[15:27 4/7/2014 exu040.tex] LogCom: Journal of Logic and Computation Page: 1 1–22 Dynamic consequence for soft information OLIVIER ROY, Department of Philosophy, Universität Bayreuth, Germany E-mail: Olivier.Roy@uni-bayreuth.de OLE THOMASSEN HJORTLAND, Chair of Logic and Philosophy of Language, LMU Munich, Germany Abstract This article looks at so-called dynamic consequence relations for models of soft information change. We provide a sound, complete calculus for one-step soft dynamic consequence relations. We then study a generalization to sequences of updates, for which we show a number of valid and invalid structural rules. Keywords: Dynamic epistemic logic, substructural logic, soft information. Dynamic consequence relations are consequence relations generated by concrete information update procedures. Here we study soft information updates, which are updates that are reversible and not necessarily truthful. We provide a sound and complete axiomatization for dynamic consequence relations generated by a large class of such updates. The calculus covers one-shot updates. It describes the effect of a ψ-type of update in contexts where φ holds. Towards the end of the article we generalize this calculus to cover sequences of information updates, also known as iterated revisions. We show a number of valid inferences in that calculus, and use them to compare dynamic and classical consequence relations. 1 Why dynamic consequence relations Rational belief change can be seen as licensing specific kinds of inferences, that is, inducing a specific kind of consequence relation. In the words of [12], belief change and non-classical inferences are 'two sides of the same coin'. This idea, by now widely accepted in default logic, also underlies the notion of dynamic consequence. This article focuses on one specific framework for the study of information change in multiagent settings: dynamic epistemic logic (DEL). See [26] and [22] for an in-depth presentation. The gist of DEL is to enrich propositional modal languages with operators that describe the effects or consequences of certain 'epistemic actions' in a given situation. Epistemic actions are events that only affect the information available in a given situation, such as observing certain states of affairs or learning about them from a truthful and trusted source.1 Dynamic consequences for DEL were first proposed by [21] in the context of public announcement logic. Public announcement logic contains formula [!φ]ψ , to be read 'ψ holds after the announcement of φ'. One can use these formulas to define a consequence relation as follows: φ1,...,φn |=dynψ iff M,w |=[!φ1]...[!φn]ψ for all pointed models M,w. 1Epistemic actions are usually distinguished from 'ontic events', i.e. events, like turning on the lights, which change some non-informational facts. Ontic events have also been studied in DEL, cf. [23]. We do not study them here. © The Author, 2014. Published by Oxford University Press. All rights reserved. For Permissions, please email: journals.permissions@oup.com doi:10.1093/logcom/exu040 Journal of Logic and Computation Advance Access published July 7, 2014 at U niversitetsbiblioteket i B ergen on A pril 4, 2015 http://logcom .oxfordjournals.org/ D ow nloaded from [15:27 4/7/2014 exu040.tex] LogCom: Journal of Logic and Computation Page: 2 1–22 2 Dynamic consequence for soft information Dynamic consequence relations are interesting for a number of reasons. First, they are substructural. van Benthem [21] shows for instance that structural rules like cut, contraction and weakening fail for the relation just defined. We see in Section 5 that this also holds true for dynamic consequence relations for soft information changes. Such consequence relations therefore fall within the broad family of sub-structural logic [19]. It is well known that these logics can be given an abstract 'informational interpretation', and that this interpretation can be used to argue for or against certain structural rules-see e.g. [16]. But the strength of such arguments depends on the plausibility of the notion of information and information flow that are used in interpreting substructural systems. This can be hard to assess in the abstract. Dynamic consequence circumvents this difficulty by arguing the other way around. It starts with a concrete, well-understood belief update procedure, and uses this to construct a consequence relation that turns out to be sub-structural. For dynamic consequence relations the motivation to 'go sub-structural' comes from the plausibility of the underlying belief update mechanism. Dynamic consequence relations are also contributing to our understanding of information change in social interaction. The standard technique to axiomatize validities in DEL is to use so-called reduction axioms, which show us how to analyse, compositionally, the effects of epistemic action in terms of what holds in the initial situation.2 Recently, cut-free, complete Gentzen-style systems for DEL have also been developed, using labelled sequents or display logic [2, 13, 14]. These approaches say little about the valid logical operations on the epistemic actions themselves. These operations are conceptually important. Different epistemic actions are different types of learning events,3 for example a public announcement that φ to a group vs a private announcement of φ to all members of that group. Valid and invalid inferences about epistemic actions show how these different learning events relate to one another. Structural rules in dynamic consequences make these inferences explicit, and thus unveil logical principles governing the dynamic of information that were hitherto implicit or not well studied. Finally, echoing Cordón-Franco et al. [8], it should be noted that dynamic consequence relations can be used for reasoning about protocols in multi-agent systems and that they put DEL in a historical perspective. They indeed indicate that dynamic consequence relations constitute a return to the original motivation behind dynamic semantics [20, 27], which arguably lies at the source of DEL. 2 State of the art and present contribution The study of dynamic consequence is young but lively. van Benthem's [21] work on dynamic consequence relations, sketched above, primarily treats changes in hard information induced by public announcements. To repeat, the dynamic consequence relation that he studies takes the following form: φ1,...,φn |=dynψ iff M,w |=[!φ1]...[!φn]ψ for all pointed models M,w. He showed that despite violating cut, monotonicity and contraction, this particular type of dynamic consequence relation can be completely axiomatized by a set of rules reminiscent of rules found in the literature on non-monotonic reasoning. This work has set the agenda. Cordón-Franco et al. [8] have generalized van Benthem's approach to allow for public announcements in specific informational contexts. To be more precise, their main object of study is a relation defined as follows: ,φ1,...,φn |=dynψ iff M,w |=[!φ1]...[!φn]ψ for all pointed models M,w such that M,w |=. 2Of course, not all DEL-like systems are prone to this reduction technique. Well-known examples are public announcements in S5 + Common knowledge [5] and the logic of 'epistemic protocols' [24]. 