Non-Monotonic S4F Standpoint Logic (Extended Version with Proofs)
📝 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.
💡 Analysis
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.
📄 Content
Standpoint logic is a modal logic-based formalism for representing multiple diverse (and potentially conflicting) viewpoints within a single framework. Its main appeal derives from its conceptual simplicity and its attractive properties:
In the presence of conflicting information, standpoint logic sacrifices neither consistency nor logical conclusions about the shared understanding of common vocabulary (Gómez Álvarez and Rudolph 2021). The underlying idea is to start from a base logic (originally propositional logic; Gómez Álvarez and Rudolph 2021) and to enhance it with two modalities pertaining to what holds according to certain standpoints. There, a standpoint is a specific point of view that an agent or other entity can take, and that has a bearing on how the entity understands and employs a given logical vocabulary (that may at the same time be used by other entities with a potentially different understanding). The two modalities are, respectively: □sϕ, expressing “it is unequivocal [from the point of view s] that ϕ”; and its dual ♢ s ϕ, where “it is conceivable [from the point of view s] that ϕ”.
Standpoint logic escapes global inconsistency by keeping conflicting pieces of knowledge separate, yet avoids duplication of vocabulary and in this way conveniently keeps portions of common understanding readily available. It has historic roots within the philosophical theory of supervaluationism (Bennett 2011), which explains semantic variability “by the fact that natural language can be interpreted in many different yet equally acceptable ways, commonly referred to as precisifications” (Gómez Álvarez and Rudolph 2021).
In our work, such semantic commitments can be made on the basis of incomplete knowledge using a form of default reasoning. Consequently, in our logic each standpoint embodies a consistent (but possibly partial) point of view, potentially using non-monotonic reasoning (NMR) to arrive there. This entails that the overall formalism becomes nonmonotonic with respect to its logical conclusions.
Several non-monotonic formalisms that could be employed for default reasoning within standpoints come to mind, and obvious criteria for selection among the candidates are not immediate. We choose to employ the nonmonotonic modal logic S4F (Segerberg 1971;Schwarz and Truszczyński 1994), which is a very general formalism that subsumes several other NMR languages, decidedly allowing the possibility for later specialisation via restricting to proper fragments. In this way, we obtain standpoint versions of default logic (Reiter 1980) (see also Theorem 1 below), answer set programming (Gelfond and Lifschitz 1991), and abstract argumentation (Dung 1995), all as corollaries of our general approach. The usefulness of non-monotonic S4F for knowledge representation and especially non-monotonic reasoning has been aptly demonstrated by Schwarz and Truszczyński (1994) (among others), but seems to be underappreciated in the literature to this day.
We illustrate the logic we propose by showcasing a worked example in standpoint default logic, a standpoint variant of Reiter’s default logic (1980), where defaults and definite knowledge can be annotated with standpoint modalities. In Theorem 1, defaults are of the standard form, namely ϕ : ψ 1 , . . . , ψ n /ξ where (as usual) if the prerequisite ϕ is established and there is no evidence to the contrary of the justifications ψ 1 , . . . , ψ n , then the consequence ξ is concluded. An extension is a deductively closed set of formulas representing one possible belief set derived by maximally applying the defaults. A sentence follows credulously if some extension entails it, and sceptically if all extensions do. Example 1. Ovulation disorders are among the leading causes of female infertility. Their origins and diagnostics vary, and medical communities do not agree on a unified diagnosis or treatment. For example, community D1 typically attributes ovulation disorders to polycystic ovary syndrome arXiv:2511.10449v2 [cs.AI] 16 Nov 2025 (PCOS), while community D2 often sees functional hypothalamic amenorrhea (FHA) as their main source (unless the patient is pregnant, Preg). The initial treatment for PCOS, as generally accepted in the overall medical community M -including its subcommunities D1 and D2 -involves hormone therapy (Horm); however, this should be avoided in FHA, as it may be ineffective and could mask the underlying issue. 1This can be formalised as a standpoint default theory: 1 yields □D1(PCOS∧Horm) and □D2(FHA∧¬Horm), so these conclusions follow sceptically. The patient or physician may choose a treatment based on the reputation and trust attributed to the community from which the conclusion derives. If it is later learned that the patient is in fact pregnant -D O 2 := D O 1 ∪ {□ * Preg} -then, □D2(FHA ∧ ¬Horm) is withdrawn, whereas □D1(PCOS ∧ Horm) remains.
Compare this with related logics: A plain default theory is obtained by dropping the standpoint modalities
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