Boole's syllogistic: Difference between revisions
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[[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.]] | [[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. | '''[[Boolean logic]]'''<ref>{{cite web |title=Boole Publishes The Mathematical Analysis of Logic {{!}} Research Starters {{!}} EBSCO Research |url=https://www.ebsco.com/research-starters/history/boole-publishes-mathematical-analysis-logic |website=ebsco.com |access-date=30 August 2025 |language=en}}</ref> 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. | 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. | ||
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* [[Boolean logic]] | * [[Boolean logic]] | ||
* [[Propositional logic]] | * [[Propositional logic]] | ||
* | * [[list of Boolean algebra topics]] | ||
==References== | ==References== | ||
{{reflist}} | {{reflist}} | ||
{{DEFAULTSORT:Boole's Syllogistic}} | {{DEFAULTSORT:Boole's Syllogistic}} | ||
Latest revision as of 16:57, 30 August 2025
Template:Short description Template:One source
In the Venn diagrams black areas are empty and red areas are nonempty.
The faded arrows and faded red areas apply in traditional logic.
Boolean logic[1] 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.
See also
References
- ↑ Script error: No such module "citation/CS1".