3For this terminology, see e.g. [10] and the references therein. at U niversitetsbiblioteket i B ergen on A pril 4, 2015 http://logcom .oxfordjournals.org/ D ow nloaded from [15:27 4/7/2014 exu040.tex] LogCom: Journal of Logic and Computation Page: 3 1–22 Dynamic consequence for soft information 3 This is the 'local' variant of the contextualized dynamic consequence relation. The authors [8] also study the properties of a global version of this relation, which uses model validity instead of truth at pointed models. They have shown that the rules used by van Benthem fail in this more general setting, for both local and global versions of the relation. Following a different trend, Aucher [1, 2] studied dynamic consequences of the following form: φ1;φ2 φ3 These should be read 'in every situation where φ1 holds, updating by a φ2-type of learning event will result in a situation where φ3 holds.' Aucher has shown that the set of valid sequents of that form, defined for general dynamic epistemic logic and arbitraty epistemic attitudes, can be completely axiomatized. This is the framework that is adopted in the present article, so we present it in greater detail. The formal definition can be found in Section 4. The three-place sequents used by Aucher are rigid. They connect pairs of single formulas to single formulas. The left-most formula describes the initial situation in which the update is performed. The second formula describes the type of learning event that is taking place. The formula on the right of the turnstile describes the situation after the update. This framework thus captures a fragment of the set of DEL-validities for a one-shot update. Furthermore, as we shall see below, the two formulas on the left describe different types of structures. The first and the third formulas are interpreted according to standard Kripke models for multi-modal logics. The second one describes so-called 'event models'. Again, we give further details on this terminology below. What matters here is that these two formulas, because they describe different kinds of objects, might be from different languages, so rules like contraction or exchange have no straightforward correspondent in this setup. Some form of weakening can be expressed, but not the standard rule, i.e. adding a formula to the left or the right side. Both Aucher's and Cordón-Franco et al.'s sequents are contextual dynamic consequence relations. They take as argument formula(s) that describe the initial situation in which the update is performed. In Aucher's framework, this is always a single formula or, equivalently, a finite set thereof, while Cordón-Franco et al. allows for infinite sets of such formulas. As mentioned, the three-place sequents are suited to talking about one-shot updates. van Benthem and Cordón-Franco et al., on the other hand, explicitly address the iterated case. This facilitates the expression and the study of classical structural properties, and both van Benthem and Cordón-Franco et al. do indeed focus on identifying valid and invalid ones. Another important difference is that van Benthem and Cordón-Franco et al. study dynamic consequences for specific update rules: hard and soft public announcements. The consequence relation defined by Aucher builds on full DEL, and thus covers a large variety of information update mechanisms. The set of valid rules identified by Aucher has been proved to axiomatize completely dynamic consequence for DEL. The present article contributes to this literature by studying dynamic consequences for a large class of soft information update mechanisms. Our underlying dynamic framework is the soft information counterpart of DEL, developed by Baltag and Smets [3]. Its models and language are those of beliefs and conditional beliefs. As such it is more specific than DEL, which works for arbitrary multi-modal systems. The update rule is different, however. While DEL uses the so-called product update rule, Baltag and Smets have proposed a 'lexicographic' update procedure, which gives priority to the information carried by the incoming (soft) epistemic action. They have shown that this update rule embeds all the most well-known update rules for soft information. These are central to the theory of multi-agent belief revision [6, 9, 22]. It is thus important to study the dynamic consequence relations that they generate, which is what we do here. We provide a sound and complete axiomatization at U niversitetsbiblioteket i B ergen on A pril 4, 2015 http://logcom .oxfordjournals.org/ D ow nloaded from [15:27 4/7/2014 exu040.tex] LogCom: Journal of Logic and Computation Page: 4 1–22 4 Dynamic consequence for soft information for soft information dynamics in the three-place sequents framework just discussed. We then study its generalization to sequences of information update. For that case we focus on identifying which classical structural rules hold and which fail. 3 Lexicographic update for soft information Here we introduce the framework for studying soft information dynamics developed in [3] or, as the authors put it, a general theory of multi-agent belief revision. We only give the bare essentials, before moving on to dynamic consequence relations. For more details and examples the reader should consult [3] and [22]. 3.1 Static models and language We are interested in the dynamics of so-called soft informational attitudes. These are attitudes, like beliefs, which are revisable, might be mistaken, and might not be fully introspective.4 Our starting point is thus a static modal language expressive enough to encode conditional beliefs, i.e. modalities of the form Bφi ψ , to be read 'conditional on φ, agent i believes that ψ'. Mainly for technical reasons, we work in the language LS ('S' for 'State model'), in which conditional beliefs are definable. Throughout we assume a finite set A of agents and denote its elements i,j, etc. Let P be a countable set of atomic propositions. φ :=p∈P | ¬φ | φ∧φ | [∼i]φ | [≤i]φ We write 〈∼i〉φ for ¬[∼i]¬φ and 〈≤i〉φ for ¬[≤i]¬φ. Formulas of the form [∼i]φ and [≤i]φ should be read, respectively, as 'agent i knows that φ' and 'agent i safely believes that φ'. We write 〈∼i〉φ for ¬[∼i]¬φ and 〈≤i〉φ for ¬[≤i]¬φ. Conditional beliefs Bφi ψ are then defined as follows: Bφi ψ⇔df 〈∼i〉φ→〈∼i〉(φ∧[≤i](φ→ψ)) This language is interpreted in plausibility models, which are Kripke structures [7] equipped with a collection of partial pre-orders ≤i and a valuation V assigning to each state w∈W a subset of a given set of atomic propositions. Definition 3.1 Let P be a countable set of propositions and A a set of agents. A plausibility model M is a tuple 〈W ,{≤i}i∈A,V〉 where: • W is a non-empty set of states. • for each i∈A, ≤i is a well-founded,5 reflexive and transitive relation on W . • V :W →P(P) is a valuation function. Plausibility frames are models minus the valuation. The reader familiar with the literature on belief revision or non-monotonic logic will recognize these models as the usual preferential models-see 4By 'introspective' we mean attitudes that do not validate the so-called positive and negative instrospection axioms. Positive introspection is the '4' axiom, which states that if an agent believes φ, then she believes that she believes φ. Negative introspection is the '5' axiom, which states that if an agent does not believe φ, then she believes that she does not believe φ. 5Well-foundedness is only needed to ensure that conditional beliefs are well defined. at U niversitetsbiblioteket i B ergen on A pril 4, 2015 http://logcom .oxfordjournals.org/ D ow nloaded from [15:27 4/7/2014 exu040.tex] LogCom: Journal of Logic and Computation Page: 5 1–22 Dynamic consequence for soft information 5 e.g. [15] and references therein. This is what they are, with the proviso that this is a multi-agent setting. In this framework, the epistemic accessibility relation usually associated with hard attitudes like knowledge is definable. Definition 3.2 Let M be a plausibility model. The epistemic accessibility relation ∼i for agent i is defined as follows: • w∼i w′ iff w≤i w′ or w′ ≤i w. Call the set [w]i ={w′ :w′ ∼i w} the similarity class of w. Definition 3.3 Let M be a plausibility model. The similarity class [w]i for a state w is locally connected whenever every state w′,w′′ ∈[w]i are comparable, i.e. either w′ ≤i w′′ or the other way around. A plausibility model is locally connected if all its similarity classes are. It is easy to check that ∼i is an equivalence relation on locally connected models. In this article our attention is confined to such models. The truth conditions for the modalities then go as follows. • M,w |=[≤i]φ iff M,w |=φ for all w′ ≥i w. • M,w |=[∼i]φ iff M,w |=φ for all w′ ∼i w. Validity on models and frames, and classes thereof, is defined as usual. Each [≤i] is a KT modality. Well-foundedness and transitivity can be enforced by the Löb axiom. Each [∼i] is an S5 modality. Interaction between [≤i] and [∼i] can be captured by standard inclusion axioms. Local connectedness is enforced by the following [3]: [∼i](φ∨[≤i]ψ)∧[∼i](ψ∨[≤i]φ)→[∼i]φ∨[∼i]ψ (LC) Putting all this together we get: Theorem 3.4 Multi-agent KTL + S5 together with the relevant inclusion axioms and (LC) is sound and complete with respect to the class of locally connected plausibility frames. In what follows, we write S φ when φ is a theorem of multi-agent KTL + S5 together with the relevant inclusion axioms and (LC). 3.2 Event models and language The modelling of epistemic action developed in [3] follows the same methodology as for DEL. Soft epistemic actions are encoded in so-called event models. These are Kripke structures equipped with a collection of pre-orders, where the elements of its domain are thought of as basic events or programs. Definition 3.5 Let E be a finite set of propositional atoms. An event model E is a tuple 〈E,{≤Ei }i∈A,VE 〉 where: • E is a non-empty, finite set of events. • for each i∈A, ≤Ei is a well-founded, reflexive, transitive and locally connected relation on E. at U niversitetsbiblioteket i B ergen on A pril 4, 2015 http://logcom .oxfordjournals.org/ D ow nloaded from [15:27 4/7/2014 exu040.tex] LogCom: Journal of Logic and Computation Page: 6 1–22 6 Dynamic consequence for soft information • VE :E →Event is a valuation function such that V (e) is a singleton for all e∈E. We write e∼=i e′ whenever e≤i e′ and e′ ≤i e. An event frame is an event model minus the valuation. This presentation, taken from [1], differs from the usual one. In DEL, event models are equipped with a precondition function instead of a propositional valuation. This function is meant to specify the conditions of executability of events or epistemic actions. However, to define the dynamic consequence relation we need a language to describe event models and types of epistemic actions. To do that it is more convenient to work with a propositional valuation. This valuation is constrained to re-capture the precondition function. Definition 3.6 Let E be a finite set of atomic propositions. A precondition function is a function pre :E →LS that assigns to each e∈E a formula of LS . From now on, when we talk about an event model E we will implicitly assume a precondition function on a given finite set of atomic propositions E associated with E . As just mentioned, the dynamic consequence relation requires a language to talk about event models. Unlike the language for state models, this one is equipped with a Kripke modality for the strict sub-relation of the plausibility order ≤i. The reason for that is the lexicographic update rule. It makes explicit reference to some events being strictly more plausible than others, and this is reflected in the set of valid sequents that we want to capture. The language LE is constructed as follows. To simplify the notation we also use e,e′,... to refer to elements of the set E of atomic propositions. φ :=e∈E | ¬φ | φ∧φ | [∼i]Eφ | [≤i]Eφ | [<i]Eφ We write 〈∼i〉Eφ for ¬[∼i]E¬φ, 〈≤i〉Eφ for ¬[≤i]E¬φ and 〈<i〉Eφ for ¬〈<i〉E¬φ. Truth for this language is defined as usual, and so are the different notions of validity. Definition 3.7 Let E,e be an event model. • E,e |=[≤i]φ iff E,e′ |=φ for all e′ ≥i e. • E,e |=[<i]φ iff E,e′ |=φ for all e′>i e. • E,e |=[∼i]φ iff E,e′ |=φ for all e′ ∼i e. The validities for the weak and strict boxes [≤i] and [<i] and their proper interrelation over the class of event frames is completely axiomatized by the systemLP presented in [25, p.93], again together with the Löb axiom to take care of well-foundedness. The interaction between these modalities and the S5 box [∼i] is a simple inclusion. It can easily be dealt with. We assume it when we writeLP +S5. The constraint on the valuation function is captured by a set ExE of axioms of the following form, one for each e∈E. e↔¬ ∨ E−e (Excl) where E−e is the set of atoms in E except e. Theorem 3.8 Multi-agent LP +S5+ExE+(LC) is sound and complete with respect to the class of event frames. In what follows, we write E φ when φ is a theorem of multi-agent LP +S5+ExE. at U niversitetsbiblioteket i B ergen on A pril 4, 2015 http://logcom .oxfordjournals.org/ D ow nloaded from [15:27 4/7/2014 exu040.tex] LogCom: Journal of Logic and Computation Page: 7 1–22 Dynamic consequence for soft information 7 3.3 Lexicographic update The lexicographic update rule takes pairs of plausibility and event models M,w and E,e and return the updated model M⊗E where the domain is the set of pairs (w,e) such that M,w satisfies the precondition of e, written M,w |=pre(e) and the valuation is taken directly from M, i.e V ′(w,e)= V (w). The adjective 'lexicographic' comes from the update rule for the pre-orders ≤i, which gives priority to the events: Definition 3.9 Let E be an event model and pre a precondition function on E. The lexicographic update M⊗E = 〈W ′,≤′i,V ′〉 is defined as follows: • W ′ ={(w,e)|M,w |=pre(e) for the unique e such that E,e |=e}. • (w,e)≤′i (w′,e′) iff either e<e′ and w∼i w′ or e∼=i e′ and w≤w′. • V ′(w,e)=V (w). 4 Dynamic consequence for lexicographic update In this section, we build up to one of the two main contributions of this article: a sound and complete system of rules for the soft dynamic consequence relation. We first give a semantic definition of this consequence relation. Then we introduce a set of axioms and rules, and show that they are sound. The proof that they are also complete uses the same technique as in [1], namely a detour via 'canonical formulas' for plausibility and event models. We introduce and prove basic facts about these formulas first, and then progress to the completeness result. 