definition 5, M ∈ MSALmj−1

(Γ). By (†) and Definition

5, M 6∈ MSALmj(Γ), whence also M 6∈ MSALm

i(Γ) — a contradiction.

In the remainder we will often restrict our focus to a specific class of super-posed adaptive logics, namely of sequences of the type 〈ALm

(1),ALm

(2), . . .〉. Inwhat follows we write SALm

(i) for the superposed adaptive logic characterized by

the sequence 〈ALm

(1), . . . ,ALm

(i)〉, and SALmc for the superposed adaptive logic

characterized by 〈ALm

(i)〉i∈I .From Theorem 12 below, we can infer that the semantics of SALm

c is equiv-alent to that of the logic ALm

⊏characterized by the triple 〈LLL, 〈Ωi〉i∈I ,m〉.

Theorem 12 MSALmc(Γ) = MALm

⊏(Γ).

Proof. (MSALmc(Γ) ⊆ MALm

⊏(Γ)) Assume that M ∈ MSALm

c(Γ)−MALm

⊏(Γ).

By Definition 5, M ∈ MLLL+(Γ). Hence by Definition 1 and Definition 2,there is an i ∈ I and an M ′ ∈ MLLL+(Γ) such that (1) for every j < i:Ab(M ′) ∩ Ωj = Ab(M) ∩ Ωj , and (2) Ab(M ′) ∩ Ωi ⊂ Ab(M) ∩ Ωi. It followsthat (1)’ for every j < i: Ab(M ′) ∩ Ω(j) = Ab(M) ∩ Ω(j) and (2)’ Ab(M ′) ∩Ω(i) ⊂ Ab(M) ∩ Ω(i). By Definition 5, M ∈ MSALm

(i−1)(Γ). But then also

M ′ ∈ MSALm(i−1)

(Γ), which implies that M 6∈ MSALm(i)(Γ). By Definition 5,

M 6∈ MSALmc(Γ) — a contradiction.

(MALm⊏(Γ) ⊆ MSALm

c(Γ)) Assume that M ∈ MLLL+(Γ)−MSALm

c(Γ).

Take the smallest i ∈ I for which M 6∈ MSALm(i)(Γ). By Definition 5, there is

an M ′ ∈ MSALm(i−1)

(Γ) such that Ab(M ′) ∩ Ω(i) ⊂ Ab(M) ∩ Ω(i). Note that

M ′ ∈ MLLL+(Γ). Also, it follows that there is a k ≤ i such that (1) for everyj ∈ I, j < k: Ab(M ′) ∩Ωj = Ab(M) ∩Ωj , and (2) Ab(M ′) ∩Ωk ⊂ Ab(M) ∩Ωk.But then M 6∈ MALm

⊏(Γ) by Definitions 1 and 2.

3.2 Intersection of Selections

Similar to superpositions of ALs, hierarchic ALs are characterized on the ba-sis of sequences of flat adaptive logics: 〈AL1,AL2, . . .〉. In the hierarchic casehowever, the logics in the sequence have to fulfill the condition that Ωi ⊆ Ωi+1.Thus, the sequences that characterize hierarchical adaptive logics using the mini-mal abnormality strategy can be written as 〈ALm

(1),ALm

(2), . . .〉. Their semanticsis characterized by the following set of models:

Definition 6 MHALm(Γ) =⋂

i∈I MALm(i)(Γ).

As for SALmc , we can prove that the semantics of every logic HALm is

equivalent to that of ALm⊏, where the latter is defined by 〈LLL, 〈Ωi〉i∈I ,m〉:

Theorem 13 MHALm(Γ) = MALm⊏(Γ)

10

Proof. (MHALm(Γ) ⊆ MALm⊏(Γ)) Suppose M ∈ MLLL+(Γ)−MALm

⊏(Γ). By

Definition 1 and Definition 2, there is an i ∈ I and anM ′ ∈ MLLL+(Γ) such that(1) for every j < i: Ab(M ′)∩Ωj = Ab(M)∩Ωj , and (2) Ab(M ′)∩Ωi ⊂ Ab(M)∩Ωi. It follows that (2)’ Ab(M ′) ∩ Ω(i) ⊂ Ab(M) ∩ Ω(i). Thus M 6∈ MALm

(i)(Γ),

whence by Definition 6, M 6∈ MHALm(Γ).(MALm

⊏(Γ) ⊆ MHALm(Γ)) Suppose M ∈ MLLL+(Γ)−MHALm(Γ). Hence

there is an i ∈ I: M 6∈ MALm(i)(Γ), whence by Definition 2, there is an M ′ ∈

MLLL+(Γ): Ab(M ′)∩Ω(i) ⊂ Ab(M)∩Ω(i). It follows that there is a k ≤ i suchthat (1) for every j < k: Ab(M ′) ∩ Ωj = Ab(M) ∩ Ωj , and (2) Ab(M ′) ∩ Ωk ⊂Ab(M) ∩ Ωk. Thus M 6∈ MALm

⊏(Γ).

3.3 Some Corollaries

The following corollary is one of the central results presented in this paper. Itshows that the semantics of hierarchic adaptive logics and logics in the ALm

⊏-

format define the same consequence relation. Moreover, there is a class of su-perposed adaptive logics for which the semantics is equivalent to that of HALm

and ALm⊏

as well. This class of superposed adaptive logics is characterized bythe same sequences of flat adaptive logics as hierarchical adaptive logics, namelysequences of the form 〈ALm

(1),ALm

(2), . . .〉. The fact that three different pathsof devising selection semantics for prioritized logics lead to the same semanticconsequence relation demonstrates the centrality, robustness and usefulness ofthe latter.

Corollary 1 MALm⊏(Γ) = MSALm

c(Γ) = MHALm(Γ). Hence, Γ |=ALm

⊏A iff

Γ |=SALmcA iff Γ |=HALm A.

In the remainder of this section, let PAL ∈ ALm⊏,SALm

c ,HALm. ByFact 3 and Corollary 1, we have:

Corollary 2 MULL(Γ) ⊆ MPAL(Γ).

Since ALm⊏

is sound and complete with respect to |=ALm⊏

(see Theorem 1),the corollary equips us with alternative semantic selection procedures for ALm

⊏:

Corollary 3 Each of the following holds:

1. If Γ ⊢ALm⊏A, then Γ |=PAL A.

2. Where Γ ⊆ W: if Γ |=PAL A, then Γ ⊢ALm⊏A.

Theorems 2, 12 and 13 imply that the Strong Reassurance property holdsfor SALm

c and HALm:

Corollary 4 If M ∈ MLLL+(Γ)−MPAL(Γ), then there is an M ′ ∈ MPAL(Γ)such that Ab(M ′) ⊏ Ab(M).

4 Two Consequence Relations

In this section, we provide a definition of CnSALm(Γ) and CnHALm(Γ). We willshow that the SALm-consequence relation is both reflexive and closed underLLL+. Next, we will show that each logic in the specific class of logics SALm

c

11

is sound with respect to the semantic characterization from Section 3.1. We willrecapitulate the soundness result for HALm from [20] and derive from this thatALm

⊏is always at least as strong asHALm. Finally, we will show that whenever

it is possible in view of the lower limit logic and Γ, to falsify all abnormalitiesup to a certain level i, then each of SALm, HALm and ALm

⊏will do so.

4.1 Superpositions of Consequence Relations

Before we turn to the definition of SALm, recall that this format is more generalthan SALm

c – see Section 3.1. The consequence relation of SALm, obtained bythe superposition of the logics 〈ALm

i〉i∈I , is defined as follows:

Definition 7 Let SALm0 = LLL+. For every i ∈ I, let

CnSALmi(Γ) = CnALm

i(. . . (CnALm

2(CnALm

1(Γ))) . . .)

Moreover, let CnSALm(Γ) = lim infi→~I

CnSALmi(Γ) = lim sup

i→~ICnSALm

i(Γ) =

⋃i∈I CnSALm

i(Γ).13

Fact 8 CnSALm1= CnSALm

(1)= CnALm

1= CnALm

(1).

Let us first consider some properties that hold for SALm in general. Sinceevery logic ALm

iis reflexive, we immediately get:

Fact 9 Each of the following holds:

1. For every i ∈ I, Γ ⊆ CnSALmi(Γ).

2. Γ ⊆ CnSALm(Γ). (Reflexivity)3. For every i ∈ I: CnSALm

i−1(Γ) ⊆ CnSALm

i(Γ).

The following lemma requires a bit more explanation. As Theorem 3 states,every logic ALm

iis closed under LLL+. However, this theorem is restricted to

the case where Γ ⊆ W.14 In order to generalize it to SALm, we first need toestablish the LLL+-closure of ALm, for a specific kind of premise sets Γ ⊆ W+:

Lemma 2 If Γ = CnLLL+(Γ), then CnALm(Γ) = CnLLL+(CnALm(Γ)).

Proof. That CnALm(Γ) ⊆ CnLLL+(CnALm(Γ)) is immediate in view of the re-flexivity of LLL+. For the other direction, suppose that (†) Γ = CnLLL+(Γ) andA ∈ CnLLL+(CnALm(Γ)). Hence A is true in every M ∈ MLLL+(CnALm(Γ)).By the soundness of LLL+, (‡) A is true in every LLL+-model of CnALm(Γ).By Definition 2 and the soundness of ALm, every ALm-model of Γ is an LLL+-model of CnALm(Γ). Hence by (‡), Γ |=ALm A. By (†) and Theorem 5,Γ ⊢ALm A.

