Possibility Semantics

TL;DR

Possibility Semantics reframes classical and modal semantics by replacing classical possible worlds with partially specified possibilities represented as posets and their regular opens, or as compact regular opens on UV-spaces. It develops dual representations for propositions (regular opens on posets and UV-spaces), yields choice-free, forcing-inspired constructions (MacNeille completions and canonical extensions), and extends to first-order and modal languages (including relational, neighborhood, and quasi-normal variants) with soundness and completeness results in ZF. The framework yields robust, constructively grounded semantics that can capture extended languages (e.g., propositional quantification, inquisitive logic) and overcome limitations of world-based approaches, while preserving deep algebraic/topological dualities with Boolean algebras and BAOs. These developments have foundational significance forlogic, offering rich semantic tools, alternative completeness results, and broad applicability to forcing, interval semantics, and provability logic in a choice-free setting.$

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.

Figures (12)

• Figure 1: world vs. possibility representations
• Figure 2: four ways of building a possibility frame from a BA
• Figure 3: The $\sqsubseteq$-back condition of p-morphisms, where solid arrows represent the relation $\sqsubseteq$ and dotted arrows represent the function $h$.
• Figure 4: realizations of MacNeille completion and canonical extension
• Figure 5: the up-$\boldsymbol{R}$ condition. Given $x'R_iy'$, we may go up in the first coordinate to any $x$ above $x'$ to obtain $xR_iy'$.

Theorems & Definitions (239)

• proof
• proof
• proof
• Example 3.1.1
• Example 3.1.2
• Example 3.2.1
• Example 3.2.2
• Theorem 3.2.3
• proof : Sketch of 2
• Definition 3.2.4