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Canonical static concept record

Semantic consequence

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Summary

A sentence $\varphi$ is a semantic consequence of a theory $T$ when every structure satisfying $T$ also satisfies $\varphi$ .

Record metadata

Canonical name: Semantic consequence
Concept ID: semantic_consequence
Content version: 2.3.0
Primary field: Mathematics
Fields: Mathematics
Primary domain: Logic
Domains: Logic
Alternate terminology: No alternate terminology is currently authored for this record.
Concept type: Semantic relation

Carrier(s)

first-order language
theory $T$ and sentence $\varphi$

Data

satisfaction relation over structures

Axioms / constraints

$T\models\varphi$ iff every model of $T$ is a model of $\varphi$

Canonically induces

semantic closure of a theory

Notes

Semantic consequence is distinct from syntactic derivability until a soundness or completeness theorem relates them for a specified proof calculus.

Concept sources

Wikipedia — Logical consequence — Logical consequence · encyclopedic reference · source ID wp-logical-consequence
nLab — sequent — sequent · mathematical reference · source ID nlab-sequent
Open Logic Project — Models and Theories — Open Logic Project: Models and Theories · open textbook · source ID openlogic-model-theory

Incoming relations (arrows to this concept)

Each relation below ends at this concept.

First-order theory Semantic consequence

Permalink to relation

This is an authored directed relation from the source endpoint to the target endpoint.

Relation ID: first_order_theory_to_semantic_consequence
Relation type: Canonical construction canonical-construction
Direction: source → target
Endpoint roles: source: Canonically constructs; target: Canonically constructed from
Authored annotation: take semantic closure

Authored explanation

Relative to the fixed language and its standard structures, a theory determines the sentences true in all of its models.

How to interpret this relation type

Apply a standard functorial or canonical construction whose output is not merely a reduct of the input and is not generally an equivalent presentation of the same object.

Relation sources

Wikipedia — Logical consequence — Logical consequence · encyclopedic reference · source ID wp-logical-consequence
nLab — sequent — sequent · mathematical reference · source ID nlab-sequent
Open Logic Project — Models and Theories — Open Logic Project: Models and Theories · open textbook · source ID openlogic-model-theory

Satisfaction relation Semantic consequence

Permalink to relation

This is an authored directed relation from the source endpoint to the target endpoint.

Relation ID: logic_satisfaction_relation_to_semantic_consequence
Relation type: Theory component theory-component
Direction: source → target
Endpoint roles: source: Contributes to; target: Contains theory component
Authored annotation: supplies model-by-model truth

Authored explanation

Semantic consequence is defined by universal satisfaction: $\Gamma\models\varphi$ when every structure satisfying all sentences in $\Gamma$ also satisfies $\varphi$ .

How to interpret this relation type

The source supplies a substantive formal ingredient, dynamical sector, law, field content, or mechanism of the target theoretical framework.

Relation sources

Open Logic Project — Models and Theories — Open Logic Project: Models and Theories · open textbook · source ID openlogic-model-theory

Outgoing relations (arrows from this concept)

Each relation below starts at this concept.

No direct relations are authored in this direction.

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Semantic consequence

Summary

Record metadata

Carrier(s)

Data

Axioms / constraints

Canonically induces

Notes

Concept sources

Incoming relations (arrows to this concept)

First-order theory Semantic consequence

Authored explanation

How to interpret this relation type

Relation sources

Satisfaction relation Semantic consequence

Authored explanation

How to interpret this relation type

Relation sources

Outgoing relations (arrows from this concept)