Journal of Applied Logic 11 (2013) 147–168Contents lists available at SciVerse ScienceDirect Journal of Applied Logic www.elsevier.com/locate/jal Two adaptive logics of norm-propositions Mathieu Beirlaen ∗, Christian Strasser Centre for Logic and Philosophy of Science, Ghent University, Blandijnberg 2, 9000 Ghent, Belgium a r t i c l e i n f o a b s t r a c t Article history: Available online 9 February 2013 Keywords: Deontic logic Normative gaps Normative conflicts Adaptive logic Defeasible reasoning We present two defeasible logics of norm-propositions (statements about norms) that (i) consistently allow for the possibility of normative gaps and normative conflicts, and (ii) map each premise set to a sufficiently rich consequence set. In order to meet (i), we define the logic LNP, a conflictand gap-tolerant logic of norm-propositions capable of formalizing both normative conflicts and normative gaps within the object language. Next, we strengthen LNP within the adaptive logic framework for non-monotonic reasoning in order to meet (ii). This results in the adaptive logics LNPr and LNPm, which interpret a given set of premises in such a way that normative conflicts and normative gaps are avoided 'whenever possible'. LNPr and LNPm are equipped with a preferential semantics and a dynamic proof theory. © 2013 Elsevier B.V. All rights reserved. 1. Introduction In this section, we introduce in an informal way the key concepts studied in this paper. Formally precise characterizations follow later on. 1.1. Normative conflicts and normative gaps Ideally, sets of norms issued by agents, authorities, legislators, etc. are both consistent and complete. In our everyday practice, however, such sets often contain normative conflicts and normative gaps. A normative conflict occurs when two or more norms are issued that are mutually unsatisfiable. The existence of such conflicts is motivated as follows by Alchourrón and Bulygin: Even one and the same authority may command that p and that not p at the same time, especially when a great number of norms are enacted on the same occasion. This happens when the legislature enacts a very extensive statute, e.g. a Civil Code, that usually contains four to six thousand dispositions. All of them are regarded as promulgated at the same time, by the same authority, so that there is no wonder that they sometimes contain a certain amount of explicit or implicit contradictions [3, pp. 112–113]. Normative conflicts do not always consist of conflicting commands or obligations. They also arise where both an obligation to do something and a (positive) permission not to do it are promulgated [1,3,14,49]. The adaptive logics presented in this paper are able to adequately deal with both normative conflicts and normative gaps. We say that a set of norms contains a normative gap with respect to A if A is neither positively permitted nor forbidden nor obliged. For a defense of the existence of normative gaps, see e.g. [2, Chapters 7, 8], [15]. * Corresponding author. Tel.: +32 9 264 39 79; fax: +32 9 264 41 87. E-mail addresses: mathieu.beirlaen@UGent.be (M. Beirlaen), christian.strasser@UGent.be (C. Strasser).1570-8683/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jal.2013.02.001 148 M. Beirlaen, C. Strasser / Journal of Applied Logic 11 (2013) 147–168Note that the formulation refers to positive permissions (also, strong permissions), i.e. permissions that are either explicitly stated as such, or permissions that are derivable from other explicitly stated permissions or obligations. This is to be distinguished from so-called weak or negative permissions: A is weakly permitted in case A is not forbidden. Would we replace "positive permission" by "weak permission" in the definition of normative gaps then the concept would be vacuous since each A is either forbidden or not forbidden (and hence, weakly permitted). The practical use of the distinction between positive and negative permission can be illustrated by means of the legal principle nullum crimen sine lege. According to this principle anything which is not forbidden is permitted.1 Alternatively, the principle states that a negative permission to do A implies a positive permission to do A. Typically, the nullum crimen principle is understood as a rule of closure permitting all the actions not prohibited by penal law [2, pp. 142–143]. We return to this principle in Section 2.1. We will in the remainder of the paper tacitly assume that in case A is obliged then A is positively permitted. In this case, there is a normative gap with respect to A iff A is neither positively permitted nor forbidden. Another way to think about normative gaps is in terms of normative determination: A is normatively determined if and only if A is either positively permitted or forbidden, which is to say that there is no normative gap with respect to A.2 We say that a set of norms is normatively complete if all of its norms are normatively determined, i.e. if there are no gaps with respect to any of its norms. From the existence of incomplete legal systems, Bulygin concludes that legal gaps are perfectly possible: It is not true that all legal systems are necessarily complete. The problem of completeness is an empirical, contingent, question, whose truth depends on the contents of the system. So legal gaps due to the silence of the law (. . .) are perfectly possible [15, p. 28]. 1.2. Norm-propositions and their formal representation In ordinary language, normative sentences exhibit a characteristic ambiguity. The very same words may be used to enunciate a norm (give a prescription) and to make a normative statement (description) [47, pp. 104–106]. In deontic logic, it is important to carefully distinguish between this prescriptive and descriptive use of norms. When interpreted prescriptively, a formula of the form OA means something like "you ought to do A", or "it ought to be that A", and a formula of the form PA means something like "you may do A", or "it is permitted that A".3 When interpreted descriptively, a formula of the form OA [PA] means something like "there is a norm to the effect that A is obligatory [permitted]". Thus, in our descriptive reading a formula PA always denotes a strong permission. Following [47], we take the term norm to denote the prescriptive, and norm-proposition to denote the descriptive interpretation of normative statements.4 According to Alchourrón and Bulygin [1–3], any perceived harmony between norms and norm-propositions in deontic logic is merely apparent. Instead of using the same calculus of deontic logic for reasoning with both norms and normpropositions, we need two separate logics: a logic of norms and a logic of norm-propositions. This paper is concerned with the characterization of a logic of norm-propositions. In formal language normative conflicts are expressed by formulas such as OA ∧O not A in case two obligations conflict, and OA ∧ P not A in case an obligation conflicts with a permission. We call a conflict of the former kind an OO-conflict, and a conflict of the latter kind an OP-conflict. Normative gaps occur if neither PA nor O not A is the case. A full formal characterization of normative gaps is presented after the definition of our formal language. As pointed out above, the permission in question is a strong permission. Weak permissions may be simply defined as the modal dual to O: by not O not A. The latter expresses that "there is no norm to the effect that not A is obliged" and hence it expresses the descriptive meaning of a weak permission. However, we need an independent permission operator P in order to express strong permissions. From PA we cannot infer not O not A due to the possible existence of an OP-conflict. Similarly we cannot, vice versa, infer PA from not O not A since, despite the absence of a norm that expresses that not A is obliged, A may not be positively permitted.5 In the remainder we show how each of the concepts presented in this introductory section is formalized and treated by the logics defined later on in this paper. 1 Legal philosophers also refer to this principle as the sealing legal principle. We thank an anonymous referee for pointing this out. 2 The notion of normative determination is adopted from [48]. 3 Until our formal language is defined, we use brackets "" and "" for denoting formulas. 4 Von Wright [47] and Åqvist [6] cite Ingemar Hedenius as the first philosopher to note the distinction between norms and norm-propositions. According to Hedenius, norms are "genuine", and norm-propositions are "spurious" deontic sentences [25]. The distinction between norms and norm-propositions was later also drawn – among others – by Wedberg [50], Stenius [41], Alchourrón [1], and Hansson [23] (see also [6]). 5 See [2,47] for further arguments against the equivalence of PA and not O not A in a descriptive setting. M. Beirlaen, C. Strasser / Journal of Applied Logic 11 (2013) 147–168 1491.3. The structure of the paper This paper is structured as follows. In Section 2 we define the Logic of Norm-Propositions LNP. This logic is sufficiently expressive to formalize both normative conflicts and normative gaps without having to resort to the meta-language. Inside the scope of its deontic operators, LNP makes use of a paraconsistent and paracomplete negation connective for dealing with normative conflicts and normative gaps. As a result of the weakness of this negation connective, LNP is not powerful enough for capturing many intuitive normative inferences. We deal with this problem in Section 3, where we strengthen LNP within the adaptive logics framework for non-monotonic reasoning. This results in two adaptive logics which interpret a given premise set 'as consistently and as completely as possible'. In Section 4 we equip the logics defined in Section 3 with a proof theory, and prove some further meta-theoretical results. In Section 5, we compare our logics to other approaches taken up in the literature on norm-propositions and on conflicting norms. As the scope of our analysis is entirely conceptual/philosophical, and as space is limited, we illustrate the workings of the logics presented in this paper by abstract examples rather than by 'real-life' cases. 2. A negation-weakened foundation: the logic LNP 2.1. Syntax In the setting of norm-propositions, negation behaves differently depending on whether it occurs inside or outside the scope of an operator O or P. Outside the scope of a deontic operator, negation behaves classically. A formula not Op is read as "it is not the case that there is a norm to the effect that p is obligatory". Under this reading, not Op is incompatible with Op: Op and not Op cannot both be the case. Moreover, one of Op or not Op must hold: either there is a norm to the effect that p is obligatory, or there is not. Things change when we turn to negations inside the scope of O or P. Here, both Op and O not p are verified by the same set of norm-propositions if this set contains an OO-conflict with respect to p. Moreover, neither Pp nor O not p are verified by a given set of norm-propositions that contains a normative gap with respect to p. Given the standard characterizations of O and P, this means that – inside the scope of O or P – both the consistency and the completeness constraint for negation fail in some instances: P(p ∧ not p) is true in the case of a normative conflict, and O(p ∨ not p) is false in the case of a normative gap. The logic LNP is defined in such a way that it respects this distinction: outside the scope of a deontic operator, only the classical negation connective "¬" occurs. Inside the scope of a deontic operator, LNP makes use of the connective "∼", which is a paraconsistent and paracomplete "negation" connective, i.e. it invalidates both (A ∧ ∼A) ⊃ B (Ex Contradictione Quodlibet) and A ∨ ∼A (Excluded Middle).6 Where Wa = {p,q, r, . . .