Theorem 14 Where Γ ⊆ W, each of the following holds:

1. for every i ∈ I, CnSALmi(Γ) = CnLLL+(CnSALm

i(Γ)).

2. CnSALm(Γ) = CnLLL+(CnSALm(Γ)). (LLL-closure)

13Note that the sequence 〈CnSALmi(Γ)〉i∈I converges to its limes inferior resp. to its limes

superior due to the fact that the sequence is monotonic (see Fact 9.3).14There are Γ ⊆ W+ for which it fails – see [21, Chapter 2, Section 8] for an example.

12

Proof. Ad 1. (i = 1) Immediate in view of Theorem 3 and Fact 8.(i ⇒ i + 1) By Definition 7, CnSALm

i+1(Γ) = CnALm

i+1(CnSALm

i(Γ)). Hence

by the induction hypothesis and Lemma 2,

CnSALmi+1

(Γ) = CnLLL+(CnALmi+1

(CnSALmi(Γ))) (1)

By (1) and Definition 7, CnSALmi+1

(Γ) = CnLLL+(CnSALmi+1

(Γ)).

Ad 2. That CnSALm(Γ) ⊆ CnLLL+(CnSALm(Γ)) is immediate in view ofthe reflexivity of LLL+. Suppose that A ∈ CnLLL+(CnSALm(Γ)). By Def-inition 7 and the compactness of LLL+, there is an i ∈ I such that A ∈CnLLL+(

⋃j≤i CnSALm

i(Γ)), whence by Fact 9.3, A ∈ CnLLL+(CnSALm

i(Γ)).

By item 1, A ∈ CnSALmi(Γ), whence by Definition 7, A ∈ CnSALm(Γ).

For the proof of Theorem 15 below, we need to establish a specific propertyof the ALm

≺ -semantics. It states that if every member of a set Γ′ is true in allALm

≺ -models of Γ, then MALm≺(Γ ∪ Γ′) = MALm

≺(Γ):

Lemma 3 If Γ′ ⊆ A | Γ |=ALm≺A, then MALm

≺(Γ ∪ Γ′) = MALm

≺(Γ).15

Proof. Suppose (†) Γ′ ⊆ A | Γ |=ALm≺A.

(MALm≺(Γ ∪ Γ′) ⊆ MALm

≺(Γ)) Consider an M ∈ MALm

≺(Γ ∪ Γ′). By Def-

inition 2, M ∈ MLLL+(Γ ∪ Γ′) and hence M ∈ MLLL+(Γ). Assume thatM 6∈ MALm

≺(Γ). By Theorem 2, there is an M ′ ∈ MALm

≺(Γ) such that

Ab(M ′) ≺ Ab(M). However, in view of (†), M ′ A for every A ∈ Γ′, whencealso M ′ ∈ MLLL+(Γ ∪ Γ′). By Definition 2, M 6∈ MALm

≺(Γ ∪ Γ′) — a contra-

diction.(MALm

≺(Γ) ⊆ MALm

≺(Γ ∪ Γ′)) Consider an M ∈ MALm

≺(Γ). By (†), M A

for every A ∈ Γ′. By Definition 2, M is an LLL+-model of Γ. We thus obtainthat M is a LLL+-model of Γ ∪ Γ′. Assume that M 6∈ MALm

≺(Γ ∪ Γ′). By

Theorem 2, there is an M ′ ∈ MLLL+(Γ ∪ Γ′): Ab(M ′) ≺ Ab(M). Hence M ′ ∈MLLL+(Γ). By Definition 2, M 6∈ MALm

≺(Γ) — a contradiction.

The preceding obervations allow us to establish the main result of this sec-tion. This concerns the more specific class of logics SALm

c , defined in terms ofthe sequence of logics 〈ALm

(i)〉i∈I . In the remainder, we prove that these logicsare sound with respect to the sequential superposition of semantic selectionsfrom Section 3.1, whence by Corollary 1 they are also sound with respect to theother semantic selections featured in this paper.16

Theorem 15 Each of the following holds for every i ∈ I:

1. For every M ∈ MSALm(i)(Γ), Ab(M) ∩ Ω(i) ∈ Φ(i)(CnSALm

(i−1)(Γ))

2. If A ∈ CnSALm(i)(Γ), then Γ |=SALm

(i)A.

Proof. (i = 1) Item 1 is immediate in view of Fact 8 and Theorem 6.1; item 2is immediate in view of Fact 6, Fact 8 and the Soundness of ALm

(1).

(i⇒ i+ 1) Ad 1. By the induction hypothesis (2.), Fact 7 and Theorem 12respectively, CnSALm

(i)(Γ) ⊆ A | Γ |=SALm

(i)A ⊆ A | Γ |=SALm

(i+1)A = A |

15This lemma generalizes Lemma 34 from [24] – there it was shown that the consequent ofthe lemma holds for ALm

⊏whenever Γ′ ⊆ CnALm

⊏(Γ).

16Logics in SALm-format are not in general sound with respect to their semantics – see[21, Chapter 3, Section 3] for an example.

13

Γ |=ALm⊏(i+1)

A. By Lemma 3, MALm⊏(i+1)

(Γ) = MALm⊏(i+1)

(Γ ∪ CnSALm(i)(Γ)),

whence by the Reflexivity of SALm

(i),MALm⊏(i+1)

(Γ) = MALm⊏(i+1)

(CnSALm(i)(Γ)).

By Theorem 12,

MSALm(i+1)

(Γ) = MSALm(i+1)

(CnSALm(i)(Γ)) (2)

Suppose (†) M ∈ MSALm(i+1)

(Γ). By (2), M ∈ MSALm(i+1)

(CnSALm(i)(Γ)). By

Definition 5, M is an LLL+-model of CnSALm(i)(Γ), whence by Fact 1, Ab(M)∩

Ω(i+1) is a choice set of Σ(i+1)(CnSALm(i)(Γ)).

Suppose Ab(M) ∩ Ω(i+1) 6∈ Φ(i+1)(CnSALm(i)(Γ)). By Definition 3, there is a

choice set ψ of Σ(i+1)(CnSALm(i)(Γ)), such that ψ ⊂ Ab(M) ∩ Ω(i+1). By Fact

2, there is an LLL+-model M ′ of CnSALm(i)(Γ) such that Ab(M ′) ∩ Ω(i+1) ⊆ ψ,

whence Ab(M ′) ∩ Ω(i+1) ⊂ Ab(M) ∩ Ω(i+1). Note that since M ∈ MSALm(i)(Γ),

by Lemma 1, M ′ ∈ MSALm(i)(Γ). But then by Definition 5, M 6∈ MSALm

(i+1)(Γ),

which contradicts (†).

Ad 2. Suppose A ∈ CnSALm(i+1)

(Γ), whence by Definition 7,

A ∈ CnALm(i+1)

(CnSALm(i)(Γ)). By Theorem 9, for every ϕ ∈ Φ(i+1)(CnSALm

(i)(Γ)),

there is a ∆ ⊂ Ω(i+1) such that (†) CnSALm(i)(Γ) ⊢LLL+ A ∨Dab(∆) and

∆ ∩ ϕ = ∅. By Theorem 14.1, (‡) A ∨Dab(∆) ∈ CnSALm(i)(Γ) for every such ∆.

Let M ∈ MSALm(i+1)

(Γ). By item 1, there is a ϕ ∈ Φ(i+1)(CnSALm(i)(Γ)) such

that Ab(M)∩Ω(i+1) = ϕ. By Definition 5, M ∈ MSALm(i)(Γ). By the induction

hypotheses,M B for everyB ∈ CnSALm(i)(Γ), whence by (‡),M A ∨Dab(∆)

for a ∆ ⊂ Ω(i+1) such that ϕ ∩ ∆ = ∅. But then, since Ab(M) ∩ Ω(i+1) = ϕ,M ¬Dab(∆), whence M A.

Note that if A ∈ CnSALmc(Γ), then by Definition 7, there is an i ∈ I such

that A ∈ CnSALm(i)(Γ). Also, by Definition 5, if Γ |=SALm

(i)A for an i ∈ I, then

Γ |=SALmcA. Hence in view of Theorem 15.2, we immediately obtain:

Corollary 5 If A ∈ CnSALmc(Γ), then Γ |=SALm

cA.

By Theorems 1.2 and 12, we obtain:

Corollary 6 Where Γ ⊆ W: CnSALmc(Γ) ⊆ CnALm

⊏(Γ)

4.2 Unions of Consequence Sets

In the notational conventions from the current paper – see also Section 3.3 –,we can define the HALm-consequence relation as follows:

Definition 8 CnHALm(Γ) = CnLLL+(⋃

i∈I CnALm(i)(Γ))

We refer to [20] for illustrations of how this combination leads to a prioritizedtreatment of the sets of abnormalities 〈Ωi〉i∈I . Theorem 14 from that paperstates that HALm is sound with respect to the semantics defined in Section3.2:

14

Theorem 16 If A ∈ CnHALm(Γ), then Γ |=HALm A.

By Corollary 1 and Theorem 1.2, we immediately have:

Corollary 7 Where Γ ⊆ W: CnHALm(Γ) ⊆ CnALm⊏(Γ).

