Adaptive Logic as a Modal Logic

Patrick Allo Adaptive Logic as a ModalLogic

Abstract. Modal logics have in the past been used as a unifying framework for the mini-

mality semantics used in defeasible inference, conditional logic, and belief revision. The

main aim of the present paper is to add adaptive logics, a general framework for a wide

range of defeasible reasoning forms developed by Diderik Batens and his co-workers, to the

growing list of formalisms that can be studied with the tools and methods of contemporary

modal logic. By characterising the class of abnormality models, this aim is achieved at the

level of the model-theory. By proposing formulae that express the consequence relation

of adaptive logic in the object-language, the same aim is also partially achieved at the

syntactical level.

Keywords: Adaptive logic, Modal logic, Preference logic, Nonmonotonic inference.

1. Introduction

Adaptive logics have evolved from systems for handling inconsistent premises[4–6] to a general framework for all kinds of defeasible reasoning. The stan-dard format developed in [8] subsumes all adaptive logics for non-prioritisedpremise-sets. Modal logics have gone through a similar evolution. Theywere originally conceived as an analysis of alethic modalities, but have nowbecome the privileged language to reason about all kinds of relational struc-tures [10, p. xii].

One field where modal logics have been used as a unifying framework isin the analysis of what [14] describes as the different “faces of minimality”in defeasible inference, conditional logic, and belief revision. Modal transla-tions of so-called minimality semantics are found in [11], and more recentlyin [16]. Given the hypothesis that the standard format of adaptive logic issufficiently general to incorporate most (if not all) forms of defeasible infer-ence [9, Chap. 1], it is natural to ask whether the consequence relation ofadaptive logic can also be embedded in a modal language. Such a modalreconstruction can be given because the consequence relations for adaptivelogics can be formalised as formula-preferential systems in the sense of [1].

Presented by Heinrich Wansing; Received October 23, 2011

Studia Logica (2012)DOI: 10.1007/s11225-012-9403-1 c© Springer Science+Business Media B.V. 2012

P. Allo

That is, models are ordered on the basis of the formulae they verify, andthe ordering is used to select a subset of the models of the premises. Thissubset is then used to define the semantic consequence relation. By definingKripke-style models wherein states are ordered in a similar way, we can thenselect states in the same manner as we selected models.

Insights and results laid out in the present paper are essentially prepara-tory. The long term aim of this project is to integrate adaptive consequencerelations into dynamic doxastic logics (in the style of [2]). The short termaim is more modest, as we merely wish to understand how the Kripke-stylemodels that are needed to obtain a faithful modal reconstruction of adap-tive consequence relations differ from the Kripke-style models that have beenused to reformulate other types of minimality semantics.The present paper is primarily concerned with the model-theoretic aspects ofthe modal embedding of adaptive logic. Sections 2 and 3 contain the prelim-inaries: The standard format of adaptive logic is described (Sect. 2), and thedesiderata and formal features of the target systems for a modal reconstruc-tion are reviewed (Sect. 3). The core of the reconstruction is in Sect. 4, andproofs for the correctness of the construction are given in Sect. 5. A specificexample follows in Sect. 6. The one exception to the purely model-theoreticapproach to the modal reconstruction of adaptive logic is the discussionof modal formulae that express the adaptive consequence relation. This isthe topic of Sect. 7. A final section (Sect. 8) situates the results that wereobtained in a broader context, and lists what remains to be done.

2. Adaptive Logics: Purpose and Standard Format

Adaptive logics were initially introduced as a non-monotonic strengthen-ing of standard paraconsistent logics. These so-called inconsistency-adaptivelogics are meant to overcome the main flaws of paraconsistent reasoning;for instance, the inability to account for intuitively acceptable uses of thedisjunctive syllogism. Inconsistency-adaptive logics solve this problem byinvalidating the instances of such argument forms rather than the argumentform itself. The functioning of adaptive logics has two sides: a formula-preferential semantics and a dynamic proof-theory. The former is the focusof the present paper. The latter is more closely connected to the core aim ofadaptive logics; namely, the explication of actual reasoning processes. A gen-eralisation of how inconsistency adaptive logics are defined can be appliedto many reasoning processes, and is characterised by the standard formatfor adaptive logics.

Adaptive Logic as a Modal Logic

2.1. Standard Format

An adaptive logic is characterised by a triple that comprises:

1. A Tarski-logic referred to as the lower limit logic (LLL),

2. a set of formulae Ω characterised by a logical form and referred to asthe set of abnormalities, and

3. a criterion, referred to as the adaptive strategy, which (in its model-the-oretic form) selects models of premise-sets that are no more abnormalthan what is actually required by that premise-set.

Adaptive consequence relations are defined over a base-language L0. To sim-plify meta-theoretical proofs, this language is customarily extended with aclassical negation (¬), disjunction (∨), and absurdity-constant (⊥). I shallrefer to the resulting language as L1.

The lower limit logic (LLL) is a logic that is defined over the language L1,enjoys all the usual Tarskian properties, is compact and includes classicallogic. An LLL-model of a premise-set Γ is defined as an L1-model of Γ. Wesay that ϕ is an LLL-consequence of Γ (formally: Γ |=LLL ϕ) iff ϕ is verifiedby all LLL-models of Γ.

The set of abnormalities Ω is a set of L1-formulae characterised by a (pos-sibly restricted) logical form that contains at least one logical connective.

Adaptive consequence relations are obtained by only considering LLL-models of the premises that do not verify more members of Ω than needed.There are multiple non-equivalent criteria to ensure this. We only considertwo of them: the minimal abnormality strategy and the reliability strategy.

The comparison of LLL-models is formalised in terms of their abnormalparts.

Definition 1. (Abnormal Part of a Model) Ab(M) = {ϕ ∈ Ω : M � ϕ}.A first way of selecting models by looking at their abnormal parts pro-

ceeds by selecting only the normal models.

Definition 2. (Normal Model) An LLL-model M of Γ is a normal modeliff its abnormal part Ab(M) is empty.

A more interesting selection is the minimal abnormality strategy, whichselects only those LLL-models (of a premise-set) that are minimal relativeto an ordering of models based on their abnormal parts:

Definition 3. (Minimally Abnormal Model) An LLL-model M of Γ is min-imally abnormal iff there is no LLL-model M′ of Γ such that Ab(M′) ⊂Ab(M).

P. Allo

Because the ordering of the models is not total, the above selection ofmodels is not the only sensible one. A more cautious alternative to the onegiven above is the reliability strategy.

Definition 4. (Dab-formulae and (minimal)Dab-consequences) A disjunc-tion of abnormalities Dab(Δ) is the disjunction of the members of a finiteΔ ⊆ Ω. Dab-consequences of Γ, are the Dab-formulae verified by all LLL-models of Γ. A Dab-consequence Dab(Δ) is minimal iff there is no Θ ⊂ Δsuch that Dab(Θ) is also a Dab-consequence.

Definition 5. (Unreliable Formulae) If Dab(Δ1), Dab(Δ2), . . . are the min-imal Dab-consequences of Γ, then U(Γ) = Δ1∪Δ2∪· · · is the set of formulaethat are unreliable with respect to Γ.

Definition 6. (Reliable Model) An LLL-model M of Γ is reliable iffAb(M) ⊆ U(Γ).

The following proposition, which we shall use later on, is a corollary ofLemma 4 from [8] and relates the abnormalities verified by minimally abnor-mal models to the set of unreliable formulae.

Proposition 1. If Γ has LLL-models, then ω ∈ Ω belongs to U(Γ) iff it isverified by some minimally abnormal model of Γ.

