Hierarchic Adaptive Logics - Semantic Scholar

Hierarchic Adaptive Logics∗ Frederik Van De Putte Centre for Logic and Philosophy of Science Ghent University Blandijnberg 2, 9000 Gent, Belgium [email protected] Phone number:

0492 87 54 62

March 21, 2011

Abstract This paper discusses the proof theory, semantics and meta-theory of a class of adaptive logics, called hierarchic adaptive logics. Their specific characteristics are illustrated throughout the paper with the use of one exemplary logic HKx , an explicans for reasoning with prioritized belief bases. A generic proof theory for these systems is defined, together with a less complex proof theory for a subclass of them. Soundness and a restricted form of completeness are established with respect to a nonredundant semantics. It is shown that all hierarchic adaptive logics are reflexive, have the strong reassurance property and that a subclass of them is a fixed point for a broad class of premise sets. Finally, they are compared to a different yet related class of adaptive logics.

Keywords: combined adaptive logics; prioritized belief bases

1

Introduction

Flat adaptive logics are a well-studied class of logics, developed to explain various non-monotonic reasoning methods.1 I will give a brief characterization of these logics and provide an example of them in Section 2. A general format for these logics has been put forward, called the “Standard Format of Adaptive Logics”[3, 5]. If we define a logic in this format, the proof theory and semantics for this system, together with many important meta-theoretic properties are immediately available. ∗ I am greatly indebted to Christian Straßer and Diderik Batens for their comments to previous drafts, and for the fruitful discussions that helped to settle some central issues in this paper. 1 I refer to [10, p.2] for a recent overview of the many different adaptive logics developed so far, and their applications to problems in philosophy and artificial intelligence. Unpublished papers in the reference section (and many others) are available from the internet address http://logica.UGent.be/centrum/writings/.

1

This paper focuses on a certain combination of flat adaptive logics, and the properties of the resulting combined logics. A combined adaptive logic is especially well-suited to explain reasoning with preferential or prioritized premises.2 However, a combined approach also turned out to be useful to strengthen the inductive consequences from a set of data, as was argued in [4]. Finally, to capture the many-sidedness of actual human reasoning, a combination of logics often seems necessary – e.g. the combination of a logic that deals with (possibly inconsistent) background knowledge, and of a different logic that allows one to derive explanations for a given fact. The range of logics that may be combined into one sensible system, and the ways to combine them, seem nearly infinite. This holds for so-called Tarski-logics3 , but it holds equally well for non-monotonic logics such as adaptive logics.4 Hence the need for a holistic approach to the combinatorics themselves. The most common structure of combined adaptive logics is that of a superposition of logics. Where AL1 , AL2 , . . . are the flat logics and Γ is a premise set, the consequence set for the combined logic CAL is approximately defined as follows: CnCAL (Γ) = . . . CnAL3 (CnAL2 (CnAL1 (Γ)) . . . where the right “. . .” denotes a (possibly infinite) sequence of right brackets. This kind of combination has many meta-theoretic advantages – e.g. reassurance is easily provable. A universal proof theory for a subclass of these systems was presented in [1]. I will concentrate on a different way in which adaptive logics can be combined, i.e. one where several adaptive logics are applied to the same premise set Γ. This combination looks as follows: CnCAL0 (Γ) = CnAL1 (Γ) ∪ CnAL2 (Γ) ∪ CnAL3 (Γ) ∪ . . . Although applications of this sort of combination were already mentioned in previous work [4], its meta-theoretic properties were not yet investigated in detail. In this paper, I will present formal restrictions that turn the basic idea into a non-trivial and closed consequence relation and illustrate the resulting format by one such logic (Section 3). In the spirit of Universal Logic, I will provide a proof theory, semantics and further meta-theoretic results for all logics that are combined in this manner (sections 5, 6 and 7). In Section 8, I will compare these logics to a related class of flat adaptive logics. The final section summarizes my results and topics for further research.

2

Flat Adaptive Logics

2.1

The Standard Format

In this section, the building blocks for the rest of the paper are presented: the flat adaptive logics defined in standard format. I will only explain the general 2 See

Section 3 where this is explained and illustrated with an example. [12] for a thorough up-to-date study of the current research on combining and splitting monotonic logics. 4 See [5, Chapter 6] for an overview of combinations of adaptive logics. This is an unpublished book about Adaptive Logics by Diderik Batens, of which seven chapters are available on the internet at the moment I am writing this, including Chapter 6 where combined adaptive logics are discussed. 3 See

2

characteristics, and refer to [3] for details and meta-theoretic proofs. A flat adaptive logic ALx is defined by a triple: 1. 2.

3.

A lower limit logic LLL: a reflexive, transitive, monotonic and compact logic that has a characteristic semantics and contains CL (Classical Logic) A set of abnormalities Ω: a set of formulas , characterized by a (possibly restricted) logical forms F which contains at least one logical symbol; or a union of such sets An adaptive strategy: Reliability or Minimal Abnormality

As is clear from 1., all viable candidates for an LLL are Tarski-logics and are assumed to be sound and complete with respect to their semantics. Anything that is LLL-derivable from Γ, is ALx -derivable from Γ. Where ALx captures a reasoning method for a certain context, LLL expresses the logical certainties within this context. The logic ALx strengthens its LLL by adding a (defeasible) assumption, i.e. that formulas that have the logical form of an abnormality are false. ˇ ∀. ˇ The language ˇ, ∧ ˇ , ⊃, ˇ ∃, The classical connectives of LLL are noted as ¬ ˇ, ∨ of LLL without the classical connectives is denoted by L; for most practical applications the premise set of the adaptive logic is stated in this language. The language of LLL with the classical connectives is denoted by L+ and the set of closed formulas of L+ by W+ . The adaptive logic is a function: ℘(W+ ) → ℘(W+ ).5 The set of abnormalities Ω represents those formulas that the adaptive logic assumes to be false unless the premises run counter to this assumption. This vague phrase is turned into a precise criterion by the adaptive strategy (x), that can be either Reliability (r) or Minimal abnormality (m). The categorization of an adaptive logic as “flat” refers to the fact that in a certain sense, all abnormalities in Ω are treated equally by this logic. This will be specified during the course of this paper, when the distinction is made between flat and hierarchic adaptive logics. In the literature, flat adaptive logics are often referred to as simple adaptive logics, or as adaptive logics in standard format. Before we come to the ALx -semantics, we first need a few extra definitions. Where L is a logic with a characteristic semantics, we say that M is a L-model of Γ iff M is a L-model and it validates every member of Γ. ML Γ denotes the set of L-models of Γ. As defined below, the set of adaptive models of Γ is always a subset of the set of LLL-models of Γ. Throughout this paper, all formulas in W+ are assumed to be finite strings. A Dab-formula Dab(∆) is the classical disjunction of the members of ∆ ⊂ Ω, whence ∆ is a finite set. Dab(∆) is a minimal Dab-consequence of Γ iff Γ `LLL Dab(∆) and there is no ∆0 ⊂ ∆ for which Γ `LLL Dab(∆0 ). Where Dab(∆1 ), Dab(∆2 ), . . . are the S minimal Dab-consequences of Γ, let Σ(Γ) = {∆1 , ∆2 , . . .}. Let U (Γ) = Σ(Γ). We say that U (Γ) is the set of unreliable formulas with respect to Γ. Finally, where M is a LLL-model, Ab(M ) is the set {B | B ∈ Ω, M |= B}. Definition 1 A LLL-model M of Γ is reliable iff Ab(M ) ⊆ U (Γ). 5 See [5, Chapter 4] for a more lengthy discussion on the role of the classical connectives in adaptive logics.

3

Definition 2 Γ |=ALr A iff A is verified by all reliable models of Γ. Definition 3 A LLL-model M of Γ is minimally abnormal iff there is no LLLmodel M 0 of Γ such that Ab(M 0 ) ⊂ Ab(M ). Definition 4 Γ |=ALm A iff A is verified by all minimally abnormal models of Γ. Although the above definition of Γ |=ALm A is more direct, we can also define the semantics of Minimal Abnormality in terms of the minimal Dabconsequences of Γ. This requires some notational preparation. Where Σ is a set of sets, we say that ϕ is a choice set of Σ iff for every ∆ ∈ Σ, ϕ ∩ ∆ 6= ∅. ϕ is a minimal choice set of Σ iff there is no choice set ψ of Σ such that ψ ⊂ ϕ. The following is proven in [5, Chapter 5]: Lemma 1 If Σ is a set of finite sets, then Σ has minimal choice sets. Φ(Γ) is the set of minimal choice sets of Σ(Γ). Note that since all the members of Σ(Γ) are finite, Φ(Γ) 6= ∅ for every Γ ⊆ W by Lemma 1. It is proven in [3] that M is a minimally abnormal model iff Ab(M ) ∈ Φ(Γ) (see Theorem 8 below). The proof theory of adaptive logics reflects the dynamic character of nonmonotonic reasoning methods. Every ALx -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 a finite condition ∆ ⊆ Ω. Where Γ is the set of premises, the inference rules are given by PREM

If A ∈ Γ:

RU

If A1 , . . . , An `LLL B:

RC

.. . A

ˇ Dab(Θ) If A1 , . . . , An `LLL B ∨

.. . ∅

A1 .. .

∆1 .. .

An B

∆n ∆1 ∪ . . . ∪ ∆n

A1 .. .

∆1 .. .

An B

∆n ∆1 ∪ . . . ∪ ∆n ∪ Θ

A stage of a proof can be seen as a sequence of lines, obtained by the application of the above rules. A proof is simply a chain of stages. Every proof starts off with stage 1. Adding a line to a proof by applying one of the rules of inference brings the proof to its next stage, which is the sequence of all lines written so far. In view of the inference rules, the condition and the justification of any line l are necessarily finite. Dab(∆) is a Dab-formula at stage s of a proof, iff it is the second element of a line of the proof with an empty condition. Dab(∆) is a minimal Dab-formula at stage s iff there is no other Dab-formula Dab(∆0 ) at stage s for which ∆0 ⊂ ∆. Where Dab(∆1 ), Dab(∆2 ), . . . are the minimal S Dab-formulas at stage s of a proof, let Σs (Γ) = ∆1 ∪ ∆2 ∪ . . .. Us (Γ) = Σs (Γ) and Φs (Γ) is the set of minimal choice sets of Σs (Γ). 4

