TY - JOUR
AU - Velázquez-Quesada, Fernando, R
AB - Abstract Belief revision is concerned with belief change fired by incoming information. Despite the variety of frameworks representing it, most revision policies share one crucial feature: incoming information outweighs current information and hence, in case of conflict, incoming information will prevail. However, if one is interested in representing the way actual humans revise their beliefs, one might not always want for the agent to blindly believe everything they are told. This manuscript presents a semantic approach to non-prioritized belief revision. It uses plausibility models for depicting an agent’s beliefs, and model operations for displaying the way beliefs change. The first proposal, semantically-based screened revision, compares the current model with the one the revision would yield, accepting or rejecting the incoming information depending on whether the ‘differences’ between these models go beyond a given threshold. The second proposal, semantically-based gradual revision, turns the binary decision of acceptance or rejection into a more general setting in which a revision always occurs, with the threshold used rather to choose ‘the right revision’ for the given input and model. 1 Introduction Belief revision ([22, 45, 52], among many others) is concerned with belief change fired by incoming information. It emerged as a proper research field when philosophical traditions dealing with both the requirements of rational belief change and the mechanisms by which scientific theories develop [30] converged with computer-science oriented approaches on database updates [20, 35] and deontic studies focused on derogations in legal codes [1]. The seminal work by Alchourrón et al. [2], the AGM approach, is considered the birth of the field; since then, many of its concerns and ideas have been proved relevant in philosophy, computer science, learning theory and other fields. Given this process’ importance, it is not surprising that many frameworks have been proposed for representing it. An important difference among them has been the representation of beliefs themselves, with some proposals following a syntactic approach (using either a plain set of formulas or else a deductively closed one; 45) and some others using semantic structures (e.g. ordinal functions [48], a system of spheres [25] or a plausibility relation [5, 6, 12, 18]).1 Still, a more fundamental one has been the principles followed by the belief change mechanism. Indeed, different revision policies have been proposed over time, with the 27 belief change operations described in Rott [47] (albeit including also operations for belief expansion and belief contraction) being a good example of the existing diversity. Despite of this variety, most revision policies share one crucial feature: incoming information outweighs current information, so that in case of conflict, the latter will prevail. This is not accidental. In fact, this success requirement is, among the AGM postulates for rational belief revision [2], the first: any successful revision act with a consistent formula |$\chi$| as the new information (a |$\chi$|-revision) should make |$\chi$| part of the agent’s beliefs. The postulate is reasonable when one considers the original situations targeted by belief revision. Indeed, if a new observation contradicts a scientific theory, the theory should be adapted to account for it; if new information is entered into a database, the database should not return old data in future searches; if a law changes, subsequent decisions (and, in some cases, previous ones, as with retroactive decisions) should be (re) ruled by its most recent version. However, if one is interested in representing the way actual humans revise their beliefs, one might not want for them to blindly accept everything they are told. First, information might come from unreliable sources, and thus the agent should be able to judge the reliability of the received news. Maybe more importantly, humans tend to be self-righteous [26], assuming most of the time that their own beliefs are the correct ones, and thus likely to reject information that contradicts them.2 Some approaches have studied revision operators that do not enforce the success postulate. One of the earliest is non-prioritized belief revision [28], which represents beliefs as a set of formulas closed under logical consequence, and under which new information will be accepted only if it has more ‘epistemic value’ than the original beliefs it might contradict. This notion of ‘epistemic value’ is defined sometimes by a set of core beliefs (the incoming |$\chi$| will be accepted only if it is consistent with such set: the screened revision of [39]) and sometimes by a set of credible formulas (the incoming |$\chi$| will be accepted only if it belongs to such set: the credibility-limited revision of [29]). Still, not much work in this direction has been done using richer belief representations.3 In this regard, an appealing one is the mentioned plausibility model, a relational (possible worlds) structure in which the relation is interpreted as describing the plausibility order the agent assigns to the situations she considers possible, and in which beliefs are defined as what is the case in the most plausible worlds.4 The model provides a natural description of how likely it is for an input to be accepted, as it represents not only what the agent believes, but also the ordering she assigns to those other possibilities that are currently not among the most plausible (i.e. believed) ones. Thus, if a formula is not currently believed (i.e. it fails in at least one of the most plausible worlds), the model can still tell us ‘how difficult’ it would be for the agent to come to believe it (i.e. how much the model would need to change to have only worlds satisfying it on the top of the ordering). Similar models are used by the improvement operators of Konieczny and Pino Pérez [36] and Konieczny et al. [37], under which the incoming formula does not need to be believed after the improvement, but its plausibility will always be increased.5 This work proposes a semantic approach to non-prioritized revision. The key idea is to make use of a semantically defined distance between models in order to decide, in a first alternative, whether a revision will go through, and in a second alternative, which revision policy will be performed.6 This way, the decision of whether/how to perform the revision depends only on how much the agent’s epistemic state would need to change in order to accept the incoming information (and also on the agent’s ‘openness’ to modifying her beliefs), leaving out ‘external’ factors such as the core beliefs or the credible formulas mentioned before. For the semantic structures representing beliefs, this text uses the discussed plausibility models. For the representation of belief change, it follows the dynamic epistemic logic approach (DEL; [7, 19]), under which an act of revision changes the original plausibility ordering. Here is an example of the intuitive idea. Example 1.1 Consider the plausibility model below, with each possible world (i.e. each node) being a full propositional description of the way the world could be (given in terms of the atoms that hold at it), and with the relation representing the agent’s plausibility ordering (reflexive and transitive edges omitted): The plausibility ordering is |$w_1 < w_2 <w_3$|⁠, with |$w_1$| being the least plausible world and |$w_3$| the most plausible one (thus, the agent believes |$p \land q$|⁠). Intuitively, it is more difficult to accept new information when integrating it into our beliefs would change the way we see the world in a drastic way. In the given model, we see the order would change less if it became |$w_1 < w_3 < w_2$| than if it became |$w_2 < w_3 < w_1$|⁠: the first would need to shift |$w_2$| up once, while the second would need to shift |$w_1$| up twice. In other words, |$\lnot p \land \lnot q$| is more likely to be accepted (as it is true in the second-most-plausible |$w_2$|⁠) than |$\lnot p \land q$| (as it is true the least plausible |$w_1$|⁠). The paper is structured as follows. Section 2 defines the basics of the plausibility framework for belief revision (plausibility model, three well-known revision policies and a formal language for describing both the agent’s epistemic state and the way it changes). Section 3 introduces the first variant of the proposal, fixing a revision policy, and using a notion of distance between models to compare the model that would result from the revision with the original one. If they are not ‘too different’, the revision will go through; otherwise, no change will be made. Section 4 introduces the second variant of this proposal, using the notion of distance between models not for deciding whether a revision will take place, but rather for deciding which revision policy will be applied. Natural candidates come among families of parametrized revision policies, two of which are introduced. Section 6 summarizes the contributions, also listing directions for further research. 2 Beliefs and Belief Revision in Plausibility Models This section recalls the formal definition of a plausibility model [5, 6, 12, 18], a language for describing it, and some well-known belief revision policies in this setting. Throughout the text, let |${\mathtt{P}}$| be a countable set of atomic propositions, and let |${\mathtt{N}}$| be a finite set of special atoms called nominals (with |${\mathtt{P}} \cap{\mathtt{N}} = \varnothing$|⁠). Definition 2.1 (Plausibility model). A (single-agent) plausibility model (⁠|${\mathscr{PM}}$|⁠) is a tuple |$M = {\left \langle{W, \leq , V} \right \rangle }$| in which (i)|$W \neq \varnothing$| is a finite set of worlds (also denoted by |${\mathcal{D}_{{M}}}$|⁠), (ii)|${\leq } \subseteq (W \times W)$| is a total preorder on |$W$|⁠, the agent’s plausibility relation over |$W$| (⁠|$u \leq v$| is read as ‘|$u$| is at most as plausible as |$v$|’) and (iii)|$V:{\mathtt{P}} \cup{\mathtt{N}} \to \wp (W)$| is a valuation function, returning the set of worlds in which each atom and nominal holds, with each nominal assumed to be true at exactly one world (i.e. |$i \in{\mathtt{N}}$| implies |${\vert{V(i)} \vert } = 1$|⁠).78 The class of all plausibility models is denoted by |$\mathcal{M}$|⁠. A pair |$(M, w)$| with |$M$| a plausibility model and |$w \in{\mathcal{D}_{{M}}}$| is called a plausibility state (⁠|${\mathscr{PS}}$|⁠). Here are further useful definitions. For |$u,v \in W$|⁠, − |$u < v$| (‘|$u$| is less plausible than |$v$|’) |$\textrm{iff}_{\textit{def}}$||$u \leq v$| and |$v \not \leq u$|⁠; − |$u \sim v$| (‘|$u$| and |$v$| are comparable’) |$\textrm{iff}_{\textit{def}}$||$u \leq v$| or |$v \leq u$| (note that, due to the plausibility order being total, |$u \sim v$| holds for all |$u,v \in W$|⁠); − |$u \simeq v$| (‘|$u$| and |$v$| are equally plausible’) |$\textrm{iff}_{\textit{def}}$||$u \leq v$| and |$v \leq u$|⁠; − |$L \subseteq W$| is a layer if and only if it is a maximum set of equally plausible worlds, i.e. a set such that |$(L \times L) \subseteq{\leq }$| (so all elements of |$L$| are |$\leq$|-related), but |$L \subset L^{\prime }$| implies |$(L^{\prime } \times L^{\prime }) \not \subseteq{\leq }$| (no superset of |$L$| satisfies such property). − the set of |$\leq$|-maximum elements in |$U \subseteq W$| is given by $$\begin{equation*}\textrm{Mx}{U}:= {\left\{ {u \in U \mid v \leq u \;\textrm{for all}\; v \in U} \right\}}.\end{equation*}$$ A plausibility model is then a collection of one or more layers, with the layers themselves ordered according to their plausibility. The nominals, borrowed from hybrid logic (see, e.g. [3] and [11, Section 7.3]), are used as a natural tool for ‘naming’ worlds, and will be useful for axiomatising the revisions that will be introduced in later sections.9 These structures can be described by different languages; here is the one this manuscript will use. Definition 2.2 (Language |${\mathcal{L}_{i}}$|⁠). Formulas |$\varphi , \psi$| of the language |${\mathcal{L}_{i}}$| are given by $$\begin{equation*}\varphi, \psi::= i \mid p \mid \lnot \varphi \mid \varphi \lor \psi \mid{{\langle{\scriptstyle \leq} \rangle}{\varphi}} \mid{{\langle{\scriptstyle <}\rangle} {\varphi}} \mid{{\langle{\scriptstyle \sim} \rangle}{\varphi}}\end{equation*}$$ with |$i \in{\mathtt{N}}$| and |$p \in{\mathtt{P}}$|⁠. Propositional constants (⁠|$\top , \bot$|⁠), other Boolean connectives (⁠|$\land , {\rightarrow }, {\leftrightarrow }$|⁠) and dual modal operators (⁠|${{[{\scriptstyle \leq}]}{}}, {{[{\scriptstyle <}]}{}}, {{[{\scriptstyle \sim }]}{}}$|⁠) are defined as usual (⁠|${{[{\scriptstyle \leq}]}{\varphi }}:= \lnot{{\langle{\scriptstyle \leq} \rangle }{\lnot \varphi }},\ {{[{\scriptstyle <}]}{\varphi }}:= \lnot{{\langle{\scriptstyle <}\rangle } {\lnot \varphi }}$| and |${{[{\scriptstyle \sim}]}{\varphi }}:= \lnot{{\langle{\scriptstyle \sim} \rangle }{\lnot \varphi }}$| for the latter). For the semantic interpretation, let |$(M, w)$| be a |${\mathscr{PS}}$| with |$M = {\left \langle{W, \leq , V} \right \rangle }$|⁠. The cases for negations and disjunctions are as usual; for the rest, $$\begin{equation*}\begin{array}{lll} (M, w) \Vdash p &\textrm{iff}_{def} & w \in V(p) \\ (M, w) \Vdash i &\textrm{iff}_{def} & {\left\{ {w} \right\}} = V(i) \\ (M, w) \Vdash{{\langle{\scriptstyle \leq} \rangle}{\varphi}}&\textrm{iff}_{def} & \textrm{there is}\ u \in W \ \textrm{such that}\ w \leq u \ \textrm{and}\ (M, u) \Vdash \varphi \\ (M, w) \Vdash{{\langle{\scriptstyle <}\rangle} {\varphi}} &\textrm{iff}_{def} & \textrm{there is}\ u \in W \ \textrm{such that}\ w < u \ \textrm{and}\ (M, u) \Vdash \varphi \\ (M, w) \Vdash{{\langle{\scriptstyle \sim} \rangle}{\varphi}}&\textrm{iff}_{def} & \textrm{there is}\ u \in W \ \textrm{such that}\ w \sim u \ \textrm{and}\ (M, u) \Vdash \varphi \\\end{array}\end{equation*}$$ Given a |${\mathscr{PM}}$||$M$| and a formula |$\varphi \in{\mathcal{L}_{i}}$|⁠, the truth-set of |$\varphi$| at |$M$| (the set of those worlds in |$M$| where |$\varphi$| holds) is defined as $$\begin{equation*}{{[\kern{-1.5pt}[}{\varphi} {]\kern{-1.5pt}]}^{{M}}}:= {\left\{ {w \in W \mid (M, w) \Vdash \varphi} \right\}}\end{equation*}$$ A formula |$\varphi$| is valid (notation: |$\Vdash \varphi$|⁠) if and only if |$(M, w) \Vdash \varphi$| for all worlds |$w \in{\mathcal{D}_{{M}}}$| of all |${\mathscr{PM}}$|s |$M$|⁠. All that is left is to define the notion of belief. As sketched before, the agent believes a given formula |$\varphi$| (⁠|${\mathop{B}{\varphi }}$|⁠) if and only if it holds in all the most plausible worlds. More precisely, given a |${\mathscr{PM}}$||$M$| and |$w \in{\mathcal{D}_{{M}}}$|⁠, $$\begin{equation*}(M, w) \Vdash B\varphi \quad \textrm{iff}_{def} \quad{\operatorname{Mx}({\mathcal{D}_{M}})} \subseteq{{[\kern{-1.5pt}[}{\varphi} {]\kern{-1.5pt}]}^{{M}}}.\end{equation*}$$ Given |$\leq$|’s properties, it is not hard to see [5] that, within |${\mathcal{L}_{i}}$|⁠, beliefs can be defined by abbreviation: $$\begin{equation*}\Vdash{\mathop{B}{\varphi}}\, {\leftrightarrow}\,{{\langle{\scriptstyle \leq} \rangle}{{[{\scriptstyle \leq}]}\varphi}}.\end{equation*}$$ More interestingly, the modality for the strict plausibility relation allows for other alternatives. The set of most plausible words in a |${\mathscr{PM}}$| is syntactically characterised by |${{[{\scriptstyle <}]}{\bot }}$| (i.e. |${\operatorname{Mx}({\mathcal{D}_{M}})} = {{[\kern{-1.5pt}[}{{[{\scriptstyle <}]}\bot } {]\kern{-1.5pt}]} ^{{M}}}$| for any |${\mathscr{PM}}$||$M$|⁠); hence, $$\begin{equation*}\Vdash{\mathop{B}{\varphi}}\, {\leftrightarrow}\,{{[{\scriptstyle \sim}]}{({[{\scriptstyle <}]}\bot \rightarrow \varphi)}}.\end{equation*}$$ In fact, all layers can be characterised syntactically. Define $$\begin{equation*}{\operatorname{L}^{{M}}_{{0}}}:= {\operatorname{Mx}({\mathcal{D}_{M}})}, \qquad{\operatorname{L}^{{M}}_{{j+1}}}:= {\operatorname{Mx}({\mathcal{D}_{M} \setminus \bigcup_{k=0}^{j} \operatorname{L}^{M}_{k}})}.\end{equation*}$$ Then, by taking $$\begin{equation*}\lambda_0:= {{[{\scriptstyle <}]}{\bot}} \qquad\textrm{and}\qquad \lambda_{j+1}:= {{\langle{\scriptstyle <}\rangle} {\lambda_j}} \land \lnot{{\langle{\scriptstyle <}\rangle} {{\langle{\scriptstyle <}\rangle} \lambda_j}},\end{equation*}$$ it follows that |${\operatorname{L}^{{M}}_{{j}}} = {{[\kern{-1.5pt}[}{\lambda _j} {]\kern{-1.5pt}]} ^{{M}}}$|⁠, for every |$j \in{\mathbb{N}}$|⁠. Thus, the language can characterise further belief notions, as the qualitative degree of beliefs [25, 48], looking for what holds ‘from some layer up’. In this single-agent setting, a notion of knowledge (⁠|${\mathop{K}{\varphi }}$|⁠) can be defined as truth in all epistemic possibilities, as usual: $$\begin{equation*}{\mathop{K}{\varphi}}:= {{[{\scriptstyle \sim}]}{\varphi}}.