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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 .
"Is Logic Empirical?" is the title of two articles (one by Hilary Putnam and another by Michael Dummett) [1] [2] that discuss the idea that the algebraic properties of logic may, or should, be empirically determined; in particular, they deal with the question of whether empirical facts about quantum phenomena may provide grounds for revising classical logic as a consistent logical rendering of ...
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.
(box distributivity) (): This rule allows you to do modus ponens inside the provability operator. If it is provable that A implies B, and A is provable, then B is provable. If it is provable that A implies B, and A is provable, then B is provable.
An infinitary logic is a logic that allows infinitely long statements and/or infinitely long proofs. [1] The concept was introduced by Zermelo in the 1930s. [2] Some infinitary logics may have different properties from those of standard first-order logic. In particular, infinitary logics may fail to be compact or complete.
A non-classical logic that attempts to capture the peculiarities of quantum mechanics, challenging traditional logical principles such as the law of excluded middle and distributivity. Quine's dictum The principle that "To be is to be the value of a variable", emphasizing ontological commitment in terms of quantification and the variables of ...