Logical Consequence¶
A logic-relative relation licensing a formula as following from premises under a specified semantic or derivational criterion.
Core Idea¶
Logical consequence is a relation, relative to a specified formal language and logic, between a set of premises \(\Gamma\) and a candidate conclusion \(\varphi\). Semantically, \(\Gamma\vDash\varphi\) can mean every selected model satisfying \(\Gamma\) also satisfies \(\varphi\). Proof-theoretically, \(\Gamma\vdash\varphi\) can mean a chosen calculus derives \(\varphi\) from \(\Gamma\). These are different determinations; equivalence requires soundness and completeness theorems for the particular calculus and semantics, not merely the shared word “consequence.”[ref-08eedff428e6][ref-325ec7ac272b][^ref-6bbeff23a85b]
This is a meta-level judgment about formulas, not the object-language conditional \(A\to B\) or an episode of human reasoning. The proposed strict parent is live prime Relation: a fixed consequence relation specifies which premise-set/conclusion pairs are licensed under its own criterion.[ref-6bbeff23a85b][ref-23865c39f73c]
Scope of Application¶
In classical first-order model semantics, \(\{\forall x(P(x)\to Q(x)),P(a)\}\vDash Q(a)\) because every model satisfying both premises satisfies the conclusion. The language, classical interpretation of logical terms and admissible model class are part of this assertion; a countermodel would refute it. This illustrative formula instantiates Tarski's model criterion, rather than being quoted from his paper.[^ref-08eedff428e6]
In Gentzen-style calculi, consequence is determined by derivations from assumptions using selected inference figures. One can derive the corresponding \(Q(a)\) with suitable instantiation and detachment rules, but proof-based and model-based judgments are identified only when matching theorems are known.[^ref-325ec7ac272b] In KLM preferential logic, the bird/penguin example shows a formal consequence relation where adding an exception premise can withdraw a prior flight conclusion; classical monotonicity is therefore not universal.[^ref-6bbeff23a85b]
Clarity¶
Before asserting a consequence, specify: the formal language and logic, the premise collection, the conclusion formula, and the licensing criterion. For \(\vDash\), identify models or interpretations; for \(\vdash\), identify a calculus and derivation rules. Tarski emphasizes that satisfaction is language-relative and even the division of logical from extra-logical vocabulary affects what reinterpretations count.[ref-08eedff428e6][ref-325ec7ac272b]
An inference rule is one proof step schema; a derivation is a structured sequence or tree of steps; a consequence relation comprises all licensed premise/conclusion pairs under its chosen standard. Material implication is a formula inside the language, not the same meta-level relation. The redirected “logical implication” and “model-theoretic consequence” titles therefore remain distinct-sense holds rather than automatic aliases.[ref-325ec7ac272b][ref-6bbeff23a85b]
Manages Complexity¶
The relation turns many individual arguments into membership questions: search for a countermodel, produce a derivation, or inspect the preferred premise-compatible worlds. It also localizes disagreement. Two formal systems may use the same surface formulas yet disagree because they select different models, rules or structural properties.[ref-08eedff428e6][ref-325ec7ac272b][^ref-6bbeff23a85b]
This simplification has limits. A proof relation and a semantic relation can diverge; one cannot import a soundness or completeness result from another system. A monotone consequence relation preserves conclusions under added premises, but a preferential nonmonotonic relation may retract them to accommodate exceptions. These are relation-specific tradeoffs, not flaws in the generic schema.[ref-08eedff428e6][ref-6bbeff23a85b]
Abstract Reasoning¶
For each case, place the premise collection and target conclusion on the two sides of a typed judgment, then apply the declared criterion. In classical semantics, look for an admissible model making all premises true and conclusion false. In a proof calculus, check whether a derivation exists; failure to find one is not proof of semantic failure without completeness. The classical example and KLM's bird/penguin case map the same premise–conclusion–criterion roles but produce different behavior under premise expansion.[ref-08eedff428e6][ref-325ec7ac272b][^ref-6bbeff23a85b]
The portable skeleton is live prime Relation; the domain-specific content is formula language, logical vocabulary and formal licensing. Monotonicity, reflexivity and cut must be established for a chosen relation rather than assumed from the label.[ref-23865c39f73c][ref-6bbeff23a85b]
Knowledge Transfer¶
Transfer the question format—under this logic and criterion, does \(\Gamma\) license \(\varphi\)?—from Tarski's models to Gentzen's proofs and KLM's preferential worlds. Do not transfer the exact model class, proof rules or monotonicity automatically. Prime Deductive Reasoning and live Inference Rule are related but concern a reasoning process and a local proof schema, not the entire meta-level relation.[ref-08eedff428e6][ref-325ec7ac272b][^ref-6bbeff23a85b]
The six Wikipedia request surfaces remain provenance, not six accepted aliases: “Entailment” and “Consequence relation” are plausible but unreviewed broad terms; “Follows from” and “Logical conclusion” are context-sensitive; “Model-theoretic consequence” is narrower; and “Logical implication” may mean a connective. A separate vocabulary review must resolve them.[ref-08eedff428e6][ref-6bbeff23a85b]
[^ref-08eedff428e6]: A. Tarski, “On the Concept of Logical Consequence”, original 1936 essay in English translation, printed pp.410–420; scan inspected 2026-10-01 with OCR limitations in the central pages. [^ref-325ec7ac272b]: G. Gentzen, “Investigations into Logical Deduction”, English translation/reprint, PDF pp.2–5, inspected 2026-10-01. [^ref-6bbeff23a85b]: S. Kraus, D. Lehmann and M. Magidor, “Nonmonotonic Reasoning, Preferential Models and Cumulative Logics”, Artificial Intelligence 44 (1990), author-hosted copy, abstract and §§1.1–1.3, inspected 2026-10-01. [^ref-23865c39f73c]: Encyclopedia of Abstractions, live prime Relation, Core Idea and Structural Signature, inspected 2026-10-01.
Relationships to Other Abstractions¶
Current abstraction Logical Consequence Domain-specific
Parents (1) — more general patterns this builds on
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Logical Consequence is a kind of Relation Prime
Logical consequence is a formal relation between a premise collection and a conclusion.
Hierarchy path (1) — routes to 1 parentless root
- Logical Consequence → Relation
Neighborhood in Abstraction Space¶
Logical Consequence sits in a moderately populated region (43rd percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.
Family — Logical Semantics & Many-Valued Systems (11 abstractions)
Nearest neighbors
- Formal Theory — 0.87
- Peirce's Law — 0.87
- Sequent — 0.87
- Propositional logic — 0.87
- Lindström quantifier — 0.86
Computed from structural-signature embeddings · 2026-10-08