} is a denumerable set of atomic propositions, we define W¬ as the 〈¬,∨,∧,⊃,≡〉-closure of Wa , and W∼ as the 〈∼,∨,∧,⊃,≡〉-closure of Wa . Let: WO := O〈W∼〉 ∣∣ P〈W∼〉 ∣∣ ¬〈WO〉 ∣∣ 〈WO〉 ∨ 〈WO〉 ∣∣ 〈WO〉 ∧ 〈WO〉 ∣∣ 〈WO〉 ⊃ 〈WO〉 ∣∣ 〈WO〉 ≡ 〈WO〉 Note that we do not allow for nested occurrences of the modal operators in our language. The set W of well-formed formulas of LNP is defined as the 〈¬,∨,∧,⊃,≡〉-closure of W¬ ∪WO . Since the denotation of formulas is no longer ambiguous now that our language W is defined, we skip the -marks in the remainder of the paper. For future reference, we also define the set W l = {A,∼A | A ∈Wa} of ∼-literals. Both normative conflicts and normative gaps are expressible in the object language W . A normative conflict occurs relating to a formula A ∈W∼ whenever we can derive one of OA ∧ O∼A or OA ∧ P∼A. A normative gap occurs relating to A whenever we can derive ¬PA ∧ ¬O∼A, i.e. whenever there is no norm to the effect that A is permitted or forbidden. The P-operator functions as an operator for positive permission. A proposition A is said to be negatively permitted if there is no obligation to the contrary, i.e. if ¬O∼A. The nullum crimen principle can be formalized as an axiom schema: (NC) ¬O∼A ⊃ PA Clearly, (NC) a priori excludes the possibility of normative gaps. That is why it is invalidated by any gap-tolerant logic of norm-propositions. 6 "∼" as defined below is actually a "dummy" connective rather than a negation connective: it has no properties at all, except that it validates de Morgan's laws. However, in Sections 3 and 4 we show that "∼" functions as a negation connective in the adaptive extensions of the logic LNP. 150 M. Beirlaen, C. Strasser / Journal of Applied Logic 11 (2013) 147–1682.2. Semantics LNP is characterizable within a Kripke-style semantics with a set of worlds or points W and a designated or 'actual' world w0 ∈ W . In w0, negation is defined classically by means of the connective "¬". In the other worlds, negation is defined by the paraconsistent and paracomplete connective "∼".7 An LNP-model is a tuple 〈W , w0, R, v0, v〉, where R = {w0} × (W \ {w0}) is a non-empty accessibility relation, and v0 : Wa × {w0} → {0,1} and v : W l × (W \ {w0}) → {0,1} are assignment functions. v0 assigns truth values to atomic propositions at the actual world w0. Since all logical connectives (including negation) behave classically at this world, truth values for complex formulas can be defined in terms of a valuation function in the usual way. The situation is slightly different for other worlds. In the latter, the ∼-connective does not behave classically and truth values are assigned to all ∼-literals, i.e. all atomic propositions p and their ∼-negation ∼p. Let w ∈ W , w ′ ∈ W \{w0}. Then the valuation v M : (W×{w0})∪(W∼×W \{w0}) → {0,1}, associated with the model M , is defined by (C0) where A ∈ Wa, v M(A, w0) = 1 iff v0(A, w0) = 1 (Cl) where A ∈ W l, v M(A, w ′) = 1 iff v(A, w ′) = 1 (C¬) v M(¬A, w0) = 1 iff v M(A, w0) = 0 (C∼∼) v M(∼∼A, w ′) = 1 iff v M(A, w ′) = 1 (C∼⊃) v M(∼(A ⊃ B), w ′) = 1 iff v M(A ∧ ∼B, w ′) = 1 (C∼∧) v M(∼(A ∧ B), w ′) = 1 iff v M(∼A ∨ ∼B, w ′) = 1 (C∼∨) v M(∼(A ∨ B), w ′) = 1 iff v M(∼A ∧ ∼B, w ′) = 1 (C∼≡) v M(∼(A ≡ B), w ′) = 1 iff v M((A ∨ B) ∧ (∼A ∨ ∼B), w ′) = 1 (C⊃) v M(A ⊃ B, w) = 1 iff v M(A, w) = 0 or v M(B, w) = 1 (C∧) v M(A ∧ B, w) = 1 iff v M(A, w) = v M(B, w) = 1 (C∨) v M(A ∨ B, w) = 1 iff v M(A, w) = 1 or v M(B, w) = 1 (C≡) v M(A ≡ B, w) = 1 iff v M(A, w) = v M(B, w) (CO) v M(OA, w0) = 1 iff v M(A, w ′) = 1 for every w ′ such that R w0 w ′ (CP) v M(PA, w0) = 1 iff v M(A, w ′) = 1 for some w ′ such that R w0 w ′ (C0) and (Cl) simply take over the values of the assignment functions v0 and v respectively. (C¬) determines truth values for the classical negation connective "¬" in w0. (C∼∼)–(C∼≡) guarantee that de Morgan's laws hold for "∼" in accessible worlds. Where A ∈ Wa , the interpretation of ∼A is provided directly by the assignment function v . Where A is a complex formula, its negation ∼A can be reduced to simpler constituents in view of (C∼∼)–(C∼≡). (C⊃)–(C≡) determine truth values for the other classical connectives ⊃, ∧, ∨, and ≡ in all worlds, and (CO) and (CP) define the deontic operators O and P in the usual way. A semantic consequence relation for LNP is defined in terms of truth preservation at the actual world. An LNP-model M verifies A (M  A) iff v M(A, w0) = 1. Where Γ ⊆ W , M is an LNP-model of Γ iff M is an LNP-model and M  A for all A ∈ Γ . |LNP A iff all LNP-models verify A, and Γ |LNP A iff all LNP-models of Γ verify A. In Section 2.4, we discuss the workings of LNP in more detail and provide some illustrations. But first we define its syntactic consequence relation. 2.3. Axiomatization and meta-theory Inside the scope of O and P, we want to allow for the consistent possibility of contradictions and gaps. In order to do so, we make use of the propositional fragment of the logic CLoNs (cf. footnote 7). CLoNs is defined by adding de Morgan's laws for "∼" to the positive fragment of classical (propositional) logic CL8: (A∼∼) ∼∼A ≡ A (A∼⊃) ∼(A ⊃ B) ≡ (A ∧ ∼B) (A∼∧) ∼(A ∧ B) ≡ (∼A ∨ ∼B) (A∼∨) ∼(A ∨ B) ≡ (∼A ∧ ∼B) (A∼≡) ∼(A ≡ B) ≡ ((A ∨ B) ∧ (∼A ∨ ∼B)) 7 The semantic clauses for accessible worlds are inspired by those for (the propositional fragment of) Batens' paraconsistent and paracomplete logic CLoNs, a variation on the paraconsistent logic CLuNs as found in e.g. [10]. CLoNs is defined in Appendix A. 8 An axiomatization of the positive fragment of CL (i.e. CL without negation) is contained in Appendix A. Note that it validates full Modus Ponens. M. Beirlaen, C. Strasser / Journal of Applied Logic 11 (2013) 147–168 151Except for de Morgan's laws, "∼" has no properties at all. The logic LNP is fully axiomatized by CL (with the classical negation connective "¬") plus: (K) O(A ⊃ B) ⊃ (OA ⊃ OB) (D) OA ⊃ PA (NEC∼) If CLoNs A then  OA (KP) O(A ⊃ B) ⊃ (PA ⊃ PB) (OD) O(A ∨ B) ⊃ (OA ∨ PB) (PD) P(A ∨ B) ⊃ (PA ∨ PB) We write Γ LNP A to denote that A is LNP-derivable from Γ , and LNP A to denote that A is LNP-derivable from the empty premise set. LNP resembles SDL in the sense that it contains (K), (D), and a necessitation rule. However, it is non-standard in the sense that its necessitation rule (NEC∼) is defined in terms of theoremhood in CLoNs instead of theoremhood in CL. Moreover, in LNP the permission operator P is not definable in terms of the obligation operator O. Instead, the P-operator is characterized by the axiom schemata (KP), (OD), and (PD), all of which also hold in SDL. The axiom schemata (O-AND) and (P-AND) are derivable in LNP (their derivability is shown in Fact 1 in Appendix B): (O-AND) OA,OB LNP O(A ∧ B) (P-AND) OA,PB LNP P(A ∧ B) Theorem 1 (Soundness of LNP). If Γ LNP A, then Γ |LNP A. Theorem 2 (Strong completeness of LNP). If Γ |LNP A, then Γ LNP A. Proofs of Theorems 1 and 2 are contained in Appendix C. 2.4. Discussion LNP allows for the consistent possibility of normative conflicts and normative gaps, and invalidates deontic explosion: Op ∧ O∼p LNP Oq (1) Op ∧ P∼p LNP Oq (2) ¬Pp ∧ ¬O∼p LNP Oq (3) In accordance with the discussion in Section 1.2, the following interdependencies between the Oand P-operators are invalid in LNP: Pp LNP ¬O∼p (4) ¬Pp LNP O∼p (5) Op LNP ¬P∼p (6) ¬Op LNP P∼p (7) (4)–(7) correspond to the characterization of the P-operator as an operator for positive permission. (4) fails in the presence of an OP-conflict Pp ∧O∼p. (5) fails in the presence of a gap ¬Pp ∧¬O∼p. (6) fails in the presence of a conflict Op ∧P∼p, and (7) fails in the presence of a gap ¬P∼p ∧ ¬Op. The conflictand gap-tolerance of LNP, as well as the non-interdefinability of its Oand P-operators, all depend crucially on the paraconsistency and paracompleteness of the "∼"-connective. However, the very weak characterization of "∼" also causes the LNP-invalidity of the following inferences: O(p ∨ q),O∼q LNP Op (8) O(p ∨ q),O(∼p ∨ q) LNP Oq (9) O(p ⊃ q),O∼q LNP O∼p (10) Indeed, except for de Morgan's laws LNP invalidates all classically valid inferences that somehow depend on the properties of the ∼-connective, e.g. the Disjunctive Syllogism or Contraposition rules. (8) is invalid because the possibility of an OOconflict Oq ∧ O∼q cannot be excluded. In that case, Op need not follow from the premises O(p ∨ q) and O∼q. Likewise, (9) is invalid since Oq need not follow from O(p ∨ q) and O(∼p ∨ q) in the presence of an OO-conflict Op ∧ O∼p. 152 M. Beirlaen, C. Strasser / Journal of Applied Logic 11 (2013) 147–168(10) fails (i) in the case of a normative conflict relating to q or (ii) in the case of a normative gap relating to p. Suppose that O(p ⊃ q) and O∼q are true at the actual world. Then p ⊃ q and ∼q are true at all accessible worlds. In case (i), both q and ∼q are true in at least one accessible world. In this world, p ⊃ q is automatically true in view of (C⊃), and ∼p need not be true. In case ∼p is false at an accessible world, we have a model in which O∼p is false at the actual world. In case (ii), both p and ∼p will be false in at least one accessible world. Again we have a model in which O∼p is false at the actual world. For similar reasons all of the following 'variants' of (8)–(10) are invalid in LNP: O(p ∨ q), P∼q LNP Pp (11) P(p ∨ q), O∼q LNP Pp (12) O(p ∨ q),P(∼p ∨ q) LNP Pq (13) P(p ∨ q), O(∼p ∨ q) LNP Pq (14) O(p ⊃ q), P∼q LNP P∼p (15) P(p ⊃ q), O∼q LNP P∼p (16) O(p ⊃ q) LNP O(∼q ⊃ ∼p) (17) P(p ⊃ q) LNP P(∼q ⊃ ∼p) (18) In spite of the rationale behind their invalidity (i.e. the possibility of normative conflicts/gaps), all of (8)–(18) have some intuitive appeal. In real life, we tend to assume that norms behave consistently and that propositions are normatively regulated. Normative conflicts and normative gaps are anomalies. We rely on inferences like (8)–(18) in our everyday reasoning processes, albeit in a defeasible way. It seems then, that LNP is too weak to account for our normative reasoning. Inferences like (8)–(18) should only be blocked once we can reasonably assume that one of the norm-propositions needed in the inference behaves abnormally, i.e. that there might be a conflict or gap relating to this norm-proposition. Note that this reasoning process is non-monotonic: new premises may provide the information that there is a conflict or gap relating to some norm-proposition that was previously deemed to behave normally. Consider, for instance, the inference from O(p ∨ q) and O∼p to Oq. This inference is intuitive assuming that there is no normative conflict relating to p. If, however, we obtain the new information that there is a normative conflict relating to p, then the inference should be blocked, since we do not want to rely on conflicted norm-propositions in deriving new information. In the next section, we strengthen LNP in a non-monotonic fashion in order to overcome the problems mentioned here, and to make formally precise the idea of 'assuming' norm-propositions to behave 'normally'. 