4.1 Semantic dynamic consequence relation Dynamic consequence relations for lexicographic update are defined as follows: Definition 4.1 (Soft Dynamic Consequence Relation Semantic Version) Let φ1 and φ3 be formulas of LS and φ2 a formula of LE . A soft dynamic sequent is an object of the following form φ1;φ2 |=φ3 We say that such a sequent is valid whenever for all M,w and E,e, if M,w |=φ1 ∧pre(e) and E,e |=φ2 then (M⊗E,(w,e)) |=φ3 So we are working with three-place sequents in the same fashion as in [1]. We emphasize again that the second formula on the left, φ2, is not of the same language as the two other formulas φ1 and φ3. In other words, in this framework there is no straightforward version of exchange between φ1 and φ2, or contraction for that matter. Some form of cut can be formulated here, but we shall return to this in Section 5. This dynamic consequence relation is contextualized in the same sense as those studied by [8]. The rule takes as first argument a formula, φ1, which describes the initial situation in which the update takes place. This could of course be equivalently formulated by taking a finite set of formulas in the first argument set. We do that in Section 5, when the connection with substructural systems will be made more explicit. Unlike [8], however, we do not consider initial situations described by infinite sets of formulas. This dynamic consequence relation takes as second argument general types of epistemic actions or learning events. A sequent of the form φ1;φ2 |=φ3 is valid iff any update of type φ2 in an initial at U niversitetsbiblioteket i B ergen on A pril 4, 2015 http://logcom .oxfordjournals.org/ D ow nloaded from [15:27 4/7/2014 exu040.tex] LogCom: Journal of Logic and Computation Page: 8 1–22 8 Dynamic consequence for soft information Table 1. Axioms and rules for the dynamic consequence relation . In R3–R5 it must be the case that Lψi →φi for either L=S and i=1,3 or L=E and i=2. ⊥;φ2 φ3 A1 φ1;⊥φ3 A2 φ1;φ2  A3 p;φ2 p A4¬p;φ2 ¬p A5 ¬Pre(e);e⊥ A6 φ1;φ2 φ3 φ1;φ2 φ4 φ1;φ2 φ3 ∧φ4 R1 φ1;φ2 φ3 ψ1;φ2 φ4 φ1 ∨ψ1;φ2 φ3 R2 φ1;φ2 φ3 φ1;ψ2 φ4 φ1;φ2 ∨ψ2 φ3 R2′ φ1;φ2 φ3 ψ1;φ2 φ3 R3 φ1;φ2 φ3 φ1;ψ2 φ3 R4 φ1;φ2 ψ3 φ1;φ2 φ3 R5 φ1;φ2 φ3 φ4;φ5 φ3 [∼i]φ1 ∧[≤i]φ4;[<i]Eφ2 ∧[≤i]Eφ5 [≤i]φ3 (R6) φ1;φ2 φ3 〈∼i〉(φ1 ∧pre(e));〈<i〉E (φ2 ∧e)〈≤i〉φ3 (R7) φ1;φ2 φ3 〈≤i〉(φ1 ∧pre(e));〈≤i〉E (φ2 ∧e)〈≤i〉φ3 (R8) φ1;φ2 φ3 [∼i]φ1;[∼i]Eφ2 [∼i]φ3 (R9) φ1;φ2 φ3 〈∼i〉(φ1 ∧pre(e));〈∼i〉E (φ2 ∧e)〈∼i〉φ3 (R10) situation where φ1 holds results in a φ3-situation. So our object of study here is inferences valid for large collections of soft information updates, as opposed to inferences holding for specific epistemic actions, such as public announcements or conservative upgrades. 4.2 The Gentzen-style system The set of soft dynamic sequents is completly axiomatized by the set of axioms and rules presented in Table 1. Most of them are directly imported from [1]. At the level of axioms, for instance, the set of validities is the same for DEL and its soft version, which we consider in this article. Epistemic actions do not change the truth value of atoms in either framework (A4, A5), and a pointed event model is executable in a pointed plausibility model whenever the latter satisfies the preconditions of the former (A6). Soft dynamic consequence also inherits from DEL rules to introduce conjunction to the right (R1), and disjunction to the left (R2, R2'). The rules R3 to R5 are also ported directly from [1], but a few remarks are in order. Each of these three rules makes explicit reference to valid implications in the "underlying logic". Take for instance R3: φ1;φ2 φ3 ψ1;φ2 φ3 R3 at U niversitetsbiblioteket i B ergen on A pril 4, 2015 http://logcom .oxfordjournals.org/ D ow nloaded from [15:27 4/7/2014 exu040.tex] LogCom: Journal of Logic and Computation Page: 9 1–22 Dynamic consequence for soft information 9 This rule holds with the proviso that Sψ1 →φ1. This might sound suspicious to the prooftheoretically inclined reader. Are we not trying to capture the set of soft-DEL valid formulas? No, not all of them. To see why, one must bear in mind that we are axiomatizing here the set of valid soft dynamic sequents. These, in turn, correspond to a fragment of the dynamic extension of the language for safe belief and knowledge. Take a soft dynamic sequent φ1;φ2 |=φ3. Let E,e be any pointed event model such that E,e |=φ2. The sequent is valid iff for all pointed models M,w, we have that M,w |=φ1 →[E,e]φ3 So axiomatizing soft dynamic consequence is axiomatizing a fragment of the validities in LS . The key operator of this fragment is the one for lexicographic update. Its behaviour presupposes the complete logic of the respective modal operators for static and event models. So using valid implications from these underlying logics in the Gentzen-style system is not circular. The system builds on these logics, as opposed to trying to capture them. Rules R3–R5 are also important because they are as close to Weakening as one can get in the three-place sequents we are working with. The classical form of weakening allows one to add formulas on the left, and also on the right if one allows for multiple conclusions. We cannot do this in our three-place sequents. What we can do is replace existing formulas with logically weaker or stronger ones, depending on whether they occur on the right or on the left side of the sequent. Take R3 again. It states that if one can prove the sequent φ1;φ2 φ3, then one can prove the overall logically weaker sequent ψ1;φ2 φ3 by replacing φ1 in the antecedent with any logically stronger formula ψ1. By logically stronger we mean that the latter implies the former in the underlying logic. Of course, the specificity of our axiomatization lies in the rules R6–R10, which reflects the behaviour of the lexicographic update rule. The two-premise rule R6 introduces boxes in front of the different arguments in the sequent. Dually, the pair R7 and R8 allows the introduction of diamonds. This reflects the disjunctive character of lexicographic update. Take the diamond-introduction rules, for instance. Informally they can be read as follows. Suppose any update by a φ2-type of epistemic action in a situation where φ1 holds will result in a situation where φ3. When will a φ3-state (w′,e) be ≤i-reachable from another state (w,e) after an update? By assumption and the lexicographic update rule we know that this can happen in two cases. The first one is when there was a φ1 state w′ comparable to w and a φ2-event e′ strictly more plausible than e, under the assumption that e′ was executable in w′. This is what R7 states. The second one is when e′ was a φ2-state equi-plausible to e and w′ a φ1 at least as plausible as w, with the same proviso regarding preconditions. R8 covers this case because equi-plausibility implies comparability. See the proof of soundness below for details. The same idea works dually for R6. The reader can check that, because of the way lexicographic update is set up, it collapses to the standard DEL product update for the hard information modality [∼i]. It is therefore not surprising that the rules regimenting the introduction of that modality to the right are exactly as in [1]. With these informal explanations in mind, we now proceed to show soundness of these rules. Observation 4.1 All axioms and rules in Table 1 are sound. Proof. We only prove the cases for R6–R10. Soundness for the axioms and for R1–R5 proceeds exactly as in [1]. R6 Suppose M,w |=[∼i]φ1 ∧[≤i]φ4 and E,e |=[<i]Eφ2 ∧[≤i]Eφ4. It is sufficient to show that M⊗E,(w,e) |=[≤i]φ3. If the M,w |=pre(e) then we are done. Suppose then that M,w |= at U niversitetsbiblioteket i B ergen on A pril 4, 2015 http://logcom .oxfordjournals.org/ D ow nloaded from [15:27 4/7/2014 exu040.tex] LogCom: Journal of Logic and Computation Page: 10 1–22 10 Dynamic consequence for soft information pre(e).Take any (w1,e1)∈M⊗E such that (w1,e1)≥i (w,e).This can happen in two cases. First, it can be that w1 ∼i w and e1>i e. Because M,w |=[∼i]φ1 and E,e |=[<i]Eφ2, respectively, we get M,w1 |=φ1 and E,e1 |=φ2. But since φ1;φ2 |=φ3, we get that M⊗E,(w1,e1) |=φ3. The other case is when e1 ∼=i e and w1 ≤i w. Since M,w |=[≤i]φ4 and E,e |=[≤i]Eφ5 we get M,w1 |=φ4 and E,e1 |=φ5. But then since φ4;φ5 |=φ3, we get M⊗E,(w1,e1) |=φ3 again. So we have just shown that the latter holds for any (w1,e1)≥i (w,e), which means M⊗E,(w,e) |=[≤i]φ3. R7 (Sketch) E,e |=〈<i〉E (φ2 ∧e′) gives us the e′ where φ2 (and e′ hold) such that e<i e′ and similarly for the required w′ from M,w |=〈∼i〉(φ1 ∧pre(e′)). So the pair (w′,e′) turn out to be related to (w,e) in the updated model. But then we can use φ1;φ2 |=φ3 to conclude that M⊗E,(w′,e′) |= φ3. R8 (Sketch) E,e |=〈≤i〉E (φ2 ∧e′) gives us the e′ where φ2 and e′ hold such that e≤i e′, and M,w |= 〈≤i〉(φ1 ∧pre(e′)) the w′i ≥w that satisfy