This immediately implies that the consequence relation for minimalabnormality is at least as strong as the consequence relation for reliabil-ity. The following generic example illustrates why the reliability-strategy ismore cautious than the minimal-abnormality strategy.

Example 1. Let Γ be a premise-set, and ω1 and ω2 be two abnormalitiessuch that for some adaptive consequence relation AL with a logic L as itslower limit logic we have Γ |= ω1 ∨ ω2 while Γ �|= ω1 and Γ �|= ω2. Assume,without loss of generality, that M, M′ and M′′ are L-models of Γ such thatAb(M) = {ω1}, Ab(M′) = {ω2} and Ab(M′′) = {ω1, ω2}. In that case, Mand M′ are minimally abnormal (as well as reliable) models of Γ, whereasM′′′ is a reliable but not a minimally abnormal model of Γ.

Definition 7. (Semantic Consequence for Minimal Abnormality) Γ |∼m ϕiff ϕ is verified in all minimally abnormal models M of Γ.

Definition 8. (Semantic Consequence for Reliability) Γ |∼r ϕ iff ϕ is ver-ified in all reliable models M of Γ.

Definition 9. (Upper Limit Logic) Γ |=ULL ϕ iff ϕ is verified by all normalmodels M of Γ.

Both |∼m and |∼r reduce to |=ULL whenever Γ has normal models.


A meta-theoretical property of adaptive logics that should be mentionedat this point is the property of strong-reassurance.

Theorem 1. (Strong Reassurance for Minimal Abnormality) If M is anLLL-model of Γ that is not minimally abnormal, then there is a minimallyabnormal model M′ of Γ such that Ab(M′) ⊂ Ab(M).

What this property ensures is that there are no infinite paths of lessand less abnormal models of a premise-set (cfr. the so-called limit-assump-tion form conditional logic. For a general discussion see [1] and [7]). Theproperty of reassurance, which is a consequence of strong reassurance, saysthat whenever a premise-set has LLL-models, there is also a selection ofminimally abnormal models of that premise-set.

3. What Kind of Reconstruction?

The main difference between the standard format of adaptive logic and itsintended reformulation within a modal logic is that the former is based onthe selection of LLL-models of Γ, whereas the latter is based on the selec-tion of states in the truth-set of Γ in a class of Kripke-style models. In thissection we describe a specific type of model that is particularly well-suitedfor the purpose at hand, and propose two correctness conditions.

3.1. Preference Models and Preference Languages

Since preference languages have already been used to reduce the conditionalsof a basic conditional logic to a combination of two unary modalities [16], weextend the language L1 (which we used to define LLL-models) accordinglyand use preference models as the intended class of models for that language.

Definition 10. (Basic Preference Language) The set of formulae of thebasic preference language L2 is defined by:

ϕ ::= ψ| ⊥ |¬ϕ|ϕ1 ∨ ϕ2|Uϕ|♦�ϕ

with ψ ranging over the formulae of a base-language L0 (or a set of proposi-tional atoms). The dual modal operators E and �� are standardly defined.

Definition 11. (Preference Frame) A preference frame is a pair F = (S,)with S a non-empty set of states, and a pre-order on S.

Definition 12. (Preference Models) A preference model is a three-tupleM = (S,, ‖·‖M) with (S,) a preference frame, and ‖·‖M : Form �→ P(S)a valuation-function (with Form the set of L0-formulae).

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Satisfaction is standardly defined. We only mention the clauses for themodalities.

– M, s � Uϕ iff M, s′ � ϕ for all s′ ∈ S,

– M, s � Eϕ iff M, s′ � ϕ for some s′ ∈ S,

– M, s � ��ϕ iff M, s′ � ϕ for all s′ s ∈ S,

– M, s � ♦�ϕ iff M, s′ � ϕ for some s′ s ∈ S.

To facilitate the reference to (possibly infinite) semantic propositions we liftthe identity s ∈ ‖ϕ‖M iff M, s � ϕ to arbitrary sets of L1-formulae. Whenϕ is true at all states in a model, we write M � ϕ instead of M, s � ϕ.

A natural extension of the bi-modal language L2 consists in the additionof the strict preference operators ♦≺ and �≺ with the following satisfac-tion-clauses:

– M, s � �≺ϕ iff M, s′ � ϕ for all s′ ≺ s ∈ S,

– M, s � ♦≺ϕ iff M, s′ � ϕ for some s′ ≺ s ∈ S.

Where the requirement that ≺ be a strict sub-relation of is captured by:

s ≺ s′ iff s s′ and s′ � s (≺-adequacy)

When it comes to reconstructing the minimal conditional logic, all that needsto be done is to come up with a formula of the preference-language that isequivalent to ϕ > ψ (with > the conditional). As explained in [11] and [16],this is achieved by the following formula:1

U(ϕ → ♦�(ϕ ∧ ��(ϕ → ψ))) (ESI)

A reconstruction based on the language and models of a basic preferencelogic isn’t sufficient for the intended reconstruction of the adaptive con-sequence relation. First, unlike conditional logics, adaptive logics are notdefined relative to any pre-ordering of models (in the standard format) orstates (in the intended reconstruction). LLL-models of Γ are ordered in termsof the abnormalities they verify. We should therefore try to achieve a simi-lar ordering of the states in a preference model. Second, the conditional isa connective, but |∼ is not. In particular, the antecedents can be sets offormulae rather than single formulae.

1The label we use for this formula refers to the fact that it captures the so-called “even-tual strict implication” condition rather than the minimality-condition for the truth of aconditional. The former approach is unavoidable whenever the limit-assumption fails. Weshall come back to this issue in Sect. 7.


3.2. Correctness Conditions

We consider two correctness conditions, a semantic and a syntactic one. Thesemantic correctness condition expressed by (AL) is only concerned with theextension of the modal reconstruction of the adaptive consequence relation.

Γ |∼x ϕ iff ‖Γ‖xM ⊆ ‖ϕ‖M for all M ∈ M (AL)

with x ∈ {m, r}, and ‖Γ‖xM ⊆ ‖Γ‖M.

The syntactical correctness condition bears on additional definability andexpressivity issues. Namely, for every Γ and ϕ of the language L1 thereshould be a ψ (not in L1) such that (AL*) holds for x ∈ {m, r}.

‖Γ‖xM ⊆ ‖ϕ‖M iff M � ψ (AL*)

In combination with (AL) this amounts to the requirement that there shouldbe a formula ψ of the language L2 (or some further extension of that lan-guage) that is globally valid (i.e. true at all states in all models of M) iff ϕis an adaptive consequence of Γ.

4. Abnormality Models

As a preliminary, we define what it means for a class of preference modelsto generate a certain logic.

Definition 13. Where M is a class of preference models and L a logicdefined over a language L, we say that the class of models M generates thelogic L iff

‖Γ‖M ⊆ ‖ϕ‖M for all M ∈ M iff Γ |=L ϕ

for all Γ ∪ {ϕ} that only contain L-formulae.

For now, we are primarily interested in the L1-fragments of the modallogics generated by M.

Definition 14. (Abnormality Models) An abnormality model is a 3-tupleM = (S,Ω, ‖·‖M) where S is a set of states, Ω the set of abnormalities, and‖·‖M a valuation-function. We define a function AbM : S �→ P(S), and abinary relation over S in accordance with the following clauses:

1. AbM(s) = {ω ∈ Ω : s ∈ ‖ω‖M}2. s s′ =⇒ AbM(s) ⊆ AbM(s′)

3. AbM(s) ⊆ AbM(s′) =⇒ s s′ and restrict the valuation such as tocomply with a last clause:

P. Allo

4. For every proposition ‖Γ‖M ⊆ S and every s ∈ ‖Γ‖M, if for someΔ ⊂ AbM(s), we have Γ ∪ {¬ϕ : ϕ ∈ Ω \ Δ} �|=LLL⊥, then there is ans′ ∈ ‖Γ‖M such that AbM(s′) = Δ.