Definition 5 ALr -Marking: a line l is marked at stage s iff, where ∆ is its condition, ∆ ∩ Us (Γ) 6= ∅. Definition 6 ALm -Marking: a line l with formula A is marked at stage s iff, where its condition is ∆: (i) there is no ϕ ∈ Φs (Γ) such that ϕ ∩ ∆ = ∅, or (ii) for a ϕ ∈ Φs (Γ), there is no line on which A is derived on a condition Θ for which Θ ∩ ϕ = ∅. If a line that has as its second element A is marked at stage s, this indicates that according to our best insights at this stage, A cannot be considered derivable. If the line is unmarked at stage s, we say that A is derivable at stage s of the proof. As a line may be marked at stage s, unmarked at a later stage s0 and marked again at a still later stage, we also define a stable notion of derivability. Where p is a proof, an extension of this proof is every longer proof p0 that contains the lines occurring in p in the same order. Hence putting lines in front of a proof, inserting them somewhere in the proof, or simply adding them at the end of the proof may all result in an extension of the proof. Of course, it is required that the justification of every line only refers to preceding lines in the proof.6 Definition 7 A is finally derived from Γ on line l of a stage s iff (i) A is the second element of line l, (ii) line l is unmarked at stage s, and (iii) every extension of the stage in which line l is marked may be further extended in such a way that line l is unmarked again. Definition 8 Γ `ALx A iff A is finally derived on a line of an ALx -proof from Γ. Note that in order to be finally derivable, A must be derived on a line l, where l ∈ N. This means that for every formula that is finally derivable from Γ, it can be finally derived in a finite proof from Γ. The following theorems are important for the rest of this paper and were proven in [3] – I give the original lemmata, theorems and corollaries between square brackets: Theorem 1 Γ `ALx A iff Γ |=ALx A. (Soundness and Completeness) [Corr. 2, Th. 9] Theorem 2 Γ ⊂ CnALx (Γ). (Reflexivity) [Th. 11.2] x

x

Theorem 3 If M ∈ MLLL − MAL , then there is an M ∈ MAL such that Γ Γ Γ 0 Ab(M ) ⊂ Ab(M ). (Strong Reassurance) [Th. 4, Th. 5] Theorem 4 CnALx (CnALx (Γ)) = CnALx (Γ). (Fixed Point) [Th. 11.6, Th. 11.7] Theorem 5 CnALr (Γ) ⊆ CnALm (Γ). [Th. 11.1] ˇ Dab(∆) and ∆∩U (Γ) = ∅. Theorem 6 Γ `ALr A iff for a ∆ ⊂ Ω: Γ `LLL A ∨ [Th. 6] Theorem 7 Γ `ALm A iff for every ϕ ∈ Φ(Γ), there is a ∆ ⊂ Ω such that ˇ Dab(∆). [Th. 8] ϕ ∩ ∆ = ∅ and Γ `LLL A ∨ Theorem 8 M is a ALm -model of Γ iff (M is a LLL-model of Γ and Ab(M ) ∈ Φ(Γ)). [Lemma 4] 6 For

more details on the notion of a dynamic proof, see [5, Chapter 5].

5

An Example: Kx1

2.2

Let me clarify the proof theory of adaptive logics by a simple example. The lower limit logic I will use is Kripke’s minimal modal logic K on the basis of the standard modal language LM , axiomatized by the propositional fragment of CL together with the following axioms: K (A ⊃ B) ⊃ (A ⊃ B) RN if ` A then ` A As usually, define ♦A = ¬¬A. Let W M denote the set of closed formulas in LM , and W a the set of atoms (sentential letters and their negations). K-models are defined as pointed-Kripke frames with the standard valuations. The adaptive logic Kr1 is defined by the following triple: 1. 2. 3.

LLL = K Ω = {♦A ∧ ¬A | A ∈ W a } the Reliability Strategy

Recall that in Section 2, Dab-formulas where defined as classical disjunctions of the members of Ω. Hence a Dab-formula can only be written down with the ˇ . This has both meta-theoretic and philosophical aid of the checked symbol ∨ reasons – see [5, Chapter 4] for more details. However, for reasons of space, and since the disjunction of K behaves classically, I now introduce the convention that in all subsequent examples, Dab-formulas are simply disjunctions of abnormalities. Kr1 is intended to explicate reasoning from plausible knowledge. It is but a variation on an existing approach to the handling of belief revision, within the adaptive logics programme.7 That a formula A is plausible, is expressed by ♦A. We thus translate the original set of plausible statements Γ into Γ♦ = {♦A | A ∈ Γ}. We may also reason from certainties together with plausible knowledge, such that a premise set contains formulas of the form ♦A and A, where A is a non-modal formula. The adaptive logic enables one to derive A from ♦A, in case that ♦A ∧ ¬A does not occur in a minimal Dab-consequence of Γ. This makes sense in view of the fact that our plausible knowledge can be contradicted by other plausible knowledge, or by certainties. Consider the following premise set: Γ = {♦p, ♦q, ♦r, ¬p ∨ ¬q}. We start a Kr1 -proof from Γ by writing down the premises: 1 2 3 4

♦p ♦q ♦r ¬p ∨ ¬q

∅ ∅ ∅ ∅

PREM PREM PREM PREM

Note that the fourth element is ∅, indicating that premises are derived on the empty condition. We may now derive p from line 1, using RU and RC: 4 5 6

p ∨ ¬p p ∨ (♦p ∧ ¬p) p 7A

RU 1,4; RU 5; RC

∅ ∅ {♦p ∧ ¬p} X7

similar logic based on Feys’ modal logic T is presented in [9].

6

For the time being, ignore the X7 at line 6. At stage 6 of the proof, there are no unreliable formulas: U6 (Γ) = ∅, and p is derived on an unmarked line. However, we may immediately add the following line: 7

(♦p ∧ ¬p) ∨ (♦q ∧ ¬q)

1,2,4; RU ∅

This means that U7 (Γ) = {♦p ∧ ¬p, ♦q ∧ ¬q}. As a consequence line 6 is marked at stage 7 of the proof, which is indicated by the X7 . Moreover, since Γ 0K ♦q ∧ ¬q, the line will be marked in every extension of the proof. On the other hand we can apply RC to derive r on an unmarked line: 8 r ∨ ¬r 9 r ∨ (♦r ∧ ¬r) 10 r

RU 3,8; RU 9; RC

∅ ∅ {♦r ∧ ¬r}

Line 10 is unmarked in every extension of the proof, since the only minimal Dab-consequence from Γ is the formula on line 7, hence no new Dab-formula can render ♦r ∧ ¬r unreliable. The difference between Reliability and Minimal Abnormality can also be clarified by the above example. Minimal Abnormality is a little stronger: Γ `Km 1 p ∨ q, while Γ 0Kr1 p ∨ q. The K-models of Γ that verify ♦p ∧ ¬p and ♦q ∧ ¬q are not minimally abnormal, since there are models that verify only one of both abnormalities. This implies that there must be a Km 1 -proof in which p ∨ q is finally derived. Since the choice of the strategy only affects the marking definition in the proof theory, we may simply continue the preceding proof to achieve this goal: 11 12 13 14

p∨q q ∨ (♦q ∧ ¬q) q p∨q

6; RU 2; RU 12; RC 13; RU

{♦p ∧ ¬p} ∅ {♦q ∧ ¬q)}X14 {♦q ∧ ¬q)}

Note that throughout the stages 11-14, Φs (Γ) remains the same, that is, Φs (Γ) = Φ(Γ) = {{♦p ∧ ¬p}, {♦q ∧ ¬q}}. Let us call the choice sets ϕ1 and ϕ2 respectively. From stage 11 to stage 13, line 11 is marked. That is, as long as p ∨ q is not derived on a line with condition ∆ such that ∆ ∩ ϕ1 = ∅, the marking definition stipulates that line 11 is marked. However, at stage 14, line 11 is unmarked, because at that stage of the proof we know that p ∨ q is true both when ♦p ∧ ¬p is false, and when ♦q ∧ ¬q is false. Line 13 will however be marked in every extension of the proof, hence q is treated in exactly the same way as p.

3 3.1

Hierarchic Adaptive Logics A Precise Definition

I will now consider combinations of two or more flat adaptive logics, denoted by AL1 , AL2 , . . .. Unless stated differently, the results in this paper hold for combinations of both finitely and infinitely many flat adaptive logics. Henceforth, 7

I use the index set I ⊆ N to range over the set of flat adaptive logics that are used in the combined adaptive logic. I will narrow down the focus to combinations of logics that have the same lower limit logic LLL and the same strategy. Although the union of the consequence sets of adaptive logics with a different lower limit logic or strategy may have interesting applications, this case is much harder to deal with from a universal, meta-theoretic point of view. In the remainder, Ω1 , Ω2 , . . . refer to the respective sets of abnormalities of the flat logics ALx1 , ALx2 , . . .. Let us return to the basic idea from the introduction: the combination of flat adaptive logics through a union of their respective consequence sets. A first problem this combination faces is that the consequence set is not always closed under LLL. Let me briefly explain this. For a logic L to be closed under another logic L0 , the following must hold: (†) CnL0 (CnL (Γ)) = CnL (Γ) As explained in Section 2.1, LLL delivers the unconditional rules for every logic ALxi . This means that these rules are also unconditionally valid in CAL. It would therefore be strange that the application of these rules to CnCAL (Γ) would result in a yet different consequence set.8 There is also a more technical reason why CAL should be LLL-closed. Note that if (†) does not hold for LLL, then the CAL-models of Γ (if any) are not LLL-models of Γ (on the assumption that every CAL-model of Γ validates every member of CnCAL (Γ)). Hence if we want to characterize the combined adaptive logic by a semantics that is somehow similar in spirit to the AL-semantics, it has to be closed under LLL. However, as defined in Section 1, CnCAL0 is not closed under LLL. It is rather easy to produce an example for which (†) fails – Section 5 contains one. An obvious way out is to redefine the combination, and thus obtain CAL00 : CnCAL00 (Γ) = CnLLL (CnCAL0 (Γ)) For CAL00 , obviously (†) holds with L0 = LLL, since LLL is a Tarski-logic. This relates to a second problem: can we assure that CnCAL0 (Γ) is not LLL-trivial, unless Γ is itself LLL-trivial? The answer to this question depends on the specific sets of abnormalities that are involved in the definition. If for i, j ∈ I, neither Ωi ⊆ Ωj , nor Ωj ⊆ Ωi , the answer is negative. ˇ B}. Suppose To see why, suppose A ∈ Ωi − Ωj and B ∈ Ωj − Ωi , Γ = {A ∨ moreover that A and B are both LLL-contingent, hence Γ is not LLL-trivial. ˇ B is not But according to the definition of a Dab-consequence in Section 2, A ∨ a Dab-consequence of Γ for either of the two logics ALri and ALrj . This implies that ¬ ˇ A, B ∈ CnALri (Γ) and ¬ ˇ B, A ∈ CnALrj (Γ). Hence the union of both consequence sets will be LLL-trivial. It is therefore required that (‡) for all i, j ∈ I, either Ωi ⊆ Ωj or Ωj ⊆ Ωi To further simplify the meta-theory and notation, it will be assumed that the following stronger requirement holds:9 8 Every

flat adaptive logic is also closed under its lower limit logic – see [5, Theorem 5.6.1]. that (‡) reduces to (‡0 ) in case I is finite. In the infinite case, (‡) suffices to avoid a trivial consequence set whenever Γ is not LLL-trivial – see [5, Chapter 6] – but the Theorem of Strong Reassurance from Section 7.3 does not hold in general. 9 Note

8

(‡0 ) Ω1 ⊂ Ω2 ⊂ Ω3 ⊂ . . . S The resulting logic is different from a flat adaptive logic with Ω = i∈I . The combination imposes a hierarchy on the total set of abnormalities, and most often yields a larger consequence set than its flat counterpart – see Section 8 for more details. This combination is called a hierarchic adaptive logic, and henceforth referred to as HALx . Let us now look at the full-blown definition of CnHALx (Γ). It is stipulated that hΩi ii∈I is a sequence of sets of formulas given by a (possibly restricted) logical form, and that for all i ∈ I, Ωi ⊂ Ωi+1 . Where i ∈ I and x ∈ {Reliability, Minimal Abnormality}, ALxi is defined by (i) LLL, (ii) Ωi and (iii) the Strategy x. Define HALx as follows: S Definition 9 CnHALx (Γ) = CnLLL ( i∈I CnALxi (Γ)) The proof theory and semantics that yield this consequence set will be defined in the next two sections. Soundness and completeness will be discussed with reference to membership of the consequence set. Note that, as for flat adaptive logics, we may define a hierarchic adaptive logic by a triple, where the second element is a sequence of sets of abnormalities: hLLL, hΩi ii∈I , xi, where x ∈ {r, m}.