φ1 and would survive the update with e′. Now, if the relation between e and e′ does not go the other way around, the argument follows the same line as for R7. If it does, then (w′,e′) are related to (w,e) in the updated model, from which we can conclude M⊗E,(w′,e′) |=φ3 from φ1;φ2 |=φ3. R9 Suppose M,w |=[∼i]φ1 and E,e |=[∼i]Eφ2. Take any (w1,e1)∈M⊗E such that (w,e)∼i (w1,e1), i.e. either (w,e)≤i (w1,e1) or (w1,e1)≤i (w,e). Consider the first case. There are two sub-cases. First e<i e1 and w∼i w1. From the latter we get directly that M,w |=φ1. From the latter it follows that e∼i e1, and so M,e |=φ2. But since φ1;φ2 |=φ3, we get that M⊗E,(w1,e1) |=φ3. The second sub-case is when e∼=i e1 and w≤i w1. Each entails its respective weaker ∼i, from which the desired conclusion follows. Now back up one level. The second case is when (w1,e1)≤i (w,e). Again, there are two sub-cases. They follow exactly the same steps because the relation ∼i abstracts away from the direction of ≤i. R10 Take any M,w and E,e such that M,w |=pre(e), M,w |=〈∼i〉(φ1 ∧pre(e′)) and E,e |=〈∼i〉E (φ2 ∧e′). We have to show that M⊗E,(w,e) |=〈∼i〉φ3. Since M,w |=pre(e) we know that the pair (w,e) is in the updated model. We know furthermore that there is a w′ such that w∼i w′ and M,w′ |=φ∧pre(e′). So (w′,e′) is in the updated model too. Do we have (w,e)∼i (w′,e′)? There are a number of cases to consider. We can group them in two families. First, whenever e<i e′ or the other way around we get the desired (w,e)∼i (w′,e′). In that case, because w∼i w′ the update rule gives priority to events. The remaining group of cases is when e∼=i e′. But then the update rule reverts to the order of the relation in the state model, but whatever that is we get (w,e)∼i (w′,e′). But then we are done, for since φ1;φ2 |=φ3 we know that M⊗E,(w′,e′) |=φ3 and so M⊗E,(w,e) |=〈∼i〉φ3.  4.3 Completeness for the Gentzen-style system In this section, we prove that the set of axioms and rules in Table 1 completely axiomatize the set of valid soft dynamic sequents. First we define formally what a derivable soft dynamic sequent is. Definition 4.2 A soft dynamic sequent is derivable, written φ1,φ2 φ3 , iff it is either an instance of one of the axioms A1-A7 or it follows from such an axiom by a finite number of applications of the rules R1–R10. at U niversitetsbiblioteket i B ergen on A pril 4, 2015 http://logcom .oxfordjournals.org/ D ow nloaded from [15:27 4/7/2014 exu040.tex] LogCom: Journal of Logic and Computation Page: 11 1–22 Dynamic consequence for soft information 11 As in [1], the completeness proof proceeds in two steps. First we prove that all valid sequents formed of so-called 'canonical formulas' are derivable. Then we show that this result extends to arbitrary soft dynamic sequents. So we need to explain first what these canonical formulas are. 4.3.1 Canonical formulas and cover modality Canonical modal formulas have been known at least since Kit Fine's 1975 paper. The formulation we take here comes from Larry Moss [17]. We present it independently of the specific modal operators in LS and LE . The facts we mention about them hold in general. We thus speak abstractly of a language L and of boxes [Ri] and diamonds 〈Ri〉 for i∈A and R∈{∼i,≤i,∼Ei ,≤Ei ,<Ei }. Similarly, when we use theorems of the logic of plausibility or event models, we omit the subscripts on S and E unless ambiguity can arise. We only present proofs that are in any interesting way different from those in [17]. Definition 4.2 Let P be a finite subset of P and π a subset of P. Let ,Ri be finite subsets of L, with R in either {∼i,≤i,}i∈A or {∼Ei ,≤Ei ,<Ei }i∈A, depending on the models at hand. Then: π := ∧ p∈π p∧ ∧ p∈P−π ¬p ∇Ri :=[Ri] ∨ φ∈ φ∧ ∧ φ∈ 〈R〉φ π • ∧ Ri Ri :=π∧ ∧ Ri (∇RiRi ) The 'bullet' (•) and 'nabla' (∇) notations are from [28]. We use them for convenience. Canonical formulas are built inductively. The intuition behind the construction is to view canonical formulas as descriptions of pointed models. Canonical formulas of level 0 describe the propositional valuation up to the given finite subset P. Canonical formulas of level n≥1 list all and only the valuations accessible in n step from a given state/valuation. The construction is essentially the same for epistemic-plausibility and for event models. The only difference is for the base case. Definition 4.3 (Canonical formula of LS) Let P be a non-empty finite subset of P. (1) S0P ={π | π⊆P}. (2) Sn+1P ={π • ∧ Ri R i | π⊆P,Ri ⊆SnP}. Call any formula δn ∈SnP a canonical LS-P-formula of depth n. Definition 4.4 (Canonical formula of LE ) Recall that E is already a finite set, so we start with that. (1) S0P ={π | π is a singleton subset of E}. (2) Sn+1P ={π • ∧ Ri R i | π⊆P,Ri ⊆SnP}. Call any formula δn ∈SnP a canonical LE -P-formula of depth n. Except when ambiguity can arise we drop the prefix 'LS/E -P' and just say 'canonical formulas.' at U niversitetsbiblioteket i B ergen on A pril 4, 2015 http://logcom .oxfordjournals.org/ D ow nloaded from [15:27 4/7/2014 exu040.tex] LogCom: Journal of Logic and Computation Page: 12 1–22 12 Dynamic consequence for soft information Lemma 4.5 Let M,w be an epistemic-plausibility model and P a finite set of atomic propositions. Then for all n there is exactly one δ∈SnP such that M,w |=δ. The same holds, mutatis mutandis, for event models. Proof. Essentially the same as in [17], except for a minor detail in the case of event models.  Now we can state the crucial fact about canonical formulas. The argument only uses normality of the modal operator at hand. So we omit it. Lemma 4.6 Let φ be a formula of LS of modal depth n, and let P(φ)={p∈P :p is a sub-formula of φ}. • For all δ∈SnP(φ), either δ→φ or δ→¬φ. • S ∨SnP(φ) δ The same holds, mutatis mutandis, for event models. The main point of the previous lemma is to get us the following corollary, which will be crucial in lifting the completeness result for canonical formulas to arbitrary soft dynamic sequents. We omit the proof, as it is essentially the same as that in [1]. The 'S possibly empty' proviso is to take care of the case where φ is not satisfiable, following the usual convention that ∨∅=⊥. Corollary 4.7 Let φ be a formula of LS of modal depth n, and let P(φ)={p :p is a sub-formula of φ}. Then there is a S ⊆SnP(φ), with S possibly empty, such that : φ↔ ∨ δ∈S δ 4.3.2 Canonical formulas and n-bisimilarity Bisimulation is a core concept for the model theory of modal logic, cf. [7]. So-called n-bisimulation is a finite version of that notion. Canonical formulas are closely connected to n-bisimilarity. This will come in handy below, so we present them in detail. Definition 4.8 (n-P-Bisimulation) Let P⊆P. Two plausibility models M,w and M′,v are n-P-bisimular, written (M,w)↔n(M′,w′) whenever there is a sequence ↔n ⊆ ...