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.
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Notes
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After Frege, Peano and Boole there is increasingly less interest in syllogistic logic, and so with the canonical Aristotelian square of opposition. See [16, especially 177–198, 404–427]. Boole [6, 7] in particular marks the turning point in logic, beginning with an expanded formulation of Aristotelian syllogistic logic, enhanced especially by William Hamilton to a four term logic, which Boole then converts to algebraic formulation at exactly the transitional midpoint between Aristotelian-Scholastic and contemporary symbolic logic.
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[20, 105–106]. On 105, McLaughlin rightly asks: ‘Writers have also criticized the traditional process called ‘inversion,’ which licenses the inference from ‘All S is P’ to ‘Some non-S is not P’—in our terms, from s[p(N)] to −(−s[p(N)]) or −s[−p(M)]. Why must it be the case that if all S is P, some non-S is not P? After all, if it were the case that everything is P, then all things S and non-S alike would be P. But when ‘All S is P’ is taken to mean that a thing’s being S warrants the conclusion that it is P, and when the problem is addressed from the point of view of OC [McLaughlin’s theory of Ordinary Conditionals], the inference becomes entirely natural. Because s[p(N)] means only that the presence of s ensures the presence of p.’ McLaughlin’s final remark suggests that N plays no role in the meaning of s[p(N)], which would need to be explained. McLaughlin says that s[p(N)] means that the ‘presence’ of s ensures the ‘presence’ of p, which is patently a metaphysical relation of ontic dependence or supervenience. It is a metaphysical relation, if ‘presence’ means as it appears ‘existence’, anyway, surely not just presence to us at a particular time, rather than the expected logical or semantic relation by which an inversion at least of an AEIO proposition might first be explicated. McLaughlin’s interpretation supports an interesting and plausible interpretation by which, for example, the meaning of the A proposition, ‘All whales are mammals’, is directly bound to specific truth conditions by which the presence of whales ensures the presence of mammals, and, indeed, the presence of any whale ensures the presence of a mammal.
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Lukasiewicz [12] does not even consider AEIO inversions in his second edition of Aristotle’s Syllogistic from the Standpoint of Modern Formal Logic. Among the few available sources for the topic is [11]; a valuable study in its own right that is not concerned with the particular questions that I have been investigating. Prior [22, 23] discusses negative terms in Boethius’s syllogistic logic, citing the latter’s Introductio ad Syllogismos Categoricos, but does not mention inversions by that name. See especially [23, Part II, Chapter II—Categorical Forms with Negative, Complex, and Quantified Terms, 126–156; including §1 Negative Terms in Boethius and de Morgan, 126–134].
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[15, v].
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[15, 147].
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[15, 118].
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[15, 119].
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Keynes is explicit about the logical equivalence or equipollence of the E′ (E(V)) and A and between I′ (I(V)) and O inversions with their canonical categorical propositional forms. Johnson [14, (vol. I), 140] defines the inversions equivalent, using a short horizontal bar above subject and predicate terms to indicate their internal ‘negations’ or complements. Having presented the equivalences, however, he appears to deny that there are any equivalences between the inversions and the standard AEIO propositions, when he concludes immediately thereafter on the same page, that the enumeration provides ‘a list of eight distinct [Johnson’s emphasis] (i.e. non-equipollent) [Johnson’s parenthetical clarification] general categoricals, as an extension of the usual four.’ This certainly seems false, at least from a more contemporary logical perspective, as well as being at odds with Keynes’ understanding.
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Acknowledgements
I am grateful to an anonymous commentator for useful suggestions and criticisms in preparing the final version of this essay. Special thanks are due to my research assistant Sebastian Elliker at Universität Bern and Tina Marie Jacquette for digitalizing my hand-drawn diagrams.
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Jacquette, D. (2012). Thinking Outside the Square of Opposition Box. In: Béziau, JY., Jacquette, D. (eds) Around and Beyond the Square of Opposition. Studies in Universal Logic. Springer, Basel. https://doi.org/10.1007/978-3-0348-0379-3_5
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