Aristotle's Proofs Through the Impossible in Prior Analytics 1.15

Abstract

In Prior Analytics 1.15, Aristotle attempts to give a proof through the impossible of Barbara, Celarent, Darii, and Ferio with an assertoric first premiss, a contingent second premiss, and a possible conclusion. These proofs have been controversial since antiquity. I shall show that they are valid, and that Aristotle is able to explain them by relying on two meta-syllogistic lemmas on the nature of possibility interpreted as syntactic consistency. It will turn out that Aristotle's proofs are not of the intended schemata. I shall investigate some of the results that the impact of this reconstruction on the modal syllogistic has: the relationship between Aristotle's syllogistic and the logics of relevance; the value of Aristotle's requirement that universal affirmative propositions be taken ‘absolutely’; the destruction of many Aristotelian proofs; the recovery of certain principles of modal opposition from a charge of inconsistency.

Keywords:

• Aristotle
• Prior Analytics
• modal syllogistic
• possibility
• relevance

Notes

1 Henceforth, for Greek authors I shall use Liddell et al. 1940's standard abbreviations. For Latin authors, I shall employ abbreviations that are easy to decode.

2 I adopt the medieval convention of indicating universal affirmative propositions with the letter a, universal negative propositions with the letter e, particular affirmative propositions with the letter i, and particular negative propositions with the letter o. I use the following notation to denote assertoric, apodeictic, contingent (or two-sided possible) and possible (or one-sided possible) propositions, respectively: ${\bullet }_{\mathrm{X}}$, ${\bullet }_{\mathrm{N}}$, ${\bullet }_{\mathrm{Q}}$, ${\bullet }_{\mathrm{M}}$; $\bullet \in \{a;e;i;o\}$. Accordingly, I speak of $a_{\mathrm{X}}$-propositions, and analogously for the other combinations. And when I speak of, e.g. Barbara XQM, I mean the schema Barbara in which the first premiss is an assertoric proposition, the second a contingent proposition, and the conclusion a possible proposition. In the case of assertoric schemata, I shall drop the Xs.

3 πρῶτον δὲ λϵκτέον ὅτι ϵἰ τοῦ Α ὄντος ἀνάγκη τὸ Β ϵἶναι, καὶ δυνατοῦ ὄντος τοῦ Α δυνατὸν ἔσται καὶ τὸ Β ἐξ ἀνάγκης. ἔστω γὰρ οὕτως ἐχόντων τὸ μὲν ἐφ' ᾧ τὸ Α δυνατόν, τὸ δ' ἐφ' ᾧ τὸ Β ἀδύνατον. ϵἰ οὖν τὸ μὲν δυνατόν, ὅτϵ δυνατὸν ϵἶναι, γένοιτ' ἄν, τὸ δ' ἀδύνατον, ὅτ' ἀδύνατον, οὐκ ἂν γένοιτο, ἅμα δ' ϵἴη τὸ Α δυνατὸν καὶ τὸ Β ἀδύνατον, ἐνδέχοιτ' ἂν τὸ Α γϵνέσθαι ἄνϵυ τοῦ Β, ϵἰ δὲ γϵνέσθαι, καὶ ϵἶναι· τὸ γὰρ γϵγονός, ὅτϵ γέγονϵν, ἔστιν. δϵῖ δὲ λαμβάνϵιν μὴ μόνον ἐν τῇ γϵνέσϵι τὸ ἀδύνατον καὶ δυνατόν, ἀλλὰ καὶ ἐν τῷ ἀληθϵύϵσθαι καὶ ἐν τῷ ὑπάρχϵιν, καὶ ὁσαχῶς ἄλλως λέγϵται τὸ δυνατόν· ἐν ἅπασι γὰρ ὁμοίως ἕξϵι. ἔτι τὸ ὄντος τοῦ Α τὸ Β ϵἶναι, οὐχ ὡς ἑνός τινος ὄντος τοῦ Α τὸ Β ἔσται δϵῖ ὑπολαβϵῖν· οὐ γὰρ ἔστιν οὐδὲν ἐξ ἀνάγκης ἑνός τινος ὄντος, ἀλλὰ δυοῖν ἐλαχίστοιν, οἷον ὅταν αἱ προτάσϵις οὕτως ἔχωσιν ὡς ἐλέχθη κατὰ τὸν συλλογισμόν. ϵἰ γὰρ τὸ Γ κατὰ τοῦ Δ, τὸ δὲ Δ κατὰ τοῦ Ζ, καὶ τὸ Γ κατὰ τοῦ Ζ ἐξ ἀνάγκης· καὶ ϵἰ δυνατὸν ἑκάτϵρον, καὶ τὸ συμπέρασμα δυνατόν. ὥσπϵρ οὖν ϵἴ τις θϵίη τὸ μὲν Α τὰς προτάσϵις, τὸ δὲ Β τὸ συμπέρασμα, συμβαίνοι ἂν οὐ μόνον ἀναγκαίου τοῦ Α ὄντος ἅμα καὶ τὸ Β ϵἶναι ἀναγκαῖον, ἀλλὰ καὶ δυνατοῦ δυνατόν. Τούτου δὲ δϵιχθέντος, φανϵρὸν ὅτι ψϵύδους ὑποτϵθέντος καὶ μὴ ἀδυνάτου καὶ τὸ συμβαῖνον διὰ τὴν ὑπόθϵσιν ψϵῦδος ἔσται καὶ οὐκ ἀδύνατον. οἷον ϵἰ τὸ Α ψϵῦδος μέν ἐστι μὴ μέντοι ἀδύνατον, ὄντος δὲ τοῦ Α τὸ Β ἔστι, καὶ τὸ Β ἔσται ψϵῦδος μὲν οὐ μέντοι ἀδύνατον. ἐπϵὶ γὰρ δέδϵικται ὅτι ϵἰ τοῦ Α ὄντος τὸ Β ἔστι, καὶ δυνατοῦ ὄντος τοῦ Α ἔσται τὸ Β δυνατόν, ὑπόκϵιται δὲ τὸ Α δυνατὸν ϵἶναι, καὶ τὸ Β ἔσται δυνατόν· ϵἰ γὰρ ἀδύνατον, ἅμα δυνατὸν ἔσται τὸ αὐτὸ καὶ ἀδύνατον (APr. 1.15, 34a5–33).