3. Two adaptive extensions For any A ∈Wa , the classical negation connective "¬" satisfies the following semantic conditions at the actual world: (†) If v M(A, w0) = 1, then v M(¬A, w0) = 0, (‡) If v M(A, w0) = 0, then v M(¬A, w0) = 1. (†) guarantees the consistency of A: A and ¬A cannot both be true at w0. (‡) imposes a completeness condition on A: at least one of A and ¬A is true at w0. As is clear from the LNP-semantics, (†) and (‡) fail for "∼" at accessible worlds. Instead of (†) and (‡), only the weaker conditions (†′) and (‡′) hold for "∼" at a world w ∈ W \ {w0}: (†′) If v M(A, w) = 1, then either v M(∼A, w) = 0 or v M(A ∧ ∼A, w) = 1, (‡′) If v M(A, w) = 0, then either v M(∼A, w) = 1 or v M(A ∨ ∼A, w) = 0. In view of the semantic clauses for LNP it is easily checked that whenever a normative conflict occurs relating to a proposition p, the formula p ∧ ∼p is true at some accessible world. In the case of an OP-conflict Op ∧ P∼p or O∼p ∧ Pp, this follows in view of (CO), (CP), and (C∧). In the case of an OO-conflict Op ∧ O∼p, it follows in view of (CO), (C∧) and the non-emptiness of the accessibility relation. In a similar fashion, we can check that whenever a normative gap occurs relating to p, the formula p ∨ ∼p is false at some accessible world. Suppose, for instance, that ¬Op ∧ ¬P∼p is true at w0. Then by (C¬), both Op and P∼p are false at w0. By (CO), there is a world w such that R w0 w and v M(p, w) = 0. By (CP), ∼p too is false at this world: v M(∼p, w) = 0. By (C∨), v M(p ∨ ∼p, w) = 0. M. Beirlaen, C. Strasser / Journal of Applied Logic 11 (2013) 147–168 153Normative conflicts create truth-value gluts, whereas normative gaps create truth-value gaps at accessible worlds.9 Suppose now that we label such gluts and gaps as abnormal, and that we try to interpret our worlds as normally as possible. Then, in view of (†′) and (‡′), normal behavior corresponds to the satisfaction of the consistency and completeness demands (†) and (‡) for "∼" at accessible worlds. The adaptive logics LNPr and LNPm defined in this section exploit the above idea in making the assumption that normpropositions behave 'normally' unless and until we find out that they are involved in some normative conflict or gap. 3.1. Semantic characterization of LNPr and LNPm Adaptive logics in the so-called standard format from [7] are characterized as triples, consisting of: (1) A lower limit logic (LLL): a compact, reflexive, transitive, and monotonic logic that contains CL and has a characteristic semantics. (2) A set of abnormalities Ω: a set of formulas, characterized by a (possibly restricted) logical form F; or a union of such sets. (3) An adaptive strategy: reliability or minimal abnormality. The LLL of an adaptive logic (AL) in standard format is its monotonic base; everything derivable by means of the LLL is derivable by means of the AL. The AL extends the LLL by interpreting abnormalities as false "as much as possible". The formal disambiguation of this idea is relative to the adaptive strategy used by the AL. At the moment, two adaptive strategies are included in the standard format: the reliability strategy and the minimal abnormality strategy. We use LNPx as a generic name for the logics LNPr and LNPm . The former uses the reliability strategy, whereas the latter uses minimal abnormality (hence the superscripts r and m). Both of these logics have LNP as their LLL. Moreover, LNPr and LNPm share the same set of abnormalities Ω = Ω1 ∪ Ω2, where Ω1 = {P(A ∧ ∼A) | A ∈ Wa} and Ω2 = {¬O(A ∨ ∼A) | A ∈Wa}. Ω1 is the set of atomic gluts true at some accessible world. Note that, in view of the validity of de Morgan's laws for "∼", more complex gluts can be reduced to (disjunctions of) atomic gluts by the LLL, e.g. if v M((p ∨ q) ∧ ∼(p ∨ q), w) = 1, then v M((p ∧ ∼p) ∨ (q ∧ ∼q), w) = 1. Consequently, whenever some LNP-model verifies an OOor OP-conflict, it also validates an abnormality in the set Ω1. In view of the LNP-semantics, p ∨∼p is false at some accessible world whenever ¬O(p ∨∼p) is true at the actual world. Thus Ω2 is the set of atomic gaps true at some accessible world. Again, complex instances of gaps are LNP-reducible to a (disjunction of) atomic gap(s), e.g. if v M((p ∨ q) ∨ ∼(p ∨ q), w) = 0, then v M((p ∨ ∼p) ∧ (q ∨ ∼q), w) = 0. Hence whenever some LNP-model verifies a normative gap, it also validates an abnormality in the set Ω2. For any atomic proposition p, the Ω2-abnormality ¬O(p ∨ ∼p) expresses that there is an accessible world in which neither p nor ∼p is verified, whereas the Ω1-abnormality P(p ∧ ∼p) expresses that there is an accessible world in which both p and ∼p are verified. Thus, in LNP both gluts and gaps in accessible worlds constitute abnormalities. In view of the discussion at the beginning of this section, this means that both normative conflicts and normative gaps constitute abnormalities in LNP. Semantically, adaptive logics proceed by selecting a subset of their LLL-models.10 This selection makes use of the abnormal part of an LNP-model, i.e. the set of all abnormalities verified by it. The abnormal part of an LNP-model M is defined as Ab(M) = {A ∈ Ω | M LNP A}. The minimal abnormality strategy selects all LNP-models of a premise set Γ which have a minimal abnormal part (with respect to set-inclusion). Definition 1. An LNP-model M of Γ is minimally abnormal iff there is no LNP-model M ′ of Γ such that Ab(M ′) ⊂ Ab(M). The semantic consequence relation of the logic LNPm is defined by selecting the minimally abnormal LNP-models: Definition 2. Γ |LNPm A iff A is verified by all minimally abnormal LNP-models of Γ . Before we can define the semantic consequence relation for LNPr , we need some more terminology. Where  is a finite, non-empty set of abnormalities, the disjunction ∨  is called a Dab-formula and is written as Dab(). A Dab-formula Dab() is a Dab-consequence of Γ if it is LNP-derivable from Γ ; Dab() is a minimal Dab-consequence of Γ if it is a Dab-consequence of Γ and there is no ′ such that Dab(′) is a Dab-consequence of Γ and ′ ⊂ . The set of formulas that are unreliable with respect to Γ , denoted by U (Γ ), is defined by 9 A truth-value glut for p occurs when both p and ∼p are assigned the value 1; a truth-value gap for p occurs when neither p nor ∼p are assigned the value 1. 10 Besides adaptive logics many other formal frameworks make use of semantic selections. Logics that make use of such semantic selection functions are said to have a preferential semantics, e.g. [30,40]. 154 M. Beirlaen, C. Strasser / Journal of Applied Logic 11 (2013) 147–168Table 1 Accessible worlds for Γ1 and Γ3. Grey cells indicate propositions that behave abnormally in w . (a) Accessible worlds for Γ1 w p ∼p q ∼q (1) 1 0 1 0 (2) 1 0 1 1 (3) 1 1 0 0 (4) 1 1 0 1 (5) 1 1 1 0 (6) 1 1 1 1 (b) Accessible worlds for Γ3 w p ∼p q ∼q (1) 0 0 0 1 (2) 0 0 1 1 (3) 0 1 0 1 (4) 0 1 1 1 (5) 1 0 1 1 (6) 1 1 1 1 Definition 3. Where Dab(1),Dab(2), . . . are the minimal Dab-consequences of Γ , U (Γ ) = 1 ∪ 2 ∪ * * * is the set of formulas that are unreliable with respect to Γ . Where Ab(M) is defined as above, we can now select the reliable models and define the semantic consequence relation for LNPr: Definition 4. An LNP-model M of Γ is reliable iff Ab(M) ⊆ U (Γ ). Definition 5. Γ |LNPr A iff A is verified by all reliable models of Γ . The fact that the set of LNPx-models of Γ is a subset of the set of LNP-models of Γ immediately ensures that LNPx strengthens LNP. Theorem 3. If Γ |LNP A, then Γ |LNPx A. Where MLNPΓ , MmΓ , and MrΓ denote the set of LNP-models, minimally abnormal LNP-models, resp. reliable LNP-models of Γ , we also know that: Theorem 4 (Strong reassurance). If M ∈MLNPΓ −MmΓ , then there is an M ′ ∈MmΓ such that Ab(M ′) ⊂ Ab(M). If M ∈MLNPΓ −MrΓ , then there is an M ′ ∈MrΓ such that Ab(M ′) ⊂ Ab(M). Theorem 4 is shown generically for adaptive logics in standard format as Corollary 1 in [7]. 3.2. Some illustrations In this section, we illustrate the semantics of LNPr and LNPm . We begin with a simple example and expand it for illustrating the non-monotonicity of LNPr and LNPm . We end with an example that illustrates the difference between both logics. Example 1. Let Γ1 = {Op,O(∼p ∨ q)}. Then, for all LNP-models M of Γ1, M, w0 | Op and M, w0 | O(∼p ∨ q). By (CO), M, w | p and M, w | ∼p ∨q for all worlds w such that R w0 w . The possible truth values for p,∼p, q, and ∼q at accessible worlds in M are depicted in Table 1(a). Let R(w0) abbreviate the set of worlds w ∈ W \ {w0} such that R w0 w . Then each w ∈ R(w0) is of one of types (1)–(6). If at least one w ∈ R(w0) is of one of types (3)–(6), then, by (C∧) and (CP), M, w0 | P(p ∧∼p), and P(p ∧∼p) ∈ Ab(M). Similarly, if at least one w ∈ R(w0) is of type (2) or type (6), then P(q ∧ ∼q) ∈ Ab(M). Moreover, if some w ∈ R(w0) is of type (3), then, by (C∨), (CO) and (C¬), M, w0 | ¬O(q ∨ ∼q), and ¬O(q ∨ ∼q) ∈ Ab(M). If, however, all worlds w ∈ R(w0) are of type (1), then M verifies no abnormalities relating to p or q. In view of Definition 1, only models for which all worlds w ∈ R(w0) are of type (1) qualify as minimally abnormal LNP-models of Γ1. Note that, for all type (1)-worlds w ∈ R(w0), M, w | q. By (CO), M, w0 | Oq. By Definition 2, Γ1 |LNPm Oq. Since Γ1 has LNP-models M of which all accessible worlds w ∈ R(w0) are such that, for all A ∈ Wa , M, w | A ∧ ∼A and M, w | A ∨ ∼A, we can conclude that Γ1 has LNP-models M such that Ab(M) = ∅. It follows that Γ1 has no minimal Dab-consequences. In view of Definition 3, U (Γ1) = ∅. By Definition 4, Ab(M) = ∅ for all reliable LNP-models M of Γ1. Again, only models for which all worlds w ∈ R(w0) are of type (1) qualify as reliable LNP-models of Γ1. By Definition 5, Γ1 |LNPr Oq. Example 2. Let Γ2 = {Op,O(∼p ∨q),O∼p}. It is easily checked that Γ2 |LNP P(p ∧∼p). Consequently, all LNP-models verify this abnormality, including the minimally abnormal and reliable ones. Hence all accessible worlds in all LNP-models of Γ2 are of one of types (3)–(6) in Table 1(a). Since P(p ∧ ∼p) is the only Dab-consequence of Γ2, the selected LNPx-models for M. Beirlaen, C. Strasser / Journal of Applied Logic 11 (2013) 147–168 155Table 2 Accessible worlds for Γ4. w p ∼p q ∼q r ∼r (1) 1 0 1 0 1 0 (2) 1 0 1 0 1 1 (3) 1 0 1 1 1 0 (4) 1 0 1 1 1 1 (5) 1 1 1 0 1 0 (6) 1 1 1 0 1 1 (7) 1 1 1 1 0 0 (8) 1 1 1 1 0 1 (9) 1 1 1 1 1 0 (10) 1 1 1 1 1 1 both strategies are those which verify exactly this abnormality, i.e. models of which all accessible worlds are of type (4) or (5). In all of these models, p,∼p ∨ q, and ∼p are true at all accessible worlds. Since q need not be true at some of these worlds, Γ2 has LNPx-models in which Oq is false. Hence Γ2 |LNPx Oq. Note that Examples 1 and 2 illustrate the non-monotonicity of LNPx: adding the premise O∼p to Γ1 blocks the derivation of Oq. Example 3. Let Γ3 = {O(p ⊃ q),O∼q}, and let M be an LNP-model of Γ3. The possible truth values for p,∼p, q, and ∼q at accessible worlds in M are depicted in Table 1(b). If at least one w ∈ R(w0) is of one of types (1) or (2), then ¬O(p ∨ ∼p) ∈ Ab(M). If at least one w ∈ R(w0) is of one of types (2), (4), (5) or (6), then P(q ∧ ∼q) ∈ Ab(M). Only if all w ∈ R(w0) are of type (3) it is possible that Ab(M) = ∅. In view of Definition 1, only models of which all w ∈ R(w0) are of type (3) qualify as minimally abnormal models. But then M, w0 | O∼p, and, by Definition 2, Γ3 |LNPm O∼p. It is safely left to the reader to check that, in view of Definitions 4 and 5, Γ3 |LNPr O∼p. Example 4. Let Γ4 = {O(p ∧ q),O(∼(p ∨ q) ∨ r),P(∼p ∨ ∼q)}, and let M be an LNP-model of Γ4. By (CO) we know that, for all w ∈ R(w0) in M , M, w | p ∧ q and M, w | ∼(p ∨ q) ∨ r. Hence every w ∈ R(w0) is of one of types (1)–(10) depicted in Table 2. By (CP), we also know that there is at least one world w such that w ∈ R(w0) and M, w | ∼p ∨ ∼q. Thus, w cannot be of type (1) or type (2). If w is of type (3), then P(q ∧ ∼q) ∈ Ab(M). If w is of type (5), then P(p ∧ ∼p) ∈ Ab(M). It is easily checked that if w is of type (4), (6), (7), (8), (9), or (10), then M validates more than one abnormality, i.e. either {P(p ∧ ∼p)} ⊂ Ab(M) or {P(q ∧ ∼q)} ⊂ Ab(M). In general, it follows by Definition 1 that M only qualifies as a minimally abnormal LNP-model of Γ4 if either w is of type (3) and all w ′ ∈ R(w0) \ {w} are of type (1) or type (3), or w is of type (5) and all w ′ ∈ R(w0) \ {w} are of type (1) or type (5). Hence if M is minimally abnormal, then all accessible worlds in M are of type (1), type (3), or type (5). But then, by (CO), M, w0 | Or and, by Definition 2, Γ4 |LNPm Or. Since at least one accessible world w in M is of types (3)–(10), it follows by (C∧), (CP), and (C∨) that M, w0 | P(p∧∼p)∨P(q∧∼q) (for any model M of Γ4). On the other hand, there exist models M of Γ4 such that M, w0 | P(p∧∼p), and there exist models M of Γ4 such that M, w0 | P(q ∧ ∼q). Thus, it follows that P(p ∧ ∼p) ∨ P(q ∧ ∼q) is a minimal Dab-consequence of Γ4. By Definition 3, P(p ∧ ∼p),P(q ∧ ∼q) ∈ U (Γ4). Suppose now that all w ∈ R(w0) are of type (8), and that, for all A ∈Wa \ {p,q, r}, M, w | A ∧∼A and M, w | A ∨∼A. Then it is easily verified that the only abnormalities verified by M are P(p ∧ ∼p) and P(q ∧ ∼q). Thus, Ab(M) ⊆ U (Γ4). By Definition 4, M is reliable. However, M, w0 | Or. Thus, by Definition 5, Γ4 |LNPr Or. Example 4 illustrates that there are premise sets Γ ⊆W and formulas A ∈W such that Γ |LNPr A and Γ |LNPm A. The inverse does not hold: in Section 4.3 we show that in general the logic LNPm is strictly stronger than LNPr . We come back to the intuitive differences between both strategies in Section 4.2. 