4.3 Normal Premise Sets Revisited

In Section 2.3, we saw that whenever Γ ∪ Ω¬ has LLL+-models, then ALm≺ is

identical to its upper limit logic ULL. As we will see below, this result canbe generalized to sequential superpositions of ALs, as well as to hierarchic ALs– see Theorem 19 below. However, in the case of prioritized ALs, one mayalso wonder whether a slightly stronger property holds. That is, suppose thatfor some i ∈ I, it is possible to verify all members of Γ, yet also falsify allabnormalities up to level i. In that case, it seems a desirable property for aprioritized logic PAL that CnLLL+(Γ ∪ Ω¬

(i)) ⊆ CnPAL(Γ) – in other words,that the prioritized logic indeed considers all the members of Ω(i) to be false.

To formally express this property, we introduce the concepts of normality atlevel i, resp. up to level i:

Definition 9 Γ is normal at level i iff Γ ∪Ω¬i has LLL+-models. Γ is normal

up to level i iff Γ ∪ Ω¬(i) has LLL+-models.

The following is immediate in view of Definition 9:

Fact 10 If Γ is normal up to level i, then each of the following holds:

1. Γ is normal at level j, for every j ≤ i

2. Γ ∪ Ω¬(j) is normal at level j + 1, for every j < i

3. CnLLL+(Γ ∪ Ω¬(j)) is normal at level j + 1, for every j < i

In the remainder, we use ULLi to refer to the upper limit logic of ALm

i, i.e.

ULLi is the monotonic logic that trivializes all abnormalities of level i. Likewise,ULL(i) denotes the upper limit logic of ALm

(i) and trivializes all abnormalities

up to level i. Finally, let PAL ∈ ALm⊏,SALm

c ,HALm.

Theorem 17 If Γ is normal up to level i, then CnULL(i)(Γ) ⊆ CnPAL(Γ).

Proof. (PAL = ALm⊏) This is Theorem 25 in Appendix A.3.17

(PAL = SALmc ) Suppose Γ is normal up to level i. We show by an in-

duction that for all j ≤ i, CnSALmj(Γ) = CnLLL+(Γ ∪ Ω¬

(j)) — the rest followsimmediately.

(j = 1) By Fact 10.1 and the supposition, Γ is normal at level 1. By Fact 8and Theorem 11 respectively, CnSALm

1(Γ) = CnALm

1(Γ) = CnLLL+(Γ ∪ Ω¬

1 ).(j ⇒ j + 1): By the induction hypothesis, CnSALm

j(Γ) = CnLLL+(Γ ∪

Ω¬(j)). By Fact 10.3 and the supposition, CnLLL+(Γ ∪ Ω¬

(j)) is normal at level

j + 1. But then by Theorem 11, CnSALmj+1

(Γ) = CnALmj+1

(CnSALmj(Γ)) =

CnALmj+1

(CnLLL+(Γ∪Ω¬(j))) = CnULLj+1

(CnLLL+(Γ∪Ω¬(j))) = CnLLL+(CnLLL+(Γ∪

Ω¬(j)) ∪ Ω¬

j+1) = CnLLL+(Γ ∪ Ω¬(j) ∪ Ω¬

j+1) = CnLLL+(Γ ∪ Ω¬(j+1)).

17Where Γ ⊆ W, this theorem can be proven by semantic means. For the more general case(Γ ⊆ W+), the proof of this theorem requires reference to syntactic notions, whence we putit in the appendix.

15

(PAL = HALm) Immediate in view of the fact that CnALm(i)(Γ) ⊆ CnHALm(Γ)

(see Definition 8), Definition 9, and Theorem 11.

Theorem 18 CnPAL(Γ) ⊆ CnULL(Γ).

Proof. Suppose A ∈ CnPAL(Γ). By the soundness of PAL (see Theorem1.1, Corollary 6 and Theorem 16 respectively), Γ |=PAL A. By Corollary 2,Γ |=ULL A. By the completeness of ULL, A ∈ CnULL(Γ).

In view of Theorem 17, Theorem 18 and the monotonicity and compactnessof LLL+, the proof of the following can be safely left to the reader:

Theorem 19 If Γ is normal, then CnPAL(Γ) = CnULL(Γ).

5 Equivalence Results

In this section, we establish the third major result we promised in the intro-duction, i.e. that given certain weak conditions, the logics SALm

c and HALm

(defined from the sequence of flat ALs 〈ALm

(i)〉i∈I) are complete and equivalent

to ALm⊏

(defined by the triple 〈LLL+, 〈Ωi〉i∈I ,m〉).

5.1 The Basic Criteria for Equivalence

Note that the following is the case:

Theorem 20 Where Γ ⊆ W and PAL ∈ SALmc ,HALm: if

MLLL+(CnPAL(Γ)) = MPAL(Γ) (3)

then

1. Γ |=PAL A iff A ∈ CnPAL(Γ), and2. CnPAL(Γ) = CnALm

⊏(Γ).

Proof. Ad 1. (⇒) If Γ |=PAL A then A is true in everyM ∈ MLLL+(CnPAL(Γ)).By the completeness of LLL+ and the fact that CnPAL(Γ) is closed under LLL+

(in case PAL = SALmc see Theorem 14.2, in case PAL = HALm this holds by

definition), A ∈ CnPAL(Γ). (⇐) See Corollary 5, resp. Theorem 16.Ad 2. Immediate in view of item 1, Corollary 1 and the soundness and

completeness of ALm⊏.

Equation (3) expresses that the set of PAL-models is characterized by meansof the PAL-consequence set: the models of the prioritized adaptive logic areexactly those LLL+-models that verify the PAL-consequences. This is a centralcriterion since it is sufficient for both, the soundness and completeness of PAL

(point 1.), and for the equivalence of the consequence relation of the threeprioritized adaptive logics that are presented in this paper (point 2.).

The criteria for soundness and equivalence are defined by means of sets ofcomplements of minimal choice sets. Where ≺ ∈ ⊂,⊏, let cΦ≺(Γ) = Ω−ϕ |ϕ ∈ Φ≺(Γ). Likewise, let cΦ⊏(i)(Γ) = Ω(i) − ϕ | ϕ ∈ Φ⊏(i)(Γ) and letcΦ(i)(Γ) = Ω(i) − ϕ | ϕ ∈ Φ(i)(Γ).

In Sections 5.2 and 5.3 we will give syntactic criteria in terms of these sets,that warrant (3) for SALm

c , resp. HALm. But first, let us show that this holdsfor flat ALs and ALs in AL⊏-format.

16

Lemma 4 Where Γ ⊆ W: if cΦ≺(Γ) has no infinite minimal choice sets, thenMLLL+(CnALm

≺(Γ)) = MALm

≺(Γ).

Proof. Suppose cΦ≺(Γ) has no infinite minimal choice sets. That MALm≺(Γ) ⊆

MLLL+(CnALm≺(Γ)) is immediate in view of Definition 2 and the soundness of

ALm≺ . So assume that M ∈ MLLL+(CnALm

≺(Γ))−MALm

≺(Γ). By Theorem 6,

for every ϕ ∈ Φ≺(Γ), there is an Aϕ ∈ Ω − ϕ such that M Aϕ. Note thatAϕ | ϕ ∈ Φ≺(Γ) is a choice set of cΦ≺(Γ)). Hence by the supposition, thereis a finite Θ ⊆ Aϕ | ϕ ∈ Φ≺(Γ), such that Θ is a choice set of cΦ≺(Γ). Itfollows by Theorem 6 that Γ |=ALm

≺¬ ∧Θ, and hence by Theorem 1.2, also

Γ ⊢ALm≺¬ ∧Θ. But then M 6∈ MLLL+(CnALm

≺(Γ)) — a contradiction.

Lemma 5 If MLLL+(CnALm≺(Γ)) = MALm

≺(Γ), then cΦ≺(Γ) has no infinite

minimal choice sets.

Proof. Let Θ be an infinite minimal choice set of cΦ≺(Γ). Assume there isno LLL+-model of CnALm

≺(Γ) ∪ Θ. By the compactness of LLL+, there is a

finite Aj | j ∈ J ⊂ Θ such that CnALm≺(Γ) |=LLL+ ∨j∈J ¬Aj . Hence by

the soundness18 of ALm≺ , every ALm

≺ -model of Γ falsifies an Aj (j ∈ J). ByTheorem 6, for every ϕ ∈ Φ≺(Γ), there is a j ∈ J such that Aj 6∈ ϕ. But thenAj | j ∈ J is a choice set of cΦ≺(Γ) — a contradiction to the minimality ofΘ. So there is a LLL+-model M of CnALm

≺(Γ) ∪Θ.

Assume M ∈ MALm≺(Γ). By Theorem 6, there is a ϕ ∈ Φ≺(Γ) such that

Ab(M) = ϕ. However, since Θ is a choice set of cΦ≺(Γ), there is a A ∈ (Ω−ϕ)∩Θ— a contradiction. Hence M ∈ MLLL+(CnALm

≺(Γ))−MALm

≺(Γ).

Corollary 8 Where Γ ⊆ W: cΦ≺(Γ) has no infinite minimal choice sets iffMLLL+(CnALm

≺(Γ)) = MALm

≺(Γ).

Where ≺ = ⊂, the same result can be obtained for a specific class of premisesets Γ ⊆ W+:

Lemma 6 Where Γ = CnLLL+(Γ): cΦ(Γ) has no infinite minimal choice setsiff MLLL+(CnALm(Γ)) = MALm(Γ).