\end{equation*}$$ 2.1 Some existing belief revision policies A plausibility model depicts the plausibility order the agent assigns to her epistemic possibilities, with her beliefs given by what holds in all most plausible ones. Then, changes in beliefs can be represented by changes in plausibility [5, 6, 18]. In particular, the act of revising beliefs in order to accept a given |$\chi \in{\mathcal{L}_{i}}$| can be understood as an operation that puts |$\chi$|-worlds at the top of the plausibility order. Of course, there are several ways in which such a new order can be defined, but each one of them can be seen as a different policy for revising beliefs. The definitions below are some of the best-known possibilities. Definition 2.3 Let |$M = {\left \langle{W, \leq , V} \right \rangle }$| be a |${\mathscr{PM}}$|⁠; let |$\chi$| be a formula. (i) The radical|$\chi$|-revision operation (called moderate in [40] and lexicographic in [47]) yields the |${\mathscr{PM}}$| model |${{{M}}^{{\Uparrow }_{{\chi }}}} = {\langle{W, \mathrel{{\leq }^{{\Uparrow }_{\chi }}}, V} \rangle }$|⁠, with the new plausibility ordering given by $$\begin{equation*}{{\mathrel{{{\leq}}^{{\Uparrow}_{{\chi}}}}}}:= \left( {\leq} \setminus ({{[\kern{-1.5pt}[}{\chi}{]\kern{-1.5pt}]}^{{M}}} \times{{[\kern{-1.5pt}[}{\lnot\chi} {]\kern{-1.5pt}]}^{{M}}}) \right) \,\cup\, \left( {\sim} \cap ({{[\kern{-1.5pt}[}{\lnot\chi} {]\kern{-1.5pt}]}^{{M}}} \times{{[\kern{-1.5pt}[}{\chi} {]\kern{-1.5pt}]}^{{M}}}) \right),\end{equation*}$$ thus reversing arrows going from |$\chi$|-worlds to |$\lnot \chi$|-worlds. Intuitively, all |$\chi$|-worlds become more plausible than all |$\lnot \chi$|-worlds, and within the two zones, the old ordering remains unchanged.10 (ii) The conservative|$\chi$|-revision operation ([46], called natural in [13] and minimal conditional in [14]) yields the |${\mathscr{PM}}$| model |${{{M}}^{{\uparrow }_{{\chi }}}} = {\langle{W, \mathrel{{\leq }^{{\uparrow }_{\chi }}}, V} \rangle }$|⁠, with the new plausibility ordering given by $$\begin{equation*}{{\mathrel{{{\leq}}^{{\uparrow}_{{\chi}}}}}}:= \left( {\leq} \setminus \big({\operatorname{Mx}({{[\kern{-1.5pt}[} \chi {]\kern{-1.5pt}]}^{M}})} \times{{[\kern{-1.5pt}[}{\lnot\chi} {]\kern{-1.5pt}]}^{{M}}} \big) \right) \,\cup\, \left( {\sim} \cap \big( {{[\kern{-1.5pt}[}{\lnot\chi} {]\kern{-1.5pt}]}^{{M}}} \times{\operatorname{Mx}({{[\kern{-1.5pt}[} \chi {]\kern{-1.5pt}]}^{M}})} \big) \right),\end{equation*}$$ thus reversing arrows from the most plausible |$\chi$|-worlds (those in |${\operatorname{Mx}({{[\kern{-1.5pt}[} \chi {]\kern{-1.5pt}]} ^{M}})}$|⁠) to |$\lnot \chi$|-worlds (within a comparability class). Intuitively, the most plausible worlds satisfying |$\chi$| become more plausible than all other worlds, and within the two zones, the old ordering remains unchanged.11 (iii) Define |${\prec }:= {<} \cap{\left \{ {(u,v) \in W \times W \mid \forall w \in W.(u \leq w \leq v \Rightarrow (u \simeq w \textrm{or} w \simeq v))} \right \}}$|⁠, so |$u \prec v$| if and only if |$u$| is on the layer immediately below that of |$v$|⁠. The improvement|$\chi$|-revision operation (here given in a ‘plausibility’ version; cf. the original [36]) yields the |${\mathscr{PM}}$| model , with the new plausibility ordering given by thus removing arrows going from |$\chi$|-worlds to |$\lnot \chi$|-worlds when they were equally-plausible (hence breaking ties between |$\lnot \chi$|-worlds and |$\chi$|-worlds in favour of the latter), and adding arrows from |$\lnot \chi$|-worlds to |$\chi$|-worlds when the latter were immediately below the former. Intuitively, the operation shifts every |$\chi$|-world ‘one layer up’. In all cases, the operation does yield a total preorder, so |${{{M}}^{{\Uparrow }_{{\chi }}}},\ {{{M}}^{{\uparrow }_{{\chi }}}}$| and are all indeed plausibility models. Here is an example of how these revision policies work. Example 2.4 Consider the following |${\mathscr{PM}}$||$M$|⁠, with each world containing exactly the atoms true at it, and with an arrow from |$w$| to |$u$| indicating |$w \leq u$| (unless stated otherwise, reflexive and transitive arrows are omitted from diagrams). (i) A radical|$p$|-revision makes all |$p$|-worlds more plausible than all |$\lnot p$|-worlds, keeping the old ordering within both zones; thus, |${{{M}}^{{\Uparrow }_{{p}}}}$| is given by (ii) A conservative|$p$|-revision makes the most plausible |$p$|-worlds the most plausible overall, keeping the rest of the ordering as before; thus, |${{{M}}^{{\uparrow }_{{p}}}}$| is (iii) An improvement|$p$|-revision shifts |$p$|-worlds one layer up; thus, is In order to describe the effects of these policies, one can extend the provided language with modalities that, following the DEL approach, are semantically interpreted over the models that result from their respective operations. Definition 2.5 (Language). The language |${\mathcal{L}_{i}}$| is extended with three modalities, |${\mathop{\langle{\chi }{\Uparrow }\rangle }{{}}},\ {\mathop{\langle{\chi }{\uparrow }\rangle }{{}}}$| and , for |$\chi$| a formula, resulting in the languages |${\mathcal{L}_{i,{\Uparrow }}}$|⁠, |${\mathcal{L}_{i,\uparrow }}$| and , respectively. As for their semantic interpretation, given a |${\mathscr{PM}}$||$M$| and |$w \in{\mathcal{D}_{{W}}}$|⁠, Axiom systems. The validities of formulas in |${\mathcal{L}_{i}}$| (the ‘static’ language) with respect to plausibility models are characterised by the axiom system of Table 1. The system combines the axioms and rules of a language over |${\mathscr{PM}}$|s that uses modalities |${{\langle{\scriptstyle \leq} \rangle }{}}$| and |${{\langle{\scriptstyle \sim} \rangle }{}}$| (first two blocks; [5]) with axioms and rules that characterise |${{\langle{\scriptstyle <}\rangle } {}}$| as a normal modality standing for the strict version of the relation of the modality for |${{\langle{\scriptstyle \leq} \rangle }{}}$| (third block; [8]). The axioms on the fourth block take advantage of the fact that |${\mathcal{L}_{i}}$| already contains a global modality (⁠|${{\langle{\scriptstyle \sim} \rangle }}$|⁠) in order to characterise the behaviour of nominals [16, Table 5.3]: the axiom on the left indicates that every nominal |$i$| corresponds to some world, and the one on the right indicates that the world named by each nominal is unique. Table 1. Axiom system for |${\mathcal{L}_{i}}$| w.r.t. |${\mathscr{PM}}$| |$\vdash \varphi$| for |$\varphi$| an instance of a tautology . From |$\vdash \varphi$| and |$\vdash \varphi \, {\rightarrow }\, \psi$| derive |$\vdash \psi$| . |$\vdash{{[{\scriptstyle \leq} ]}{(\varphi \rightarrow \psi )}}\, {\rightarrow }\, ({{[{\scriptstyle \leq} ]}{\varphi }}\, {\rightarrow }\, {{[{\scriptstyle \leq} ]}{\psi }})$| |$\vdash{{[{\scriptstyle \sim} ]}{(\varphi \rightarrow \psi )}}\, {\rightarrow }\, ({{[{\scriptstyle \sim} ]}{\varphi }}\, {\rightarrow }\, {{[{\scriptstyle \sim} ]}{\psi }})$| |$\vdash{{\langle{\scriptstyle \leq} \rangle }{\varphi }}\, {\leftrightarrow }\,\lnot{{[{\scriptstyle \leq} ]}{\lnot \varphi }}$| |$\vdash{{\langle{\scriptstyle \sim} \rangle }{\varphi }}\, {\leftrightarrow }\,\lnot{{[{\scriptstyle \sim} ]}{\lnot \varphi }}$| From |$\vdash \varphi$| derive |$\vdash{{[{\scriptstyle \leq} ]}{\varphi }}$| From |$\vdash \varphi$| derive |$\vdash{{[{\scriptstyle \sim} ]}{\varphi }}$| |$\vdash \varphi \, {\rightarrow }\, {{\langle{\scriptstyle \leq} \rangle }{\varphi }}$| |$\vdash \varphi \, {\rightarrow }\, {{\langle{\scriptstyle \sim} \rangle }{\varphi }}$| |$\vdash{{\langle{\scriptstyle \leq} \rangle }{{\langle{\scriptstyle \leq} \rangle }\varphi }}\, {\rightarrow }\, {{\langle{\scriptstyle \leq} \rangle }{\varphi }}$| |$\vdash{{\langle{\scriptstyle \sim} \rangle }{{\langle{\scriptstyle \sim} \rangle }\varphi }}\, {\rightarrow }\, {{\langle{\scriptstyle \sim} \rangle }{\varphi }}$| |$\vdash \varphi \, {\rightarrow }\, {{[{\scriptstyle \sim} ]}{{\langle{\scriptstyle \sim} \rangle }\varphi }}$| |$\vdash ({{\langle{\scriptstyle \sim} \rangle }{\varphi }} \land{{\langle{\scriptstyle \sim} \rangle }{\psi }})\, {\rightarrow }\, \left ( {{\langle{\scriptstyle \sim} \rangle }{(\varphi \land{\langle{\scriptstyle \leq} \rangle }\psi )}} \lor{{\langle{\scriptstyle \sim} \rangle }{(\psi \land{\langle{\scriptstyle \leq} \rangle }\varphi )}} \right )$| |$\vdash{{\langle{\scriptstyle \leq} \rangle }{\varphi }}\, {\rightarrow }\, {{\langle{\scriptstyle \sim} \rangle }{\varphi }}$| |$\vdash{{[{\scriptstyle <}]}{(\varphi \rightarrow \psi )}}\, {\rightarrow }\, ({{[{\scriptstyle <}]}{\varphi }}\, {\rightarrow }\, {{[{\scriptstyle <}]}{\psi }})$| |$\vdash{{\langle{\scriptstyle <}\rangle } {\varphi }}\, {\rightarrow }\, {{\langle{\scriptstyle \leq} \rangle }{\varphi }}$| |$\vdash{{\langle{\scriptstyle <}\rangle } {\varphi }}\, {\leftrightarrow }\,\lnot{{[{\scriptstyle <}]}{\lnot \varphi }}$| |$\vdash{{\langle{\scriptstyle \leq} \rangle }{{\langle{\scriptstyle <}\rangle } \varphi }}\, {\rightarrow }\, {{\langle{\scriptstyle <}\rangle } {\varphi }}$| From |$\vdash \varphi$| derive |$\vdash{{[{\scriptstyle <}]}{\varphi }}$| |$\vdash{{\langle{\scriptstyle <}\rangle } {{\langle{\scriptstyle \leq} \rangle }\varphi }}\, {\rightarrow }\, {{\langle{\scriptstyle <}\rangle } {\varphi }}$| |$\vdash (\varphi \land{{\langle{\scriptstyle \leq} \rangle }{\psi }})\, {\rightarrow }\, \left ( {{\langle{\scriptstyle <}\rangle } {\psi }} \lor{{\langle{\scriptstyle \leq} \rangle }{(\psi \land{\langle{\scriptstyle \leq} \rangle }\varphi )}} \right )$| |$\vdash{{\langle{\scriptstyle \sim} \rangle }{i}}$| |$\vdash{{\langle{\scriptstyle \sim} \rangle }{(i \land \varphi )}}\, {\rightarrow }\, {{[{\scriptstyle \sim} ]}{(i \rightarrow \varphi )}}$| |$\vdash \varphi$| for |$\varphi$| an instance of a tautology . From |$\vdash \varphi$| and |$\vdash \varphi \, {\rightarrow }\, \psi$| derive |$\vdash \psi$| . |$\vdash{{[{\scriptstyle \leq} ]}{(\varphi \rightarrow \psi )}}\, {\rightarrow }\, ({{[{\scriptstyle \leq} ]}{\varphi }}\, {\rightarrow }\, {{[{\scriptstyle \leq} ]}{\psi }})$| |$\vdash{{[{\scriptstyle \sim} ]}{(\varphi \rightarrow \psi )}}\, {\rightarrow }\, ({{[{\scriptstyle \sim} ]}{\varphi }}\, {\rightarrow }\, {{[{\scriptstyle \sim} ]}{\psi }})$| |$\vdash{{\langle{\scriptstyle \leq} \rangle }{\varphi }}\, {\leftrightarrow }\,\lnot{{[{\scriptstyle \leq} ]}{\lnot \varphi }}$| |$\vdash{{\langle{\scriptstyle \sim} \rangle }{\varphi }}\, {\leftrightarrow }\,\lnot{{[{\scriptstyle \sim} ]}{\lnot \varphi }}$| From |$\vdash \varphi$| derive |$\vdash{{[{\scriptstyle \leq} ]}{\varphi }}$| From |$\vdash \varphi$| derive |$\vdash{{[{\scriptstyle \sim} ]}{\varphi }}$| |$\vdash \varphi \, {\rightarrow }\, {{\langle{\scriptstyle \leq} \rangle }{\varphi }}$| |$\vdash \varphi \, {\rightarrow }\, {{\langle{\scriptstyle \sim} \rangle }{\varphi }}$| |$\vdash{{\langle{\scriptstyle \leq} \rangle }{{\langle{\scriptstyle \leq} \rangle }\varphi }}\, {\rightarrow }\, {{\langle{\scriptstyle \leq} \rangle }{\varphi }}$| |$\vdash{{\langle{\scriptstyle \sim} \rangle }{{\langle{\scriptstyle \sim} \rangle }\varphi }}\, {\rightarrow }\, {{\langle{\scriptstyle \sim} \rangle }{\varphi }}$| |$\vdash \varphi \, {\rightarrow }\, {{[{\scriptstyle \sim} ]}{{\langle{\scriptstyle \sim} \rangle }\varphi }}$| |$\vdash ({{\langle{\scriptstyle \sim} \rangle }{\varphi }} \land{{\langle{\scriptstyle \sim} \rangle }{\psi }})\, {\rightarrow }\, \left ( {{\langle{\scriptstyle \sim} \rangle }{(\varphi \land{\langle{\scriptstyle \leq} \rangle }\psi )}} \lor{{\langle{\scriptstyle \sim} \rangle }{(\psi \land{\langle{\scriptstyle \leq} \rangle }\varphi )}} \right )$| |$\vdash{{\langle{\scriptstyle \leq} \rangle }{\varphi }}\, {\rightarrow }\, {{\langle{\scriptstyle \sim} \rangle }{\varphi }}$| |$\vdash{{[{\scriptstyle <}]}{(\varphi \rightarrow \psi )}}\, {\rightarrow }\, ({{[{\scriptstyle <}]}{\varphi }}\, {\rightarrow }\, {{[{\scriptstyle <}]}{\psi }})$| |$\vdash{{\langle{\scriptstyle <}\rangle } {\varphi }}\, {\rightarrow }\, {{\langle{\scriptstyle \leq} \rangle }{\varphi }}$| |$\vdash{{\langle{\scriptstyle <}\rangle } {\varphi }}\, {\leftrightarrow }\,\lnot{{[{\scriptstyle <}]}{\lnot \varphi }}$| |$\vdash{{\langle{\scriptstyle \leq} \rangle }{{\langle{\scriptstyle <}\rangle } \varphi }}\, {\rightarrow }\, {{\langle{\scriptstyle <}\rangle } {\varphi }}$| From |$\vdash \varphi$| derive |$\vdash{{[{\scriptstyle <}]}{\varphi }}$| |$\vdash{{\langle{\scriptstyle <}\rangle } {{\langle{\scriptstyle \leq} \rangle }\varphi }}\, {\rightarrow }\, {{\langle{\scriptstyle <}\rangle } {\varphi }}$| |$\vdash (\varphi \land{{\langle{\scriptstyle \leq} \rangle }{\psi }})\, {\rightarrow }\, \left ( {{\langle{\scriptstyle <}\rangle } {\psi }} \lor{{\langle{\scriptstyle \leq} \rangle }{(\psi \land{\langle{\scriptstyle \leq} \rangle }\varphi )}} \right )$| |$\vdash{{\langle{\scriptstyle \sim} \rangle }{i}}$| |$\vdash{{\langle{\scriptstyle \sim} \rangle }{(i \land \varphi )}}\, {\rightarrow }\, {{[{\scriptstyle \sim} ]}{(i \rightarrow \varphi )}}$| Open in new tab Table 1. Axiom system for |${\mathcal{L}_{i}}$| w.r.t. |${\mathscr{PM}}$| |$\vdash \varphi$| for |$\varphi$| an instance of a tautology . From |$\vdash \varphi$| and |$\vdash \varphi \, {\rightarrow }\, \psi$| derive |$\vdash \psi$| . |$\vdash{{[{\scriptstyle \leq} ]}{(\varphi \rightarrow \psi )}}\, {\rightarrow }\, ({{[{\scriptstyle \leq} ]}{\varphi }}\, {\rightarrow }\, {{[{\scriptstyle \leq} ]}{\psi }})$| |$\vdash{{[{\scriptstyle \sim} ]}{(\varphi \rightarrow \psi )}}\, {\rightarrow }\, ({{[{\scriptstyle \sim} ]}{\varphi }}\, {\rightarrow }\, {{[{\scriptstyle \sim} ]}{\psi }})$| |$\vdash{{\langle{\scriptstyle \leq} \rangle }{\varphi }}\, {\leftrightarrow }\,\lnot{{[{\scriptstyle \leq} ]}{\lnot \varphi }}$| |$\vdash{{\langle{\scriptstyle \sim} \rangle }{\varphi }}\, {\leftrightarrow }\,\lnot{{[{\scriptstyle \sim} ]}{\lnot \varphi }}$| From |$\vdash \varphi$| derive |$\vdash{{[{\scriptstyle \leq} ]}{\varphi }}$| From |$\vdash \varphi$| derive |$\vdash{{[{\scriptstyle \sim} ]}{\varphi }}$| |$\vdash \varphi \, {\rightarrow }\, {{\langle{\scriptstyle \leq} \rangle }{\varphi }}$| |$\vdash \varphi \, {\rightarrow }\, {{\langle{\scriptstyle \sim} \rangle }{\varphi }}$| |$\vdash{{\langle{\scriptstyle \leq} \rangle }{{\langle{\scriptstyle \leq} \rangle }\varphi }}\, {\rightarrow }\, {{\langle{\scriptstyle \leq} \rangle }{\varphi }}$| |$\vdash{{\langle{\scriptstyle \sim} \rangle }{{\langle{\scriptstyle \sim} \rangle }\varphi }}\, {\rightarrow }\, {{\langle{\scriptstyle \sim} \rangle }{\varphi }}$| |$\vdash \varphi \, {\rightarrow }\, {{[{\scriptstyle \sim} ]}{{\langle{\scriptstyle \sim} \rangle }\varphi }}$| |$\vdash ({{\langle{\scriptstyle \sim} \rangle }{\varphi }} \land{{\langle{\scriptstyle \sim} \rangle }{\psi }})\, {\rightarrow }\, \left ( {{\langle{\scriptstyle \sim} \rangle }{(\varphi \land{\langle{\scriptstyle \leq} \rangle }\psi )}} \lor{{\langle{\scriptstyle \sim} \rangle }{(\psi \land{\langle{\scriptstyle \leq} \rangle }\varphi )}} \right )$| |$\vdash{{\langle{\scriptstyle \leq} \rangle }{\varphi }}\, {\rightarrow }\, {{\langle{\scriptstyle \sim} \rangle }{\varphi }}$| |$\vdash{{[{\scriptstyle <}]}{(\varphi \rightarrow \psi )}}\, {\rightarrow }\, ({{[{\scriptstyle <}]}{\varphi }}\, {\rightarrow }\, {{[{\scriptstyle <}]}{\psi }})$| |$\vdash{{\langle{\scriptstyle <}\rangle } {\varphi }}\, {\rightarrow }\, {{\langle{\scriptstyle \leq} \rangle }{\varphi }}$| |$\vdash{{\langle{\scriptstyle <}\rangle } {\varphi }}\, {\leftrightarrow }\,\lnot{{[{\scriptstyle <}]}{\lnot \varphi }}$| |$\vdash{{\langle{\scriptstyle \leq} \rangle }{{\langle{\scriptstyle <}\rangle } \varphi }}\, {\rightarrow }\, {{\langle{\scriptstyle <}\rangle } {\varphi }}$| From |$\vdash \varphi$| derive |$\vdash{{[{\scriptstyle <}]}{\varphi }}$| |$\vdash{{\langle{\scriptstyle <}\rangle } {{\langle{\scriptstyle \leq} \rangle }\varphi }}\, {\rightarrow }\, {{\langle{\scriptstyle <}\rangle } {\varphi }}$| |$\vdash (\varphi \land{{\langle{\scriptstyle \leq} \rangle }{\psi }})\, {\rightarrow }\, \left ( {{\langle{\scriptstyle <}\rangle } {\psi }} \lor{{\langle{\scriptstyle \leq} \rangle }{(\psi \land{\langle{\scriptstyle \leq} \rangle }\varphi )}} \right )$| |$\vdash{{\langle{\scriptstyle \sim} \rangle }{i}}$| |$\vdash{{\langle{\scriptstyle \sim} \rangle }{(i \land \varphi )}}\, {\rightarrow }\, {{[{\scriptstyle \sim} ]}{(i \rightarrow \varphi )}}$| |$\vdash \varphi$| for |$\varphi$| an instance of a tautology . From |$\vdash \varphi$| and |$\vdash \varphi \, {\rightarrow }\, \psi$| derive |$\vdash \psi$| . |$\vdash{{[{\scriptstyle \leq} ]}{(\varphi \rightarrow \psi )}}\, {\rightarrow }\, ({{[{\scriptstyle \leq} ]}{\varphi }}\, {\rightarrow }\, {{[{\scriptstyle \leq} ]}{\psi }})$| |$\vdash{{[{\scriptstyle \sim} ]}{(\varphi \rightarrow \psi )}}\, {\rightarrow }\, ({{[{\scriptstyle \sim} ]}{\varphi }}\, {\rightarrow }\, {{[{\scriptstyle \sim} ]}{\psi }})$| |$\vdash{{\langle{\scriptstyle \leq} \rangle }{\varphi }}\, {\leftrightarrow }\,\lnot{{[{\scriptstyle \leq} ]}{\lnot \varphi }}$| |$\vdash{{\langle{\scriptstyle \sim} \rangle }{\varphi }}\, {\leftrightarrow }\,\lnot{{[{\scriptstyle \sim} ]}{\lnot \varphi }}$| From |$\vdash \varphi$| derive |$\vdash{{[{\scriptstyle \leq} ]}{\varphi }}$| From |$\vdash \varphi$| derive |$\vdash{{[{\scriptstyle \sim} ]}{\varphi }}$| |$\vdash \varphi \, {\rightarrow }\, {{\langle{\scriptstyle \leq} \rangle }{\varphi }}$| |$\vdash \varphi \, {\rightarrow }\, {{\langle{\scriptstyle \sim} \rangle }{\varphi }}$| |$\vdash{{\langle{\scriptstyle \leq} \rangle }{{\langle{\scriptstyle \leq} \rangle }\varphi }}\, {\rightarrow }\, {{\langle{\scriptstyle \leq} \rangle }{\varphi }}$| |$\vdash{{\langle{\scriptstyle \sim} \rangle }{{\langle{\scriptstyle \sim} \rangle }\varphi }}\, {\rightarrow }\, {{\langle{\scriptstyle \sim} \rangle }{\varphi }}$| |$\vdash \varphi \, {\rightarrow }\, {{[{\scriptstyle \sim} ]}{{\langle{\scriptstyle \sim} \rangle }\varphi }}$| |$\vdash ({{\langle{\scriptstyle \sim} \rangle }{\varphi }} \land{{\langle{\scriptstyle \sim} \rangle }{\psi }})\, {\rightarrow }\, \left ( {{\langle{\scriptstyle \sim} \rangle }{(\varphi \land{\langle{\scriptstyle \leq} \rangle }\psi )}} \lor{{\langle{\scriptstyle \sim} \rangle }{(\psi \land{\langle{\scriptstyle \leq} \rangle }\varphi )}} \right )$| |$\vdash{{\langle{\scriptstyle \leq} \rangle }{\varphi }}\, {\rightarrow }\, {{\langle{\scriptstyle \sim} \rangle }{\varphi }}$| |$\vdash{{[{\scriptstyle <}]}{(\varphi \rightarrow \psi )}}\, {\rightarrow }\, ({{[{\scriptstyle <}]}{\varphi }}\, {\rightarrow }\, {{[{\scriptstyle <}]}{\psi }})$| |$\vdash{{\langle{\scriptstyle <}\rangle } {\varphi }}\, {\rightarrow }\, {{\langle{\scriptstyle \leq} \rangle }{\varphi }}$| |$\vdash{{\langle{\scriptstyle <}\rangle } {\varphi }}\, {\leftrightarrow }\,\lnot{{[{\scriptstyle <}]}{\lnot \varphi }}$| |$\vdash{{\langle{\scriptstyle \leq} \rangle }{{\langle{\scriptstyle <}\rangle } \varphi }}\, {\rightarrow }\, {{\langle{\scriptstyle <}\rangle } {\varphi }}$| From |$\vdash \varphi$| derive |$\vdash{{[{\scriptstyle <}]}{\varphi }}$| |$\vdash{{\langle{\scriptstyle <}\rangle } {{\langle{\scriptstyle \leq} \rangle }\varphi }}\, {\rightarrow }\, {{\langle{\scriptstyle <}\rangle } {\varphi }}$| |$\vdash (\varphi \land{{\langle{\scriptstyle \leq} \rangle }{\psi }})\, {\rightarrow }\, \left ( {{\langle{\scriptstyle <}\rangle } {\psi }} \lor{{\langle{\scriptstyle \leq} \rangle }{(\psi \land{\langle{\scriptstyle \leq} \rangle }\varphi )}} \right )$| |$\vdash{{\langle{\scriptstyle \sim} \rangle }{i}}$| |$\vdash{{\langle{\scriptstyle \sim} \rangle }{(i \land \varphi )}}\, {\rightarrow }\, {{[{\scriptstyle \sim} ]}{(i \rightarrow \varphi )}}$| Open in new tab The axiom system for validities involving the dynamic modalities (⁠|${\mathop{\langle{\chi }{\Uparrow }\rangle }{{}}},\ {\mathop{\langle{\chi }{\uparrow }\rangle }{{}}}$| and ) is built via the DEL technique of recursion (reduction) axioms: valid formulas and validity-preserving rules indicating how to translate a formula with occurrences of the model-changing modality (a formula in the ‘dynamic’ language; |${\mathcal{L}_{i,{\Uparrow }}}$|⁠, |${\mathcal{L}_{i,\uparrow }}$| and here) into a provably equivalent one without them (a formula in the ‘basic’ ‘static’ language; |${\mathcal{L}_{i}}$| here). Soundness follows from the validity and validity-preserving properties of the new axioms and rules (a formula and its translation are semantically equivalent); completeness follows from the completeness of the axiom system for the basic language, as the recursion axioms define a recursive validity-preserving translation from the ‘dynamic’ language into the ‘static’ one. The reader is referred to [19, Chapter 7] (cf. [51], in particular its Theorem 11) for an extensive explanation of this technique. Theorem 2.6 The recursion axioms on Table 2 provide, together with the axioms on Table 1, a sound and complete axiom system for formulas in |${\mathcal{L}_{i,{\Uparrow }}}$| w.r.t. plausibility models. Proof. The text below provides an abridged version for readers not familiar with the recursion axioms technique. The |${\left \{ {i,{\langle{\scriptstyle <}\rangle }} \right \}}$|-less fragment of the system is known (see, e.g. [49]); the novel parts are the recursion axiom for nominals (straightforward, as the operation does not affect the atomic valuation) and the recursion axiom for |${{\langle{\scriptstyle <}\rangle } {}}$|⁠. Soundness follows from the validity and validity-preserving properties of these axioms. In particular, the validity of the axioms on the left column reflects the fact that the operation does not change the atomic valuation (first and second), that it is functional (third and fourth) and that it does not change the model’s domain (fifth). The axioms also reflect the fact that the semantic interpretation of the operation’s associated modality, |${\mathop{\langle{\chi }{\Uparrow }\rangle }{{}}}$|⁠, does not require a precondition.12 On the right column, the first axiom reflects the three cases in the definition of the new relation (Footnote 10). For its validity, let |$(M,w)$| be a |${\mathscr{PS}}$|⁠. |${\boldsymbol{(\Rightarrow )}}$| Suppose |$(M, w) \Vdash{\mathop{\langle{\chi }{\Uparrow }\rangle }{{{\langle{\scriptstyle \leq} \rangle }\varphi }}}$|⁠; then, |$({{{M}}^{{\Uparrow }_{{\chi }}}}, w) \Vdash{{\langle{\scriptstyle \leq} \rangle }{\varphi }}$|⁠, i.e. there is |$u \in W$| such that |$w\, {\mathrel{{{\leq }}^{{\Uparrow }_{{\chi }}}}} u$| and |$({{{M}}^{{\Uparrow }_{{\chi }}}}, u) \Vdash \varphi$|⁠, with the latter equivalent to |$(M, u) \Vdash{\mathop{\langle{\chi }{\Uparrow }\rangle }{{\varphi }}}$|⁠. If |$(M,w) \Vdash \lnot \chi$|⁠, then either |$w \leq u$|⁠, or else |$(M,u) \Vdash \chi$|⁠. In the first case, |$(M,w) \Vdash \lnot \chi \land{{\langle{\scriptstyle \leq} \rangle }{\mathop{\langle \chi{\Uparrow }\rangle }{\varphi }}}$|⁠. In the second case, |$(M,w) \Vdash \lnot \chi \land{{\langle{\scriptstyle \sim} \rangle }{(\chi \land \mathop{\langle \chi{\Uparrow }\rangle }{\varphi })}}$|⁠. If |$(M,w) \Vdash \chi$|⁠, then necessarily |$(M,u) \Vdash \chi$| and |$w \leq u$|⁠; hence, |$(M,w) \Vdash{{\langle{\scriptstyle \leq} \rangle }{(\chi \land \mathop{\langle \chi{\Uparrow }\rangle }{\varphi })}}$|⁠. |${\boldsymbol{(\Leftarrow )}}$| If |$(M, w) \Vdash{{\langle{\scriptstyle \leq} \rangle }{(\chi \land \mathop{\langle \chi{\Uparrow }\rangle }{\varphi })}} \lor (\lnot \chi \land{{\langle{\scriptstyle \leq} \rangle }{\mathop{\langle \chi{\Uparrow }\rangle }{\varphi }}}) \lor (\lnot \chi \land{{\langle{\scriptstyle \sim} \rangle }{(\chi \land \mathop{\langle \chi{\Uparrow }\rangle }{\varphi })}})$|⁠, then in all disjuncts there is a world |$u$| satisfying |${\mathop{\langle{\chi }{\Uparrow }\rangle }{{\varphi }}}$| (and thus satisfying |$\varphi$| in |${{{M}}^{{\Uparrow }_{{\chi }}}}$|⁠); moreover, in each case, the additional conditions guarantee that |$u$| also satisfies |$w\, {\mathrel{{{\leq }}^{{\Uparrow }_{{\chi }}}}} u$|⁠; hence, |$(M, w) \Vdash{\mathop{\langle{\chi }{\Uparrow }\rangle }{{{\langle{\scriptstyle \leq} \rangle }\varphi }}}$|⁠. The validity of the novel axiom, that for |${{\langle{\scriptstyle <}\rangle } {}}$|⁠, can be proved with a reasoning similar to the one for the |${{\langle{\scriptstyle \leq} \rangle }{}}$|-case. Completeness follows from the completeness of the axiom system for |${\mathcal{L}_{i}}$|⁠, as the recursion axioms define a validity-preserving translation from |${\mathcal{L}_{i,{\Uparrow }}}$| to |${\mathcal{L}_{i}}$|⁠, with the rule of at the bottom of Table 2 taking care of formulas with more than one occurrence of |${\mathop{\langle{\chi }{\Uparrow }\rangle }{{}}}$| (an ‘inside-out’ approach, eliminating always the deepest occurrence of the modality). More details can be found in the above mentioned references. Table 2. Recursion axioms for |${\mathcal{L}_{i,{\Uparrow }}}$| w.r.t. |${\mathscr{PM}}$| |$\vdash{\mathop{\langle{\chi }{\Uparrow }\rangle }{{p}}} {\;\leftrightarrow \;} p$| $$\vdash{\mathop{\langle{\chi }{\Uparrow }\rangle }{{{\langle{\scriptstyle \leq} \rangle }\varphi }}} {\;\leftrightarrow \;} \bigvee \left \{ \begin{array}{l} {{\langle{\scriptstyle \leq} \rangle }{(\chi \land \mathop{\langle \chi{\Uparrow }\rangle }{\varphi })}}, \\ \lnot \chi \land{{\langle{\scriptstyle \leq} \rangle }{\mathop{\langle \chi{\Uparrow }\rangle }{\varphi }}}, \\ \lnot \chi \land{{\langle{\scriptstyle \sim} \rangle }{(\chi \land \mathop{\langle \chi{\Uparrow }\rangle }{\varphi })}}\end{array} \right \}$$ |$\vdash{\mathop{\langle{\chi }{\Uparrow }\rangle }{{i}}} {\;\leftrightarrow \;} i$| |$\vdash{\mathop{\langle{\chi }{\Uparrow }\rangle }{{\lnot \varphi }}} {\;\leftrightarrow \;} \lnot{\mathop{\langle{\chi }{\Uparrow }\rangle }{{\varphi }}}$| |$\vdash{\mathop{\langle{\chi }{\Uparrow }\rangle }{{(\varphi \lor \psi )}}} {\;\leftrightarrow \;} ({\mathop{\langle{\chi }{\Uparrow }\rangle }{{\varphi }}} \lor{\mathop{\langle{\chi }{\Uparrow }\rangle }{{\psi }}})$| |$\vdash{\mathop{\langle{\chi }{\Uparrow }\rangle }{{{\langle{\scriptstyle \sim} \rangle }\varphi }}} {\;\leftrightarrow \;} {{\langle{\scriptstyle \sim} \rangle }{\mathop{\langle \chi{\Uparrow }\rangle }{\varphi }}}$| $$\vdash{\mathop{\langle{\chi }{\Uparrow }\rangle }{{{\langle{\scriptstyle <}\rangle } \varphi }}} {\;\leftrightarrow \;} \bigvee \left \{ \begin{array}{l} {{\langle{\scriptstyle <}\rangle } {(\chi \land \mathop{\langle \chi{\Uparrow }\rangle }{\varphi })}}, \\ \lnot \chi \land{{\langle{\scriptstyle <}\rangle } {\mathop{\langle \chi{\Uparrow }\rangle }{\varphi }}}, \\ \lnot \chi \land{{\langle{\scriptstyle \sim} \rangle }{(\chi \land \mathop{\langle \chi{\Uparrow }\rangle }{\varphi })}}\end{array} \right \}$$ From |$\vdash \psi _1\, {\leftrightarrow }\,\psi _2$| derive |$\vdash \varphi \, {\leftrightarrow }\,\varphi [\psi _2/\psi _1]$|⁠, with |$\varphi [\psi _2/\psi _1]$| any formula obtained by replacing one or more non-dynamic occurrences of |$\psi _1$| in |$\varphi$| with |$\psi _2$|⁠, with the non-dynamic occurrences of |$\psi _1$| being those that are not inside the angle brackets |$`{\mathop{\langle{} \rangle }}^{\prime }$| of the dynamic modality |${\mathop{\langle{\chi }{\Uparrow }\rangle }{{}}}$|⁠. |$\vdash{\mathop{\langle{\chi }{\Uparrow }\rangle }{{p}}} {\;\leftrightarrow \;} p$| $$\vdash{\mathop{\langle{\chi }{\Uparrow }\rangle }{{{\langle{\scriptstyle \leq} \rangle }\varphi }}} {\;\leftrightarrow \;} \bigvee \left \{ \begin{array}{l} {{\langle{\scriptstyle \leq} \rangle }{(\chi \land \mathop{\langle \chi{\Uparrow }\rangle }{\varphi })}}, \\ \lnot \chi \land{{\langle{\scriptstyle \leq} \rangle }{\mathop{\langle \chi{\Uparrow }\rangle }{\varphi }}}, \\ \lnot \chi \land{{\langle{\scriptstyle \sim} \rangle }{(\chi \land \mathop{\langle \chi{\Uparrow }\rangle }{\varphi })}}\end{array} \right \}$$ |$\vdash{\mathop{\langle{\chi }{\Uparrow }\rangle }{{i}}} {\;\leftrightarrow \;} i$| |$\vdash{\mathop{\langle{\chi }{\Uparrow }\rangle }{{\lnot \varphi }}} {\;\leftrightarrow \;} \lnot{\mathop{\langle{\chi }{\Uparrow }\rangle }{{\varphi }}}$| |$\vdash{\mathop{\langle{\chi }{\Uparrow }\rangle }{{(\varphi \lor \psi )}}} {\;\leftrightarrow \;} ({\mathop{\langle{\chi }{\Uparrow }\rangle }{{\varphi }}} \lor{\mathop{\langle{\chi }{\Uparrow }\rangle }{{\psi }}})$| |$\vdash{\mathop{\langle{\chi }{\Uparrow }\rangle }{{{\langle{\scriptstyle \sim} \rangle }\varphi }}} {\;\leftrightarrow \;} {{\langle{\scriptstyle \sim} \rangle }{\mathop{\langle \chi{\Uparrow }\rangle }{\varphi }}}$| $$\vdash{\mathop{\langle{\chi }{\Uparrow }\rangle }{{{\langle{\scriptstyle <}\rangle } \varphi }}} {\;\leftrightarrow \;} \bigvee \left \{ \begin{array}{l} {{\langle{\scriptstyle <}\rangle } {(\chi \land \mathop{\langle \chi{\Uparrow }\rangle }{\varphi })}}, \\ \lnot \chi \land{{\langle{\scriptstyle <}\rangle } {\mathop{\langle \chi{\Uparrow }\rangle }{\varphi }}}, \\ \lnot \chi \land{{\langle{\scriptstyle \sim} \rangle }{(\chi \land \mathop{\langle \chi{\Uparrow }\rangle }{\varphi })}}\end{array} \right \}$$ From |$\vdash \psi _1\, {\leftrightarrow }\,\psi _2$| derive |$\vdash \varphi \, {\leftrightarrow }\,\varphi [\psi _2/\psi _1]$|⁠, with |$\varphi [\psi _2/\psi _1]$| any formula obtained by replacing one or more non-dynamic occurrences of |$\psi _1$| in |$\varphi$| with |$\psi _2$|⁠, with the non-dynamic occurrences of |$\psi _1$| being those that are not inside the angle brackets |$`{\mathop{\langle{} \rangle }}^{\prime }$| of the dynamic modality |${\mathop{\langle{\chi }{\Uparrow }\rangle }{{}}}$|⁠. Open in new tab Table 2. Recursion axioms for |${\mathcal{L}_{i,{\Uparrow }}}$| w.r.t. |${\mathscr{PM}}$| |$\vdash{\mathop{\langle{\chi }{\Uparrow }\rangle }{{p}}} {\;\leftrightarrow \;} p$| $$\vdash{\mathop{\langle{\chi }{\Uparrow }\rangle }{{{\langle{\scriptstyle \leq} \rangle }\varphi }}} {\;\leftrightarrow \;} \bigvee \left \{ \begin{array}{l} {{\langle{\scriptstyle \leq} \rangle }{(\chi \land \mathop{\langle \chi{\Uparrow }\rangle }{\varphi })}}, \\ \lnot \chi \land{{\langle{\scriptstyle \leq} \rangle }{\mathop{\langle \chi{\Uparrow }\rangle }{\varphi }}}, \\ \lnot \chi \land{{\langle{\scriptstyle \sim} \rangle }{(\chi \land \mathop{\langle \chi{\Uparrow }\rangle }{\varphi })}}\end{array} \right \}$$ |$\vdash{\mathop{\langle{\chi }{\Uparrow }\rangle }{{i}}} {\;\leftrightarrow \;} i$| |$\vdash{\mathop{\langle{\chi }{\Uparrow }\rangle }{{\lnot \varphi }}} {\;\leftrightarrow \;} \lnot{\mathop{\langle{\chi }{\Uparrow }\rangle }{{\varphi }}}$| |$\vdash{\mathop{\langle{\chi }{\Uparrow }\rangle }{{(\varphi \lor \psi )}}} {\;\leftrightarrow \;} ({\mathop{\langle{\chi }{\Uparrow }\rangle }{{\varphi }}} \lor{\mathop{\langle{\chi }{\Uparrow }\rangle }{{\psi }}})$| |$\vdash{\mathop{\langle{\chi }{\Uparrow }\rangle }{{{\langle{\scriptstyle \sim} \rangle }\varphi }}} {\;\leftrightarrow \;} {{\langle{\scriptstyle \sim} \rangle }{\mathop{\langle \chi{\Uparrow }\rangle }{\varphi }}}$| $$\vdash{\mathop{\langle{\chi }{\Uparrow }\rangle }{{{\langle{\scriptstyle <}\rangle } \varphi }}} {\;\leftrightarrow \;} \bigvee \left \{ \begin{array}{l} {{\langle{\scriptstyle <}\rangle } {(\chi \land \mathop{\langle \chi{\Uparrow }\rangle }{\varphi })}}, \\ \lnot \chi \land{{\langle{\scriptstyle <}\rangle } {\mathop{\langle \chi{\Uparrow }\rangle }{\varphi }}}, \\ \lnot \chi \land{{\langle{\scriptstyle \sim} \rangle }{(\chi \land \mathop{\langle \chi{\Uparrow }\rangle }{\varphi })}}\end{array} \right \}$$ From |$\vdash \psi _1\, {\leftrightarrow }\,\psi _2$| derive |$\vdash \varphi \, {\leftrightarrow }\,\varphi [\psi _2/\psi _1]$|⁠, with |$\varphi [\psi _2/\psi _1]$| any formula obtained by replacing one or more non-dynamic occurrences of |$\psi _1$| in |$\varphi$| with |$\psi _2$|⁠, with the non-dynamic occurrences of |$\psi _1$| being those that are not inside the angle brackets |$`{\mathop{\langle{} \rangle }}^{\prime }$| of the dynamic modality |${\mathop{\langle{\chi }{\Uparrow }\rangle }{{}}}$|⁠. |$\vdash{\mathop{\langle{\chi }{\Uparrow }\rangle }{{p}}} {\;\leftrightarrow \;} p$| $$\vdash{\mathop{\langle{\chi }{\Uparrow }\rangle }{{{\langle{\scriptstyle \leq} \rangle }\varphi }}} {\;\leftrightarrow \;} \bigvee \left \{ \begin{array}{l} {{\langle{\scriptstyle \leq} \rangle }{(\chi \land \mathop{\langle \chi{\Uparrow }\rangle }{\varphi })}}, \\ \lnot \chi \land{{\langle{\scriptstyle \leq} \rangle }{\mathop{\langle \chi{\Uparrow }\rangle }{\varphi }}}, \\ \lnot \chi \land{{\langle{\scriptstyle \sim} \rangle }{(\chi \land \mathop{\langle \chi{\Uparrow }\rangle }{\varphi })}}\end{array} \right \}$$ |$\vdash{\mathop{\langle{\chi }{\Uparrow }\rangle }{{i}}} {\;\leftrightarrow \;} i$| |$\vdash{\mathop{\langle{\chi }{\Uparrow }\rangle }{{\lnot \varphi }}} {\;\leftrightarrow \;} \lnot{\mathop{\langle{\chi }{\Uparrow }\rangle }{{\varphi }}}$| |$\vdash{\mathop{\langle{\chi }{\Uparrow }\rangle }{{(\varphi \lor \psi )}}} {\;\leftrightarrow \;} ({\mathop{\langle{\chi }{\Uparrow }\rangle }{{\varphi }}} \lor{\mathop{\langle{\chi }{\Uparrow }\rangle }{{\psi }}})$| |$\vdash{\mathop{\langle{\chi }{\Uparrow }\rangle }{{{\langle{\scriptstyle \sim} \rangle }\varphi }}} {\;\leftrightarrow \;} {{\langle{\scriptstyle \sim} \rangle }{\mathop{\langle \chi{\Uparrow }\rangle }{\varphi }}}$| $$\vdash{\mathop{\langle{\chi }{\Uparrow }\rangle }{{{\langle{\scriptstyle <}\rangle } \varphi }}} {\;\leftrightarrow \;} \bigvee \left \{ \begin{array}{l} {{\langle{\scriptstyle <}\rangle } {(\chi \land \mathop{\langle \chi{\Uparrow }\rangle }{\varphi })}}, \\ \lnot \chi \land{{\langle{\scriptstyle <}\rangle } {\mathop{\langle \chi{\Uparrow }\rangle }{\varphi }}}, \\ \lnot \chi \land{{\langle{\scriptstyle \sim} \rangle }{(\chi \land \mathop{\langle \chi{\Uparrow }\rangle }{\varphi })}}\end{array} \right \}$$ From |$\vdash \psi _1\, {\leftrightarrow }\,\psi _2$| derive |$\vdash \varphi \, {\leftrightarrow }\,\varphi [\psi _2/\psi _1]$|⁠, with |$\varphi [\psi _2/\psi _1]$| any formula obtained by replacing one or more non-dynamic occurrences of |$\psi _1$| in |$\varphi$| with |$\psi _2$|⁠, with the non-dynamic occurrences of |$\psi _1$| being those that are not inside the angle brackets |$`{\mathop{\langle{} \rangle }}^{\prime }$| of the dynamic modality |${\mathop{\langle{\chi }{\Uparrow }\rangle }{{}}}$|⁠. Open in new tab Theorem 2.7 The recursion axioms on Table 3 provide, together with the axioms on Table 1, a sound and complete axiom system for formulas in |${\mathcal{L}_{i,\uparrow }}$| w.r.t. plausibility models. Table 3. Recursion axioms for |${\mathcal{L}_{i,\uparrow }}$| w.r.t. |${\mathscr{PM}}$| |$\vdash{\mathop{\langle{\chi }{\uparrow }\rangle }{{p}}} {\;\leftrightarrow \;} p$| $$\vdash{\mathop{\langle{\chi }{\uparrow }\rangle }{{{\langle{\scriptstyle \leq} \rangle }\varphi }}} {\;\leftrightarrow \;} \bigvee \left \{ \begin{array}{l} \lnot \lambda ^{\chi }_0 \land{{\langle{\scriptstyle \leq} \rangle }{\mathop{\langle \chi{\uparrow }\rangle }{\varphi }}}, \\{{\langle{\scriptstyle \sim} \rangle }{(\lambda ^{\chi }_0 \land \mathop{\langle \chi{\uparrow }\rangle }{\varphi })}}\end{array} \right \}$$13 |$\vdash{\mathop{\langle{\chi }{\uparrow }\rangle }{{i}}} {\;\leftrightarrow \;} i$| |$\vdash{\mathop{\langle{\chi }{\uparrow }\rangle }{{\lnot \varphi }}} {\;\leftrightarrow \;} \lnot{\mathop{\langle{\chi }{\uparrow }\rangle }{{\varphi }}}$| |$\vdash{\mathop{\langle{\chi }{\uparrow }\rangle }{{(\varphi \lor \psi )}}} {\;\leftrightarrow \;} ({\mathop{\langle{\chi }{\uparrow }\rangle }{{\varphi }}} \lor{\mathop{\langle{\chi }{\uparrow }\rangle }{{\psi }}})$| $$\vdash{\mathop{\langle{\chi }{\uparrow }\rangle }{{{\langle{\scriptstyle <}\rangle } \varphi }}} {\;\leftrightarrow \;} \bigvee \left \{ \begin{array}{l} \lnot \lambda ^{\chi }_0 \land{{\langle{\scriptstyle <}\rangle } {\mathop{\langle \chi{\uparrow }\rangle }{\varphi }}}, \\ \lnot \lambda ^{\chi }_0 \land{{\langle{\scriptstyle \sim} \rangle }{(\lambda ^{\chi }_0 \land \mathop{\langle \chi{\uparrow }\rangle }{\varphi })}}\end{array} \right \}$$ |$\vdash{\mathop{\langle{\chi }{\uparrow }\rangle }{{{\langle{\scriptstyle \sim} \rangle }\varphi }}} {\;\leftrightarrow \;} {{\langle{\scriptstyle \sim} \rangle }{\mathop{\langle \chi{\uparrow }\rangle }{\varphi }}}$| From |$\vdash \psi _1\, {\leftrightarrow }\,\psi _2$| derive |$\vdash \varphi \, {\leftrightarrow }\,\varphi [\psi _2/\psi _1]$|⁠, with |$\varphi [\psi _2/\psi _1]$| any formula obtained by replacing one or more non-dynamic occurrences of |$\psi _1$| in |$\varphi$| with |$\psi _2$|⁠, with the non-dynamic occurrences of |$\psi _1$| being those that are not inside the angle brackets |$`{\mathop{\langle{} \rangle }}^{\prime }$| of the dynamic modality |${\mathop{\langle{\chi }{\uparrow }\rangle }{{}}}$|⁠. |$\vdash{\mathop{\langle{\chi }{\uparrow }\rangle }{{p}}} {\;\leftrightarrow \;} p$| $$\vdash{\mathop{\langle{\chi }{\uparrow }\rangle }{{{\langle{\scriptstyle \leq} \rangle }\varphi }}} {\;\leftrightarrow \;} \bigvee \left \{ \begin{array}{l} \lnot \lambda ^{\chi }_0 \land{{\langle{\scriptstyle \leq} \rangle }{\mathop{\langle \chi{\uparrow }\rangle }{\varphi }}}, \\{{\langle{\scriptstyle \sim} \rangle }{(\lambda ^{\chi }_0 \land \mathop{\langle \chi{\uparrow }\rangle }{\varphi })}}\end{array} \right \}$$13 |$\vdash{\mathop{\langle{\chi }{\uparrow }\rangle }{{i}}} {\;\leftrightarrow \;} i$| |$\vdash{\mathop{\langle{\chi }{\uparrow }\rangle }{{\lnot \varphi }}} {\;\leftrightarrow \;} \lnot{\mathop{\langle{\chi }{\uparrow }\rangle }{{\varphi }}}$| |$\vdash{\mathop{\langle{\chi }{\uparrow }\rangle }{{(\varphi \lor \psi )}}} {\;\leftrightarrow \;} ({\mathop{\langle{\chi }{\uparrow }\rangle }{{\varphi }}} \lor{\mathop{\langle{\chi }{\uparrow }\rangle }{{\psi }}})$| $$\vdash{\mathop{\langle{\chi }{\uparrow }\rangle }{{{\langle{\scriptstyle <}\rangle } \varphi }}} {\;\leftrightarrow \;} \bigvee \left \{ \begin{array}{l} \lnot \lambda ^{\chi }_0 \land{{\langle{\scriptstyle <}\rangle } {\mathop{\langle \chi{\uparrow }\rangle }{\varphi }}}, \\ \lnot \lambda ^{\chi }_0 \land{{\langle{\scriptstyle \sim} \rangle }{(\lambda ^{\chi }_0 \land \mathop{\langle \chi{\uparrow }\rangle }{\varphi })}}\end{array} \right \}$$ |$\vdash{\mathop{\langle{\chi }{\uparrow }\rangle }{{{\langle{\scriptstyle \sim} \rangle }\varphi }}} {\;\leftrightarrow \;} {{\langle{\scriptstyle \sim} \rangle }{\mathop{\langle \chi{\uparrow }\rangle }{\varphi }}}$| From |$\vdash \psi _1\, {\leftrightarrow }\,\psi _2$| derive |$\vdash \varphi \, {\leftrightarrow }\,\varphi [\psi _2/\psi _1]$|⁠, with |$\varphi [\psi _2/\psi _1]$| any formula obtained by replacing one or more non-dynamic occurrences of |$\psi _1$| in |$\varphi$| with |$\psi _2$|⁠, with the non-dynamic occurrences of |$\psi _1$| being those that are not inside the angle brackets |$`{\mathop{\langle{} \rangle }}^{\prime }$| of the dynamic modality |${\mathop{\langle{\chi }{\uparrow }\rangle }{{}}}$|⁠. Open in new tab Table 3. Recursion axioms for |${\mathcal{L}_{i,\uparrow }}$| w.r.t. |${\mathscr{PM}}$| |$\vdash{\mathop{\langle{\chi }{\uparrow }\rangle }{{p}}} {\;\leftrightarrow \;} p$| $$\vdash{\mathop{\langle{\chi }{\uparrow }\rangle }{{{\langle{\scriptstyle \leq} \rangle }\varphi }}} {\;\leftrightarrow \;} \bigvee \left \{ \begin{array}{l} \lnot \lambda ^{\chi }_0 \land{{\langle{\scriptstyle \leq} \rangle }{\mathop{\langle \chi{\uparrow }\rangle }{\varphi }}}, \\{{\langle{\scriptstyle \sim} \rangle }{(\lambda ^{\chi }_0 \land \mathop{\langle \chi{\uparrow }\rangle }{\varphi })}}\end{array} \right \}$$13 |$\vdash{\mathop{\langle{\chi }{\uparrow }\rangle }{{i}}} {\;\leftrightarrow \;} i$| |$\vdash{\mathop{\langle{\chi }{\uparrow }\rangle }{{\lnot \varphi }}} {\;\leftrightarrow \;} \lnot{\mathop{\langle{\chi }{\uparrow }\rangle }{{\varphi }}}$| |$\vdash{\mathop{\langle{\chi }{\uparrow }\rangle }{{(\varphi \lor \psi )}}} {\;\leftrightarrow \;} ({\mathop{\langle{\chi }{\uparrow }\rangle }{{\varphi }}} \lor{\mathop{\langle{\chi }{\uparrow }\rangle }{{\psi }}})$| $$\vdash{\mathop{\langle{\chi }{\uparrow }\rangle }{{{\langle{\scriptstyle <}\rangle } \varphi }}} {\;\leftrightarrow \;} \bigvee \left \{ \begin{array}{l} \lnot \lambda ^{\chi }_0 \land{{\langle{\scriptstyle <}\rangle } {\mathop{\langle \chi{\uparrow }\rangle }{\varphi }}}, \\ \lnot \lambda ^{\chi }_0 \land{{\langle{\scriptstyle \sim} \rangle }{(\lambda ^{\chi }_0 \land \mathop{\langle \chi{\uparrow }\rangle }{\varphi })}}\end{array} \right \}$$ |$\vdash{\mathop{\langle{\chi }{\uparrow }\rangle }{{{\langle{\scriptstyle \sim} \rangle }\varphi }}} {\;\leftrightarrow \;} {{\langle{\scriptstyle \sim} \rangle }{\mathop{\langle \chi{\uparrow }\rangle }{\varphi }}}$| From |$\vdash \psi _1\, {\leftrightarrow }\,\psi _2$| derive |$\vdash \varphi \, {\leftrightarrow }\,\varphi [\psi _2/\psi _1]$|⁠, with |$\varphi [\psi _2/\psi _1]$| any formula obtained by replacing one or more non-dynamic occurrences of |$\psi _1$| in |$\varphi$| with |$\psi _2$|⁠, with the non-dynamic occurrences of |$\psi _1$| being those that are not inside the angle brackets |$`{\mathop{\langle{} \rangle }}^{\prime }$| of the dynamic modality |${\mathop{\langle{\chi }{\uparrow }\rangle }{{}}}$|⁠. |$\vdash{\mathop{\langle{\chi }{\uparrow }\rangle }{{p}}} {\;\leftrightarrow \;} p$| $$\vdash{\mathop{\langle{\chi }{\uparrow }\rangle }{{{\langle{\scriptstyle \leq} \rangle }\varphi }}} {\;\leftrightarrow \;} \bigvee \left \{ \begin{array}{l} \lnot \lambda ^{\chi }_0 \land{{\langle{\scriptstyle \leq} \rangle }{\mathop{\langle \chi{\uparrow }\rangle }{\varphi }}}, \\{{\langle{\scriptstyle \sim} \rangle }{(\lambda ^{\chi }_0 \land \mathop{\langle \chi{\uparrow }\rangle }{\varphi })}}\end{array} \right \}$$13 |$\vdash{\mathop{\langle{\chi }{\uparrow }\rangle }{{i}}} {\;\leftrightarrow \;} i$| |$\vdash{\mathop{\langle{\chi }{\uparrow }\rangle }{{\lnot \varphi }}} {\;\leftrightarrow \;} \lnot{\mathop{\langle{\chi }{\uparrow }\rangle }{{\varphi }}}$| |$\vdash{\mathop{\langle{\chi }{\uparrow }\rangle }{{(\varphi \lor \psi )}}} {\;\leftrightarrow \;} ({\mathop{\langle{\chi }{\uparrow }\rangle }{{\varphi }}} \lor{\mathop{\langle{\chi }{\uparrow }\rangle }{{\psi }}})$| $$\vdash{\mathop{\langle{\chi }{\uparrow }\rangle }{{{\langle{\scriptstyle <}\rangle } \varphi }}} {\;\leftrightarrow \;} \bigvee \left \{ \begin{array}{l} \lnot \lambda ^{\chi }_0 \land{{\langle{\scriptstyle <}\rangle } {\mathop{\langle \chi{\uparrow }\rangle }{\varphi }}}, \\ \lnot \lambda ^{\chi }_0 \land{{\langle{\scriptstyle \sim} \rangle }{(\lambda ^{\chi }_0 \land \mathop{\langle \chi{\uparrow }\rangle }{\varphi })}}\end{array} \right \}$$ |$\vdash{\mathop{\langle{\chi }{\uparrow }\rangle }{{{\langle{\scriptstyle \sim} \rangle }\varphi }}} {\;\leftrightarrow \;} {{\langle{\scriptstyle \sim} \rangle }{\mathop{\langle \chi{\uparrow }\rangle }{\varphi }}}$| From |$\vdash \psi _1\, {\leftrightarrow }\,\psi _2$| derive |$\vdash \varphi \, {\leftrightarrow }\,\varphi [\psi _2/\psi _1]$|⁠, with |$\varphi [\psi _2/\psi _1]$| any formula obtained by replacing one or more non-dynamic occurrences of |$\psi _1$| in |$\varphi$| with |$\psi _2$|⁠, with the non-dynamic occurrences of |$\psi _1$| being those that are not inside the angle brackets |$`{\mathop{\langle{} \rangle }}^{\prime }$| of the dynamic modality |${\mathop{\langle{\chi }{\uparrow }\rangle }{{}}}$|⁠. Open in new tab Proof. Soundness is as in the proof of Theorem 2.6, with the validity of the axioms on the left column as discussed there. On the right column, the first axiom reflects the two cases in the definition of the new relation (Footnote 11), with |$\lambda ^{\chi }_0:= \chi \land{{[{\scriptstyle <}]}{\lnot \chi }}$| characterising a model’s most plausible |$\chi$|-worlds.14 For its validity, let |$(M,w)$| be a |${\mathscr{PS}}$|⁠. |${\boldsymbol{(\Rightarrow )}}$| Suppose |$(M, w) \Vdash{\mathop{\langle{\chi }{\uparrow }\rangle }{{{\langle{\scriptstyle \leq} \rangle }\varphi }}}$|⁠; then, |$({{{M}}^{{\uparrow }_{{\chi }}}}, w) \Vdash{{\langle{\scriptstyle \leq} \rangle }{\varphi }}$|⁠, i.e. there is |$u \in W$| such that |$w\, {\mathrel{{{\leq }}^{{\uparrow }_{{\chi }}}}} u$| and |$({{{M}}^{{\uparrow }_{{\chi }}}}, u) \Vdash \varphi$| (with the latter equivalent to |$(M, u) \Vdash{\mathop{\langle{\chi }{\uparrow }\rangle }{{\varphi }}}$|⁠). From |${\mathrel{{{\leq }}^{{\uparrow }_{{\chi }}}}}$|’s definition, |$w\, {\mathrel{{{\leq }}^{{\uparrow }_{{\chi }}}}} u$| implies two possibilities: either |$w \leq u$| and |$(M, w) \Vdash \lnot \lambda ^{\chi }_0$|⁠, or else |$w \sim u$| and |$(M, u) \Vdash \lambda ^{\chi }_0$|⁠. In the first case, |$(M, w) \Vdash \lnot \lambda ^{\chi }_0\land{{\langle{\scriptstyle \leq} \rangle }{\mathop{\langle \chi{\uparrow }\rangle }{\varphi }}}$|⁠. In the second case, |$(M, w) \Vdash{{\langle{\scriptstyle \sim} \rangle }{(\lambda ^{\chi }_0 \land \mathop{\langle \chi{\uparrow }\rangle }{\varphi })}}$|⁠. |${\boldsymbol{(\Leftarrow )}}$| If |$(M, w) \Vdash (\lnot \lambda ^{\chi }_0 \land{{\langle{\scriptstyle \leq} \rangle }{\mathop{\langle \chi{\uparrow }\rangle }{\varphi }}}) \lor{{\langle{\scriptstyle \sim} \rangle }{(\lambda ^{\chi }_0 \land \mathop{\langle \chi{\uparrow }\rangle }{\varphi })}}$|⁠, then in all disjuncts there is a world |$u$| satisfying |${\mathop{\langle{\chi }{\uparrow }\rangle }{{\varphi }}}$| (and thus satisfying |$\varphi$| in |${{{M}}^{{\uparrow }_{{\chi }}}}$|⁠); moreover, in each case, the additional conditions guarantee that |$u$| also satisfies |$w\, {\mathrel{{{\leq }}^{{\uparrow }_{{\chi }}}}} u$|⁠; hence, |$(M, w) \Vdash{\mathop{\langle{\chi }{\uparrow }\rangle }{{{\langle{\scriptstyle \leq} \rangle }\varphi }}}$|⁠. Still on the right column, the second axiom states the way < changes. For its validity, |${\boldsymbol{(\Rightarrow )}}$| suppose |$(M, w) \Vdash{\mathop{\langle{\chi }{\uparrow }\rangle }{{{\langle{\scriptstyle <}\rangle } \varphi }}}$|⁠; then, |$({{{M}}^{{\uparrow }_{{\chi }}}}, w) \Vdash{{\langle{\scriptstyle <}\rangle } {\varphi }}$|⁠, i.e. there is |$u \in W$| such that |$w {\mathrel{{{<}}^{{\uparrow }_{{\chi }}}}} u$| and |$({{{M}}^{{\uparrow }_{{\chi }}}}, u) \Vdash \varphi$| (with the latter equivalent to |$(M, u) \Vdash{\mathop{\langle{\chi }{\uparrow }\rangle }{{\varphi }}}$|⁠). Necessarily we have |$(M,w) \Vdash \lnot \lambda ^{\chi }_0$|⁠. From there, either |$w<u$| or |$(M, u) \Vdash \lambda ^{\chi }_0$|⁠. In the first case, we have |$(M,w) \Vdash \lnot \lambda ^{\chi }_0 \land{{\langle{\scriptstyle <}\rangle } {\mathop{\langle \chi{\uparrow }\rangle }{\varphi }}}$|⁠. In the second, |$M,u\Vdash (\lambda ^{\chi }_0\land{\mathop{\langle{\chi }{\uparrow }\rangle }{{\varphi }}})$|⁠, and thus |$M,w\Vdash{{\langle{\scriptstyle \sim} \rangle }(}\lambda ^{\chi }_0\land{\mathop{\langle{\chi }{\uparrow }\rangle }{{\varphi }}})$|⁠. |${\boldsymbol{(\Leftarrow )}}$| If the disjunction on the right-hand side of the axiom holds at |$(M, w)$|⁠, then in all cases the described world |$u$| satisfying |${\mathop{\langle{\chi }{\uparrow }\rangle }{{\varphi }}}$| (and thus satisfying |$\varphi$| in |${{{M}}^{{\uparrow }_{{\chi }}}}$|⁠) also satisfies both |$w\, {\mathrel{{{\leq }}^{{\uparrow }_{{\chi }}}}} u$| and |$u {\mathrel{{{\not \leq }}^{{\uparrow }_{{\chi }}}}} w$|⁠; hence, |$(M, w) \Vdash{\mathop{\langle{\chi }{\uparrow }\rangle }{{{\langle{\scriptstyle <}\rangle } \varphi }}}$|⁠. The argument for completeness is similar to that in the proof of Theorem 2.6. Theorem 2.8 The recursion axioms on Table 4 provide, together with the axioms on Table 1, a sound and complete axiom system for formulas in |${\mathcal{L}_{i,{\uparrow}}}$| w.r.t. plausibility models. Proof. Again, soundness follows from the validity and validity-preserving properties of the recursion axioms. The validity of the first five is as discussed in the previous cases. For the validity of the sixth, the difficulty is to identify the |$\simeq$|-‘levels’; this will be done by using the formulas |$\lambda _i$| defined before (page 5) that characterise the layers in a given model. To get an idea of what the axioms are saying, here is what happens during an improvement operation: in the diagram, |$w_{\lnot \chi }$| and |$u_{\lnot \chi }$| are worlds in |${{[\kern{-1.5pt}[}{\lnot \chi } {]\kern{-1.5pt}]} ^{{M}}}$|⁠, and |$w_{\chi }$| and |$u_{\chi }$| are worlds in |${{[\kern{-1.5pt}[}{\chi } {]\kern{-1.5pt}]} ^{{M}}}$|⁠. Take |$w_{\chi }$| and note how the worlds that are |$\leq$|-reachable from it at are the worlds satisfying |$\lnot \chi$| at |$M$| that were strictly above it, together with those satisfying |$\chi$| at |$M$| that were at its same level or above. In other words, at , a world |$u$| is |$\leq$|-reachable from a world |$w$| that satisfied |$\chi$| at |$M$| if and only if |$u$| satisfies either both |$(M,u) \Vdash \lnot \chi$| and |$w < u$|⁠, or else both |$(M,u) \Vdash \chi$| and |$w \leq u$|⁠. Now take |$w_{\lnot \chi }$|⁠, and note how the worlds that are |$\leq$|-reachable from it at are the worlds satisfying |$\lnot \chi$| at |$M$| that were at its same level or above, together with those satisfying |$\chi$| at |$M$| that were at most one level below it. In other words, at , a world |$u$| is |$\leq$|-reachable from a world |$w$| that satisfied |$\lnot \chi$| at |$M$| if and only if |$u$| satisfies either |$w \leq u$|⁠, or else both |$(M, u) \Vdash \chi$| and |$u \in{\operatorname{L}^{{\mathcal{D}_{M}}}_{{j+1}}}$|⁠, for |$w$| being in the |$j$|th layer at |$M$| (i.e. |$w \in{\operatorname{L}^{{\mathcal{D}_{M}}}_{{j}}}$|⁠). The argument for the validity of the sixth axiom is analogous. First, at , a world |$u$| is <-reachable from a world |$w$| that satisfied |$\chi$| at |$M$| (say, |$w_{\chi }$|⁠) if and only if |$u$| satisfies either |$(M,u) \Vdash \lnot \chi$| while being at least two layers above |$w$| at |$M$|⁠, or else |$(M, u) \Vdash \chi$| while being strictly above |$w$| at |$M$|⁠. Second, at , a world |$u$| is <-reachable from a world |$w$| that satisfied |$\lnot \chi$| at |$M$| (say, |$w_{\lnot \chi }$|⁠) if and only if |$u$| satisfies either |$(M, u) \Vdash \lnot \chi$| while being strictly above |$w$| at |$M$|⁠, or else |$(M, u) \Vdash \chi$| while being at |$w$|’s level or above at |$M$|⁠. The argument for completeness is similar to the previous cases. Table 4. Recursion axioms for w.r.t. |$\mathscr{PM}$| . From |$\vdash \psi _1 \leftrightarrow \psi _2$| derive |$\vdash \varphi \leftrightarrow \varphi [\psi _2/\psi _1]$|⁠, with |$\varphi [\psi _2/\psi _1]$| any formula obtained by replacing one or more non-dynamic occurrences of |$\psi _1$| in |$\varphi$| with |$\psi _2$|⁠, with the non-dynamic occurrences of |$\psi _1$| being those that are not inside the angle brackets |$`\langle \ \rangle ^{\prime}$| of the dynamic modality . From |$\vdash \psi _1 \leftrightarrow \psi _2$| derive |$\vdash \varphi \leftrightarrow \varphi [\psi _2/\psi _1]$|⁠, with |$\varphi [\psi _2/\psi _1]$| any formula obtained by replacing one or more non-dynamic occurrences of |$\psi _1$| in |$\varphi$| with |$\psi _2$|⁠, with the non-dynamic occurrences of |$\psi _1$| being those that are not inside the angle brackets |$`\langle \ \rangle ^{\prime}$| of the dynamic modality Open in new tab Table 4. Recursion axioms for w.r.t. |$\mathscr{PM}$| . From |$\vdash \psi _1 \leftrightarrow \psi _2$| derive |$\vdash \varphi \leftrightarrow \varphi [\psi _2/\psi _1]$|⁠, with |$\varphi [\psi _2/\psi _1]$| any formula obtained by replacing one or more non-dynamic occurrences of |$\psi _1$| in |$\varphi$| with |$\psi _2$|⁠, with the non-dynamic occurrences of |$\psi _1$| being those that are not inside the angle brackets |$`\langle \ \rangle ^{\prime}$| of the dynamic modality . From |$\vdash \psi _1 \leftrightarrow \psi _2$| derive |$\vdash \varphi \leftrightarrow \varphi [\psi _2/\psi _1]$|⁠, with |$\varphi [\psi _2/\psi _1]$| any formula obtained by replacing one or more non-dynamic occurrences of |$\psi _1$| in |$\varphi$| with |$\psi _2$|⁠, with the non-dynamic occurrences of |$\psi _1$| being those that are not inside the angle brackets |$`\langle \ \rangle ^{\prime}$| of the dynamic modality Open in new tab A final word on the axiom system for (Table 4). It is not a ‘pure’ axiom system because one of its axioms uses model information (the |${\vert{\mathcal{D}_{M}} \vert }$| on the right-hand side of the recursion axiom for |${{\langle{\scriptstyle \leq} \rangle }}$|⁠); still, the system works, in particular when there is a fixed bound on the cardinality of the domain (and thus a fixed bound on the maximum number of layers the models can have). Pure axiomatisations can be found when |${\mathcal{L}_{i}}$| is extended with tools that allow further modalities, as that for the converse relation |$\geq$|⁠, or that for the equal plausibility relation |$\simeq$|⁠.15 3 Deciding whether to revise As mentioned in the introduction, there have been syntax-based proposals for non-prioritized belief revision. They work by performing the revision only when the incoming information has more epistemic value than the original beliefs it might contradict, with the notion of epistemic value defined in terms of either a set of core beliefs or else a set of credible formulas. The use of plausibility models allows a semantic alternative; suppose there is a threshold|$\tau \in{\mathbb{R}}$|⁠, indicating the agent’s ‘openness’ to changing her epistemic state. She will go through the revision process only when ‘the amount of change’ that accepting |$\chi$| implies does not exceed this threshold; in any other scenario, the change is too drastic for her, and thus no revision will occur. In other words, the agent will revise her beliefs only when her actual epistemic state and the one that would result from the revision are ‘not too different’. For a proper formalisation of this idea, the notion of distance between models (or, more properly, distance between relations; see [17]) will be useful; here are two possibilities. Definition 3.1 (Distance between relations). Take |$S_1, S_2 \subseteq U \times U$|⁠. − The Kemeny distance (also called the swap distance, [33]; cf. [27, 34]) measures the minimum number of pairs that should be added/removed from one relation in order to get the other. By defining the symmetric difference between |$S_1$| and |$S_2$| (the pairs in one relation but not in the other) as |$S_1\, {\mathop{\varDelta }} S_2:= (S_1\setminus S_2) \cup (S_2 \setminus S_1)$|⁠, the Kemeny distance is given by $$\begin{equation*}{d_{K}}(S_1, S_2):= {\vert{S_1 \mathop{\varDelta} S_2} \vert}.