Logica Universalis

, Volume 10, Issue 2–3, pp 171–189 | Cite as

Leibniz’s Logic and the “Cube of Opposition”

  • Wolfgang LenzenEmail author


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.

Mathematics Subject Classification

01A45 03A05 03B20 


Square of opposition theory of the syllogism concept logic Leibniz quantification of the predicate individual concepts 


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Copyright information

© Springer International Publishing 2016

Authors and Affiliations

  1. 1.University of OsnabrückOsnabrückGermany

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