4 Cf. Smith 1989, p. 131; Ebert and Nortmann 2007, p. 548; Rosen and Malink 2012, p. 182.

5 Cf. Malink and Rosen 2013, pp. 957–960.

6 Cf. Hintikka 1973, p. 186; McClelland 1981, pp. 141–142; van Rijen 1989, pp. 21–23; Patterson 1995, pp. 155, 157, 161, 166; Fait 1999, pp. 141–142; Striker 2009, p. 143; Rosen and Malink 2012, pp. 182–187; Malink and Rosen 2013, p. 960.

7 Malink and Rosen 2013, p. 982; cf. also Mignucci 2002a, p. 168; Nortmann 2006, p. 380; Smith 2016, p. 64.

8 Other commentators have suggested that in APr. 1.15, ‘possible’ indicates semantic compatibility or consistency: cf. Ross 1949, p. 339; McCall 1963, p. 90; Angelelli 1979, pp. 196–197; Smith 1989, p. 132; Striker 2009, p. 145; Crubellier 2014, p. 264. But none of these scholars has properly (and formally) elaborated on this idea in connection with Aristotle's discussion at 34a5–33.

9 τϵθέντος δ' ὑπάρχϵιν, οὐδὲν ἔσται διὰ τοῦτ' ἀδύνατον· (APr. 1.13, 32a19–20).

10 By ‘⊢’, I indicate a consequence relation obtaining between a set of propositions and a single proposition. Regardless of what Aristotle says at APr. 1.1, 24b19 (the conclusion of a syllogism is ‘different from the things posited’), I take this consequence relation to be reflexive. On the relations of modal contradictoriness and incompatibility, cf. Malink 2013b, pp. 197–199.

11 Cf. also APr. 1.23, 40b33–37; 2.2, 53b16–20, 23–24; 2.11, 94a21–22, 24–27.

12 Cf. Tredennick 1938, p. 268 n. a; Malink 2013a, p. 221.

13 Cf. Ebert and Nortmann 2007, p. 549; Rosen and Malink 2012, pp. 183–185 (‘For example, the conclusion “some horses are not animals” follows from the premisses “some horses are sick ” and “no animals are sick”. The premisses are separately possible, yet the conclusion is not possible’).

14 The two ‘necessary’ (ἀναγκαίου at 34a23 and ἀναγκαῖον at 34a24) clearly express necessitas consequentis.

15 Cf., e.g. APo. 1.6, 75a1–11, 1.30, 87b22–25; Metaph. Δ 5, 1015b6–9.

16 Cf. Ross 1949, p. 339; Smith 1989, p. 131; Rosen and Malink 2012, p. 185.

17 Cf. McCall 1963, p. 90; Angelelli 1979, p. 196; Tricot 1983, p. 74 n. 1; Mueller and Gould 1999, p. 39; Rosen and Malink 2012, p. 185; Malink and Rosen 2013, p. 963; Crubellier 2014, p. 264; similarly Ross 1949, p. 341; Smith 1989, pp. 131–132; Patterson 1995, p. 159; Ebert and Nortmann 2007, p. 552; Rini 2011, pp. 147–148.

18 Cf. Alexander in APr. 185.17–20 Wallies; Tredennick 1938, p. 270 n. a; Ross 1949, p. 339; Mignucci 1969, p. 323 n. 10; Striker 2009, p. 145.

19 Cf. Mignucci 1969, p. 322 n. 9; Rosen and Malink 2012, p. 185; Malink and Rosen 2013, p. 963.

20 Cf. Malink and Rosen 2013, p. 955; Smith 2016, p. 61.

21 Other reconstructions that rely on propositional modal logic (cf., e.g. Schmidt 2008, p. 81) are not entirely faithful to Aristotle's text.

