using resolution. Known as idempotency of entailment in classical logic. Exchange, where two members on the same side of a sequent may be swapped. Symbolically:...

for the conclusion. Linear logic, which lacks monotonicity and idempotency of entailment. Contraction Exchange rule Substructural logic No-cloning theorem...

of excluded middle and double negation elimination Law of noncontradiction, and the principle of explosion Monotonicity of entailment and idempotency...

rejects the law of the excluded middle, double negation elimination, and part of De Morgan's laws; Linear logic rejects idempotency of entailment as well; Paraconsistent...

{\displaystyle {\overline {\alpha ,\alpha \vdash \beta }}} Rule of contraction (or idempotency of entailment) (aka no-deleting theorem) α , α , γ ⊢ β _ {\displaystyle...

entailment Idempotency of entailment Commutativity of conjunction De Morgan duality – every logical operator is dual to another Term logic Outline of logic...

Morgan, and idempotence inferences (for conjunction and disjunction). Furthermore, inconsistency-robust proof of negation holds for entailment: (A⇒(B∧¬B))⊢¬A...

value of one function of these data given another function of these data does depend on θ, but the very definition of sufficiency given above entails that...

multiplication and addition operations of a field GF(2), and as in any field they obey the distributive law.) Idempotency: no Monotonicity: no Truth-preserving:...

{\displaystyle (a\lor (b\equiv c))\equiv ((a\lor b)\equiv (a\lor c))} Idempotency: a ∨ a ≡ a {\displaystyle a\lor a\equiv a} Monotonicity: ( a → b ) →...

the conjunction of its propositions, the succedent as the disjunction of its propositions, and the sequent itself as the entailment of the succedent by...

and order (entailment). Both conjunction and disjunction are associative, commutative and idempotent in classical logic, most varieties of many-valued...

Associativity x ∧ (y ∧ z) = (x ∧ y) ∧ z Commutativity x ∧ y = y ∧ x Idempotency x ∧ x = x A meet-semilattice ⟨ S , ∧ ⟩ {\displaystyle \langle S,\land...

{\mathcal {I}}}C_{i}\in {\mathfrak {S}}[\mathbf {c} ]} . Notice that, by idempotency [K3], one may succinctly write S [ c ] = im ⁡ ( c ) {\displaystyle {\mathfrak...

tautologies of MTL. The notion of global entailment (or: global consequence) is defined for MTL as follows: a set of formulae Γ entails a formula A (or:...

10th edition. Wadsworth Publishing. p. 392. Beth, Evert W.; "Semantic entailment and formal derivability", series: Mededlingen van de Koninklijke Nederlandse...