Proof. (⇒) Immediate in view of the proof for Lemma 4 – replace Theorem 1.2by Theorem 5. (⇐) Immediate in view of Lemma 5.

The above results are of crucial importance for the completeness and equiv-alence results of both SALm

c and HALm, which we shall present subsequently.The following additional lemmas will also be useful in the remainder:

Lemma 7 For every ϕ ∈ Φ⊏(i)(Γ), there is a ψ ∈ Φ⊏(Γ) such that ψ∩Ω(i) = ϕ.

Proof. Case 1. Γ is not LLL+-satisfiable. In that case, Γ ⊢LLL+ A for everyA ∈ Ω, whence Φ⊏(i)(Γ) = Ω(i) and Φ⊏(Γ) = Ω. Hence the lemma followsimmediately.

Case 2. Γ is LLL+-satisfiable. Suppose ϕ ∈ Φ⊏(i)(Γ) for an i ∈ I. ByTheorem 6.2, there is an M ∈ MALm

⊏(i)(Γ) such that Ab(M) ∩ Ω(i) = ϕ. Note

that M ∈ MLLL+(Γ). If M ∈ MALm⊏(Γ), then by Theorem 6.1, Ab(M) ∈

18Note that every ALm≺-model of Γ is an LLL+-model.

17

Φ⊏(Γ), whence the lemma follows immediately. So suppose M 6∈ MALm⊏(Γ).

Then by Theorem 2, there is an M ′ ∈ MALm⊏(Γ) such that Ab(M ′) ⊏ Ab(M).

Assume (†) Ab(M ′) ∩ Ω(i) 6= Ab(M) ∩ Ω(i). In view of Definitions 1 and 4,there is a j ≤ i such that Ab(M ′) ⊏(j) Ab(M). By Fact 5.3, Ab(M ′) ⊏(i) Ab(M).But then M 6∈ MALm

⊏(i)(Γ) — a contradiction. Hence (†) fails: Ab(M ′)∩Ω(i) =

Ab(M) ∩ Ω(i). Since by Theorem 6.1, Ab(M ′) ∈ Φ⊏(Γ), the lemma followsimmediately.

Lemma 8 For every ϕ ∈ Φ⊏(Γ), ϕ ∩ Ω(i) ∈ Φ⊏(i)(Γ).

Proof. Assume that ϕ ∈ Φ⊏(Γ), but ϕ ∩ Ω(i) 6∈ Φ⊏(i)(Γ). Note that since ϕ

is a choice set of Σ(Γ), ϕ ∩ Ω(i) is a choice set of Σ(i)(Γ). Hence there is aψ ∈ Φ⊏(i)(Γ) such that ψ⊏(i) ϕ. By Lemma 7, there is a ψ′ ∈ Φ⊏(Γ) such thatψ′ ∩ Ω(i) = ψ. But then by Fact 5.2, ψ′ ⊏ ϕ — a contradiction.

Corollary 9 Φ⊏(i)(Γ) = ϕ ∩ Ω(i) | ϕ ∈ Φ⊏(Γ).

Lemma 9 If cΦ⊏(Γ) has no infinite minimal choice sets, then for every i ∈ I,cΦ⊏(i)(Γ) has no infinite minimal choice sets.

Proof. Let Θ be an infinite minimal choice set of cΦ⊏(i)(Γ). By Corollary 9,

(†1) Θ is a minimal choice set of Ω(i) − (ϕ ∩ Ω(i)) | ϕ ∈ Φ⊏(Γ) = Ω(i) − ϕ |ϕ ∈ Φ⊏(Γ)

Assume that for some ϕ ∈ Φ⊏(i)Γ, Ω(i) − ϕ = ∅. But then ϕ = Ω(i) andwhence Φ⊏(i) = Ω(i). Hence cΦ⊏(i)(Γ) = ∅ which is a contradiction to theminimality of Θ. Thus:

(†2) for all ϕ ∈ Φ⊏(i)Γ, Ω(i) − ϕ 6= ∅

By (†1) and (†2), for all ϕ ∈ Φ⊏(Γ), Ω(i) − ϕ 6= ∅. By (†1) and since Ω(i) − ϕ ⊆Ω− ϕ, Θ is a choice set of cΦ⊏(Γ).

Assume there is a Θ′ ⊂ Θ which is a choice set of cΦ⊏(Γ). Since Θ ⊆ Ω(i),also (†3) Θ

′ ⊂ Ω(i). Note that (†4) for each ϕ ∈ Φ⊏(Γ), ((Ω−ϕ)− (Ω(i) −ϕ))∩Ω(i) = ∅. By (†3) and (†4), Θ

′ is a choice set of cΦ⊏(i)(Γ), which contradicts theminimality of Θ. Hence Θ is a minimal choice set of cΦ⊏(Γ).

5.2 Restricted Completeness and Equivalence for SALm

c

The basic completeness/equivalence criterion for SALmc reads as follows:

(⋆SALmc) cΦ⊏(Γ) has no infinite minimal choice sets

In the remainder, we will show that ⋆SALmc

is equivalent to equation (3)from Theorem 20, for SALm

c .

Lemma 10 If for all i ∈ I, MLLL+(CnSALmi(Γ)) = MSALm

i(Γ), then

MLLL+(CnSALm(Γ)) = MSALm(Γ)

Proof. We have: MLLL+(CnSALm(Γ)) = MLLL+(⋃

i∈I CnSALmi(Γ)) =⋂

i∈I MLLL+(CnSALmi(Γ)) =

⋂i∈I MSALm

i(Γ) = MSALm(Γ).

18

Lemma 11 Where Γ ⊆ W: if Γ satisfies ⋆SALmc, then MLLL+(CnSALm(Γ)) =

MSALm(Γ).

Proof. Suppose Γ ⊆ W and Γ satisfies ⋆SALmc. Thus, cΦ⊏(Γ) has no infinite

minimal choice sets.If Γ is not LLL+-satisfiable, then by Fact 9.1 and the monotonicity of LLL+,

CnSALm(i)(Γ) is not LLL+-satisfiable for every i ∈ I. Also, by Definition 5,

Γ is not SALm

(i)-satisfiable for every i ∈ I. Hence MLLL+(CnSALm(i)(Γ)) =

MSALm(i)(Γ) = ∅, whence the lemma follows immediately. So suppose that Γ is

LLL+-satisfiable. We will prove by induction that for every i ∈ I,MLLL+(CnSALm(i)(Γ)) =

MSALm(i)(Γ), whence by Lemma 10, the property follows immediately.

(i = 1) By the supposition, Fact 4 and Lemma 9, cΦ(1)(Γ) has no infiniteminimal choice sets. Hence by Lemma 4, MLLL+(CnALm

(1)(Γ)) = MALm

(1)(Γ).

The rest follows immediately in view of Facts 6 and 8.(i ⇒ i + 1) Let Γ′ = CnSALm

(i)(Γ). By Definition 2, MALm

(i+1)(Γ′) =

M ∈ MLLL+(Γ′) | there is no M ′ ∈ MLLL+(Γ′) such that Ab(M ′)∩Ω(i+1) ⊂Ab(M) ∩ Ω(i+1). By the induction hypotheses and Definition 5,

MALm(i+1)

(Γ′) = MSALm(i+1)

(Γ) (4)

By Theorem 12 and (4), MALm(i+1)

(Γ′) = MALm⊏(i+1)

(Γ). By Theorem 6.2, we

obtain that Φ(i+1)(Γ′) = Φ⊏(i+1)(Γ), whence also

cΦ(i+1)(Γ′) = cΦ⊏(i+1)(Γ) (5)

By the supposition and Lemma 9, cΦ⊏(i+1)(Γ) has no infinite minimal choicesets. Hence in view of (5), cΦ(i+1)(Γ′) has no infinite minimal choice sets. ByTheorem 14.1 and Lemma 6,

MLLL+(CnALm(i+1)

(Γ′)) = MALm(i+1)

(Γ′) (6)

Hence in view of Definition 7, MLLL+(CnSALm(i+1)

(Γ)) = MALm(i+1)

(Γ′). By (4),

MLLL+(CnSALm(i+1)

(Γ)) = MSALm(i+1)

(Γ).

Lemma 12 Where Γ ⊆ W: if MLLL+(CnSALmc(Γ)) = MSALm

c(Γ) then Γ

satisfies ⋆SALmc.

Proof. Suppose cΦ⊏(Γ) has an infinite minimal choice set. By Lemma 5,MLLL+(CnALm

⊏(Γ)) 6= MALm

⊏(Γ). By the soundness of ALm

⊏, MALm

⊏(Γ) ⊆

MLLL+(CnALm⊏(Γ)). It follows that there is anM ∈ MLLL+(CnALm

⊏(Γ))−MALm

⊏(Γ).

By Corollary 6 and the monotonicity of LLL+, M ∈ MLLL+(CnSALmc(Γ)). By

Corollary 1, M 6∈ MSALmc(Γ).

Corollary 10 Where Γ ⊆ W: Γ satisfies ⋆SALmc

iff MLLL+(CnSALmc(Γ)) =

MSALmc(Γ).

In view of Theorem 20, we immediately obtain:19

19SALmc is not complete for all premise sets – we refer to Appendix C.3 for a counterex-

ample. Notably, this example also illustrates that in some cases, HALm may yield moreconsequences that SALm

c .

