Venn’s Syllogistic and a Certain Notational Convention
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.
References
Dubakov D.V., Markin V.I.: Sistema sillogistiki s iskhodnymi konstantami, sootvetstvuyushchimi krugovym diagrammam, „Trudy nauchno-issledovatel’skogo seminara Logicheskogo tsentra Instituta filosofii RAN”, Vyp. XVIII, Moskva 2007 [Д.В. Дубаков, В.И. Маркин, Система силлогистики с исходными константами, соответствующими круговым диаграммам, „Труды научно-исследовательского семинара Логического центра Института философии РАН”, Вып. XVIII, Москва 2007].
Łukasiewicz Jan: Elementy logiki matematycznej, skrypt autoryzowany (opracowany przez M. Presburgera), Warszawa 1929 [Reprint wydany przez Wydawnictwo Naukowe UAM – Poznań 2008].
Markin V.I., Formal’nyye rekonstruktsii sillogistiki Venna, „Vestnik Moskoskogo Universiteta”, Ser. 7. Filosofiya [В.И. Маркин, Формальные реконструкции силлогистики Венна, „Вестник Москоского Университета”, Сер. 7. Философия] 1 (2011), s. 63-73.
Venn John: Symbolic logic, London: Macmillan and Co. 1881.
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