§1.1.23Ὅτι μὲν οὖν οἱ ἐν τούτοις τοῖς σχήμασι συλλογισμοὶ τελειοῦνταί τε διὰ τῶν ἐν τῷ πρώτῳ σχήματι καθόλου συλλογισμῶν καὶ εἰς τούτους ἀνάγονται, δῆλον ἐκ τῶν εἰρημένων·
That the syllogisms in these figures, then, are both completed through the universal syllogisms in the first figure and reduced to them, is clear from what has been said.
ὅτι δʼ ἁπλῶς πᾶς συλλογισμὸς οὕτως ἕξει, νῦν ἔσται φανερόν, ὅταν δειχθῇ πᾶς γινόμενος διὰ τούτων τινὸς τῶν σχημάτων.
But that simply every syllogism will be related in this way will now be manifest, when it is shown that every [syllogism] comes about through one of these figures.
Ἀνάγκη δὴ πᾶσαν ἀπόδειξιν καὶ πάντα συλλογισμὸν ἢ ὑπάρχον τι ἢ μὴ ὑπάρχον δεικνύναι, καὶ τοῦτο ἢ καθόλου ἢ κατὰ μέρος, ἔτι ἢ δειπτικῶς ἢ ἐξ ὑποθέσεως.
Indeed, it is necessary that every demonstration and every syllogism shows either that something belongs or that it does not belong, and this either universally or particularly, and furthermore either ostensively or from a hypothesis.
τοῦ δʼ ἐξ ὑποθέσεως μέρος τὸ διὰ τοῦ ἀδυνάτου.
And a part of that from a hypothesis is that through the impossible.
πρῶτον οὖν εἴπωμεν περὶ τῶν δεικτικῶν· τούτων γὰρ δειχθέντων φανερὸν ἔσται καὶ ἐπὶ τῶν εἰς τὸ ἀδύνατον καὶ ὅλως τῶν ἐξ ὑποθέσεως.
First, then, let us speak about ostensive syllogisms; for when these have been shown, it will be manifest also in the case of those leading to the impossible and generally those from a hypothesis.
Εἰ δὴ δέοι τὸ Α κατὰ τοῦ Β συλλογίσασθαι ἢ ὑπαρχον ἢ μὴ ὑπάρχον, ἀνάγκη λαβεῖν τι κατά τινος.
If, then, it should be required to syllogize that A belongs or does not belong to B, it is necessary to take something of something.
εἰ μὲν οὖν τὸ Α κατὰ τὸ Β ληφθείη, τὸ ἐξ ἀρχῆς ἔσται εἰλημμένον.
If, then, A should be taken of B, the original question will have been taken.
εἰ δὲ κατὰ τοῦ Γ, τὸ δὲ Γ κατὰ μηδενός, μηδʼ ἄλλο κατʼ ἐκείνου, μηδὲ κατὰ τοῦ ἕτερον, οὐδεὶς ἔσται συλλογισμός· τῷ γὰρ ἕν καθʼ ἑνὸς ληφθῆναι οὐδὲν συμβαίνει ἐξ ἀνάγκης.
But if [A should be taken] of Γ, and Γ of nothing, and nothing else of Γ, and nothing else of another [thing], there will be no syllogism; for by one thing being taken of one thing, nothing results of necessity.
ὥστε προσληπτέον καὶ ἑτέραν πρότασιν.
So another premise must be taken in addition.
ἐὰν μὲν οὖν ληφθῇ τὸ κατʼ ἄλλου ἢ ἄλλο κατὰ τοῦ Α, ἢ κατὰ τοῦ Γ ἕτερον, εἶναι μὲν συλλογισμὸν οὐδὲν κωλύει, πρός μέντοι τὸ Β οὐκ ἔσται διὰ τῶν εἰλημμένων.
If, then, [A] should be taken of another, or another of A, or another of Γ, nothing prevents there being a syllogism, but indeed there will be none in relation to B through the assumed premises.
