Boole's syllogistic
Encyclopedia
Boolean logic
is a system of 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 values.
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.
Boolean logic
Boolean algebra is a logical calculus of truth values, developed by George Boole in the 1840s. It resembles the algebra of real numbers, but with the numeric operations of multiplication xy, addition x + y, and negation −x replaced by the respective logical operations of...
is a system of syllogistic
Syllogism
A syllogism is a kind of logical argument in which one proposition is inferred from two or more others of a certain form...
logic
Logic
In philosophy, Logic is the formal systematic study of the principles of valid inference and correct reasoning. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, semantics, and computer science...
invented by 19th-century British mathematician George Boole
George Boole
George Boole was an English mathematician and philosopher.As the inventor of Boolean logic—the basis of modern digital computer logic—Boole is regarded in hindsight as a founder of the field of computer science. Boole said,...
, which attempts to incorporate the "empty set", that is, a class of non-existent entities, such as round squares, without resorting to uncertain truth values.
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
Square of opposition
In the system of Aristotelian logic, the square of opposition is a diagram representing the different ways in which each of the four propositions of the system are logically related to each of the others...
, only the contradictory relationships remain intact.
See also
- Boolean logicBoolean logicBoolean algebra is a logical calculus of truth values, developed by George Boole in the 1840s. It resembles the algebra of real numbers, but with the numeric operations of multiplication xy, addition x + y, and negation −x replaced by the respective logical operations of...
- Propositional logic
- list of Boolean algebra topics