In the above definition the Clauses (1) and (2) warrant that is an order-ing that only depends on the abnormalities true at a state, while Clause (3)ensures that every difference (and identity) in abnormality between statesis captured by the order-relation . Finally, Clause (4) forces the presenceof “sufficiently normal states” in every proposition.2

Because abnormality models give rise to preference-models (the definitionof ensures that it is a pre-order) that comply with an additional restric-tion on the admissible valuations on a preference frame (which technicallymakes it a general frame) we should verify whether the bi-conditional (forformulae in L1) in Definition 13 is preserved when the restrictions imposedby the clauses of Definition 14 are applied.

Proposition 2. Where M1 and M2 are, respectively, the class of all pref-erence models and the class of all abnormality models, both classes generatethe same logic over the language L1 (equivalently: the logics generated byboth classes have the same propositional fragment).

Proof. We need to prove that ‖Γ‖M ⊆ ‖ϕ‖M (where Γ∪{ϕ} contains onlyL1-formulae) holds for all M ∈ M1 iff it holds for all M ∈ M2.=⇒ Immediate in view of the fact that M2 ⊂ M1.⇐= We prove the contrapositive. Assume that ‖Γ‖M �⊆ ‖ϕ‖M for someM ∈ M1. That is, some s ∈ ‖Γ‖M does not verify ϕ. If M is also in M2 theresult is trivial, so we only consider the case where M isn’t an abnormalitymodel. If M isn’t an abnormality model because it fails to satisfy Clauses(1), (2), or (3), it is easily seen that there is an abnormality model M′ thatonly differs from M with respect to the extension of . Since Γ and ϕ are L1-formulae, it immediately follows that some s ∈ ‖Γ‖′

M won’t verify ϕ either.If M isn’t an abnormality model because it (also) fails to satisfy Clause (4),we only need to extend it to a model M′ = (S′,′,Ω, AbM′ , ‖·‖M′) thatdoes satisfy Clause (4) (and, if necessary, also the remaining clauses). Sincethis can be done in such a way that M is a sub-model of M′ (in particular:S ⊂ S′, and for all p we have ‖p‖M = S ∩ ‖p‖M′), the fact that ϕ and Γ do

2An alternative to Clause (4) would be the following: Whenever Γ �|=LLL⊥, we have‖Γ‖M �= ∅ for all abnormality models. This would ensure that for every maximally non-trivial set of L1-formulae the corresponding infinite proposition is non-empty in everyabnormality model. This approach is dismissed for being unnecessarily restrictive.


not contain modal formulae suffices to ensure that for any s ∈ ‖Γ‖M thatdoes not verify ϕ there is an s ∈ ‖Γ‖′

M that does not verify ϕ either.

Proposition 3. If M is a class of preference models that generates a logicL over the language L1, then for all M ∈ M it holds that s ∈ ‖Γ‖M iff theL1-theory of M, s (the set of all L1-formulae true at s) is the L1-theory ofsome L-model of Γ (the set of all L1-formulae verified by M).

Proof. If M generates L, we have ‖Γ‖M ⊆ ‖ϕ‖M for all M ∈ M iff Γ |=L ϕ.By substituting ⊥ for ϕ we obtain Γ �|=L⊥ iff ‖Γ‖M is non-empty for someM ∈ M, which shows that there is a model M of Γ iff Γ is also satisfiablein a model M ∈ M.⇐= Let Θ be the L1-theory of some L-model M of Γ. From the above, itfollows that Θ can be satisfied in some M ∈ M and thus that some s in thatmodel will verify all members of Θ. Since Γ ⊆ θ this s is also in ‖Γ‖M.=⇒ Let Θ be the L1-theory of M, s with s ∈ ‖Γ‖M. From the above itfollows that Θ must have an L-model M, while Γ ⊆ Θ warrants that Mis also an L-model of Γ. Hence, we have an L-model of Γ that verifies allmembers of Θ.

To find out whether the class of abnormality models complies with (AL),we now need to define the different ways of selecting states in terms of .As a preliminary, we define the relation of abnormality-comparability.

Definition 15. (Ab-comparable) The states s and s′ are Ab-comparable(formally, s ∼ s′) iff s s′ or s′ s.

Note that the relation of Ab-comparability is not an equivalence-relation;it is reflexive and symmetric, but not transitive.

Definition 16. (Minimally abnormal states) The set of minimally abnor-mal states in ‖Γ‖M is given by:

‖Γ‖mM = {s ∈ ‖Γ‖M : (s′ ∈ ‖Γ‖M & s ∼ s′) =⇒ s s′)}

The definition of minimally abnormal states follows quite closely the orig-inal definition of minimally abnormal models. The only difference lies in theuse of an abnormality pre-ordering rather than a strict pre-order (but seeProposition 8 for an alternative). For the definition of reliable states wefollow a different approach.

Definition 17. (Reliable States) The set of reliable states in ‖Γ‖M isgiven by:

P. Allo

‖Γ‖rM = {s ∈ ‖Γ‖M : ∀ω ∈ Ω(s ∈ ‖ω‖M ⇒ ∃s′ ∈ ‖Γ‖m

M & s′ ∈ ‖ω‖M)}To see how the above characterisation relates to an order-theoretic char-

acterisation in terms of , consider first the following definition.

Definition 18. (Minimal Upper Bound) Where 〈W,〉 is a pre-ordered setand X a subset of W , we say that u is an upper-bound of X in 〈W,〉 iffu ∈ W and x u for all x ∈ X. An upper-bound u of X in 〈W,〉 isminimal iff there’s no upper-bound u′ of X in 〈W,〉 such that u′ u andu � u′.

The following negative result shows that the set of reliable states cannotbe given a direct order-theoretic formulation in terms of .

Proposition 4. Let M = (S,Ω, ‖·‖M) be an abnormality model.

1. s ∈ ‖Γ‖rM �=⇒ s u for some minimal upper-bound u of an X ⊆ ‖Γ‖m

M

in ‖Γ‖M.

2. s ∈ ‖Γ‖rM �⇐= s u for some minimal upper-bound u of an X ⊆ ‖Γ‖m

M

in ‖Γ‖M.

Proof. (Ad1.) Let ‖Γ‖M = {s1, s2, s3} with Ab(s1) = {ω1, ω2}, Ab(s2) ={ω3, ω4}, and Ab(s3) = {ω1, ω2, ω3}. It is clear that s1 and s2 are the min-imally abnormal states in ‖Γ‖M and that s3 is a reliable state, while thereis no upper-bound u of any subset of {s1, s2} such that s3 u.(Ad2.) Consider a premise-set Γ = {ω1 ∨ω2, (ω1 ∧ω2) → ω3}, and let M bean abnormality model such that for some s, s′ ∈ ‖Γ‖M we have Ab(s) = ω1

and Ab(s′) = ω2, and thus s, s′ ∈ ‖Γ‖mM. Since Γ ∪ {ω1, ω2,¬ω3} cannot be

satisfied, there is no s′′ ∈ ‖Γ‖M such that Ab(s′′) = {ω1, ω2}. Hence, if u isa minimal upper-bound of {s1, s2} in ‖Γ‖M, then ω3 ∈ Ab(u). Yet, u u,but u is not a reliable state.