3.2

An Example: HKr

In Section 2.2, we saw an example of the flat logic Kx that helps us cope with (possibly inconsistent or false) plausible knowledge. However, in real life, we often distinguish between different degrees of plausibility. A lengthy discussion of the issue can be found in [11], and adaptive logics for the consequence relations from that paper were presented in [19]. A hierarchic variant of Kx results in a similar consequence relation. To express the plausibility degree of a piece of information, sequences of diamonds are used: ♦♦ . . . ♦A. The longer the sequence, the less plausible the information. A sequence of i diamonds will be abbreviated by ♦i . If we start from a prioritized belief base Ψ = {Γ0 , Γ1 , Γ2 , . . .}, where the subscript of each Γ indicates its priority degree and Γ1 is the set of the most plausible beliefs, let Γ♦ = {♦i A | A ∈ Γi } The HKx -consequence set of Γ is given by the following recursive definition: For every i ∈ N, define Ωi : {♦j A ∧ ¬A | A ∈ W a , j ≤ i} For every i ∈ N, define Kxi by 1. The lower limit logic K 2. The set of abnormalities Ωi 3. The strategy x Define CnHKx = CnK (CnKx1 (Γ) ∪ CnKx2 (Γ) ∪ . . .) Note that by the definition above, Ω1 ⊂ Ω2 ⊂ . . .. More generally, HKx is defined in the format of Definition 9, hence it is a hierarchic adaptive logic. Some results can give a first hint at how this logic works. Consider the set Γ = {♦p, ♦♦q, ♦♦r, ¬p ∨ ¬r}. The following is provable for both x = r and x = m: 9

Γ `HKx Γ `HKx Γ `HKx Γ 0HKx

p q p∧q r

These results will be commented upon in the Section 5. Note that in all subsequent sections, HKx refers to a particular logic, whereas HALx is used as a variable for any logic that is constructed according to Definition 9.

4

Some Important Lemmas

Several proofs in subsequent sections rely on one crucial lemma and a number of other lemmas that can be easily derived from it. To facilitate the reading of the paper, their proofs will be presented in this separate section. Readers who are only interested in the main results of the paper may skip them. To prepare for the crucial lemma, one lemma about minimal choice sets need to be established first: Lemma 2 Where (1) Σ = {∆1 , ∆2 , . . .} is a set of sets and (2) ϕ is a choice set of Σ: (3) for every A ∈ ϕ, there is a ∆i ∈ Σ for which ∆i ∩ ϕ = {A} iff (4) ϕ is a minimal choice set of Σ. Proof. Suppose (1) and (2) hold. (⇒) Suppose (3) holds, and consider a ϕ0 ⊂ ϕ and a B ∈ ϕ, B 6∈ ϕ0 . By (3), there is a ∆i ∈ Σ for which ∆i ∩ ϕ = {B} and hence ∆i ∩ ϕ0 = ∅. This implies that ϕ0 is not a choice set of Σ. As a result, ϕ is a minimal choice set of Σ. (⇐) Suppose there is a B ∈ ϕ such that, for no ∆i ∈ Σ, ϕ ∩ ∆i = {B}. In that case for every ∆i for which B ∈ ∆i , there is a C ∈ ϕ such that C ∈ ∆i . Hence ϕ − {B} is a choice set of Σ, hence ϕ is not a minimal choice set of Σ. Let me introduce some notational conventions to facilitate the proofs. Where i ∈ I, a Dabi -formula Dab(∆) is a classical disjunction of the members of a ∆ ⊂ Ωi . Dab(∆) is a minimal Dabi -consequence of Γ iff ∆ ⊂ Ωi , Γ `LLL Dab(∆) and there is no ∆0 ⊂ ∆ for which Γ `LLL Dab(∆0 ). Where Dab(∆1 ), Dab(∆2 ), . . . are the minimal Dabi -consequences of Γ, let Σi (Γ) = {∆1 , ∆2 , . . .} and define Φi (Γ) as the set of minimal choice sets of Σi (Γ). Furthermore, for every i ∈ I, i > 1, we define Ωi = Ωi − Ωi−1 and we stipulate that Ω1 = Ω1 . Note that for each ∆ ∈ Σi+1 (Γ) − Σi (Γ), ∆ ∩ Ωi+1 6= ∅. Let ϕ ∈ Φi (Γ) and  i+1 let Φϕ (Γ) be the set of minimal choice sets of ∆ ∩ Ωi+1 | ∆ ∈ Σi+1 (Γ) − Σi (Γ), ϕ ∩ ∆ = ∅ .10 i+1

Lemma 3 Where i ∈ I: for all ϕ ∈ Φi (Γ) and all ϕ0 ∈ Φϕ (Γ), ϕ ∪ ϕ0 ∈ Φi+1 (Γ). Proof. Let ∆ ∈ Σi+1 (Γ). Suppose ∆ ∩ ϕ = ∅. Then ∆ ∈ / Σi (Γ) since ϕ ∈ Φi (Γ). i+1 i Hence, ∆ ∈ Σ (Γ) − Σ (Γ). In this case ∆ ∩ Ωi+1 6= ∅. Hence ϕ0 ∩ ∆ 6= ∅, i+1 since ϕ0 ∈ Φϕ . Hence ϕ ∪ ϕ0 is a choice set of Σi+1 (Γ). 10 The precise definition of this set greatly benefited from some comments by Christian Straßer. He also contributed to the proof for Lemma 3.

10

By the right-left direction of Lemma 2 and the fact that ϕ ∈ Φi (Γ), for every A ∈ ϕ there is a ∆ ∈ Σi (Γ) such that ∆ ∩ ϕ = {A}. Moreover, for all these ∆, ϕ0 ∩ ∆ = ∅, since ϕ0 ⊆ Ωi+1 . Finally, Σi (Γ) ⊆ Σi+1 (Γ), which gives us: (1) for every A ∈ ϕ there is a ∆ ∈ Σi+1 (Γ) such that ∆ ∩ (ϕ ∪ ϕ0 ) = {A}. From the right-left direction of Lemma 2: for every A ∈ ϕ0 , there is a Θ such that Θ ∩ ϕ0 = {A}, where Θ = ∆ ∩ Ωi+1 for a ∆ ∈ Σi+1 (Γ). Since ϕ0 ⊆ Ωi+1 , i+1 ∆ ∩ ϕ0 = {A}. Moreover, in view of the definition of Φϕ , ∆ ∩ ϕ = ∅. Hence we have: (2) for every A ∈ ϕ0 , there is a ∆ ∈ Σi+1 (Γ) such that ∆ ∩ (ϕ ∪ ϕ0 ) = {A}. By (1) and (2): for every A ∈ ϕ ∪ ϕ0 , there is a ∆ ∈ Σi+1 (Γ) such that ∆ ∩ (ϕ ∪ ϕ0 ) = {A}. By the left-right direction of Lemma 2, ϕ ∪ ϕ0 is a minimal choice set of Σi+1 (Γ), hence ϕ ∪ ϕ0 ∈ Φi+1 (Γ). Lemma 4 For all i ∈ I and for every ϕ ∈ Φi (Γ), there is a ψ ∈ Φi+1 (Γ) for which ψ ∩ Ωi = ϕ. i+1

Proof. Suppose ϕ ∈ Φi (Γ). Let ϕ0 be an arbitrary element in Φϕ . Note that ϕ0 ⊆ Ωi+1 . Define ψ = ϕ ∪ ϕ0 . The lemma follows immediately in view of Lemma 3. Throughout the rest of this paper,SI will refer to the flat adaptive logic ALx that is defined by (i) LLL, (ii) Ω = i∈I Ωi , (iii) the strategy x ∈ {r, m}. The notions of a minimal Dab-consequence, Ab(M ), U (Γ) and Φ(Γ) are defined as in Section 2. It is important to observe that Lemma 4 holds no matter how we define the hierarchic adaptive logic. More specifically, it does not depend on the way the Ωi s are constructed, as long as requirement (‡0 ) from Section 3 holds. Moreover, this requirement implies that Ωi ⊂ Ω for every i ∈ I, and that Ωi ⊂ Ωj for every i, j ∈ I such that i < j. Hence we can derive: Lemma 5 For all i ∈ I and for every ϕ ∈ Φi (Γ), there is a ψ ∈ Φ(Γ) for which ψ ∩ Ωi = ϕ. Lemma 6 For all i, j ∈ I, j > i and for every ϕ ∈ Φi (Γ), there is a ψ ∈ Φj (Γ) for which ψ ∩ Ωi = ϕ. A last lemma concerns the cardinality of Φi (Γ) in view of Φ(Γ): Lemma 7 For every i ∈ I, the cardinality of Φi (Γ) is never bigger than that of Φ(Γ). Proof. Let i ∈ I and consider a ϕ, ψ ∈ Φi (Γ), with ϕ 6= ψ. By Lemma 5, there is a ϕ0 ∈ Φ(Γ) : ϕ0 ∩ Ωi = ϕ and a ψ 0 ∈ Φ(Γ) : ψ 0 ∩ Ωi = ψ. As a result, ϕ0 6= ψ 0 .

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5

The Proof Theory

Can we establish a unified proof theory for CnHALx (Γ), notwithstanding the fact that it is the result of the application of several adaptive logics to Γ, that each have their own separate proof theory? As the header of this section indicates, we can. This will be possible through an extension of the existing proof theory for flat adaptive logics. In Section 5.1, this generic proof theory for both the Reliability and the Minimal Abnormality Strategy is spelled out; in Section 5.2, the adequacy of this proof theory with respect to Definition 9 is proven. After this general approach, a shorter and intuitively appealing proof theory for HALr is discussed in Section 5.3.

5.1

Starred Marking

The proof theory of HALx contains the inference rules PREM, RU and RC of ALx . One extra inference rule RU* will be introduced to complete the proof theory for HALx . For reasons that will become clear in this section, the application of RU and RC are restricted to a subclass of preceding lines in the proof, such that the following holds: Restriction 1: RU and RC are only applied to lines with a condition ∆ ⊂ Ω. Let us first assume that these are the only inference rules for a HAL-proof from Γ. Consider the Γ from Section 3.2, and consider a HKr - proof from Γ: 1 2 3 4 5 6 7

♦p ♦♦q ♦♦r ¬p ∨ ¬r p ∨ (♦p ∧ ¬p) p (♦p ∧ ¬p) ∨ (♦♦r ∧ ¬r)

PREM PREM PREM PREM 1; RU 5; RC 1,3,4; RU

∅ ∅ ∅ ∅ ∅ {♦p ∧ ¬p} ∅

On line 6, we have derived p on the condition ♦p ∧ ¬p. However, this condition occurs in a minimal Dab-formula on line 7. So we may ask ourselves: should line 6 be marked? Recall the definition of Kr1 . The set of abnormalities for this logic is Ω1 = {♦A ∧ ¬A | A ∈ W a }. This indicates that the formula on line 7 is not a Dabformula for the logic Kr1 . More generally, we can prove that ♦p ∧ ¬p does not occur in any Dab-consequence Dab(∆) of Γ, with ∆ ⊂ Ω1 .11 Hence Γ `Kr1 p, and therefore also Γ `HKr p. The marking definition requires a classification of sets of unreliable formulas in view of the respective flat adaptive logics ALx1 , ALx2 , . . .. As in the previous section, a Dabi -formula is the classical disjunction of members of Ωi . Dab(∆) is a minimal Dabi -formula at stage s iff there is no Dabi -formula Dab(∆0 ) at 11 Roughly speaking, the proof goes as follows. Let Θ be the set of all sentential letters minus r. Let Θ¬♦¬ = {¬♦¬A | A ∈ Θ}. Note that  = Γ ∪ Θ ∪ Θ¬♦¬ ∪ {¬♦r} is K-satisfiable. Every K-model that validates all the members of , falsifies every member of Ω1 . Hence every Dab-consequence Dab(∆) with ∆ ⊂ Ω1 is false in this K-model of Γ.

