Indexed Semantics and Its Application in Modelling Interactive Unawareness
Yanjing Wang
Abstract:
This thesis aims to propose an alternative Kripke-like semantics for multi-agent doxastic/epistemic logics. The approach is based on the intuition that when thinking about others' information states, we actually have some ``imaginary agents" in our mind, who may differ from the actual persons in the possibility relations they have. By including extra possibility relations for each imaginary agent, we can explicitly represent such imaginations of agents about each other in the so called Indexed Models. Accordingly, the Indexed Semantics we developed interprets nested-modality formulas context-dependently to capture the intuition about the imaginary agents. In this indexed semantics, a modal operator has different explicit meanings under different scopes of other operators.
By adopting this semantics, the modelling of multi-agent information states becomes much easier. We can have a desired non-redundant indexed model for many situations, while it wouldn't be possible for the classic semantics. By taking the advantage of such non-redundant indexed model, we give an intuitive and succinct way for modelling interactive unawareness under some constraints. Due to the context-dependent feature of our semantics, the rule of generalization and uniform substitution no longer preserve validity on arbitrary class of indexed frames. This also gives us the freedom to have more subtle axioms in the logic. Some complete logic systems are provided for different purposes.