Proposition 5. ‖Γ‖mM ⊆ ‖Γ‖r

M

Proof. Immediate in view of Definition 17.

Proposition 6. If the restriction of to ‖Γ‖M is connected (for everys, t ∈ ‖Γ‖M we have s t or t s), then ‖Γ‖m

M = ‖Γ‖rM.

Proof. Assume that the restriction of to ‖Γ‖M is connected. In view ofProposition 5, we only need to show ‖Γ‖r

M ⊆ ‖Γ‖mM.

Let s, s′ be two states in ‖Γ‖mM. Since s ∼ s′ holds by assumption, it follows

from Definition 16 that s s′ and s′ s. In other words, all minimally


abnormal states in ‖Γ‖M have the same abnormal part, and only minimallyabnormal states can be reliable.

To see why the converse doesn’t hold, just consider a model where ‖Γ‖mM

and ‖Γ‖M coincide, while some s, t ∈ ‖Γ‖M have different and hence incom-parable abnormal parts.

To conclude this section, we briefly consider abnormality models with astrict abnormality ordering.

Definition 19. (Strict Abnormality Models) Where (S,Ω, ‖·‖M) is anabnormality model, we define a binary relation ≺ over S such that:

1. AbM(s) ⊂ AbM(s′) =⇒ s ≺ s′

2. s ≺ s′ =⇒ AbM(s) ⊂ AbM(s′)

Proposition 7. (≺-adequacy of strict abnormality-ordering) M = (S,Ω,‖·‖M) is a strict abnormality model iff: (i) it satisfies clauses (2) and (3)from Definition 14, and (ii) ≺ is the strict sub-relation of that satisfies≺-adequacy.

Proof. We only need to prove that AbM(s) ⊂ AbM(s′) ⇔ s ≺ s′ holdsfor all states iff ≺ is the strict sub-relation of that satisfies ≺-adequacy.This follows immediately from that fact that (i) we already have AbM(s) ⊆AbM(s′) ⇔ s s′, and (ii) ⊂ is itself the strict sub-relation of ⊆ such thatA ⊂ B iff A ⊆ B and B �⊆ A.

The strict ordering ≺ can be used to formulate an alternative defini-tion of minimally abnormal states by requiring that a state s is mini-mally abnormal in ‖Γ‖M iff there is no state s′ in ‖Γ‖M such that s′ ≺ s.As shown below, the alternative approach is equivalent to that of Defini-tion 16.

Proposition 8. A state s ∈ ‖Γ‖M is a minimally abnormal state iff thereis no s′ ∈ ‖Γ‖M such that s′ ≺ s.

Proof. The following claims are all equivalent:

1. For all s′ ∈ ‖Γ‖ that are Ab-comparable with s we have s s′.

2. For no s′ ∈ ‖Γ‖ we have s′ s and s � s′.

3. For no s′ ∈ ‖Γ‖ we have s′ ≺ s.

The crucial step is the equivalence of (2) and (3), which holds in virtue of≺-adequacy.

P. Allo

5. Correctness and Related Results

We first establish that versions of the properties of strong reassurance andreassurance (cfr. Theorem 1) do hold within the class of abnormality mod-els. The main reason why these results carry over to Abnormality Modelsis that Clause (4) from Definition 14 ensures that whenever there is a states ∈ ‖Γ‖M and an LLL-model M of Γ such that Ab(M) ⊂ Ab(s), there isalso an s′ ∈ ‖Γ‖M such that Ab(s′) = Ab(M).

Theorem 2. (Strong Reassurance for Abnormality Models) Whenever s ∈‖Γ‖M \ ‖Γ‖m

M, there is an s′ ∈ ‖Γ‖M such that s′ ∈ ‖Γ‖mM and s′ s.

Proof. Recall that in virtue of Proposition 3 every s ∈ ‖Γ‖M will have thesame L1-theory as some LLL-model of Γ.

Let M1 be the LLL-model with the same L1-theory as some s ∈ ‖Γ‖M \‖Γ‖m

M. Since s isn’t -minimal in ‖Γ‖M, there should be si’s in ‖Γ‖M suchthat si s and s � si. Since for every such si there must be an LLL-modelof Γ with the same L1-theory, M1 isn’t a minimally abnormal model of Γ.By Theorem 1, there must then be a minimally abnormal model M2 of Γsuch that Ab(M2) ⊂ Ab(M1). But then by Clause (4) of Definition 14 oneof the states in virtue of which s isn’t included in ‖Γ‖m

M must have the sameL1-theory as M2. Call this state sm. By Clause (3) it follows that sm sand by Clause (2) it follows that s � sm. What remains to be shown isthat sm is also -minimal in ‖Γ‖M and thus a member of ‖Γ‖m

M. Assume,for reductio, that there is an s′ ∈ ‖Γ‖M such that s′ sm and sm � s′.In view of clauses (1) and (3), this implies that AbM(s′) ⊂ AbM(sm), andbecause there should be an LLL-model M3 of Γ that verifies exactly thesame L1-formulae as s′, we would also have that Ab(M3) ⊂ Ab(M2). Thelatter is impossible if M2 is a minimally abnormal model of Γ.

Corollary 2.1. (Reassurance for Abnormality Models) If ‖Γ‖M is non-empty, then ‖Γ‖m

M is also non-empty.

We can now start with the actual proof of the correctness of our recon-struction of adaptive logic. We first show that (AL) holds for minimal abnor-mality.

Theorem 3. (Semantic Correctness for Minimal Abnormality) If M is theclass of all abnormality models that generates a logic with LLL as its L1-fragment, then (AL) holds relative to the adaptive consequence relation |∼m

with LLL as its lower limit logic.

Proof. The left-to-right direction follows immediately from Lemma 3.1,while the right-to-left direction follows from Lemma 3.2.


Lemma 3.1. For all s ∈ S we have s ∈ ‖Γ‖mM iff there is a minimally

abnormal model M of Γ such that Ab(M) = AbM(s).

Proof. =⇒ Assume that s ∈ ‖Γ‖mM. In view of Proposition 3 we only need

to prove that the LLL-model of Γ with the same L1-theory as s is itself aminimally abnormal model of Γ. Call this model M1. Assume, for reductio,that there is an M2 such that Ab(M2) ⊂ Ab(M1). This would mean that forsome Δ ⊂ AbM(s) we have Γ∪{¬ϕ : ϕ ∈ Ω\Δ} �|=LLL⊥, but then by Clause(4) of Definition 14 there should be an s′ ∈ ‖Γ‖M such that AbM(s′) = Δ.This implies that s′ ∈ ‖Γ‖M and AbM(s′) ⊂ AbM(s), and thus also s′ sand s � s′. The latter two contradict the assumption that s is a minimallyabnormal state in ‖Γ‖M.⇐= Immediate from the proof of Theorem 2.

Lemma 3.2. For every minimally abnormal model M of Γ, there is an abnor-mality model M such that for some s ∈ ‖Γ‖m

M we have Ab(M) = AbM(s).

Proof. Immediate in view of (the proof of) Theorem 2 and the fact thatall L1-formulae verified by some LLL-model can be satisfied at some pointin a preference model.

Theorem 4. (Semantic Correctness for Reliability) If M is the class of allabnormality models that generates a logic with LLL as its L1-fragment, then(AL) holds relative to the adaptive consequence relation |∼r with LLL as itslower limit logic.

Proof. The left-to-right direction follows immediately from Lemma 4.2,while the right-to-left direction follows from Lemma 4.3.

Lemma 4.1. For all s ∈ ‖Γ‖M, the state s is in ‖Γ‖rM iff AbM(s) ⊆ U(Γ).

