Logical consequence

Logical consequence is a relation between premises and a conclusion in which the conclusion must be true whenever the premises are true, under specified interpretations or formal rules.

Semantic consequence: A consequence relation in which every interpretation satisfying the premises also satisfies the conclusion. It captures the truth-preserving definition across interpretations.
Premise: A proposition assumed or asserted as a basis for an argument or proof. Consequence asks what must follow when these assumptions hold.
Material implication: The classical truth-functional connective that is false only when its antecedent is true and its consequent false. A true material conditional does not by itself establish a consequence relation.
Natural deduction: A proof system organized around introduction and elimination rules for logical connectives. Its derivations display how conclusions follow from temporary assumptions.
Compactness theorem: A result stating that a set of first-order sentences has a model if every finite subset has a model. Compactness constrains which consequence relations first-order semantics can support.
Syntactic consequence: A consequence relation in which a conclusion follows from premises by specified rules of inference. It defines consequence through derivations rather than interpretations.
Conclusion: A proposition an argument or proof aims to establish from its premises. It is the statement whose truth the relation tests.
Logical entailment: A relation in which one or more propositions support or necessitate another, with usage varying by field. The phrase is often used interchangeably, though some traditions distinguish it from formal consequence.
Sequent calculus: A proof system whose judgments represent inferential relations between collections of formulas. Sequents make premise-to-conclusion consequence explicit in proof rules.
Interpolation theorem: A theorem stating that certain derivable implications admit an intermediate formula using only shared vocabulary. Interpolation reveals structural limits on what can mediate a consequence.

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