⊆ ↔0 of relations ↔i ⊆ W ×W ′ such that, for all k+1≤n, w′ ∈W , v′ ∈W ′ and Ri ∈{≤i,∼i} (1) w↔nv; (2) If w′ ↔0v′ then (V (w′)∩P)= (V ′(v′)∩P). (3) If w′ ↔k+1v′ if w′Riw′′ then there is a v′′ ∈W ′ such that v′R′iv′′ and w′′ ↔kv′′. (4) Same from W ′ to W . This definition is adapted to event models in the obvious way. Now suppose that two plausibility or event models satisfy the same canonical formula of degree n. We show now that this is both necessary and sufficient to guarantee that these models are n-bisimilar. It is well-known that bisimulation and nbisimulation are equivalence relations between models. Canonical formulas capture this equivalence syntactically. at U niversitetsbiblioteket i B ergen on A pril 4, 2015 http://logcom .oxfordjournals.org/ D ow nloaded from [15:27 4/7/2014 exu040.tex] LogCom: Journal of Logic and Computation Page: 13 1–22 Dynamic consequence for soft information 13 Fact 4.9 Let P⊆fin P. Then t.f.a.e: (1) There is a δ∈SPn such that M,w |=δ and M′,w′ |=δ. (2) M,w↔Pn M′,w′ And similarly for event models. Proof. The direction from 2 to 1 follows from the well-known invariance of modal formulas of bounded depths under n-bisimulation and the Lemma 4.5 that each state in a model satisfies exactly one formula in SPn . For the other direction take the δ∈SnP that is true in both pointed models. We define the n-P-bisimulation as follows. First set ↔n ={(w,w′)}. Then, for (n>k ≥0), define ↔k as v↔kv′ iff there is a δ′ ∈SPk such that M,v |=δ′ and M′,v′ |=δ′. Now we have to check that this is indeed an n-bisimulation between w and w′. We get trivially w↔nw′. The case of w↔0v′ follows from the definition of canonical formulas. Now suppose that v↔k+1v′ for k>0 and there is a v1 such that v≤i v1. Take δ∈Sk+1i such that M,v |=δ. By definition of δ we know that there is (exactly) one δ′ ∈Ski such that M,v |=〈≤i〉δ′ and M,v1 |=δ′. But by definition of δ we know that M′,v′ |=〈≤i〉δ′ too. But then there is a v′ ≤i v′1 such that M′,v′ |=δ′, which means by definition that M,v1 ↔M′,v′. The argument is the same for the other relations, for the other direction, and for event models.  Another important fact about n-bisimulations is that they are preserved by lexicographic update. Take two n-bisimilar pointed event models that can be executed in two n-bisimilar pointed plausibility models. The results of their respective lexicographic update will also be n-bisimilar. Fact 4.10 Take two state models such that (M,w)↔n+N (M′,w′) and event models executable in M,w such that (E,e)↔n(E ′,e′) with N =max{md(pre(e)) :e∈E}. Then (M⊗E,(w,e))↔n(M′⊗E ′,(w′,e′)). Proof. We define the n-bisimulation component-wise: for 0≤k ≤n let (w,e)↔k(w′,e′) iff w↔k+N w′ and e↔ie′. Now we have to show that this is an n-bisimulation. Since valuation is preserved in the updated model, we know that ↔0 preserves propositional equivalence. Now take two (w,e),(w′,e′) that are related k-bisimilar for 0<k ≤n, and suppose that there is a (w,e)≤i (v,f ). This can happen in two cases. First, w∼i v and f <i e. Since w↔k+N w′ we know that there is a v′ ∼i w′ such that v′ ↔(i+K)−1v. Since the pair v,f is in the updated model we know furthermore that M,v |=pre(f ). But observe that since k>0, by our choice of N we can conclude that M′,v′ |=pre(f ) from v′ ↔(k+N)−1v. On the other hand, since e↔ke′ we know that there is an e′< f ′ such that f ′ ↔k−1f . Since M′,v′ |=pre(f ) the pair (v′,f ′) will be in the updated model. But now e′< f ′ and w′ ∼i v′ give us together that (w′,e′)≤i (v′,f ′), and v′ ↔(k+N)−1v with f ′ ↔k−1f give (v,f )↔k−1(v′,f ′). The second case is when e∼=i f and w≤i v. As before, we have to find the right pair (v′,f ′) that is k-bisimilar to (v,f ) and ≤i-related to (w′,e′). We get to v′ in the same way as in the previous case. Now for f ′. Since e∼=i f we know that e≤i f . Because e↔ke′ there must be a f ′ such that e′ ≤i f ′ and e↔k−1e′. From there the argument proceeds as before, except when it comes to establishing that (w′,e′)≤i (v′,f ′). Here we make a case distinction. Suppose we also have f ′ ≤i e′, then the second case of the lexicographic update rule applies and we conclude that (w′,e′)≤i (v′,f ′). If not, then what at U niversitetsbiblioteket i B ergen on A pril 4, 2015 http://logcom .oxfordjournals.org/ D ow nloaded from [15:27 4/7/2014 exu040.tex] LogCom: Journal of Logic and Computation Page: 14 1–22 14 Dynamic consequence for soft information we have is e′<i f ′. But then since w′ ≤i v′ we are back in the first case and can conclude that e′ ≤i f ′ as well.  4.3.3 Proof of the completeness theorem We are almost ready to start the completeness argument itself. We only need the following preparatory Lemma, which serves as a bridge in the completeness proof. The Lemma essentially states that the failure of a dynamic consequence δ1;δ2 |=φ with canonical formulas in the antecendent can be strengthened to a valid dynamic sequent that derives the negation of φ. Lemma 4.11 Let P′ be a subset of E and δ2 ∈SP′n . Take furthermore φ∈LS with md(φ)≤n, and set P=P(φ)∪ {P(pre(e)) :e∈P′}. Take N =max{md(pre(e)) :e∈P′}, δ1 ∈Sn+NP , and write pre(δ2) for pre(e′) with e′ such that δ2 →e′.6 Then t.f.a.e.: • δ1;δ2 |=φ • δ1;δ2 |=¬φ and |=δ1 →pre(δ2) Proof. The direction from (2) to (1) is just unpacking the definitions. For the converse, suppose there is a M,w |=δ1 and E,e |=δ2 such that M⊗E,(w,e) |=¬φ. Take any other models M′,w′ |=δ1 and E ′,e′ |=δ2. By Lemma 4.9 we know that they are, respectively, n+Nand n-bisimilar to M,w and E,e. But then by Fact 4.10 and Lemma 4.9 again we get that M′⊗E ′,(w′,e′) |=¬φ. |=S δ1 →pre(δ2) follows from our choice of δ1 and Lemma 4.6.  We are now ready to start the completeness theorem. As mentioned, we prove it first for the canonical formula, and then lift the result to arbitrary soft dynamic sequents. Theorem 4.12 (Completeness for canonical formulas) Let δ1 and δ2 be as in Lemma 4.11. Take δ3 ∈SnP. Then: If δ1;δ2 |=δ3 then δ1;δ2 δ3 Proof. By induction on n. The basic case is the same as in [1]. Induction step. We prove the contrapositive. Suppose that δ1;δ2 δ3. By the shape of δ3 and Rule R1 one of the following must be the case: δ1;δ2 π (4.1) δ1;δ2  ∧ Ri (∇RiRi ) (4.2) Only the second case differs from the argument in [1]. By the shape of ∇RiRi , using R1 again, one of the following must hold for some Ri and Ri ⊆Sn−1P : (1) δ1;δ2  〈Ri〉δ′3 for some δ′3 ∈Ri (2) δ1;δ2  [Ri]∨Ri So there are two cases to consider, depending on whether the failure of the dynamic consequence comes from ≤i or ∼i. Each has two sub-cases. We consider them in turn. 6Recall that by construction there will always be a unique such e′. at U niversitetsbiblioteket i B ergen on A pril 4, 2015 http://logcom .oxfordjournals.org/ D ow nloaded from [15:27 4/7/2014 exu040.tex] LogCom: Journal of Logic and Computation Page: 15 1–22 Dynamic consequence for soft information 15 (1) Ri =∼i. Both sub-cases here are exactly as in Aucher's [2] paper. (2a.) Ri =≤i δ1;δ2  〈≤i〉δ′3 for some δ′3 ∈≤ii . Take such a δ′3. We have two (similar) intermediate goals. First we want to show that δ′1;δ′2 |=¬δ′3 for any δ′1 ∈∼iδ1 ⊆Sn−1P and δ′2 ∈ <i δ2 ⊆Sn−1P . Take such a δ′2. By construction we have that E δ′2 →e for a unique e. Now take a δ′1 ∈∼iδ1 . If S δ′1 →¬pre(e) then δ′1;δ′2 |=¬δ3 follows by the same argument as in Aucher's paper. So now suppose that S δ′1 →pre(e). By our choice of δ′1 and δ′2, applying R3 and R4 twice, we first get that that 〈∼i〉δ′1;〈<i〉δ′2  〈≤i〉δ′3 and then 〈∼i〉(δ′1 ∧pre(e));〈<i〉(δ′2 ∧e)  〈≤i〉δ′3. So now we can use R7 and conclude δ′1;δ′2 δ′3. By our induction hypothesis we then get δ′1;δ′2 |=δ′3, and so Lemma 4.11 delivers δ′1;δ′2 |=¬δ′3. We are shown that for arbitrary δ′1 and δ′2. So we have reached our first intermediate goal. Now we show basically the same as in the previous paragraph, but for a different set of formulas in the first and second argument positions. We want to show that δ′1;δ′2 |=¬δ′3, but this time for any δ′1 ∈≤iδ1 ⊆Sn−1P and δ′2 ∈ ≤i δ2 ⊆Sn−1P . The argument runs just as before, the only difference being that in the case that S δ′1 →pre(e), after the applications R3 and R4 we use R8 to conclude δ′1;δ′2 δ′3. From there the induction hypothesis and Lemma 4.11 give us our second intermediate goal, namely that δ′1;δ′2 |=¬δ′3 for any δ′1 ∈≤iδ1 ⊆Sn−1P and δ′2 ∈≤iδ2 ⊆Sn−1P . Recall that we proved that δ′1;δ′2 |=¬δ′3 for any δ′1 ∈∼iδ1 ⊆Sn−1P and δ′2 ∈ <i δ2 ⊆Sn−1P . So we get ∨  ∼i δ1 ; ∨  <i δ2 |=¬δ′3 by soundness of R2 and R2′. By the same argument, this time using our second intermediate goal, we conclude that ∨  ≤i δ1 ; ∨  ≤i δ2 |=¬δ′3 So now we are able to