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Possible-World Semantics

Interpret expressions relative to possible worlds and use those world-indexed values to give truth conditions.

Version
v1 · 2026-10-31 · History
Domain-specific #
14557

Core Idea

Possible-world semantics interprets expressions relative to possible worlds or world-like points. A sentence’s pattern of truth across those points can be represented as a function from worlds to truth values. A model and its evaluation clauses determine results relative to the selected language and interpretation.[1]

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Checking Every Pretend World

Imagine a big set of pretend worlds, each a different way things could be. To check if a sentence is true, you look at it in each world. Saying something 'must' be true means it's true in every world you're allowed to look at, and 'might' means it's true in at least one. That way of figuring out meaning is possible-world semantics.

Truth Across Possible Worlds

Possible-world semantics is a method in logic and language study for working out when sentences are true. Instead of judging a sentence only for the real world, you judge it at many 'worlds' — different ways things could be. A model says what's true at each world and how to figure out complicated sentences from simpler ones. In the most common version, 'necessarily P' is true at a world if P is true at every world that can be 'reached' from it, and 'possibly P' is true if P is true at at least one reachable world. Other versions use different rules.

World-Relative Truth Conditions

Possible-world semantics is a framework that interprets expressions relative to alternative worlds or evaluation points. A model says what each expression refers to at each world and how to evaluate complex expressions from those pieces. A sentence's pattern of truth values across worlds can be packaged as its intension — a function from worlds to true/false, or equivalently the set of worlds where it's true. In the standard version for modal logic, a model has a set of worlds W, an accessibility relation R saying which worlds can 'see' which, and a valuation V for basic sentences. Then 'necessarily φ' is true at w when φ is true at every world accessible from w, and 'possibly φ' when φ is true at some accessible world. Other theories use different setups, so not every possible-world semantics needs a single accessibility relation.

 

Possible-world semantics interprets expressions relative to alternative worlds or world-like evaluation points. A model assigns each expression an extension at each world and specifies how complex expressions are evaluated compositionally from those assignments. A sentence's world-indexed truth values can be represented as its intension, a function from worlds to truth values or equivalently the set of worlds at which it is true. It is thus a semantic framework that determines truth conditions, not merely a claim that alternatives exist. In standard relational (Kripke) semantics for normal propositional modal logic, a model M = ⟨W, R, V⟩ consists of a nonempty set of points W, an accessibility relation R on W and a valuation V of atomic sentences. At a point w, □φ is true exactly when φ is true at every R-accessible point, and ◇φ is true exactly when φ is true at some R-accessible point. Other world-based theories use different parameters and evaluation rules, so a single binary accessibility relation is not a requirement of possible-world semantics in general.

Scope of Application

In ordinary relational modal semantics, a model ⟨W,R,V⟩ supplies worlds, accessibility and an atomic valuation. At w, □φ holds when φ holds at every accessible point, and ◇φ when it holds at at least one. World-based semantics also includes intensional treatments whose additional indices and selection rules need not be the same relational model.[2][1]

Clarity

A world is one point; an accessibility relation is one possible component; the full semantics also interprets expressions and specifies evaluation rules. Truth at one world, truth throughout one model, and validity across all valuations on a frame have different scopes. The model does not by itself decide what worlds are metaphysically.[2][1]

Manages Complexity

The method lets an analyst vary alternatives, valuations and interpretation rules separately. It does not itself choose which alternatives or semantic clauses fit an application.

Abstract Reasoning

In a toy model where w reaches only u, and p is false at w but true at u, □p and ◇p are both true at w. If w also reaches itself, □p becomes false while ◇p stays true. The contrast illustrates the standard relational clauses, not every world-based semantics.[2]

Knowledge Transfer

In epistemic models, alternatives can represent what an agent cannot rule out; in intensional semantics, a function tracks an expression’s extension across worlds. Both are world-indexed interpretations, but the particular indices and clauses are application-specific.[3][1]

Neighborhood in Abstraction Space

Possible-World Semantics sits in a sparse region of the domain-specific corpus (97th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Unclustered & Miscellaneous (2835 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-31

References

[1] Menzel, Christopher, “Possible Worlds”, Stanford Encyclopedia of Philosophy, §§1.1–1.2; directly inspected scholarly exposition of possible-world interpretation and intensions. registry ↩a ↩b ↩c ↩d

[2] Garson, James, “Modal Logic”, Stanford Encyclopedia of Philosophy, §§6–7, directly inspected scholarly exposition of relational frames, valuations, and modal clauses. registry ↩a ↩b ↩c

[3] Rendsvig, Rasmus, John Symons, and Yanjing Wang, “Epistemic Logic”, Stanford Encyclopedia of Philosophy, §2.4, directly inspected exposition of information-compatible alternatives. registry ↩