Editing Boole's syllogistic (section)

{{Short description|Mathematical argument}}
{{Unreferenced|date=December 2009}}
[[File:Square of opposition, set diagrams.svg|thumb|Square of opposition<br>In the [[Venn diagram]]s black areas are [[empty set|empty]] and red areas are nonempty.<br>The faded arrows and faded red areas apply in traditional logic.]]
'''[[Boolean logic]]''' is a system of [[syllogism|syllogistic]] [[logic]] invented by 19th-century British mathematician [[George Boole]], 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]]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.