use soundness of R6 and the distribution of normal boxes over conjunctions to conclude that [∼i] ∨  ∼i δ1 ∧[≤i] ∨  ≤i δ1 ;[<i] ∨  <i δ2 ∧[≤i] ∨  ≤i δ2 |=[≤i]¬δ′3 But observe that we have the following S δ1 →[∼i] ∨  ∼i δ1 ∧[≤i] ∨  ≤i δ1 E δ2 →[<i]( ∨  <i δ2 )∧[≤i] ∨  ≤i δ2 S ¬〈≤i〉δ′3 →¬δ3 So we can apply R3 and R4 on the left side and R5 on the right side to conclude that: δ1;δ2 |=¬δ3 from which by Lemma 4.11 we get the required conclusion. at U niversitetsbiblioteket i B ergen on A pril 4, 2015 http://logcom .oxfordjournals.org/ D ow nloaded from [15:27 4/7/2014 exu040.tex] LogCom: Journal of Logic and Computation Page: 16 1–22 16 Dynamic consequence for soft information (2b.) Ri =≤i, and δ1;δ2  [≤i]∨≤ii . The proof is essentially the converse of the previous case. Using R3 and R4 we obtain that [∼i] ∨  ∼i δ1 ∧[≤i] ∨  ≤i δ1 ;[<i] ∨  <i δ2 ∧[≤i] ∨  ≤i δ2  [≤i] ∨  ≤i i So using R6 we conclude that one of the following must be the case: ∨  ∼i δ1 ; ∨  <i δ2  ∨  ≤i i ∨  ≤i δ1 ; ∨  ≤i δ2  ∨  ≤i i Let us consider them in turn. By the rule for introduction of ∨ on the left side we know that there must be δ′1 ∈∼iδ1 , δ′2 ∈ <i δ2 such that: δ′1;δ′2  ∨  ≤i i But we can show something stronger, namely that this holds for δ′1 and δ′2 such that δ′1 → pre(e) for the unique e such that δ′2 →e. The argument for that is the same as in Aucher's paper. Now we can use our induction hypothesis to get: δ′1;δ′2 |= ∨  ≤i i and using Lemma 4.11 we conclude that δ′1;δ′2 |=¬ ∨  ≤i i But then (soundness of) R7 gets us 〈∼i〉(δ′1 ∧pre(e));〈<i〉(δ2 ∧e)′ |=〈≤i〉¬ ∨  ≤i i by pulling out the negation on the right side we get 〈∼i〉(δ′1 ∧pre(e));〈<i〉(δ2 ∧e)′ |=¬[≤i] ∨  ≤i i But now observe that |=δ1 →〈∼i〉(δ′1 ∧pre(e)), |=δ2 →〈<i〉(δ′2 ∧e) and |=¬[≤i] ∨  ≤i i →¬δ3. So using strengthening on the left and weakening on the right we conclude that δ1;δ2 |=δ3, as desired. The case where ∨  ≤i δ1 ;∨≤iδ2  ∨  ≤i i follows the same steps, except we use R8 instead.  Corollary 4.13 (Soundness and Completeness for ) Let φ1,φ3 ∈LS and φ2 ∈LE . Then: φ1;φ2 |=φ3 iff φ1;φ2,φ3 Proof. Same argument as in [1].  at U niversitetsbiblioteket i B ergen on A pril 4, 2015 http://logcom .oxfordjournals.org/ D ow nloaded from [15:27 4/7/2014 exu040.tex] LogCom: Journal of Logic and Computation Page: 17 1–22 Dynamic consequence for soft information 17 5 Generalization and structural properties This section takes a closer look at valid and invalid structural rules for soft dynamic sequents. This is the second main contribution of this article. To do so we have to generalize soft dynamic sequents so they also cover sequences of updates. Before we present this generalization, however, we recall briefly the structural rules that we shall consider, and explain why the generalization is necessary. 5.1 Tarski consequence relation and soft dynamic sequents Standard consequence relations satisfy a number of structural conditions. A consequence relation  |=φ is a Tarski consequence relation whenever it satisfies the following conditions, for any finite set of formula , and formulas φ, ψ and χ :7 • if ,φ,φ |=ψ , then ,φ |=ψ [contraction] • if ,φ,ψ, |=χ , then ,ψ,φ, |=χ [permutation, a.k.a. exchange] • φ |=φ [reflexivity, a.k.a. inclusion] • if  |=φ, then ,ψ |=φ [monotonicity, a.k.a. weakening] • if  |=φ and ,φ |=ψ , then , |=ψ [transitivity, a.k.a. cut] The set of valid soft dynamic sequents does not constitute a Tarski consequence relation. Soft dynamic sequents are triples of single formulae (a state formula, an event formula and a state formula). The result is that there is no way to formulate contraction, and no straightforward way to formulate exchange. Some form of exchange might be valid modulo with a pair of translations τ1 :LE →LS , τ2 :LS →LE : • if φ;ψ |=χ , then τ1(ψ);τ2(φ) |=χ But lexicographic update ignores the valuation of atomic propositions in event models, so we conjecture that exchange will turn out to be invalid for any meaningful way to spell out τ1 and τ2. Reflexivity can be formulated for state formulas. • φ1;φ2 |=φ1 But this is invalidated by the same kind of 'Moore sentences' that cause failure of reflexivity for dynamic consequences for public announcements [21]. As mentioned above, monotonicity or weakening is also not expressible in soft dynamic sequents, but rules like R3 capture it through reference to the logical strength in the logic of plausibility and event models. Conceptually this can be seen as a surprise. Soft attitudes like conditional beliefs are non-monotonic. But the semantic definition of valid soft dynamic sequents is a universal quantification: for all M,w and E,e, if M,w |=φ1 and E,e |=φ3, then (M⊗E,(w,e)) |=φ3. Such universal quantifications are monotonic. Finally, there is a version of cut that can be formulated in soft dynamic sequents: • if φ1;φ2 |=φ3 and φ3;φ2 |=ψ , then φ1;φ2 |=ψ [contractive cut] The rule is "contractive" in that it not only cuts the middle term φ3, it also suppresses one occurrence of the φ2-type of update. 7The permutation rule below is strong enough to entail associativity. Although we do not explore it here, a more finegrained approach would involve a restricted permutation rule together with an explicit rule for associativity. We owe this point to an anonymous referee. at U niversitetsbiblioteket i B ergen on A pril 4, 2015 http://logcom .oxfordjournals.org/ D ow nloaded from [15:27 4/7/2014 exu040.tex] LogCom: Journal of Logic and Computation Page: 18 1–22 18 Dynamic consequence for soft information Fact 5.1 Contractive cut is not valid for soft dynamic sequents Proof. (Sketch) Counterexamples are easy to devise using the following observation. From the first premise we know that after an update by a φ2-type of epistemic action in a φ1-initial situation we always end up in a situation where φ3 holds. But to use the second premise and reach a ψ-type of situation, as required to establish the consequent of the rule, we need an additional update: a φ2-type of epistemic action. There is no certainty that the first update will suffice.  The failure of contractive cut shows why a generalization to sequences of updates is indeed of interest. As we shall see below, multiplicative cut, viz., cut that brings with it the second occurrence of φ2, is valid for generalized soft dynamic sequents. Generalizing to sequences of updates really shows the sub-structural character of soft dynamic sequents. 5.2 Generalized soft dynamic sequents We want to generalize soft dynamic sequents in such a way that the structural rules just mentioned can be expressed more naturally. To do this we allow for sequences of update formulas in the second argument. Conceptually and formally, this is the important difference from the sequents studied so far. We also allow finite sets of formulas to occur in the first argument. This, of course, is equivalent to what we had before, viz., a single formula in the first argument. We do this to make generalized soft dynamic sequents notationally closer to consequence relations, as studied in substructural logic. Definition 5.1 (Dynamic Consequence Relation – Generalized) Let  be a finite set of formulas of LS and be a finite sequence (ψ1,...,ψn) of formulas of LE . A generalized soft dynamic sequent (GSDS) is an object of the form: ; |=φ A GSDS is valid iff for all M,w and sequence ((E1,e1),...(En,en)) of length n, if M,w |=∧ and for all 1≤ i≤n we have that Ei,ei |=ψi then ((M⊗E1)⊗ ...)