Abstract
After giving a short summary of the traditional theory of the syllogism, it is shown how the square of opposition reappears in the much more powerful concept logic of Leibniz (1646–1716). Within Leibniz’s algebra of concepts (which may be regarded as an “intensional” counterpart of the extensional Boolean algebra of sets), the categorical forms are formalized straightforwardly by means of the relation of concept-containment plus the operator of concept-negation as ‘S contains P’ and ‘S contains Not-P’, ‘S doesn’t contain P’ and ‘S doesn’t contain Not-P’, respectively. Next we consider Leibniz’s version of the so-called Quantification of the Predicate which consists in the introduction of four additional forms ‘Every S is every P’, ‘Some S is every P’, ‘Every S isn’t some P’, and ‘Some S isn’t some P’. Given the logical interpretation suggested by Leibniz, these unorthodox propositions also form a Square of Opposition which, when added to the traditional Square, yields a “Cube of Opposition”. Finally it is shown that besides the categorical forms, also the non-categorical forms can be formalized within an extension of Leibniz’s logic where “indefinite concepts” X, Y, Z\({\ldots}\) function as quantifiers and where individual concepts are introduced as maximally consistent concepts.
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References
Leibniz’s works
German Academy of Science et al. (eds.), Gottfried Wilhelm Leibniz, Sämtliche Schriften und Briefe, Series VI, Philosophische Schriften, Darmstadt (1930), Berlin (1962) ff
Couturat, L. (ed.): Opuscules et fragments inédits de Leibniz. Felix Alcan, Paris (1903) (reprint Hildesheim (Olms) 1961)
Schupp, F. (ed.): Generales Inquisitiones de Analysi Notionum et Veritatum—Allgemeine Untersuchungen über die Analyse der Begriffe und Wahrheiten. Meiner, Hamburg (1982)
Gerhardt, C.I (ed.): Die philosophischen Schriften von G. W. Leibniz. vol 7, Berlin (1849-1863) (reprint Hildesheim (Olms) 1962).
Leibniz—Logical Papers, ed. and translated by G. H. R. Parkinson. Clarendon Press, Oxford (1966)
Other Works
Kauppi, R.: Über die Leibnizsche Logik, Helsinki. Acta Philos. Fenn (1960)
Lenzen W.: Glauben, Wissen und Wahrscheinlichkeit—Systeme der epistemischen Logik. Springer, Vienna (1980)
Lenzen W.: Zur extensionalen und ‘intensionalen’ Interpreta-tion der Leibnizschen Logik. Studia Leibnitiana 15, 129–148 (1983)
Lenzen W.: Leibniz und die Boolesche Algebra. Studia Leibnitiana 16, 187–203 (1984)
Lenzen W: ‘Non est’ non est ‘est non’—Zu Leibnizens Theorie der Negation. Studia Leibnitiana 18, 1–37 (1986)
Lenzen W.: On Leibniz’s essay ‘Mathesis rationis’ (Critical Edition and Commentary). Topoi 9, 29–59 (1990)
Quine W.V.O.: From a logical point of view. Harper & Row, New York (1953)
Swoyer C.: Leibniz on intension and extension. Noûs 29, 96–114 (1995)
Thom P.: The syllogism. Philosophia, Munich (1981)
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Lenzen, W. Leibniz’s Logic and the “Cube of Opposition”. Log. Univers. 10, 171–189 (2016). https://doi.org/10.1007/s11787-016-0143-2
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DOI: https://doi.org/10.1007/s11787-016-0143-2