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

James T. Oswald, Brandon Rozek, Thomas Macaulay Ferguson

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