Abstract
It has been known at least since Abelard (12th century) that the classic Square of Opposition suffers from so-called undue existential import (UEI) in that this system of predicate logic collapses when the class denoted by the restrictor predicate is empty. It is usually thought that this mistake was made by Aristotle himself, but it has now become clear that this is not so: Aristotle did not have the Conversions but only one-way entailments, which ‘saves’ the Square. The error of UEI was introduced by his later commentators, especially Apuleius and Boethius. Abelard restored Aristotle’s original logic. After Abelard, some 14th- and 15th-century philosophers (mainly Buridan and Ockham) meant to save the Square by declaring the O-corner true when the restrictor class is empty. This ‘leaking O-corner analysis’, or LOCA, was taken up again around 1950 by some American philosopher-logicians, who now have a fairly large following. LOCA does indeed save the Square from logical disaster, but modern analysis shows that this makes it impossible to give a uniform semantic definition of the quantifiers, which thus become ambiguous—an intolerable state of affairs in logic. Klima (Ars Artium, Essays in Philosophical Semantics, Medieval and Modern, Institute of Philosophy, Hungarian Academy of Sciences, Budapest, 1988) and Parsons (in Zalta (ed.), The Stanford Encyclopedia of Philosophy, http://plato.standford.edu/entries/square/, 2006; Logica Univers. 2:3–11, 2008) have tried to circumvent this problem by introducing a ‘zero’ element into the ontology, standing for non-existing entities and yielding falsity when used for variable substitution. LOCA, both without and with the zero element, is critically discussed and rejected on internal logical and external ontological grounds.
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Notes
- 1.
The standard double bracket notation is used for predicate extensions: 〚F〛 stands for all elements in the universe of entities ENT for which the predicate F delivers truth.
- 2.
The eight sentence types used are A (All F is G), I (Some F is G), their external negations ¬ A and ¬ I, their internal negations A * and I * (All/Some F is not G), and their external and internal negations ¬ A * and ¬ I * (Seuren, [8]: 31–37). The old E-corner is thus split up into ¬ I and A * and the old O-corner into ¬ A and I *. ¬ I and A * and ¬ A and I * can thus be distinguished in systems where the members of each pair are not equivalent.
- 3.
“Ockham’s Summa Logicae (The Logic Handbook), written ca. 1323, is a manifesto masquerading as a textbook” (King, [2]: 243).
- 4.
It is difficult to find a 14th-century text actually stating LOCA, instead of merely playing with the idea. One such text is Ockham, Summa Logicae, beginning of Chap. II.3 (http://individual.utoronto.ca/pking/resources/Ockham?Summa_logicae.txt):
<Q>uandoque sufficit quod subiectum indefinitae vel particularis negativae pro nullo supponat. (<S>ometimes it suffices for the truth of a negative indefinite or particular proposition that the subject refers to nothing.)
For ample commentary and a full text of the passage in question, see Seuren ([8]: 163–166).
- 5.
For explicit exposés of Valuation Space (VS) modeling and corresponding polygonal representations of logical systems, see Seuren ([8]: 46–50; [9]: Ch. 6). Intuitively, the notions are clear enough. A VS model splits up the universe U of possible situations into classes of situations in which each sentence type of a logical system is true. The valuation space /S/ of a sentence S is the set of situations in which S is true. For example, in Fig. 2, /A/ = {1}, /I/ = {1, 2}, etc. A polygonal representation is a polygon with as many vertices as there are sentence types in the system (without vacuous repetitions of negations). Lines between vertices stand for logical relations (equivalence or =, entailment or >, contradictoriness or CD, contrariety or C, subcontrariety or SC).
- 6.
When the logic turns presuppositional, the hyphen will make a logical difference. As Aristotle already observed, the only generalization that can be made for a predicate P and its negative counterpart not-P (or unP, or whatever morphological or lexical means the language has available for creating negative predicates) is that they make for contrary sentences, which cannot both be true, but can both be false. A relation of contradiction between sentences made with P and not-P is restricted to those predicates that are nongradable and have no presuppositional preconditions, such as existent and nonexistent. In presuppositional logic, this means that the predicate-internal negation can only be presupposition-preserving. But the not in not P, without the hyphen, can be the radical presupposition-canceling not that yields truth in cases of presupposition failure.
- 7.
A formula like ∀x:F(x) | [G(x)] is read as ‘for all x such that x is an F, G(x)’, and ∃x:F(x) | [G(x)] as ‘for some x such that x is an F, G(x)’. In this restricted quantification notation, the part between the colon and the upright stroke is the range of the restricted quantifier (∀x or ∃x) and the part after the upright stroke is the argument. It should be noted that the use of a first-order logical language in no way implies the use of the a particular first-order logical system. When the Russellian language is used here, it is because it has been found satisfactory for the present purpose. The language of restricted quantification is used when it is found useful for that purpose. A given formal logical language can be used with different semantic interpretations according to the logical system under discussion. The important thing is that the set-theoretical basis underlying any logical system is made explicit in a convenient, consistent and unambiguous way.
- 8.
Klima’s own notation is slightly different, but amounts to the same. I skip a discussion of Klima’s elaborate formal and semantic description of his system, as it easily reduces to restricted quantification.
- 9.
Parsons has a third quantifier no, but, as far as I can make out, this no is equivalent with not some, in his system.
- 10.
Parsons seems to make use of the ploy used by Odysseus in Homer’s Odyssey, when he escaped from the Cyclops Polyphemus, whose single eye he had just pierced with a red-hot club, by telling the Cyclops that his name was ‘No-one’. When the blinded Polyphemus called on his fellow Cyclopes for help and revenge, he told them that No-one had pierced his eye, which made the other Cyclopes think that Polyphemus had been punished by the gods. In an epilogue called ‘epicycle’, Parsons discusses the philosophical question of the reification of ‘nothing’, quoting various ancient, medieval and modern authors. But he does not come to any clear conclusion.
- 11.
This is not the place to elaborate the presuppositional solution to UEI. For an up-to-date and full discussion, see Seuren ([8]: Ch. 10).
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Seuren, P.A.M. (2012). Does a Leaking O-Corner Save the Square?. 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_9
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