Abstract
Public announcement logic is a logic that studies epistemic updates. In this paper, we propose a sound and complete labelled natural deduction system for public announcement logic with the common knowledge operator (PAC). The completeness of the proposed system is proved indirectly through a Hilbert calculus for PAC known to be complete and sound. We conclude with several discussions regarding the system including some problems of the system in attaining normalisation and subformula property.
Highlights
An agent’s knowledge over a proposition may be updated when new information is given.Public announcement logic (PAL) is a logic that formalises the notion of this epistemic updates.The common knowledge operator on the other hand is an operator that attempts to formalise the notion of common or mutual knowledge among a group of agents
All known proof systems for the logic with the interaction of public announcement and common knowledge are formulated as Hilbert calculi for example the Hilbert calculus for public announcement logic with common knowledge (PAC) of [6]
We propose a labelled natural deduction for public announcement logic with common knowledge (NPAC)
Summary
We write Γ A to mean that there is a derivation Π of a labelled or relational formula A in which all undischarged assumptions belong to the set of labelled or relational formula Γ.
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