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The Horn theory of Boole’s partial algebras. (English) Zbl 1288.03037

The paper under review concerns one of the two components of Boole’s algebra of logic, namely his algebra of classes. Boole claimed that, with one exception, the laws of his algebra of classes agreed with the laws of the common algebra of numbers – the one exception was the idempotent law for variables. The paper augments T. Hailperin’s efforts (cf. his book [Boole’s logic and probability. A critical exposition from the standpoint of contemporary algebra, logic and probability theory. Amsterdam etc.: North-Holland Publishing Company (1976; Zbl 0352.02002); second edition (1986; Zbl 0611.03001)]) to place Boole’s algebra of logic on a solid footing. The authors use Horn sentences to give a modern formulation of the principle that Boole adopted in 1854 as the foundation for his algebra of logic – this principle is called in the present paper The Rule of 0 and 1.

MSC:

03G05 Logical aspects of Boolean algebras
01A55 History of mathematics in the 19th century
03-03 History of mathematical logic and foundations
03B05 Classical propositional logic
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References:

[1] An investigation of the laws of thought on which are founded the mathematical theories of logic and probabilities (1958)
[2] The Cambridge and Dublin Mathematical Journal 3 pp 183– (1848)
[3] The mathematical analysis of logic, being an essay towards a calculus of deductive reasoning (1951)
[4] Algebra der Logik I (1890)
[5] DOI: 10.1090/S0002-9947-1979-0522263-8 · doi:10.1090/S0002-9947-1979-0522263-8
[6] Modules over commutative regular rings (1967)
[7] Pure logic, or the logic of quality apart from quantity: with remarks on Boole’s system and on the relation of logic and mathematics (1864)
[8] Boole’s logic and probability (1976)
[9] DOI: 10.1090/S0002-9947-1913-1500960-1 · doi:10.1090/S0002-9947-1913-1500960-1
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