4. Proof theory and meta-theory for LNPr and LNPm In this section, we provide a proof theory for the logics LNPr (Section 4.1) and LNPm (Section 4.2). We present the proof theory by means of an example, the premise set Γ4 from Section 3.2. We end the section with some meta-theoretical properties of both logics. 4.1. Proof theory for LNPr A line in an annotated LNPx-proof consists of a line number, a formula, a justification and a condition. The justification consists of a (possibly empty) list of line numbers (from which the formula is derived) and of the name of a rule. A condition is a finite set of abnormalities. A line with formula A and condition Θ is read as "A is derived on the assumption that none 156 M. Beirlaen, C. Strasser / Journal of Applied Logic 11 (2013) 147–168of the abnormalities in Θ is true". The presence of a condition is part of what makes an adaptive proof dynamic. As we will explicate further below, the idea is that lines whose assumptions are not tenable are deactivated by means of marking them. Similarly, new information may render an assumption of a marked line tenable in which case the line is reactivated by means of removing the marking. The effect of the marking definition is that, at every stage11 of the proof, certain lines may be marked whereas others are unmarked. Whether or not a line is marked depends only on the condition of the line and on the minimal Dab-formulas (cf. Definition 6) that have been derived in the proof. The rules of inference of LNPx reduce to three generic rules. Where Γ is a set of premises, and where A  abbreviates that A occurs in the proof on the condition , the simplest inference rule is the premise introduction rule PREM: PREM If A ∈ Γ : ... ... A ∅ PREM simply states that, at any line of a proof, a premise may be introduced on the empty condition. For instance, we can start an LNPr-proof for Γ4 by entering the premises via PREM: 1 O(p ∧ q) PREM ∅ 2 O(∼(p ∨ q) ∨ r) PREM ∅ 3 P(∼p ∨ ∼q) PREM ∅ The second rule of inference of LNPx is the unconditional rule RU: RU If A1, . . . , An LNP B: A1 1 ... ... An n B 1 ∪ * * * ∪ n Whenever A1, . . . , An LNP B and A1, . . . , An occur in the proof on the conditions 1, . . . ,n respectively, then RU may be used to derive B in the proof, on the condition 1 ∪ * * * ∪n . If for each i  n, Ai is derived on the assumption that each abnormality in i is false, then B is derivable on the assumption that each abnormality in 1 ∪ * * * ∪ n is false since its derivation relies on each Ai and hence also on the assumptions made in order to derive Ai . For instance, since O(p ∧ q) LNP Op, we can derive Op in our example by applying RU to line 1: 4 Op 1; RU ∅ By (O-AND), we can aggregate lines 2 and 4. Hence line 5 follows via RU: 5 O ((∼(p ∨ q) ∨ r) ∧ p) 2,4;RU ∅ As ((∼(p ∨ q) ∨ r) ∧ p) ⊃ (r ∨ (p ∧ ∼p)) is a CLoNs-theorem, line 6 follows from line 5 by (NEC∼) and (K). Thus, via RU: 6 O ( r ∨ (p ∧ ∼p)) 5;RU ∅ 7 Or ∨ P(p ∧ ∼p) 6;RU ∅ Line 7 is obtained by applying (OD) to line 6. Note that at this line we have derived the formula Or in disjunction with a member of Ω . In interpreting a set of formulas 'as normally as possible' adaptive proofs feature a third rule of inference that allows us to move abnormalities to the condition. This rule is called the conditional rule (RC). Intuitively, moving an abnormality to the condition of a line corresponds to making the (defeasible) assumption that this abnormality is false. RC is defined by RC If A1, . . . , An LNP B ∨ Dab(Θ) A1 1 ... ... An n B 1 ∪ * * * ∪ n ∪ Θ 11 A stage of a proof is a sequence of lines and a proof is a sequence 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. M. Beirlaen, C. Strasser / Journal of Applied Logic 11 (2013) 147–168 157If A1, . . . , An LNP B ∨ Dab(Θ) and A1, . . . , An occur in a proof on the conditions 1, . . . ,n respectively, then by RC we can infer B on the condition 1 ∪ * * * ∪ n ∪ Θ . RC is the only rule that allows for the introduction of new conditions in an adaptive proof. We can now continue our proof as follows: 8 Or 7; RC {P(p ∧ ∼p)} At line 8, the abnormality derived in disjunction with Or at line 7 is moved to the condition by means of an application of RC. At stage 8 of the proof, Or is considered derived. Intuitively, line 8 can be interpreted as "Or is derived on the assumption that P(p ∧ ∼p) is false". In the proof so far, lines 4–7 serve a purely explanatory purpose; their aim is to show that Or ∨ P(p ∧ ∼p) is an LNPconsequence of lines 1 and 2. However, since indeed O(p ∧ q),O(∼(p ∨ q) ∨ r) LNP Or ∨ P(p ∧ ∼p), we could have skipped these lines and applied RC immediately to lines 1 and 2. In a fashion analogous to the derivation of Or ∨ P(p ∧ ∼p) above, we can show that O(p ∧ q),O(∼(p ∨ q) ∨ r) LNP Or ∨ P(q ∧ ∼q). Hence we can apply RC immediately to lines 1 and 2: 9 Or 1,2; RC {P(q ∧ ∼q)} Consider now the premises at lines 1 and 3. By (P-AND), we can derive P((p ∧ q) ∧ (∼p ∨ ∼q)). Since CLoNs ((p ∧ q) ∧ (∼p ∨ ∼q)) ⊃ ((p ∧ ∼p) ∨ (q ∧ ∼q)), it follows by (NEC∼) and (KP) that P((p ∧ ∼p) ∨ (q ∧ ∼q)). By (PD), we obtain the Dab-formula P(p ∧∼p)∨P(q∧∼q). In view of the derivability of this formula, it seems rather bold to assume that P(q∧∼q) is false, or to assume that P(p ∧ ∼p) is false. That is why, according to the marking definition for the reliability strategy, all lines at which at least one of these assumptions was made will become marked once we have derived the disjunction P(p ∧ ∼p) ∨ P(q ∧ ∼q). Formulas occurring at lines that are marked are no longer considered to be derived. The marking definition for the reliability strategy proceeds in terms of the minimal Dab-formulas derived at a stage of the proof: Definition 6. Dab() is a minimal Dab-formula at stage s iff, at stage s, Dab() is derived on the condition ∅, and no Dab(′) with ′ ⊂  is derived on the condition ∅. Using this terminology, we can define the set of unreliable formulas relative to a stage of the proof: Definition 7. Where Dab(1),Dab(2), . . . are the minimal Dab-formulas derived at stage s, U s(Γ ) = 1 ∪ 2 ∪ * * * . Definition 8 (Marking for reliability). Where  is the condition of line i, line i is marked at stage s iff  ∩ Us(Γ ) = ∅. Returning to our example, we can now continue the proof as follows (we repeat the proof from line 8 on): 8 Or 6; RC {P(p ∧ ∼p)}10 9 Or 1,2; RC {P(q ∧ ∼q)}10 10 P(p ∧ ∼p) ∨ P(q ∧ ∼q) 1,3; RU ∅ The formula P(p ∧ ∼p) ∨ P(q ∧ ∼q) is a minimal Dab-formula at stage 10. Both its disjuncts are in the set of unreliable formulas at this stage: U10(Γ4) = {P(p ∧ ∼p),P(q ∧ ∼q)}. In view of Definition 8, this causes the marking of lines 8 and 9 at stage 10 (hence the checkmark sign). Consequently, the formula Or is no longer considered derived at stage 10 of the proof. The marking definition proceeds in terms of the minimal Dab-formulas that are derived at a certain stage. It is clear that marking is a dynamic matter: a line may be unmarked at a stage s, marked at a later stage s′ and again unmarked at an even later stage s′′ . This is why a more stable notion of derivability is needed than mere derivability at an unmarked line: Definition 9. A is finally derived from Γ at line i of a proof at finite stage s iff (i) A is the second element of line i, (ii) line i is not marked at stage s, and (iii) every extension of the proof in which line i is marked can be further extended in such a way that line i is unmarked. This definition may be interpreted in terms of a two-person game where the task of the proponent is to finally derive A and the opponent tries to prevent the latter (see [8]). First of all, our proponent needs to produce a finite proof in which A is derived at an unmarked line l. Now our opponent has the chance to attack and hence to thwart the proponent's intentions by means of extending the proof in such a way that l is marked. However, our proponent has a chance to defend herself if she is able to further extend the proof such that l is unmarked again. The proponent has a winning strategy in case she has a response to every possible attack by the opponent. In this case A is finally derivable. The derivability relation of LNPx is defined with respect to the notion of final derivability: 158 M. Beirlaen, C. Strasser / Journal of Applied Logic 11 (2013) 147–168Definition 10. Γ LNPx A (A is finally LNPx-derivable from Γ ) iff A is finally derived at a line of an LNPx-proof from Γ . Theorem 5 (Proof invariance). If Γ LNPx A, then every finite LNPx-proof from Γ can be extended in such a way that A is finally derived in it. See [7] for the proof of Theorem 5. Since Or is not derived at an unmarked line in our example, and since there is no way to extend the proof from Γ4 in such a way that Or is derived on an unmarked line, we know by Definitions 9 and 10 that Γ4 LNPr Or. 4.2. Proof theory for LNPm An LNPm-proof in many ways resembles an LNPr-proof. The only difference between the proof theories of both logics is that they make use of a different marking criterion. The marking definition for minimal abnormality requires some more terminology. A choice set of Σ = {1,2, . . .} is a set that contains one element out of each member of Σ . A minimal choice set of Σ is a choice set of Σ of which no proper subset is a choice set of Σ . Where Dab(1),Dab(2), . . . are the minimal Dab-formulas that are derived at stage s, Φs(Γ ) is the set of minimal choice sets of {1,2, . . .