Ontology and Logic: The Case of Scholastic and Late-Scholastic Theory of Relations

Abstract

This paper investigates the reason (or reasons) why, in the tradition of Western philosophy, a logic of relations was developed only in the second half of the nineteenth century. To this end, it moves along two different but interconnected paths: on the one hand, it attempts to reconstruct the main views concerning the ontology of relations during the middle ages; on the other, it focuses on the treatment of so-called oblique terms (termini obliqui) in the logical works of some preeminent authors belonging to the scholastic and late-scholastic tradition. From the ontological point of view, realists and nominalist both denied that polyadic expressions of the language (spoken, written or mental) signify polyadic properties in the world extra. Some authors, such as Peter Auriol, claimed that polyadic expressions signify something merely mental (as contrasted with ‘real’), thus recognizing, even though in a limited ontological domain, the existence of full-fledged relations, that is, of ‘things’ simultaneously ‘inhering’ in a plurality of subjects. These authors too, however, were unable to produce a logic of relations analogous to that developed in the nineteenth century by De Morgan. If we look at the treatment of so-called ‘oblique inferences’ in the scholastic and late-scholastic tradition, we find that the validity of the inferences in which oblique terms (relations) are involved, does not depend on specific properties of relations but on some general principles on which the syllogistic theory is based. This is true even of the logical work of Juan Caramuel y Lobkowitz (1606–1682), a Spanish Catholic scholastic, who wrote extensively on oblique inferences and who is credited with having introduced a kind of ‘proto-logic’ of relations. Thus, whereas medieval thinkers, from the ontological point of view, attributed to relations understood as polyadic properties a very poor role in the ‘fabric of the world’, from the strictly logical point of view, they made no attempt to develop a systematic analysis of the main properties governing inferences composed by relational sentences. In the case of relations, thinkers belonging to the scholastic and late scholastic tradition did not develop something analogous to the theory of consequences: in other words, they did not possess a logic of relations.

Keywords:

• Medieval logic
• oblique terms and oblique syllogisms
• theory of relations
• Auriol
• Buridan
• Caramuel
• De Morgan
• Ockham

ACKNOWLEDGEMENTS

I wish to express my gratitude to John Marenbon, Michael Beaney and to an anonymous referee for their very helpful comments on an earlier version of this essay.

Notes

¹On this point and the related notion of ‘aggregate’ cf. Alessandro Conti's entry ‘Walter Burley', The Stanford Encyclopedia of Philosophy (SEP).

²Cf. below the quotation from Ludovicus Carbo's Introduction to Logic.

³See Conti's entry on SEP quoted above, footnote 1.

⁴See Henninger 1999; Jeffrey Brower's entry ‘Medieval Theories of Relations', The Stanford Encyclopedia of Philosophy (SEP).

⁵According to contemporary linguistics, the word ‘Every’ in a categorical sentence of the general form ‘Every S is P’ is a determiner expressing a relation between the two classes corresponding to S and P. In their turn, ‘S’ and ‘P’ can be considered, respectively, as the left- and the right-argument. Thus, in the sentence ‘Every horse is an animal’, ‘Every’ is the determiner, ‘horse’ the left-argument and ‘animal’ the right-argument. ‘Every’, as regards monotonicity, is a determiner monotone decreasing on the left and monotone increasing on the right. This means that from Every A is B one may infer Every C is B, provided that C is a subset of A; and one may infer Every A is C from Every A is B, provided that C is a superset of B. In traditional syllogistic, the so-called dictum de omni (et nullo) expresses quite fairly the main properties of monotonicity of the quantifiers. On distribution, dictum and monotonicity, see Makinson, Remarks; Sanchez-Valencia, Studies; Parsons, Articulating Medieval Logic, 45–50 in particular.

⁶As Kahled El-Rouayheb has shown (Relational Syllogisms and the History of Arabic Logic, 900-1900), from around 900 to 1200, some authors belonging to the Arabic tradition were well aware of the difficulties implied by attempting to express relational inferences in form of an Aristotelian syllogism. Since the thirteenth century onwards, Arabic thinkers continued to discuss the problem of relational inferences still maintaining a logical framework largely inspired by traditional syllogistic doctrines. These discussions, for some aspects, are quite similar to those that took place in the United Kingdom in the period immediately preceding De Morgan's work. Yet there is no logician belonging to the Arabic milieu known for having developed a logic of relations analogous to that of De Morgan and Peirce.

⁷This does not mean that, during the middle ages, the Elements were unknown. In the first half of the twelfth century Adelard of Bath translated into Latin an Arabic manuscript of the Elements. Adelard's edition was followed by several other translations made from Arabic sources and by the translation of Isḥāq ibn Ḥunain's version made by Gerardo da Cremona. In the thirteenth century, Campano da Novara prepared a text of Euclid's work, which was later employed by Luca Pacioli and Nicolò Tartaglia for their editions. Medieval commentaries to the Elements, however, with the exception of those belonging to the Arabic tradition, were usually quite poor. In the second half of the sixteenth century, the Elements was translated into modern languages: into Italian (1543), German (1562), French (1564), English (1570), Spanish (1576) and Dutch (1606). In the same period, Greek manuscripts began to circulate in Europe and Bartolomeo Zamberti translated the Elements into Latin directly from the Greek (1505), thus opening a new era in the philological reception of the text. In 1533 Simon Grynaeus published the editio princeps of the Greek text. In this sense I feel authorized to speak of a proper ‘rediscovery’ of Euclid in the Western World. See Vincenzo De Risi, ‘The Development of Euclidean Axiomatic’.