The principle of distributivity states that the algebraic

distributive law In mathematics, the distributive property of binary operations is a generalization of the distributive law, which asserts that the equality x \cdot (y + z) = x \cdot y + x \cdot z is always true in elementary algebra. For example, in elementary ...

is valid, where both

logical conjunction In logic, mathematics and linguistics, ''and'' (\wedge) is the Truth function, truth-functional operator of conjunction or logical conjunction. The logical connective of this operator is typically represented as \wedge or \& or K (prefix) or ...

and

logical disjunction In logic, disjunction (also known as logical disjunction, logical or, logical addition, or inclusive disjunction) is a logical connective typically notated as \lor and read aloud as "or". For instance, the English language sentence "it is ...

are distributive over each other so that for any

proposition A proposition is a statement that can be either true or false. It is a central concept in the philosophy of language, semantics, logic, and related fields. Propositions are the object s denoted by declarative sentences; for example, "The sky ...

s ''A'', ''B'' and ''C'' the equivalences :

A \land (B \lor C) \iff (A \land B) \lor (A \land C)

and :

A \lor (B \land C) \iff (A \lor B) \land (A \lor C)

hold. The principle of distributivity is valid in

classical logic Classical logic (or standard logic) or Frege–Russell logic is the intensively studied and most widely used class of deductive logic. Classical logic has had much influence on analytic philosophy. Characteristics Each logical system in this c ...

, but both valid and invalid in

quantum logic In the mathematical study of logic and the physical analysis of quantum foundations, quantum logic is a set of rules for manipulation of propositions inspired by the structure of quantum theory. The formal system takes as its starting p ...

. The article " Is Logic Empirical?" discusses the case that quantum logic is the correct, empirical logic, on the grounds that the principle of distributivity is

inconsistent In deductive logic, a consistent theory is one that does not lead to a logical contradiction. A theory T is consistent if there is no formula \varphi such that both \varphi and its negation \lnot\varphi are elements of the set of consequences o ...

with a reasonable interpretation of quantum phenomena.

References

Abstract algebra Principles Propositional calculus {{mathlogic-stub