οὐδʼ ὅταν τὸ Γ ἑτέρῳ, κἀκεῖνο ἄλλῳ, καὶ τοῦτο ἑτέρῳ, μὴ συνάπτῃ δὲ πρὸς τὸ Β, οὐδʼ οὕτως ἔσται πρὸς τὸ Β συλἀογισμός.
Nor when Γ [belongs] to another, and that to another, and this to another, but it does not connect to B, not even in this way will there be a syllogism in relation to B.
ὅλως γὰρ εἴπομεν ὅτι οὐδεὶς οὐδέποτε ἔσταισυλλογισμὸς ἄλλου κατʼ ἄλλου μὴ ληφθέντος τινός μέσου, ὃ πρὸς ἑκάτερον ἔχει πως ταῖς κατηγορίαις·
For in general we said that there will never be any syllogism of one thing of another unless some middle is assumed, which is related to each in some way by the predications.
ὁ μὲν· γὰρ συλλογισμὸς ἀπλῶς ἐκ προτάσεων ἐστιν, ὁ δὲ πρὸς τόδε συλλογισμὸς ἐκ τῶν πρὸς τόδε προτάσεων, ὁ δὲ τοῦδε· πρός τόδε διὰ τῶν τοῦδε πρός τόδε προτάσεων.
For the syllogism simply is from premises, but the syllogism in relation to this [particular thing] is from premises in relation to this, and the syllogism of this in relation to this is through premises of this in relation to this.
ἀδύνατον δέ πρός τὸ Β λαβεῖν πρότασιν μηδὲν μήτε κατηγοροῦντας αὐτοῦ μήτʼ ἀπαρνουμένους, ἢ πάλιν τοῦ Α πρὸς τὸ Β μηδὲν κοινὸν λαμβάνοντας ἀλλ’ ἑκατέρου ἴδια ἄττα κατηγοροῦντας ἢ ἀπαρνουμένους.
But it is impossible to assume a premise in relation to B while neither affirming nor denying anything of it, or again, by assuming nothing common to A and B but rather affirming or denying some peculiar things of each.
ὥστε ληπτέον τι μέσον ἀμφοῖν, ὃ συνάψει τὰς κατηγορίας, εἴπερ ἔσται τοῦδε πρός τόδε συλλογισμός.
So we must assume some middle between both, which will connect the predications, if indeed there is to be a syllogism of this in relation to this.
εἰ οὖν ἀνάγκη μέν τι λαβεῖν πρὸς ἄμφω κοινόν, τοῦτο δʼ ἐνδέχεται τριχῶς (ἢ γὰρ τὸ τοῦ Γ καὶ τὸ Γ τοῦ Β κατηγορήσαντας, ἢ τὸ Γ κατʼ ἀμφοῖν, ἢ ἄμφω κατὰ τοῦ Γ), ταῦτα δʼ ἐστὶ τὰ εἰρημένα σχήματα, φανερόν ὅτι πάντα συλλογισμὸν ἀνάγκη γίνεσθαι διὰ τούτων τινὸς τῶν σχημάτων.
If, then, it is necessary to assume something common to both, and this is possible in three ways (for either by predicating A of Γ and Γ of B, or Γ of both, or both of Γ), and these are the figures mentioned, it is clear that every syllogism must necessarily come about through one of these figures.
ὁ γὰρ αὐτὸς λόγος καὶ εἰ διὰ πλειόνων συνάπτοι πρός τὸ Β. ταὐτὸ γὰρ ἔσται σχῆμα καὶ ἐπὶ τῶν πολλῶν.
For the same account holds also if it should connect to B through more [terms]; for the figure will be the same even in the case of many.
Ὅτι μὲν οὖν οἱ δεικτικοὶ περαίνονται διὰ τῶν προειρημένων σχημάτων, φανερόν· ὅτι δὲ καὶ οἱ εἰς τὸ ἀδύνατον, δῆλον ἔσται διὰ τούτων.
That the ostensive [syllogisms], then, are concluded through the aforementioned figures is clear; but that also those leading to the impossible [are concluded so] will be manifest through these.
