Proof theory for the logics of bringing-it-about: Ability, coalitions and means-end relationship

Abstract

The logic of bringing-it-about (BIAT) aims to capture a notion of agency in which actions are analysed in terms of their results: ‘An agent does something’ means that the agent brings it about that something takes place. Our starting point is the basic BIAT logic as introduced by Elgesem in the ‘90s: this logic contains only a modal operator to express BIAT statements by single agents. Several extensions have been proposed by Elgesem himself and others, notably with the capability operator, coalitions of agents and means-end BIAT statements (i.e. of the form ‘the agent does B by doing A’). We first propose a variant of the neighbourhood semantics, called bi-neighbourhood semantics, for the basic BIAT logic and the mentioned extensions, in which a world is equipped by a set of pairs or neighbourhoods. Differently from the semantics defined in the literature, this reformulation is well suited for countermodel construction. We then introduce modular hypersequent calculi for all logics considered in this work. Our calculi enjoy the fundamental property of cut admissibility, from which it follows their completeness with respect to the axiomatization. Moreover, our calculi provide at the same time a decision procedure, as well as the first practical countermodel extraction procedure: from a single failed proof it is possible to build directly a finite countermodel of the formula under verification in the bi-neighbourhood semantics. By this last result, we obtain constructive proofs of the semantic completeness of the calculi and consequently of the finite model property for all logics.