}. Definition 11 (Marking for minimal abnormality). Where A ∈W is derived at line i of a proof from Γ on a condition , line i is marked at stage s iff (i) there is no ′ ∈ Φs(Γ ) such that ′ ∩  = ∅, or (ii) for some ′ ∈ Φs(Γ ), there is no line at which A is derived on a condition Θ for which ′ ∩ Θ = ∅. We now return to our example but, instead of an LNPr-proof, we suppose that the proof we started in Section 4.1 is an LNPm-proof from Γ4. We repeat the proof from line 8 on: 8 Or 7; RC {P(p ∧ ∼p)} 9 Or 1,2; RC {P(q ∧ ∼q)} 10 P(p ∧ ∼p) ∨ P(q ∧ ∼q) 1,3; RU ∅ In the LNPm-proof from Γ4, the set Φ10(Γ4) of minimal choice sets of Γ4 at stage 10 consists of the sets {P(p ∧ ∼p)} and {P(q ∧ ∼q)}. In view of Definition 11, lines 8 and 9 remain unmarked. The different behavior of the logics LNPr and LNPm for this example is explained by considering the intuitions behind the reliability and minimal abnormality strategies. According to the reliability strategy, a formula is deemed 'suspicious' and is subsequently marked whenever it is derived on a line of which the condition intersects with some disjunct of a minimal Dab-formula. The minimal abnormality strategy is a tad less cautious. In this example, the latter strategy takes only one of the disjuncts P(p ∧ ∼p) and P(q ∧ ∼q) to be true, although of course we do not know which one. If, on the one hand, P(p ∧ ∼p) is true, then P(q ∧ ∼q) is safely considered false. Hence Or is derivable in view of line 9. If, on the other hand, P(q ∧ ∼q) is true, then P(p ∧ ∼p) is safely considered false. Hence Or is derivable in view of line 8. As for LNPr , the notion of final derivability and the syntactic consequence relation of LNPm are provided by Definitions 9 and 10. Thus, by these definitions, Γ4 LNPm Or. 4.3. Further (meta-)theoretical properties Formulating adaptive logics in the standard format has the advantage that a rich meta-theory is immediately available for this format. Generic proofs of Theorems 6–9 below can be found in [7]. Any adaptive logic in standard format is sound and complete with respect to its semantics: Theorem 6. Γ LNPx A iff Γ |LNPx A. Since LNPx is defined within the standard format for adaptive logics, the proof of Theorem 6 follows immediately by Corollary 2 and Theorem 9 from [7]. In case no Dab-formulas are derivable from a premise set by means of the lower limit logic, it is safe to consider all abnormalities as false. As a consequence, the adaptive logic will then yield the same consequence set as the logic that interprets all abnormalities as false (or equivalently, as the logic that fully validates the inference rules whose application the adaptive logic only allows conditionally). This logic is called the upper limit logic (ULL) of an adaptive logic. The ULL of LNPx is obtained by adding to LNP the axiom schemata (U1) and (U2), which trivialize all members of Ω1 and Ω2 respectively: M. Beirlaen, C. Strasser / Journal of Applied Logic 11 (2013) 147–168 159(U1) P(A ∧ ∼A) ⊃ B (U2) ¬O(A ∨ ∼A) ⊃ B ULL is related to LNP as set out by the Derivability Adjustment Theorem: Theorem 7. Γ ULL A iff (there is a  ⊆ Ω for which Γ LNP A ∨ Dab() or Γ LNP A). The set of Dab-consequences derivable from the premise set determines the amount to which the LNPx-consequence set will resemble the ULL-consequence set. This is why adaptive logicians say that LNPx adapts itself to a premise set. LNPx will always be at least as strong as LNP and maximally as strong as ULL: Theorem 8. CnLNP(Γ ) ⊆ CnLNPx (Γ ) ⊆ CnULL(Γ ). In view of Theorem 11 from [7], it follows immediately that: Corollary 1. CnLNP(Γ ) ⊆ CnLNPr (Γ ) ⊆ CnLNPm (Γ ) ⊆ CnULL(Γ ). If Γ is normal, i.e. if Γ has no Dab-consequences, then we can even prove a stronger result: Theorem 9. If Γ is normal, then CnLNPx (Γ ) = CnULL(Γ ). The reader may have noticed that ULL trivializes both gluts and gaps at accessible worlds, thus promoting "∼" to a fully classical negation connective. It should come as no surprise then, that ULL is just SDL in disguise. Where Γ ⊆ W , define Γ ¬ by replacing every A ∈ Γ by π(A), where π(A) is the result of replacing every occurrence of "∼" in A by "¬". Then: Theorem 10. Γ ULL A iff Γ ¬ SDL π(A). A proof outline for Theorem 10 is contained in Appendix D. At this point, it is useful to come back to the nullum crimen principle (NC) as defined in Section 2.1. In LNPx-proofs, we can derive the formula PA from the formula ¬O∼A on the condition that {¬O(A ∨∼A)} is false. Suppose that this condition is satisfied, i.e. that there is no normative gap relating to A. Then it follows by LNPx that PA, and we have derived a strong or positive permission from a weak or negative one. As such, (NC) is conditionally applicable in LNPx: if A is not prohibited, then it is permitted on the condition that there is no normative gap relating to A. 5. Related work 5.1. Alchourrón and Bulygin In [1–4], Alchourrón and Bulygin present a logic of norm-propositions that is built 'on top' of a logic of norms.12 A normproposition "there exists a norm to the effect that A is permitted" is formalized as NPA, where the operator N behaves like a quantifier over the norm PA. The latter formula (without N) is read simply as "A is permitted". Only obligations and permissions can occur inside the scope of the N-operator; formulas of the form NA where A is not of the form OB or PB are not well-formed formulas of NL. Alchourrón and Bulygin's logic of norms is just SDL. Their logic of norm-propositions NL extends SDL by adding to it the axiom schema (NK) and the rule (NRM): (NK) N(A ⊃ B) ⊃ (NA ⊃ NB) (NRM) If  A ⊃ B then  NA ⊃ NB In NL, OO-conflicts are formulas of the form NOA ∧NO¬A. Similarly, OP-conflicts are formulas of the form NOA ∧NP¬A. As opposed to normative conflicts, normative gaps cannot be expressed in the object language of NL. Instead, Alchourrón and Bulygin define a normative gap as a situation in which, for some CL-formula A, we cannot derive NPA nor NO¬A, i.e. NL NPA ∨ NO¬A. Normative conflicts and gaps are treated consistently in NL. Where A and B are well-formed NLformulas13: 12 Alchourrón and Bulygin's logic of norm-propositions is inspired by Rescher's assertion logic from [37]. 13 Alchourrón and Bulygin allow for iterated/nested deontic and normative operators. Nothing in principle prevents the occurrence of such nestings in LNPx . This requires some modifications of the language W and of the sets Ω1 and Ω2 such that e.g. PP(p ∧ ∼p) is also considered an abnormality. 160 M. Beirlaen, C. Strasser / Journal of Applied Logic 11 (2013) 147–168NOA ∧ NO¬A NL B (19) NOA ∧ NP¬A NL B (20) NL NPA ∨ NO¬A (21) However, the following variants of deontic explosion are valid in NL: NOA ∧ NO¬A NL NOB (22) NOA ∧ NP¬A NL NOB (23) With Alchourrón, Bulygin, and von Wright, we agree that "experience seems to testify that mutually contradictory norms may co-exist within one and the same legal order – and also that there are a good many "gaps" in any such order" [49, p. 32]. But if conflicting normative propositions indeed often coexist within a normative order, then deontic explosion should be avoided by any logic of normative propositions. No judge will agree that a normative order containing one or more conflicts contains norms to the effect that anything whatsoever is obligatory. Hence (22) and (23) cause serious problems for NL. (22) and (23) follow by applications of (NRM) and (NK) to the SDL-theorems  OA ⊃ (O∼A ⊃ OB) and  OA ⊃ (P∼A ⊃ OB) respectively. This led von Wright to questioning the presupposition of SDL by NL [49, footnote 2]. As opposed to NL, LNPx is not built 'on top' of the CL-based logic SDL. Although LNP contains full CL, its 'deontic' formulas make use of the much weaker logic CLoNs inside the scope of the Oand P-operator. This way, LNPx avoids deontic explosion. Interestingly, Alchourrón and Bulygin point out that under the assumptions of consistency and completeness, the logic of norm-propositions is 'isomorphic' to SDL: if we dismiss the possibility of normative conflicts and normative gaps, the differences between both logics disappear [1,4]. In Section 4.3 we proved this isomorphism for LNPx by showing that for normal (consistent and complete) premise sets, LNPx is just as strong as SDL. 5.2. Input/output logic In input/output logic (I/O logic), norms are represented as ordered pairs of formulas (a, x), where each coordinate of a pair is a CL-formula.14 The body of such a pair constitutes an input consisting of some condition or factual situation. The head constitutes an output representing what the norm tells us to be desirable/obligatory/permitted in that situation. A normative order or system is a set G of input/output pairs. G is seen as a 'transformation device' in which CL functions as its 'secretarial assistant' [34, p. 2]. In [31], Makinson and van der Torre define various operations of the form out(G, A) for making up the output of G given a set A containing factual information (input). In [32], the authors add constraints to these systems for dealing with contrary-to-duty scenarios and conflicting norms. In [33], the framework is extended for dealing with permissions. Constrained I/O logics make use of maximally consistent subsets. In doing so, they avoid explosion when dealing with conflicting conditional obligations, even if e.g. the norms (a, x) and (a,¬x) tell us that both x and ¬x are obligatory under the same circumstances. The treatment of obligation–permission conflicts by constrained I/O logics is less straightforward. In [42], Stolpe noted that the constrained systems deontically explode when facing a conflict between an obligation (a, x) and a positive permission (a,¬x).15 Stolpe's solution to this problem is to treat positive permissions as derogations: "a positive permission suspends, annuls or obstructs a covering prohibition, thereby generating a corresponding set of liberties" [42, p. 99]. Stolpe's solution creates an asymmetry between obligations and permissions. In obligation–obligation conflicts, both norms may still be of equal importance. In obligation–permission conflicts however, the permission always overrides the obligations it is in conflict with. Although certainly of interest in legal contexts, where the concept of derogation is a very important one, we doubt that all obligation–permission conflicts can be dealt with in this way. In the literature on I/O logic, normative gaps are left unmentioned. However, it seems possible to model gaps in this framework. For instance, we could say that there is a normative gap relating to proposition x in circumstances a if neither the obligations to do x or ¬x, nor the positive permissions to do x or ¬x are in the output of a given set of norms. One drawback seems to be that, whichever I/O operation we pick, both the obligation to do x ∨ ¬x and the positive permission to do x ∨ ¬x will always be in the output set. This is due to the closure of the output set under classical logic. Furthermore, as with Alchourrón and Bulygin's approach, normative gaps cannot be modeled at the object level in I/O logic. Another difference between I/O logic and LNPx is that for I/O operations the input is restricted to simple norm-bases, i.e. sets of input–output pairs. More complex formulas such as disjunctions between norms or negated norms cannot be fed into the system. LNPx is more flexible in this sense, since it can easily deal with premise sets containing formulas such as ¬Op, Oq ∨ Pr, etc. 14 The framework of I/O logic was initially developed for dealing with conditional norms. We do not discuss its merits as a conditional logic here. Instead, we focus on issues related to conflictand gap-tolerance. For a discussion of the representation of conditional norms in I/O logic, see [51]. 