22 Διωρισμένων δὴ τούτων ὑπαρχέτω τὸ Α παντὶ τῷ Β, τὸ δὲ Β παντὶ τῷ Γ ἐνδϵχέσθω· ἀνάγκη οὖν τὸ Α παντὶ τῷ Γ ἐνδέχϵσθαι ὑπάρχϵιν. μὴ γὰρ ἐνδϵχέσθω, τὸ δὲ Β παντὶ τῷ Γ κϵίσθω ὡς ὑπάρχον· τοῦτο δὲ ψϵῦδος μέν, οὐ μέντοι ἀδύνατον. ϵἰ οὖν τὸ μὲν Α μὴ ἐνδέχϵται παντὶ τῷ Γ, τὸ δὲ Β παντὶ ὑπάρχϵι τῷ Γ, τὸ Α οὐ παντὶ τῷ Β ἐνδέχϵται· γίνϵται γὰρ συλλογισμὸς διὰ τοῦ τρίτου σχήματος. ἀλλ' ὑπέκϵιτο παντὶ ἐνδέχϵσθαι ὑπάρχϵιν. ἀνάγκη ἄρα τὸ Α παντὶ τῷ Γ ἐνδέχϵσθαι· ψϵύδους γὰρ τϵθέντος καὶ οὐκ ἀδυνάτου τὸ συμβαῖνόν ἐστιν ἀδύνατον (APr. 1.15, 34a34–b2).

23 Similarly Ross 1949, p. 338; Crubellier 2014, p. 264; Smith 2016, p. 61.

24 There is a third option, which I find untenable: von Kirchmann 1877, p. 79, Hamelin 1920, p. 204 and Tredennick 1938, 270 n. d submit that $B{a}_{\mathrm{Q}}C$ is changed into $B{a}_{\mathrm{X}}C$.

25 Cf. Alexander in APr. 175.26–28, 175.32, 176.7–9, 176.18–19, 186.10–14 W.; similarly Mignucci 1969, 324 n. 13; Angelelli 1979, p. 196; Tricot 1983, 74 n. 3; Mueller and Gould 1999, p. 39; Ebert and Nortmann 2007, pp. 562, 566–567.

26 Cf. Malink and Rosen 2013, p. 963–964; similarly Patterson 1995, pp. 163–164 (although it is not clear to me how he connects the possibility of $B{a}_{\mathrm{X}}C$ with its actual holding), Ebert and Nortmann 2007, p. 565.

27 Cf. Malink and Rosen 2013, 964 n. 19. On the de re interpretation, P is – for example – $i_{\mathrm{Q}}$-predicated of S just if there is at least an x such that S is said of x and P is contingently said of x. This interpretation is contrasted with the de dicto interpretation, according to which P is $i_{\mathrm{Q}}$-predicated of S just if it is contingent that there is at least an x such that both S and P are said of x.

28 Passages which suggest the de re interpretation are APr. 1.13, 32b29–30; 1.17, 37a22–24; 1.17, 37a5–8. This attribution is far from attaining a state of unanimous acceptance, as the de re interpretation of modal propositions exposes a controversy surrounding all modal conversions, rendering them prima facie invalid. Consider, e.g. the following counterexample to $e_{\mathrm{N}}$-conversion. Suppose that ‘sleeping’ is $e_{\mathrm{X}}$-predicated of ‘cat’. Then, every sleeping thing is necessarily not a cat. Hence, ‘cat’ is $e_{\mathrm{N}}$-predicated of ‘sleeping’. However, it is not the case that every cat is necessarily not asleep. Hence, ‘sleeping’ is not $e_{\mathrm{N}}$-predicated of ‘cat’. Consider also the following counterexample to $i_{\mathrm{Q}}$-conversion. Suppose that ‘awake’ is $i_{\mathrm{Q}}$-predicated of ‘horse’. Then, there is at least one horse which is contingently awake. However, it is not the case that there is at least one awake thing which is contingently a horse. Hence, ‘horse’ is not $i_{\mathrm{Q}}$-predicated of ‘sleeping’. Regrettably, the space available does not allow for a thorough exploration of this intricate matter: in Zanichelli 2022, an endeavour is undertaken to validate all modal conversions within the framework of the de re interpretation of modal propositions.

29 Cf. Ross 1949, pp. 338–339; Mignucci 1969, 324 nn. 11, 13; Angelelli 1979, pp. 195–196; Patterson 1995, p. 160; Striker 2009, p. 146.

30 Cf. van Rijen 1989, pp. 197–198; Thom 1996, p. 134; Ebert and Nortmann 2007, pp. 461–462; cf. also Malink 2013b, p. 185.

31 Cf. Patterson 1995, pp. 162–163.

32 Whilst operating within the problematic syllogistic, Aristotle might have obtained a proof through the impossible of Bocardo NXN; such a proof could rely on the relation of contradictoriness holding between $e_{\mathrm{M}}$- and $i_{\mathrm{N}}$-propositions, on that holding between $a_{\mathrm{M}}$- and $o_{\mathrm{N}}$-propositions, and on the validity Barbara MXM. Although Aristotle never discusses this schema, it is plausible that he would accept it as valid; at least it is valid de re. Thus, in the problematic syllogistic, Bocardo NXN can be reassessed as an imperfect syllogism. This lends justification to the view that in 1.15, Aristotle uses Bocardo–NXN as a derivable rule of inference of the problematic syllogistic. Alternatively, Malink and Rosen 2013, pp. 964–965, 968–970 propose to take the contradictory of an $a_{\mathrm{M}}$-proposition to be (not an $o_{\mathrm{N}}$-proposition, but) the negation of the $a_{\mathrm{M}}$-proposition; as a result, the subordinate inference turns out to be a contraposed inference in Barbara MXM.