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stage s such that ∆0 ⊂ ∆. Where Dab(∆1 ), Dab(∆2 ), S . . . are the minimal Dabi formulas at stage s, Σis (Γ) = {∆1 , ∆2 , . . .}, Usi (Γ) = Σis (Γ) and Φis (Γ) is the set of minimal choice sets of Σis (Γ). Note that Σ1s (Γ) ⊆ Σ2s (Γ) ⊆ . . . ⊆ Σs (Γ). Hence we have obtained many Usi (Γ)’s and Φis (Γ)’s instead of just one. Let us return to our example. In view of the above definitions, U71 (Γ) = ∅, and U72 (Γ) = {♦p ∧ ¬p, ♦♦r ∧ ¬r}. The next thing we have to do, is to assure that the marking happens in view of the right logic and hence, for the example above, in view of the right Usi (Γ). If line 6 is marked in view of U 2 (Γ), our problem remains. More specifically, since we only need a condition ∆ ⊂ Ω1 to derive p on line 6, we should not consider the members of U72 (Γ). The following definition takes care of this: Definition 10 HALr -Marking: a line l is marked at stage s iff, where ∆ is its condition, ∆ ⊂ Ωi and ∆ 6⊂ Ωi−1 : ∆ ∩ Usi (Γ) 6= ∅. According to this marking definition, line 6 is not marked at stage 7, nor is it marked in any extension of the proof. In a similar spirit, we adjust the marking definition for minimal abnormality: Definition 11 HALm -Marking: a line l with formula A is marked at stage s iff, where its condition is ∆, ∆ ⊂ Ωi and ∆ 6⊂ Ωi−1 : (i) there is no ϕ ∈ Φis (Γ) such that ϕ ∩ ∆ = ∅, or (ii) for a ϕ ∈ Φis (Γ), there is no line on which A is derived on a condition Θ ⊂ Ωi for which Θ ∩ ϕ = ∅. With this new marking definition, Γ `ALxi A for an i ∈ I iff A is finally derived on a line l of a HALx -proof from Γ, and the condition of line l is a subset of Ω. This is stated by Corollary 1 below. However, we are not home yet. Consider the following continuation of the proof from Γ: 8 q ∨ (♦♦q ∧ ¬q) 9 q 10 p ∧ q

2; RU 8; RC 6,9; RU

∅ {♦♦q ∧ ¬q} {♦p ∧ ¬p, ♦♦q ∧ ¬q}X

Consider line 10. Note that the condition on this line is not a subset of Ω1 , though it is a subset of Ω2 . Hence this line should be marked in view of U 2 (Γ), and the formula p ∧ q is not finally derivable. However, both p and q are finally derivable in the proof. This is the example I promised in Section 3: it shows that CnKr1 (Γ) ∪ CnKr2 (Γ) is not K-closed. Nevertheless, HKr was defined in such a way that this problem is avoided. On the proof theoretic level, we therefore need to add one extra inference rule, and a new marking definition for lines that are inferred by this rule: RU*

If A1 , . . . , An `LLL B:

i1 .. .

A1 .. .

∆1 .. .

in j

An B

∆n ∗

Definition 12 RU*-Marking: starting from l = 1: a line l is marked at stage s iff, where “i1 , . . . , in ; RU ∗” is its justification, line i1 or . . . or line in is marked at stage s. 13

The restriction to the application of RU and RC avoids that the condition ∗ is intermixed with conditions of the form ∆ ⊂ Ω. Although this would not hinder the marking definition, it would severely complicate the proofs and the meta-theory. Note that whether a line l is RU*-marked depends on whether the lines from which line l was derived are themselves marked. Those lines can be either HALx -marked or RU*-marked. The general idea is that if the lines i1 , . . . , in are derived on an unmarked line, every LLL-consequence of A1 , . . . , An can be derived on an unmarked line in an extension of the proof, by applications of RU*. The interpretation of a marked (unmarked) line at stage s remains identical, whether it is marked in view of Definition 10, 11 or 12: the second element of an unmarked line is considered as derived at that stage, the second element of a marked line is considered as not derived at that stage. To illustrate the new inference rule and marking definition, let me continue the proof: 11 p ∧ q 12 r 13 p ∧ r

6,9; RU* ∗ 3; RC {♦♦r ∧ ¬r} X12−13 6,12; RU* ∗ X13

Final derivability is defined in exactly the same manner as for flat adaptive logics (see Definition 7 and Definition 8, ALx is replaced by HALx ). This completes the proof theory of HALx .

5.2

The Adequacy of Final HALx -Derivability

The proofs of the various lemmas and theorems that lead to Theorem 11 below were obtained by slight variations on proofs for the Soundness and Completeness of ALx in [3]. Lemma 8 There is a HALx -proof from Γ that contains a line at which A is ˇ Dab(∆). derived on a condition ∆ ⊂ Ω iff Γ `LLL A ∨ Proof. (⇒) Suppose A is derived on line l of a HALx -proof from Γ, on the condition ∆ ⊂ Ω. In view of Restriction 1, line l is obtained by applications of PREM, RU and RC. In view of the left-right direction of Lemma 1 from [3], ˇ Dab(∆). (⇐) Suppose Γ `LLL A ∨ ˇ Dab(∆). In view of the right-left Γ `LLL A ∨ x direction of Lemma 1 from [3], there is a AL -proof in which A is derived on the condition ∆ by applications of PREM, RU and RC. It immediately follows that this is a HALx -proof from Γ as well. S Define U i (Γ) = Σi (Γ).12 Note that since Σ1 (Γ) ⊆ Σ2 (Γ) ⊆ . . . ⊆ Σ(Γ), U 1 (Γ) ⊆ U 2 (Γ) ⊆ . . . ⊆ U (Γ). Lemma 9 If A is finally derived on line l of a HALr -proof from Γ, where ∆ ⊂ Ωi , ∆ 6⊂ Ωi−1 is the condition of line l, then ∆ ∩ U i (Γ) = ∅. Proof. Suppose the antecedent holds, but ∆ ∩ U i (Γ) 6= ∅. Hence there is a minimal Dabi -formula Dab(Θ) : Θ ∩ ∆ 6= ∅. Hence there is an extension of the proof in which Dab(Θ) is derived on the empty condition. Where s is the 12 The

set Σi (Γ) was defined on page 10.

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last stage of this extension, Θ ⊆ Usi (Γ). Hence ∆ ∩ Usi (Γ) 6= ∅ and line l is marked in view of Definition 10. Since Dab(Θ) is a minimal Dabi -consequence of Γ, Θ ⊆ Usi0 (Γ) for all subsequent stages s0 , and thereby ∆ ∩ Usi0 (Γ) 6= ∅. This means that in every further extension of the proof, line l remains marked. By Definition 7 this contradicts the antecedent. Theorem 9 A is finally derived on a line l of a HALr -proof from Γ, where ∆ ⊂ Ω is the condition of line l iff Γ `ALri A for an i ∈ I. Proof. (⇒) Suppose A is finally derived on a line l of a HALr -proof from Γ, ˇ Dab(∆). where ∆ ⊂ Ω is the condition of line l. By Lemma 8, (1) Γ `LLL A ∨ Since ∆ is finite, there is an i ∈ I: ∆ ⊂ Ωi , ∆ 6⊂ Ωi−1 . By Lemma 9, (2) ∆ ∩ U i (Γ) = ∅. By (1), (2) and Theorem 6, Γ `ALri A. (⇐) Suppose Γ `ALri A for an i ∈ I. By Theorem 6, there is a ∆ ⊂ Ωi : ˇ Dab(∆) and ∆ ∩ U i (Γ) = ∅. By Lemma 8, there is a HALr -proof Γ `LLL A ∨ from Γ in which A is derived at a line l on a condition ∆ ⊂ Ωi ⊂ Ω. Take the j ≤ i for which ∆ ⊂ Ωj , ∆ 6⊂ Ωj−1 . Note that since U j (Γ) ⊆ U i (Γ), ∆ ∩ U j (Γ) = ∅. In every extension of the proof in which line l is marked, a Dabj -formula Dab(Θ) is derived on the condition ∅, such that Θ ∩ ∆ 6= ∅. As ∆ ∩ U j (Γ) = ∅, there is a Θ0 ⊂ (Θ − ∆) for which Γ `LLL Dab(Θ0 ) — in other words, Dab(Θ) is not a minimal Dab-consequence of Γ. . Hence we can further extend this extension such that Dab(Θ0 ) is derived on the empty condition, and as a result line l is unmarked again. By Definition 7, A is finally derived on line l. Theorem 10 A is finally derived on a line l of a HALm -proof from Γ, where ∆ ⊂ Ω is the condition of line l iff Γ `ALm A for an i ∈ I. i Proof. (⇒) Suppose the antecedent holds. Since ∆ is finite, there is an i ∈ I: ∆ ⊂ Ωi , ∆ 6⊂ Ωi−1 . By Definitions 7 and 8, line l is unmarked, and every extension of the proof in which line l is marked may be further extended in such a way that line l is unmarked. Suppose we extend the proof to a stage s by deriving every minimal Dabi -consequence of Γ on the condition ∅, whence Φis (Γ) = Φi (Γ).13 In view of Definition 11, line l is unmarked iff the extended proof has a further extension such that (i) ∆ ∩ ϕ = ∅ for a ϕ ∈ Φi (Γ) and (ii) for every ϕ ∈ Φi (Γ), there is a line that has A as its formula and a Θ ⊂ Ωi as its condition such that Θ ∩ ϕ = ∅. (ii) holds iff for every ϕ ∈ Φi (Γ), there ˇ Dab(Θ). By Theorem 7, is a Θ ⊂ Ωi such that ϕ ∩ Θ = ∅ and Γ `LLL A ∨ Γ `ALm A. i (⇐) Suppose Γ `ALm A for an i ∈ I. By Theorem 7: (†) for every ϕ ∈ Φi (Γ), i ˇ Dab(∆). Where these there is a ∆ ⊂ Ωi such that ϕ ∩ ∆ = ∅ and Γ `LLL A ∨ conditions are ∆1 , ∆2 , . . ., define Θ = ∆1 ∪ ∆2 ∪ . . .. Take the j ≤ i for which Θ ⊂ Ωj , Θ 6⊂ Ωj−1 . By Lemma 6, for every ϕ ∈ Φj (Γ), there is a ϕ0 ∈ Φi (Γ) such that ϕ ⊆ ϕ0 . By (†) this implies that for every ϕ ∈ Φj (Γ), there is a ∆ ⊂ Ωj ˇ Dab(∆). Θ 6⊂ Ωj−1 implies that there is a such that ϕ ∩ ∆ = ∅ and Γ `LLL A ∨ ˇ Dab(∆k ). condition, say ∆k , for which ∆k ⊂ Ωj , ∆k 6⊂ Ωj−1 while Γ `LLL A ∨ We can infer that there is a HALm -proof from Γ in which (i) every minimal Dabj -consequence of Γ is derived on the condition ∅, (ii) for every ϕ ∈ Φj (Γ), A is derived on a condition ∆ ⊂ Ωj for which ∆ ∩ ϕ = ∅ and (iii) on a line l of the 13 For

the definition of Φi (Γ), see page 10.

