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Demonstration differs from definition.[1] First, we have to investigate definition, whether it is either one method in virtue of which the definitions of all existing things can be offered, or many. And if [it is] one, which one; but if many, how many in number and what kinds. And when by means of a description or a definition we discover so many and such things, another investigation again appears to us: which methods we have to make use of with regard to which objects. And when we discover this, since definition is by genera and differentiae, [we have to establish] which the genus of the object that is set before is and which the differentiae are, in order that we can offer its definition while combining the coherent differentiae by genus. There is another difficulty too. For in whatever something is present primarily, this also belongs to by itself. And [the expression] "by itself" means "what is according to substance".[2] But even though there is no something by itself any more, this also belongs to in a primary way. For animal belongs by itself to man, although not in a primary sense. Therefore, even though there is no man, there is animal, so animal belongs to man not in a primary sense. Similarly figure also [belongs] by itself to triangle, although not primarily. But a demonstration in the strict sense arises from what belongs to by itself and in a primary sense. And without qualification,[a demonstration is what] confirms the caused effects out of causes and the particulars out of the universals. And [a demonstration] confirming causes out of caused effects is called "conclusive proof";[3] for instance, to show that the moon is spherical out of its lights. For if it were shown that it gives light in this manner because it is spherical, that would be a demonstration in the strict sense. For what is caused would be shown out of a cause. But we have not made use of this mode of proof in this case, because what is caused is clearer to us than the cause. A conclusive demonstration is also to discover and understand that there is fire because there is smoke.[4] And if what is caused absolutely follows the cause, this is called "conclusive proof"; and a conclusive proof is an irrefutable sign.[5] But if [what is caused] did not absolutely follow [the cause], the sign would not be called conclusive proof anymore, because [in that case] it is not irrefutable, such as it happens in the case [of a woman] who is pale because she has given birth to a child. For having given birth to a child is not absolutely concomitant with paleness.[6] A demonstration is a deductive method through deductive arguments,[7] and it exists when a deductive argument arises from true and primary premises.

Ἀπόδειξις: ἀπόδειξις ὅρου διαφέρει. δεῖ πρῶτον ζητῆσαι περὶ ὅρου, ἆρα μία ἐστὶ μέθοδος, καθ' ἣν πάντων τῶν ὄντων δυνατὸν τοὺς ὅρους ἀποδοῦναι, ἢ πολλαί. καὶ εἰ μία, τίς αὕτη: εἰ δὲ πολλαὶ, πόσαι τὸν ἀριθμὸν καὶ ποῖαι. καὶ ὅταν εὕρωμεν, τοσαίδε καὶ τοιαίδε, ὑπογραφῇ ἢ ὁρισμῷ, ἄλλη πάλιν ἡμᾶς διαδέχεται ζήτησις, ποίαις τῶν μεθόδων ἐπὶ ποίων πραγμάτων χρηστέον. καὶ ὅταν εὕρωμεν τοῦτο, ἐπειδὴ ὁ ὁρισμὸς ἐκ γενῶν ἐστι καὶ διαφορῶν, τί τοῦ προκειμένου πράγματος τὸ γένος καὶ τίνες αἱ διαφοραί, ἵνα συμπλέξαντες τὰς συστατικὰς διαφορὰς τῷ γένει, ἀποδῶμεν αὐτοῦ τὸν ὁρισμόν. ἔστι δὲ καὶ ἑτέρα χαλεπότης. ᾥτινι γὰρ πρώτως τι ἐνυπάρχει, τοῦτο καὶ καθ' αὑτὸ ὑπάρχει. καθ' αὑτὸ δέ ἐστι τὸ κατ' οὐσίαν. οὐκέτι δὲ καὶ εἴ τι καθ' αὑτὸ ὑπάρχει, τοῦτο καὶ πρώτως ὑπάρχει. τὸ γὰρ ζῷον τῷ ἀνθρώπῳ καθ' αὑτὸ μὲν ὑπάρχει, οὐ πρώτως δέ: διὸ κἂν μὴ ᾖ ἄνθρωπος, ζῷόν ἐστιν, ὥστε οὐ πρώτως τῷ ἀνθρώπῳ τὸ ζῷον ὑπάρχει. ὁμοίως δὲ καὶ τὸ σχῆμα τῷ τριγώνῳ καθ' αὑτὸ μέν, οὐ μέντοι πρώτως. ἡ δὲ κυρίως ἀπόδειξις ἐκ τῶν καθ' αὑτὸ καὶ πρώτως ὑπαρχόντων γίνεται. καὶ ἁπλῶς, ἡ ἐκ τῶν αἰτίων τὰ αἰτιατὰ πιστουμένη καὶ ἐκ τῶν καθόλου τὰ μερικώτερα. ἡ δὲ ἐκ τῶν αἰτιατῶν τὰ αἴτια πιστουμένη τεκμηριώδης ἀπόδειξις λέγεται: οἷον τὸ δεῖξαι ἐκ τῶν φωτισμῶν τῆς σελήνης ὅτι σφαιρική. εἰ γὰρ ἐδείκνυτο, διότι σφαιρική, διὰ τοῦτο φωτίζεσθαι οὕτως, τοῦτο ἂν εἴη κυρίως ἀπόδειξις. ἐκ γὰρ τοῦ αἰτίου τὸ αἰτιατὸν ἐδείκνυτ' ἄν. οὐ κεχρήμεθα δὲ ἐνταῦθα τῇ δείξει ταύτῃ, διότι τὸ αἰτιατὸν πρὸς ἡμᾶς σαφέστερόν ἐστι τοῦ αἰτίου. τεκμηριώδης δὲ ἀπόδειξις καὶ τὸ ἐκ τοῦ καπνοῦ τὸ πῦρ εὑρεῖν καὶ κατανοῆσαι. καὶ ἂν μὲν πάντως ἕπηται τὸ αἰτιατὸν τῷ αἰτίῳ, τεκμήριον τοῦτο λέγεται. τεκμήριον δέ ἐστι τὸ ἄλυτον σημεῖον. ἂν δὲ μὴ πάντως ἕποιτο, σημεῖον μὲν ἂν λέγοιτο, τεκμήριον δὲ οὐκέτι, διότι μηδὲ ἄλυτόν ἐστι: οἷον ὡς τὸ, ὠχρά ἐστιν, ἐπειδὴ τέτοκεν: οὐ πάντως γὰρ τῇ ὠχρότητι παρακολουθεῖ τὸ τετοκέναι. ἀπόδειξίς ἐστι μέθοδος διὰ συλλογισμῶν συλλογιστική, ὅταν ἐξ ἀληθῶν καὶ πρώτων ὁ συλλογισμὸς γίνηται.

