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Absorption is a valid argument form and rule of inference of propositional logic. [ 1 ] [ 2 ] The rule states that if P {\displaystyle P} implies Q {\displaystyle Q} , then P {\displaystyle P} implies P {\displaystyle P} and Q {\displaystyle Q} .
The absorption law does not hold in many other algebraic structures, such as commutative rings, e.g. the field of real numbers, relevance logics, linear logics, and substructural logics. In the last case, there is no one-to-one correspondence between the free variables of the defining pair of identities.
Therefore (Mathematical symbol for "therefore" is ), if it rains today, we will go on a canoe trip tomorrow". To make use of the rules of inference in the above table we let p {\displaystyle p} be the proposition "If it rains today", q {\displaystyle q} be "We will not go on a canoe today" and let r {\displaystyle r} be "We will go on a canoe ...
A proof system is formed from a set of rules chained together to form proofs, also called derivations. Any derivation has only one final conclusion, which is the statement proved or derived. If premises are left unsatisfied in the derivation, then the derivation is a proof of a hypothetical statement: "if the premises hold, then the conclusion ...
Where "" is a metalogical symbol representing "can be replaced in a proof with." In strict terminology, ( ( P ∧ Q ) → R ) ⇒ ( P → ( Q → R ) ) {\displaystyle ((P\land Q)\to R)\Rightarrow (P\to (Q\to R))} is the law of exportation, for it "exports" a proposition from the antecedent of ( P ∧ Q ) → R {\displaystyle (P\land Q)\to R} to ...
The absorption law is the only defining identity that is peculiar to lattice theory. A bounded lattice can also be thought of as a commutative rig without the distributive axiom. By commutativity, associativity and idempotence one can think of join and meet as operations on non-empty finite sets, rather than on pairs of elements.
The same is true about distributivity of conjunction over disjunction and disjunction over conjunction, as well as for the absorption law. In classical logic and some varieties of many-valued logic, conjunction and disjunction are dual, and negation is self-dual, the latter is also self-dual in intuitionistic logic.
The expression "mathematical proof" is used by lay people to refer to using mathematical methods or arguing with mathematical objects, such as numbers, to demonstrate something about everyday life, or when data used in an argument is numerical. It is sometimes also used to mean a "statistical proof" (below), especially when used to argue from data.