\end{equation*}$$ − The Jaccard distance (cf. the Jaccard index of similarity; [32]) measures dissimilarity between sets, and it is given by the ratio of the size of the relations’ symmetric difference to their union, i.e. $$\begin{equation*}d_{J}(S_1, S_2):= \frac{{\vert{S_1 \mathop{\varDelta} S_2} \vert}}{{\vert{S_1 \cup S_2} \vert}}.\end{equation*}$$ Over finite relations (and hence over the finite-domain models this proposal works with), both distances are metrics: for any |$S_1, S_2, S_3 \subseteq (U \times U)$|⁠, they satisfy (i) non-negativity: |$d(S_1, S_2) \geq 0$|⁠; (ii) symmetry: |$d(S_1, S_2) = d(S_2, S_1)$|⁠; (iii) reflexivity and identity of indiscernibles: |$d(S_1, S_2) = 0$| if and only if |$S_1 = S_2$|⁠; (iv) subadditivity: |$d(S_1, S_3) \leq d(S_1, S_2) + d(S_2, S_3)$|⁠. These notions of distance between relations define notions of distance between models with the same domain in the obvious way. Note that |$(S_1 {\mathop{\varDelta }} S_2) \subseteq S_1 \cup S_2$| holds for every |$S_1, S_2 \subseteq (U \times U)$|⁠. Thus, the Jaccard distance is a normalised notion that always lie within the interval |$[0,1]$|⁠, regardless of the actual involved relations. However, note how the more arrows are added to the new model, the less they individually add to the distance; this is because the value of |${\vert{S_1 \cup S_2} \vert }$| also increases. Whether or not these two properties are desirable depends on the precise application in mind. An alternative to keep the distance normalised while avoiding the drop in the edges’ ‘value’ as they increase in number is to simply take |${\vert{U \times U} \vert }$| as the denominator. The following example shows models with different plausibility orderings over the four valuations for atoms |$p$| and |$q$|⁠, discussing their distances with the models that would result from each one of the presented |$\lnot p$|-revisions. Example 3.2 In Table 5, each column provides a model representing a doxastic attitude towards |$p$| (note: an agent has a strong belief on a given |$\varphi$| if and only if all |$\varphi$|-worlds are strictly more plausible than all |$\lnot \varphi$|-worlds), then indicating the distance (both Kemeny and Jaccard) between the model and the one that would result from a |$\lnot p$|-revision with each one of the described policies. In the cases of radical and conservative revision, the distances obtained match the intuition: the stronger |$p$| is believed, the larger the distance to the |$\lnot p$|-revised model is, and thus the more difficult it should be to accept |$\lnot p$|⁠. This also gives an idea of the value the threshold should have, depending on the distance that will be used, and how credulous we want the agent to be. The improvement operator, however, does not follow this trend. Because radical and conservative revisions bring worlds satisfying the input formula to the ‘top’ of the plausibility order, the corresponding distances have some correlation to ‘how high up’ (i.e. how plausible) those worlds already were to begin with. In the case of improvement, on the other hand, what is measured is rather ‘how spread out’ those worlds are. For this reason, improvement might not be the best revision policy to work with here. Still, it proves to be an interesting policy when seen as a weaker form of radical revision, as the next section will discuss. Table 5. |${\mathscr{PM}}$|s and their distances with the ones resulting from different policies for a |$\lnot p$|-revision Doxastic . |${\mathop{B}{p}}$| . |${\mathop{B}{p}}$| . |$\lnot{\mathop{B}{p}} \land \lnot{\mathop{B}{\lnot p}}$| . |${\mathop{B}{\lnot p}}$| . |${\mathop{B}{\lnot p}}$| . attitude . (strong belief) . . . . (strong belief) . Initial model |$M$| |${d_{K}}(M, {{{M}}^{{\Uparrow }_{{\lnot p}}}})$| 8 6 4 4 0 |${d_{K}}(M, {{{M}}^{{\uparrow }_{{\lnot p}}}})$| 4 2 1 0 0 1 2 3 1 0 |$d_{J}(M, {{{M}}^{{\Uparrow }_{{\lnot p}}}})$| 0.8 0.67 0.44 0.5 0 |$d_{J}(M, {{{M}}^{{\uparrow }_{{\lnot p}}}})$| 0.5 0.29 0.13 0 0 0.14 0.25 0.33 0.11 0 Doxastic . |${\mathop{B}{p}}$| . |${\mathop{B}{p}}$| . |$\lnot{\mathop{B}{p}} \land \lnot{\mathop{B}{\lnot p}}$| . |${\mathop{B}{\lnot p}}$| . |${\mathop{B}{\lnot p}}$| . attitude . (strong belief) . . . . (strong belief) . Initial model |$M$| |${d_{K}}(M, {{{M}}^{{\Uparrow }_{{\lnot p}}}})$| 8 6 4 4 0 |${d_{K}}(M, {{{M}}^{{\uparrow }_{{\lnot p}}}})$| 4 2 1 0 0 1 2 3 1 0 |$d_{J}(M, {{{M}}^{{\Uparrow }_{{\lnot p}}}})$| 0.8 0.67 0.44 0.5 0 |$d_{J}(M, {{{M}}^{{\uparrow }_{{\lnot p}}}})$| 0.5 0.29 0.13 0 0 0.14 0.25 0.33 0.11 0 Open in new tab Table 5. |${\mathscr{PM}}$|s and their distances with the ones resulting from different policies for a |$\lnot p$|-revision Doxastic . |${\mathop{B}{p}}$| . |${\mathop{B}{p}}$| . |$\lnot{\mathop{B}{p}} \land \lnot{\mathop{B}{\lnot p}}$| . |${\mathop{B}{\lnot p}}$| . |${\mathop{B}{\lnot p}}$| . attitude . (strong belief) . . . . (strong belief) . Initial model |$M$| |${d_{K}}(M, {{{M}}^{{\Uparrow }_{{\lnot p}}}})$| 8 6 4 4 0 |${d_{K}}(M, {{{M}}^{{\uparrow }_{{\lnot p}}}})$| 4 2 1 0 0 1 2 3 1 0 |$d_{J}(M, {{{M}}^{{\Uparrow }_{{\lnot p}}}})$| 0.8 0.67 0.44 0.5 0 |$d_{J}(M, {{{M}}^{{\uparrow }_{{\lnot p}}}})$| 0.5 0.29 0.13 0 0 0.14 0.25 0.33 0.11 0 Doxastic . |${\mathop{B}{p}}$| . |${\mathop{B}{p}}$| . |$\lnot{\mathop{B}{p}} \land \lnot{\mathop{B}{\lnot p}}$| . |${\mathop{B}{\lnot p}}$| . |${\mathop{B}{\lnot p}}$| . attitude . (strong belief) . . . . (strong belief) . Initial model |$M$| |${d_{K}}(M, {{{M}}^{{\Uparrow }_{{\lnot p}}}})$| 8 6 4 4 0 |${d_{K}}(M, {{{M}}^{{\uparrow }_{{\lnot p}}}})$| 4 2 1 0 0 1 2 3 1 0 |$d_{J}(M, {{{M}}^{{\Uparrow }_{{\lnot p}}}})$| 0.8 0.67 0.44 0.5 0 |$d_{J}(M, {{{M}}^{{\uparrow }_{{\lnot p}}}})$| 0.5 0.29 0.13 0 0 0.14 0.25 0.33 0.11 0 Open in new tab 3.1 Screened revision The simplest way to use distances between models when it comes to revision operators is the threshold approach described before. Take a threshold |$\tau$|⁠, indicating the agent’s ‘openness’ to changing her epistemic state. If ‘the amount of change’ implied by accepting the incoming |$\chi$| does not go above |$\tau$|⁠, the agent will revise her beliefs; otherwise, no revision will occur. This requires not only to have the threshold, but also to fix the revision policy that will be used, in order to know ‘how drastic’ of a change accepting the new information would make. Definition 3.3 Let |$M = {\left \langle{W, \leq , V} \right \rangle }$| be a |${\mathscr{PM}}$|⁠; let |$\chi$| be a formula. Let |${{{M}}^{\operatorname{R}_{{\chi }}}} = {\langle{W, \mathrel{{\leq }^{\operatorname{R}_{\chi }}}, V} \rangle }$| be model that results from applying a |$\chi$|-revision with one of the discussed policies, let |${d}:(\mathcal{M} \times \mathcal{M}) \to{\mathbb{R}}$| be a function returning the distance between two plausibility models (see Definition 3.1), and let |$\tau \in{\mathbb{R}}$| be the threshold. The |$\tau$|-screened|$\chi$|-revision operation yields the |${\mathscr{PM}}$| model |${{{M}}^{{\veebar }_{{\tau }}^{{\chi }}}} = {\langle{W, \mathrel{{\leq }^{{\veebar }^{\chi }_{\tau }}}, V} \rangle }$|⁠, with the new plausibility ordering defined in the following way. $$\begin{equation*}{\mathrel{{{\leq}}^{{\veebar}^{{\chi}}_{{\tau}}}}}:= \left\{ \begin{array}{ll} \leq & {\text{{if}}\;} {d}(M, {{{M}}^{\operatorname{R}_{{\chi}}}})> \tau \\{\mathrel{{{\leq}}^{\operatorname{R}_{{\chi}}}}} & {\text{{otherwise}}\;} ({\text{{i.e. if}}\;} {d}(M, {{{M}}^{\operatorname{R}_{{\chi}}}}) \leq \tau)\end{array} \right.\end{equation*}$$ For the language, |${\mathcal{L}_{i}}$| is extended with a modality |${\mathop{\langle{\veebar }^{{\chi }}_{{\tau }}\rangle }{{}}}$| with |$\chi$| a formula and |$\tau$| in |${\mathbb{R}}$|⁠. For its semantic interpretation, given a |${\mathscr{PM}}$||$M$| and |$w \in{\mathcal{D}_{{W}}}$|⁠, $$\begin{equation*}(M, w) \Vdash{\mathop{\langle{\veebar}^{{\chi}}_{{\tau}}\rangle}{{\varphi}}}\quad \textrm{iff}_{def}\quad ({{{M}}^{{\veebar}_{{\tau}}^{{\chi}}}}, w) \Vdash \varphi\end{equation*}$$Axiom system. The strategy for an axiomatisation of the new modality is straightforward. A formula of the form |${\mathop{\langle{\veebar }^{{\chi }}_{{\tau }}\rangle }{{\varphi }}}$| simply checks whether applying the chosen revision policy would go above the given threshold, evaluating |$\varphi$| in either |$M$| (in case it does) or else in |${{{M}}^{\operatorname{R}_{{\chi }}}}$| (in case it does not). The discussed revision policies have been already axiomatised, so one simply needs to find the tools to turn $$\begin{equation*}{\mathop{\langle{\veebar}^{{\chi}}_{{\tau}}\rangle}{{\varphi}}} {\;\leftrightarrow\;} \left( (`{d}(M, {{{M}}^{\operatorname{R}_{{\chi}}}})> \tau^{\prime} \land \varphi) \lor (`{d}(M, {{{M}}^{\operatorname{R}_{{\chi}}}}) \leq \tau^{\prime} \land{\mathop{\langle\operatorname{R}_{{\chi}}\rangle}{{\varphi}}}) \right)\end{equation*}$$ into a proper formula. Now, the characterisation of |${d}(M, {{{M}}^{\operatorname{R}_{{\chi }}}})> \tau$| and |${d}(M, {{{M}}^{\operatorname{R}_{{\chi }}}}) \leq \tau$| is, in standard modal languages, not straightforward. The discussed distances rely on the symmetric difference between two relations, i.e. the pairs in one relation but not in the other. Thus, in order to characterise such distances syntactically, one needs to be able to distinguish (syntactically) one possible world from another. One possibility is to assume that, in the models under study, each world can be uniquely characterised by some propositional formula.16 However, working only with such models might be a strong restriction; asking for each world to have different propositional content would limit the high-order beliefs the agent might have. Here is where the use of nominals pays off. As mentioned, they provide a natural tool for ‘naming’ worlds, allowing us to distinguish whether a pair in a given relation has been either preserved or removed by the operation. Thus, in case of the Kemeny distance, they make the following possible. Proposition 3.4 Let |$M = {\left \langle{W, \leq , V} \right \rangle }$| be a |${\mathscr{PM}}$| and take |$\tau \in{\mathbb{R}}$|⁠. Denote by |$i_w$| the nominal corresponding to each world |$w \in W = {\{ {w_1, \ldots , w_{\vert W \vert }} \}}$| (so |${{[\kern{-1.5pt}[}{i_{w}} {]\kern{-1.5pt}]} ^{{M}}} = {\left \{ {w} \right \}}$| for each |$w \in W$|⁠). Define the formula |${\alpha ^{>{\tau }}_{R_{\chi }}}$| as $$\begin{equation*}{\alpha^{>{\tau}}_{R_{\chi}}}:= \bigvee_{{\left\{ {T \subseteq W \times W \,\boldsymbol{\mid}\, \vert T \vert = \tau+1} \right\}}} \; \bigwedge_{(u,v) \in T} \; \left( \begin{array}{c} {{\langle{\scriptstyle \sim} \rangle}{(i_u \land{\langle{\scriptstyle \leq} \rangle}i_v \land \mathop{\langle\operatorname{R}_{\chi}\rangle}{\lnot{\langle{\scriptstyle \leq} \rangle}i_v})}} \\ \lor \\{{\langle{\scriptstyle \sim} \rangle}{(i_v \land{\langle{\scriptstyle <}\rangle} i_u \land \mathop{\langle\operatorname{R}_{\chi}\rangle}{\lnot{\langle{\scriptstyle <}\rangle} i_u})}} \\\end{array} \right)\end{equation*}$$ Then, $$\begin{equation*}{d_{K}}(M, {{{M}}^{\operatorname{R}_{{\chi}}}})> \tau{\qquad\textrm{iff}\qquad} {{[\kern{-1.5pt}[}{\alpha^{>\tau}_{R_{\chi}}} {]\kern{-1.5pt}]}^{{M}}} = {\mathcal{D}_{{M}}} \quad \left( {\quad\textrm{iff}\quad} {{[\kern{-1.5pt}[}{\alpha^{>\tau}_{R_{\chi}}} {]\kern{-1.5pt}]}^{{M}}} \neq\varnothing \right).\end{equation*}$$ Proof. Suppose |${d_{K}}(M, {{{M}}^{\operatorname{R}_{{\chi }}}})> \tau$|⁠. This holds if and only if there are at least |$\tau +1$| different pairs in |$W \times W$| that appear in the relation of one model but not in the relation of the other. More precisely, |${d_{K}}(M, {{{M}}^{\operatorname{R}_{{\chi }}}})> \tau$| holds if and only if there is a large enough set |$T = {\left \{ {(u_1, v_1), \ldots , (u_{\tau +1}, v_{\tau +1}} \right \}} \subseteq (W \times W)$| such that every |$(u_i, v_i) \in T$| satisfies (i) either |$u_i \leq v_i$| and |$v_i {\mathrel{{{<}}^{\operatorname{R}_{{\chi }}}}} u_i$| (so |$(u_i, v_i)$| is in |$\leq$| but not in |${\mathrel{{{\leq }}^{\operatorname{R}_{{\chi }}}}}$|⁠) or else, (ii)|$v_i < u_i$| and |$u_i\, {\mathrel{{{\leq }}^{\operatorname{R}_{{\chi }}}}} v_i$| (so |$(u_i, v_i)$| is not in |$\leq$| but it is in |${\mathrel{{{\leq }}^{\operatorname{R}_{{\chi }}}}}$|⁠).17 When taking into account that each world |$w \in W$| is syntactically characterised by a formula |$i_w$|⁠, this is exactly what the formula |${\alpha ^{>{\tau }}_{R_{\chi }}}$| expresses. Thus, when such set |$T$| exists, |${\alpha ^{>{\tau }}_{R_{\chi }}}$| holds in all possible worlds in |$M$|⁠,18 i.e. |${{[\kern{-1.5pt}[}{\alpha ^{>\tau }_{R_{\chi }}} {]\kern{-1.5pt}]} ^{{M}}} = {\mathcal{D}_{{M}}}$|⁠. Corollary 3.5 Assume the Kemeny distance is used in Definition 3.3. Then, $$\begin{equation*}\Vdash{\mathop{\langle{\veebar}^{{\chi}}_{{\tau}}\rangle}{{\varphi}}} {\;\leftrightarrow\;} \big( ({\alpha^{>{\tau}}_{R_{\chi}}} \land \; \varphi) \lor (\lnot{\alpha^{>{\tau}}_{R_{\chi}}} \land{\mathop{\langle\operatorname{R}_{{\chi}}\rangle}{{\varphi}}}) \big).\end{equation*}$$Coda: a modal measurement of the distance between two models. As mentioned, the discussed notions of distance between models rely on the difference between the models’ relations: the pairs in one but not in the other. This is the reason why the axiomatisation of the operation has relied on nominals, special kind of atoms that are true at exactly one world of the model. Still, there are alternative strategies, an important one being the opposite one: instead of looking for a language that ‘matches’ the given notion of distance, one might want to look for a notion of distance that ‘matches’ the language. Indeed, the strategy used above has been to rely on a language that can describe the symmetric difference between the relations of two models. However, it can be argued that one should stick to |${\mathcal{L}}$| (the ‘nominal-less’ fragment of |${\mathcal{L}_{i}}$|⁠), looking instead for a notion of distance that relies on it. For such a notion of distance, a crucial property should be that the distance between two models |$M$| and |$M^{\prime }$| is 0 if and only if they cannot be distinguished by a formula of the given language. In the modal setting of this paper, this is asking that |${d}(M,M^{\prime }) = 0$| if and only if the two models are |${\mathcal{L}}$|-bisimilar, i.e. if and only if every world of one model can be put in correspondence with a world of the other, and correspondent worlds satisfy exactly the same formulas in |${\mathcal{L}}$|⁠. Notions of distance satisfying this intuitive requirement (albeit for different languages) have been already proposed in the literature, and they have been also used for providing revision policies. An earlier example is [38], which works with the propositional case (thus taking models to be propositional valuations, i.e. possible worlds), and identifies an epistemic state with a belief set, the closed-under-logical-consequence set of formulas the agent believes. The paper’s main insight is that, by assuming |$d$| to be a distance between any two models, one may then define the result of revising a theory |$T$| with a formula |$\chi$| as the theory of all those models of |$\chi$| that are closest, in terms of |$d$|⁠, to the set of models of |$T$|⁠. The paper proposes different abstract (pseudo-)distances, then studying the properties of the revisions they define. Another proposal using semantic notions of distance, now within a modal framework, is provided in [4, Section 4.4]. Because of the chosen setting, models are now pointed relational models (i.e. relational states), and the distance between them is defined in terms of a variation of an |$n$|-bisimulation [11].19 The distance between two pointed models is defined as the largest |$i$| for which |$i$|-bisimilarity between them holds; from this, a number of similarity-related concepts are derived, as distance between two sets of worlds, similarity between two pointed models and similarity between sets of pointed models. The latter is then used to define a |$\varphi$|-ordering among pointed models, |$\leq _{\varphi }$|⁠, with ‘|$(M, w)$| [being] closer to [|$\varphi$|] than |$(M^{\prime },w^{\prime })$| when its degree of similarity with the models of [|$\varphi$|] is higher than the degree of similarity of |$(M^{\prime },w^{\prime })$| with the models of [|$\varphi$|]’ [4, page 81]. With such |$\varphi$|-similarity ordering defined, the set of models that result from revising |$\varphi$| with the incoming information |$\chi$| is the set of |$\leq _{\varphi }$|-minimal models of |$\chi$|⁠.20 4 Deciding how to revise The screened revision of the previous section compares the agent's current epistemic state with the one that would emerge if she accepted the incoming information. If the ‘amount of effort’ it would take to revise the beliefs does not exceed the agent’s willingness to change, the revision will be performed; otherwise, no change will occur. This is a reasonable representation of a non-prioritized belief revision, and yet the binary nature of the decision might be too drastic: the agent might be faced with a choice between, say, a radical |$\chi$|-revision (leading her, intuitively, to a strong belief on |$\chi$|⁠) or a plain rejection of the incoming information. More to the point, the ‘yes’/‘no’ options treat incoming information for which the amount of change falls just above the threshold in exactly the same way as information whose acceptance would require a ‘full-scale revolution’. Having only acceptance and rejection as options has another counter-intuitive result. Imagine, for instance, that the agent is told multiple times an information whose acceptance falls only slightly above her threshold. Under the proposed screened revision, repetition will not make any difference, as each time the information will be rejected, leaving the model as it was before. However, one can argue that repetition should make a difference.21 In such cases, one might want for the agent to start believing the information after hearing it several times, even though she may not have accepted it if it was announced to her in a more isolated manner. This section presents another approach to non-prioritized belief revision, one in which the final outcome still depends on both the initial epistemic state and the incoming information, but in which the choice is not whether to revise, but rather how to do it. Indeed, it is possible to use a notion of ‘strength’ among revision policies, combined with the distances and thresholds that have been considered, to choose how strongly we want the agent to revise her belief for each input. However, the strength of two revision policies are not always directly comparable: some of them differ on which worlds they will shift (e.g. all |$\chi$|-worlds in the radical case vs the most plausible |$\chi$|-worlds for the conservative approach), some other differ on how much the worlds will be shifted (e.g. one level in the improvement operator vs |$\bot _{\chi } - {\intercal _{{\lnot \chi }}} + 1$| in the radical case, with |$\bot _{\chi }$| the number of the layer in which the least plausible |$\chi$|-worlds appear, and |${\intercal _{{\lnot \chi }}}$| that in which the most plausible |$\lnot \chi$|-worlds show up).22 The following two subsections explore two of these alternatives, thus obtaining a notion of comparative strengths among revision policies. 