The problems generated by the Aristotelian Ontological Square can be grouped into extensional ones on the one hand (dealing with the extent of these divisions), and intensional ones (dealing with their interpretation), on the other. The extensional problems concern the sufficiency and necessity of the division of entities provided by the Square. The intensional problems are related to the interpretation of the extent (what counts as an existent in what sense) and criteria (what does it mean for something to be in/said of or not to be in/said of a subject) of the fourfold division.
The problem of the sufficiency of the division is whether it really comprehends all entities, or perhaps there are others that cannot be placed in any of the four domains of the Square. The problem of the necessity of the division is whether it contains perhaps more than what is needed for classifying all entities, that is to say, whether it contains some nonentities.
The first, naturally emerging Aristotelian suspicion concerning the Square should be that about its necessity. After all, the division is supposed to contain universals,
Substance, Accident and Modes. Table 1 The Aristotelian Ontological Square
Whereas Aristotle denies the existence of universal entities. If, therefore, the Square contains universals, and universals are not entities, then it seems that the Square has to contain some nonentities, i. e., it contains more than is necessary for the classification of all entities, for all entities are either particular substances or particular accidents, but there are no universal substances or accidents among real existents.
The problem of sufficiency, however, is generated by considerations concerning entities that somehow would not seem to fit into Aristotle’s fourfold division. A case in point is provided by the significata of propositions, described most poignantly by the anonymous author of the twelfth-century tract Ars Burana as being ‘‘extra-predicamental,’’ i. e., as not belonging to any of the ten Aristotelian categories, namely, the category of substance, and the nine categories of accidents (De Rijk 1967:357-359). Earlier on, Abelard’s dicta were also assigned by him a peculiar place, apparently outside the Aristotelian Square. And later authors, continuing in the tradition of assigning propositions their significata as distinct from the significata or supposita of their categorematic terms, would also place them outside the divisions of the Square: thus enuntiabilia as conceived by thirteenth-century authors, or the real propositions of Walter Burley, or the complexe significabilia of Adam Wodeham or Gregory of Rimini, not being identifiable with either substances or accidents, were placed in their own, separate category (Nuchelmans 1973, 1980).
The problem with all these additions is that since Aristotle’s division was provided in terms of contradictory criteria it was supposed to be an exhaustive and mutually exclusive division of everything there is.
This way of putting the problem, however, directly leads us to the intensional problems of the Square. Aristotle’s opening words in the relevant passage indicate that his division is supposed to cover all existents. However, depending on the interpretation of what we take to be existent and in what sense, different items will be taken to fall within the realm of existents to be divided by the Square.
Taking his cue from Boethius’ remarks concerning the subject matter of Aristotle’s Categories, almost
A millennium later Thomas of Vio Cajetan characterized the entities to be considered here in the following way:
Cajetan’s interpretation of the subject matter of the Categories provides an elegant solution to both problems with the Square posed above. Since the entities to be considered here are not only mind-independent real beings, but any objects of our simple concepts, universals fit into the Square, even if there are no mind-independent entities existing in a universal manner, insofar as universals are beings of reason, having some foundation in reality. For the same reason, however, enuntiabilia are ruled out, insofar as they are the objects not of simple, but of complex concepts, namely, of complex thoughts formed by the judgment-forming intellect.
Clearly, Cajetan’s solution is able to accommodate beings of reason, because it presupposes the Thomistic interpretation of what and how being is divided in the Aristotelian Square, namely, the extension ofan analogical notion into the extensions of its analogata. According to this doctrine, the extent of the Aristotelian Square should cover both beings in an absolute sense, without qualification, and beings in some diminished sense, with qualification (see the entry on Being in this volume).
This is, in fact, the basis of Aquinas’ understanding of the Aristotelian idea of inherence, that is, an accident’s being-in a subject. For an accident to be is nothing but for its subject to be informed by it, or, conversely, for it to be in its subject: accidentis esse est inesse (‘‘for an accident to be is for it to be in (a subject)’’). This is precisely why on Aquinas’ conception an accident cannot be said to be in the same sense as a substance. When we say that an accident, say, the whiteness of a sheet of paper, exists, the act of being signified by the predicate of this predication is not the act of being of this sheet of paper without qualification (for that would be the substantial act of being of this sheet), but the act of being of the sheet with respect to its whiteness; it is not the being of the sheet absolutely, rather, it is the sheet’s being white. So, the act of being of the whiteness of this sheet is nothing but an act of being of the sheet, although, of course, it is just an accidental act of being of the sheet: the sheet may continue in its own existence even if its whiteness perishes, say, when the sheet is dipped in black ink.
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