19

Corollary 11 Where Γ ⊆ W: if Γ satisfies ⋆SALmc

, then each of the followingholds:

1. A ∈ CnSALmc(Γ) iff Γ |=SALm

cA

2. CnSALmc(Γ) = CnALm

⊏(Γ)

5.3 Restricted Completeness and Equivalence for HALm

The following (restricted) completeness result for HALm was proven in [20]:whenever Φ(Γ) is finite for a Γ ⊆ W, then A ∈ CnHALm(Γ) iff Γ |=HALm A.In the following, we further generalize this result to all premise sets that satisfythe following criterion:

(⋆HALm) for every i ∈ I, cΦ(i)(Γ) has no infinite minimal choice sets

Lemma 13 Where Γ ⊆ W: if Γ satisfies ⋆HALm , then MLLL+(CnHALm(Γ)) =MHALm(Γ).

Proof. Suppose the antecedent holds. By Lemma 8, it follows that (†) for everyi ∈ I, MLLL+(CnALm

(i)(Γ)) = MALm

(i)(Γ). By Definition 8, (†) and Definition

6 consecutively, we have MLLL+(CnHALm(Γ)) = MLLL+(⋃

i∈I CnALm(i)(Γ)) =

⋂i∈I MLLL+(CnALm

(i)(Γ)) =

⋂i∈I MALm

(i)(Γ) = MHALm(Γ).

Unlike for SALmc , the right-left direction of the above lemma fails – we

refer to Appendix C.4 for a counterexample. By Theorem 20, we immediatelyobtain:20

Corollary 12 Where Γ ⊆ W: if Γ satisfies ⋆HALm , then each of the followingholds:

1. A ∈ CnSALmc(Γ) iff Γ |=SALm

cA

2. CnSALmc(Γ) = CnALm

⊏(Γ)

5.4 Some Weaker Completeness and Equivalence Criteria

In the preceding, we saw two sufficient syntactic criteria for the completeness andequivalence results for SALm

c , resp. HALm. As we will now show, several morestraightforward criteria can be listed, each of which imply that either one or bothof the conditions for equivalence are obeyed. Hence in concrete applications,there are various ways to establish that e.g. CnSALm

c(Γ) = CnHALm(Γ) =

CnALm⊏(Γ), or that Γ |=HALm A iff A ∈ CnHALm(Γ). The following is proven

in Appendix B:

Theorem 21 Each of the following holds for every Γ ⊆ W:

1. Σ(Γ) is finite iff every ϕ ∈ Φ(Γ) is finite.2. If every ϕ ∈ Φ(Γ) is finite, then Φ(Γ) is finite.3. If Φ(Γ) is finite, then for all i ∈ I, Φ(i)(Γ) is finite.4. If Φ(i)(Γ) is finite for every i ∈ I, then Γ satisfies ⋆HALm .5. If Φ(Γ) is finite, then Φ⊏(Γ) is finite.6. If Φ⊏(Γ) is finite, then for all i ∈ I, Φ⊏(i)(Γ) is finite.

20As shown in Appendix C.2, unrestricted completeness fails for HALm.

20

7. If Φ⊏(Γ) is finite, then Γ satisfies ⋆SALmc.

8. Γ satisfies ⋆SALmc

iff for no i ∈ I, cΦ⊏(i)(Γ) has infinite minimal choicesets.

Σ(Γ) is finite

f.a. ϕ ∈ Φ(Γ),ϕ is finite

Φ(Γ) is finitef.a. i ∈ I,

Φ(i)(Γ) is finiteΓ satisfies⋆HALm

Φ⊏(Γ) is finite f.a. i ∈ I,Φ⊏(i) is finite

Γ satisfies⋆SALm

c

f.a. i ∈ I, cΦ⊏(i)(Γ)has no infinite

minimal choice sets

Figure 1: Syntactic criteria for completeness and equivalence

Figure 1 illustrates the relation between the criteria listed in Theorem 21and the criteria ⋆SALm

cand ⋆HALm . In Appendix C.4, we show by concrete

examples that the converses of items 2-7 in Theorem 21 fail.

6 Some Additional Results

Some properties which were hitherto not proven for SALmc and HALm, follow

almost immediately from the soundness results together with Corollaries 11 and12. These properties are:

Cumulative Transitivity: where Γ′ ⊆ CnL(Γ) : CnL(Γ ∪ Γ′) ⊆ CnL(Γ)

Fixed Point: CnL(Γ) = CnL(CnL(Γ))

The Deduction Theorem: If B ∈ CnL(Γ ∪ A), then A ⊃B ∈ CnL(Γ)

Each of these was proven to hold for L = ALm⊏

in [24]. We will now show thata restricted version of them can be easily derived for SALm

c andHALm, in viewof the soundness and equivalence results from the two preceding sections. In theremainder, let PAL ∈ HALm,SALm

c . The following Corollary summarizesCorollaries 6 and 7:

Corollary 13 Where Γ ⊆ W: CnPAL(Γ) ⊆ CnALm⊏(Γ).

Theorem 22 Where Γ ⊆ W and Γ satisfies ⋆PAL: if Γ′ ⊆ CnPAL(Γ), thenCnPAL(Γ ∪ Γ′) ⊆ CnPAL(Γ). (Restricted Cumulative Transitivity)

Proof. Suppose the antecedent holds. By the soundness of PAL and Corollary1, Γ′ ⊆ A | Γ |=PAL A = A | Γ |=ALm

⊏A. Hence by Corollary 1 and Lemma

21

3:MPAL(Γ) = MALm

⊏(Γ) = MALm

⊏(Γ ∪ Γ′) = MPAL(Γ ∪ Γ′) (7)

Suppose that A ∈ CnPAL(Γ∪Γ′). By the soundness of PAL, A is true in everyM ∈ MPAL(Γ ∪ Γ′). Hence by (7), A is true in every M ∈ MPAL(Γ). Since Γobeys ⋆PAL, it follows that A ∈ CnPAL(Γ).

Theorem 23 Where Γ ⊆ W and Γ satisfies ⋆PAL: CnPAL(Γ) = CnPAL(CnPAL(Γ)).(Restricted Fixed Point)

Proof. Suppose the antecedent holds. (CnPAL(Γ) ⊆ CnPAL(CnPAL(Γ))) Im-mediate in view of the reflexivity of PAL.

(CnPAL(CnPAL(Γ)) ⊆ CnPAL(Γ)) By the reflexivity of PAL, CnPAL(Γ) =Γ∪CnPAL(Γ). But then CnPAL(CnPAL(Γ)) = CnPAL(Γ∪CnPAL(Γ)), whenceby the restricted Cumulative Transitivity ofPAL, CnPAL(CnPAL(Γ)) ⊆ CnPAL(Γ).

Theorem 24 Where Γ ⊆ W, Γ satisfies ⋆PAL and A ∈ W: if B ∈ CnPAL(Γ∪A), then A ⊃B ∈ CnPAL(Γ). (Restricted Deduction Theorem)

Proof. Suppose the antecedent holds. By Corollary 13, B ∈ CnALm⊏(Γ ∪ A).

Hence since the Deduction Theorem holds for ALm⊏, A ⊃B ∈ CnALm

⊏(Γ). By

Corollary 11.2 (for SALmc ), resp. Corollary 12.2 (for HALm) and the supposi-

tion, A ⊃B ∈ CnPAL(Γ).

7 In Conclusion

Let us briefly recapitulate the main results of this paper. We have shown thatthe semantic selections defined by ALm

⊏, SALm

c and HALm lead to an identicalset of models, whence these logics define the same semantic consequence relation.We have shown that SALm

c is sound with respect to its semantics, and that bothHALm and SALm

c are weaker than ALm⊏. Finally, we have established criteria

for the completeness of SALmc , resp. HALm, which immediately imply their

equivalence to ALm⊏. These facts allowed us to easily prove some additional

properties of SALmc and HALm.

On the basis of these results, future research may further extend the metathe-ory of prioritized ALs, e.g. by showing whether one can prove a restricted cau-tious monotonicity theorem for SALm

c and HALm as well.21 Also, the currentresults may help in the establishment of a unified and generic proof theory forsequential superpositions and more specifically, logics in the SALm

c -format.

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22

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APPENDIX

A Some Syntactic Proofs

In this section, we define a generic proof theory for logics in ALm

≺ -format – as before,≺ is a metavariable for both ⊂ and ⊏. We do so without further comments – for illus-trations and philosophical motivations, we refer to [24, Section 2]. Subsequently, weprove two Theorems which call for syntactic meta-proofs and were therefore ommittedfrom the main text. Both meta-proofs rely only on theorems stated in Section 2.

A.1 The Proof Theory of ALm

≺

Every ALm≺ -proof consists of lines that have four elements: a line number i, a formula

A, a justification (consisting of a series of line numbers and a derivation rule) and acondition ∆ ⊆ Ω. Where Γ is the set of premises, the inference rules are given by:

PREM If A ∈ Γ:...

...

A ∅

RU If A1, . . . , An ⊢LLL+ B: A1 ∆1

......

An ∆n

B ∆1 ∪ . . . ∪ ∆n

RC If A1, . . . , An ⊢LLL+ B ∨Dab(Θ) A1 ∆1

......

An ∆n

B ∆1 ∪ . . . ∪ ∆n ∪ Θ

24

A stage of a proof is a sequence of lines, obtained by the application of the aboverules. A proof is a sequence of stages. Every proof starts off with stage 1. Adding aline to a proof by applying one of the rules of inference brings the proof to a successorstage, which is the sequence of all lines written so far. An extension of a proof at stages is simply the same proof at a later stage s′.