33 Cf. Waitz 1844/1886, p. i. 411; Malink and Rosen 2013, p. 965. It can be shown that the inference is valid on the de re interpretation of possible propositions.

34 The inference of showing from a hypothesis (in syllogisms through the impossible) follows this pattern: the contradictory of the conclusion of a syllogism through the impossible – this is, strictly speaking, the hypothesis – is assumed, and becomes the starting point of a subordinate inference; if this inference leads to an impossible result, then it is possible to infer the conclusion of the syllogism through the impossible (cf., e.g. APr. 1.23, 41a24–26; 41a28–30). The inference can be represented schematically as follows:

If the hypothesis υ – which may occur in a proof leading to β and in a proof leading to ${\text{β}}^{\mathrm{\perp }\mathrm{\perp }}$ – leads to an impossible result whereby β and ${\text{β}}^{\mathrm{\perp }\mathrm{\perp }}$ hold together, ${\upsilon }^{\mathrm{\perp }}$ can be inferred, and any numbers $j,k\ge 1$ of occurrences of υ in the two subordinate proofs can be closed at the inference; a numerical label is used to indicate when an occurrence of υ is closed (cf. von Plato 2016, p. 328–329). As I shall show in Subsection 3.3, Aristotle is committed to rejecting the application of the inference with vacuous discharge of the hypothesis, i.e. if j or k = 0.

35 Similarly Ross 1949, pp. 338–339. Notice that the reasoning with which I credit Aristotle is different from the invalid ones (gratuitously) attributed to him by Becker 1933, pp. 52–54 – $A{a}_{\mathrm{X}}B$ is not possible with respect to $\left\{A{o}_{\mathrm{N}}C\right\}\cup \left\{B{a}_{\mathrm{X}}C\right\}$; since $\left\{B{a}_{\mathrm{X}}C\right\}\cup \left\{A{a}_{\mathrm{X}}B\right\}$ is possible, $\left\{A{a}_{\mathrm{X}}B\right\}\cup \left\{A{o}_{\mathrm{N}}C\right\}$ must be impossible – and Tredennick 1938, 270 n. d – $A{o}_{\mathrm{N}}B$ follows from $A{o}_{\mathrm{N}}C$ and $B{a}_{\mathrm{X}}C$, but $A{o}_{\mathrm{N}}B$ is impossible with respect to $A{a}_{\mathrm{X}}B$; since $B{a}_{\mathrm{X}}C$ is possible with respect to $A{o}_{\mathrm{N}}B$, $A{o}_{\mathrm{N}}C$ must be impossible with respect to $A{o}_{\mathrm{N}}B$ (both Becker and Tredennick's arguments are rendered according to the ‘syntactic’ interpretation I have been rehearsing). Patterson 1995, pp. 158–161, 171 argues that Aristotle's reasoning is invalid because the separate possibility of $A{o}_{\mathrm{N}}C$ and $B{a}_{\mathrm{X}}C$ does not grant their joint possibility, and so neither the possibility of the conclusion which can be inferred from them, i.e. $A{o}_{\mathrm{N}}B$, for only their joint possibility can justify the possibility of $A{o}_{\mathrm{N}}B$; however, Aristotle has not established such joint possibility. On my interpretation, Aristotle does not establish the joint possibility of $A{o}_{\mathrm{N}}C$ and $B{a}_{\mathrm{X}}C$, nor he asserts the possibility of $A{o}_{\mathrm{N}}B$.

36 Cf. Angelelli 1979, p. 196.

37 Cf. Malink and Rosen 2013, pp. 965–968.

38 τϵθέντος τοῦ ψϵύδους ὁ συλλογισμὸς γίνϵται διὰ τοῦ πρώτου σχήματος (APr. 1.7, 29a35–36).

39 Cf. APr. 1.29, 45b9–11; 2.14, 62b29–30.

40 Other reconstructions silently commit Aristotle to derive $A{a}_{\mathrm{M}}C$ from $A{a}_{\mathrm{X}}B$, $B{a}_{\mathrm{Q}}C$, with the assumption of $B{a}_{\mathrm{X}}C$ left open: cf. Ebert and Nortmann 2007, pp. 565–566; Rini 2011, p. 153; Malink and Rosen 2013, p. 966.

41 The string aaaa indicates that all four propositions are a-propositions, the number ‘1’ indicates that the schema is in the I figure, and ‘XXQM’ indicates the modality of each proposition. The claim that $aaaa.1.\mathrm{X}\mathrm{X}\mathrm{Q}\mathrm{M}$ is in the I figure is not controversial: once we have got rid of the redundant $a_{\mathrm{Q}}$-premiss, the schema is at once reduced to Barbara XXM. Indeed, von Kirchmann 1877, p. 77–78 objects to Aristotle that at 34a34–b2, he has actually given a proof of Barbara XXM.

42 We usually find other passages being brought up as evidence that Aristotle's syllogistic is a logic of relevance, e.g. SE 5, 167b21–36; APr. 2.17. I do not address them here.

