The Ways to Wisdom

Aquinas Day by Day: Logic of Arguments: How the Three Parts of a Demonstrative Science Work


Aquinas’ Topic: Logic of Arguments: How the Three Parts of a Demonstrative Science Work


Aquinas’ Text: Expositio libri Posteriorum , Bk. 1, lec. 18

Here Br. Thomas explains how the three parts of a demonstrative sceince work—its subject, principles and attributes.

(76a42) [Aristotle] shows how the demonstrative sciences make use of the principles laid down. First, about common principles, he says that “it suffices to accept each of those” common principles, as it pertains to the subject genus with which the science is concerned. For geometry does this not by taking the common principles laid down in their generality, but only about magnitudes; and arithmetic only about numbers. For the geometer then can reach his conclusions by saying: if equal magnitudes be taken from equal magnitudes, what remains are equal, just as if he were to say: if equals are taken from equals, the remainders are equal. The same must also be said about numbers.

Second (76b2), he shows how the demonstrative sciences make use of proper principles. And he says that proper principles are what are pre-supposed to exist in the sciences, namely, the subjects about which a science inquires into the things that are in it essentially.  For example, arithmetic considers unities, and geometry considers “signs,” that is, points, and lines. For these sciences pre-suppose these things to be and to be “this,” that is, they pre-suppose about them “that they are and “what they are”. But about the attributes [passionibus] these sciences pre-suppose “what each signifies.” For example, arithmetic pre-supposes what “even” is and what “odd” is, and what a “square” or “cubic number” are. And geometry pre-supposes what “rational” means about lines. For a line is called “rational” about which we can give a ratio to a given line. Every such line is commensurable with a given line; but one which is incommensurable is called “irrational” or “surd.” Likewise, geometry supposes what is a “reflex” or a “curved.” Now these sciences demonstrate concerning all the above-mentioned attributes “that they are” by means of common principles and principles demonstrated from the common principles. And what has been said about geometry and arithmetic should also be understood of astronomy.

“For every demonstrative science is concerned with three things”: (1) one of them is the subject genus whose essential attributes are investigated; (2) a second are the common axioms from which, as from “first” premisses, it demonstrates; (3) and the third are the “attributes” about which each science pre-supposes “what their names signify.”

Tertio, ibi: sufficiens autem est etc., ostendit quomodo praemissis principiis scientiae demonstrativae utantur. Et primo quidem de communibus dicit quod sufficiens est accipere unumquodque istorum communium, quantum pertinet ad genus subiectum, de quo est scientia. Idem enim faciet geometria, si non accipiat praemissum principium commune in sua communitate, sed solum in magnitudinibus, et arithmetica in solis numeris. Ita enim poterit concludere geometria, si dicat: si ab aequalibus magnitudinibus aequales auferas magnitudines, quae remanent sunt aequales; sicut si diceret: si ab aequalibus aequalia demas, quae remanent sunt aequalia. Et similiter dicendum est de numeris.

Secundo, ibi: sunt autem propria etc., ostendit qualiter demonstrativae scientiae utantur propriis principiis, dicens quod propria principia sunt quae supponuntur esse in scientiis, scilicet subiecta, circa quae scientia speculatur ea quae per se insunt eis. Sicut arithmetica considerat unitates, et geometria considerat signa, idest puncta et lineas. Praedictae enim supponunt esse et hoc esse, idest supponunt de eis, et quia sunt et quid sunt. Sed de passionibus supponunt praedictae scientiae quid significet unaquaeque; sicut arithmetica supponit quid est par, et quid est impar, aut quid est numerus quadratus aut cubicus; et geometria supponit quid est rationale in lineis. Dicitur enim linea rationalis, de qua possumus ratiocinari per lineam datam: huiusmodi autem est omnis linea commensurabilis lineae datae; quae vero est ei non commensurabilis, vocatur irrationalis vel surda. Similiter et geometria supponit quid est reflexum aut curvum. Sed praedictae scientiae demonstrant de omnibus praedictis passionibus quod sint per principia communia, et ex illis principiis, quae demonstrantur ex communibus. Et quod dictum est de geometria et arithmetica, intelligendum est etiam de astrologia. Omnis enim scientia demonstrativa est circa tria: quorum unum est genus subiectum, cuius per se passiones scrutantur; et aliud est communes dignitates, ex quibus sicut ex primis demonstrat; tertium autem passiones, de quibus unaquaeque scientia accipit quid significent.

[Introductions and translations © R.E. Houser]

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