15

proof, A is derived on the condition ∆k . In view of Definition 11 and Definition 7, A is finally derived on line l of this proof. Corollary 1 A is finally derived on a line l of a HALx -proof from Γ, where ∆ ⊂ Ω is the condition of line l iff Γ `ALxi A for an i ∈ I. To understand the importance of the following theorem, it is important to keep in mind that we defined CnHALx (Γ) in Section 3 as a function of the different sets CnALxi (Γ) and the lower limit logic – hence independent of the proof theory spelled out in the current section. Theorem 11 Γ `HALx A iff A ∈ CnHALx (Γ). Proof. First, note that the following two statements are equivalent by Definition 9 and the compactness of LLL: (1) A ∈ CnHALx (Γ) (2) There are B1 , . . . , Bn such that (2a) (2b)

Γ `ALxi B1 , . . . , Γ `ALxin Bn and 1 {B1 , . . . , Bn } `LLL A

(⇒) Suppose Γ `HALx A, hence by Definition 8, there is a HALx -proof p from Γ in which A is finally derived on a line l. If the condition of line l is a ∆ ⊂ Ω, we immediately have (2) (with n = 1) by Corollary 1, hence A ∈ CnHALx (Γ). If the condition of line l is ∗, we can show by an induction on the length of the justification of line l: there are B1 , . . . , Bn : {B1 , . . . , Bn } `LLL A and for every Bi (with i ∈ {1, . . . , n}), Bi is finally derived in p, on a condition ∆ ⊂ Ω. By Corollary 1, we obtain (2) and hence A ∈ CnHALx (Γ). (⇐) Suppose A ∈ CnHALx (Γ). From (2), by Corollary 1: for every j ∈ {1, . . . , n}: Bj is finally derived on a line lj of a finite HALx -proof pj from Γ, where ∆ ⊂ Ω is the condition of line lj . Where p t p0 denotes the concatenation of p and p0 , there is a HALx -proof from Γ: p1 t. . .tpn . In this proof, B1 , . . . , Bn are finally derived.14 By the rule RU*, we can add one line l to this proof at which A is derived from B1 , . . . , Bn on the condition ∗. Since by Definition 12, the marking of line l depends solely on the marking of lines l1 , . . . , ln , every extension of the proof in which line l is marked can be further extended such that line l is unmarked again. By Definition 8, Γ `HALx A.

5.3

The Shortcut for Reliability

There is a more perspicuous proof theory for HALr as well. As a first indication of how this works, consider again the Γ from Section 3.2: Γ = {♦p, ♦♦q, ♦♦r, ¬p∨ ¬r}. Recall that the minimal Dab-formula (♦p ∧ ¬p) ∨ (♦♦r ∧ ¬r) implied that ♦p ∧ ¬p ∈ U 2 (Γ), and therefore we needed the extra rule RU* and the new marking definition, to allow for the derivation of p ∧ q on an unmarked line. However, as ♦p ∧ ¬p 6∈ U 1 (Γ), ¬ ˇ (♦p ∧ ¬p) is a Kr1 -consequence of Γ, and hence r it is also a HK -consequence of Γ. 14 Note that this proof is an extension of every proof p , where j ∈ {1, . . . , n}. See page 5 j where this was explained. From Definition 7, it follows that if a formula is finally derived in a proof, it is also finally derived in all extensions of this proof.

16

Let me generalize this insight. Recall from the previous section that Ωi = Ωi − Ωi−1 , where Ω1 = Ω1 . Note that by restriction (‡0 ) above, for every i ∈ I, Ωi is non-empty. Note also that in the example of Section 3.2, Ωi = {♦i A ∧ ¬A | A ∈ W a }. If B ∈ Ωi and B 6∈ U i (Γ), then Γ `HKr ¬ ˇ B. That is, we can derive ¬ ˇ B in a Kri -proof from Γ on the condition {B}, and {B} ∩ U i (Γ) = ∅. In other words, whether or not B ∈ U j (Γ) for a j > i, does not matter for the hierarchic logic. As long as B 6∈ U i (Γ), it will be considered reliable. Still in other words, as long as B ∈ Ωi does not occur in a minimal Dabi -consequence, it remains a reliable abnormality. Note that by definition, B cannot occur in a minimal Dabk -consequence for any k < i. For every ∆ ⊂ Ω, define ∆0 = ∆∩Ωi , with i ∈ I such that ∆ ⊂ Ωi , ∆ 6⊂ Ωi−1 . Where Dab(∆1 ), Dab(∆2 ), . . . are the minimal Dab-consequences of Γ, define the set of unreliable formulas for the logic HALr : U ? (Γ) = ∆01 ∪ ∆02 ∪ . . . By this definition, B ∈ U ? (Γ) iff B ∈ Ωi and B occurs in a minimal Dabi consequence of Γ. Here we can observe again that the logic imposes a “hierarchy” on the whole set Ω. This hierarchy can be represented by the following sequence: Ω1 , Ω2 , . . .. If the logic is forced by the premises to choose between the abnormality A ∈ Ωi and the abnormality B ∈ Ωi+j , it will consider B as unreliable. For example, consider Γ0 = {♦p, ♦♦q, ¬p ∨ ¬q}. Note that either ♦p ∧ ¬p or ♦♦q∧¬q has to be true in view of these premises and the logic K. Informally, this means that either a very plausible belief (p) has to be given up, or a slightly less plausible belief (q). The hierarchic logic HKr then chooses for the least harmful of these two options: Γ `HKr ♦♦q ∧ ¬q and also Γ `HKr p. ˇ B, A ∈ Ωi and B ∈ Ωi+j , However, this does not mean that if Γ `LLL A ∨ then A is necessarily reliable. For example, where the logic is HKr , and Γ00 = {♦p, ♦2 q, ♦3 r, ¬p ∨ ¬q, ¬q ∨ ¬r}, (♦2 q ∧ ¬q) ∨ (♦3 r ∧ ¬r) is a minimal Dabconsequence of Γ00 . Nevertheless, q is still considered unreliable, in view of the minimal Dab-consequence (♦p ∧ ¬p) ∨ (♦2 q ∧ ¬q). In other words, only p will be freed from suspicion by the hierarchic logic. It is fairly easy to obtain a new proof theory for HALr , relying on these considerations. The inference rules for this proof theory are exactly the same as for the flat logic ALr – see Section 2. Where Dab(∆1 ), Dab(∆2 ), . . . are the minimal Dab-formulas at stage s, define the set of unreliable formulas at stage s as follows: Us? (Γ) = ∆01 ∪ ∆02 ∪ . . . Definition 13 HALr ?-Marking: a line l is marked at stage s iff, where ∆ is its condition, ∆ ∩ Us? (Γ) 6= ∅. Again, final derivability is given by Definition 7 and an adjustment of Definition 8 – replace ALx by HALr ?. This gives us the consequence relation Γ `?HALr A. Note that this proof theory is significantly less complex than the more general one from Section 5.1. No extra inference rules or marking definitions are required; a small correction to the definition of the set of unreliable formulas suffices. The reader may wonder why I called this proof theory a “shortcut” proof theory. The reason is fairly straightforward: there are Γ, A for which A can be 17

proven in fewer steps in a HALr ?-proof from Γ, than in a HALr -proof from Γ. An example is the one from Section 5. If we consider the proof in that section as a HALr ?-proof, the formula p ∧ q is finally derived on line 10. On the other hand, a HALr -proof from Γ may be more instructive, in that it shows us which formulas follow by one of the logics ALri . The general conclusion one should draw is that there are two distinct proof theories for HALr , and that each one has its own merits. In the remainder of this section, it is proven that Γ `?HALr A iff A ∈ CnHALr (Γ). ˇ Dab(∆) for a Lemma 10 For every i ∈ I: if A ∈ CnALri (Γ), then Γ `LLL A ∨ ? ∆ ⊂ Ω and ∆ ∩ U (Γ) = ∅. Proof. Suppose (i) A ∈ CnALri (Γ), with i ∈ I. By Theorem 6, (ii) Γ `LLL ˇ Dab(∆) for a ∆ ⊂ Ωi and ∆ ∩ U i (Γ) = ∅. Since Ωi ⊂ Ω, ∆ ⊂ Ω. Suppose A∨ (iii) ∆ ∩ U ? (Γ) 6= ∅. By the definition of U ? (Γ), there is a minimal Dabconsequence Dab(Θ), with Θ ⊂ Ωj and Θ 6⊂ Ωj−1 , such that (Θ ∩ Ωj ) ∩ ∆ 6= ∅. As a result, Θ ∩ ∆ 6= ∅ and j ≤ i. This implies that Dab(Θ) is a minimal Dabi -consequence of Γ. This implies that ∆ ∩ U i (Γ) 6= ∅, which contradicts (ii). Lemma 11 For every i ∈ I and every finite ∆ ⊂ Ωi : if ∆ ∩ U ? (Γ) = ∅, then Γ `HALr ¬ ˇ Dab(∆). Proof. Suppose (i) ∆∩U ? (Γ) = ∅ for a finite ∆ ⊂ Ωi . Suppose (ii) ∆∩U i (Γ) 6= ∅. This implies that there is a minimal Dabi -consequence of Γ, say Dab(Θ), such that ∆ ∩ Θ 6= ∅. Since ∆ ⊂ Ωi , Θ 6⊂ Ωi−1 . By the definition of U ? (Γ), Θ ∩ Ωi ⊆ U ? (Γ). As a result, ∆ ∩ U ? (Γ) 6= ∅, which contradicts (i). Hence (ii) is false: ∆ ∩ U i (Γ) = ∅. This implies by Definition 1 that every ALri -model falsifies every member of ∆, hence ¬ ˇ Dab(∆) ∈ CnALri (Γ). By Definition 9 and the reflexivity of LLL, ¬ ˇ Dab(∆) ∈ CnHALr (Γ). ˇ Dab(∆) for a finite ∆ ⊂ Ω such Theorem 12 A ∈ CnHALr (Γ) iff Γ `LLL A ∨ ? that ∆ ∩ U (Γ) = ∅. Proof. (⇒) Suppose A ∈ CnHALr (Γ). By the compactness of LLL, there are B1 , . . . , Bn such that B1 ∈ CnALri (Γ), . . . , Bn ∈ CnALrin (Γ) and {B1 , . . . , Bn } `LLL 1 ˇ Dab(∆j ) for a ∆j ⊂ Ω A. By Lemma 10: for every j ∈ {1, . . . , n}: Γ `LLL Bj ∨ ˇ Dab(∆1 ∪ . . . ∪ ∆n ) for which ∆ ∩ U ? (Γ) = ∅. By CL-properties, Γ `LLL A ∨ where ∆1 ∪ . . . ∪ ∆n is a subset of Ω, and (∆1 ∪ . . . ∪ ∆n ) ∩ U ? (Γ) = ∅. (⇐) ˇ Dab(∆) for a ∆ and (ii) ∆ ∩ U ? (Γ) = ∅. For every Suppose that (i) Γ `LLL A ∨ i ∈ I, define ∆i = ∆ ∩ Ωi . Since ∆ is finite, ∆ = ∆1 ∪ . . . ∪ ∆n for an n ∈ I. By Lemma 11, Γ `HALr ¬ ˇ Dab(∆i ) for every i ∈ {1, . . . , n}. By CL-properties, Γ `HALr A. Lemma 12 There is a HALr ?-proof from Γ that contains a line on which A ˇ Dab(∆). is derived on the condition ∆ ⊂ Ω iff Γ `LLL A ∨ Proof. Immediate in view of Lemma 1 from [3] and the fact that the inference rules of a ALx -proof are identical to the inference rules of a HALr ?-proof. Lemma 13 A is finally derived at line l of a HALr ?-proof from Γ, iff, where ∆ is the condition of line l, ∆ ∩ U ? (Γ) = ∅. 18