43 Cf. Alexander in Top. 13.25–14.2 Wallies; Ammonius in APr. 30.18–25 Wallies; Frede 1974, pp. 22–23; Barnes 1980, p. 168–169, 172; Thom 1981, pp. 27–31; Cavini 1991, pp. 29–30; Mignucci 2002b, pp. 8, 11–12; Woods and Irvine 2004, pp. 53–55, 65 (according to them, the condition expressed by ‘for the fact that these things are’ (34b29) makes syllogismhood also a linear – i.e. each premiss is used only once – and hence relevant – i.e. all the premisses are used – consequence relation); Nasti de Vincentis 2009, p. 133; Cavini 2011, p. 126; Malink 2015, 286 n. 60; Castagnoli 2016, pp. 5–11; Smith 2020; Malink 2021, pp. 404a–b, 565a.

44 Cf. Steinkrüger 2015, pp. 1428–1434.

45 Cf. Smiley 1973, pp. 140, 143; von Plato 2016, pp. 328–329; Dyckhoff  2019, p. 198.

46 Cf. Alexander in APr. 21.25–28 W.; Ammonius in APr. 30.9–18 W.; Keyt 2009, p. 36; Malink 2015, pp. 286–293; cf. also Mignucci 1969, 190 n. 21. An immediate objection to my reply is Top. 8.11, 161b28–30, where Aristotle says that a syllogism can be criticised ‘if [it reaches its conclusion] when some things are taken away; for sometimes they assume more things than those necessary, so that the syllogism comes to be not for the fact that these things are’ (ϵἰ ἀφαιρϵθέντων τινῶν· ἐνίοτϵ γὰρ πλϵίω λαμβάνουσι τῶν ἀναγκαίων, ὥστϵ οὐ τῷ ταῦτ´ ϵἶναι γίνϵται ὁ συλλογισμός). Notice that Aristotle is not committed to saying that valid inferences containing superfluous premisses fail to be syllogisms; rather, he appears to be conceding that redundant inferences are syllogisms even before they are purged of unnecessary premises; they just fail to be ‘for the fact that these things are’. My tentative reply to the objection: it is possible that in Top. 8.11, the clause ‘for the fact that these things are’ expresses a further meaning than to require that all the premisses necessary for the inference of the conclusion be explicitly stated (see APr. 1.1, 24b20–22), for it also means that the premisses must provide a relevant reason for inferring the conclusion. Then, it is possible to submit that APr. and Top. assign different meanings to the same clause, ‘for the fact that these things are’.

47 Aristotle speaks of issues arising in writing and asking questions (see 47a16, 47a18). The act of asking questions is performed by the questioner in a dialectical debate (cf. Smith 1989, p. 161; Striker 2009, p. 214; Malink 2015, p. 288). The first reference is unclear: following Pacius 1597b, p. 185a, Striker 2009, p. 214 thinks that Aristotle might have the mathematical arguments treated in APr. 1.24 in mind (cf. Malink 2015, pp. 288–293).

48 ϵἶτα σκοπϵῖν ποτέρα ἐν ὅλῳ καὶ ποτέρα ἐν μέρϵι, καί, ϵἰμὴ ἄμφω ϵἰλημμέναι ϵἶϵν, αὐτὸν τιθέναι τὴν ἑτέραν. ἐνίοτϵ γὰρ τὴν καθόλου προτϵίναντϵς τὴν ἐν ταύτῃ οὐ λαμβάνουσιν, οὔτϵ γράφοντϵς οὔτ' ἐρωτῶντϵς· ἢ ταύτας μὲν προτϵίνουσι, δι' ὧν δ' αὗται πϵραίνονται, παραλϵίπουσιν, ἄλλα δὲ μάτην ἐρωτῶσιν. σκϵπτέον οὖν ϵἴ τι πϵρίϵργον ϵἴληπται καὶ ϵἴ τι τῶν ἀναγκαίων παραλέλϵιπται, καὶ τὸ μὲν θϵτέον τὸ δ' ἀφαιρϵτέον, ἕως ἂν ἔλθῃ ϵἰς τὰς δύο προτάσϵις· ἄνϵυ γὰρ τούτων οὐκ ἔστιν ἀναγαγϵῖν τοὺς οὕτως ἠρωτημένους λόγους. (APr. 1.32, 47a13–22).

49 Pace Mignucci 2002b, p. 9.

50 φανϵρὸν οὖν ὡς ἐν ᾧ λόγῳ συλλογιστικῷ μὴ ἄρτιαί ϵἰσιν αἱ προτάσϵις δι' ὧν γίνϵται τὸ συμπέρασμα τὸ κύριον (ἔνια γὰρ τῶν ἄνωθϵν συμπϵρασμάτων ἀναγκαῖον ϵἶναι προτάσϵις), οὗτος ὁ λόγος ἢ οὐ συλλϵλόγισται ἢ πλϵίω τῶν ἀναγκαίων ἠρώτηκϵ πρὸς τὴν θέσιν (APr. 1.25, 42a35–40).

51 Cf. Pacius 1597a, 240 nn. e–f; Striker 2009, p. 185; von Plato 2016, p. 336. According to Alexander, Aristotle adds ‘syllogistic’ because there are also inductive inferences which result from premisses, but not necessarily two (see in APr. 281.6–8 W.; cf. also Mignucci 1969, 436 n. 13). Philoponus apparently takes it to be synonymous with ‘syllogism’ (see in APr. 263.6–7 Wallies).

