The principle of distributivity states that the algebraic

distributive law In mathematics, the distributive property of binary operations generalizes 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 arithmetic, ...

is valid, where both

logical conjunction In logic, mathematics and linguistics, And (\wedge) is the truth-functional operator of logical conjunction; the ''and'' of a set of operands is true if and only if ''all'' of its operands are true. The logical connective that represents thi ...

and

logical disjunction In logic, disjunction is a logical connective typically notated as \lor and read aloud as "or". For instance, the English language sentence "it is raining or it is snowing" can be represented in logic using the disjunctive formula R \lor ...

are distributive over each other so that for any

proposition In logic and linguistics, a proposition is the meaning of a declarative sentence. In philosophy, " meaning" is understood to be a non-linguistic entity which is shared by all sentences with the same meaning. Equivalently, a proposition is the no ...

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, 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 field takes as its starting point an observ ...

. 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 classical deductive logic, a consistent theory is one that does not lead to a logical contradiction. The lack of contradiction can be defined in either semantic or syntactic terms. The semantic definition states that a theory is consistent ...

with a reasonable interpretation of quantum phenomena.

References

Abstract algebra Principles Propositional calculus {{mathlogic-stub