Extension (predicate logic)¶

The set of ordered tuples for which a predicate is true under a particular interpretation.

Version: v2 · 2026-09-19 · History
Domain-specific #: 4483
Domain group: Humanities
Origin domain: Linguistics & Semiotics
Subdomain: Formal Semantics → Linguistics & Semiotics

Core Idea¶

Extension depends on a model and assignment where relevant, contrasts with intension and changes across possible worlds or times when the interpretation changes. A predicate’s interpretation acts as a truth-valued function on n-tuples, and collecting exactly the tuples mapped to true yields an n-ary relation that determines its extensional truth conditions. The abstraction is therefore identified by a declared carrier, a transformation or constraint over that carrier, and an invariant that tells an analyst whether the named structure is genuinely present.

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The Pile of Red Things

Think of a word like "is red." Its extension is the pile of all the things in your toy box that really are red. If you swap in a different toy box, the pile can change, even though the word still means the same thing.

Everything a Word Fits

In logic, a describing word like "is tall" or "is taller than" is called a predicate. Its extension is the collection of exactly the things, or pairs of things, it is true of. For "is taller than," the extension is a list of ordered pairs like (Sam, Mia), and order matters, because Sam being taller than Mia is not the same as Mia being taller than Sam. The extension depends on which world or situation you are looking at, so it can change even when the meaning of the word stays the same.

A Predicate's Truth Set

In predicate logic, a model fixes a domain of objects and interprets each predicate symbol. The extension of an n-place predicate is the set of ordered n-tuples from the domain that the predicate is true of—for example, the set of all pairs (a, b) where a loves b. Equivalently, you can think of the interpretation as a function that sends each tuple to true or false; the extension is exactly the tuples sent to true. Extension contrasts with intension, which is closer to the meaning or concept: two predicates can have the same extension in one model but different intensions. Because the extension depends on the model (and on variable assignments where relevant), it can vary across possible worlds or times.

Given a formal language and a model with domain D, each n-ary predicate symbol P receives an interpretation. That interpretation can be viewed as a characteristic function from Dⁿ to truth values, and the extension of P is the set {⟨d₁,…,dₙ⟩ ∈ Dⁿ : P is satisfied by that tuple}—the n-ary relation of exactly the tuples mapped to true. The function and set views are equivalent, and the extension fixes P's extensional truth conditions: an atomic formula is true just when the tuple of its arguments' denotations lies in the extension. For open formulas, extension also depends on a variable assignment. Extension must be distinguished from intension, which in possible-worlds semantics is roughly a function from worlds or times to extensions; coextensive predicates may differ in intension. Since extensions are relative to an interpretation, they change across models, worlds, or times when the interpretation changes. Identifying an extension correctly requires stating the language, model, domain, arity, and satisfaction conditions explicitly.

Scope of Application¶

Extension (predicate logic) belongs to formal semantics and is useful where the analyst can specify the typed formal semantics carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets, then evaluate the formal language and model, domain of discourse, n-ary predicate symbol and interpretation, ordered n-tuples, satisfaction or truth condition, set-builder extension, characteristic-function equivalence and distinction from intension and variation across interpretations are explicit. The scope is broad within that domain but bounded by the need for the formal language and model, domain of discourse, n-ary predicate symbol and interpretation, ordered n-tuples, satisfaction or truth condition, set-builder extension, characteristic-function equivalence and distinction from intension and variation across interpretations are explicit.

Clarity¶

The abstraction clarifies a crowded vocabulary by making the formal language and model, domain of discourse, n-ary predicate symbol and interpretation, ordered n-tuples, satisfaction or truth condition, set-builder extension, characteristic-function equivalence and distinction from intension and variation across interpretations are explicit the center of the account. A claim should name the carrier, the governing operation or relation, the applicable assumptions, and the recognition test.

Manages Complexity¶

Without the abstraction, an analyst must reason directly over many local details: the carrier roles, admissibility assumptions, competing conventions, derived invariants, boundary cases, and proof or validation obligations specific to Extension (predicate logic). Extension (predicate logic) compresses them into the roles in the structural signature. That compression permits comparison across instances without erasing the variables that determine validity. It also exposes which details may be varied safely and which are constitutive.

Abstract Reasoning¶

Identify the carrier. State what the elements, states, objects, or observations are: the typed formal semantics carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets. Reject examples whose alleged carrier belongs to a different problem.
Lock the constitutive rule. Express the formal language and model, domain of discourse, n-ary predicate symbol and interpretation, ordered n-tuples, satisfaction or truth condition, set-builder extension, characteristic-function equivalence and distinction from intension and variation across interpretations are explicit independently of one notation or implementation.

Knowledge Transfer¶

Knowledge transfers strongly among subfields of formal semantics because they reuse the typed formal semantics carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets, A predicate’s interpretation acts as a truth-valued function on n-tuples, and collecting exactly the tuples mapped to true yields an n-ary relation that determines its extensional truth conditions., and type the carrier, state every parameter and convention in the definition, test that the formal language and model, domain of discourse, n-ary predicate symbol and interpretation, ordered n-tuples, satisfaction or truth condition, set-builder extension, characteristic-function equivalence and distinction from intension and variation across interpretations are explicit, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.

Relationships to Other Abstractions¶

Current abstraction Extension (predicate logic) Domain-specific

Parents (1) — outgoing typed relations

Extension (predicate logic) is a kind of Relation Prime

It remains its own entry because its identity is fixed by the formal language and model, domain of discourse, n-ary predicate symbol and interpretation, ordered n-tuples, satisfaction or truth condition, set-builder extension, characteristic-function equivalence and distinction from intension and variation across interpretations.

Hierarchy path (1) — routes to 1 parentless root

Extension (predicate logic) → is a kind of Relation

Neighborhood in Abstraction Space¶

Extension (predicate logic) sits in a crowded region of the domain-specific corpus (4^th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Truth, Meaning & Argumentation (37 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-25