52 Cf. also Striker 2009, p. 185. The non-redundancy of the disjunction at 42a38–40 is my objection to Castagnoli 2016, p. 15's remark: ‘one can argue that the fault identified here is that the superfluous proposition(s) conceded end(s) up not even being included in the syllogism – and not that the argument uses more premisses than needed while still being syllogistic’.

53 Cf. Prawitz 1965, p. 85; Tennant 1987, pp. 670–671; Tennant 2002, pp. 302, 328.

54 ἐγχωρϵῖ δὲ καὶ διὰ τοῦ πρώτου σχήματος ποιῆσαι τὸ ἀδύνατον, θέντας τῷ Γ τὸ Β ὑπάρχϵιν. ϵἰ γὰρ τὸ Β παντὶ τῷ Γ ὑπάρχϵι, τὸ δὲ Α παντὶ τῷ Β ἐνδέχϵται, κἂν τῷ Γ παντὶ ἐνδέχοιτο τὸ Α. ἀλλ' ὑπέκϵιτο μὴ παντὶ ἐγχωρϵῖν (APr. 1.15, 34b2–6).

55 Similarly Waitz 1844/1886, p. i. 411.

56 Cf. Becker 1933, 57 n. 26; Ross 1949, p. 339; similarly Striker 2009, p. 147. Alexander (in APr. 188.7–17 W.) apparently argues that the second proof of Barbara XQM is not through the impossible, because the impossible is not inferred from the contradictory of the expected conclusion. Crubellier 2014, p. 265 presupposes that at 34b6 (point [4]), ‘not’ applies to ‘of all’ rather than to the whole copula ‘possibly of all’; thus, he thinks that Aristotle can be committed to taking ‘$A$ possibly holds not of all $C$’ to be the contradictory of ‘$A$ possibly holds of all $C$’; since this is improbable, he is tempted to agree with Becker and Ross. Cf. also Mueller and Gould 1999, 192 n. 116; Zekl 1998, 537 n. 78.

57 Cf. Mignucci 1972, pp. 15–16.

58 Cf. Thom 1996, p. 79; Fait 1999, p. 143 (who however follows Colli 1955, pp. 855–858 in taking the conclusion of the schema to be proved to be contingent); Striker 2009, p. 147. Analogously Hamelin 1920, 205 n. 1, Morrow 1951, pp. 131, Tricot 1983, 75 n. 4, although they think that the schema proved at 34b2–6 is Felapton NXN; but this is unlikely: cf. Malink and Rosen 2013, 969 n. 27.

59 Patterson 1995, p. 165 suggests that $A{a}_{\mathrm{M}}B$ is inferred from $A{a}_{\mathrm{X}}B$ qua false but not impossible; however, Aristotle does not say that $A{a}_{\mathrm{M}}B$ is false but not impossible.

60 Similarly Patterson 1995, p. 165.

61 Cf. Angelelli 1979, p. 197; (possibly) Smith 1989, p. 132; Ebert and Nortmann 2007, pp. 571–572.

62 Cf. Patterson 1995, p. 165.

63 A similar reconstruction is proposed by Malink and Rosen 2013, pp. 972–973.

64 Mignucci 1969, 325 n. 14 would criticise both reconstructions by arguing that Aristotle neither uses nor introduces $a_{\mathrm{X}}\to a_{\mathrm{M}}$, and that the modal syllogistic does not accommodate inferences that contain a one-sided possible premiss (cf. also Colli 1955, p. 859). However, as I have argued, at 34a40–41 Aristotle seems to use an instance of $a_{\mathrm{X}}\to a_{\mathrm{M}}$; he also seems to accept $e_{\mathrm{X}}\to e_{\mathrm{M}}$ at APr. 1.16, 36a15–17. And I think that the exclusion of inferences that contain a one-sided possible premiss from the modal syllogistic is arbitrary; in fact, the proof at 34b2–7 can be taken to be an exception.

65 Πάλιν ἔστω στϵρητικὴ πρότασις καθόλου ἡ Α Β, καὶ ϵἰλήφθω τὸ μὲν Α μηδϵνὶ τῷ Β ὑπάρχϵιν, τὸ δὲ Β παντὶ ἐνδϵχέσθω ὑπάρχϵιν τῷ Γ. τούτων οὖν τϵθέντων ἀνάγκη τὸ Α ἐνδέχϵσθαι μηδϵνὶ τῷ Γ ὑπάρχϵιν. μὴ γὰρ ἐνδϵχέσθω, τὸ δὲ Β τῷ Γ κϵίσθω ὑπάρχον, καθάπϵρ πρότϵρον. ἀνάγκη δὴ τὸ Α τινὶ τῷ Β ὑπάρχϵιν· γίνϵται γὰρ συλλογισμὸς διὰ τοῦ τρίτου σχήματος· τοῦτο δὲ ἀδύνατον. ὥστ' ἐνδέχοιτ' ἂν τὸ Α μηδϵνὶ τῷ Γ· ψϵύδους γὰρ τϵθέντος ἀδύνατον τὸ συμβαῖνον. οὗτος οὖν ὁ συλλογισμὸς οὐκ ἔστι τοῦ κατὰ τὸν διορισμὸν ἐνδϵχομένου, ἀλλὰ τοῦ μηδϵνὶ ἐξ ἀνάγκης (αὕτη γάρ ἐστιν ἡ ἀντίφασις τῆς γϵνομένης ὑποθέσϵως· ἐτέθη γὰρ ἐξ ἀνάγκης τὸ Α τινὶ τῷ Γ ὑπάρχϵιν, ὁ δὲ διὰ τοῦ ἀδυνάτου συλλογισμὸς τῆς ἀντικϵιμένης ἐστὶν φάσϵως) (APr. 1.15, 34b19–31).

