Nonclassical logic in the context of "Boolean algebra"

The material conditional (also known as material implication) is a binary operation commonly used in logic. When the conditional symbol ${\displaystyle \to }$ is interpreted as material implication, a formula ${\displaystyle P\to Q}$ is true unless ${\displaystyle P}$ is true and ${\displaystyle Q}$ is false.

Material implication is used in all the basic systems of classical logic as well as some nonclassical logics. It is assumed as a model of correct conditional reasoning within mathematics and serves as the basis for commands in many programming languages. However, many logics replace material implication with other operators such as the strict conditional and the variably strict conditional. Due to the paradoxes of material implication and related problems, material implication is not generally considered a viable analysis of conditional sentences in natural language.

In logic, disjunction (also known as logical disjunction, logical or, logical addition, or inclusive disjunction) is a logical connective typically notated as ${\displaystyle \lor }$ and read aloud as "or". For instance, the English language sentence "it is sunny or it is warm" can be represented in logic using the disjunctive formula ${\displaystyle S\lor W}$, assuming that ${\displaystyle S}$ abbreviates "it is sunny" and ${\displaystyle W}$ abbreviates "it is warm".

In classical logic, disjunction is given a truth functional semantics according to which a formula ${\displaystyle \phi \lor \psi }$ is true unless both ${\displaystyle \phi }$ and ${\displaystyle \psi }$ are false. Because this semantics allows a disjunctive formula to be true when both of its disjuncts are true, it is an inclusive interpretation of disjunction, in contrast with exclusive disjunction. Classical proof theoretical treatments are often given in terms of rules such as disjunction introduction and disjunction elimination. Disjunction has also been given numerous non-classical treatments, motivated by problems including Aristotle's sea battle argument, Heisenberg's uncertainty principle, as well as the numerous mismatches between classical disjunction and its nearest equivalents in natural languages.