A distinctive feature of adaptive proofs is the marking definition. At every stageof a proof, this definition determines for each line in the proof whether it is markedor not. If a line that has as its second element A is marked at stage s, this indicatesthat according to our best insights at this stage, A cannot be considered derivable. Ifthe line is unmarked at stage s, we say that A is derived at stage s of the proof. Toprepare for the marking definition, we need some more conventions.

Where ∅ 6= ∆ ⊂ Ω, Dab(∆) is a Dab-formula at stage s of a proof iff it is thesecond element of a line at stage s with an empty condition. Dab(∆) is a minimalDab-formula at stage s iff there is no other Dab-formula Dab(∆′) at stage s for which∆′ ⊂ ∆. Where Dab(∆1), Dab(∆2), . . . are the minimal Dab-formulas at stage s ofa proof, let Σs(Γ) = ∆1,∆2, . . .. Let Φ≺

s (Γ) be the set of ≺-minimal choice sets ofΣs(Γ).

Definition 10 ALm

≺ -Marking: a line l with formula A is marked at stage s iff, whereits condition is ∆: (i) there is no ϕ ∈ Φ≺

s (Γ) such that ϕ ∩ ∆ = ∅, or (ii) fora ϕ ∈ Φ≺

s (Γ), there is no line on which A is derived on a condition Θ for whichΘ ∩ ϕ = ∅.

Put differently: a line with formula A is unmarked at stage s iff its condition hasan empty intersection with at least one ϕ ∈ Φ≺

s (Γ), and for every ψ ∈ Φ≺

s (Γ), thereis a line on which A is derived on a condition ∆ such that ∆ ∩ ψ = ∅. As a line maybe marked at stage s, unmarked at a later stage s′ and marked again at a still laterstage s′′, we also define a stable notion of derivability.

Definition 11 A is finally derived from Γ on line l of a finite stage s iff (i) A is thesecond element of line l, (ii) line l is unmarked at stage s, and (iii) every extension ofthe proof at stage s, in which line l is marked may be further extended in such a waythat line l is unmarked again.

Definition 12 Γ ⊢ALm≺A iff A is finally derived on a line of an ALm

≺ -proof from Γ.

A.2 A Specific Kind of Adequacy for ALm

Lemma 14 Φ(Γ) 6= ∅.

Proof. Case 1: Γ has no LLL+-models. By the completeness of LLL+, Γ ⊢LLL+ A

for every A ∈ Ω, whence Σ(Γ) = A | A ∈ Ω and hence Φ(Γ) = Ω 6= ∅.Case 2: Γ has LLL+-models. By Theorem 2, Γ has ALm-models. By Theorem

6.2, Φ(Γ) = Ab(M) |M ∈ MALm(Γ) 6= ∅.

Lemma 15 For every finite ∆ ⊂ Ω: Γ |=LLL+ Dab(∆) iff Γ |=ALm Dab(∆).22

Proof. (⇒) Immediate in view of the fact that every ALm-model of Γ is an LLL+-model of Γ — see Definition 2. (⇐) Suppose Γ |=ALm Dab(∆). Let M ∈ MLLL+(Γ).IfM ∈ MALm(Γ), it follows by the supposition thatM Dab(∆). IfM ∈ MLLL+(Γ)−MALm(Γ),then by Theorem 2, there is an M ′ ∈ MALm(Γ) such that Ab(M ′) ⊂ Ab(M). In viewof the supposition, M ′

Dab(∆), whence M ′ A for an A ∈ ∆. It follows immedi-

ately that also M A, whence M Dab(∆).

22This lemma is a semantic variant of the one for Theorem 10 in [3], but generalized toevery Γ ⊆ W+.

25

Lemma 16 If Γ |=ALm≺A, then for every ϕ ∈ Φ≺(Γ), there is a ∆ ⊆ Ω−ϕ such that

Γ |=LLL+ A ∨Dab(∆).

Proof. Suppose Γ |=ALm≺A. Assume that there is a ϕ ∈ Φ≺(Γ) for which there is no

∆ ⊆ Ω − ϕ such that Γ |=LLL+ A ∨Dab(∆). This implies that Γ is LLL+-satisfiable.By the compactness of LLL+ there is an M ∈ MLLL+(Γ) for which M 6 A andM 6 Dab(∆) for all ∆ ⊆ Ω−ϕ. Hence, M 6 B for all B ∈ Ω−ϕ. Hence, Ab(M) ⊆ ϕ.By Theorem 6.2 there is an M ′ ∈ MALm

≺(Γ) for which Ab(M ′) = ϕ. Since M ′ is

≺-minimally abnormal, Ab(M) = Ab(M ′). Hence, by Theorem 6.1, M ∈ MALm≺

(Γ).This is a contradiction since M 6 A and Γ |=ALm

≺A.

Proof of Theorem 5. Note that if Γ ⊢ALm A, then Γ |=ALm A by the (unrestricted)soundness of ALm. For the other direction, suppose that (†) Γ = CnLLL+(Γ) and(‡) Γ |=ALm A. By Lemma 14, there is a ϕ ∈ Φ(Γ). By (‡) and Lemma 16, thereis a ∆ ⊆ Ω − ϕ such that Γ |=LLL+ A ∨Dab(∆). By the completeness of LLL+,Γ ⊢LLL+ A ∨Dab(∆), whence by (†), A ∨Dab(∆) ∈ Γ.

We start an ALm-proof from Γ as follows: (a) introduce the premise A ∨Dab(∆)on line 1; (b) derive A on line 2, using the rule RC, on the condition ∆. Let s2 be thestage consisting of these two lines.

Suppose line 2 is marked at stage s2. This implies that A ∨Dab(∆) is a Dab-formula, whence also A is a Dab-formula. But then, by (‡) and Lemma 15, Γ |=LLL+ A.By the completeness of LLL+, Γ ⊢LLL+ A whence by (†), A ∈ Γ. By the reflexivityof ALm, A ∈ CnALm(Γ).

Suppose line 2 is not marked at stage s2. Suppose moreover that, in an extensionof the proof, line 2 is marked. In view of the preceding, we may further extend theextended proof, such that (c) every minimal Dab-consequence of Γ is derived in it, and(d) for every ϕ′ ∈ Φ(Γ), A is derived on a condition ∆′ that has an empty intersectionwith ϕ′. Let s be the stage of the second extension. In view of (c), Φs(Γ) = Φ(Γ).Hence in view of (d), line 2 is unmarked at stage s. But then, by Definition 11, A isfinally derived at line 2, whence by Definition 12, A ∈ CnALm(Γ).

A.3 Normal Premise Sets Revisited: ALm

⊏

Lemma 17 If Γ is normal up to level i, then there is no ∆ ⊂ Ω(i) such that Γ ⊢LLL+

Dab(∆).

Proof. Suppose the antecedent holds, and let ∆ be a finite subset of Γ. By the suppo-sition, there is an M ∈ MLLL+(Γ ∪ Ω¬

(i)), whence M 6 Dab(∆). By the soundness of

LLL+, Γ 0LLL+ Dab(∆).

Lemma 18 If Γ is normal up to level i, then at every stage s of a proof from Γ,ϕ ∩ Ω(i) = ∅ for every ϕ ∈ Φ⊏

s (Γ).

Proof. Suppose the antecedent holds and let s be a stage of a proof from Γ. Assumethat for a ϕ ∈ Φ⊏

s (Γ), ϕ ∩ Ω(i) 6= ∅. Let ψ =⋃Θ − Ω(i) | Θ ∈ Σ

(i)s (Γ). By the

supposition and Lemma 17, every Θ ∈ Σ(i)s (Γ) is such that Θ − Ω(i) 6= ∅, whence ψ

is a choice set of Σ(i)s (Γ). However, since ψ ∩ Ω(i) = ∅, it follows that ψ ⊏ ϕ — a

contradiction.

Theorem 25 If Γ is normal up to level i, then CnULL(i)(Γ) ⊆ CnALm

⊏(Γ).

Proof. Suppose the antecedent holds and A ∈ CnULL(i)(Γ), whence Γ ∪ Ω¬

(i) ⊢LLL+

A. By the compactness of LLL+, there are B1, . . . , Bn ∈ Γ and there is a finite∆ ⊂ Ω(i) such that B1, . . . , Bn ∪ ∆¬ ⊢LLL+ A. By the Deduction Theorem,B1, . . . , Bn ⊢LLL+ A ∨Dab(∆).

26

Let p be an ALm

⊏ -proof from Γ, obtained by (i) introducing all the premises Bi

(i ≤ n) and (ii) deriving A on the condition ∆ from these premises, by the rule RC.Let l be the line on which A is derived.

Assume that l is marked. It follows that there is a ϕ ∈ Φ⊏

s (Γ) such that ϕ∩∆ 6= ∅,whence ϕ ∩ Ω(i) 6= ∅. By Lemma 18, Γ is not normal up to level i — a contradiction.By the same reasoning, it follows that in every extension of p, line l remains unmarked.Hence A is finally derived in p, whence A ∈ CnALm

⊏(Γ).

B Criteria for Equivalence

For the proof of Theorem 21, we will rely on two facts and a lemma about minimalchoice sets. The first fact was proven in [17] (Lemma 3.2.4), the second is an immediateconsequence of Theorem 12 and Definition 5.

Fact 11 If every ϕ ∈ Φ(Γ) is finite, then Φ(Γ) is finite.

