Chapter III.—Demonstration Defined.

Similarly, also, all men will admit that demonstration is discourse,1879    It is necessary to read λόγον here, though not in the text, on account of ἐκπορίζοντα which follows; and as εὔλογον εἷναι λόγον occurs afterwards, it seems better to retain δὔλογον than to substitute λόγον for it. agreeable to reason, producing belief in points disputed, from points admitted.

Now, not only demonstration and belief and knowledge, but foreknowledge also, are used in a twofold manner. There is that which is scientific and certain, and that which is merely based on hope.

In strict propriety, then, that is called demonstration which produces in the souls of learners scientific belief. The other kind is that which merely leads to opinion. As also, both he that is really a man, possessing common judgment, and he that is savage and brutal,—each is a man. Thus also the Comic poet said that “man is graceful, so long as he is man.” The same holds with ox, horse, and dog, according to the goodness or badness of the animal. For by looking to the perfection of the genus, we come to those meanings that are strictly proper. For instance, we conceive of a physician who is deficient in no element of the power of healing, and a Gnostic who is defective in no element of scientific knowledge.

Now demonstration differs from syllogism; inasmuch as the point demonstrated is indicative of one thing, being one and identical; as we say that to be with child is the proof of being no longer a virgin. But what is apprehended by syllogism, though one thing, follows from several; as, for example, not one but several proofs are adduced of Pytho having betrayed the Byzantines, if such was the fact. And to draw a conclusion from what is admitted is to syllogize; while to draw a conclusion from what is true is to demonstrate.

So that there is a compound advantage of demonstration: from its assuming, for the proof of points in question, true premisses, and from its drawing the conclusion that follows from them. If the first have no existence, but the second follow from the first, one has not demonstrated, but syllogized. For, to draw the proper conclusion from the premisses, is merely to syllogize. But to have also each of the premisses true, is not merely to have syllogized, but also to have demonstrated.

And to conclude, as is evident from the word, is to bring to the conclusion. And in every train of reasoning, the point sought to be determined is the end, which is also called the conclusion. But no simple and primary statement is termed a syllogism, although true; but it is compounded of three such, at the least,—of two as premisses, and one as conclusion.

Now, either all things require demonstration, or some of them are self-evident. But if the first, by demanding the demonstration of each demonstration we shall go on ad infinitum; and so demonstration is subverted. But if the second, those things which are self-evident will become the starting points [and fundamental grounds] of demonstration.

In point of fact, the philosophers admit that the first principles of all things are indemonstrable. So that if there is demonstration at all, there is an absolute necessity that there be something that is self-evident, which is called primary and indemonstrable.

Consequently all demonstration is traced up to indemonstrable faith.1880    [We begin, that is, with axioms: and he ingeniously identifies faith with axiomatic truth. Hence the faith not esoteric.]

It will also turn out that there are other starting points for demonstrations, after the source which takes its rise in faith,—the things which appear clearly to sensation and understanding. For the phenomena of sensation are simple, and incapable of being decompounded; but those of understanding are simple, rational, and primary. But those produced from them are compound, but no less clear and reliable, and having more to do with the reasoning faculty than the first. For therefore the peculiar native power of reason, which we all have by nature, deals with agreement and disagreement. If, then, any argument be found to be of such a kind, as from points already believed to be capable of producing belief in what is not yet believed, we shall aver that this is the very essence of demonstration.

Now it is affirmed that the nature of demonstration, as that of belief, is twofold: that which produces in the souls of the hearers persuasion merely, and that which produces knowledge.

If, then, one begins with the things which are evident to sensation and understanding, and then draw the proper conclusion, he truly demonstrates. But if [he begin] with things which are only probable and not primary, that is evident neither to sense nor understanding, and if he draw the right conclusion, he will syllogize indeed, but not produce a scientific demonstration; but if [he draw] not the right conclusion, he will not syllogize at all.

Now demonstration differs from analysis. For each one of the points demonstrated, is demonstrated by means of points that are demonstrated; those having been previously demonstrated by others; till we get back to those which are self-evident, or to those evident to sense and to understanding; which is called Analysis. But demonstration is, when the point in question reaches us through all the intermediate steps. The man, then, who practices demonstration, ought to give great attention to the truth, while he disregards the terms of the premisses, whether you call them axioms, or premisses, or assumptions. Similarly, also, special attention must be paid to what suppositions a conclusion is based on; while he may be quite careless as to whether one choose to term it a conclusive or syllogistic proposition.

