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Venn’s Syllogistic and a Certain Notational Convention
Eugeniusz Wojciechowski
Abstract
John Venn in his Formal Logic (1881) constructed a certain system of syllogistic, which is one of implementations of the idea of the quantification of predicates. An interesting reconstruction of this system was proposed by V.I. Markin (2011). Markin makes use of five primary functors {aa, ai, ia, ii, e}. The elementary expressions SaaP, SaiP, SiaP, SiiP and SeP are respectively read as: all S is all P, all S is some P, some S is all P, some S is some P and no S is any P.
Markin gives the axiom system for the system. He also proposes the rules of translation of its formulas into the language of classical syllogistic of Łukasiewicz’s axiom system {SaS, SiS, MaPΛSaM → SaP, MaPΛMiS → SiP} and the rules of reverse translation.
This formulation of Venn’s syllogistic can be simplified, including the strong understanding of particular-affirmative sentences (SιP) and by adopting the following notational convention:
SφψP / SφPΛPψS SφPΛPψS / SφψP for φ,ψ{a,ι}.
A new axiom system for Venn’s syllogistic is proposed here. The logical relations between Venn’s sylogistic (SV) and the Łukasiewicz’s system (SL) are examined. A system (SI) has been formulated with a strong understanding of particular affirmative sentences. The proof that systems SI and SL are equivalent is given.
Keywords:
syllogistic, Venn’s syllogistic, strong understanding of particular affirmative sentences, certain notational conventionDetails
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