Pointwise intersection in neighbourhood modal logic

We study the logic of neighbourhood models with pointwise intersection, as a means to characterize multi-modal logics. Pointwise intersection takes us from a set of neighbourhood sets $\mathcal{N}_i$ (one for each member $i$ of a set $G$, used to interpret the modality $\square_i$) to a new neighbourhood set $\mathcal{N}_G$, which in turn allows us to interpret the operator $\square_G$. Here, $X$ is in the neighbourhood for $G$ if and only if $X$ equals the intersection of some $\mathcal{Y} = \{Y_i \mid i\in G\}$. We show that the notion of pointwise intersection has various applications in epistemic and doxastic logic, deontic logic, coalition logic, and evidence logic. We then establish sound and strongly complete axiomatizations for the weakest logic characterized by pointwise intersection and for a number of variants, using a new and generally applicable technique for canonical model construction.Comment: Submitted to Advances in Modal Logic 201

Free choice permission in STIT

We argue for a new approach to free choice permission in the context of a-temporal STIT logic. According to our analysis, an agent has a free choice permission w.r.t. two propositions phi and psi iff (a) the agent is permitted to see to phi Lambda inverted left perpendicular psi and (b) the agent is permitted to see to psi Lambda inverted left perpendicular phi The primitive notion of permission we use is the dual of one of Horty's operators for "ought to do" from (Horty, 2001). We argue that the approach improves on existing proposals in various ways

Adaptive logic characterizations of input/output logic

We translate unconstrained and constrained input/output logics as introduced by Makinson and van der Torre to modal logics, using adaptive logics for the constrained case. The resulting reformulation has some additional benefits. First, we obtain a proof-theoretic (dynamic) characterization of input/output logics. Second, we demonstrate that our framework naturally gives rise to useful variants and allows to express important notions that go beyond the expressive means of input/output logics, such as violations and sanctions

Pooling Modalities and Pointwise Intersection : Axiomatization and Decidability

We establish completeness and the finite model property for logics featuring the pooling modalities that were introduced in Van De Putte and Klein (Pooling modalities and pointwise intersection: semantics, expressivity, and applications). The definition of our canonical models combines standard techniques with a so-called "puzzle piece construction", which we first illustrate informally. After that, we apply it to the weakest classical logics with pooling modalities and investigate the technique's potential for the axiomatization of stronger logics, obtained by imposing well-known frame conditions on the models

Adaptive logics: a parametric approach

Adaptive logics (ALs) in standard format are defined in terms of a monotonic core logic L, a distinct set of 'abnormal' formulas Omega and a strategy, which can be either reliability or minimal abnormality. In this article we we ask under which conditions the consequence relation of two ALs that use the same strategy are identical, and when one is a proper subrelation of the other. This results in a number of sufficient (and sometimes necessary) conditions on L and Omega which apply to all ALs in standard format. In addition, we translate our results to the closely related family of default assumption consequence relations

Splitting and relevance : broadening the scope of Parikh's concepts

When our current beliefs face a certain problem e.g. when we receive new information contradicting them, then we should not remove beliefs that are not related to this problem. This principle is known as "minimal mutilation" or "conservativity" [21]. To make it formally precise, Rohit Parikh [32] defined a Relevance axiom for (classical) theory revision, which is based on the notion of a language splitting. I show that both concepts can and should be applied in a much broader context than mere revision of theories in the traditional sense. First, I generalize their application to belief change in general, and strengthen the axiom of relevance in order to make it fully syntax-independent. This is done by making use of the least letter-set representation of a set of formulas [27]. Second, I show that the logic underlying both concepts need not be classical logic and establish weak sufficient conditions for both the finest splitting theorem from [25] and the least letter-set theorem from [27]. Both generalizations are illustrated by means of the paraconsistent logic CLuNs and compared to ideas from [14, 36, 24]

Expressivity results for deontic logics of collective agency

We use a deontic logic of collective agency to study reducibility questions about collective agency and collective obligations. The logic that is at the basis of our study is a multi-modal logic in the tradition of *stit* ('sees to it that') logics of agency. Our full formal language has constants for collective and individual deontic admissibility, modalities for collective and individual agency, and modalities for collective and individual obligations. We classify its twenty-seven sublanguages in terms of their expressive power. This classification enables us to investigate reducibility relations between collective deontic admissibility, collective agency, and collective obligations, on the one hand, and individual deontic admissibility, individual agency, and individual obligations, on the other