Proof. =⇒ Assume that s is a reliable state in ‖Γ‖M. By Lemma 3.1 thereis for every t ∈ ‖Γ‖m

M a minimally abnormal model M of Γ with the sameL1-theory as t. By Proposition 1 every abnormality verified by such an M isan unreliable formula. Since AbM(t) = Ab(M), we have AbM(t) ⊆ U(Γ) forevery t ∈ ‖Γ‖m

M. Finally, by Definition 17, since every abnormality true at smust also be true at some t ∈ ‖Γ‖m

M we obtain AbM(s) ⊆ U(Γ) as required.⇐= Assume for reductio that AbM(s) ⊆ U(Γ) and s ∈ ‖Γ‖M, but s �∈ ‖Γ‖r

M.By Definition 17 this means that M, s � ω for some ω ∈ Ω that is not ver-ified by any s in ‖Γ‖m

M. But then, because by Lemma 3.1 s has the sameL1-theory as some minimally abnormal model of Γ, it follows by Proposition1 that ω �∈ U(Γ), which contradicts AbM(s) ⊆ U(Γ).

Lemma 4.2. For all s ∈ S we have s ∈ ‖Γ‖rM iff there is a reliable model M

of Γ such that Ab(M) = AbM(s).

P. Allo

Proof. =⇒ Assume that s ∈ ‖Γ‖rM. By Lemma 4.1, it then follows that

AbM(s) ⊆ U(Γ). Let M be an LLL-model of Γ that verifies exactly the sameL1-formulae as s. It then holds that Ab(M) = AbM(s) ⊆ U(Γ), and thus byDefinition 5 that M is a reliable model of Γ.⇐= We prove the contrapositive. Assume that s �∈ ‖Γ‖r

M. By Lemma 4.1, itthen follows that some ω ∈ AbM(s) does not belong to U(Γ). Let M be theLLL-model of Γ with the same L1-theory as s. It then holds that because ωis also verified by M, we have that Ab(M) �⊆ U(Γ), and thus by Definition5 that M isn’t a reliable model of Γ.

Lemma 4.3. For every reliable model M of Γ, there is an abnormality modelM such that for some s ∈ ‖Γ‖r

M we have Ab(M) = AbM(s).

Proof. Let M1 be an arbitrary reliable model of Γ and M a preferencemodel where some s ∈ ‖Γ‖M has the same L1-theory as M1. We then needto prove s ∈ ‖Γ‖r

M. From our construction, it follows that (i) Ab(M1) =AbM(s), (ii) Ab(M1) ⊆ U(Γ), and thus AbM(s) ⊆ U(Γ). Our result thenfollows by Lemma 4.1.

6. An Example: Inconsistency Adaptive Logic

We give a detailed account of the modal reconstruction of the inconsistencyadaptive logic CLuNm. The paraconsistent logic CLuN is used as a lowerlimit logic of CLuNm. It is defined over a language L0 with the follow-ing connectives {∼,∨,∧,⊃} (with ∼ a paraconsistent negation), which weextend to a language L1 with the remaining classical connectives (classicalnegation and absurdity constant).

The axiomatic presentation of the L0-fragment of CLuN is obtainedby adding excluded middle (ϕ∨ ∼ ϕ) to the positive fragment of propo-sitional classical logic. Except for the assignment-function (which rangesover both atomic propositions and negated formulae) and the clause forthe paraconsistent negation (v∼), the semantic characterisation is classical.

vM(∼ ϕ) = 1 iff vM(ϕ) = 0 or v(∼ ϕ) = 1 (v∼)

The set of abnormalities is the set Ω = {ϕ∧ ∼ ϕ : ϕ ∈ Form} of all explicitcontradictions. The combination of CLuN as a lower limit logic, and Ω asthe set of abnormalities suffices to conclude that the normal models of apremise-set are just its classical models, and that the upper limit logic is


classical propositional logic. As suggested by the notation, the strategy isminimal abnormality.3

We start by defining the class of abnormality models that generates theL0-fragment of the paraconsistent logic CLuN. The first step is to mod-ify the standard assignment-function for CLuN to obtain a mapping frompairs of propositional constants and states to truth-values, and from pairsof negated L0-formulae and states to truth-values:

– v : (Prop ∪ Form∼) × S �→ {0, 1},

We then modify the standard CLuN valuation-function accordingly:

– vM(ϕ, s) = 1 iff v(ϕ, s) = 1, for ϕ ∈ Prop,

– vM(∼ ϕ, s) = 1 iff vM(ϕ, s) = 0 or v(∼ ϕ, s) = 1,

– vM(ϕ1 ∨ ϕ2, s) = 1 iff vM(ϕ1, s) = 1 or vM(ϕ2, s) = 1,

– vM(ϕ1 ∧ ϕ2, s) = 1 iff vM(ϕ1, s) = 1 and vM(ϕ2, s) = 1,

– vM(ϕ1 ⊃ ϕ2, s) = 1 iff vM(ϕ1, s) = 0 or vM(ϕ2, s) = 1.

Given an abnormality model M = (S,Ω, ‖·‖M), we can then require that‖·‖M extends vM by stipulating that for all L0-formulae we have:

‖ϕ‖M = {s ∈ S : vM(ϕ, s) = 1}The classical behaviour of ¬ and ⊥ is ensured by the usual satisfaction-clauses. As before, we use ‖·‖M to refer to the truth-set of any L1-formulaor set of L1-formulae in a preference model M, and therefore lift the aboveidentity to the level of arbitrary sets of formulae.

‖Γ‖M = {s ∈ S : ϕ ∈ Γ ⇒ vM(ϕ, s) = 1}In the previous section we already showed that (AL) holds for both min-imal abnormality and reliability whenever the class of abnormality mod-els generates a logic with the lower limit logic as its L1-fragment. Spe-cifically, it follows from Theorem 3 that when a class M of abnormalitymodels generates a logic with CLuN as its propositional fragment, we havethat

Γ |∼ CLuNmϕ iff ‖Γ‖mM ⊆ ‖ϕ‖M for all M ∈ M (∗)

As a result, we only need the following proposition to prove (∗).

3The propositional fragment of CLuN is described in [3]; it’s first-order formulationand the adaptive logics based on it can be found in [5] and [9, Chap. 2].

P. Allo

Proposition 9. Where vM is a CLuN valuation-function, and M the classof abnormality models such that s ∈ ‖ϕ‖M iff vM(ϕ, s) = 1, then M gener-ates the L1-fragment of the logic CLuN.

Proof. =⇒ We prove the contrapositive. Assume that ‖Γ‖M �⊆ ‖ϕ‖M forsome M ∈ M. Let s be a state in M such that s is in ‖Γ‖M, but not in‖ϕ‖M. Since for all M ∈ M we have that s ∈ ‖ϕ‖M iff vM(ϕ, s) = 1, this isequivalent to saying that for some vM we have vM(ψ, s) = 1 for all ψ ∈ Γwhile vM(ϕ, s) �= 1. Let v′ be an assignment-function for CLuN such thatv′(ϕ) = 1 iff v(ϕ, s) = 1 for all ϕ ∈ Prop ∪ Form∼. By induction on the com-plexity of the formulae we can show that there should be a CluN valuationv′

M such that v′M(ψ) = 1 for all ψ ∈ Γ while v′

M(ϕ) �= 1, which gives usΓ �|=CLuN ϕ as required.⇐= We prove the contrapositive. Assume that Γ �|=CLuN ϕ, and thus thatfor some CLuN-valuation vM that extends v we have vM(ψ) = 1 for allψ ∈ Γ while vM(ϕ) �= 1. Let v′ : (Prop ∪ Form∼) × S �→ {0, 1} be an assign-ment-function such that v′(ϕ, s) = 1 iff v(ϕ) = 1 for all ϕ ∈ Prop ∪ Form∼.By induction on the complexity of the formulae we can show that there is avaluation v′

M : Form×S �→ {0, 1} (as defined above) such that v′M(ψ, s) = 1

for all ψ ∈ Γ while v′M(ϕ, s) �= 1. Consequently s is in ‖Γ‖M, but not in

‖ϕ‖M, which gives us ‖Γ‖M �⊆ ‖ϕ‖M.

