H. P. Grice's Disjunctum

DISJUNCTUM – Disjunction. Logic Also called an alternation. A complex proposition of the form “p or q,” where p and q are component propositions and are called disjuncts. The connective “or” has an inclusive sense and an exclusive sense. In its inclusive sense, sometimes called inclusive disjunction, “p or q” means “p or q or both.” In propositional calculus, such a disjunction is symbolized as “p ∨ q.” It is true if p is true, if q is true, or if both disjuncts are true; it is false if and only if both disjuncts are false. From the premise p or q and the premise not p, we may infer the conclusion q, and this is called a disjunctive syllogism. In its exclusive sense, sometimes called exclusive disjunction, “p or q” means “either p or q but not both.” In propositional calculus, it is symbolized as “[(p ∨ q) ∧ ~ (p ∧ q)].” It is true if p and q have opposite truth-values and is false if they have the same truth-values. “We may take next disjunction, ‘p or q’. This is a function whose truth-value is truth when p is true and also when q is true, but is falsehood when both p and q are false.” Russell, Introduction to Mathematical Philosophy