Modeling $$\mathscr {C}^{0}$$ C 0 Family Logics for Artificial Intelligence: Doxastic-Temporal Logics for Reasoning About Goals

Abstract

We introduce the $$\mathscr {C}^{0}$$ C 0 family of logics, which include temporalized modal operators for belief and hyperintensional modal operators for obligations and goals. We motivate the $$\mathscr {C}^{0}$$ C 0 family as extended doxastic fragments of the $$\mathcal {DCEC}$$ DCEC family of logics, which are cognitive calculi designed for theory-of-mind reasoning among multiple artificial agents. In the literature, $$\mathcal {DCEC}$$ DCEC family logics are defined exclusively using proof-theoretic semantics. In this work we provide a model theory for the $$\mathscr {C}^{0}$$ C 0 family of logics which constitutes the first steps towards providing a model theory for the $$\mathcal {DCEC}$$ DCEC cognitive calculi family as a whole. We investigate the fragment relationships between both the $$\mathscr {C}^{0}$$ C 0 family and the $$\mathcal {DCEC}$$ DCEC family, produce a model theory for the $$\mathscr {C}^{0}$$ C 0 family and prove important results establishing completeness for all $$\mathscr {C}^{0}$$ C 0 family logics and establish soundness for $$\mathscr {C}^{0}$$ C 0 fragments without time.