This entry draws largely on Philoponus, Commentary on Aristotle's de anima (28.7-18, 29.23-28, 20.12-13 and 20-26, 31.15-20 Hayduck), with a final sentence from Alexander of Aphrodisias, Commentaries on Aristotle's Topica 2.22-23.
[1] A demonstration is, in fact, a certain type of syllogism, for a deductive argument (or "syllogism") is a demonstration when the premises from which the deduction starts are true and primary, or are such that our knowledge of them has originally come through premises which are primary and true (see Topica 100a27-29). In more technical contexts Aristotle distinguishes ὅρος from ὁρισμος , the former meaning "term" and the latter "definition". In the present case both words are used indiscriminately, something more or less usual in Aristotle himself. For the Aristotelian definition by proximate genus and specific differentia, see Metaphysics 1037b25-1038a35 and Topica 103b14-19.
[2] As usual, the word οὐσία is hard to render. In this context it probably means "essence", so the expression τὸ κατ' οὐσίαν should be rendered "the essential". In the Suda example, animal belongs "by itself" (which, in accordance with the previous distinction, is the same as "essentially") to man.
[3] τεκμήριον and σημεῖον are Aristotle's technical terms. A τεκμήριον is characterized as being "a necessary or always existing sign" or "a necessary sign" (Rhetoric 1357b4ff.). When people think it is impossible to refute an argument, they think they are offering a τεκμήριον , as though the matter at issue were shown and concluded (Rhetoric 1357b5-10). On the other hand, according to Aristotle, a σημεῖον (sign) is meant to be a demonstrative proposition (πρότασις ἀποδεικτική ), either necessary or reputable; for anything such that when it is another thing is, or when it has come into being the other has come into being before or after, is a sign of the other's being or having coming into to being (An.Pr. 70a6-9; Oxford Translation). That is to say, given two propositions (p, q) indicating two events, p is said to be a sign of q if and only if p's verification implies q's verification. The fact that a sign is included within the domain of what "is reputable" or within the domain of "common beliefs" (ἔνδοξα ) shows that a sign, insofar as it is refutable, is fallible. For instance, if someone were to state that since Socrates was wise and just, it is a sign that the wise are just. Although this is a sign, it is refutable (the example is Aristotle's; Rhetoric 1357b12-14). So a sign differs from a τεκμήριον , because the former is a fallible type of demonstration while the latter is not. From the reasons offered above, a plausible translation of τεκμήριον is "conclusive proof". For further discussion on this issue see Mignucci 1969, 722.
[4] These lines are literally reproduced in tau 245, and the example is probably taken from Sextus Empiricus, Pyrrhoniae hypotyposes 2.100. The case of fire and smoke is an example of what Sextus calls "recollective sign", i.e. the one that "having been observed evidently together with the thing it signifies, at the same time as it makes an impression on us [...] it leads us to recall the thing which has been observed together with it and is not now making an evident impression on us, as in the case of smoke and fire" (Translation Barnes and Annas; see also Sextus, Adversis mathematicos 8.157). In the example, of course, smoke is the "recollective sign" of the presence of fire.
[5] This is, once more, Aristotle's position (see Rhetoric 1357b16-17, 1403a13-15.
[6] The example is taken from Aristotle, An.Pr. 70a13-16; 20-24; 35-38. The account offered by Aristotle is the following: the proof that a woman is pregnant because she is pale is meant to come through the middle figure. For since paleness follows pregnant women and is a concomitant of this woman, people suppose it has been proved that she is pregnant (An.Pr. 70a20-23).
[7] That is to say "syllogisms" (συλλογισμοί ; see here note 1). On the scholarly controversy with regard to the relation between demonstration and syllogistic, see Barnes (1994) xv-xviii.

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