For I assert that these two things must be attended to by the man who would demonstrate—to assume true premisses, and to draw from them the legitimate conclusion, which some also call “the inference,” as being what is inferred from the premisses.

Now in each proposition respecting a question there must be different premisses, related, however, to the proposition laid down; and what is advanced must be reduced to definition. And this definition must be admitted by all. But when premisses irrelevant to the proposition to be established are assumed, it is impossible to arrive at any right result; the entire proposition—which is also called the question of its nature—being ignored.

In all questions, then, there is something which is previously known,—that which being self-evident is believed without demonstration; which must be made the starting point in their investigation, and the criterion of apparent results.

Ὡσαύτως δὲ καὶ τὴν ἀπόδειξιν πάντες ἄνθρωποι ὁμολογήσαιεν [ἂν] λόγον εἶναι τοῖς ἀμφισβητουμένοις ἐκ τῶν ὁμολογουμένων ἐκπορίζοντα τὴν πίστιν. οὐ μόνον δὲ ἀπόδειξις καὶ πίστις καὶ γνῶσις, ἀλλὰ καὶ πρόγνωσις λέγεται διχῶς, ἣ μὲν ἐπιστημονική τε καὶ βεβαία, ἄλλη δὲ μόνον ἐλπιστική. κυριώτατα μὲν οὖν ἀπόδειξις λέγεται ἡ τὴν ἐπιστημονικὴν πίστιν ἐντιθεῖσα ταῖς τῶν μανθανόντων ψυχαῖς, δοξαστικὴ δὲ ἡ ἑτέρα, ὡς καὶ ἄνθρωπος ὁ μὲν ὄντως ἄνθρωπος ὁ τὰς κοινὰς φρένας κεκτημένος, ὃ δὲ ἄγριος καὶ θηριώδης. οὕτω τοι καὶ ὁ κωμικὸς ἔλεγεν· ὡς χαρίεν ἐστὶν ἄνθρωπος, ἔστ' ἂν ἄνθρωπος ᾖ. καὶ ἐπὶ βοὸς καὶ ἵππου καὶ κυνὸς ὁ αὐτὸς λόγος παρὰ τὴν ἀρετὴν τοῦ ζῴου ἢ κακίαν. εἰς γὰρ τὴν τοῦ γένους τελειότητα βλέποντες ἐπὶ τὰ κυριώτατα τῶν σημαινομένων ἐρχόμεθα. αὐτίκα ἰατρὸν νοοῦμεν ᾧ μηδὲν τῆς ἰατρικῆς δυνάμεως ἐνδεῖ, γνωστικὸν δὲ ᾧ μηδὲν λείπει τῆς ἐπιστημονικῆς εἰδήσεως. Καὶ διαφέρει ἔνδειξις συλλογισμοῦ ᾗ τὸ μὲν ἐνδεικνύμενον ἑνός ἐστι δηλωτικὸν ἓν ὑπάρχον καὶ αὐτό, ὡς τοῦ μηκέτι εἶναι παρθένον ἔνδειξιν εἶναι τὸ κυεῖν φαμεν, τὸ δὲ συλλογισμῷ λαμβανόμενον ἓν ὑπάρχον ἕπεται πλείοσιν, ὡς τοῦ Πύθωνα προδιδόναι