7. Expressing Adaptive Consequence

To comply with the second correctness criterion (AL*) we need to find for-mulae ϕ and ψ whose satisfaction-conditions are equivalent to, respectively,‖Γ‖m

M ⊆ ‖ϕ‖M and ‖Γ‖rM ⊆ ‖ϕ‖M. Because the latter are global conditions

on the models, the corresponding satisfaction-conditions should be global aswell. The latter fact can be illustrated by first considering a simpler example.We can express ‖Γ‖M ⊆ ‖ϕ‖M as the strict implication

U

⎛⎝ ∧

γi∈Γ

γi → ϕ

⎞⎠ . (STRICT)

Since this formula starts with a universal modality it expresses a global con-dition. It is important to keep the following in mind: (STRICT) counts asa formula of the language L2 only when Γ is finite. When Γ is infinite, itcan only be expressed in a language that allows for countably infinite con-junctions. Yet, since lower limit logics have to be compact, there has to bea finite subset Δ of Γ such that M � U(

∧γi∈Δ γi → ϕ) iff ‖Γ‖n

M ⊆ ‖ϕ‖M.


7.1. Minimal Abnormality

We now consider two different options that correspond to the minimal abnor-mality strategy. The first one is a version of (ESI)

U

( ∧γi∈Γ

γi → ♦�( ∧

γi∈Γ

γi ∧ ��( ∧

γi∈Γ

γi → ϕ

))), (ESI*)

and only makes use of the universal modality and the modalities for the pre-order . The second one, by contrast, uses the box-operator for the strictorder ≺ (see e.g. [12, p. 37])

U

⎛⎝⎛⎝ ∧

γi∈Γ

γi ∧ �≺¬∧

γi∈Γ

γi

⎞⎠→ ϕ

⎞⎠ . (MIN)

The need to express premise-sets as the conjunction of all the premises canbe problematic because the failure of compactness for adaptive logics blocksthe detour via finite premise-sets we used for (STRICT). As a consequence,(ESI*) as well as (MIN) can either only be used to express adaptive conse-quence for finite premise-sets, or they have to be formulated in a suitableinfinitary extension of the modal language L2. We provisionally restrict our-selves to finite premise-sets.

(ESI*) is indirect in the sense that it does not refer to minimally abnor-mal states; it even works when there are no such states. (MIN), by contrast,directly captures the idea that ϕ needs to be true at all states in ‖Γ‖m

M.The former is preferable when there is no guarantee that there are alwaysminimally abnormal states, but given Corollary 2.1 this is not a concern.

We show that within the class of strict abnormality models both versionsare equivalent by proving that they both comply with (AL*).

Theorem 5. (Syntactic Correctness for Minimal Abnormality (I)) For allstrict abnormality models M, the formula

U

(( ∧γi∈Γ

γi ∧ �≺¬∧

γi∈Γ

γi

)→ ϕ

)

is verified by all s in M iff ‖Γ‖mM ⊆ ‖ϕ‖M.

Proof. We only need to prove that M, s �∧

γi∈Γ γi ∧ �≺¬∧γi∈Γ iff s ∈‖Γ‖m

M. The first conjunct is true iff s is in ‖Γ‖M. The second conjunct istrue iff every s′ ≺ s falsifies at least one member of Γ, which by Proposition8 is necessary and sufficient for s to be in ‖Γ‖m

M whenever it is in ‖Γ‖M.

P. Allo

Theorem 6. (Syntactic Correctness for Minimal Abnormality (II)) For all(strict) abnormality models M, the formula

U

( ∧γi∈Γ

γi → ♦�( ∧

γi∈Γ

γi ∧ ��( ∧

γi∈Γ

γi → ϕ

)))

is verified by all s in M iff ‖Γ‖mM ⊆ ‖ϕ‖M.

Proof. ⇐= Assume that ‖Γ‖mM ⊆ ‖ϕ‖M. If this is trivially satisfied because

‖Γ‖mM is empty, then we know by Corollary 2.1 that ‖Γ‖M is empty as well.

But then we know that∧

γi∈Γ γi is false at all states, and hence the impli-cation within the scope of U is true at all states. The more interesting caseis when ‖Γ‖m

M is non-empty. Since the implication within the scope of U isagain vacuously true at all states that do not belong to ‖Γ‖M, we only needto check that the same implication is true at all states in ‖Γ‖M. Let s be anarbitrary state in ‖Γ‖M. We need to show that ♦�(

∧γi∈Γ γi∧��(

∧γi∈Γ γi →

ϕ)) is true at s. That is, there must be an s′ ∈ ‖Γ‖M such that s′ s,and

s′ � ��( ∧

γi∈Γ

γi → ϕ

)(†)

We first show that (†) holds for all s′ ∈ ‖Γ‖mM, and then show that for any

s ∈ ‖Γ‖M there is such an s′ for which s′ s holds.Let s′′ be an arbitrary state such that s′′ s′. If s′′ is also in ‖Γ‖m

M,then the implication within the scope of �� is true in virtue of ‖Γ‖m

M ⊆‖ϕ‖M. If s′′ isn’t in ‖Γ‖m

M, then it cannot be in ‖Γ‖M either. Two casesneed to be considered. Either we have s′ s′′, which contradicts s′′ �∈‖Γ‖m

M; or we have s′ � s′′ which contradicts s′ ∈ ‖Γ‖mM. The falsity of

Γ at s′′ makes the implication within the scope of �� vacuously true ats′′.

We need to consider two separate cases to show that for any s ∈ ‖Γ‖M,there is always an s′ ∈ ‖Γ‖m

M such that s′ s. If s is itself already in ‖Γ‖mM,

the result follows immediately from s s. If s isn’t in ‖Γ‖mM, then we only

need to appeal to Theorem 2.=⇒ To prove the contrapositive we assume ‖Γ‖m

M �⊆ ‖ϕ‖M. Let s bea state in ‖Γ‖m

M that doesn’t verify ϕ. Note first that since s doesn’tverify

∧γi∈Γ γi → ϕ, it follows in virtue of s s that s doesn’t ver-

ify ��(∧

γi∈Γ γi → ϕ) either. But for the same reason, it won’t verify♦�(

∧γi∈Γ γi ∧ ��(

∧γi∈Γ γi → ϕ)), and yet verify Γ. Since U is a global

modality, we only need one state where the implication within its scope isfalse for the strict implication to be false everywhere.


The (restricted) strict implications expressed by (STRICT), (ESI*), and(MIN) can also be seen as conditional box-operators. It is useful to intro-duce some specific notation that reflects this. We shall henceforth write [Γ]ϕas shorthand for (STRICT), and (because they’re equivalent in the presentcontext) write [Γ]mϕ as shorthand for (ESI*) as well as for (MIN). For rea-sons that shall become clear below, it is also useful to consider their duals〈Γ〉ϕ and 〈Γ〉mϕ.

