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Extension (predicate logic)

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

Version
v1 · 2026-09-08 · History
Domain-specific #
4483
Origin domain
formal semantics
Subdomain
formal semantics

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.

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

  1. 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. 2. 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

Local relationship map for Extension (predicate logic)Parents appear above the current abstraction, mutual partners to the right, and children below. Node labels state whether each abstraction is prime or domain-specific; colors identify relation types.Extension(predicate logic)DOMAINPrime abstraction: Relation — is a kind ofRelationPRIME

Current abstraction Extension (predicate logic) Domain-specific

Parents (1) — more general patterns this builds on

  • Extension (predicate logic) is a kind of Relation Prime

    The proposed strict upward parent is prime:relation.

Hierarchy path (1) — routes to 1 parentless root

  • Extension (predicate logic) → Relation

Neighborhood in Abstraction Space

Extension (predicate logic) sits in a crowded region of the domain-specific corpus (7th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Formal Logic & Type Theory (34 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-09-08