66 Thus, I interpret the ἀνάγκη at 34b23 as expressing necessitas consequentis. Cf. Patterson 1995, p. 182; Thom 1996, p. 80; Striker 2009, p. 148; Rini 2011, 159 n. 3.

67 Cf. Striker 2009, p. 125; Rini 2011, pp. 101–102; Malink and Rosen 2013, pp. 974–975.

68 Cf. van Rijen 1989, p. 197; Patterson 1995, pp. 86–87; Striker 2009, p. 148.

69 See Subsection 3.1.2. Whilst operating within the problematic syllogistic, Aristotle might have obtained a proof through the impossible of Disamis NXN; such a proof could rely on the relation of contradictoriness holding between $e_{\mathrm{M}}$- and $i_{\mathrm{N}}$-propositions, on that holding between $a_{\mathrm{M}}$- and $o_{\mathrm{N}}$-propositions, and on the validity Celarent MXM. Although Aristotle never discusses this schema, it is plausible that he would accept it as valid; at least it is valid de re. Thus, in the problematic syllogistic, Disamis NXN can be reassessed as an imperfect syllogism. This lends justification to the view that in 1.15, Aristotle uses Disamis–NXN as a derivable rule of inference of the problematic syllogistic. Alternatively, Malink and Rosen 2013, pp. 975–976 propose to take the contradictory of an $e_{\mathrm{M}}$-proposition to be (not an $i_{\mathrm{N}}$-proposition, but) the negation of the $e_{\mathrm{M}}$-proposition; as a result, the subordinate inference turns out to be a contraposed inference in Celarent MXM.

70 It can be shown that the inference is valid on the de re interpretation of possible propositions.

71 Angelelli 1979, pp. 196–197 suggests that $B{a}_{\mathrm{X}}C$ should be taken to be compossible with other assumptions, so that the corresponding set is consistent, and that the addition of $A{i}_{\mathrm{N}}C$ makes this set impossible, i.e. inconsistent. In order to go from the impossibility of $\left\{A{e}_{\mathrm{X}}B\right\}\cup \left\{A{i}_{\mathrm{N}}C\right\}\cup \left\{B{a}_{\mathrm{X}}C\right\}$ to the assertion that $A{e}_{\mathrm{M}}C$ follows from $\{A{e}_{\mathrm{X}}B,B{a}_{\mathrm{Q}}C\}$, Angelelli relies on the following reasoning: if $A{i}_{\mathrm{N}}C$ makes the aforementioned set impossible, then $\{A{e}_{\mathrm{X}}B,B{a}_{\mathrm{X}}C,A{a}_{\mathrm{M}}C\}$ is consistent; $B{a}_{\mathrm{X}}C$ entails $B{a}_{\mathrm{M}}C$, so $\{A{e}_{\mathrm{X}}B,B{a}_{\mathrm{M}}C,A{a}_{\mathrm{M}}C\}$ is consistent; if $A{e}_{\mathrm{X}}B$ and $B{a}_{\mathrm{Q}}C$ are true together, then $A{e}_{\mathrm{X}}B$ and $B{a}_{\mathrm{M}}C$ are true together; hence, if $A{e}_{\mathrm{X}}B$ and $B{a}_{\mathrm{Q}}C$ are true together, then $A{e}_{\mathrm{M}}C$ can be true. Several parts of this argument are not supported by 34a5–33, and most importantly, it actually fails to show that the expected conclusion follows from the premisses.

72 Δϵῖ δὲ λαμβάνϵιν τὸ παντὶ ὑπάρχον μὴ κατὰ χρόνον ὁρίσαντας, οἷον νῦν ἢ ἐν τῷδϵ τῷ χρόνῳ, ἀλλ' ἁπλῶς· διὰ τοιούτων γὰρ προτάσϵων καὶ τοὺς συλλογισμοὺς ποιοῦμϵν, ἐπϵὶ κατά γϵ τὸ νῦν λαμβανομένης τῆς προτάσϵως οὐκ ἔσται συλλογισμός· (APr. 1.15, 34b7–11).

73 Cf. Malink 2013b, pp. 233–234.

74 Cf. Malink 2013b, p. 238.

75 Cf. Stephanus in Int. 6.30–32 Hayduck; Pacius 1597a, 88 nn. c–d; Ackrill 1963, p. 115; Whitaker 1996, p. 68; Weidemann 2015, p. 164.

76 Ammonius in Int. 29-12–17 Busse (cf. also Blank 1996, 148 n. 150); Waitz 1844/1886, p. 327; Cooke 1938, p. 117. Boethius (in Int. Sec. Ed. 51.3–52.9 Meiser) considers three possibilities: (1) when ‘to be’ is used in the existential sense, it is used ἁπλῶς, whereas when it is used to indicate something present, it is used κατὰ χρόνον; (2) when one refers to the present, one is speaking ἁπλῶς, whereas when one refers to the past or future, one is speaking κατὰ χρόνον; (3) when ‘to be’ is left temporally indeterminate, it is used ἁπλῶς, whereas when ‘now’, ‘yesterday’, or ‘tomorrow’ are added to ‘to be’, ‘to be’ is used κατὰ χρόνον.

