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The principle of distributivity is valid in classical logic, but both valid and invalid in quantum logic. 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 with a reasonable interpretation of quantum phenomena. [1]
Distributivity is a property of some logical connectives of truth-functional propositional logic. The following logical equivalences demonstrate that distributivity is a property of particular connectives. The following are truth-functional tautologies.
Modern philosophers reject quantum logic as a basis for reasoning, because it lacks a material conditional; a common alternative is the system of linear logic, of which quantum logic is a fragment. Mathematically, quantum logic is formulated by weakening the distributive law for a Boolean algebra, resulting in an orthocomplemented lattice .
In the mathematical area of order theory, there are various notions of the common concept of distributivity, applied to the formation of suprema and infima.Most of these apply to partially ordered sets that are at least lattices, but the concept can in fact reasonably be generalized to semilattices as well.
The term "Boolean algebra" honors George Boole (1815–1864), a self-educated English mathematician. He introduced the algebraic system initially in a small pamphlet, The Mathematical Analysis of Logic, published in 1847 in response to an ongoing public controversy between Augustus De Morgan and William Hamilton, and later as a more substantial book, The Laws of Thought, published in 1854.
Both conjunction and disjunction are associative, commutative and idempotent in classical logic, most varieties of many-valued logic and intuitionistic logic. The same is true about distributivity of conjunction over disjunction and disjunction over conjunction, as well as for the absorption law.
Order theory is a branch of mathematics that studies various kinds of objects (often binary relations) that capture the intuitive notion of ordering, providing a framework for saying when one thing is "less than" or "precedes" another.
In propositional logic, associativity is a valid rule of replacement for expressions in logical proofs. Within an expression containing two or more occurrences in a row of the same associative operator, the order in which the operations are performed does not matter as long as the sequence of the operands is not changed. That is (after ...