|Aquinas Day by Day: Logic of Arguments: Using “Universals” in Demonstrations|
Aquinas’ Topic: Logic of Arguments: Using “Universals” in Demonstrations
Church Calendar: St. Joseph
Aquinas’ Text: Expositio libri Posteriorum , Bk. 1, lec. 11
Here Br. Thomas gives some very practical advice for uncovering the universal appropriate to a given demonstrative argument.
Then (74a33) [Aristotle] assigns the reason for what he said before, first asking when one knows universally and absolutely, if one who knows in way mentioned before does not know universally. And he responds that it is clear that if the essence of triangle in general and of each of its species, whether taken separately or all taken together, were the same, then one would know universally and absolutely about triangle, when he knew about some one of its species or about all of them together. But if the essence is not the same, then it will not be the same thing but something different to know triangle and to know each of its species. And by knowing about the species one does not know triangle as triangle.
Then (74a35) he gives advice abaout how to understand a universal properly. He says that whether something belongs to triangle as triangle or to isosceles as isosceles, and when that about which there is a demonstration is “first” and universal as “this,” that is, according to the subject posited, all this will be clear from what I shall say.
For whenever something is removed but what is called the universal still remains, then one must understand that it is not the first universal of that subject. For example, when isosceles or bronze is removed from triangle, there still remains that it has three angles equal to two right angles. Consequently, to have three angles equal to two right angles is not a “first” universal of isosceles or bronze triangle. But if “figure” is removed, having three such angles is removed, and also if you remove “bounded,” which is more universal than figure, since a figure is something enclosed by a bound or bounds. But having three such angles does not belong “first” to figure or to bounded, because it does not belong to them universally.
To what then does it belong “first”? Clearly to triangle, because it belongs to others, both superiors and inferiors, as triangles. For it belongs to figure to have three such angles because a triangle is a kind of figure, and likewise it belongs to isosceles, because it is a triangle, and it is about triangle that “having three such angles” is demonstrated. Consequently, it is to triangle that the universal belongs “first.”
Deinde cum dicit: quando igitur non novit etc., assignat rationem praedictorum, quaerens quando aliquis cognoscat universaliter et simpliciter, ex quo praedicto modo cognoscens non cognoscit universaliter. Et respondet manifestum esse quod, si eadem esset ratio trianguli in communi et uniuscuiusque specierum eius seorsum acceptae aut omnium simul acceptarum, tunc universaliter et simpliciter nosceret de triangulo, quando sciret de aliqua specie eius vel de omnibus simul. Si vero non est eadem ratio, tunc non erit idem cognoscere triangulum in communi et singulas species eius; sed est alterum. Et cognoscendo de speciebus, non cognoscitur de triangulo secundum quod est triangulus.
Deinde cum dicit: utrum autem etc., dat documentum quo proprie possit accipi universale, dicens quod, utrum aliquid sit trianguli secundum quod est triangulus, aut isoscelis, secundum quod est isosceles, et quando id cuius est demonstratio sit primum et universale, secundum hoc, idest secundum aliquod subiectum positum; manifestum est ex hoc quod dicam.
Quandocumque enim, remoto aliquo, adhuc remanet illud quod assignatur universale, sciendum est quod non est primum universale illius. Sicut, remoto isoscele vel aeneo triangulo, remanet quod habeat tres angulos, scilicet duobus rectis aequales. Unde patet quod habere tres angulos aequales duobus rectis non est universale primum, neque isoscelis, neque aenei trianguli. Remota autem figura non remanet habere tres, nec etiam, remoto termino, qui est superius ad figuram, cum figura sit, quae termino vel terminis clauditur; sed tamen non primo convenit neque figurae, neque termino, quia non convenit eis universaliter.
Cuius ergo erit primo? Manifestum est quod trianguli, quia secundum triangulum inest aliis, tam superioribus, quam inferioribus: ideo enim competit figurae habere tres, quia triangulus est quaedam figura; et similiter isosceli, quia triangulus est, et de triangulo habere tres universaliter demonstratur. Unde eius est universale primum.
[Introductions and translations © R.E. Houser]