|Aquinas Day By Day|
Aquinas’s topic: logic of arguments: common and proper scientific principles
Aquinas’s text: Expositio libri Posteriorum , Bk. 1, lec. 5
Following Aristotle closely, Br. Thomas distinguishes common scientific principles, the axioms, from proper scientific principles.
Because the Philosopher stated above that demonstration is from “first and immediate” premisses but has not as yet made them clear, he now sets out to identify them. … He says therefore first (72a15), that there are two kinds of immediate principles of a syllogism: one is called a “thesis” and since it is not demonstrated it is said to be immediate. And “the student,” that is, the one being instructed in the demonstrative science, “does not have to have it,” that is, conceive it in his mind or assent to it. The other is called an “axiom” or “maxim,” which anyone who is being instructed must have in his mind and assent to. It is clear that there are such principles, as is shown in Metaphysics 4 about the principle “affirmation and negation are not simultaneously true,” for no one can believe the contrary of this in his mind, even if he should say so vocally. To such principles we give the aforesaid name “axiom” or “maxim” because the certainty of such principles works to manifest other things.
To clarify this division one must understand that every proposition whose predicate is included within the definition of its subject is “immediate” and “self-evident” as far as the proposition itself is concerned. However, the terms of some of these propositions are such that they are understood by everyone, for example, “being” and “one” and other notions that belong to being as being; for “being [ens]” is the first concept of the intellect. Consequently, it is necessary that such propositions, not only in themselves but also in relation to us, are held to be, as it were, self-evident. Examples are “It does not happen that the same thing is and is not,” and “The whole is greater than its part,” and others like these. Consequently, all the sciences take principles of this kind from metaphysics whose task it is to consider being absolutely and what belongs to being.
By contrast, there are some immediate propositions whose terms are not known by everyone. Consequently, although their predicate is included in the definition of the subject, yet because the definition of the subject is not known to all, it is not necessary that such propositions be admitted by all. For example, the proposition, “All right angles are equal,” is in itself a proposition which is “self-evident” or “immediate,” because equality falls in the definition of a right angle. For a right angle is one which a straight line forms when it meets another straight line in such a way that the angles on each side are equal. Therefore, such principles are received as a kind of thesis laid down. And there is another mode, in which certain propositions are called “hypotheses.” For there are some propositions which can be proven only through the principles of another science. Therefore, it is necessary that they be hypothesized in the one science, even though they are proven through the principles of the other science. For example, drawing one straight line from one point to another is a hypothesis in geometry but is proven in natural philosophy, by showing that there is one straight line between any two points.
Quia superius philosophus dixerat quod demonstratio est ex primis et immediatis, et haec ab ipso nondum manifestata erant, ideo intendit ista notificare. … Deinde cum dicit: immediati autem etc. dividit immediatum principium. Et circa hoc duo facit: primo dividit; secundo subdividit; ibi: positiones autem quaedam et cetera. Dicit ergo primo quod immediatum principium syllogismi duplex est. Unum est quod dicitur positio, quam non contingit demonstrare et ex hoc immediatum dicitur; neque tamen aliquem docendum, idest qui doceri debet in demonstrativa scientia, necesse est habere, idest mente concipere sive ei assentire. Aliud vero est, quod dicitur dignitas vel maxima propositio, quam necesse est habere in mente et ei assentire quemlibet, qui doceri debet. Et manifestum est quod quaedam principia talia sunt ut probatur in IV metaphysicae de hoc principio, quod affirmatio et negatio non sunt simul vera, cuius contrarium nullus mente credere potest etsi ore proferat. Et in talibus utimur nomine praedicto, scilicet dignitatis vel maximae propositionis, propter huiusmodi principiorum certitudinem ad manifestandum alia.
Ad huius autem divisionis intellectum sciendum est quod quaelibet propositio, cuius praedicatum est in ratione subiecti, est immediata et per se nota, quantum est in se. Sed quarundam propositionum termini sunt tales, quod sunt in notitia omnium, sicut ens, et unum, et alia quae sunt entis, in quantum ens: nam ens est prima conceptio intellectus. Unde oportet quod tales propositiones non solum in se, sed etiam quoad omnes, quasi per se notae habeantur. Sicut quod, non contingit idem esse et non esse; et quod, totum sit maius sua parte: et similia. Unde et huiusmodi principia omnes scientiae accipiunt a metaphysica, cuius est considerare ens simpliciter et ea, quae sunt entis.
Quaedam vero propositiones sunt immediatae, quarum termini non sunt apud omnes noti. Unde, licet praedicatum sit de ratione subiecti, tamen quia definitio subiecti non est omnibus nota, non est necessarium quod tales propositiones ab omnibus concedantur. Sicut haec propositio: omnes recti anguli sunt aequales, quantum est in se, est per se nota sive immediata, quia aequalitas cadit in definitione anguli recti. Angulus enim rectus est, quem facit linea recta super aliam rectam cadens, ita quod ex utraque parte anguli reddantur aequales. Et ideo, cum quadam positione recipiuntur huiusmodi principia.
Est et alius modus, quo aliquae propositiones suppositiones dicuntur. Sunt enim quaedam propositiones, quae non possunt probari nisi per principia alterius scientiae; et ideo oportet quod in illa scientia supponantur, licet probentur per principia alterius scientiae. Sicut a puncto ad punctum rectam lineam ducere, supponit geometra et probat naturalis; ostendens quod inter quaelibet duo puncta sit linea media.
[Introductions and translations © R.E. Houser]