Fact 12 MALm⊏

(Γ) = MSALmc

(Γ) =⋂

i∈IMSALm

(i)(Γ) =

⋂i∈I

MALm⊏(i)

(Γ).

Lemma 19 If Σ is a finite set of sets, then Σ has no infinite minimal choice sets.

Proof. Let Σ = Θi | i ≤ n and let ϕ be an infinite choice set of Σ. For every i ≤ n,let Ai be an arbitrary element of ϕ ∩ Θi, and let ϕ′ = A1, . . . , An. Note that sinceϕ′ is finite, ϕ′ ⊂ ϕ. Since ϕ′ is a choice set of Σ, ϕ is not a minimal choice set of Σ.

Proof of Theorem 21. Let Γ ⊆ W. Ad 1. (⇐) Immediate in view of Lemma 19. (⇒)Suppose that every ϕ ∈ Φ(Γ) is finite. By Fact 11, Φ(Γ) is finite, whence also

⋃Φ(Γ)

and ℘(⋃

Φ(Γ)) are finite. As stated in [3, Theorem 11.5],⋃

Σ(Γ) =⋃

Φ(Γ).23 Itfollows that Σ(Γ) ⊆ ℘(

⋃Φ(Γ)), whence Σ(Γ) is finite.

Ad 2. This is Fact 11.Ad 3. See Lemma 7 from [20].Ad 4 and 7. Immediate in view of Lemma 19.Ad 5. Immediate in view of Theorem 7.Ad 8. (⇒) This is Lemma 9. (⇐) Suppose that for no i ∈ I, cΦ⊏(i)(Γ) has infinite

minimal choice sets. Hence (†) for every i ∈ I, MLLL+(CnALm⊏(i)

(Γ)) = MALm⊏(i)

(Γ).

By (†), Fact 5.4 and the monotonicity of LLL+, for every i ∈ I, MLLL+(CnALm⊏

(Γ)) ⊆MALm

⊏(i)(Γ). By Fact 12, MLLL+(CnALm

⊏(Γ)) ⊆ MALm

⊏(Γ). Also, by the sound-

ness of ALm⊏ , MLLL+(CnALm

⊏(Γ)) ⊇ MALm

⊏(Γ). Hence MLLL+(CnALm

⊏(Γ)) =

MALm⊏

(Γ). By Lemma 5, cΦ⊏(Γ) has no infinite minimal choice sets.

C Some Negative Results

In this section, we define the prioritized adaptive logics K2m⊏ , SK2m

c and HK2mc , and

use them to prove some negative results about the respective formats ALm

⊏ , SALm

c

and HALm. The concrete systems are merely presened for argumentative purposes –we refer to [24, Section 3] for more details and examples.

23This is a well-established fact in the study of flat ALs. However, it is usually spelled outin terms of the set of unreliable formulas U(Γ), which is identical to what we call

⋃Σ(Γ).

27

C.1 Some Particular Prioritized Adaptive Logics

Several ALs have been developed to explicate reasoning with prioritized belief bases– see [8], [27] and [26], and [23, Section 6]. The ALs that deal with such belief basestypically use a certain logical operator or a sequence of such operators to express thata belief has a certain degree of plausibility. We will define three such systems, one ineach of the three generic formats, in order to prove the negative results we promisedin Section 1.

We restrict the logic to the propositional level. We use the standard modallanguage LM of Kripke’s minimal normal modal logic K . As usually, we define♦A = ¬¬A. Let WM denote the set of modal wffs, and W l the set of literals (sen-tential letters and their negations). To express the plausibility degree of a piece ofinformation, sequences of diamonds are used: ♦♦ . . .♦A. The longer the sequence, theless plausible the information. A sequence of i diamonds will be abbreviated by ♦

i –♦0 denotes the empty string.

We will define prioritized logics that allow for the defeasible inference from ♦iA

(where i ∈ N) to A. This is done by defining “A is plausible (to degree i), but false”as an abnormality (of rank i). Where A ∈ W l, let !iA abbreviate ♦

iA ∧ ¬A. Everyset of abnormalities ΩK

i (i ∈ N) is defined as follows:

ΩK

i = !iA | A ∈ W l

Let in the remainder ΩK

(i) = ΩK1 ∪ . . . ∪ ΩK

i for all i ∈ N. Every flat adaptive logicKm

(i) is defined by the triple 〈K,Ω(i),m〉. We will often restrict ourselves to systemsthat only consider two degrees of plausibility, whence the number 2 in the name ofthe logics. By the triple characterization, we can readily define the following threesystems:

1. the ALm⊏ -logic K2m

⊏ defined by the triple 〈K, 〈ΩK1 ,Ω

K2 〉,m〉

2. the superposition-logic SK2mc defined by the sequence of logics 〈Km

(1),Km

(2)〉3. the hierarchic logic HK2m

c defined by the sequence of logics 〈Km

(1),Km

(2)〉

Before we prove the negative results for HK2mc and SK2m

c , let us introduce someconventions that will lighten notation in the remainder. Where i ∈ N, we say thatDab(∆) is a Dabi-consequence of Γ iff ∆ ⊆ Ω(i). Where Dab(∆1), Dab(∆2), . . . are the

minimal Dabi-consequences of Γ, we use Φ(i)(Γ) to refer to the set of ⊂-minimal choicesets of Σ(i)(Γ) = ∆1,∆2, . . .. We use Φ⊏i(Γ) to refer to the ⊏

K-minimal choice setsof Σ(i)(Γ), where the order ⊏

K is spelled out as in Definition 1, but replacing eachΩi by ΩK

i . Finally, where M ∈ MK+ , let Ab(i)(M) = A ∈ Ω(i) | M A. Slightlyabusing notation, we write M ∆ to denote that M A for every A ∈ ∆. Where∆ ⊆ W, ∆¬ = ¬A | A ∈ ∆.

C.2 The Incompleteness Result for HK2m

c

As a result of Corollary 12, whenever Γ satisfies ⋆HALm , then it holds that A ∈CnHK2m

c(Γ) iff Γ |=HK2m

cA. We will now give an example that shows that HALm is

not in general complete with respect to its semantics.Let Γ1 = Γ1

1 ∪ Γ21 ∪ Γ3

1 ∪ Γ41, where

Γ11 = !1pi∨!2qj | i, j ∈ N, i ≥ j

Γ21 = !2qi∨!2qj | i, j ∈ N, i 6= j

Γ31 = !2qi∨!2r | i ∈ N

Γ41 = s∨!2r

Lemma 20 Φ⊏2(Γ1) = !2qi | i ∈ N.

Proof. First of all, note that Φ(2)(Γ1) = Υ1 ∪ Υ2, where

Υ1 = !2qi | i ∈ N

28

Υ2 = ϕk | k ∈ N = !2qi, !1pj , !

2r | i ∈ N− k, j ≥ k | k ∈ N

By Theorem 7, Φ⊏2(Γ1) ⊆ Φ(2)(Γ1). Note that for every ϕ ∈ Υ2, !2qi | i ∈ N ⊏K ϕ.

Hence Φ⊏2(Γ1) = !2qi | i ∈ N.

By Theorem 21.7, we can derive:

Corollary 14 cΦ⊏(2)(Γ1) has no infinite minimal choice sets.

Lemma 21 Γ1 |=HK2mcs.

Proof. We prove that Γ1 |=K2m⊏s – the rest is immediate in view of Corollary 1.

By Lemma 20 and Theorem 6, for every M ∈ MK2m⊏

(Γ), Ab(M) = !2qi | i ∈ N.

But then for every such M , M 6 !2r, whence in view of Γ41, M s.

To show that s is not in the HK2mc -consequence set of Γ1, we will need a slightly

longer proof. Note that there is no Θ ⊂ ΩK

(1) such that Γ1 ⊢K+ Dab(Θ). Hence Γ1 is

normal with respect to ΩK

(1). By Theorem 11, we have:

Lemma 22 CnKm(1)

(Γ1) = CnK+(Γ1 ∪ ¬A | A ∈ ΩK

(1)).

Lemma 23 s 6∈ CnHK2mc

(Γ1).

Proof. Suppose s ∈ CnHK2mc

(Γ1). By Definition 8, CnKm(1)

(Γ1) ∪ CnKm(2)

(Γ1) ⊢K+

s. By Lemma 22, CnK+(Γ1 ∪ ΩK

(1)¬) ∪ CnKm

(2)(Γ1) ⊢K+ s. Since K+ is mono-

tonic, transitive and reflexive, we can derive that Γ1 ∪ ΩK

(1)¬ ∪ CnKm

(2)(Γ1) ⊢K+ s.

Since Km

(2) is reflexive, ΩK

(1)¬ ∪ CnKm

(2)(Γ1) ⊢K+ s. By the compactness of K+,

Θ¬ ∪ CnKm(2)

(Γ1) ⊢K+ s for a finite Θ ⊂ ΩK

(1). But then, by the deduction theo-

rem, CnKm(2)

(Γ1) ⊢K+ s ∨Dab(Θ). By Theorem 3, Γ1 ⊢Km(2)

s ∨Dab(Θ).

Since Θ is finite, there is a k ∈ N such that, for every l ≥ k: !1pl 6∈ Θ. LetM ∈ MK+(Γ1) be such that each of the following holds:24

(C1) Ab(2)(M) = ϕk

(C2) M 6 s

By Theorem 6 and Lemma 20, M ∈ MKm(2)

(Γ1). By (C1), M 6 Dab(Θ), whence

by (C2), also M 6 s ∨Dab(Θ). By the soundness of Km

(2), Γ1 6⊢Km(2)

s ∨Dab(Θ) — a

contradiction.

