Relevant logic

Relevant logic is a family of nonclassical logics that requires premises and conclusions to share a meaningful connection, blocking certain inferences accepted by classical logic.

Relevance logic: A logic in which implication requires a relevant connection between antecedent and consequent. It is the central strand of relevant logic, studied through its distinctive implication connective.
Clarence Irving Lewis: An American philosopher and logician who developed strict implication as an alternative to material implication. His challenge to classical implication helped prepare the ground for relevance-based approaches.
Classical logic: A formal logic governed by principles such as excluded middle and unrestricted classical implication. Relevant logic modifies classical consequence where premise-conclusion connection is absent.
Entailment and relevance: The philosophical problem of whether logical consequence requires a substantive connection between premises and conclusion. This is the motivating issue that relevant logic makes precise.
Intuitionistic logic: A constructive logic that accepts a proposition only when evidence or a proof establishes it. It rejects classical principles for constructive reasons, not primarily to enforce premise-conclusion relevance.
Entailment: A relation in which one set of propositions guarantees another proposition. Relevant logic constrains entailment so premises cannot be disconnected from conclusions.
Wilfrid Sellars: An American philosopher whose work on implication and inference influenced the development of relevant logic. His proposals helped shape early discussions of relevance in entailment.
Material implication: The classical connective defined as false only when its antecedent is true and consequent false. Its truth conditions allow conditionals that relevant implication rejects.
Inconsistent mathematics: The study of mathematical theories that contain contradictions without treating every statement as provable. Relevant logics can provide consequence relations for theories that retain inconsistent assumptions.
Modal logic: A family of logics that adds operators for necessity, possibility, or related modes of truth. Modal logic expands what can be expressed, whereas relevant logic chiefly restricts implication and consequence.

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