15 Translated to the I/O setting, deontic explosion ensues from a given input if – under certain circumstances invoked by the input – everything becomes obligatory in the output. M. Beirlaen, C. Strasser / Journal of Applied Logic 11 (2013) 147–168 1615.3. SDL-weakened modal logics The logics LNP and LNPx are SDL-weakened conflict-tolerant deontic logics. In the literature on conflicting norms in deontic logic, many different weakenings of SDL have been proposed for dealing with conflicting norms. The most popular approach for tolerating normative conflicts is to weaken SDL by invalidating the aggregation principle (O-AND), e.g. [17,39,52,18]. Non-monotonic systems that validate only certain applications of (O-AND) were presented in [28,26,22,35].16 An alternative approach for weakening SDL suggested by Goble in [19,20] restricts the modal inheritance principle (RM): (RM) If  A ⊃ B then  OA ⊃ OB The result is a family of monotonic SDL-weakened systems capable of consistently accommodating OO-conflicts. Nonmonotonic adaptive extensions of this logic were presented in [43,45]. A third way of weakening SDL consists in giving up the Ex Contradictione Quodlibet schema: (ECQ) (A ∧ ¬A) ⊃ B The result is a paraconsistent deontic logic. This approach was taken up in [16,36]. Non-monotonic adaptive paraconsistent deontic logics are presented in [13,12]. The systems defined in this paper are 'paraconsistent' in the sense that they restrict (ECQ) for the negation connective "∼". None of the SDL-weakened systems mentioned in this section were designed for reasoning in the presence of normative gaps. Consequently, these logics do not provide a satisfactory treatment of normative gaps. Moreover, in systems in which a permission operator is characterized, the latter is always treated as the dual of the obligation-operator, i.e. PA ≡ ¬O¬A is valid. 5.4. Strengthened SDL-based logics In [29], Kooi and Tamminga deal with conflicting norms by enriching SDL so as to be capable of distinguishing between various sources and interest groups in view of which norms arise. Moreover, following [27] they equip their system with modal stit-operators for dealing with the difficult notion of (moral) agency. Similarly, we could try to deal with conflicting norms by imposing a preference ordering on our obligations and permissions, e.g. [24]. Such extensions are very successful in increasing the expressive power of SDL, but they are unable to consistently allow for all normative conflicts. Remember from Section 1.1 that conflicts may arise between norms promulgated at the same time, by the same authority. It is not difficult to see how we could extend this type of reasoning to norms of the same hierarchical foot, addressed at the same group of people, etc. so that in the end we need a logic that invalidates at least some SDL-theorems if we want to deal with all instances of normative conflicts. A weakness of the systems devised in this paper is that they are not very expressive. Relativizing the deontic operators to individual/group operators is relatively straightforward (this can be done by defining a set of operators OI and PI where I is an index set representing some (group of) agent(s)), but it is less clear how to extend LNPx in a way that it satisfactorily treats e.g. conditional norms or the notion of agency. Moreover, the strengthened SDL-based logics discussed here point to a trade-off between complexity and degrees of conflict-tolerance in dealing with conflicts in deontic logic. On the one hand, many normative conflicts can be dealt with by making explicit the different sources they arise from or the different levels of priority attributed to them. On the other hand, if we want our logic to be fully conflict-tolerant we need to weaken SDL and allow only for the defeasible validity of some of its inferences. The latter option is technically more involving and computationally more complex.17 6. Conclusion and outlook We presented two non-monotonic logics for reasoning with norm-propositions in the presence of normative conflicts and normative gaps. The logics LNPr and LNPm interpret a given premise set 'as consistently and as completely as possible'. LNPr uses a slightly more cautious strategy than LNPm . LNPr and LNPm are equipped with a well-defined semantics and proof theory. Due to their characterization within the standard framework for adaptive logics, soundness and completeness properties are guaranteed, as are many of their meta-theoretical properties. As opposed to other systems devised for dealing with norm-propositions, the logics defined in this paper make use of a formal language in which all necessary distinctions can be made already at the object level. This is realized by making use of a classical negation connective outside, and a paraconsistent and paracomplete negation connective inside the scope of the operators O and P. 16 Although the (constrained) I/O logics from Section 5.2 are presented in a non-modal framework, these too are non-monotonic non-aggregative systems in the sense that for two obligations (a, x) and (a, y) fed to the logic, the output does not always contain x ∧ y, e.g. if x and y are logically incompatible. 17 For some results and a discussion on the computational complexity of adaptive logics, see [11,46]. 162 M. Beirlaen, C. Strasser / Journal of Applied Logic 11 (2013) 147–168Two possible drawbacks of the non-classical SDL-weakened approach taken up here are that (i) due to their nonmonotonicity the resulting logics are highly complex from the computational point of view; and (ii) the systems defined in this paper are not very expressive. In response to (i), we conjecture that in order to model actual human reasoning, a complex logic is what we need (see e.g. [11]). In response to (ii), we point to some existing work on more expressive adaptive deontic logics [12,44,43] and hope to provide more results in the future. Acknowledgements Research for this paper was supported by subventions from the Research Foundation – Flanders (FWO Vlaanderen). The authors are indebted to Tjerk Gauderis, Joke Meheus, Frederik van de Putte, and two anonymous referees for their valuable comments and suggestions on earlier drafts of this paper. Appendix A. CLoNs and the positive fragment of CL Syntactically, the positive fragment of CL is defined by Modus Ponens (MP) and the following axiom schemata: (A⊃1) A ⊃ (B ⊃ A) (A⊃2) (A ⊃ (B ⊃ C)) ⊃ ((A⊃B) ⊃ (A ⊃ C)) (A⊃3) ((A ⊃ B) ⊃ A) ⊃ A (A∧1) (A ∧ B) ⊃ A (A∧2) (A ∧ B) ⊃ B (A∧3) A ⊃ (B ⊃ (A ∧ B)) (A∨1) A ⊃ (A ∨ B) (A∨2) B ⊃ (A ∨ B) (A∨3) (A ⊃ C) ⊃ ((B ⊃ C) ⊃ ((A∨B) ⊃ C)) (A≡1) (A ≡ B) ⊃ (A ⊃ B) (A≡2) (A ≡ B) ⊃ (B ⊃ A) (A≡3) (A ⊃ B) ⊃ ((B ⊃ A) ⊃ (A ≡ B)) In order to obtain full CL, all one has to do is to add the axiom schemata (A¬1) and (A¬2) below to its positive fragment. (A¬1) (A ⊃ ¬A) ⊃ ¬A (A¬2) A ⊃ (¬A ⊃ B) CLoNs is defined by adding to the positive fragment of CL the axiom schemata (A∼∼)–(A∼≡) from Section 2.3. CLoNsderivability for a formula A ∈W∼ from a set of formulas Γ ⊆W∼ is denoted by Γ CLoNs A. If A ∈W∼ is CLoNs-derivable from the empty set, then CLoNs A. Semantically, CLoNs-models are associated with an assignment function v : W l → {0,1}. The valuation function v M for the CLoNs-model M is defined by adjusting clauses (Cl) and (C∼∼)–(C≡) so that references to accessible worlds are dropped, e.g. (Cl) where A ∈ W l, v M(A) = 1 iff v(A) = 1 (C∼∼) v M(∼∼A) = 1 iff v M(A) = 1 and so on. A CLoNs-model M verifies A (M  A) iff v M(A) = 1. Where Γ ⊆ W∼ , M is a CLoNs-model of Γ iff M is a CLoNs-model and M  A for all A ∈ Γ . |CLoNs A iff all CLoNs-models verify A, and Γ |CLoNs A iff all CLoNs-models of Γ verify A. CLoNs is similar to Anderson and Belnap's logic of first-degree entailment (FDE) from [5] in the sense that its negation connective has no properties except for de Morgan's laws. Unlike FDE however, CLoNs validates full Modus Ponens. Appendix B. Some facts about LNP and CLoNs The following theorems will come in handy for the proof of Theorem 2. Let in the remainder L ∈ {CLoNs,LNP}: Theorem 11. L is reflexive, transitive and monotonic.18 18 Where CnL(Γ ) denotes the consequence set of some premise set Γ for L, L is reflexive iff, for all premise sets Γ , Γ ⊆ CnL(Γ ); it is transitive iff, for all sets of wffs Γ and Γ ′ , if Γ ′ ⊆ CnL(Γ ) then CnL(Γ ∪ Γ ′) ⊆ CnL(Γ ); and it is monotonic iff, for all sets of wffs Γ and Γ ′ , CnL(Γ ) ⊆ CnL(Γ ∪ Γ ′). M. Beirlaen, C. Strasser / Journal of Applied Logic 11 (2013) 147–168 163Theorem 12. L is compact (if Γ L A then Γ ′ L A for some finite Γ ′ ⊆ Γ ). Theorem 13. If Γ L B and A ∈ Γ , then Γ − {A} L A ⊃ B (Generalized Deduction Theorem for L). The proofs of Theorems 11–13 are straightforward and safely left to the reader. Fact 1. (i) OA,OB LNP O(A ∧ B), (ii) OA,PB LNP P(A ∧ B), (iii) LNP (OA ∧ OB) ⊃ O(A ∧ B), (iv) LNP (OA ∧ PB) ⊃ P(A ∧ B), (v) LNP P(A ⊃ A), (vi) If CLoNs A′ ⊃ A then A ⊃ B CLoNs A′ ⊃ B, (vii) If CLoNs B ⊃ B ′ then A ⊃ B CLoNs A ⊃ B ′ , (viii) CLoNs (A ∨ (A ⊃ B)) ≡ (((A ∨ (A ⊃ B)) ⊃ B) ⊃ B), (ix) CLoNs (A ⊃(A ⊃ B))⊃(A ⊃ B), (x) CLoNs A ∨ (A ⊃ B). Proof. Ad (i). Suppose OA and OB . By (A∧3), CLoNs A ⊃ (B ⊃ (A ∧ B)). By (NEC∼), it follows that LNP O(A ⊃ (B ⊃ (A ∧ B))). By (K), LNP OA ⊃ O(B ⊃ (A ∧ B)). By (MP), O(B ⊃ (A ∧ B)). By (K), OB ⊃ O(A ∧ B). By (MP), O(A ∧ B). Ad (ii). Suppose OA and PB . By (A∧3), CLoNs A ⊃ (B ⊃ (A ∧ B)). By (NEC∼), LNP O(A ⊃ (B ⊃ (A ∧ B))). By (K), LNP OA ⊃ O(B ⊃ (A ∧ B)). By (MP), O(B ⊃ (A ∧ B)). By (KP), PB ⊃ P(A ∧ B). By (MP), P(A ∧ B). Ad (iii)–(iv). Immediate in view of (i), (ii), and Theorem 13. Ad (v). Since A ⊃ A is a theorem of the positive fragment of CL, it is also a CLoNs-theorem. By (NEC∼), LNP O(A ⊃ A). By (D), LNP P(A ⊃ A). Ad (vi). Suppose CLoNs A′ ⊃ A. By (A⊃2), CLoNs (A′ ⊃(A ⊃ B))⊃((A′ ⊃ A)⊃(A′ ⊃ B)). By (A⊃1) and (MP), A ⊃ B CLoNs A′ ⊃(A ⊃ B). The rest follows by multiple applications of (MP). Ad (vii). The proof is similar and left to the reader. Ad (viii). Left-to-right: By (MP), (A ∨ (A ⊃ B)) ⊃ B, A ∨ (A ⊃ B) CLoNs B . The rest follows by Theorem 13. Right-to-left: By (A⊃1), (†) CLoNs B ⊃ (A ⊃ B). By (A∨2), (‡) CLoNs (A ⊃ B) ⊃ (A ∨ (A ⊃ B)). Altogether, by (†), (‡), (vii) and (MP), CLoNs B ⊃ (A ∨ (A ⊃ B)). Hence, by (vii), CLoNs ((A ∨ (A ⊃ B) ⊃ B) ⊃ B) ⊃ ((A ∨ (A ⊃ B) ⊃ B) ⊃ (A ∨ (A ⊃ B))). By (A⊃3), CLoNs ((A ∨ (A ⊃ B) ⊃ B) ⊃ (A ∨ (A ⊃ B))) ⊃ (A ∨ (A ⊃ B)). Hence, again by (vii), ((A ∨ (A ⊃ B) ⊃ B) ⊃ B) ⊃ (A ∨ (A ⊃ B)). Ad (ix). By (MP), A, A ⊃(A ⊃ B) CLoNs A ⊃ B . By (MP), A, A ⊃(A ⊃ B) CLoNs B . By Theorem 13, A ⊃(A ⊃ B) CLoNs A ⊃ B , CLoNs (A ⊃(A ⊃ B))⊃(A ⊃ B). Ad (x). By (A∨1), CLoNs A ⊃ (A ∨ (A ⊃ B)). By (vi), (A ∨ (A ⊃ B)) ⊃ B CLoNs A ⊃ B . By Theorem 13, CLoNs ((A ∨ (A ⊃ B)) ⊃ B) ⊃ (A ⊃ B). By (A∨2), CLoNs (A ⊃ B) ⊃ (A ∨ (A ⊃ B)). Hence, by (vii), CLoNs ((A ∨ (A ⊃ B)) ⊃ B) ⊃ (A ∨ (A ⊃ B)). By (viii), CLoNs (A ∨ (A ⊃ B)) ≡ (((A ∨ (A ⊃ B)) ⊃ B) ⊃ B). Thus, by (vii), CLoNs ((A ∨ (A ⊃ B)) ⊃ B) ⊃ (((A ∨ (A ⊃ B)) ⊃ B) ⊃ B). By (ix) and (MP), CLoNs ((A ∨ (A ⊃ B)) ⊃ B) ⊃ B . By (viii), (A≡2), and (MP), CLoNs A ∨ (A ⊃ B).  Appendix C. Proofs of Theorems 1 and 2 In order to simplify the notation in the following meta-proofs we define R(w) = {w ′ | R w w ′}. Proof of Theorem 1. Let in the following M = 〈W , w0, R, v0, v〉 be an LNP-model. It is easy to check that all CL-axiom schemata hold at w0 in M due to (C0), (C¬), and (C⊃)–(C≡). Similarly, (†) where w ∈ W \ {w0}, all CLoNs-axiom schemata hold at w in M due to (Cl) and (C∼∼)–(C≡). Ad (NEC∼). Let |CLoNs A. By (CO), (†) and the definition of R , v M(OA, w0) = 1. Ad (K). Suppose M  O(A ⊃ B). By (CO) and (C⊃), for all w ∈ R(w0), v M(A, w) = 0 or v M(B, w) = 1. Suppose M  OA, then for all w ∈ R(w0), v M(A, w) = 1. Hence, for all w ∈ R(w0), v M(B, w) = 1. Thus by (CO), M  OB . Hence, by (C⊃), M  OA ⊃ OB . Altogether, by (C⊃), M  O(A ⊃ B) ⊃ (OA ⊃ OB). 