Andreas Herzig

We study the extension of public announcement logic PAL by public assignments, which we call PALA. Just as in the case of PAL, the standard procedure for deciding PALA validity, i.e. the use of so-called reduction axioms to translate PALA formulae into formulae in epistemic logic EL, may lead to exponential growth. In this paper, we show that such a price is not mandatory, for we provide a polynomial translation of PALA into EL. This is based on abbreviations of subformulae by new propositional letters. Such optimal translation also enables us to show the computational complexity of the problem of deciding PALA validity, which turns out to be coNP-complete in the single-agent case and PSPACE-complete in the multiagent case.

In this paper we investigate a logic for modelling individual and collective acceptances that is called acceptance logic. The logic has formulae of the form $A_{Gx} \phi $ reading 'if the agents in the set of agents G identify themselves with institution x then they together accept that φ'. We extend acceptance logic by two kinds of dynamic modal operators. The first kind are public announcements of the form x!ψ, meaning that the agents learn that ψ is the case in context x. Formulae of the form [x!ψ]φ mean that φ is the case after every possible occurrence of the event x!ψ. Semantically, public announcements diminish the space of possible worlds accepted by agents and sets of agents. The announcement of ψ in context x makes all ¬ψ-Worlds inaccessible to the agents in such context. In this logic, if the set of accessible worlds of G in context x is empty, then the agents in G are not functioning as members of x, they do not identify themselves with x. In such a situation the agents in G may have the possibility to join x. To model this we introduce here a second kind of dynamic modal operator of acceptance shifting of the form G:x↑ψ. The latter means that the agents in G shift (change) their acceptances in order to accept ψ in context x. Semantically, they make ψ-worlds accessible to G in the context x, which means that, after such operation, G is functioning as member of x (unless there are no ψ-worlds). We show that the resulting logic has a complete axiomatization in terms of reduction axioms for both dynamic operators. In the paper we also show how the logic of acceptance and its dynamic extension can be used to model some interesting aspects of judgement aggregation. In particular, we apply our logic of acceptance to a classical scenario in judgment aggregation, the so-called 'doctrinal paradox' or 'discursive dilemma' (Pettit, Philosophical Issues 11: 268-299, 2001; Kornhauser and Sager, Yale Law Journal 96: 82-117, 1986)