〈Γ〉ϕ ↔ E

⎛⎝ ∧

γi∈Γ

γi ∧ ϕ⎞⎠

〈Γ〉mϕ ↔ E

⎛⎝ ∧

γi∈Γ

γi ∧ ��

⎛⎝ ∧

γi∈Γ

γi → ♦�

⎛⎝ ∧

γi∈Γ

γi ∧ ϕ⎞⎠⎞⎠⎞⎠

〈Γ〉mϕ ↔ E

⎛⎝ ∧

γi∈Γ

γi ∧ �≺

⎛⎝¬

∧γi∈Γ

γi

⎞⎠ ∧ ϕ

⎞⎠

7.2. Reliability

Finding a formula that corresponds to the reliability strategy is more cum-bersome. The main reason is that from a model-theoretic viewpoint theselection of reliable states is less straightforward—it either requires a refer-ence to unreliable formulae or it requires a reference to states that do notverify any abnormality that isn’t also verified by some minimally abnor-mal state. By contrast, from a proof-theoretic viewpoint reliability is muchmore natural. The two-step procedure described below partly exploits thisproof-theoretic naturalness.4

Formulae of the form

〈Γ〉∧

ωi∈Δ

ωi

(with Δ ⊆ Ω) can be used to express that all abnormalities in Δ are verifiedby some state in ‖Γ‖M. Similarly, formulae of the form

∧ωi∈Δ

〈Γ〉mωi

(again, with Δ ⊆ Ω) can then be used to express that each abnormality in Δis verified by some minimally abnormal state in ‖Γ‖M. By putting the two

4What I primarily refer to is the following theorem: Γ |∼r ϕ iff Γ |=LLL ϕ ∨ Dab(Δ)and U(Γ) ∩ Δ = ∅. See Theorem 7 in [8].

P. Allo

together we can express an important feature of the abnormalities that areverified by reliable states in ‖Γ‖M; namely the fact that at most all abnor-malities verified by some minimally abnormal state in ‖Γ‖M are verified byreliable states in ‖Γ‖M.

To state this fact in its full generality, we need all instances of the fol-lowing two formulae (for Γ a set of L1-formulae and Δ ⊆ Ω)

( ∧ωi∈Δ

〈Γ〉mωi ∧ 〈Γ〉( ∧

ωi∈Δ

ωi ∧∧

θ∈Ω\Δ

¬θi

))→ 〈Γ〉r

∧ωi∈Δ

ωi (R1)

〈Γ〉r∧

ωi∈Δ

→∧

ωi∈Δ

〈Γ〉mωi (R2)

Remark also that in view of Proposition 4, we cannot use the simpler uniqueversion given below:( ∧

ωi∈Δ

〈Γ〉mωi ∧ 〈Γ〉∧

ωi∈Δ

ωi

)↔ 〈Γ〉r

∧ωi∈Δ

ωi

As before, this requires the use of infinitely long conjunctions. Fortunately,if we’re only interested in finite premise-sets, we can restrict our attentionto instances where Δ is finite as well. More exactly, we can make use ofthe following finitary restriction: Let at(Γ) be the set of all atomic formulaein Γ, and FormΓ the set of L1 formulae that can be obtained from at(Γ)and the connectives of L1. Use this set to define ΩΓ as Ω ∩ FormΓ. If Γ isfinite, at(Γ) will be finite as well. For many lower limit logics (e.g. classicallogic, but also some paraconsistent and paracomplete logics) a restriction toa finite stock of atoms will suffice to ensure that ΩΓ contains only finitelymany non-equivalent abnormalities as well. If, by contrast, the lower limitlogic is a modal logic with infinitely many non-equivalent modalities (forinstance K or T [see 13, p. 56], but also the paraconsistent logic CluN when∼ is seen as a modality), the required finitary restriction cannot be achievedby merely restricting the set of atoms. In these cases, we need to define theset FormΓ

n ⊂ FormΓ of formulae in FormΓ with modal degree ≤ n, with nthe modal degree of Γ.5 Using a standard result in modal logic, we can thenshow that for any n the set Ω ∩ FormΓ

n will again only contain finitely manynon-equivalent abnormalities.6

5The modal degree of a formula is inductively defined as follows: (i) deg() = 0, (ii)deg(ϕ∧ψ) = max(deg(ϕ), deg(ψ)), (iii) deg(♦ϕ) = deg(ϕ) + 1. The modal degree of a setof formulae Γ is the maximum of the degree of its members.

6See for instance Proposition 2.29 on p. 74 in [10].


The above results hold a fortiori for each Δ ⊆ ΩΓ and Δ ⊆ ΩΓn, and thus

for each finite Γ, we will only have to consider (finitely many) instances of(R1) and (R2) that contain finite conjunctions of abnormalities. Since thereare countably many finite premise-sets, there will only be countably manyinstances of (R1) and (R2).7

We now show that the conjunction of (R1) and (R2) gives us the rightanswer to the question which abnormalities are verified by reliable states.

Proposition 10. For all abnormality models M, and finite Δ ⊆ Ω we have

M �( ∧

ωi∈Δ

〈Γ〉mωi ∧ 〈Γ〉( ∧

ωi∈Δ

ωi ∧∧

θ∈Ω\Δ

¬θi

))

only if Δ is verified by some s ∈ ‖Γ‖rM.

Proof. The formula∧

ωi∈Δ〈Γ〉mωi is globally true in M iff every ωi ∈Δ is verified by some minimally abnormal state in ‖Γ‖M. Likewise,〈Γ〉(∧ωi∈Δ ωi ∧ ∧θ∈Ω\Δ ¬θi) is globally true in a model iff there is ans ∈ ‖Γ‖M such that Δ = Ab(s). Let Δ′ be the set of all abnormalitiesthat are verified by some minimally abnormal state in ‖Γ‖M. Since Δ ⊆ Δ′

and s falsifies all abnormalities in Ω \ Δ, it also falsifies all abnormalities inΩ \ Δ′. By Definition 17, this implies s ∈ ‖Γ‖r

M.

Proposition 11. For all abnormality models M, and finite Δ ⊆ Ω we have

M �∧

ωi∈Δ

〈Γ〉mωi

if Δ is verified by some s ∈ ‖Γ‖rM.

Proof. Immediate in view of Definition 17.

The conjunction of (R1) and (R2) is not sufficient to express ‖Γ‖rM ⊆

‖ϕ‖M. This is because it allows us to derive facts about which abnormalitieshold at reliable states, but remains silent about other formulae verified atthose states. The following equivalence closes this gap:8

[Γ]rϕ ↔(

[Γ]

(ϕ ∨

( ∨ωi∈Δ

ωi

))∧ ¬〈Γ〉r

( ∨ωi∈Δ

ωi

)), (R3)

7Non-modal logics where a finite supply of atoms isn’t sufficient to obtain a finite setof non-equivalent formulae require a different strategy. This lies beyond the scope of thepresent paper.

8Remark that (R3) does not imply that [Γ]r and 〈Γ〉r would fail to be duals.

P. Allo

Roughly, the right hand-side of (R3) states that ‖Γ‖M ⊆ (‖ϕ‖M ∪‖ω1‖M∪ · · · ‖ωn‖M) and ‖Γ‖r

M∩(‖ω1‖M∪ · · · ‖ωn‖M) = ∅, for {ω1, . . . , ωn}=Δ ⊆ Ω (cfr. Lemma 7.1 below). This is all we need to prove (AL*) for reli-ability.

Theorem 7. (Syntactic Correctness for Reliability) For all abnormalitymodels M we have ‖Γ‖r

M ⊆ ‖ϕ‖M iff(

[Γ]

(ϕ ∨

( ∨ωi∈Δ

ωi

))∧ ¬〈Γ〉r

( ∨ωi∈Δ

ωi

))

is globally true in M.

