Towards Metalogical Systematisation of Deontic Action Logics Based on Boolean Algebra

Trypuz, Robert; Kulicki, Piotr

doi:10.1007/978-3-642-14183-6_11

Cited by 12 publications

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“…However, as stated in [31], this axiom induces some problems. Let us, for example, consider two actions smoke and driving.…”

Section: Segerberg's Deontic Logicmentioning

confidence: 99%

“…A problematic issue with the first version of obligation (as already noted in [31]) is that strict refinements of forbidden actions are forbidden and obligatory at the same time, that is:…”

Section: Segerberg's Deontic Logicmentioning

confidence: 99%

“…We distinguish between two kinds of logics; first, those logics that interpret deontic operators as sets of events/outcomes that fulfill these operators, among these logics we can cite those of Castro and Maibaum [7], R. van der Meyden [22] and Fiadeiro and Maibaum [9] as well as the work of Trypuz and Kulicki [31] enriching Segerberg's logic to obtain a more appealing version of obligation.…”

Section: Contemporary Deontic Action Logics and Boolean Algebramentioning

confidence: 99%

“…Segerberg used boolean algebra to give the semantics of deontic operators; in [7,31] a variation of this approach is taken: the set of action letters is considered finite and therefore the underlying algebra of actions becomes atomic. Atomic boolean algebras have some good properties, from the topological point of view, the atoms allow us to refer to the points of the underlying space: there is a one-to-one mapping between the set of atoms of a boolean algebra and the set of its maximal ideals (or ultrafilters); the maximal ideals (or ultrafilters) can be thought of as points of the field of sets which is isomorphic to the boolean algebra (by the Stone theorem).…”

Section: Deontic Logics Based On Atomic Boolean Algebrasmentioning

confidence: 99%

“…We may use the atoms to introduce some further axioms. In the following we analyze the possible extensions of BOD, we follow the ideas of [31] to classify the systems.…”

Section: Deontic Logics Based On Atomic Boolean Algebrasmentioning

confidence: 99%

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Deontic Logics Based on Boolean Algebra

Castro

Kulicki

2013

Outstanding Contributions to Logic

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Deontic logic is devoted to the study of logical properties of normative predicates such as permission, obligation and prohibition. Since it is usual to apply these predicates to actions, many deontic logicians have proposed formalisms where actions and action combinators are present. Some standard action combinators are action conjunction, choice between actions and not doing a given action. These combinators resemble boolean operators, and therefore the theory of boolean algebra offers a well-known mathematical framework to study the properties of the classic deontic operators when applied to actions. In his seminal work, Segerberg uses constructions coming from boolean algebras to formalize the usual deontic notions. Segerberg's work provided the initial step to understand logical properties of deontic operators when they are applied to actions. In the last years, other authors have proposed related logics. In this chapter we introduce Segerberg's work, study related formalisms and investigate further challenges in this area. IntroductionThe so-called boolean operators (or, and, not) are commonly used in ordinary language as basic connectors in phrases to put together propositions, subjects and verbs. George Boole in his famous text An Investigation of the Laws of Thought [5] was one of the first mathematicians (if not the first) to study the mathematical properties of these connectors, his work is considered a cornerstone of modern logic, and can be thought of as capturing some universal laws of logic. One of the main contributions [15,16]) have been used to study the mathematical properties of logics by means of algebras. A boolean algebra is made up of a non-empty set of elements, binary operators +, ×, the unary operator − and two distinguished constants 0 and 1. Several (complete) axiomatizations of boolean algebras have been proposed in the literature; the following axiomatization comes from [12].• −0 = 1 and 0 = −1 (Zero and One laws).• x × 0 = 0 and x + 1 = 1 (Absorption of zero and one laws).This set of axioms is not the smallest one possible, but it exposes the standard properties of boolean algebras. It is straightforward to see that these properties are true for set intersection, set union and set complement in any field of sets. One may think of logical propositions such as it is raining or the wall is white as elements of a boolean algebra; and therefore the boolean operators allow us to construct more complicate statements, such as: it is raining or it is sunny; the wall is not white; it is raining and the wall is white. As a consequence, propositional logic can be seen as a boolean algebra, the formal technique to connect both worlds is called Lindenbaum-Tarski algebra, which is a boolean algebra made up of equivalence classes of sentences and the corresponding operations [29]. Two useful concepts that we will use through this chapter are those of ideal and filter; an ideal I of a boolean algebra B is a non-empty set I ⊆ B satisfying the following conditions:1. If x ∈ I and y ∈ I, ...

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“…However, as stated in [31], this axiom induces some problems. Let us, for example, consider two actions smoke and driving.…”

Section: Segerberg's Deontic Logicmentioning

confidence: 99%

“…A problematic issue with the first version of obligation (as already noted in [31]) is that strict refinements of forbidden actions are forbidden and obligatory at the same time, that is:…”

Section: Segerberg's Deontic Logicmentioning

confidence: 99%

Section: Contemporary Deontic Action Logics and Boolean Algebramentioning

confidence: 99%

Section: Deontic Logics Based On Atomic Boolean Algebrasmentioning

confidence: 99%

“…We may use the atoms to introduce some further axioms. In the following we analyze the possible extensions of BOD, we follow the ideas of [31] to classify the systems.…”

Section: Deontic Logics Based On Atomic Boolean Algebrasmentioning

confidence: 99%

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