164 M. Beirlaen, C. Strasser / Journal of Applied Logic 11 (2013) 147–168Ad (D). Suppose M  OA. Hence for all w ∈ R(w0), v M(A, w) = 1 (by (CO)). By the non-emptiness of R , there is a w ∈ R(w0) for which v M(A, w) = 1. By (CP), M  PA. By (C⊃), M  OA ⊃ PA. Ad (KP). Suppose M  O(A ⊃ B). By (CO) and (C⊃), (‡) for all w ∈ R(w0), v M(A, w) = 0 or v M(B, w) = 1. Suppose M  PA. Then, by (CP) there is a w ∈ R(w0) for which v M(A, w) = 1. Hence, by (‡), there is a w ∈ R(w0) such that v M(B, w) = 1. Thus, by (CP) M  PB and, by (C⊃), M  PA ⊃ PB . Altogether, by (C⊃), M  O(A ⊃ B) ⊃ (PA ⊃ PB). Ad (OD). Suppose M  O(A ∨ B). By (CO) and (C∨), (	) for all w ∈ R(w0), v M(A, w) = 1 or v M(B, w) = 1. Suppose M  PB . By (CP), for all w ∈ R(w0), v M(B, w) = 0. By (	), for all w ∈ R(w0), v M(A, w) = 1. Thus, by (CO), M  OA. Hence, by (C∨), M  OA ∨ PB . Altogether, by (C⊃), M  O(A ∨ B) ⊃ (OA ∨ PB). Ad (PD). This is similar to the previous case and is left to the reader. We now know that all axiom schemata and rules of LNP are semantically valid. That Γ LNP A implies Γ |LNP A can now be shown via the usual induction on the length of the proof of A. This is safely left to the reader.  Let in the remainder Wc be the LNP-deductively closed and maximally LNP-non-trivial subsets of W .19 Moreover, let W ∼c be the CLoNs-deductively closed subsets Γ of W∼ where Γ is prime, i.e. for each A ∨ B ∈ Γ either A ∈ Γ or B ∈ Γ . For the completeness proof of LNP, we make use of the following lemmas20: Lemma 1. If  ∈ Wc, then  is prime. Proof. Suppose that, for a  ∈ Wc , A ∨ B ∈  and A /∈  and B /∈ . Then, since  is maximally LNP-non-trivial,  ∪ {A} is trivial and  ∪ {B} is trivial. Then, for any C ∈ W ,  ∪ {A} LNP C and  ∪ {B} LNP C . Then, by Theorem 13,  LNP A ⊃ C and  LNP B ⊃ C . But then, by (MP) and (A∨3),  LNP (A ∨ B) ⊃ C . Since A ∨ B ∈ , since by (MP)  LNP C , and since  is LNP-deductively closed, C ∈ . This contradicts the supposition. Hence if A ∨ B ∈ , then A ∈  or B ∈ .  Where Γ ∈ Wc and A ∈ W∼ , we will use the following abbreviations: ΓO = {B | OB ∈ Γ }, Γ AO = ΓO ∪ {A}, ΓP = {B | PB /∈ Γ }, ∨ΓP = {∨Θ | Θ ⊆ ΓP ,Θ is finite} and ∨Γ BP = {∨Θ | Θ ⊆ ΓP ∪ {B},Θ is finite}. Lemma 2. Let Γ ∈ Wc. (i) If C ∈ CnCLoNs(ΓO ) then OC ∈ Γ . (ii) Where PA ∈ Γ , if C ∈ CnCLoNs(Γ AO ) then PC ∈ Γ . Proof. Ad (i). Suppose that ΓO CLoNs C . Then Γ ′ CLoNs C for some finite Γ ′ ⊆ ΓO (given the compactness of CLoNs). Hence, CLoNs (∧Γ ′) ⊃ C by Theorem 13. Thus, LNP O((∧Γ ′) ⊃ C) by (NEC∼). By (K), CLoNs O∧Γ ′ ⊃ OC . By the deductive closure of Γ , the fact that Γ ′ ⊆ Γ and Fact 1(i), O∧Γ ′ ∈ Γ . By (MP), OC ∈ Γ . Ad (ii). Suppose that Γ AO CLoNs C . Then Γ ′ CLoNs C for some finite Γ ′ ⊆ Γ AO (given the compactness of CLoNs). Then Γ ′ ∪ {A} CLoNs C by the monotonicity of CLoNs. Then CLoNs (∧Γ ′ ∧ A) ⊃ C by Theorem 13. Then LNP O((∧Γ ′ ∧ A) ⊃ C) by (NEC∼). Then LNP P(∧Γ ′ ∧ A) ⊃ PC by (KP) and (MP). By the supposition, {OB | B ∈ Γ ′} ⊆ Γ and PA ∈ Γ . Given the deductive closure of Γ and LNP (O(∧Γ ′) ∧ PA)⊃P(∧Γ ′ ∧ A) (which follows from Fact 1(ii)), it follows that P(∧Γ ′ ∧ A) ∈ Γ . Hence PC ∈ Γ , since Γ is deductively closed and LNP P(∧Γ ′ ∧ A) ⊃ PC .  Lemma 3. Let Γ ∈ Wc. (i) Where PA ∈ Γ , ∨ΓP ∩ CnCLoNs(Γ AO ) = ∅. (ii) Where B /∈ ΓO , ∨Γ BP ∩ CnCLoNs(ΓO ) = ∅. Proof. Ad (i). Let C = ∨Θ where Θ = {C1, . . . , Cn} ⊆ ΓP . Suppose C ∈ CnCLoNs(Γ AO ) then by Lemma 2(ii), P∨Θ ∈ Γ . Hence, by (PD), ∨n i=1 PCi ∈ Γ . Hence, since Γ is prime, there is an i ∈ {1, . . . ,n} for which PCi ∈ Γ and hence Ci /∈ ΓP , – a contradiction. Ad (ii). Let C = ∨Θ where Θ = {C1, . . . , Cn} ⊆ (ΓP ∪ {B}). Suppose C ∈ CnCLoNs(ΓO ). By Lemma 2(i), O∨Θ ∈ Γ . Assume that, where i ∈ {1, . . . ,n}, all Ci ∈ ΓP . By (D), P∨Θ ∈ Γ . By (PD), ∨ni=1 PCi ∈ Γ . Hence, since Γ is prime, there is an i ∈ {1, . . . ,n} such that PCi ∈ Γ and hence Ci /∈ ΓP , – a contradiction. Hence there is a non-empty J ⊆ {1, . . . ,n} such that for each j ∈ J , C j = B . Hence, by (OD), OB ∨ P∨{1,...,n}\ J Ci ∈ Γ . Thus, by (PD), OB ∨ ∨{1,...,n}\ J PCi ∈ Γ . Since B /∈ ΓO and since Γ is prime, there is an i ∈ {1, . . . ,n} \ J such that PCi ∈ Γ and hence Ci /∈ ΓP , – a contradiction.  Lemma 4. Let Γ ∈ Wc. 19 Where WL is the set of wffs of L, Γ is L-trivial iff CnL(Γ ) = WL , Γ is L-deductively closed iff CnL(Γ ) = Γ , and Γ is maximally L-non-trivial iff it is L-non-trivial and all supersets Γ ′ ⊃ Γ are L-trivial. 20 The proof of Lemma 4 is inspired by the proof of Lemma 1.7.1 from [9]. M. Beirlaen, C. Strasser / Journal of Applied Logic 11 (2013) 147–168 1651. Where PA ∈ Γ , there is a  ⊆W∼ for which (i) Γ AO ⊆ , (ii) ∨ΓP ∩  = ∅, and (iii)  ∈ W ∼c . 2. Where B /∈ ΓO , there is a  ⊆W∼ for which (i) ΓO ⊆ , (ii) ∨Γ BP ∩  = ∅, and (iii)  ∈ W ∼c . Proof. Let 〈Γ O,Γ P〉 ∈ {〈Γ AO ,∨ ΓP 〉, 〈ΓO ,∨ Γ BP 〉}. Where 〈B1, B2, . . .〉 is a list of the members of W∼ , define 0 = CnCLoNs(Γ O) and  = 0 ∪ 1 ∪ * * * where i+1 = { CnCLoNs(i ∪ {Bi+1}) if Γ P ∩ CnCLoNs(i ∪ {Bi+1}) = ∅ i otherwise Ad (i). This holds by the construction and the reflexivity of CLoNs. Ad (ii). By Lemma 3 0 ∩ Γ P = ∅. The rest follows by the construction. Ad (iii). We first show that  is CLoNs-deductively closed. Suppose there is a Bi /∈  such that  CLoNs Bi . Then, by the construction of , there is a D ∈ Γ P such that  ∪ {Bi} CLoNs D and hence by Theorem 13,  CLoNs Bi ⊃ D . However, by (MP) also  CLoNs D . By the compactness of CLoNs there is a i for which i CLoNs D . By the construction i = CnCLoNs(i) and whence D ∈ i . Hence, D ∈ , – a contradiction with (ii). We now show that  is prime. Suppose A1 ∨ A2 ∈ . Assume A1, A2 /∈ . Hence, by the construction of , ∪{A1} CLoNs D1 and ∪{A2} CLoNs D2 for some D1, D2 ∈ Γ P . By Theorem 13,  CLoNs A1 ⊃ D1 and  CLoNs A2 ⊃ D2. By some simple propositional manipulations,  CLoNs (A1 ∨ A2) ⊃ (D1 ∨ D2). By (MP),  CLoNs D1 ∨ D2 and hence D1 ∨ D2 ∈ . However, by the definition of Γ P , D1 ∨ D2 ∈ Γ P , – a contradiction with (ii).  Definition 12. The binary relation R ⊆ (Wc × W ∼c ) is defined as follows: RΓ  iff the following two conditions are met: (a) if OA ∈ Γ then A ∈ , and (b) if A ∈  then PA ∈ Γ . In view of the definition of R, the following holds: Lemma 5. Where Γ ∈ Wc, PA ∈ Γ iff there is a  ∈ W ∼c such that RΓ  and A ∈ . Proof. Left–right: Suppose PA ∈ Γ . Then, by Lemma 4.1, there is a  ⊆W∼ such that (i) Γ AO ⊆ , (ii) for all C ∈ ΓP , C /∈ , (iii)  ∈ W ∼c . We now show that RΓ . Ad (a): if, for some D , OD ∈ Γ then D ∈ Γ AO , hence D ∈  by (i). Ad (b): suppose PE /∈ Γ for some E ∈W∼ . Then E ∈ ΓP , hence E /∈  by (ii). Right–left: Follows directly by Definition 12.  Lemma 6. For every Γ ∈ Wc, there is a  ∈ W ∼c such that RΓ  (i.e. R is non-empty). Proof. By Fact 1(v), LNP P(A ⊃ A). Hence, P(A ⊃ A) ∈ Γ for every Γ ∈ Wc . But then, by Lemma 5, there is a  ∈ W ∼c such that RΓ  and A ⊃ A ∈ . Hence R is non-empty as required.  Lemma 7. Where Γ ∈ Wc, OA ∈ Γ iff, for all  ∈ W ∼c such that RΓ , A ∈ . Proof. Left–right: This is an immediate consequence of Definition 12. Right–left: Suppose OA /∈ Γ . Hence, A /∈ ΓO . By Lemma 4.2, there is a  ⊆W∼ for which (i) ΓO ⊆ , (ii) (ΓP ∪{A})∩ = ∅, and (iii)  ∈ W ∼c . We now show that RΓ . Ad (a): if, for some D , OD ∈ Γ then D ∈ ΓO , hence D ∈  by (i). Ad (b): suppose PE /∈ Γ for some E ∈W∼ . Then E ∈ ΓP , hence E /∈  by (ii).  Lemma 8. Where  ∈ Wc, there is an LNP-model M such that M  A for all A ∈  and M  A for all A ∈W \ . Proof. Let  ∈ Wc . We construct an LNP-model M = 〈{} ∪ W ∼c , w0,R, v0, v〉 such that: (i) w0 = , (ii) for all A ∈Wa , v0(A, w0) = 1 iff A ∈ w0, (iii) for all A ∈W l and all w ∈ W ∼c , v(A, w) = 1 iff A ∈ w . 166 M. Beirlaen, C. Strasser / Journal of Applied Logic 11 (2013) 147–168By Lemma 6, R is non-empty. We now show that: (∗) (a) for all A ∈W, v M(A, w0) = 1 iff A ∈ w0, (b) for all A ∈W∼ and all w ∈ W ∼c , v M(A, w) = 1 iff A ∈ w. The proof proceeds as usual by an induction on the complexity of A. Let w ∈ {w0} ∪ W ∼c , and A ∈ Wa . If w = w0, then, by (ii), v0(A, w0) = 1 iff A ∈ w0. By (C0), it follows that v M(A, w) = 1 iff A ∈ w . If w = w0, then, by (iii), v(A, w) = 1 iff A ∈ w . By (Cl), it follows that v M(A, w) = 1 iff A ∈ w . Hence, for all w ∈ {w0} ∪ W ∼c , v M(A, w) = 1 iff A ∈ w and (∗) is valid for all A ∈Wa . Depending on the logical form of A, we distinguish 8 cases (6 for the connectives ∼,¬,∨,∧,⊃,≡, and 2 for the modal operators O and P) and show for each of them that v M(A, w) = 1 iff A ∈ w . Case 1. Let w ∈ W ∼c . We show that v M(∼A, w) = 1 iff ∼A ∈ w . Either ∼A ∈ W l , or A has one of the forms ∼B , B ∨ C , B ∧ C , B ⊃ C , or B ≡ C (note that, since w = w0, A cannot have the form OB or PB). If ∼A ∈W l , then, by (Cl), v M(∼A, w) = 1 iff v(∼A, w) = 1. By (iii), it follows that v M(∼A, w) = 1 iff ∼A ∈ w . If A has the form ∼B , then, by (C∼∼), v M(∼∼B, w) = 1 iff v M(B, w) = 1. By the induction hypothesis, v M(∼∼B, w) = 1 iff B ∈ w . By (A∼∼), v M(∼A, w) = 1 iff ∼A ∈ w . If A has the form B ∨ C , then, by (C∼∨), v M(∼(B ∨ C), w) = 1 iff v M(∼B ∧ ∼C, w) = 1 iff [by (C∧)] v M(∼B, w) = 1 and v M(∼C, w) = 1 iff [by the induction hypothesis] ∼B ∈ w and ∼C ∈ w iff [by (A∧1), (A∧2), and (A∧3)] ∼B ∧ ∼C ∈ w iff [by (A∼∨)] ∼A ∈ w . The cases where A is of one of the forms B ∧ C , B ⊃ C , or B ≡ C are similar and left to the reader. Case 2. Let w = w0. Suppose v M(¬A, w) = 1. By (C¬), v M(A, w) = 0. By the induction hypothesis, A /∈ w . Then, since w is maximally LNP-non-trivial, w ∪ {A} is LNP-trivial and w ∪ {A} LNP ¬A. By Theorem 13, it follows that w LNP A ⊃ ¬A. Then, since w is LNP-deductively closed, A ⊃ ¬A ∈ w and, by (A¬1) and (MP), ¬A ∈ w . Suppose ¬A ∈ w . We show via reductio that A /∈ w . Suppose thus that A ∈ w . Then, by (A¬2), (MP), and since w is LNPdeductively closed, B ∈ w for any B ∈ W . This contradicts the non-triviality of w , hence A /∈ w . But then, by the induction hypothesis v M(A, w) = 0 and, by (C¬), v M(¬A, w) = 1. Case 3. Let w ∈ {w0} ∪ W ∼c . Suppose v M(A ∨ B, w) = 1. Then, by (C∨), v M(A, w) = 1 or v M(B, w) = 1. By the induction hypothesis, A ∈ w or B ∈ w . Hence, by (A∨1), (A∨2), (MP), and the fact that w is LNP-(in case w = w0)/CLoNs-(in case w ∈ W ∼c )-deductively closed, A ∨ B ∈ w . Suppose A ∨ B ∈ w . If w = w0, then, by the definition of W ∼c , A ∈ w or B ∈ w . If w = w0, then, by Lemma 1, A ∈ w or B ∈ w . By the induction hypothesis, v M(A, w) = 1 or v M(B, w) = 1. Hence, by (C∨), v M(A ∨ B, w) = 1. Case 4. Let w ∈ {w0} ∪ W ∼c . Suppose v M(A ⊃ B, w) = 1. Then by (C⊃), v M(A, w) = 0 or v M(B, w) = 1. By the induction hypothesis, A /∈ w or B ∈ w . Let now w ∈ W ∼c . If A /∈ w , then, since CLoNs A ∨ (A ⊃ B) by Fact 1(x) and since w is prime, also A ⊃ B ∈ w . If B ∈ w , then since by (A⊃1) CLoNs B ⊃ (A ⊃ B) and by (MP), also A ⊃ B ∈ w . The same argument applies to w = w0 since also LNP A ∨ (A ⊃ B), and (A⊃1) and (MP) are also valid in LNP. Suppose A ⊃ B ∈ w . By (MP), if A ∈ w then B ∈ w . By the induction hypothesis, if v M(A, w) = 1 then v M(B, w) = 1. Hence, by (C⊃), v M(A ⊃ B, w) = 1. The proof for the other classical connectives (cases 4–6) is similar and left to the reader. We proceed with the cases for O and P. Case 7. Let w = w0. By Lemma 7, OA ∈ w0 iff A ∈ w for all w such that Rw0 w . Hence, by the induction hypothesis, OA ∈ w0 iff v M(A, w) = 1 for all w such that Rw0 w . But then, by (CO), OA ∈ w0 iff v M(OA, w0) = 1. Case 8. Let w = w0. By Lemma 5, PA ∈ w0 iff A ∈ w for some w such that Rw0 w . Hence, by the induction hypothesis, PA ∈ w0 iff v M(A, w) = 1 for some w such that Rw0 w . But then, by (CP), PA ∈ w0 iff v M(PA, w0) = 1. The rest follows by (i) and (∗).  