|Aquinas Day By Day|
Aquinas’s topic: logic of arguments: criteria for demonstrative syllogisms explained
Aquinas’s text: Expositio libri Posteriorum , Bk. 1, lec. 4
Here Br. Thomas explains Aristotle’s six criteria that must be met for a demonstration to be demonstrative: the premisses must be true, first, immediate, causes, prior, and better known.
Then (71b24) [Aristotle] makes clear the definition he has laid down, as well as what immediately follows, namely, that unless these conditions are fulfilled in a demonstration it cannot produce science. Therefore, he first shows that a demonstration always proceeds from true premisses in order to produce science. For there is no scientific knowledge of that which does not exist, for example, that the diagonal is symmetrical, that is, commensurate, with the side of a quadrilateral. For quantities are said to be incommensurable which lack a common measure. These are quantities whose ratio to one another cannot be expressed in terms of one number to another number. That this necessarily holds about the diagonal of a quadrilateral and its side is plain from Euclid. Now what is not true does not exist, for “to be” and “to be true” are convertible. Therefore, what is scientifically known must be true. Consequently, the conclusion of a demonstration that produces “scientific knowing” must be true, and consequently its premises must also be true. For the true cannot be known in a scientific way from the false, even though the true can follow as a conclusion from the false, as he will show later.
Second (71b27), he shows that the demonstration is composed of first and immediate or indemonstrable premisses. For no one can have scientific knowledge unless he has a demonstration of what can be demonstrated—“and I am speaking essentially and not accidentally.” Now he says this because it might be possible to know scientifically some conclusion without having a demonstration of the premises, even were they demonstrable; because one would know it through other principles; but this would be “accidental.” Therefore, given that a demonstrator syllogizes from demonstrable, that is, mediate premises, he either has a demonstration of these premisses or he does not. If he does not, then he does not know the premisses scientifically and so does not know the conclusion through the premises. But if he does have a demonstration, then, since one may not proceed to infinity, as will become clear below, one will finally have to arrive at principles that are immediate and indemonstrable. In this way it is necessary that demonstration proceed from principles that are immediate either directly or through middles. Consequently, it is said in Topics 1 that demonstration comes from premisses that are first and true, or from statements made credible by these.
Third (71b29), he proves that the propositions in a demonstration are the causes of the conclusion, because we know scientifically when we know the causes. And from this he concludes that they are prior and better known, because every cause is by nature prior to and better known than its effect. Now the cause of a demonstrated conclusion must be better known, not just concerning knowing “what it is,” but also concerning knowing “that it is.” For the purpose of demonstrating that there is an eclipse of the sun, it is not enough to know that the moon is interposed, one must also know that the moon is interposed between the sun and the earth. And because “prior” and “better known” are said in two ways, namely, in relation to us and by nature, he consequently says that the things from which a demonstration proceeds are prior and better known absolutely and according to nature, and not in relation to us. To elucidate this point, he says that those things are “prior and better known absolutely” that are farthest from sense, such as universals; but “the prior and better known in relation to us” are things closest to sense, namely, singulars which are opposed to universals, either in the way that prior and posterior are opposite, or in the way that near and far are opposite.
Deinde, cum dicit: verum quidem etc., manifestat positam definitionem, manifestans etiam quod immediate dixerat, scilicet quod nisi praemissae conditiones demonstrationi adessent, scientiam facere non posset. Primo ergo, ostendit quod semper procedit ex veris ad hoc quod scientiam faciat: quia quod non est, non est scire; sicut diametrum esse symmetrum, idest commensurabilem lateri quadrati (dicuntur enim quantitates incommensurabiles, quarum non potest accipi aliqua mensura communis; et huiusmodi quantitates sunt, quarum non est proportio ad invicem sicut numeri ad numerum; quod de necessitate contingit de diametro quadrati et eius latere, ut patet ex X Euclidis). Quod autem non est verum, non est: nam esse et esse verum convertuntur. Oportet ergo id quod scitur esse verum. Et sic conclusionem demonstrationis, quae facit scire, oportet esse veram, et per consequens eius propositiones: non enim contingit verum sciri ex falsis, etsi concludi possit ex eis, ut infra ostendet.
Secundo, ibi: ex primis autem etc., ostendit quod demonstratio sit ex primis et immediatis, sive indemonstrabilibus. Non enim contingit aliquem habere scientiam, nisi habeat demonstrationem eorum, quorum potest esse demonstratio, et hoc dico per se, et non per accidens. Et hoc ideo dicit, quia possibile esset scire aliquam conclusionem, non habentem demonstrationem praemissorum, etiam si essent demonstrabilia: quia sciret eam per alia principia; et hoc esset secundum accidens. Detur ergo quod aliquis demonstrator syllogizet ex demonstrabilibus, sive mediatis: aut ergo habet illorum demonstrationem, aut non habet: si non habet, ergo non scit praemissa, et ita nec conclusionem propter praemissa; si autem habet, cum in demonstrationibus non sit abire in infinitum, ut infra ostendet, tandem erit devenire ad aliqua immediata et indemonstrabilia. Et sic oportet quod demonstratio ex immediatis procedat, vel statim, vel per aliqua media. Unde et in primo libro topicorum dicitur quod demonstratio est ex primis et veris, aut ex his quae per ea fidem sumpserunt.
Tertio, ibi: causas quoque etc., probat quod demonstrationis propositiones sint causae conclusionis, quia tunc scimus, cum causas cognoscimus. Et ex hoc concludit ulterius quod sint priores et notiores, quia omnis causa est naturaliter prior et notior suo effectu. Oportet autem quod causa conclusionis demonstrativae sit notior, non solum quantum ad cognitionem quid est, sed etiam quantum ad cognitionem quia est. Non enim ad demonstrandum quod eclipsis solis est, sufficit scire quod est lunae interpositio, sed oportet etiam scire quod luna interponitur inter solem et terram. Et quia prius et notius dicitur dupliciter, scilicet quoad nos, et secundum naturam; dicit consequenter quod ea, ex quibus procedit demonstratio, sunt priora et notiora simpliciter et secundum naturam, et non quoad nos. Et ad huius expositionem dicit quod priora et notiora simpliciter sunt illa, quae sunt remota a sensu ut universalia. Priora autem et notiora quoad nos sunt proxima sensui, scilicet singularia, quae opponuntur universalibus, sive oppositione prioris et posterioris, sive oppositione propinqui et remoti.
[Introductions and translations © R.E. Houser]