A Representation of Explicit Knowledge and Epistemic Indistinguishability in a Logic of Awareness

The logic of awareness, first proposed by Fagin and Halpern, addressed the problem of logical omniscience by introducing the notion of awareness and distinguishing explicit knowledge from implicit knowledge. In their framework, explicit knowledge was defined as the conjunction of implicit knowledge and awareness, each of which was represented by modal operators. Their definition, however, may derive undesirable propositions that cannot be considered explicit knowledge when Modus Ponens is applied within implicit knowledge. Hence, focusing on indistinguishability among possible worlds, dependent on awareness, we refine the definition of explicit knowledge. In our semantics, we require that the aware implicit knowledge is not necessarily explicit knowledge, though explicit knowledge must be aware as well as implicit. We employ an example of elementary geometry, where different students may or may not reach the final answer, depending on whether they are aware of learned mathematical facts. Thereafter, we formally present the syntax and the semantics of our language, named Awareness-Based Indistinguishability Logic ($\mathrm{AIL}$). We prove that $\mathrm{AIL}$ has more expressive power than the logic of Fagin and Halpern, and show that the latter is embeddable in $\mathrm{AIL}$. Furthermore, we provide an axiomatic system of $\mathrm{AIL}$ and prove its soundness and completeness.