Βυζαντίους, εἰ οὕτως ἔτυχεν, οὐχ ἕν, ἀλλὰ πλείω λαμβάνεται τὰ δηλωτικά. καὶ τὸ μὲν περαίνειν ἐξ ὁμολογουμένων συλλογίζεσθαί ἐστιν, τὸ δέ γε ἐξ ἀληθῶν τι περαίνειν ἀποδεικνύειν ἐστίν, ὥστ' εἶναι σύνθετόν τινα τῆς ἀποδείξεως τὴν ὄνησιν ἔκ τε τοῦ τὰ λαμβανόμενα πρὸς τὰ ζητούμενα ἀληθῆ λαμβάνειν κἀκ τοῦ τὸ συμπέρασμα αὐτοῖς ἀκόλουθον ἐπιφέρεσθαι. εἰ δ' ἤτοι μὴ ὑπάρχοι τὸ πρότερον ἢ μὴ ἕποιτο αὐτῷ τὸ δεύτερον, οὐκ ἀπέδειξεν μέν, συνελογίσατο δέ. τὸ μὲν γὰρ οἰκεῖον ἐπενεγκεῖν συμπέρασμα τοῖς λήμμασιν συλλογίσασθαι μόνον ἐστίν, τὸ δὲ καὶ τῶν λημμάτων ἕκαστον ὑπάρχειν ἀληθὲς οὐ συλλογίσασθαι μόνον, ἀλλὰ καὶ ἀποδεδειχέναι. περαίνειν δ' ἐστίν, ὡς καὶ ἐκ τοῦ ὀνόματος δῆλον, τὸ ἄγειν ἐπὶ τὸ πέρας. ἔστι δὲ δήπου πέρας ἐν ἑκάστῳ λόγῳ τὸ ζητούμενον, ὃ δὴ καὶ συμπέρασμα καλεῖται. οὐδεὶς δὲ ἁπλοῦς καὶ πρῶτος λόγος ὀνομάζεται συλλογισμός, κἂν ἀληθὴς ᾖ, ἀλλ' ἔστι τοὐλάχιστον ἐκ τριῶν τοιούτων σύνθετον, δυεῖν μὲν ὡς λημμάτων, ἑνὸς δὲ ὡς συμπεράσματος. Ἤτοι δὲ πάντα ἀποδείξεως δεῖται ἢ καί τινα ἐξ αὑτῶν ἐστι πιστά. ἀλλ' εἰ μὲν τὸ πρότερον, ἑκάστης ἀποδείξεως ἀπόδειξιν αἰτοῦντες εἰς ἄπειρον ἐκβησόμεθα καὶ οὕτως ἀνατραπήσεται ἡ ἀπόδειξις· εἰ δὲ τὸ δεύτερον, ταῦτα αὐτὰ τὰ ἐξ αὑτῶν πιστὰ τῶν ἀποδείξεων ἀρχαὶ γενήσονται. αὐτίκα οἱ φιλόσοφοι ἀναποδείκτους ὁμολογοῦσι τὰς τῶν ὅλων ἀρχάς. ὥστ' εἴπερ ἐστὶν ἀπόδειξις, ἀνάγκη πᾶσα πρότερον εἶναί τι πιστὸν ἐξ ἑαυτοῦ, ὃ δὴ πρῶτον καὶ ἀναπόδεικτον λέγεται. ἐπὶ τὴν ἀναπόδεικτον ἄρα πίστιν ἡ πᾶσα ἀπόδειξις ἀνάγεται. εἶεν δ' ἂν καὶ ἄλλαι τῶν ἀποδείξεων ἀρχαὶ μετὰ τὴν ἐκ πίστεως πηγήν, τὰ πρὸς αἴσθησίν τε καὶ νόησιν ἐναργῶς φαινόμενα. τὰ μὲν γὰρ πρὸς αἴσθησιν συμβάντα ἐστὶν ἁπλᾶ τε καὶ ἄλυτα, τὰ δὲ πρὸς νόησιν ἁπλᾶ τε καὶ λογικὰ καὶ πρῶτα, τὰ δὲ ἐξ αὐτῶν γεννώμενα σύνθετα μέν, οὐδὲν δ' ἧττον ἐναργῆ καὶ πιστὰ καὶ λογικώτερα τῶν πρώτων. ἀκολούθου καὶ μαχομένου οὖν * ἐστιν ἥνπερ ἰδίαν λόγου δύναμιν πεφυκυῖαν ἅπαντες ἔχομεν φύσει. ἐὰν οὖν τις εὑρεθῇ λόγος τοιοῦτος οἷος ἐκ τῶν ἤδη πιστῶν τοῖς οὔπω πιστοῖς ἐκπορίζεσθαι τὴν πίστιν δυνάμενος, αὐτὸν τοῦτον εἶναι φήσομεν οὐσίαν ἀποδείξεως. εἴρηται δὲ ὡς