4.1 $k$-improvement As it has been pointed out earlier, improvement can be seen as weaker version of radical revision: the latter moves |$\chi$|-worlds to the top of the ordering, but the former only pushes such worlds ‘one layer up’. From this perspective, it is natural to define a more general operation, called |$k$|-improvement, which takes all |$\chi$|-worlds and moves them up by |$k$| layers. Definition 4.1 (⁠|$k$|-improvement). Let |$M = {\left \langle{W, \leq , V} \right \rangle }$| be a |${\mathscr{PM}}$|⁠, with |$k \in{\mathbb{N}}$| a number between |$0$| and |${\vert{W} \vert }$|⁠. Let |$\chi$| be a formula. Define |${\prec _0}:= {\simeq },\ {\prec _1}:= {\prec }$| (see Item iii in Definition 2.3) and, for |$j \geq 1,\ {\prec _{j+1}}:= {\left \{ {(u,v) \in W \times W \mid \exists w \in W \;\textrm{such that}\; w \prec u \;\textrm{and}\; u \prec _j v} \right \}}$|⁠, so |$\prec _0$| relates equally plausible worlds, and |$u \prec _i v$| holds when |$u$| is exactly |$i$| layers below |$v$|⁠. The |$k$|-improvement|$\chi$|-revision operation yields the |${\mathscr{PM}}$| model , with the new plausibility ordering given by thus removing arrows going from |$\chi$|-worlds to |$\lnot \chi$|-worlds when the former were strictly less than |$k$| layers ‘beneath’ the latter, and adding arrows from |$\lnot \chi$|-worlds to |$\chi$|-worlds when the latter were at most |$k$| layers ‘beneath’ the former. Intuitively, the operation shifts every |$\chi$|-world ‘|$k$| layers up’. The operation yields a total preorder, so is indeed a plausibility model. Example 4.2 Consider the |${\mathscr{PM}}$||$M$| of Example 2.4. A |$2$|-improvement|$p$|-revision shifts all |$p$|-worlds two layers up; thus, is A |$1$|-improvement is the improvement operator of before. Moreover, for |$k$| large enough (more precisely, for |$k> {\bot _{{\chi }}}$|⁠), a |$k$|-improvement is exactly radical revision. Table 6. Recursion axioms for w.r.t. |$\mathscr{PM}$| . From |$\vdash \psi _1\, {\leftrightarrow }\,\psi _2$| derive |$\vdash \varphi \, {\leftrightarrow }\,\varphi [\psi _2/\psi _1]$|⁠, with |$\varphi [\psi _2/\psi _1]$| any formula obtained by replacing one or more non-dynamic occurrences of |$\psi _1$| in |$\varphi$| with |$\psi _2$|⁠, with the non-dynamic occurrences of |$\psi _1$| being those that are not inside the angle brackets |$`{\mathop{\langle{} \rangle }}^{\prime }$| of the dynamic modality . . From |$\vdash \psi _1\, {\leftrightarrow }\,\psi _2$| derive |$\vdash \varphi \, {\leftrightarrow }\,\varphi [\psi _2/\psi _1]$|⁠, with |$\varphi [\psi _2/\psi _1]$| any formula obtained by replacing one or more non-dynamic occurrences of |$\psi _1$| in |$\varphi$| with |$\psi _2$|⁠, with the non-dynamic occurrences of |$\psi _1$| being those that are not inside the angle brackets |$`{\mathop{\langle{} \rangle }}^{\prime }$| of the dynamic modality . Open in new tab Table 6. Recursion axioms for w.r.t. |$\mathscr{PM}$| . From |$\vdash \psi _1\, {\leftrightarrow }\,\psi _2$| derive |$\vdash \varphi \, {\leftrightarrow }\,\varphi [\psi _2/\psi _1]$|⁠, with |$\varphi [\psi _2/\psi _1]$| any formula obtained by replacing one or more non-dynamic occurrences of |$\psi _1$| in |$\varphi$| with |$\psi _2$|⁠, with the non-dynamic occurrences of |$\psi _1$| being those that are not inside the angle brackets |$`{\mathop{\langle{} \rangle }}^{\prime }$| of the dynamic modality . . From |$\vdash \psi _1\, {\leftrightarrow }\,\psi _2$| derive |$\vdash \varphi \, {\leftrightarrow }\,\varphi [\psi _2/\psi _1]$|⁠, with |$\varphi [\psi _2/\psi _1]$| any formula obtained by replacing one or more non-dynamic occurrences of |$\psi _1$| in |$\varphi$| with |$\psi _2$|⁠, with the non-dynamic occurrences of |$\psi _1$| being those that are not inside the angle brackets |$`{\mathop{\langle{} \rangle }}^{\prime }$| of the dynamic modality . Open in new tab Definition 4.3 (Language). The language is extended with a modality , for |$\chi$| a formula and |$k \in{\mathbb{N}}$|⁠, resulting in the language . As for its semantic interpretation, given |${\mathscr{PM}}$||$M$| and |$w \in{\mathcal{D}_{{W}}}$|⁠, Axiom system. The axiomatisation uses once again the recursion axioms technique. Theorem 4.4 The recursion axioms on Table 6 provide, together with the axioms on Table 1, a sound and complete axiom system for formulas in w.r.t. plausibility models. Proof. Again, soundness follows from the validity and validity-preserving properties of the recursion axioms. The first five are as discussed in the previous cases. The fifth makes once again use of the formulas |$\lambda _j$| (Page 6), which characterise the layers in a given model, together with the abbreviation |${{\langle{\scriptstyle <} \rangle }^{{j}}{}}$| defined inductively as |${{\langle{\scriptstyle <} \rangle }^{{0}}{\varphi }}:= {{\langle{\scriptstyle \sim} \rangle }{\varphi }},\ {{\langle{\scriptstyle <} \rangle }^{{1}}{\varphi }}:= {{\langle{\scriptstyle <}\rangle } {\varphi }}$| and |${{\langle{\scriptstyle <} \rangle }^{{j+1}}{\varphi }}:= {{\langle{\scriptstyle <} \rangle }^{{j}}{{\langle{\scriptstyle <}\rangle } \varphi }}$|⁠. To get an idea of what the axioms are saying, here is what happens during a |$k$|-improvement |$\chi$|-revision: In the above figure, |$w_{\lnot\chi}$| and |$u_{\lnot \chi }$| are worlds in |${[\kern{-1.5pt}[}{\lnot\chi}{]\kern{-1.5pt}]}^{M}$|⁠, and |$w_{\chi}$| and |$u_{\chi }$| are worlds in |${{[\kern{-1.5pt}[}{\chi } {]\kern{-1.5pt}]} ^{{M}}}$|⁠. Take |$w_{\chi }$| and look at the worlds that are |$\leq$|-reachable from it at : these are the worlds in |${{[\kern{-1.5pt}[}{\lnot \chi } {]\kern{-1.5pt}]} ^{{M}}}$| that were at least |$k$| layers above it, together with the worlds in |${{[\kern{-1.5pt}[}{\chi } {]\kern{-1.5pt}]} ^{{M}}}$| that were at its same level or above. In other words, at a world |$u$| is |$\leq$|-reachable from a world in |${{[\kern{-1.5pt}[}{\chi } {]\kern{-1.5pt}]} ^{{M}}}$| if and only if |$u$| satisfies both |$u \in{{[\kern{-1.5pt}[}{\lnot \chi } {]\kern{-1.5pt}]} ^{{M}}}$| and |$w \prec _l u$| (see Definition 4.1) for some |$l \in{\mathbb{N}}$| satisfying |$l\geqslant k$| or else both |$u \in{{[\kern{-1.5pt}[}{\chi } {]\kern{-1.5pt}]} ^{{M}}}$| and |$w \leq u$|⁠. Now take |$w_{\lnot \chi }$| and look at the worlds that are |$\leq$|-reachable from it at : these are the worlds in |${{[\kern{-1.5pt}[}{\lnot \chi } {]\kern{-1.5pt}]} ^{{M}}}$| that were at its same level or above, together with the worlds in |${{[\kern{-1.5pt}[}{\chi } {]\kern{-1.5pt}]} ^{{M}}}$| that were at most |$k$| levels below it. In other words, at , a world |$u$| is |$\leq$|-reachable from a world |$w$| in |${{[\kern{-1.5pt}[}{\lnot \chi } {]\kern{-1.5pt}]} ^{{M}}}$| if and only if |$u$| satisfies either |$w \leq u$|⁠, or else both |$u \in{{[\kern{-1.5pt}[}{\chi } {]\kern{-1.5pt}]} ^{{M}}}$| and |$u \in{\operatorname{L}^{{\mathcal{D}_{M}}}_{{l}}}$|⁠, with |$w$| being in the |$j$|th layer at |$M$|⁠, and |$l \in{\mathbb{N}}$| satisfying |$j<l \leqslant j+k$|⁠. The argument for the validity of the sixth axiom is analogous. First, at , a world |$u$| is <-reachable from a world |$w$| in |${{[\kern{-1.5pt}[}{\chi } {]\kern{-1.5pt}]} ^{{M}}}$| (say, |$w_{\chi }$|⁠) if and only if |$u$| satisfies either |$u \in{{[\kern{-1.5pt}[}{\lnot \chi } {]\kern{-1.5pt}]} ^{{M}}}$| while being at least |$k+1$| layers above |$w$| at |$M$|⁠, or else |$u \in{{[\kern{-1.5pt}[}{\chi } {]\kern{-1.5pt}]} ^{{M}}}$| while being strictly above |$w$| at |$M$|⁠. Second, at , a world |$u$| is <-reachable from a world |$w$| in |${{[\kern{-1.5pt}[}{\lnot \chi } {]\kern{-1.5pt}]} ^{{M}}}$| (say, |$w_{\lnot \chi }$|⁠) if and only if |$u$| satisfies either |$u \in{{[\kern{-1.5pt}[}{\lnot \chi } {]\kern{-1.5pt}]} ^{{M}}}$| while being strictly above |$w$| at |$M$|⁠, or else |$u \in{{[\kern{-1.5pt}[}{\chi } {]\kern{-1.5pt}]} ^{{M}}}$| while being at most |$k-1$| levels below |$w$| at |$M$|⁠. The argument for completeness is similar to the previous cases. With this new revision policy, there is now a family or revision operators of intuitively increasing strength. This difference in strength can be stated formally using the Kemeny distance of before. Theorem 4.5 Let |$M$| be a |${\mathscr{PM}}$|⁠; let |$\chi$| be a formula. The function is monotone: |$j \leq k$| implies |${d_{K}}^{\chi }(j) \leq{d_{K}}^{\chi }(k)$|⁠. In fact, for |$j$| and |$k$| in the interval |${\{{0}..{\bot _{\chi }}\}}$|⁠, the function is strictly monotone: |$j < k$| implies |${d_{K}}^{\chi }(j) < {d_{K}}^{\chi }(k)$|⁠; for the rest (⁠|$j$| and |$k$| in the interval |${\{{\bot _{\chi }}..{\vert \mathcal{D}_{M} \vert }\}}$|⁠, the function is constant: |$j < k$| implies |${d_{K}}^{\chi }(j) = {d_{K}}^{\chi }(k)$|⁠. Proof. First, we show that |${d_{K}}^{\chi }$| is increasing. Take |$j,k\in{\mathbb{N}}$| such that |$j\leqslant k$|⁠. For any |$(w,u)$| with |$w\leq u$|⁠, if we have , then we also have . For any |$(w,u)$| such that |$w\not \leq u$|⁠, if we have , then we also have . Hence all arrows that appear or disappear between |$M$| and also appear or disappear between |$M$| and , and so the Kemeny distance between |$M$| and is at least as great as the distance between |$M$| and . We now show that |${d_{K}}$| is strictly increasing on |${\{{0}..{\bot _{\chi }}\}}$|⁠. Take |$j_0<{\bot _{{\chi }}}$|⁠. At , there exists a world , and hence there also exists a world . That is, |$w_0$| is one of the most plausible worlds satisfying |$(\chi \land{{\langle{\scriptstyle \leq} \rangle }{\lnot \chi }})$| in . There also exists a world |$v$| satisfying |$\lnot \chi$| either at the same level or one level above |$w_0$|⁠. In |$M,\ w_0$| was |$j_0$| or |$j_0+1$| levels below |$v$|⁠. In the first case, |$(w_0,v)$| is in and not in . In the second, |$(v,w_0)$| is not in but it is in . Hence |${d_{K}}^{\chi }(j_0+1)>{d_{K}}^{\chi }(j_0)$|⁠. Other families of revision policies can be built in a similar way. For example, a conservative |$k$|-improvement would introduce a graduation between what could be called a conservative improvement (moving all worlds in |${\mathopen{{[\kern{-1.5pt}[} } {\lambda ^{\chi }_0} \mathopen{{]\kern{-1.5pt}]} ^{{M}}}}$| one level up23 ) and conservative revision (seen as moving all worlds in |${\mathopen{{[\kern{-1.5pt}[} } {\lambda ^{\chi }_0} \mathopen{{]\kern{-1.5pt}]} ^{{M}}}}$| exactly |${\intercal _{{\chi }}} - {\intercal _{{\lnot \chi }}} + 1$| levels up, given |${\intercal _{{\chi }}} \geqslant{\intercal _{{\lnot \chi }}}$|⁠). Analogously, but filling a different gap, a |$k$|-moderate revision would introduce a graduation between conservative revision (seen as moving all worlds in |${\mathopen{{[\kern{-1.5pt}[} } {\lambda ^{\chi }_0} \mathopen{{]\kern{-1.5pt}]} ^{{M}}}}$| to the top, keeping the relative ordering among them as before) and radical revision (moving all worlds in |${\mathopen{{[\kern{-1.5pt}[} } { \lambda ^{\chi }_0 \lor \cdots \lor \lambda ^{\chi }_{m}} \mathopen{{]\kern{-1.5pt}]} ^{{M}}}}$| to the top, keeping the relative ordering among them as before24 ). The next subsection provides details of the latter family. 4.2 $k$-moderate revision Definition 4.6 (⁠|$k$|-moderate revision). Let |$M = {\left \langle{W, \leq , V} \right \rangle }$| be a |${\mathscr{PM}}$|⁠, with |$k \in{\mathbb{N}}$| a number between |$0$| and |${\vert{W} \vert }$|⁠. Let |$\chi$| be a formula. The |$k$|-moderate|$\chi$|-revision operation yields the |${\mathscr{PM}}$| model |${{{M}}^{{\Uparrow }^{{k}}_{{\chi }}}}={\langle{W, \mathrel{{\leq }^{{\Uparrow }^{k}_{\chi }}},V} \rangle }$|⁠, with the new plausibility ordering given by $$\begin{equation*}{{\mathrel{{{\leq}}^{{\Uparrow}^{{k}}_{{\chi}}}}}} := \left( {\leq} \cap ({\overline{{\mathopen{{[\kern{-1.5pt}[}} \lambda^{\chi}_{0} \lor \cdots \lor \lambda^{\chi}_{k} \mathopen{{]\kern{-1.5pt}]}^{M}}}}} \times W) \right) \,\cup\, \left( {\sim} \cap (W \times{\mathopen{{[\kern{-1.5pt}[}} {\lambda^{\chi}_{0} \lor \cdots \lor \lambda^{\chi}_{k}} \mathopen{{]\kern{-1.5pt}]}^{{M}}}}) \right),\end{equation*}$$ thus reversing arrows going from worlds not in |${\mathopen{{[\kern{-1.5pt}[} } {\lambda ^{\chi }_{0} \lor \cdots \lor \lambda ^{\chi }_{k}} \mathopen{{]\kern{-1.5pt}]} ^{{M}}}}$| to any other world. Intuitively, the operation looks at the ordering within |${{[\kern{-1.5pt}[}{\chi } {]\kern{-1.5pt}]} ^{{M}}}$|⁠, making those worlds in the first |$k$| layers in |${{[\kern{-1.5pt}[}{\chi } {]\kern{-1.5pt}]} ^{{M}}}$| strictly more plausible than the rest of the worlds in the model, while keeping the old ordering within the two zones. Example 4.7 Consider any |${\mathscr{PM}}$||$M$|⁠. A |$k$|-moderate |$\chi$|-revision for |$k=0$| shifts the set containing the most plausible |$\chi$|-worlds to the top, thus working as a conservative revision. Moreover, generalize the definition of the layers induced by a model’s domain (page 5) to make it work for an arbitrary finite set of worlds |$U$|⁠,25 and let |${\operatorname{{LastLay}}}(U)$| denote the number of its lowest layer.26 Then, a |$k$|-moderate |$\chi$|-revision for |$k \geqslant{\operatorname{{LastLay}}}({{[\kern{-1.5pt}[}{\chi } {]\kern{-1.5pt}]} ^{{M}}})$| places all worlds in |${{[\kern{-1.5pt}[}{\chi } {]\kern{-1.5pt}]} ^{{M}}}$| over the rest (the |$\lnot \chi$|-ones) to the top while keeping the old ordering within the two zones, thus working as a radical revision. Definition 4.8 (Language). The language |${\mathcal{L}_{i}}$| is extended with a modality |${\mathop{\langle{\chi }{\Uparrow }^{{k}}\rangle }{{}}}$|⁠, for |$\chi$| a formula and |$k \in{\mathbb{N}}$|⁠, resulting in the language |${\mathcal{L}_{i, {\Uparrow }^k}}$|⁠. As for its semantic interpretation, given |${\mathscr{PM}}$||$M$| and |$w \in{\mathcal{D}_{{W}}}$|⁠, $$\begin{equation*}(M, w) \Vdash{\mathop{\langle{\chi}{\Uparrow}^{{k}}\rangle}{{\varphi}}}\quad \textrm{iff}_{def} \quad ({{{M}}^{{\Uparrow}^{{k}}_{{\chi}}}}, w) \Vdash \varphi\end{equation*}$$Axiom system. The axiomatisation of |${\mathop{\langle{\chi }{\Uparrow }^{{k}}\rangle }{{}}}$| is essentially that of |${\mathop{\langle{\chi }{\Uparrow }\rangle }{{}}}$| (Table 2), as the two operations move a set of worlds to the top of the ordering, keeping the relative position within the two zones as before. The only difference is the worlds that are shifted up: in a radical revision, those are exactly the worlds in |${{[\kern{-1.5pt}[}{\chi } {]\kern{-1.5pt}]} ^{{M}}}$|⁠; in a |$k$|-moderate revision, those are exactly the ones in |${\mathopen{{[\kern{-1.5pt}[} } {\lambda ^{\chi }_{0} \lor \cdots \lor \lambda ^{\chi }_{k}} \mathopen{{]\kern{-1.5pt}]} ^{{M}}}}$|⁠. Theorem 4.9 The recursion axioms on Table 7 provide, together with the axioms on Table 1, a sound and complete axiom system for formulas in |${\mathcal{L}_{i, {\Uparrow }^k}}$| w.r.t. plausibility models. Table 7. Recursion axioms for |${\mathcal{L}_{i, {\Uparrow }^k}}$| w.r.t. |${\mathscr{PM}}$| . $$\begin{array}{@{}l@{\qquad\qquad\qquad}l@{}}\vdash{\mathop{\langle{\chi}{\Uparrow}^{{k}}\rangle}{{p}}}{\;\leftrightarrow \;}p\vdash{\mathop{\langle{\chi}{\Uparrow}^{{k}}\rangle}{{\lnot\varphi}}}{\;\leftrightarrow\;}\lnot{\mathop{\langle{\chi}{\Uparrow}^{{k}}\rangle}{{\varphi}}}\\\vdash{\mathop{\langle{\chi}{\Uparrow }^{{k}}\rangle}{{i}}}{\;\leftrightarrow\;}i\vdash{\mathop{\langle{\chi}{\Uparrow}^{{k}}\rangle}{{(\varphi\lor\psi)}}}{\;\leftrightarrow\;}({\mathop{\langle{\chi}{\Uparrow}^{{k}}\rangle}{{\varphi}}} \lor{\mathop{\langle{\chi}{\Uparrow}^{{k}}\rangle}{{\psi}}})\end{array}$$ |$\vdash{\mathop{\langle{\chi }{\Uparrow }^{{k}}\rangle }{{{\langle{\scriptstyle \sim} \rangle }\varphi }}} {\;\leftrightarrow \;} {{\langle{\scriptstyle \sim} \rangle }{\mathop{\langle \chi{\Uparrow }^{k}\rangle }{\varphi }}}$| $$\vdash{\mathop{\langle{\chi }{\Uparrow }^{{k}}\rangle }{{{\langle{\scriptstyle \leq} \rangle }\varphi }}} {\;\leftrightarrow \;} \bigvee \left \{ \begin{array}{l} {{\langle{\scriptstyle \leq} \rangle }{((\lambda ^{\chi }_{0} \lor \cdots \lor \lambda ^{\chi }_{k}) \land \mathop{\langle \chi{\Uparrow }^{k}\rangle }{\varphi })}}, \\ \lnot (\lambda ^{\chi }_{0} \lor \cdots \lor \lambda ^{\chi }_{k}) \land{{\langle{\scriptstyle \leq} \rangle }{\mathop{\langle \chi{\Uparrow }^{k}\rangle }{\varphi }}}, \\ \lnot (\lambda ^{\chi }_{0} \lor \cdots \lor \lambda ^{\chi }_{k}) \land{{\langle{\scriptstyle \sim} \rangle }{((\lambda ^{\chi }_{0} \lor \cdots \lor \lambda ^{\chi }_{k}) \land \mathop{\langle \chi{\Uparrow }^{k}\rangle }{\varphi })}}\end{array} \right \}$$ $$\vdash{\mathop{\langle{\chi }{\Uparrow }^{{k}}\rangle }{{{\langle{\scriptstyle <}\rangle } \varphi }}} {\;\leftrightarrow \;} \bigvee \left \{ \begin{array}{l} {{\langle{\scriptstyle <}\rangle } {((\lambda ^{\chi }_{0} \lor \cdots \lor \lambda ^{\chi }_{k}) \land \mathop{\langle \chi{\Uparrow }^{k}\rangle }{\varphi })}}, \\ \lnot (\lambda ^{\chi }_{0} \lor \cdots \lor \lambda ^{\chi }_{k}) \land{{\langle{\scriptstyle <}\rangle } {\mathop{\langle \chi{\Uparrow }^{k}\rangle }{\varphi }}}, \\ \lnot (\lambda ^{\chi }_{0} \lor \cdots \lor \lambda ^{\chi }_{k}) \land{{\langle{\scriptstyle \sim} \rangle }{((\lambda ^{\chi }_{0} \lor \cdots \lor \lambda ^{\chi }_{k}) \land \mathop{\langle \chi{\Uparrow }^{k}\rangle }{\varphi })}}\end{array} \right \}$$ From |$\vdash \psi _1\, {\leftrightarrow }\,\psi _2$| derive |$\vdash \varphi \, {\leftrightarrow }\,\varphi [\psi _2/\psi _1]$|⁠, with |$\varphi [\psi _2/\psi _1]$| any formula obtained by replacing one or more non-dynamic occurrences of |$\psi _1$| in |$\varphi$| with |$\psi _2$|⁠, with the non-dynamic occurrences of |$\psi _1$| being those that are not inside the angle brackets |$`{\mathop{\langle{} \rangle }}^{\prime }$| of the dynamic modality |${\mathop{\langle{\chi }{\Uparrow }^{{k}}\rangle }{{}}}$|⁠. . $$\begin{array}{@{}l@{\qquad\qquad\qquad}l@{}}\vdash{\mathop{\langle{\chi}{\Uparrow}^{{k}}\rangle}{{p}}}{\;\leftrightarrow \;}p\vdash{\mathop{\langle{\chi}{\Uparrow}^{{k}}\rangle}{{\lnot\varphi}}}{\;\leftrightarrow\;}\lnot{\mathop{\langle{\chi}{\Uparrow}^{{k}}\rangle}{{\varphi}}}\\\vdash{\mathop{\langle{\chi}{\Uparrow }^{{k}}\rangle}{{i}}}{\;\leftrightarrow\;}i\vdash{\mathop{\langle{\chi}{\Uparrow}^{{k}}\rangle}{{(\varphi\lor\psi)}}}{\;\leftrightarrow\;}({\mathop{\langle{\chi}{\Uparrow}^{{k}}\rangle}{{\varphi}}} \lor{\mathop{\langle{\chi}{\Uparrow}^{{k}}\rangle}{{\psi}}})\end{array}$$ |$\vdash{\mathop{\langle{\chi }{\Uparrow }^{{k}}\rangle }{{{\langle{\scriptstyle \sim} \rangle }\varphi }}} {\;\leftrightarrow \;} {{\langle{\scriptstyle \sim} \rangle }{\mathop{\langle \chi{\Uparrow }^{k}\rangle }{\varphi }}}$| $$\vdash{\mathop{\langle{\chi }{\Uparrow }^{{k}}\rangle }{{{\langle{\scriptstyle \leq} \rangle }\varphi }}} {\;\leftrightarrow \;} \bigvee \left \{ \begin{array}{l} {{\langle{\scriptstyle \leq} \rangle }{((\lambda ^{\chi }_{0} \lor \cdots \lor \lambda ^{\chi }_{k}) \land \mathop{\langle \chi{\Uparrow }^{k}\rangle }{\varphi })}}, \\ \lnot (\lambda ^{\chi }_{0} \lor \cdots \lor \lambda ^{\chi }_{k}) \land{{\langle{\scriptstyle \leq} \rangle }{\mathop{\langle \chi{\Uparrow }^{k}\rangle }{\varphi }}}, \\ \lnot (\lambda ^{\chi }_{0} \lor \cdots \lor \lambda ^{\chi }_{k}) \land{{\langle{\scriptstyle \sim} \rangle }{((\lambda ^{\chi }_{0} \lor \cdots \lor \lambda ^{\chi }_{k}) \land \mathop{\langle \chi{\Uparrow }^{k}\rangle }{\varphi })}}\end{array} \right \}$$ $$\vdash{\mathop{\langle{\chi }{\Uparrow }^{{k}}\rangle }{{{\langle{\scriptstyle <}\rangle } \varphi }}} {\;\leftrightarrow \;} \bigvee \left \{ \begin{array}{l} {{\langle{\scriptstyle <}\rangle } {((\lambda ^{\chi }_{0} \lor \cdots \lor \lambda ^{\chi }_{k}) \land \mathop{\langle \chi{\Uparrow }^{k}\rangle }{\varphi })}}, \\ \lnot (\lambda ^{\chi }_{0} \lor \cdots \lor \lambda ^{\chi }_{k}) \land{{\langle{\scriptstyle <}\rangle } {\mathop{\langle \chi{\Uparrow }^{k}\rangle }{\varphi }}}, \\ \lnot (\lambda ^{\chi }_{0} \lor \cdots \lor \lambda ^{\chi }_{k}) \land{{\langle{\scriptstyle \sim} \rangle }{((\lambda ^{\chi }_{0} \lor \cdots \lor \lambda ^{\chi }_{k}) \land \mathop{\langle \chi{\Uparrow }^{k}\rangle }{\varphi })}}\end{array} \right \}$$ From |$\vdash \psi _1\, {\leftrightarrow }\,\psi _2$| derive |$\vdash \varphi \, {\leftrightarrow }\,\varphi [\psi _2/\psi _1]$|⁠, with |$\varphi [\psi _2/\psi _1]$| any formula obtained by replacing one or more non-dynamic occurrences of |$\psi _1$| in |$\varphi$| with |$\psi _2$|⁠, with the non-dynamic occurrences of |$\psi _1$| being those that are not inside the angle brackets |$`{\mathop{\langle{} \rangle }}^{\prime }$| of the dynamic modality |${\mathop{\langle{\chi }{\Uparrow }^{{k}}\rangle }{{}}}$|⁠. Open in new tab Table 7. Recursion axioms for |${\mathcal{L}_{i, {\Uparrow }^k}}$| w.r.t. |${\mathscr{PM}}$| . $$\begin{array}{@{}l@{\qquad\qquad\qquad}l@{}}\vdash{\mathop{\langle{\chi}{\Uparrow}^{{k}}\rangle}{{p}}}{\;\leftrightarrow \;}p\vdash{\mathop{\langle{\chi}{\Uparrow}^{{k}}\rangle}{{\lnot\varphi}}}{\;\leftrightarrow\;}\lnot{\mathop{\langle{\chi}{\Uparrow}^{{k}}\rangle}{{\varphi}}}\\\vdash{\mathop{\langle{\chi}{\Uparrow }^{{k}}\rangle}{{i}}}{\;\leftrightarrow\;}i\vdash{\mathop{\langle{\chi}{\Uparrow}^{{k}}\rangle}{{(\varphi\lor\psi)}}}{\;\leftrightarrow\;}({\mathop{\langle{\chi}{\Uparrow}^{{k}}\rangle}{{\varphi}}} \lor{\mathop{\langle{\chi}{\Uparrow}^{{k}}\rangle}{{\psi}}})\end{array}$$ |$\vdash{\mathop{\langle{\chi }{\Uparrow }^{{k}}\rangle }{{{\langle{\scriptstyle \sim} \rangle }\varphi }}} {\;\leftrightarrow \;} {{\langle{\scriptstyle \sim} \rangle }{\mathop{\langle \chi{\Uparrow }^{k}\rangle }{\varphi }}}$| $$\vdash{\mathop{\langle{\chi }{\Uparrow }^{{k}}\rangle }{{{\langle{\scriptstyle \leq} \rangle }\varphi }}} {\;\leftrightarrow \;} \bigvee \left \{ \begin{array}{l} {{\langle{\scriptstyle \leq} \rangle }{((\lambda ^{\chi }_{0} \lor \cdots \lor \lambda ^{\chi }_{k}) \land \mathop{\langle \chi{\Uparrow }^{k}\rangle }{\varphi })}}, \\ \lnot (\lambda ^{\chi }_{0} \lor \cdots \lor \lambda ^{\chi }_{k}) \land{{\langle{\scriptstyle \leq} \rangle }{\mathop{\langle \chi{\Uparrow }^{k}\rangle }{\varphi }}}, \\ \lnot (\lambda ^{\chi }_{0} \lor \cdots \lor \lambda ^{\chi }_{k}) \land{{\langle{\scriptstyle \sim} \rangle }{((\lambda ^{\chi }_{0} \lor \cdots \lor \lambda ^{\chi }_{k}) \land \mathop{\langle \chi{\Uparrow }^{k}\rangle }{\varphi })}}\end{array} \right \}$$ $$\vdash{\mathop{\langle{\chi }{\Uparrow }^{{k}}\rangle }{{{\langle{\scriptstyle <}\rangle } \varphi }}} {\;\leftrightarrow \;} \bigvee \left \{ \begin{array}{l} {{\langle{\scriptstyle <}\rangle } {((\lambda ^{\chi }_{0} \lor \cdots \lor \lambda ^{\chi }_{k}) \land \mathop{\langle \chi{\Uparrow }^{k}\rangle }{\varphi })}}, \\ \lnot (\lambda ^{\chi }_{0} \lor \cdots \lor \lambda ^{\chi }_{k}) \land{{\langle{\scriptstyle <}\rangle } {\mathop{\langle \chi{\Uparrow }^{k}\rangle }{\varphi }}}, \\ \lnot (\lambda ^{\chi }_{0} \lor \cdots \lor \lambda ^{\chi }_{k}) \land{{\langle{\scriptstyle \sim} \rangle }{((\lambda ^{\chi }_{0} \lor \cdots \lor \lambda ^{\chi }_{k}) \land \mathop{\langle \chi{\Uparrow }^{k}\rangle }{\varphi })}}\end{array} \right \}$$ From |$\vdash \psi _1\, {\leftrightarrow }\,\psi _2$| derive |$\vdash \varphi \, {\leftrightarrow }\,\varphi [\psi _2/\psi _1]$|⁠, with |$\varphi [\psi _2/\psi _1]$| any formula obtained by replacing one or more non-dynamic occurrences of |$\psi _1$| in |$\varphi$| with |$\psi _2$|⁠, with the non-dynamic occurrences of |$\psi _1$| being those that are not inside the angle brackets |$`{\mathop{\langle{} \rangle }}^{\prime }$| of the dynamic modality |${\mathop{\langle{\chi }{\Uparrow }^{{k}}\rangle }{{}}}$|⁠. . $$\begin{array}{@{}l@{\qquad\qquad\qquad}l@{}}\vdash{\mathop{\langle{\chi}{\Uparrow}^{{k}}\rangle}{{p}}}{\;\leftrightarrow \;}p\vdash{\mathop{\langle{\chi}{\Uparrow}^{{k}}\rangle}{{\lnot\varphi}}}{\;\leftrightarrow\;}\lnot{\mathop{\langle{\chi}{\Uparrow}^{{k}}\rangle}{{\varphi}}}\\\vdash{\mathop{\langle{\chi}{\Uparrow }^{{k}}\rangle}{{i}}}{\;\leftrightarrow\;}i\vdash{\mathop{\langle{\chi}{\Uparrow}^{{k}}\rangle}{{(\varphi\lor\psi)}}}{\;\leftrightarrow\;}({\mathop{\langle{\chi}{\Uparrow}^{{k}}\rangle}{{\varphi}}} \lor{\mathop{\langle{\chi}{\Uparrow}^{{k}}\rangle}{{\psi}}})\end{array}$$ |$\vdash{\mathop{\langle{\chi }{\Uparrow }^{{k}}\rangle }{{{\langle{\scriptstyle \sim} \rangle }\varphi }}} {\;\leftrightarrow \;} {{\langle{\scriptstyle \sim} \rangle }{\mathop{\langle \chi{\Uparrow }^{k}\rangle }{\varphi }}}$| $$\vdash{\mathop{\langle{\chi }{\Uparrow }^{{k}}\rangle }{{{\langle{\scriptstyle \leq} \rangle }\varphi }}} {\;\leftrightarrow \;} \bigvee \left \{ \begin{array}{l} {{\langle{\scriptstyle \leq} \rangle }{((\lambda ^{\chi }_{0} \lor \cdots \lor \lambda ^{\chi }_{k}) \land \mathop{\langle \chi{\Uparrow }^{k}\rangle }{\varphi })}}, \\ \lnot (\lambda ^{\chi }_{0} \lor \cdots \lor \lambda ^{\chi }_{k}) \land{{\langle{\scriptstyle \leq} \rangle }{\mathop{\langle \chi{\Uparrow }^{k}\rangle }{\varphi }}}, \\ \lnot (\lambda ^{\chi }_{0} \lor \cdots \lor \lambda ^{\chi }_{k}) \land{{\langle{\scriptstyle \sim} \rangle }{((\lambda ^{\chi }_{0} \lor \cdots \lor \lambda ^{\chi }_{k}) \land \mathop{\langle \chi{\Uparrow }^{k}\rangle }{\varphi })}}\end{array} \right \}$$ $$\vdash{\mathop{\langle{\chi }{\Uparrow }^{{k}}\rangle }{{{\langle{\scriptstyle <}\rangle } \varphi }}} {\;\leftrightarrow \;} \bigvee \left \{ \begin{array}{l} {{\langle{\scriptstyle <}\rangle } {((\lambda ^{\chi }_{0} \lor \cdots \lor \lambda ^{\chi }_{k}) \land \mathop{\langle \chi{\Uparrow }^{k}\rangle }{\varphi })}}, \\ \lnot (\lambda ^{\chi }_{0} \lor \cdots \lor \lambda ^{\chi }_{k}) \land{{\langle{\scriptstyle <}\rangle } {\mathop{\langle \chi{\Uparrow }^{k}\rangle }{\varphi }}}, \\ \lnot (\lambda ^{\chi }_{0} \lor \cdots \lor \lambda ^{\chi }_{k}) \land{{\langle{\scriptstyle \sim} \rangle }{((\lambda ^{\chi }_{0} \lor \cdots \lor \lambda ^{\chi }_{k}) \land \mathop{\langle \chi{\Uparrow }^{k}\rangle }{\varphi })}}\end{array} \right \}$$ From |$\vdash \psi _1\, {\leftrightarrow }\,\psi _2$| derive |$\vdash \varphi \, {\leftrightarrow }\,\varphi [\psi _2/\psi _1]$|⁠, with |$\varphi [\psi _2/\psi _1]$| any formula obtained by replacing one or more non-dynamic occurrences of |$\psi _1$| in |$\varphi$| with |$\psi _2$|⁠, with the non-dynamic occurrences of |$\psi _1$| being those that are not inside the angle brackets |$`{\mathop{\langle{} \rangle }}^{\prime }$| of the dynamic modality |${\mathop{\langle{\chi }{\Uparrow }^{{k}}\rangle }{{}}}$|⁠. Open in new tab 4.3 Gradual revision Any family |${\{ {\operatorname{R}^k} \}}_{k \in{\{{0}..{\ell }\}}}$| of revision operators of increasing strength (note: |$\ell$| is the strongest) allows us to define the sketched idea of gradual revision.27 Indeed, one can use the initial epistemic state |$M$| and the incoming information |$\chi$| to find an appropriate |$k$|⁠, and then apply a |${\operatorname{R}}^k\ \chi$|-revision. Two options arise naturally. Compute |${d}(M,{{{M}}^{\operatorname{R}^{{k}}_{{\chi }}}})$| for all |$k$|⁠, and choose the largest |$k$| such that the computed distance is lower than a threshold |$\tau$|⁠. Compute |${d}(M,{{{M}}^{\operatorname{R}^{{\ell }}_{{\chi }}}})$|⁠, i.e. the distance obtained when performing the strongest revision in the family, and choose the policy to be used based on that number, by splitting the range of possible distances into intervals corresponding to different revision strengths. The first option can be defined in the following manner. Definition 4.10 (Gradual revision, v1). Let |$M = {\left \langle{W, \leq , V} \right \rangle }$| be a |${\mathscr{PM}}$|⁠; let |$\chi$| be a formula, with |$\tau \in{\mathbb{R}}$| the threshold. The gradual|$\chi$|-revision operation yields the |${\mathscr{PM}}$| model |${{{M}}^{{\natural }^{{\chi }}_{{\tau }}}} = {\langle{W, \mathrel{{\leq }^{{\natural }^{\chi }_{\tau }}}, V} \rangle }$|⁠, with the new plausibility ordering given by $$\begin{equation*}{{\mathrel{{{\leq}}^{{\natural}^{{\chi}}_{{\tau}}}}}}:= {{\mathrel{{{\leq}}^{\operatorname{R}^{{k}}_{{\chi}}}}}}\end{equation*}$$ with |$k \geqslant 0$| given by $$\begin{equation*}k:= \left\{ \begin{array}{ll} 0 & {\text{{if}}\;} \tau < {d}(M, {{{M}}^{\operatorname{R}^{{0}}_{{\chi}}}}) \\ j & {\text{{if}}\;} {d}(M, {{{M}}^{\operatorname{R}^{{j}}_{{\chi}}}}) \leqslant \tau < {d}(M, {{{M}}^{\operatorname{R}^{{j+1}}_{{\chi}}}}) \\ \ell & {\text{{if}}\;} {d}(M, {{{M}}^{\operatorname{R}^{{\ell}}_{{\chi}}}}) \leqslant \tau \\\end{array} \right. .\end{equation*}$$ For the language, |${\mathcal{L}}$| is extended with a modality |${\mathop{\langle{\natural }^{{\chi }}_{{\tau }}\rangle }{{}}}$|⁠, for |$\chi$| a formula and |$\tau$| in |${\mathbb{R}}$|⁠. For its semantic interpretation, $$\begin{equation*}(M, w) \Vdash{\mathop{\langle{\natural}^{{\chi}}_{{\tau}}\rangle}{{\varphi}}} \quad \textrm{iff}_{def} \quad ({{{M}}^{{\natural}^{{\chi}}_{{\tau}}}}, w) \Vdash \varphi .\end{equation*}$$ Note how this gradual revision picks, within the chosen family, the strongest revision that does not go above the threshold (in particular, choosing the strongest within the family when the threshold allows it). On the other hand, this policy always perform a revision, as it uses the weakest policy, even when the threshold does not allow for it.28 The second option is formally stated below. Definition 4.11 (Gradual revision, v2). Let |$M = {\left \langle{W, \leq , V} \right \rangle }$| be a |${\mathscr{PM}}$|⁠; let |$\chi$| be a formula, let |$\mathtt{T}=(\tau _1,\ldots ,\tau _{\ell })$| be a family of thresholds such that |$\tau _0\geqslant \tau _1\geqslant \ldots \geqslant \tau _{\ell }=0$|⁠. The gradual|$\chi$|-revision operation yields the |${\mathscr{PM}}$| model |${{{M}}^{{\sharp }^{{\chi }}_{{\mathtt{T}}}}} = {\langle{W, \mathrel{{\leq }^{{\sharp }^{\chi }_{\mathtt{T}}}}, V} \rangle }$|⁠, with the new plausibility ordering given by $$\begin{equation*}{{\mathrel{{{\leq}}^{{\sharp}^{{\chi}}_{{\mathtt{T}}}}}}}:= {{\mathrel{{{\leq}}^{\operatorname{R}^{{k}}_{{\chi}}}}}}\end{equation*}$$ with |$k \geqslant 0$| given by $$\begin{equation*}k:= \left\{ \begin{array}{ll} 0 & {\text{{if}}\;} {d}(M, {{{M}}^{\operatorname{R}^{{\ell}}_{{\chi}}}}) \geqslant \tau_0 \\ j & {\text{{if}}\;} \tau_{j-1}> {d}(M, {{{M}}^{\operatorname{R}^{{\ell}}_{{\chi}}}}) \geqslant \tau_j \\\end{array} \right. .\end{equation*}$$ For the language, |${\mathcal{L}}$| is extended with a modality |${\mathop{\langle{\sharp }^{{\chi }}_{{\mathtt{T}}} \rangle }{{}}}$|⁠, for |$\chi$| a formula and |$\mathtt{T}$| a family of thresholds as described above. As for its semantic interpretation, $$\begin{equation*}(M, w) \Vdash{\mathop{\langle{\sharp}^{{\chi}}_{{\mathtt{T}}}\rangle}{{\varphi}}}\quad \textrm{iff}_{def} \quad ({{{M}}^{{\sharp}^{{\chi}}_{{\mathtt{T}}}}}, w) \Vdash \varphi\end{equation*}$$ In order to revert back to only having to choose one threshold, a simpler version of this would be to pick some |$\tau _0$|⁠, and have |$\tau _k:=\tau _0(1-\frac{k}{p})$| for |$0<k\leqslant \ell$|⁠. Note that the strongest revision must be performed when the computed distance is the smallest, hence the order of the thresholds. Not requiring |$\tau _{k-1}>\tau _k$| for any |$k$| allows for some further refining of which policies we wish to allow the agent to perform: if |$\tau _{k-1}=\tau _k$|⁠, the |$k$|th element of the revision family will never be performed. The choice between these two forms of gradual revision depends not only on the application in mind, but also on the chosen family of revision policies. On the one hand, when using |$k$|-improvement operators, one can argue for the second (⁠|${{\sharp }}$|⁠) for the reasons mentioned in Section 3: the measured distance makes more intuitive sense when using radical revision than when using improvement operators. On the other hand, if one were to use |$k$|-moderate revisions, the first (⁠|${{\natural }}$|⁠) might be better theoretically, to locate ‘gaps’ between layers of |$\chi$|-worlds. From a computational point of view, the second might be better, as it is more manageable to have to look at only one updated model before choosing which revision policy to go with, instead of the multiple models that need to be computed in the first option. Axiom systems. For the axiomatisation of the new modalities, the strategy is as with screened revision, focusing again on the cases that use the Kemeny distance. For the first gradual revision, the semantic interpretation of |${\mathop{\langle{\natural }^{{\chi }}_{{\tau }}\rangle }{{}}}$| simply applies a |$k$|-improvement for the appropriate |$k$|⁠. Now, recall that the formula |${\alpha ^{>{\tau }}_{R_{\chi }}}$| of the hybrid language |${\mathcal{L}_{i}}$| characterises the fact that the distance between a model and its updated version after |$\chi$|-revision using a revision policy |$R$| is strictly larger than a given threshold |$\tau$| (Proposition 3.4). Thus, $$\begin{equation*}\Vdash{\mathop{\langle{\natural}^{{\chi}}_{{\tau}}\rangle}{{\varphi}}} {\;\leftrightarrow\;} \big( ({\alpha^{>{\tau}}_{{R_{\chi}^0}}} \land{\mathop{\langle\operatorname{R}_{{\chi}}^{{0}}\rangle}{{\varphi}}}) \lor \bigvee_{k=0}^{\ell-1} (\lnot{\alpha^{>{\tau}}_{{R_{\chi}^k}}} \land{\alpha^{>{\tau}}_{{R_{\chi}^{k+1}}}} \land{\mathop{\langle\operatorname{R}_{{\chi}}^{{k}}\rangle}{{\varphi}}}) \lor (\lnot{\alpha^{>{\tau}}_{{R_{\chi}^{\ell}}}} \land{\mathop{\langle\operatorname{R}_{{\chi}}^{{\ell}}\rangle}{{\varphi}}}) \big).