⊗En),((w,e1)...,en) |=φ3. Generalized soft dynamic sequents are close to the consequence relations investigated by [21] and [8] in that they now cover sequences of updates, and are contextualized. But in contrast to [8], we keep to finite descriptions of the initial situation, i.e. finite set  in the first argument. Each of the structural rules mentioned above can now be formulated component-wise, i.e. either for state or event formulas. For instance, one can formulate two versions of exchange, here in simple, two-formula form: • if {φ,ψ}; |=χ , then {ψ,φ}; |=χ [exchange, state formulas] • if ;〈φ,ψ〉 |=χ , then ;〈ψ,φ〉 |=χ [exchange, event formulas] But, as before, there is no counterpart of these rules that crosses the semicolon. The distinction between state and event formulas remains intact. Structural rules operate within each context, i.e. to describe the situations the agents are in and to describe the types of epistemic actions that are occurring, but not across contexts. at U niversitetsbiblioteket i B ergen on A pril 4, 2015 http://logcom .oxfordjournals.org/ D ow nloaded from [15:27 4/7/2014 exu040.tex] LogCom: Journal of Logic and Computation Page: 19 1–22 Dynamic consequence for soft information 19 5.3 Structural properties of soft dynamic sequents Although the dynamic relations are not Tarski relations, they do have a number of weaker structural properties. Here we offer a brief structural analysis of the dynamic consequence relation for lexicographic update. In a sequent , |=φ the first position will now immediately have contraction and permutation as admissible rules. This follows trivially from  being a set. In addition, the consequence relation becomes monotonic for the first position. Fact 5.2 The following is valid for GSDS: ; |=φ, then ,ψ; |=φ Proof. Analogous to soundness of R3 in for soft dynamic sequents.  Generalized soft dynamic sequents are much less structural in the second position, i.e. for sequences of update. Monotonicity, exchange and contraction all fail there. Failure of exchange within sequences arises for reasons similar to those given by [21] in the case of public announcements. For a counterexample, observe that {¬pre(e)};〈e,[<i]P〉 |=⊥ is a valid GSDS, but that {¬pre(e)};〈[<i]P,e〉 |=⊥ is not. One can easily devise a case where updating first by a [<i]P makes e-types of epistemic actions-i.e. any action with precondition pre(e)-executable. This failure of exchange reflects the fact that soft information updates are order-dependent. This dependency was already well known for hard information updates, see again [21]. But for general information dynamics, this is aggravated by the fact that epistemic actions can have arbitrary conditions of executability, as the counter-example illustrates. Failure of contraction reflects the fact that iterated update with the same type of epistemic action is not reducible to a single update of that type. This is also a well-known fact in the theory of multi-agent-iterated belief revision, see e.g. [4, 18]. There are two forms of monotonicity in the second position, only one of which fails. First, there is the straightforward generalization of R4: Fact 5.3 The following monotonicity rule is valid for soft information sequents: ;	φ ;	[φi/ψi]φ R4−Gen with ψi →φi and	[φi/ψi] being the sequence resulting from substitutingψi for the ith component φi in . Proof (Sketch). Suppose ;	φ. Take a pointed plausibility model that satisfies all the formulas in . Start the sequence of update	[φi/ψi]. This sequence is the same as up to the ith step. There one updates the current plausibility model with a ψi type of (pointed) event models instead of a φi. But because ψi →φi we know that this pointed event model also satisfies φi. But then, since and	[φi/ψi] agree on subsequent updates, we know from ;	φ that φ will be true in the pointed plausibility model resulting from updating with [φi/ψi].  With sequences of formulas in the second position we can formulate a more classical form of monotonicity, namely one that adds formulas to the existing sequence. By doing so we do not move at U niversitetsbiblioteket i B ergen on A pril 4, 2015 http://logcom .oxfordjournals.org/ D ow nloaded from [15:27 4/7/2014 exu040.tex] LogCom: Journal of Logic and Computation Page: 20 1–22 20 Dynamic consequence for soft information from a logically weaker to logically stronger antecedent. This type of operation adds one update in the sequence. Fact 5.4 The following monotonicity rules are invalid for soft information sequents: ;	φ ;ψ,	φ LM ;	φ ;	,ψ φ RM Proof (Sketch). Right monotonicity is not valid because updating with an additional ψ-type of epistemic action can break the conclusion of the sequence of update . Again, examples are easy to come by. Failure of left monotonicity for contextualized dynamic consequence has already been observed by [8]. Their counter-example can be easily adapted to lexicographic upgrade.  In Section 5.1, we mentioned that contractive cut is not valid for soft dynamic sequents. We observed that that rule failed because it contracted two φ2-updates into one. The present generalization allows us to formulate a more careful, multiplicative version of cut, which turns out to be valid. Fact 5.5 The following multiplicative cut rule is valid for soft information sequents: ;	φ {φ};	′ ψ ;	,	′ ψ MCut Proof (Sketch). Take a pointed plausibility model that satisfies all the formulas in . By the leftmost premise, applying the sequence of updates will result in a pointed plausibility model that satisfies φ. But by the right-most premise, continuing the update sequences with ′ will result in a model where ψ holds.  The reader can check that the natural generalization of most of the rules in Table 1 are valid, even for the box introduction rules R6 and R9. Rules for diamond introduction are more intricate to devise with sequences of update. One needs to refer to the preconditions of the updates after the first one in a sequence . All the rules that we have considered at this point fail to do this in a valid way. The validity of multiplicative cut also shows that this generalization of soft dynamic sequents validates some rules that allow sequences of updates to be extended. Multiplicative cut concatenates sequences. It is still an open question whether there are other rules that allow one to operate on sequences in a similar fashion. For these reasons we do not know whether the set of valid generalized soft dynamic sequents is completely axiomatizable. 6 Conclusion This article has extended the study of dynamic consequence relations to a general class of soft information changes. We have provided a sound and complete calculus for one-step soft dynamic consequence relations, and given a number of valid and invalid structural rules for a generalization to sequences of updates. The complete axiomatization of the latter has been left open: this is the obvious next step for future work. It would also be interesting to look at dynamic consequences for specific update rules for soft information that have been studied in the literature, such as radical and conservative upgrade, in particular whether they validate additional structural rules. at U niversitetsbiblioteket i B ergen on A pril 4, 2015 http://logcom .oxfordjournals.org/ D ow nloaded from [15:27 4/7/2014 exu040.tex] LogCom: Journal of Logic and Computation Page: 21 1–22 Dynamic consequence for soft information 21 Acknowledgments The authors would like to thank the participants of the Prague Workshop on Sub-Structural Epistemic Logic (June 2011), and of the Munich Meeting on Sub-structural Epistemic Logic (January 2013) for very useful comments and suggestions on earlier versions of the article. They are also very grateful to the two anonymous referees of the JLC for thorough reading and extensive comments. Part of this work has been done while O.R. was affiliated at the LMU Munich. 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