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

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