\end{equation*}$$ The formula simply states the three cases of the gradual revision operator: perform the weakest revision when doing so already goes above the threshold, or perform the one for the largest |$k$| (including the strongest in the family) when the threshold allows it. For the second gradual revision, an analogous strategy applies. This time, the required formula should not state that the distance between two models is strictly above the given threshold, but rather that it is equal or larger than. An appropriate small modification to |${\alpha ^{>{\tau }}_{R_{\chi }}}$| produces the formula |${\alpha ^{\geqslant{\tau }}_{R_{\chi }}}$|⁠, with which the following holds: $$\begin{equation*}\Vdash{\mathop{\langle{\sharp}^{{\chi}}_{{\mathtt{T}}}\rangle}{{\varphi}}} {\;\leftrightarrow\;} \big( ({\alpha^{\geqslant{\tau_0}}_{{R_{\chi}^{\ell}}}} \land{\mathop{\langle\operatorname{R}_{{\chi}}^{{0}}\rangle}{{\varphi}}}) \lor \bigvee_{k=1}^{\ell} ({\alpha^{\geqslant{\tau_k}}_{{R_{\chi}^{\ell}}}} \land \lnot{\alpha^{\geqslant{\tau_{k-1}}}_{{R_{\chi}^{\ell}}}} \land{\mathop{\langle\operatorname{R}_{{\chi}}^{{k}}\rangle}{{\varphi}}}) \big)\end{equation*}$$ where |${\alpha ^{\geqslant{\tau }}_{{R_{\chi }^{\ell }}}}$| denotes the formula expressing that |${d_{K}}(M, {{{M}}^{\operatorname{R}^{{\ell }}_{{\chi }}}}) \geqslant \tau$|⁠. 5 An Application: Trust The threshold |$\tau$|⁠, a measure of the agent’s willingness to change her mind, plays a large role in both screened and gradual revision. However, few words have been spent on an important question: how is this threshold chosen? After all, this openness to change may be influenced by different factors, such as the agent’s stubbornness or her trust in the source of information. This last section discusses briefly some of the ways this threshold can reflect a notion of trust on sources of information, by making it a function of factors, which may change the agent’s openness to received announcements. Multiple sources. A first natural extension of the presented framework is to make the threshold a function of the source of the incoming information, in the case where there are several such sources. In this way, the agent may be more willing to change her mind the more trustworthy she finds the announcer. More formally, if |$S$| is a set of possible sources of information, one can have a function |$T:S \to{\mathbb{R}}$|⁠, indicating the agent’s level of trust in each source. Then, when a source |$s \in S$| announces |$\chi$|⁠, the current model |$M$| becomes |${{{M}}^{{\veebar }_{{T(s)}}^{{\chi }}}}$| (in the case of screened revision, by using the chosen revision policy) or |${{{M}}^{{\natural }^{{\chi }}_{{T(s)}}}}$| (in the case of the first gradual revision). If the revision policy calls for several thresholds (as the second gradual revision), |$T$| can become a function returning, for each source, a list of thresholds. This |$T$| can indeed be thought of as a ‘trust function’: the higher the agent’s trust in a source of information is, the more information she will accept from that source, and hence the higher the corresponding threshold will be. Specifics about how such a function can be actually defined (the number of sources, and the agent’s ‘confidence’ in each one of them) will rely on the actual application in mind. Topical trust. Another factor that may influence the agent’s willingness to change is the topic of the announcement. This topical trust can play a role on its own, as with an agent being more willing to change her beliefs about the open hours of a bank office than to change her beliefs about the origin of the universe. But it can be also combined with other elements, such as the mentioned source of information: the willingness of an agent to accept what her philosophy professor says about philosophy or what her mathematics professor says about mathematics might be greater than her willingness to accept her philosophy professor’s take on mathematics or her mathematics professor’s views on philosophy. The expertise of the agent herself might also be part of the picture, as she might be more prone to listening to external opinions about mathematics than about her own speciality, painting. Following this, the trust function from above can be extended to take into account not only the source of the information, but also its topic. For specific ways of relating a formula to a topic, several different approaches exist, but they will not be discussed here.29 Instead, suppose there is a set |$\mathcal{T}$| of topics, and a function |$\sigma :{\mathcal{L}} \to \mathcal{T}$| returning each formula’s topic. The threshold function can be redefined as |$T_{\mathcal{T}}:S \times \mathcal{T} \to{\mathbb{N}}$|⁠; then, e.g. when |$s\in S$| announces |$\chi$|⁠, the first version of the gradual policy turns a model |$M$| into the model |${{{M}}^{{\natural }^{{\chi }}_{{T_{\mathcal{T}}(s,\sigma (\chi ))}}}}$|⁠. 6 Conclusions and Future Work This paper has introduced a semantic approach to non-prioritized belief revision, basing the decision about the revision on the internal structure of the model representing the agent’s epistemic state. For this, the basic idea has been the use of a notion of distance between models. The first proposal, semanticallybased screened revision, requires a fixed revision policy, and relies on the use of a threshold that represents the agent’s willingness to change her beliefs. When the new information arrives, the current model is compared with the one that the chosen revision would produce. If the distance between the models does not exceed the threshold, the revision will be performed; otherwise, no change will occur. The second proposal, semanticallybased gradual revision, changes the perspective, moving from the binary ‘yes’/‘no’ choice of whether to revise provided by the screened revision, to a more fine-grained decision of how strong of a revision will be performed. It does so by relying on a family of revision policies with gradually increasing strength, choosing the strongest one whose application does not go above the threshold. The paper has also shown how the threshold can be acted upon to represent trust, allowing our framework to be used with logics concerned with different takes on trust, such as topical trust or trust revision. Along the way, and while working with a hybrid language with modalities for epistemic possibility (⁠|${{\langle{\scriptstyle \sim} \rangle }{}}$|⁠) and both weak and strict plausibility (⁠|${{\langle{\scriptstyle \leq} \rangle }{}}$| and |${{\langle{\scriptstyle <}\rangle } {}}$|⁠), the paper has also (i) presented sound and complete (recursion-axioms-based) axiomatisations of three well-known revision policies, radical revision, conservative revision and improvement operators, (ii) introduced two families of increasingly strong revision policies, with the extreme cases of the first (⁠|$k$|-improvement operators) being the mentioned improvement operator and radical revision, and those of the second (⁠|$k$|-moderate revision) being the mentioned conservative operator and radical revision and (iii) characterised syntactically both proposed belief changes: semanticallybased screened and semanticallybased gradual revision. The framework presented in this manuscript suggests some interesting research lines. One direction is to work not with qualitative plausibility models, but rather with quantitative structures, as the ordinal functions of [48]. Such structures provide a finer notion of ‘how much’ a given formula is believed. This is because, given two worlds |$u$| and |$v$| such that no other world is between them in terms of plausibility, an ordinal function can tell us if |$v$| is much more plausible than |$u$|⁠, or if there is hardly any difference at all. It might be interesting to apply our revision methods to this framework, adapting the distances to take into account the ‘length’ of the arrows, and using the ordinals to account for layers, in particular for the different types of improvement that have been introduced. Another direction is to expand further on the applications. The suggested ones on logics of trust are a first direction in which further work could be done. But one can also use the methods presented here to deal with group dynamics. Indeed, one can move to a multi-agent setting, and then represent actual inter-agent communication, where agents may make announcements either individually or as a group. This brings to the table not only the idea of trust in agents, but also the idea of trust in groups of agents, which allows us to account for the actual way the information shared by one group is accepted by the rest. In particular, the choice of the primitive concept raises possibilities for the definition of the other. For example, if one takes trust in agents as the basic concept, trust in a group can be seen as a function of the trust in the individuals making it up: a skeptical agent’s trust in a group of agents could be the minimal trust she has in individuals of the group, whereas a more trusting agent could take the mean or even the maximum of these individual trusts. But if one takes trust in groups as the basic concept, what should the relation of trust among related groups be? Should trust in a group be always larger or equal than trust in any of it subgroups? Or are there situations in which the addition of an agent might ruin the group’s ‘reputation’? Finally, there are further dynamics that can be investigated, in particular those of trust revision. The implementation of mechanisms through which the threshold/trust function might change over time would allow for a more fine-grained approach to trust revision, whereas existing strategies commonly add and remove trust in a binary fashion. Footnotes 1 Of course, this has a direct influence on the technical mechanism through which beliefs change. 2 Think about, e.g. confirmation bias [41, 42] and other related psychological phenomena. 3 There are, of course, semantic approaches for belief revision. The remark is rather about the use of semantic tools for non-prioritized revision. 4 A plausibility model is equivalent to a system of spheres, and it can be seen as the relational counterpart of an ordinal function. 5 Other ‘subtle’ plausibility changes are also possible [50]. 6 Semantic notions of distance have been used for belief revision, as the text will discuss. 7 The domain’s finiteness simplifies the definitions of distance between relations that are crucial to this proposal. 8 In [5] the plausibility relation is also conversely well-founded, forbidding infinite strictly ascending |$\leq$|-chains. Here the domain is finite, so this requirement follows. 9 Alternative ‘nominal-less’ options for the axiomatisations will also be discussed. 10 The new plausibility order can be defined equivalently as |${{\mathrel{{{\leq }}^{{\Uparrow }_{{\chi }}}}}}:= \left ({\leq } \cap (W \times{{[\kern{-1.5pt}[}{\chi } {]\kern{-1.5pt}]} ^{{M}}})\right ) \cup \left ({\leq } \cap ({{[\kern{-1.5pt}[}{\lnot \chi } {]\kern{-1.5pt}]} ^{{M}}} \times W)\right ) \cup \left ({\sim } \cap ({{[\kern{-1.5pt}[}{\lnot \chi } {]\kern{-1.5pt}]} ^{{M}}} \times{{[\kern{-1.5pt}[}{\chi } {]\kern{-1.5pt}]} ^{{M}}})\right ).$| 11 This new plausibility order can be defined equivalently as |${\mathrel{{{\leq }}^{{\uparrow }_{{\chi }}}}}:= ({\leq } \cap \left ({\overline{{\operatorname{Mx}({[\kern{-1.5pt}[} \chi {]\kern{-1.5pt}]} ^{M})}}} \times W)\right ) \cup \left ({\sim } \cap (W \times{\operatorname{Mx}({{[\kern{-1.5pt}[} \chi {]\kern{-1.5pt}]} ^{M}})})\right ).$| 12 Different from, e.g. the public announcement modality in public announcement logic [23, 44]. 13 With |$\lambda ^{\chi }_0:= \chi \land{{[\scriptstyle <]}{\lnot \chi }}$|⁠. 14 More precisely, |${\operatorname{Mx}({\mathopen{{[\kern{-1.5pt}[} } \chi \mathopen{{]\kern{-1.5pt}]} ^{M}}})} = {\mathopen{{[\kern{-1.5pt}[} } {\lambda ^{\chi }_0} \mathopen{{]\kern{-1.5pt}]} ^{{M}}}}$| for any |${\mathscr{PM}}$||$M$|⁠. 15 See [24] for such a language and its axiomatisation, and [50] for a definition of the improvement operator in the setting. 16 The characterising formulas should be propositional in order to name the same world in the model that results from the revision (recall: the operations do not affect atomic valuation). If a characterising formula is modal, then it might become false in ‘its world’ after the revision takes place, or it might become true in other worlds. 17 Recall that, in |${\mathscr{PM}}$|s, the relation is a total preorder. 18 This is because, besides the leading conjunction and disjunction, the main operators of the formula are the global existential quantifiers |${{\langle \scriptstyle \sim \rangle }{}}$|⁠, and thus the formula is either globally true or else globally false. 19 Intuitively, two pointed models are |$n$|-bisimilar when they have the same modal structure up to modal depth |$n-1$|⁠; this implies that they satisfy the same modal formulas at least up to a degree |$n-1$|⁠. 20 [15] extends the proposal by investigating distances between multi-agent KD45 (i.e. serial, transitive and Euclidean) relational models. 21 Repetition creates familiarity, which leads in the majority of cases to an easier acceptance. It is used not only in some forms of propaganda, but also by several advertising techniques. 22 Formally, |${\bot _{{\varphi }}}:= \max{\{ {j \in \mathbb{N} \mid {[\kern{-1.5pt}[} \varphi {]\kern{-1.5pt}]} ^{M} \cap \operatorname{L}^{M}_{j} \neq \varnothing } \}}$| and |${\intercal _{{\varphi }}}:= \min{\{ {j \in \mathbb{N} \mid {[\kern{-1.5pt}[} \varphi {]\kern{-1.5pt}]} ^{M} \cap \operatorname{L}^{M}_{j} \neq \varnothing } \}}$|⁠. 23 With |$\lambda ^{\chi }_0:= \chi \land{{[\scriptstyle <]}{\lnot \chi }}$|⁠, as defined earlier (Footnote 13). 24 For |$\lambda ^{\chi }_{j+1}:= \chi \land{{\langle \scriptstyle <\rangle } {\lambda ^{\chi }_j}} \land \lnot{{\langle \scriptstyle <\rangle } {{\langle \scriptstyle <\rangle } \lambda ^{\chi }_j}}$| and |$m:= {\vert{{[\kern{-1.5pt}[} \chi {]\kern{-1.5pt}]} ^{M}} \vert }$|⁠. Note how |${\mathopen{{[\kern{-1.5pt}[} } {\lambda ^{\chi }_j} \mathopen{{]\kern{-1.5pt}]} ^{{M}}}} = \varnothing$| for all |$j \in{\mathbb{N}}$| that go beyond the number of layers the set of |$\chi$|-worlds define. Thus, in such cases, adding |$\lambda ^{\chi }_j$| as disjuncts does not affect the resulting set |${\mathopen{{[\kern{-1.5pt}[} } { \lambda ^{\chi }_0 \lor \cdots \lor \lambda ^{\chi }_{m}} \mathopen{{]\kern{-1.5pt}]} ^{{M}}}}$|⁠. 25 That is, for |$U \subseteq{\mathcal{D}_{{M}}}$|⁠, define |$U_0:= {\operatorname{Mx}({U})}$| and |$U_{j+1}:= {\operatorname{Mx}({U \setminus \bigcup _{k=0}^{j} U_{k}})}$|⁠. 26 That is, |${\operatorname{{LastLay}}}(U):= \min{\left \{ {j \in \mathbb{N} \mid U_{j+1} = \varnothing } \right \}}$|⁠. 27 The family might be that of |$k$|-improvement operators, that of |$k$|-moderate revisions or any other suitable defined. In fact, one might not want to use all of the operators within a given ‘standard’ family, taking e.g. and so on. One could also want to work with a ‘standard’ family, and yet avoid its strongest member, or take the minimal revision to be some other than the one for |$k=0$|⁠. In a multi-agent setting, the chosen family might vary from agent to agent. 28 Whether the weakest revision has an actual effect on the agent’s epistemic state depends, of course, of which is this weakest revision policy. In some cases, one might want for the weakest revision to be no revision at all, and in some other cases, one might want to guarantee that the weakest revision would still affect the agent’s epistemic state. For this ‘weakest revision possible’, an alternative not covered by the families defined in this section is the operator presented in [43] (called refinement in [47]), which splits each layer of the original plausibility ordering, placing the |$\chi$|-worlds above the |$\lnot \chi$|-ones. 29 A simple idea is to simply look at the atoms the formula involves, thus understanding a topic as a subset of atomic propositions; a formal generalization of this idea has been presented in [10] (cf. [9, 31]). The interested reader can find more general discussions on what the topic/content of a formula is, or on what a formula is about, in [21, 53]. References [1] C.E. Alchourrón , D. Makinson . Hierarchies of regulation and their logic . In New Studies in Deontic Logic, Synthese Library , vol. 152, R. Hilpinen ed. , pp. 125 – 148 . Reidel Publishing Company , Dordrecht, The Netherlands , 1981 . doi:10.1007/978-94-009-8484-4. 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TI - A Semantic Approach to Non-prioritized Belief Revision
JF - Logic Journal of the IGPL
DO - 10.1093/jigpal/jzz045
DA - 2020-08-01
UR - https://www.deepdyve.com/lp/oxford-university-press/a-semantic-approach-to-non-prioritized-belief-revision-WrM0kdlhdH
DP - DeepDyve
ER -