Proof. (⇒) Suppose (i) A is finally derived at line l of a HALr ?-proof from Γ, where ∆ is the condition of line l. Suppose (ii) ∆ ∩ U ? (Γ) 6= ∅. By the definition of U ? (Γ), there is a minimal Dab-consequence Dab(Θ) of Γ: Θ0 ∩ ∆ 6= ∅. Hence there is an extension of the proof in which Dab(Θ) is derived on the empty condition. Where s is the last stage of this extension, Θ0 ⊆ Us? (Γ). Hence ∆ ∩ Us? (Γ) 6= ∅ and line l is marked in view of Definition 13. Since Dab(Θ) is a minimal Dab-consequence of Γ, Θ0 ⊆ Us?0 (Γ) for all subsequent stages s0 , and thereby ∆∩Us?0 (Γ) 6= ∅. This means that in every further extension of the proof, line l is marked. By Definition 7 this contradicts (i). (⇐) Suppose A is derived at line l of a HALx -proof from Γ on the condition ∆, and ∆ ∩ U ? (Γ) = ∅. In that case, we can extend the proof and derive every minimal Dab-consequence of Γ at a stage s, whence Us? (Γ) = U ? (Γ) = Us?0 (Γ) for all s0 > s. In view of Definition 13, line l is unmarked at stage s and it is unmarked in every extension of the proof, whence Γ `HALr ? A. Theorem 13 Γ `?HALr A iff A ∈ CnHALr (Γ). Proof. Relying on Theorem 12, Lemma 12, Lemma 13, and Definition 8 consecutively, the following propositions are equivalent: A ∈ CnHALr (Γ) ˇ Dab(∆) for a ∆ ⊂ Ω and ∆ ∩ U ? (Γ) = ∅ Γ `LLL A ∨ A is derived in a HALr ?-proof from Γ on a condition ∆ and ∆ ∩ U ? (Γ) = ∅ A is finally derived in a HALr ?-proof from Γ Γ `?HALr A

(1) (2) (3) (4) (5)

6

The Semantics

6.1

The Intersection of the Sets of Flat Adaptive Models

As indicated by Definitions 1 and 2, flat adaptive logics have a selective semantics: a subset of the LLL-models of Γ is selected in view of a specific criterion. The same holds for hierarchic adaptive logics. In this case, the selection is based on the selections by each of the flat adaptive logics of the combination. The resulting set of models is complete with respect to the consequence relation of ALx HALx . Where the set of ALxi -models of Γ is denoted by MΓ i , the set of HALx -models of Γ is defined as follows: T x ALx Definition 14 MHAL = i∈I MΓ i Γ Note that the following facts hold: x

ALx i

Fact 1 For every i ∈ I: MHAL ⊆ MΓ Γ

⊆ MLLL . Γ

Proof. Immediate in view of Definition 14 and Definition 2. Fact 2 Every HALm -model is a HALr -model.

19

m

Proof. Consider a M ∈ MHAL . By Definition 14, M is a ALm i -model for Γ every i ∈ I. Hence by Theorem 12 from [3], M is a ALri -model for every i ∈ I. r Hence by Definition 14, M ∈ MHAL . Γ The semantic consequence relation can now be defined in the usual way: x

Definition 15 Γ |=HALx A iff A is true in every model M ∈ MHAL Γ Theorem 14 If A ∈ CnHALx (Γ), then Γ |=HALx A. x

Proof. Suppose A ∈ CnHALx (Γ) and consider a M ∈ MHAL . By Definition Γ ALx i 14, M ∈ MΓ for every i ∈ I. By Theorem S 1, M |= CnALi (Γ) for every i ∈ I. As M is a LLL-model, M |= CnLLL ( i∈I CnALxi (Γ)). By Definition 9, M |= CnHALx (Γ), hence M |= A. From Theorem 13 and Theorem 14, we immediately get: Theorem 15 If Γ `HALx A, then Γ |=HALx A. (Soundness for HALx )

6.2

Completeness for Reliability

For the Reliability Strategy, we can prove that CnHALr (Γ) is complete with respect to the set of all HALr -models. The proof of this property relies on the following lemma: r

and iff (M ∈ MLLL Lemma 14 There is a set ∆ such that M ∈ MAL Γ Γ M |= ∆). r

. By Proof. Define ∆ = {ˇ ¬ A | A ∈ Ω, A 6∈ U (Γ)}. (⇒) Suppose M ∈ MAL Γ and (ii) Ab(M ) ⊆ U (Γ). By (ii), M |= ¬ ˇ A for Definition 1, (i) M ∈ MLLL Γ every A ∈ Ω, A 6∈ U (Γ). Hence M |= ∆. (⇐) Suppose (i) M ∈ MLLL and (ii) Γ r M |= ∆. From (ii): Ab(M ) ⊆ U (Γ). By Definition 1, M ∈ MAL . Γ In view of the above proof, we can immediately derive the following fact: Fact 3 {ˇ ¬ A | A ∈ Ω, A 6∈ U (Γ)} ⊆ CnALr (Γ). Theorem 16 If Γ |=HALr A, then A ∈ CnHALr (Γ). (Completeness for HALr ) Proof. Suppose (?) A 6∈ CnHALr (Γ). For every i ∈ I, ¬B | S define ∆i = {ˇ B ∈ Ωi , B 6∈ U i (Γ)}, where i ∈ I. Suppose (1) Γ ∪ i∈I ∆i `LLL A. By the compactness of LLL, there is a finite number of finite sets Γ0 ⊆ Γ, ∆01 ⊆ ∆1 , . . . , ∆0n ⊆ ∆n : (2) Γ0 ∪ ∆01 ∪ ∆02 ∪ . . . ∪ ∆0n `LLL A. Abbreviate the classical V + ˇ Θ. By Fact 3, for every conjunction of the members of a finite Θ ⊂ W by V V S j ∈ {1, . . . , n}, ˇ ∆0j ∈ CnALrj (Γ), hence also ˇ ∆0j ∈ i∈I CnALri (Γ). Since S every ALri is reflexive by Theorem 2, also Γ0 ∈ i∈I CnALri (Γ). By (2) and Definition 9, A ∈ CnHALr (Γ). This contradicts (?), hence (1) is false: Γ ∪ S ∆ 0 i LLL A. This implies i∈I S by the completeness of LLL that there is a LLL-model that satisfies Γ ∪ i∈I ∆i and ¬ ˇ A. By Lemma 14, this model is an ALri -model of Γ, for every i ∈ I, hence by Definition 14, it is a HALr -model of Γ. As a result, Γ 6|=HALr A.

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6.3

An Alternative for Reliability

Recall the definition of U ? (Γ) from Section 5.3. In view of the shortcut proof theory, it is possible to define an alternative semantics for HALr : the set of all HALr -models of Γ is the set of all LLL-models M for which Ab(M ) ⊆ U ? (Γ). This is established by the following theorem: r

Theorem 17 M ∈ MHAL iff (M ∈ MLLL and Ab(M ) ⊆ U ? (Γ)). Γ Γ r

Proof. (⇒) Suppose M ∈ MHAL . (1) By Fact 1, M ∈ MLLL . (2) Suppose Γ Γ ? Ab(M ) 6⊆ U (Γ). Hence M |= B such that B ∈ Ωi for an i ∈ I, and B 6∈ U ? (Γ). This implies by the definition of U ? (Γ) that there is no minimal Dabi consequence Dab(∆) of Γ, such that B ∈ ∆. Hence B 6∈ U i (Γ). This implies ALr that Ab(M ) 6⊆ U i (Γ). By Definition 1, M 6∈ MΓ i , hence by Definition 14, r r M 6∈ MHAL , which contradicts the supposition. (⇐) Suppose M 6∈ MHAL . Γ Γ r ALi LLL By Definition 14, M 6∈ MΓ for an i ∈ I. If M 6∈ MΓ , the theorem follows immediately. If M ∈ MLLL , then by Definition 1, M |= B for a B ∈ Ωi , B 6∈ Γ U i (Γ). Take the j ≤ i for which B ∈ Ωj . Since B 6∈ U i (Γ), also B 6∈ U j (Γ), and as a result, B does not occur in a minimal Dabj -consequence of Γ. Hence B 6∈ U ? (Γ). As a result, Ab(M ) 6⊆ U ? (Γ).

6.4

Restricted Completeness for Minimal Abnormality

Where Φ(Γ) is finite, a similar completeness result can be established for the Minimal Abnormality Strategy. As in Section 6.2, the proof relies on a lemma about the flat adaptive logics ALm i : Lemma 15 Where i ∈ I: if Φi (Γ) is finite, then there is a set ∆ such that ALm M ∈ MΓ i iff (M ∈ MLLL and M |= ∆). Γ Proof. Suppose Φi (Γ) is finite. Where Φi (Γ) = {ϕ1 , . . . , ϕn }, define ∆ = ˇ ...∨ ˇ¬ {ˇ ¬ A1 ∨ ˇ An | A1 ∈ Ωi − ϕ1 , . . . , An ∈ Ωi − ϕn }. (⇐) Note that for every ˇ ...∨ ˇ¬ formula B ∈ ∆, with B = ¬ ˇ A1 ∨ ˇ An , and for every ϕj ∈ Φi (Γ), there is an {Aj } ⊂ Ωi such that {Aj } ∩ ϕj = ∅ and Γ `LLL B ∨ Aj by CL-properties. m By Theorem 8 from [3], we get that Γ |=ALm B. Hence if M ∈ MAL , then Γ ALm LLL by Definition 3 and M |= ∆. (⇒) Suppose M 6∈ MΓ i . If M ∈ MΓ LLL , then by Lemma M 6∈ MΓ , the lemma follows immediately. If M ∈ MLLL Γ 4 from [3]: for no ϕj ∈ Φi (Γ), Ab(M ) = ϕj . This means that for every j ∈ ˇ ...∧ ˇ An , with {1, . . . , n}: M |= Aj for an Aj ∈ Ωi − ϕj . Hence M |= A1 ∧ A1 ∈ Ωi − ϕ1 , . . . , An ∈ Ωi − ϕn . As a result, M 6|= ∆. Lemma 15 immediately gives us the following fact: ˇ ...∨ ˇ¬ Fact 4 Where Φi (Γ) = {ϕ1 , . . . , ϕn }, {ˇ ¬ A1 ∨ ˇ An | A1 ∈ Ωi −ϕ1 , . . . , An ∈ Ωi − ϕn } ⊆ CnALm (Γ). i Note that where Φ(Γ) is finite, Φi (Γ) is finite for every i ∈ I, in view of Lemma 7. As a result, we can apply Lemma 15 and Fact 4 to construct a similar proof as the one for Theorem 16 (we simply have to adjust the definition of the ∆i s), and arrive at the following:

21

Theorem 18 If Φ(Γ) is finite and Γ |=HALm A, then A ∈ CnHALm (Γ). (Restricted Completeness for HALm ) Notwithstanding the importance of this result, especially for most concrete applications, it is possible to construct a premise set for which completeness fails. The original example is presented by Diderik Batens in his [5, Chapter 6]. I will present an adapted version of this example, based on a logic similar to the one from Section 3.2. Proposition 1 There are Γ, A such that Γ |=HALm A, whereas A 6∈ CnHALm (Γ). Proof. Consider the logic HK.2m , defined as follows: CnHK.2m (Γ) = CnK (CnKm (Γ) ∪ CnKm (Γ)) 1 2 m a where Km 1 and K2 are defined as in Section 3.2. Where A ∈ W and i ∈ N, let i i ! A abbreviate ♦ A ∧ ¬A. Define:

Π1 = {!1 pi ∨!1 pj | i, j ∈ N, i 6= j}, Π2 = {!2 qi ∨!2 qj ∨!1 pk | i, j, k ∈ N, i 6= j} Π3 = {r∨!2 qi ∨!1 pj | i, j ∈ N} Γ = Π1 ∪ Π2 ∪ Π3 A brief look at Γ shows us that all minimal Dab1 -consequences of Γ are given by Π1 , and all minimal Dab2 -consequences by Π1 ∪ Π2 . This implies that all 1 Km 1 -models of Γ falsify exactly one abnormality ! pi (all the others have to be true in every K-model of Γ). For the Km 2 -models, the picture is slightly more complex. A first subset of these verify the minimal selection of abnormalities {!1 pi | i ∈ N}. Let us call this set of models M1 . The second subset of Km 2 -models is more diverse; every model in it verifies a set ϕkj (with j, k ∈ N): ϕkj = ({!1 pi | i ∈ N} − {!1 pj }) ∪ ({!2 qi | i ∈ N} − {!2 qk }) Let call this set of models M2 . In every model M ∈ M2 , r holds: Π3 entails that if for a j, k ∈ N, !1 pj and !2 qk are false, then r must be true. Since no HKm = M2 . As a result, Γ |=HKm r. M ∈ M1 is a Km 1 -model of Γ, MΓ However, r 6∈ CnHK.2m (Γ). To see why, consider all Km 1 -consequences that are not K-consequences of Γ. These follow from two sets of formulae: {!2 qi ∨!2 qj | i, j ∈ N, i 6= j} and {r∨!2 qj | j ∈ N}. The Km 2 -consequences that are not Kconsequences of Γ, are consequences of the formulae {r∨!1 pi | i ∈ N}. Neither of these formulae, nor any conjunction of them, suffices to K-derive r. As is clear from the example above, the Minimal Abnormality Strategy sometimes results in a selection of models that carries information beyond what can be expressed by a (possibly infinite) set of finite formulas. In the example, every 1 Km 1 -model falsifies one abnormality ! pi with i ∈ N. However, there is no finite formula, nor any set of such formulas, that can express this property of the minimally abnormal models. It is precisely this mechanism that leads to the incompleteness result above. However, as is shown in [5, Chapter 6], combined logics by superposition fare no better in this respect, for the same reason.

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7

Some Meta-theoretic Properties

This section contains some interesting meta-theoretic properties of HALr and HALm . I do not pretend that this is all there is to say about hierarchic adaptive logics; these are merely arguments in favor of further research of these systems and their application to concrete forms of non-monotonic reasoning.

7.1

Reflexivity of HALx

Just as for ALx , the consequence relation HALx is reflexive: Theorem 19 For every Γ, Γ ⊆ CnHALx (Γ). Proof. Consider an i ∈ I. By Theorem 2, Γ ⊆ CnALxi (Γ), hence Γ ⊆ Since LLL is reflexive, and by Definition 9, Γ ⊆ CnHALx (Γ).

S

CnALxi (Γ).

i∈I

From Theorem 19, we can derive a fact that will be used in Section 7.4: LLL . Fact 5 MLLL CnHALx (Γ) ⊆ MΓ

7.2

Reliability versus Minimal Abnormality

The following theorem is the hierarchic counterpart of Theorem 5: Theorem 20 For every Γ, CnHALr (Γ) ⊆ CnHALm (Γ). Proof. Suppose A ∈ CnHALr (Γ). By Definition 9 and the compactness of LLL, there are B1 , . . . , Bn such that (i) {B1 , . . . , Bn } `LLL A and (ii) for every i ∈ {1, . . . , n}: Bi ∈ CnALrj for a ji ∈ I. From (ii), by Theorem 5: for every i i ∈ {1, . . . , n}: Bi ∈ CnALm (Γ) for a ji ∈ I. By (i) and by Definition 9, ji A ∈ CnHALm (Γ).

7.3

Strong Reassurance

As noted in Section 3, every logic HALx avoids triviality where possible. That is, unless Γ is LLL-trivial, it will not be HALx -trivial. This property is often referred to as “‘Reassurance”. It follows from the theorem of “Strong Reassurance”: for every LLL-model M of the premises that is not selected by the hierarchic adaptive logic, there is a selected model M 0 that beats M in view of one of the logics ALxi . Let Abi (M ) denote the set {B ∈ Ωi | M |= B}. m

m

Theorem 21 If M ∈ MLLL − MHAL , then there is an M 0 ∈ MHAL Γ Γ Γ 0 that for an i ∈ I, Abi (M ) ⊂ Abi (M ).

such

m

Proof. Suppose M ∈ MLLL − MHAL . Take the smallest i ∈ I such that M 6∈ Γ Γ ALm

ALm i

MΓ , whence (1) for all j < i, M ∈ MΓ j . By the Strong Reassurance of m 0 0 ALm i (see Theorem 3), there is a ALi -model M such that Abi (M ) ⊂ Abi (M ). 0 This implies that for all j < i, Abj (M ) ⊆ Abj (M ), whence by (1), we have: ALm j

(2) M 0 ∈ MΓ

for all j ∈ I, j ≤ i

Let ϕi = Abi (M 0 ). Note that by (2) and Theorem 8, the following holds: 23

(3) ϕi ∩ Ωj ∈ Φj (Γ) for all j ≤ i. j

For all j ∈ I, j > i, let ϕj be some arbitrary element in Φϕj−1 .15 Note that (4) for every j ∈ I, j > i, ϕj ⊆ Ωj . Consider ϕ⊕ = ϕi ∪ ϕi+1 ∪ . . .. By (4), ϕ⊕ ∩ Ωi = ϕ. In view of Lemma 3, we obtain that (5) for all j ≥ i, ϕ⊕ ∩ Ωj ∈ Φj (Γ). Together with (3), this implies: (‡) ϕ⊕ ∈ Φk (Γ) for every k ∈ I. Suppose that ϕ⊕ is not a choice set of Σ(Γ). Hence there is a minimal Dabconsequence of Γ, say Dab(Θ), such that ϕ⊕ ∩ Θ = ∅. Since Θ is finite, there is a k ∈ I: Θ ⊂ Ωk , whence (ϕ⊕ ∩ Ωk ) ∩ Θ = ∅. Since Dab(Θ) is a minimal Dabk -consequence of Γ, ϕ⊕ ∩ Ωk 6∈ Φk (Γ). This contradicts (‡). Suppose ϕ⊕ is not a minimal choice set of Σ(Γ). Since ϕ⊕ is a choice set of Σ(Γ), this implies that there is a ψ ∈ Φ(Γ) = ψ ⊂ ϕ⊕ . Hence there is a A ∈ ϕ⊕ − ψ, where A ∈ Ωk for a k ∈ I. However, since by (‡), ϕ⊕ ∩ Ωk ∈ Φk (Γ), we have by Lemma 2 that there is a minimal Dabk -consequence Dab(Θ) of Γ, for which Θ ∩ ϕ⊕ = A, hence Θ ∩ ψ = ∅. This contradicts the fact that ψ is a choice set of Σ(Γ). As a result, ϕ⊕ ∈ Φ(Γ). : Ab(M 00 ) = ϕ⊕ . By By Lemma 4 from [3], there is an M 00 ∈ MLLL Γ m Theorem 8 and (‡), this model is an ALi -model of Γ, for every i ∈ I. Hence by Theorem 14, it is a HALm -model of Γ. Furthermore, since Abi (M 00 ) = ϕ, Abi (M 00 ) ⊂ Abi (M ). By Fact 2 and Theorem 3, we immediately get: r

r

Corollary 2 If M ∈ MLLL − MHAL , then there is an M 0 ∈ MHAL such Γ Γ Γ 0 that for an i ∈ I, Abi (M ) ⊂ Abi (M ). Corollary 3 If Γ has LLL-models, it has HALx -models. (Reassurance)

7.4

Fixed point?

Proposition 2 There is a Γ for which CnHALr (CnHALr (Γ)) 6= CnHALr (Γ). Proof. Consider the hierarchic logic HKr from Section 2. Define Γ = {♦p, ♦♦q, ♦♦r, ¬p ∨ ¬q, ¬r ∨ ¬q}. There are no minimal Dab1 -formulas; the minimal Dab2 -formulas are: (♦p ∧ ¬p) ∨ (♦♦q ∧ ¬q) (♦♦r ∧ ¬r) ∨ (♦♦q ∧ ¬q) Hence U 1 (Γ) = ∅ and U 2 (Γ) = {♦p ∧ ¬p, ♦♦q ∧ ¬q, ♦♦r ∧ ¬r}. It is easy to see that Γ 0HKr r: consider the model M that verifies p, ¬q, ¬r. This model is a Kr1 -model of Γ and a Kr2 -model of Γ, hence it is a HKr -model of Γ. (Note also that U ? (Γ) = {♦♦q ∧ ¬q, ♦♦r ∧ ¬r}.) The only minimal Dab-consequence of CnHKr (Γ) is (♦♦q ∧ ¬q). That is, since ¬(♦p ∧ ¬p) is Kr1 -derivable from Γ, it is HKr -derivable from Γ, and hence ♦♦q ∧ ¬q ∈ CnHKr (Γ). Hence ♦♦r ∧ ¬r is no longer considered unreliable when the consequence relation is iterated, thus CnHKr (Γ) `HKr r. 15 This

set was defined on page 10.

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We can prove that HALm has the fixed point property, whenever Φ(Γ) is finite. The proof relies on the restricted soundness and completeness result for HALm and on Fact 5 from Section 7.1. Theorem 22 If Φ(Γ) is finite, then CnHALm (CnHALm (Γ)) = CnHALm (Γ). (Restricted Fixed Point for HALm ) Proof. If A ∈ CnHALm (Γ), then A ∈ CnHALm (CnHALm (Γ)) by Theorem 19. For the other direction, suppose Φ(Γ) is finite, whence by Theorem 18 A ∈ CnHALm (Γ) iff Γ |=HALm A. Suppose A 6∈ CnHALm (Γ). Hence there is m a model M ∈ MHAL , such that M 6|= A. By Definition 14: for every i ∈ I, Γ m ALi M ∈ MΓ , hence for no i ∈ I, M 0 ∈ MLLL : Abi (M 0 ) ⊂ Abi (M ). By Fact 5: Γ m 0 LLL 0 (1) for no i ∈ I, M ∈ MCnHALm (Γ) : Abi (M ) ⊂ Abi (M ). Since M ∈ MHAL , Γ ALm

i also (2) M ∈ MLLL CnHALm (Γ) . By (1), (2) and Definition 2: M ∈ MCnHALm (Γ) m for every i ∈ I. By Definition 14, M ∈ MHAL CnHALm (Γ) . Since M 6|= A, it follows by Theorem 14 that A 6∈ CnHALm (CnHALm (Γ)).