πάντες γὰρ οἱ διὰ τοῦ ἀδυνάτου περαίνοντες τὸ μὲν ψεῦδος συλλογίζονται, τὸ δʼ ἐξ ἀρχῆς ἐξ ὑποθέσεως δεικνύουσιν, ὅταν ἀδύνατόν τι συμβαίνῃ τῆς ἀντιφάσεως τεθείσης, οἶον ὅτι ἀσύμμετρος ἡ διάμετρος διὰ τὸ γίνεσθαι τὰ περιττὰ ἴσα τοῖς ἀρτίοις συμμέτρου τεθείσης.
For all those concluding through the impossible syllogize a falsehood, but show the original question from a hypothesis when something impossible results when the contradiction is assumed; for example, that the diagonal is incommensurable is shown because odd numbers become equal to even numbers when it is assumed to be commensurable.
τὸ μὲν οὖν ἴσα γίνεσθαι τὰ περιττὰ τοῖς ἀρτίοις συλλογίζεται, τὸ δʼ ἀσύμμετρον εἶναι τὴν διάμετρον ἐξ ὑποθέσεως δείκνυσιν, ἐπεὶ ψεῦδος συμβαίνει διὰ τὴν ἀντίφασιν.
For that odd numbers become equal to even numbers is syllogized, but that the diagonal is incommensurable is shown from a hypothesis, since a falsehood results because of the contradiction.
τοῦτο γὰρ ἦν τὸ διὰ τοῦ ἀδυνάτου συλλογίσασθαι, τὸ δεῖξαί τι ἀδύνατον διὰ τὴν ἐξ ἀρχῆς ὑπόθεσιν.
For this was to syllogize through the impossible: to show something impossible because of the original hypothesis.
ὥστʼ ἐπεὶ τοῦ ψεύδους γίνεται συλλογισμὸς δεικτικὸς ἐν τοῖς εἰς τὸ ἀδύνατον ἀπαγομένοις, τὸ δʼ ἐξ ἀρχῆς ἐξ ὑποθέσεως δείκνυται, τοὺς δὲ δεικτικοὺς πρότερον εἴπομεν ὅτι διὰ τούτων περαίνονται τῶν σχημάτων, φανερὸν ὅτι καὶ οἱ διὰ τοῦ ἀδυνάτου συλλογισμοὶ διὰ τούτων ἔσονται τῶν σχημάτων.
So since a directive (ostensive) syllogism of the falsehood comes about in those leading to the impossible, while the original question is shown from a hypothesis, and we said before that ostensive syllogisms are concluded through these figures, it is clear that also syllogisms through the impossible will be through these figures.
ὡσαύτως δὲ καὶ οἱ ἄλλοι πάντες οἱ ἐξ ὑποθέσεως· ἐν ἅπασι γὰρ ὁ μὲν συλλογισμὸς γίνεται πρὸς τὸ μεταλαμβανόμενον, τὸ δʼ ἐξ ἀρχῆς περαίνεται διʼ ὁμολογίας ἤ τινος ἄλλης ὑποθέσεως.
And in the same way also all the other [syllogisms] from a hypothesis; for in all of them the syllogism comes about in relation to the substituted [proposition], while the original question is concluded through agreement or some other hypothesis.
εἰ δὲ τοῦτʼ ἀληθές, πᾶσαν ἀπόδειξιν καὶ πάντα συλλογισμὸν ἀνάγκη γίνεσθαι διὰ τριῶν τῶν προειρημένων σχημάτων.
And if this is true, it is necessary that every demonstration and every syllogism comes about through the three aforementioned figures.
τούτου δὲ δειχθέντος δῆλον ὡς ἅπας τε συλλογισμός ἐπιτελεῖται διὰ τοῦ πρώτου σχήματος καὶ ἀνάγεται εἰς τοὺς ἐν τούτῳ καθόλου συλλογισμούς.
And when this has been shown, it is clear that every syllogism both is performed through the first figure and is reduced to the universal syllogisms in this [figure].