καὶ τὸ τῆς πίστεως καὶ τὸ τῆς ἀποδείξεως γένος διττόν, τὸ μὲν πειθὼ μόνον ταῖς ψυχαῖς τῶν ἀκουόντων, τὸ δὲ ἐπιστήμην ἀπεργαζόμενον. εἰ μὲν οὖν ἐκ τῶν πρὸς αἴσθησιν καὶ νόησιν ἐναργῶν ἄρξαιτό τις, κἄπειτα τὸ οἰκεῖον ἐπενέγκοι συμπέρασμα, ὄντως ἀποδείκνυσιν, εἰ δ' ἐξ ἐνδόξων μόνον, οὐ μὴν πρώτων γε, τουτέστιν οὔτε πρὸς αἴσθησιν οὔτε πρὸς νόησιν ἐναργῶν, εἰ μὲν οἰκεῖον ἐπιφέροι συμπέρασμα, συλλογιεῖται μέν, οὐ μὴν ἐπιστημονικήν γε ποιήσεται τὴν ἀπόδειξιν, εἰ δ' οὐκ οἰκεῖον, οὐδὲ συλλογιεῖται τὴν ἀρχήν. ∆ιαφέρει δ' ἀναλύσεως ἀπόδειξις· ἕκαστον μὲν γὰρ τῶν ἀποδεικνυμένων διά τινων ἀποδεικνυμένων ἀποδείκνυται, προαποδεδειγμένων κἀκείνων ὑφ' ἑτέρων, ἄχρις ἂν εἰς τὰ ἐξ ἑαυτῶν πιστὰ ἀναδράμωμεν ἢ εἰς τὰ πρὸς αἴσθησίν τε καὶ νόησιν ἐναργῆ, ὅπερ ἀνάλυσις ὀνομάζεται· ἀπόδειξις δέ ἐστιν, ὅταν ἀπὸ τῶν πρώτων [τις εἰς] τὸ ζητούμενον ἀφικνῆται διὰ πάντων τῶν ἐν μέσῳ. χρὴ τοίνυν τὸν ἀποδεικτικὸν ἄνδρα τῆς μὲν ἀληθείας ὡς τῶν λημμάτων πολλὴν ποιήσασθαι πρόνοιαν, τῶν δὲ ὀνομάτων ἀφροντιστεῖν, εἴτε ἀξιώματά τις ἐθέλοι καλεῖν εἴτε προτάσεις εἴτε λήμματα, ὁμοίως δὲ καὶ τοῦ τίνων ὑποκειμένων τί περαίνεται πολλὴν ὡσαύτως πεποιῆσθαι τὴν πρόνοιαν, εἴτε δὲ περαίνοντα λόγον εἴτε περαντικὸν εἴτε συλλογιστικὸν ἐθέλοι τις ὀνομάζειν αὐτόν, ἥκιστα φροντίζειν. δύο γὰρ ταῦτα ἐν ἅπασι χρῆναί φημι τὸν ἀποδεικτικὸν φυλάττειν, τὰ μὲν λήμματα ἀληθῆ λαμβάνειν, ἀκόλουθον δ' αὐτοῖς ἐπιφέρειν [τὸ] συμπέρασμα, ὅπερ τινὲς καὶ ἐπιφορὰν καλοῦσιν, τὸ ἐπιφερόμενον τοῖς λήμμασιν. Περὶ παντὸς τοῦ ζητουμένου καθ' ἕκαστον πρόβλημα διαφόρων μὲν δεῖ τῶν λημμάτων, οἰκείων δὲ τῷ προβληθέντι, καὶ [τὸ] προβληθὲν αὐτὸ εἰς λόγον μεταλαμβάνειν ἀναγκαῖον, τόν τε λόγον τοῦτον ὁμολογούμενον ἅπασιν εἶναι προσήκει· τῶν δὲ λημμάτων μὴ οἰκείων τῷ προβλήματι λαμβανομένων οὐκ ἐνδέχεται καλῶς ** οὐδὲν αὐτῷ ἐξευρεῖν ἀγνοουμένης ὅλου τοῦ προβλήματος, ὃ καὶ ζήτημα καλεῖται τῆς φύσεως. ἐν πᾶσιν οὖν τοῖς ζητουμένοις ἔστι τι προγινωσκόμενον (ὃ πάντως ἐξ ἑαυτοῦ πιστὸν ὂν ἀναποδείκτως πιστεύεται), ὃ χρὴ ποιεῖσθαι τῆς ζητήσεως αὐτῶν ὁρμητήριον καὶ τῶν εὑρῆσθαι δοκούντων κριτήριον.