Non-Monotonic S4F Standpoint Logic (Extended Version with Proofs)

TL;DR

This work introduces S4F Standpoint Logic, a general framework that combines non-monotonic modal logic with multi-viewpoint standpoints to represent heterogeneous knowledge. It develops both monotonic and non-monotonic semantics, including expansions and minimal-model semantics, and provides a rigorous complexity analysis showing that adding standpoints does not increase the computational cost relative to the base logics. A syntactic partitioning approach characterises minimal models, enabling soundness and completeness results and enabling efficient encodings via disjunctive ASP. The paper also offers a Disjunctive ASP encoding and discusses the integration of standpoint defaults, answer set programming, and argumentation frameworks, with a clear path for future work on fragments, strong equivalence, and proof systems. Overall, the results establish a unified, scalable foundation for multi-viewpoint non-monotonic reasoning with practical encoding avenues.

Abstract

Standpoint logics offer unified modal logic-based formalisms for representing multiple heterogeneous viewpoints. At the same time, many non-monotonic reasoning frameworks can be naturally captured using modal logics, in particular using the modal logic S4F. In this work, we propose a novel formalism called S4F Standpoint Logic, which generalises both S4F and standpoint propositional logic and is therefore capable of expressing multi-viewpoint, non-monotonic semantic commitments. We define its syntax and semantics and analyze its computational complexity, obtaining the result that S4F Standpoint Logic is not computationally harder than its constituent logics, whether in monotonic or non-monotonic form. We also outline mechanisms for credulous and sceptical acceptance and illustrate the framework with an example.

Figures (1)

• Figure 1: An S4F standpoint structure $\mathfrak{F}_{\textsc{o}}=\left(\Pi_{\textsc{o}},\sigma_{\textsc{o}},\tau_{\textsc{o}},\gamma_{\textsc{o}}\right)$ that is a model of theory $D^{\textsc{o}}_2$ from \ref{['example']}. Precisifications $\Pi_{\textsc{o}}=\left\{\pi_1,\ldots,\pi_8\right\}$ within a box belong to the outer (upper) or inner (lower) set of precisifications of the standpoint labelling the box, e.g. $\pi_8\in\sigma_{\textsc{o}}(\mathsf{D1})\subseteq\sigma_{\textsc{o}}(\mathsf{D2})$. (Satisfaction $\mathfrak{F}_{\textsc{o}}\Vdash\mathsf{D1}\preccurlyeq\mathsf{D2}$ is coincidental and not required by $D^{\textsc{o}}_2$.) Precisification $\pi$'s valuation is shown as a set $\gamma_{\textsc{o}}(\pi)$ of atoms below $\pi$. Atoms are abbreviated thus: $\mathtt{OvuDis}$ ($\mathtt{O}$), $\mathtt{Preg}$ ($\mathtt{Pr}$), $\mathtt{Horm}$ ($\mathtt{H}$), $\mathtt{PCOS}$ ($\mathtt{P}$), and $\mathtt{FHA}$ ($\mathtt{F}$). For example, $\mathfrak{F}_{\textsc{o}}\Vdash\mathord{\mathop{\Box}\nolimits_{\mathsf{\ast}}}(\mathtt{Pr}\land\mathtt{O})$ (as $\mathfrak{F}_{\textsc{o}},\pi\Vdash\mathtt{Pr}\land\mathtt{O}$ for all $\pi\in\Pi_{\textsc{o}}$) and $\mathfrak{F}_{\textsc{o}},\pi_5\Vdash\mathord{\mathop{\Box}\nolimits_{\mathsf{\mathsf{M}}}}\mathtt{H}$ (since $\mathfrak{F}_{\textsc{o}},\pi\Vdash\mathtt{H}$ for all $\pi\in\sigma_{\textsc{o}}(\mathsf{M})$), while $\mathfrak{F}_{\textsc{o}},\pi_1\not\Vdash\mathord{\mathop{\Box}\nolimits_{\mathsf{\mathsf{M}}}}\mathtt{H}$ (as $\mathfrak{F}_{\textsc{o}},\pi_3\not\Vdash\mathtt{H}$ with $\pi_3\in\tau_{\textsc{o}}(\mathsf{M})$), whence $\mathfrak{F}_{\textsc{o}},\pi_1\Vdash\mathord{\mathop{\Diamond}\nolimits_{\mathsf{\mathsf{M}}}}\neg\mathtt{H}$. Intuitively, from the medical standpoint $\mathsf{M}$, $\neg\mathtt{H}$ is conceivable at $\pi_1$, whereas $\mathtt{H}$ is unequivocal at $\pi_5$.

Theorems & Definitions (41)

• Example 1
• Definition 2
• Definition 3
• Definition 4
• Definition 5
• Lemma 6
• Proof 1
• Definition 7
• Definition 8
• Proposition 9