Squares and trees

The Square of Opposition

The square of opposition is a common diagram found in medieval dialectical manuscripts and is used to relate propositions to each other. It’s a sort of argument-tester which allows one to identify logical fallacies. We don’t know precisely who the first person to relate propositions in the form of a ‘square of opposition’ was, but the earliest examples are found in copies of the Peri hermeneias, a second-century text by the logician Apuleius. In this square of opposition found in Leiden, UB, BPL 25 (f. 12r), a copy of the Peri hermeneias, the proposition under examination is “All men are just”, “Omnis homo iustus est”. Click through the captions to the image to examine the different propositional relationships expressed in the square.

Buridan’s Octagon of Opposition

A particularly elaborated version of the square of opposition is found in Jean Buridan’s (c. 1301-60) Summulae de dialectica. Known as an ‘octagon of opposition’, Buridan – teacher at the Faculty of Arts at the University of Paris – here amends the traditional square to also incorporate modal syllogisms. A modal syllogism is one that uses the words ‘necessarily’, ‘possibly’, or ‘contingently’. For example, ‘just possibly applies to some men’, ‘just necessarily applies to no men’. By incorporating these extra sets of syllogisms, Buridan extended the number of relations which were possible in the traditional square. The octagon is only present in some of the manuscripts of the Summulae, so it may not have been an authorial addition, but it clearly was a useful addition for students grappling with more complex logical propositions.

The ‘Porphyrian Tree’

Porphyry, a third-century Greek philosopher, wrote the Isagoge, an introduction to Aristotle’s Categories. This work was later translated into Latin and commented upon by Boethius. One of the key contributions of this work was a treatment of the first, and most important, of Aristotle’s categories – substance. Porphyry investigated what substance consisted of by using differentiae2 – looking at the essential features which distinguished one species within a genus3 from another. This process of distinguishing between genera was often visualised in medieval manuscripts in the form of a tree diagram. The ‘Porphyrian tree’ is a schematic tool used to divide a category, genus, into its constituents, and in so doing work out what the sub-genera and species which make up this category consist of. It is a tool for thinking about what things are and so working out the essential characteristics of particular species. Although there is no evidence that Porphyry himself came up with this schematic device, it’s most commonly associated with manuscripts of his Isagoge.

We can use the Tree to identify features of other substances. For example, using this same terminology we could describe a dog as a corporeal, animate, sensate, irrational substance. A flower is a corporeal, animate, insensate, substance. A rock is an incorporeal, inanimate substance. The Porphyrian tree permits the user to identify the essential characteristics of everything in the world. All of Aristotle’s categories could be expanded into ‘trees’ using the same process of differentiation, but the category of substance was the one which was most commonly described in this form.

In this version of the Porphyrian tree the examples given of ‘homo’, ‘man’, at the bottom of the tree are Plato, Cato, and Cicero. These names would have been evocative for the medieval reader. Plato’s name signified the discipline of philosophy. Cicero’s signified rhetoric, while Cato’s Distachs was a widely-used elementary textbook for learning Latin, and so evoked the art of grammar.

The Porphyrian tree was just one example of how medieval scholars used visual tools to grapple with thinking about the nature of being, a tradition which extends back to the Carolingian period. In these two copies of Martianus Capella’s De nuptiis Philologiae et Mercurii, a small marginal diagram is appended to the section on dialectic. Circular in form, at its centre it has the word ‘homo’, ‘man’ then ‘animalitas’, ‘ratio’, ‘mortalitas’, ‘sensus’ – animality (i.e. being animate), reason, mortality and sensation – in the next circle. In the outermost circle we find the words ‘lapis’ (stone); ‘equus’ (horse); ‘arbor’ (tree) and ‘angelus’ (angel).

In this image, taken from the Breviculum of Ramon Llull, other mnemonic devices – namely ladders and towers (which permitted the presentation of partition and progression) – are used alongside a device (to the left of the image) which is reminiscent of the Porphyrian tree. Deus, God, is presented at the top of a scala natura, a ‘ladder of nature’, to the right of the pseudo-tree. To its left are depicted nine wise men, each asking a question – utrum? quid? de quo? quare? quantum? quale? quando? ubi?, quo modo, cum quo?– Whether? What? Of which? By what? How much? Which? Of what kind? When? Where? In what manner, with whom? These ten questions, which can be asked of all substances, were derived by Lllull from a work by the Islamic philosopher, Ibn Sab’in (1217/18–69/71).

Sources used for this contribution:

• Carruthers, M., The Book of Memory (Cambridge, Cambridge University Press, 2008)
• Jeauneau, E., ‘Pour le dossier d’Israël Scot’, Archives d'histoire doctrinale et littéraire du moyen âge, 52, (1985), 7-71
• Lohr, C., ‘The Arabic Background to Ramon Lull’s “Liber Chaos” (ca. 1285), Traditio, 55 (2000), 159-70
• Londy, D. and Johanson, C., Philosophia antiqua: The logic of Apuleius (Leiden, Brill, 1987)
• Parsons, T., Articulating Medieval Logic (Oxford, Oxford University Press, 2014)
• Rouse, R. H., ‘Manuscripts belonging to Richard de Fournival’, Revue d'histoire des textes, 3 (1974), 253-269
• Verboon, A., ‘The Medieval Tree of Porphyry: An Organic Structure of Logic’, in A. Worm and P. Salonis (eds), The Tree. Symbol, Allegory and Structural Device in Medieval Art and Thought (Turnhout, Brepols, 2014), 83-101