Lemma 9. Let Γ ⊆W and Γ LNP A. There is a  ⊆W such that: (i) Γ ⊆ , (ii) A /∈ , and (iii)  ∈ Wc. Proof. Where 〈B1, B2, . . .〉 is a list of the members of W , define 0 = CnLNP(Γ ) and  = 0 ∪ 1 ∪ * * * where i+1 = { CnLNP(i ∪ {Bi+1}) if A /∈ CnLNP(i ∪ {Bi+1}) i else Ad (i). This holds by the construction of  and the reflexivity of LNP. Ad (ii). This holds by the construction and since A /∈ CnLNP(Γ ). Ad (iii). Assume that B /∈  and  LNP B . Hence, by the construction of ,  ∪ {B} LNP A and whence by Theorem 13,  LNP B ⊃ A. But then by (MP),  LNP A. Thus, by the compactness of LNP and since each i = CnLNP(i), there is a i such that A ∈ i , – a contradiction to (ii). M. Beirlaen, C. Strasser / Journal of Applied Logic 11 (2013) 147–168 167Suppose B /∈ . Assume that ¬B /∈ . By the construction of  and the monotonicity of LNP,  ∪ {¬B} LNP A and whence by Theorem 13,  LNP ¬B ⊃ A. Analogously,  LNP B ⊃ A. By (A∨3),  LNP (B ∨ ¬B)⊃ A. Since CL B ∨ ¬B , also  LNP B ∨ ¬B . By (MP),  LNP A, – a contradiction to (ii). Hence, ¬B ∈ . Thus,  ∪ {B} is CL-trivial and hence also LNP-trivial.  Proof of Theorem 2. Suppose Γ LNP A. Then, by Lemma 9, there is a  ⊇ Γ such that A /∈  and  ∈ Wc . Then, by Lemma 8, there is an LNP-model M such that M  B for all B ∈ Γ and M  A. Hence Γ |LNP A.  Appendix D. Proof of Theorem 10 SDL is fully axiomatized by adding to CL the axiom schemata (K), (D), (PO), and (NEC¬): (PO) PA ≡ ¬O¬A (NEC¬) If CL A then  OA Lemma 10. If Γ ULL A then Γ ¬ SDL π(A). Proof. It easily checked that, under the transformation given in Section 4.3, all of (K), (D), (KP), (OD), (PD), (U1), and (U2) are SDL-valid. Moreover, since CLoNs is a proper fragment of CL, (NEC∼) too is valid in SDL (assuming again the transformation from Section 4.3).  Lemma 11. If Γ ¬ SDL π(A) then Γ ULL A. Proof. By the definition, ULL verifies (K) and (D). It remains to show that ULL verifies (i) all instances of PA ≡ ¬O∼A and (ii) the rule "If CL∼ A then ULL OA", where CL∼ is classical propositional logic with the negation symbol ∼. Ad (i). Left–right: By (A∧3), PA ⊃ (O∼A ⊃ (PA ∧ O∼A)). By Fact 1(iv), (PA ∧ O∼A) ⊃ P(A ∧ ∼A). Thus, by some propositional manipulations in CL, PA ⊃ (O∼A ⊃ P(A ∧ ∼A)), which is CL-equivalent to (†) PA ⊃ (¬O∼A ∨ P(A ∧ ∼A)). Suppose now that PA. By (†), ¬O∼A ∨ P(A ∧ ∼A). Moreover, by (U1), P(A ∧ ∼A) ⊃ ¬O∼A. Thus, by (MP) and some simple CLmanipulations, we obtain ¬O∼A. Right–left: By CL, O(A ∨ ∼A) ∨ ¬O(A ∨ ∼A). By (OD), O(A ∨ ∼A) ⊃ (O∼A ∨ PA). Thus, by some propositional manipulations in CL, (O∼A ∨ PA) ∨ ¬O(A ∨ ∼A). The latter formula is CL-equivalent to ¬O∼A ⊃ (PA ∨ ¬O(A ∨ ∼A)). Suppose now that ¬O∼A. By (MP), PA ∨¬O(A ∨∼A). By (U2), ¬O(A ∨∼A) ⊃ PA. Thus, by (MP) and some simple CL-manipulations, PA. Ad (ii). Note that A ∈W∼ iff π(A) ∈W¬ . Thus, where (A∼1) (A ⊃ ∼A) ⊃ ∼A (A∼2) A ⊃ (∼A ⊃ B) it follows by the definitions of CLoNs and CL that CLoNs∪{(A∼1),(A∼2)} A iff CL π(A). We show that (i) if CLoNs A, then ULL A, (ii) ULL O((A ⊃ ∼A) ⊃ ∼A), and (iii) ULL O(A ⊃ (∼A ⊃ B)). (i) In case A is a CLoNs-theorem, OA follows immediately in view of (NEC∼). (ii) (A ∨ ∼A) ⊃ ((A ⊃ ∼A) ⊃ ∼A) is an instance of the theorem (A ∨ B) ⊃ ((A ⊃ B) ⊃ B) of positive CL, thus it is a CLoNs-theorem. By (NEC∼), ULL O((A ∨ ∼A) ⊃ ((A ⊃ ∼A) ⊃ ∼A)). By (K), ULL O(A ∨ ∼A) ⊃ O((A ⊃ ∼A) ⊃ ∼A). By CL, (†) ULL O((A ⊃ ∼A) ⊃ ∼A) ∨ ¬O(A ∨ ∼A). We know by (U2) that ¬O(A ∨ ∼A) ⊃ O((A ⊃ ∼A) ⊃ ∼A). Hence, by (†) and CL, ULL O((A ⊃ ∼A) ⊃ ∼A). (iii) (A ⊃ (∼A ⊃ B)) ∨ (A ∧ ∼A) is an instance of the theorem (A ⊃ (B ⊃ C)) ∨ (A ∧ B) of positive CL, thus it is a CLoNstheorem. By (NEC∼), ULL O((A ⊃ (∼A ⊃ B)) ∨ (A ∧ ∼A)). By (OD), (‡) ULL O(A ⊃ (∼A ⊃ B)) ∨ P(A ∧ ∼A). We know by (U1) that P(A ∧ ∼A) ⊃ O(A ⊃ (∼A ⊃ B)). Hence, by (‡) and CL, ULL O(A ⊃ (∼A ⊃ B)).  Theorem 10 follows immediately by Lemmas 10 and 11. References [1] Carlos E. Alchourrón, Logic of norms and logic of normative propositions, Logique et Analyse 47 (1969) 242–268. [2] Carlos E. Alchourrón, Eugenio Bulygin, Normative Systems, Springer-Verlag, Wien/New York, 1971. [3] Carlos E. Alchourrón, Eugenio Bulygin, The expressive conception of norms, in: Risto Hilpinen (Ed.), New Studies in Deontic Logic, D. Reidel Publishing Company, Dordrecht, 1981, pp. 95–124. [4] Carlos E. Alchourrón, Eugenio Bulygin, Von Wright on deontic logic and the philosophy of law, in: P.A. Schilpp, L.E. Hahn (Eds.), The Philosophy of Georg Henrik Von Wright, Open Court Publishing Company, 1989, pp. 665–693. [5] Alan R. Anderson, Nuel D. Belnap, Entailment: The Logic of Relevance and Necessity, vol. 1, Princeton University Press, 1975. [6] Lennart Åqvist, Deontic logic, in: Dov Gabbay, Franz Guenthner (Eds.), Handbook of Philosophical Logic, vol. 8, 2nd edition, Kluwer Academic Publishers, 2002, pp. 147–264. [7] Diderik Batens, A universal logic approach to adaptive logics, Logica Universalis 1 (2007) 221–242. 168 M. Beirlaen, C. Strasser / Journal of Applied Logic 11 (2013) 147–168[8] Diderik Batens, Towards a dialogic interpretation of dynamic proofs, in: Cédric Dégremont, Laurent Keiff, Helge Rückert (Eds.), Dialogues, Logics and Other Strange Things. Essays in Honour of Shahid Rahman, College Publications, London, 2009, pp. 27–51. [9] Diderik Batens, Adaptive logics and dynamic proofs. Mastering the dynamics of reasoning, with special attention to handling inconsistency, in preparation. [10] Diderik Batens, Kristof de Clercq, A rich paraconsistent extension of full positive logic, Logique et Analyse 185 (188) (2004) 227–257. [11] Diderik Batens, Kristof De Clercq, Peter Verdée, Joke Meheus, Yes fellows, most human reasoning is complex, Synthese 166 (1) (2009) 113–131. [12] Mathieu Beirlaen, Christian Strasser, A paraconsistent multi-agent framework for dealing with normative conflicts, in: Joao Leite, Paolo Torroni, Thomas Ågotnes, Guido Boella, Leon van der Torre (Eds.), Computational Logic in Multi-Agent Systems. Proceedings of the 12th International Workshop CLIMA XII, in: Lecture Notes in Artificial Intelligence, vol. 6814, Springer-Verlag, 2011, pp. 312–329. [13] Mathieu Beirlaen, Christian Strasser, Joke Meheus, An inconsistency-adaptive deontic logic for normative conflicts, Journal of Philosophical Logic, forthcoming, http://dx.doi.org/10.1007/s10992-011-9221-3. [14] Guido Boella, Leendert van der Torre, Permissions and obligations in hierarchical normative systems, in: Proceedings of the 9th International Conference on Artificial Intelligence and Law, ICAIL'03, ACM, New York, NY, USA, 2003, pp. 109–118. [15] Eugenio Bulygin, On legal gaps, in: P. Comanducci, R. Guastini (Eds.), Ricerche Di Giurisprudenza Analitica, Giappichelli, Turin, 2003, pp. 21–28. [16] Newton Da Costa, Walter Carnielli, On paraconsistent deontic logic, Philosophia 16 (1986) 293–305. [17] Bas C. Van Fraassen, Values and the heart's command, Journal of Philosophy 70 (1973) 5–19. Reprinted in [21]. [18] Lou Goble, Multiplex semantics for deontic logic, Nordic Journal of Philosophical Logic 5 (2) (2000) 113–134. [19] Lou Goble, A proposal for dealing with deontic dilemmas, in: Alessio Lomuscio, Donald Nute (Eds.), DEON, in: Lecture Notes in Computer Science, vol. 3065, Springer, 2004, pp. 74–113. [20] Lou Goble, A logic for deontic dilemmas, Journal of Applied Logic 3 (2005) 461–483. [21] Christopher W. Gowans (Ed.), Moral Dilemmas, Oxford University Press, 1987. [22] Jörg Hansen, Conflicting imperatives and dyadic deontic logic, Journal of Applied Logic 3 (2005) 484–511. [23] Bengt Hansson, An analysis of some deontic logics, Noûs 3 (1969) 373–398. [24] Sven Ove Hansson, The Structure of Values and Norms, Cambridge University Press, 2001. [25] Ingemar Hedenius, Om Rätt och Moral, Tiden, Stockholm, 1941. [26] John Francis Horty, Moral dilemmas and nonmonotonic logic, Journal of Philosophical Logic 23 (1) (1994) 35–66. [27] John Francis Horty, Agency and Deontic Logic, Oxford University Press, 2001. [28] John Francis Horty, Reasoning with moral conflicts, Noûs 37 (2003) 557–605. [29] Barteld Kooi, Allard Tamminga, Moral conflicts between groups of agents, Journal of Philosophical Logic 37 (2008) 1–21. [30] Sarit Kraus, Daniel J. Lehmann, Menachem Magidor, Nonmonotonic reasoning, preferential models and cumulative logics, Artificial Intelligence 44 (1990) 167–207. [31] David Makinson, Leendert van der Torre, Input/output logics, Journal of Philosophical Logic 29 (2000) 383–408. [32] David Makinson, Leendert van der Torre, Constraints for input/output logics, Journal of Philosophical Logic 30 (2001) 155–185. [33] David Makinson, Leendert van der Torre, Permission from an input/output perspective, Journal of Philosophical Logic 32 (4) (2003) 391–416. [34] David Makinson, Leendert van der Torre, What is input/output logic? Input/output logic, constraints, permissions, in: Guido Boella, Leon van der Torre, Harko Verhagen (Eds.), Normative Multi-agent Systems, in: Dagstuhl Seminar Proceedings, Dagstuhl, Germany, 2007, Internationales Begegnungsund Forschungszentrum für Informatik (IBFI), Schloss Dagstuhl, Germany, 2007, No. 07122. [35] Joke Meheus, Mathieu Beirlaen, Frederik Van de Putte, Avoiding deontic explosion by contextually restricting aggregation, in: Guido Governatori, Giovanni Sartor (Eds.), DEON, 10th International Conference on Deontic Logic in Computer Science, in: Lecture Notes in Artificial Intelligence, vol. 6181, Springer, 2010, pp. 148–165. [36] Graham Priest, In Contradiction: A Study of the Transconsistent, 2nd edition, Oxford University Press, 2006. [37] Nicholas Rescher, Topics in Philosophical Logic, Reidel, Dordrecht, 1968. [38] Geoffrey Sayre-McCord, Essays on Moral Realism, Cornell University Press, New York, 1988. [39] Peter Schotch, Raymond Jennings, Non-Kripkean deontic logic, in: Risto Hilpinen (Ed.), New Studies in Deontic Logic, D. Reidel Publishing Company, Dordrecht, 1981, pp. 149–162. [40] Yoav Shoham, A semantical approach to nonmonotonic logics, in: Matthew L. Ginsberg (Ed.), Readings in Nonmonotonic Reasoning, Morgan Kaufmann Publishers, 1987, pp. 227–250. [41] Erik Stenius, The principles of a logic of normative systems, Acta Philosophica Fennica 16 (1963) 247–260. [42] Audun Stolpe, A theory of permission based on the notion of derogation, Journal of Applied Logic 8 (1) (2010) 97–113. [43] Christian Strasser, An adaptive logic framework for conditional obligations and deontic dilemmas, Logic and Logical Philosophy 19 (1–2) (2010) 95–128. [44] Christian Strasser, A deontic logic framework allowing for factual detachment, Journal of Applied Logic 9 (1) (2011) 61–80. [45] Christian Strasser, Mathieu Beirlaen, Joke Meheus, Tolerating deontic conflicts by adaptively restricting inheritance, Logique et Analyse 219 (2012) 477–506. [46] Peter Verdée, Adaptive logics using the minimal abnormality strategy are π11 -complex, Synthese 167 (2009) 93–104. [47] Georg Henrik von Wright, Norm and Action. A Logical Enquiry, Routledge and Kegan Paul, London, 1963. [48] Georg Henrik von Wright, An Essay in Deontic Logic and the General Theory of Action, North-Holland, Amsterdam, 1968. [49] Georg Henrik von Wright, Deontic logic: a personal view, Ratio Juris 12 (1) (1999) 26–38. [50] Anders Wedberg, Some problems in the logical analysis of legal science, Theoria 17 (1951) 246–275. [51] Gregory Wheeler, Marco Alberti, No revision and no contraction, Minds and Machines 21 (3) (2011) 411–430. [52] Bernard Williams, Ethical consistency, Proceedings of the Aristotelian Society (Supplementary Volumes) 39 (1965) 103–124. Reprinted in [21,38,53]. [53] Bernard Williams, Problems of the Self: Philosophical Papers 1956–1972, Cambridge University Press, 1973.