Boolean logic is a system of syllogistic

logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premise ...

invented by 19th-century British mathematician

George Boole George Boole (; 2 November 1815 – 8 December 1864) was a largely self-taught English mathematician, philosopher, and logician, most of whose short career was spent as the first professor of mathematics at Queen's College, Cork in ...

, which attempts to incorporate the "empty set", that is, a class of non-existent entities, such as round squares, without resorting to uncertain

truth value In logic and mathematics, a truth value, sometimes called a logical value, is a value indicating the relation of a proposition to truth, which in classical logic has only two possible values ('' true'' or '' false''). Computing In some pro ...

s. In Boolean logic, the universal statements "all S is P" and "no S is P" (contraries in the traditional Aristotelian schema) are compossible provided that the set of "S" is the empty set. "All S is P" is construed to mean that "there is nothing that is both S and not-P"; "no S is P", that "there is nothing that is both S and P". For example, since there is nothing that is a round square, it is true both that nothing is a round square and purple, and that nothing is a round square and ''not''-purple. Therefore, both universal statements, that "all round squares are purple" and "no round squares are purple" are true. Similarly, the subcontrary relationship is dissolved between the existential statements "some S is P" and "some S is not P". The former is interpreted as "there is some S such that S is P" and the latter, "there is some S such that S is not P", both of which are clearly false where S is nonexistent. Thus, the subaltern relationship between universal and existential also does not hold, since for a nonexistent S, "All S is P" is true but does not entail "Some S is P", which is false. Of the Aristotelian square of opposition, only the contradictory relationships remain intact.

See also