By Corollary 14, Lemma 21 and Lemma 23, we immediately have:

Theorem 26 There are Γ, A for which cΦ⊏(2)(Γ) has no infinite minimal choice setsand Γ |=HK2m

cA, but A 6∈ CnHK2m

c(Γ).

Likewise, by Lemma 21, Corollary 1 and Theorem 26, it follows that:

Theorem 27 There are Γ, A such that A ∈ CnK2m⊏

(Γ), whereas A 6∈ CnHK2mc

(Γ).

24See Lemma 20 for the definition of ϕk.

29

C.3 Incomparability of HK2m

cand SK2m

c

Lemma 24 s ∈ CnSK2mc

(Γ1).

Proof. From Lemma 21 and Corollary 1, we can infer that Γ1 |=SK2mcs. By Corollaries

11.1 and 14, it follows that s ∈ CnSK2mc

(Γ1).

By Lemma 23 and Lemma 24, we obtain the following:

Theorem 28 There are Γ, A such that A ∈ CnSK2mc

(Γ), whereas A 6∈ CnHK2mc

(Γ).

We will now prove that also the converse holds. Let Γ2 = Γ12 ∪ Γ2

2 ∪ Γ32 ∪ Γ4

2 ∪ Γ52,

where

Γ12 = !1pi ∨ !1pj | i, j ∈ N, i 6= j

Γ22 = !1pi ∨ !2tj | i, j ∈ N, i ≤ j

Γ32 = !1pi ∨ !2s | i ∈ N

Γ42 = r ∨ !1pi ∨ !2qi | i ∈ N

Γ52 = r ∨ !2s

Lemma 25 r ∈ CnHK2mc

(Γ2).

Proof. Note that Φ(2)(Γ2) = Ψ1 ∪ Ψ2, where

Ψ1 = ϕ0 = !1pi | i ∈ NΨ2 = ϕj | j ∈ N = !1pi | i ∈ N− j ∪ !2tk | k ≥ j ∪ !2s | j ∈ N

In view of Γ42, for every ϕj ∈ Ψ2, there is a Θj = !1pj , !

2qj, such that Γ2 ⊢K+

r ∨Dab(Θj) and Θj ∩ ϕj = ∅. Also, in view of Γ52, Γ2 ⊢K+ r ∨Dab(!2s), and

!2s ∩ ϕ0 = ∅. By Theorem 8, r ∈ CnKm(2)

(Γ2). Hence by Definition 8 and the

reflexivity of LLL+, r ∈ CnHK2mc

(Γ2).

By Theorem 16 and Corollary 1, we immediately obtain the following:

Lemma 26 Γ2 |=SK2mcr.

We will now prove that r is not a member of the SK2mc -consequence set of Γ1.

The proof relies on the following lemma:

Lemma 27 There is no Θ ⊂ ΩK

(2)−(!1pi | i ∈ N∪!2s), such that CnKm(1)

(Γ2) ⊢K+

r ∨Dab(Θ).

Proof. Suppose that there is a Θ ⊂ ΩK

(2) − (!1pi | i ∈ N ∪ !2s), such thatCnKm

(1)(Γ2) ⊢K+ r ∨Dab(Θ). By Theorem 3, (†) r ∨Dab(Θ) ∈ CnKm

(1)(Γ2).

Since Θ is finite, there is a k ∈ N such that each of the following holds:

(i) !2tl 6∈ Θ for every l ≥ k

(ii) !2ql 6∈ Θ for every l ≥ k

From this and the supposition, we can derive:

Θ ⊆ ΩK

(2) − (!1pi | i ∈ N ∪ !2s ∪ !2tl, !2ql | l ≥ k) (8)

Let M ∈ MK+(Γ2) be such that each of the following holds:

(C1) M !1pi for every i 6= k

(C2) M !2tl for every l ≥ k

(C3) M !2ql for every l ≥ k

(C4) M !2s(C5) M 6 r

30

(C6) M 6 A for every A ∈ ΩK

1 − !1pi | i ∈ N− k(C7) M 6 A for every A ∈ ΩK

2 − !2tl, !2ql, !

2s | l ≥ k

Note that by (C1), M Γ12; by (C1) and (C2), M Γ2

2; by (C1) and (C3), M Γ42;

finally, by (C4), M Γ32 ∪ Γ5

2. Suppose there is an M ′ ∈ MK+(Γ2) such that

Ab(1)(M ′) ⊂ Ab(1)(M). In that case, M ′ 6 !1pi for an i 6= k. But then, in viewof Γ1

1, M ′ !1pk, whence !1pk ∈ Ab(1)(M ′) − Ab(1)(M) — a contradiction. It follows

that M ∈ MKm(1)

(Γ2).

Note that by (C6) and (C7), M 6 Dab(Λ) for every Λ ⊆ ΩK

(2) − (!1pi | i ∈

N ∪ !2tl, !2ql, !

2s | l ≥ k), whence by (8), M 6 Dab(Θ). Together with (C5), thisimplies that M 6 r ∨Dab(Θ). By the completeness of Km

(1), Γ2 6⊢Km(1)

r ∨Dab(Θ) —

a contradiction.

Lemma 28 r 6∈ CnSK2mc

(Γ2).

Proof. First of all, note that Φ(1)(Γ2) = !1pi | i ∈ N−j | j ∈ N. In view of Γ42, for

every ϕ ∈ Φ(1)(Γ2), Γ2 ⊢K+ !2s ∨Dab(Θ) for a Θ ⊂ ΩK

1 such that ϕ ∩ Θ = ∅. Hence

by Theorem 8, !2s ∈ CnKm(1)

(Γ2). This implies that Φ(2)(CnK1m(Γ2)) = Ξ1 ∪ Ξ2,

where

Ξ1 = ϕ⋆ = !1pi | i ∈ N ∪ !2sΞ2 = !1pi | i ∈ N− j ∪ !2tk | k ≥ j ∪ !2s | j ∈ N

By Lemma 27, there is no Θ ⊆ ΩK

(2) − ϕ⋆, such that CnKm(1)

(Γ2) ⊢K+ r ∨Dab(Θ). By

Theorem 8, r 6∈ CnKm(2)

(CnKm(1)

(Γ2)) = CnSK2mc

(Γ2).

By Lemma 25 and Lemma 28, we immediately have:

Theorem 29 There are Γ, A such that A ∈ CnHK2mc

(Γ), whereas A 6∈ CnSK2mc

(Γ).

Since the SK2mc -semantics and the HK2m

c -semantics are equivalent, and in viewof Theorem 16, it follows immediately that SK2m

c is not complete with respect to itssemantics:

Theorem 30 There are Γ, A such that Γ |=SK2mcA, whereas A 6∈ CnSK2m

c(Γ).

Finally, by Corollary 1, Theorem 30 and the soundness and completeness of K2m⊏ :

Theorem 31 There are Γ, A such that A ∈ CnK2m⊏

(Γ), whereas A 6∈ CnSK2mc

(Γ).

C.4 The Conditions for Equivalence

In this section, we briefly show that the converses of items 2-7 of Theorem 21 fail, andthat the right-left direction of Lemma 13 fails. We will not give full proofs for theseclaims, but simply list the counterexamples and some of their most salient properties.

Let us start with Theorem 21:Ad 2. Let Θ1 = !1p∨!1qi | i ∈ N. Note that there is an infinite minimal choice

sets of Σ(Θ1), i.e. the set ϕ = !1qi | i ∈ N. Still, Φ(Θ1) = !1p, ϕ is finite.Ad 3 and 6. Let Θ2 = !ipi1∨!ipi2 | i ∈ N. Let Φ(Θ2) be the set of minimal

choice sets with respect to the flat adaptive logic Km∪ = 〈K,

⋃i∈N

ΩK

(i),m〉 and letΦ⊏(Θ2) be the set of minimal choice sets with respect to the prioritized adaptive logicKm

⊏ = 〈K, 〈ΩK

(i)〉i∈N,m〉. Note that for every i ∈ N, Σ(i)(Θ2) is finite, whence Φ(i)(Θ2)and Φ⊏(i)(Θ2) have only finitely many minimal choice sets. However, Φ(Θ2) = Φ⊏(Θ2)is infinite.

Ad 4 and 7. Let Θ3 = !1p2n∨!1p2n+1 | n ∈ N. Note that Φ(1)(Θ3) = Φ⊏(1)(Θ3) isinfinite. Nevertheless, every minimal choice set of cΦ(1)(Θ3) = cΦ⊏(1)(Θ3) is a couple:cΦ(1)(Θ3) = !1p2, !

1p3, !1p4, !1p5, . . ..

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Ad 5. Let Θ4 = !1pi∨!1pj∨!2q | i, j ∈ N, i 6= j. Let Ψ = !1pi | i ∈ N − k |k ∈ N. Note that Φ(2)(Θ4) = !2q ∪ Ψ, whereas Φ⊏(2)(Θ4) = !2q.

The last example is also a counterexample for the right-left direction of Lemma13. That is, although cΦ(2)(Θ4) has an infinite minimal choice set (i.e. the set !1pi |i ∈ N), it can be shown that MK+(CnHK2m(Θ4)) = MHK2m(Θ4).

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