Thinking Outside the Square of Opposition Box

Abstract

The graphic meaning and formal implications of the canonical Aristotelian square of opposition are favorably compared with alternative ways of representing subsyllogistic logical relations among the four categorical AEIO propositions of syllogistic logic. The canonical arrangement of AEIO propositions in the square is justified by an exhaustive survey of graphically nonequivalent alternatives. The conspicuous omission in a modified expanded Aristotelian square of inversions involving complements of subject terms is critically considered. Two of the four inversions are shown to be logically equivalent to two of the AEIO propositions, and are consequently discounted as offering no potential significant revision of the canonical square. The remaining two inversions are logically distinct from any of the AEIO propositions, and the advantages of adding them to an expanded square are explored. It is shown that the two AEIO-distinct inversions are inferentially, and as contraries, subcontraries, subalterns or contradictories, logically isolated from the AEIO propositions. From this it seems reasonable to conclude that no inversion makes a contribution to the subsyllogistic logical relations represented in an expanded version of the square. The investigation leaves the traditional Aristotelian square unchanged, but affords a deeper appreciation of the canonical square’s optimal two-dimensional diagramming of subsyllogistic logical relations among the four fixed AEIO propositions, and of why inversions of the AEIO propositions are appropriately excluded from the square. The limits and disadvantages of subsyllogistic logic are made conspicuous in the use of contemporary classical logic to translate and verify in algebraic terms all potential square of opposition relations, as a formal basis for excluding inversions, where they are otherwise manifestly excluded without explanation from the canonical square.

Keywords

AEIO propositions AEIO inversions Canonical Aristotelian square of opposition Graphic equivalences Subsyllogistic logic Surveyability 

Mathematics Subject Classification

03A05 03B05 03B10 03B20 03B65 

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Copyright information

© Springer Basel 2012

Authors and Affiliations

  1. 1.University of BernBernSwitzerland

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