However, HALm does not have the fixed point property in general. An example is the Γ from Section 6.4. We have already seen there that Γ 0HK.2m r. However, since for every i ∈ N, r∨!1 pi ∈ CnHK.2m (Γ), we have that CnHK.2m (Γ) `Km r, hence CnHK.2m (Γ) `HK.2m r. 1

7.5

Equivalent Premise Sets and Maximality of the LLL

For any logic L, we say that Γ and Γ0 are L-equivalent iff CnL (Γ) = CnL (Γ0 ). The following theorems were proven in [10]: Theorem 23 If L is a Tarski-logic weaker than ALx , and Γ and Γ0 are Lequivalent, then Γ and Γ0 are ALx -equivalent. Theorem 24 If L is a Tarski-logic weaker than ALx , and Γ and Γ0 are Lequivalent, then Γ ∪ ∆ and Γ0 ∪ ∆ are ALx -equivalent. Theorem 25 For all monotonic logics L weaker than ALx and for all Γ, CnL (Γ) ⊆ CnLLL (Γ). (Maximality of the LLL) As explained in [10], these theorems (and others) show the strength of the adaptive logic approach to non-monotonic reasoning, compared to other approaches. Since hierarchic adaptive logics were developed with the same general goal in mind, we should see whether the above listed properties hold for them as well. As shown below, this is the case. I will reverse the order of the theorems: I first prove the theorem about the maximality of the lower limit logic, and use this theorem to establish the two theorems about the equivalence of certain premise sets. To get there, we first need two definitions and two lemmas. LLL Definition 16 Where ϕ ∈ Φ(Γ), Mϕ , Ab(M ) = ϕ}. Γ = {M | M ∈ MΓ

ˇ Lemma 16 If all members of Mϕ Γ verify A, then Γ `LLL A ∨ Dab(∆) for a ∆ ⊆ (Ω − ϕ).

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Proof.16 Define Θ¬ˇ = {ˇ ¬ B | B ∈ Θ}. Suppose all members of Mϕ Γ verify A. Hence all LLL-models of Γ∪(Ω−ϕ)¬ˇ verify A. This implies by the completeness of LLL with respect to its semantics: Γ∪(Ω−ϕ)¬ˇ `LLL A. By the compactness of LLL, Γ0 ∪ (∆)¬ˇ `LLL A, for a finite Γ0 ⊆ Γ and a finite ∆ ⊆ (Ω − ϕ). By ˇ Dab(∆), and by the monotonicity of LLL, the Deduction Theorem, Γ0 ` A ∨ ˇ Dab(∆). Γ `LLL A ∨ To prepare for Theorem 26 below, we have to introduce a new set of minimal choice sets:17 Definition 17 Φ? (Γ) = {ϕ ∈ Φ(Γ) | ∀i ∈ I : ϕ ∩ Ωi ∈ Φi (Γ)}. Lemma 17 If A ∈ CnHALm (Γ), then for every ϕ ∈ Φ? (Γ), there is a ∆ ⊂ Ω: ˇ Dab(∆) and ∆ ∩ ϕ = ∅. Γ `LLL A ∨ Proof. Suppose A ∈ CnHALm (Γ) and consider a ϕ ∈ Φ? (Γ). Consider an m M ∈ Mϕ Γ . In view of Definition 17, M is a ALi -model for every i ∈ I. By m Definition 14, M is a HAL -model of Γ, whence M |= A by Theorem 14. As a result: for every M ∈ Mϕ Γ : M |= A. By Lemma 16, there is a ∆ ⊆ (Ω − ϕ): ˇ Dab(∆). Γ `LLL A ∨ Theorem 26 For all monotonic logics L weaker than HALm and for all Γ, CnL (Γ) ⊆ CnLLL (Γ). Proof. Immediate in view of (i) Lemma 17, (ii) the proof of Theorem 10 in [10]—replace Φ(Γ ∪ Γ0 ) by Φ? (Γ ∪ Γ0 ) and Theorem 4 by Lemma 17. In view of Theorem 20, we immediately obtain: Theorem 27 For all monotonic logics L weaker than HALx and for all Γ, CnL (Γ) ⊆ CnLLL (Γ). Theorem 28 If L is a Tarski-logic weaker than HALx , and Γ and Γ0 are Lequivalent, then Γ and Γ0 are HALx -equivalent. Proof. Suppose L is a Tarski-logic weaker than HALx . By Theorem 27: L is a Tarski-logic weaker than or equipowerful to LLL. By Theorem 23, for every i ∈ I: Γ and Γ0 are ALxi -equivalent. Hence for every i ∈ I: CnALxi (Γ) = CnALxi (Γ0 ). Hence by Definition 9, CnHALx (Γ) = CnHALx (Γ0 ). Theorem 29 If L is a Tarski-logic weaker than HALx , and Γ and Γ0 are Lequivalent, then Γ ∪ ∆ and Γ0 ∪ ∆ are HALx -equivalent. Proof. Suppose the antecedent holds. Since L is a Tarski-logic, Γ ∪ ∆ and Γ0 ∪ ∆ are L-equivalent. By Theorem 28, Γ ∪ ∆ and Γ0 ∪ ∆ are HALx -equivalent. 16 This 17 In

proof is a copy of a part of the proof of Theorem 13 of [8]. [5, Chapter 6], the set I define here is referred to as the set of “orderly ϕ’s”.

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8

Going Hierarchic: The Gain

Let us return to a point made in Section 3. There it was stated that hierarchic adaptive logics impose a hierarchy on the set of abnormalities Ω. The logic HKx is but one example; another example is the inductive logic HGx , that gives preference to stronger generalizations over weaker ones [4]. Yet other examples can be easily constructed, when making hierarchic variants of the combined adaptive logics in [14], [13], [15], [17], [16] and [2]. In all these cases the same basic mechanism as explained in Section 5 renders the systems their hierarchic flavor. It was also stated that hierarchic adaptive logics are often stronger than x their flat counterpart. This point S requires clarification. Recall that AL was defined by (i) LLL, (ii) Ω = i∈I Ωi and (iii) the strategy x ∈ {r, m}. First of all, where the strategy is Reliability, the hierarchic logic is always stronger than its flat nephew, as is stated by the following theorem and the subsequent proposition: Theorem 30 CnALr (Γ) ⊆ CnHALr (Γ). Proof. Suppose (finite) ∆ ⊂ Ω which ∆ ⊂ Ωi . A ∈ CnALri (Γ).

ˇ Dab(∆), for a A ∈ CnALr (Γ). By Theorem 6, Γ `LLL A ∨ such that ∆ ∩ U (Γ) = ∅. Consider the smallest i ∈ I for Since U i (Γ) ⊆ U (Γ), ∆ ∩ U i (Γ) = ∅. Hence by Theorem 6, By Definition 9 and the reflexivity of LLL, A ∈ CnHALr (Γ).

Proposition 3 There is a Γ for which CnALr (Γ) ⊂ CnHALr (Γ). Proof. Define Kr by (i) LLL = K, (ii) Ω = {♦i A∧¬A | i ∈ I, A ∈ W a } and (iii) the Reliability Strategy. Define HKr as in Section 3.2. Now recall the example from Section 3.2: Γ = {♦p, ♦♦q, ♦♦r, ¬p ∨ ¬r}. Since ♦p ∧ ¬p ∈ U (Γ), we have that Γ 0Kr p, while it was shown in Section 5 that Γ `HKr p. Hence it is the Minimal Abnormality Strategy that enforced the use of the adverb “often”. First of all, when Φ(Γ) is finite, the counterpart of Theorem 30 can be easily proven: Theorem 31 Where Φ(Γ) is finite, CnALm (Γ) ⊆ CnHALm (Γ). Proof. Suppose Φ(Γ) is finite and A ∈ CnALm (Γ). Let Φ(Γ) = {ϕ1 , . . . , ϕn }. By Theorem 7: (?) for every ϕj with j ∈ {1, . . . , n}, there is a ∆j ⊂ Ω such ˇ Dab(∆j ). Consider the smallest i ∈ I for that ∆j ∩ ϕj = ∅ and Γ `LLL A ∨ which (∆1 ∪ . . . ∪ ∆n ) ⊆ Ωi . By Lemma 5, for every ϕ ∈ Φi (Γ), there is a ϕj with j ∈ {1, . . . , n} such that ϕj ∩ Ωi = ϕ. By (?): for every ϕ ∈ Φi (Γ), there ˇ Dab(∆j ) and ∆j ∩ ϕ = ∆j ∩ (ϕj ∩ Ωi ) = is a ∆j ⊂ Ωi such that Γ `LLL A ∨ ∆j ∩ ϕj = ∅. By Theorem 7, Γ `ALim A. By Definition 9 and the reflexivity of LLL, A ∈ CnHALm (Γ). Of course, there are Γ for which CnALm (Γ) ⊂ CnHALm (Γ) – again, see the example in Section 3.2. As the proofs indicate, a formula A that is derivable from the premise set by the logic ALx , is in most cases derivable by one of the logics ALxi from the combined logic HALx . However, this does not hold in general: 27

Proposition 4 There are Γ, A: A ∈ CnALm (Γ), while A 6∈ CnHALm (Γ). For reasons of space, the example will not be presented here. The motor behind it, is that infinitely many conditions, belonging to infinitely many different Ωi ’s are necessary to derive the formula A. As a result, no logic ALm i can yield A, whereas the flat adaptive logic that takes as its set of abnormalities Ω = Ω1 ∪ Ω2 ∪ . . . does yield A. Note that this difficulty can only arise for hierarchic logics that are built up from an infinite number of flat adaptive logics – in the other cases, ALx is the last flat logic that went into the combination. Hence if there are finitely many flat adaptive logics in the combination, the hierarchic adaptive logics are always at least as strong as their flat nephews. For those who might still see Proposition 4 as a drawback of hierarchic adaptive logics, note that we can simply change Definition 9, adding CnALx (Γ) inside the brackets, to ensure that HALx is always at least as strong as ALx . This would of course result in a new logic, but it would not do any harm to the meta-theoretic results, and the proof theory and semantics of the resulting system can be easily derived from the one defined in this paper.

9

In Conclusion

As indicated in the introduction, combinations of adaptive logics are a necessary tool to accomplish the adaptive logics project: to explicate non-monotonic reasoning processes. However, they have hitherto lagged behind flat adaptive logics, when it comes to the investigation of their universal properties. Now that our picture of flat adaptive logics has been clarified in several studies [3, 7, 6, 10, 18], it has become time to turn to the combined logics, with the aid of the metatheory of the flat logics themselves. This paper is but one step in the direction of this more general goal. It shows that such investigations are not only useful from the point of view of concrete applications, but also that they can lead to fruitful new insights into the properties of (combined and flat) adaptive logics. Lemma 15 provides an example of this double dynamics: it concerns a property of flat adaptive logics that was established in order to prove a theorem about hierarchic adaptive logics. During the many discussions within the Ghent group that guided the research on this paper, we also discovered new facts about other combinations of adaptive logics.18 Further research should focus on both the hierarchic approach, and these other combinations – notoriously the combinations by superposition and the ordered fusions of adaptive logics, as described in [5, Chapter 6]. The ultimate aim of such work should be to investigate the meta-theory of these systems, but also to simplify their proof theory and semantics as much as possible.

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[5, Chapter 6] for details.

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