Idempotency of entailment

Idempotency of entailment is a property of logical systems that states that one may derive the same consequences from many instances of a hypothesis as from just one. This property can be captured by a structural rule called contraction, and in such systems one may say that entailment is idempotent if and only if contraction is an admissible rule.

Rule of contraction: from

A,C,C → B

is derived

A,C → B.

Or in sequent calculus notation,

${\displaystyle {\frac {\Gamma ,C,C\vdash B}{\Gamma ,C\vdash B}}}$

In linear and affine logic, entailment is not idempotent.

See also

• No-deleting theorem

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Classical logic

General

• Quantifiers
• Predicate
• Connective
• Tautology
• Truth tables
• Truth function
• Truth value
• Well-formed formula
• Idempotency of entailment
• Logicism
• Problem of multiple generality
• Associativity
• Distribution
• Validity
• Soundness

Classical logics

• Term
• Propositional
• First-order
• Second-order
• Higher-order

Principles

• Commutativity of conjunction
• Excluded middle
• Bivalence
• Noncontradiction
• Monotonicity of entailment
• Explosion

Rules

• De Morgan's laws
• Material implication
• Transposition
• modus ponens
• modus tollens
• modus ponendo tollens
• Constructive dilemma
• Destructive dilemma
• Disjunctive syllogism
• Hypothetical syllogism
• Absorption

Introduction	Negation Double negation Existential Universal Biconditional Conjunction Disjunction
Elimination	Double negation Existential Universal Biconditional Conjunction Disjunction

People

• Bernard Bolzano
• George Boole
• Georg Cantor
• Richard Dedekind
• Augustus De Morgan
• Gottlob Frege
• Kurt Gödel
• Hugh MacColl
• Giuseppe Peano
• Charles Sanders Peirce
• Bertrand Russell
• Ernst Schröder
• Henry M. Sheffer
• Alfred Tarski
• Willard Van Orman Quine
• Ludwig Wittgenstein
• Jan Łukasiewicz

Works

• Begriffsschrift
• Function and Concept
• The Principles of Mathematics
• Principia Mathematica
• Tractatus Logico-Philosophicus

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