Proof. Immediate in view of Lemma’s 7.1 and 7.2 for which the proofs aregiven below.

Lemma 7.1. For all abnormality models M we have ‖Γ‖M ⊆ (‖ϕ‖M ∪‖ω1‖M∪ · · · ‖ωn‖M) and ‖Γ‖r

M∩(‖ω1‖M∪ · · · ‖ωn‖M) = ∅, for {ω1, . . . , ωn}=Δ ⊆ Ω iff:

M �(

[Γ]

(ϕ ∨

( ∨ωi∈Δ

ωi

))∧ ¬〈Γ〉r

( ∨ωi∈Δ

ωi

))

Proof. We first observe that ‖Γ‖M ⊆ (‖ϕ‖M ∪ ‖ω1‖M∪ · · · ∪ ‖ωn‖M) isequivalent to M � [Γ](ϕ ∨ (

∨ωi∈Δ ωi)) in virtue of the meaning of [Γ] and

classical disjunction. Next, M � ¬〈Γ〉r(∨

ωi∈Δ ωi) is equivalent to M �[Γ]r(

∧ωi∈Δ ¬ωi), which holds iff ‖Γ‖r

M ⊆ (‖¬ω1‖M∩ · · · ∩ ‖¬ωn‖M). Thelast condition is equivalent to ‖Γ‖r

M ∩ (‖ω1‖M∪ · · · ∪ ‖ωn‖M) = ∅ in virtueof the meaning of classical negation.

Lemma 7.2. For all abnormality models M we have ‖Γ‖rM ⊆ ‖ϕ‖M iff

‖Γ‖M ⊆ (‖ϕ‖M∪‖ω1‖M∪ · · ·∪‖ωn‖M) and ‖Γ‖rM∩(‖ω1‖M∪ · · ·∪‖ωn‖M) =

∅, for {ω1, . . . , ωn} = Δ ⊆ Ω.

Proof. ⇐= Assume that for some set of abnormalities {ω1, . . . , ωn} = Δwe have ‖Γ‖M ⊆ (‖ϕ‖M ∪ ‖ω1‖M∪ · · · ∪ ‖ωn‖M) and ‖Γ‖r

M ∩ (‖ω1‖M∪ · · · ∪‖ωn‖M) = ∅. Since ‖Γ‖r

M ⊆ ‖Γ‖M, it follows by the transitivity of ⊆ that‖Γ‖r

M ⊆ (‖ϕ‖M ∪‖ω1‖M∪ · · ·∪‖ωn‖M). Yet, because no state in ‖Γ‖rM veri-

fies an abnormality ωi ∈ Δ, it follows that all states in ‖Γ‖rM must verify ϕ.

=⇒ Assume that ‖Γ‖rM ⊆ ‖ϕ‖M. Let Θ be the set of all abnormalities that

are not verified by any s in ‖Γ‖rM. Any state of ‖Γ‖r

M will verify the negationof every abnormality in Θ. Thus we obtain the following identity

‖Γ‖rM = ‖Γ‖M ∩ ‖{¬θ : θ ∈ Θ}‖M,


which we use to derive

‖{¬θ : θ ∈ Θ}‖M ∩ ‖Γ‖M ⊆ ‖ϕ‖M.

By compactness and classical logic we then obtain

‖Γ‖M ⊆ ‖ϕ‖M ∪ ‖ω1‖M∪ · · · ∪ ‖ωn‖M

with {ω1, . . . , ωn} a finite subset of Θ. To complete the proof, we just needto observe that since no abnormality in Θ is verified by any state in ‖Γ‖r

M,we have

‖Γ‖rM ∩ (‖ω1‖M∪ · · · ∪ ‖ωn‖M) = ∅

as required.

8. Final Remarks

The modal reconstruction of adaptive logic described in this paper is onlypartial. We can highlight a few valuable features and challenges of thebroader project of trying to look at adaptive logic from the perspectiveof modal logic by explaining in which senses the present reconstruction isincomplete.

(1) The present reconstruction is only partial because an axiomatisation(and matching completeness result) is missing. This is a non-trivial taskthat forces us to go beyond the basic preference language. (2) A furtherlacuna is the absence of a satisfactory way of coping with infinite premise-sets. We have already seen that the failure of compactness blocks a detourvia finite conjunctions of premises. When there is no finite subset Δ ofsome infinite premise-set Γ such that, for some adaptive strategy, Γ |∼x ϕonly if Δ |∼x ϕ, infinite conjunctions cannot be avoided if we want toexpress that ϕ is an adaptive consequence of Γ in terms of (ESI*), (MIN),or (R3). (3) Although abnormality models get the adaptive consequencerelation right, these models fail to shed a light on the dynamic proof-the-ory that distinguishes adaptive logics from other logics for defeasible infer-ence.

This brings us to the final question of what we may gain from presentingadaptive logics as a modal logic. The specific preference models developedin the present paper allow us to present logics with a non-standard seman-tics and proof-theory in a different setting, and enables one to appreciatethe specificity and richness of adaptive logics against (what many wouldconsider) a more familiar background-formalism. This should facilitate the

P. Allo

dialogue with several other formalisms like the preference logics and condi-tional logics already mentioned in Sect. 3, but also, for instance, with theconditional doxastic models from [2].

A dialogue between different systems works in more than one way. Wecould, on the one hand, just use the common formalism to compare log-ics where the ordering of the states is syntactic as well as formal in thesense that it is determined by the formulae of a certain logical form thatare verified by those states (adaptive logics) with logics where the order-ing of the states obeys fewer constraints (doxastic logics, preference logic,conditional logic).9 On the other hand, we can use the common formal-ism to apply methods from one domain in another one. This can work intwo directions. We can consider how updates would take place in abnor-mality models to model the external dynamics of adaptive logic as a kindof belief revision, but we can also import the reliability strategy to for-mulate a more cautious alternative to the standard definition of belief astruth in all most plausible states in an agent’s information partition,10

or consider doxastic logics that are based on some kind of abnormality-ordering.

Acknowledgements. The author is a postdoctoral Fellow of the ResearchFoundation – Flanders (FWO). Thanks are due to the organisers and audi-ences of the Logics for Dynamics of Information and Preferences WorkingSessions in Amsterdam (2007), the Third Workshop in the Philosophy ofInformation in Ankara (2010), and the Logic, Reasoning and RationalityConference in Ghent (2010).

9For the intermediate situation where the ordering is syntactical, but not formal see[1] and [15]. For a criticism of logics based on non-formal orderings, see [9, 5.5].

10If we use the doxastic models from [2], both strategies still coincide. This is becausedoxastic plausibility-models are locally connected; for all states s, s′ that an agent consid-ers possible, it holds that either s ≤ s′ or s′ ≤ s, and hence (what would correspond to)the reliable states are just the minimal states. Thus, rather than providing an additionaltype of belief, the move to pre-orders really leads to a splitting of notions.Note also that the sense in which the reliability strategy is more cautious than the mini-mality strategy is quite different from the sense in which, say, the notions of strong andsafe belief are more cautious than plain belief. Since it all depends on how a pre-order isused to select states, any notion of belief could (provided that states are pre-ordered basedon the formulas of a certain class they satisfy) be replaced or supplemented with a versionthat uses the reliability selection instead of the minimality selection.


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P. Allo

Patrick Allo

Centre for Logic and Philosophy of ScienceVrije Universiteit BrusselPleinlaan 2Brussels, [email protected]

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Adaptive Logic as a Modal Logic