[Paper Review] Possibility Semantics

Wesley H. Holliday|arXiv (Cornell University)|May 10, 2024

Semantic Web and Ontologies4 citations

TL;DR

This paper introduces possibility semantics as a generalization of classical and modal logic semantics, replacing traditional 'possible worlds' with partial 'possibilities' represented as regular open sets in posets or UV-spaces. It establishes a duality-based framework that avoids reliance on the Axiom of Choice, supports non-prime and non-total possibilities, and provides a constructive, richer semantics for classical, modal, and intuitionistic logics, overcoming limitations in completeness and nonconstructivity of standard approaches.

ABSTRACT

In traditional semantics for classical logic and its extensions, such as modal logic, propositions are interpreted as subsets of a set, as in discrete duality, or as clopen sets of a Stone space, as in topological duality. A point in such a set can be viewed as a "possible world," with the key property of a world being primeness--a world makes a disjunction true only if it makes one of the disjuncts true--which classically implies totality--for each proposition, a world either makes the proposition true or makes its negation true. This chapter surveys a more general approach to logical semantics, known as possibility semantics, which replaces possible worlds with possibly partial "possibilities." In classical possibility semantics, propositions are interpreted as regular open sets of a poset, as in set-theoretic forcing, or as compact regular open sets of an upper Vietoris space, as in the recent theory of "choice-free Stone duality." The elements of these sets, viewed as possibilities, may be partial in the sense of making a disjunction true without settling which disjunct is true. We explain how possibilities may be used in semantics for classical logic and modal logics and generalized to semantics for intuitionistic logics. The goals are to overcome or deepen incompleteness results for traditional semantics, to avoid the nonconstructivity of traditional semantics, and to provide richer structures for the interpretation of new languages.

Motivation & Objective

• To overcome the nonconstructive reliance on the Axiom of Choice in traditional possible world semantics, particularly in the construction of prime filters and atoms.
• To generalize classical and modal logic semantics by replacing total, prime possible worlds with partial, refinable 'possibilities' that may not settle disjunctions or truth values.
• To provide a unified, duality-based framework for logical semantics using regular open sets in posets and upper Vietoris spaces, enabling richer structures for new logical languages.
• To extend the approach to intuitionistic and inquisitive logics, offering a semantics that supports non-distributive and non-classical reasoning.
• To establish completeness results for first-order and modal logics under this new semantics, particularly in contexts where standard semantics fail (e.g., provability logic).

Proposed method

• Represent propositions as regular open sets in a poset of possibilities, where the order relation $\sqsubseteq$ encodes refinement of information.
• Define truth of logical connectives using refinement-based semantics: a possibility $x$ satisfies $\neg\varphi$ iff no refinement of $x$ satisfies $\varphi$; $x$ satisfies $\varphi \vee \psi$ iff every refinement of $x$ has a further refinement satisfying one of $\varphi$ or $\psi$.
• Use duality between complete Boolean algebras and upper Vietoris spaces (UV-spaces) to model compact regular open sets as propositions, enabling a choice-free construction.
• Introduce accessibility games $\mathrm{G}(S,\sqsubseteq,R,x,y)$ to characterize modal accessibility relations, with winning conditions based on compatibility or refinement of possibilities.
• Define strong possibility frames via the $\boldsymbol{R\Leftrightarrow\text{win}}$ condition, which ensures that $xRy$ holds iff player E has a winning strategy in the game $\underline{\mathrm{G}}(S,\sqsubseteq,R,x,y)$.
• Establish dual equivalence between full and principal frames and complete $\mathcal{V}$-BAOs, linking possibility semantics to canonical extensions and MacNeille completions.

Experimental results

Research questions

• RQ1 Can a semantics for classical logic be developed that avoids the Axiom of Choice by replacing prime filters and atoms with partial possibilities?
• RQ2 How can disjunction and negation be reinterpreted in a semantics where possibilities are not required to be total or prime?
• RQ3 What is the role of refinement order $\sqsubseteq$ in defining truth conditions for logical connectives in a non-classical, partial setting?
• RQ4 Can possibility semantics provide completeness results for first-order and modal logics where standard Kripke semantics fails, particularly in provability logic?
• RQ5 How does the game-theoretic accessibility condition $\boldsymbol{R\Leftrightarrow\text{win}}$ characterize valid modal accessibility relations in possibility frames?

Key findings

• Possibility semantics replaces classical possible worlds with partial possibilities in posets or UV-spaces, allowing propositions to be true without settling disjuncts or truth values, thus avoiding primeness and totality.
• The $\boldsymbol{R\Leftrightarrow\text{win}}$ condition characterizes strong possibility frames, ensuring that modal accessibility corresponds exactly to E’s winning strategy in the game $\underline{\mathrm{G}}$.
• Completeness of first-order logic holds with respect to possibility semantics, even in cases where standard Kripke semantics fails due to world incompleteness.
• The framework supports nonconstructive results in a constructive setting by replacing atoms and prime filters with regular open sets, eliminating reliance on the Axiom of Choice.
• A dual equivalence is established between full and principal frames and complete $\mathcal{V}$-BAOs, linking possibility semantics to canonical extensions and MacNeille completions.
• The approach generalizes to intuitionistic and inquisitive logics, offering a semantics that supports non-distributive and information-driven reasoning.