77 Cf. Philoponus in APr. 175.21–24 W.; Pacius 1597a, 189 n. a; Waitz 1844/1886, p. i. 411; Ross 1949, p. 340; Hintikka 1973, pp. 136–137; Leszl 2014, p. 38. Rini 2011, pp. 151–155 takes Aristotle to require that the first premiss of Barbara XQM be a necessary truth; similarly Nortmann 1990, p. 67 and Ebert and Nortmann 2007, pp. 557–558; 563–564.

78 Cf. Grote 1872, p. 155; Colli 1955, pp. 860–861; McCall 1963, p. 17; Barnes 1994, pp. 111–112; de Rijk 2002, p. 203; Barnes 2007, pp. 17–19; possibly Owen 1889, 114 n. 2 and Tricot 1983, 75 n. 5.

79 Cf. Hintikka 1973, pp. 136–137; Smith 1989, pp. 132–133; Crubellier 2014, p. 265.

80 Cf. Nortmann 1990, pp. 63–65.

81 Cf. Thom 1996, p. 252.

82 Cf. Becker 1933, p. 58; Buddensiek 1994, p. 71; Smith 1989, p. 133; Patterson 1995, pp. 95, 167–169; Malink and Rosen 2013, p. 962.

83 Cf. Angelelli 1979, p. 201; Crubellier 2014, p. 265.

84 Cf. Malink 2013b, pp. 235–236.

85 Cf. Hintikka 1973, pp. 80–84; Graham 1999, p. 57.

86 Cf. Malink 2013b, pp. 239–244.

87 Cf. Ross 1949, pp. 336–337.

88 Cf. Angelelli 1979, pp. 200–201; Thom 1996, pp. 78, 100, 128, 253–257.

89 Cf. Ebert and Nortmann 2007, pp. 543–576, in particular 2007, pp. 560–566. On the plausible rejection of Barbara XQX, cf. Malink and Rosen 2013, 980 n. 49.

90 Cf. Rini 2011, pp. 41–44; cf. also Tredennick 1938, 272 n. b; note 77.

91 Aristotle maintains that the kind of change with which seeds are concerned is change with respect to the category of substance: a seed is matter characterised by privation of the form of the animal or plant into which it might grow, and the seed itself does not survive the change into an animal or plant (cf. Ph. 1.7, 190b1–10, with Morison 2019, pp. 255–256; GC 1.4, 319b14–21; 2.11, 337a34–b13). Thus, as Rini 2011, pp. 155–156 herself acknowledges, this additional assumption about the nature of the subject-term of $a_{\mathrm{Q}}$-propositions in which the predicate-term is a red term commits Aristotle to considering $a_{\mathrm{Q}}$-propositions as describing both changes with respect to categories other than substance and change with respect to the category of substance.

92 Cf. Malink and Rosen 2013, pp. 980–981.

93 See Subsection 3.1.4.

94 Cf. Malink and Rosen 2013, pp. 955–957.

95 See Subsection 3.1.1.

96 To avoid the problem with NQN schemata in the I figure, Malink and Rosen 2013, p. 981 suggest to restrict the principle of necessitation to one-sided possible propositions; however, they immediately give up this strategy, as ‘an ad hoc solution with little intrinsic plausibility’. Furthermore, Kapantais and Karamanolis 2020, pp. 208–215 have laid stress on the conflict between Malink's reconstruction of Aristotle's modal syllogistic (as given in Malink 2013b) and Malink and Rosen's calculus for Aristotle's propositional modal logic in respect to the principle of necessitation.

97 Cf. APr. 1.15, 35a3–11, 35a11–20, 35b2–8.

98 Cf. APr. 1.16, 36a25–27, 35b38–36b1, 36a40–b1.

99 Cf. APr. 1.16, 36a25–27, 36a28–29.

100 Cf. APr. 1.18, 37b24–28, 37b29, 37b29–35, 38a3–4, 38a4–7.

101 Cf. APr. 1.18, 38b25–27, 38b31–35.

102 Cf. APr. 1.21, 39b10–16; 39b22–25.

103 Cf. APr. 1.21, 39b16–22; 39b22–25.

104 Cf. APr. 1.21, 39b26–31.

105 Cf. APr. 1.22, 40a11–16, 40a33–35, 40a39–b2.

106 Cf. APr. 1.22, 40a25–32, 40b3–6.

107 Cf. Malink 2013b, pp. 201–202.

108 Cf. Malink 2013b, pp. 201–202.

109 Cf. Malink 2013b, pp. 204–209.

110 Cf. also Rosen and Malink 2012, pp. 180–181.

111 This paper has benefited greatly from discussion of an earlier draft at Bologna: thanks are due to Carlotta Capuccino, Walter Cavini, Guido Gherardi, and Eugenio Orlandelli. Other individuals whom I should like to thank for suggestions are Paolo Fait, Paolo Maffezioli, Sofia Pierini, and an anonymous referee for History and Philosophy of Logic.