The Arithmetical dictum

1 For Greek authors, we use Liddell, Scott, and Jones Citation1940's standard abbreviations and standard critical editions. For Latin authors, we use abbreviations which are self-explanatory and standard critical editions. All English translations are our own, unless otherwise indicated.

2 λέγομϵν δὲ τὸ κατὰ παντὸς κατηγορϵῖσθαι ὅταν μηδὲν ᾖ λαβϵῖν τῶν τοῦ ὑποκϵιμένου καθ᾿ο῟ θάτϵρον οὐ λϵχθήσϵται· καὶ τὸ κατὰ μηδϵνὸς ὡσαύτως (Arist. APr. 1.1, 24b28–30). At 24b29, we read τῶν with A: this reading is presupposed by the translation of George the Arab (cf. Minio-Paluello Citation1957, p. 574), printed by Bekker Citation1831, p. 24, Waitz Citation1844–1886, p. i. 147, Dübner Citation1862, p. 40, Tredennick Citation1938, p. 202, and endorsed by Colli Citation1955, p. 792, Ebert and Nortmann Citation2007, p. 183, Malink Citation2008, p. 521, Crubellier Citation2014, p. 53, Ebert Citation2015, p. 360 and Morison Citation2015, p. 137 n. 52. Ross Citation1949, p. 292 – followed by Barnes Citation2007, p. 387 n. 34 – excises τοῦ ὑποκϵιμένου on the basis of Alexander's commentary (see in APr. 24.27–30 W.; cf. also Wallies Citation1917–1918, pp. 626–627); however, the reading is attested by most of the main manuscripts.

3 Cf. Patterson Citation1995, pp. 210, 213; Barnes Citation2007, pp. 397–398; Crivelli Citation2011; Malink Citation2013, pp. 34–35. There is another way of understanding Aristotle's dictum de omni et nullo. Benjamin Morison has recently argued that the dictum de omni and the dictum de nullo do not determine the semantics of A- and E-propositions, but are two rules of inference of Aristotle's syllogistic (cf. Morison Citation2015, pp. 130, 135–141, 149).

4 Cf. Malink Citation2008, pp. 523–524; Malink Citation2013, p. 36.

5 Cf. Malink Citation2013, p. 35; Crivelli Citation2015a, p. 129.

6 Cf. Barnes Citation2007, p. 389; Crivelli Citation2012, pp. 119, 121; Malink Citation2013, p. 35; von Plato Citation2016, pp. 327–328. We take the modal element expressed by ‘can be taken’ (ᾖ λαβϵῖν , 24b29) to be inessential: cf. Barnes Citation2007, p. 389; Crivelli Citation2015a, p. 129.

7 See Subsection 6.1 on page 16. Cf. Patterson Citation1993, p. 365; Barnes Citation2007, pp. 403–404; Crivelli Citation2012, p. 119; Malink Citation2013, p. 36; Crivelli Citation2015a, pp. 130–131.

8 Cf. Malink Citation2008, pp. 521–522, 524–525.

9 We borrow the notation from Casari Citation1997. The same relation is indicated as mpaw in Malink Citation2013, p. 37.

10 Cf. Malink Citation2013, pp. 36–37.

11 Cf. Malink Citation2013, p. 59.

12 ‘H μὲν οὖν ὁδὸς κατὰ πάντων ἡ αὐτὴ καὶ πϵρὶ φιλοσοφίαν καὶ πϵρὶ τέχνην ὁποιανοῦν καὶ μάθημα (Arist. APr. 1.30, 46a3–4, with Melandri Citation1965, pp. 132–133). Cf. Mignucci Citation2007, pp. 135–138; Crivelli Citation2012, p. 120; Malink Citation2013, p. 43.

13 Cf. Crivelli Citation2004, pp. 152–180; Crivelli Citation2012, pp. 123–124; cf. also Mignucci Citation2007, pp. 132–134.

14 Cf. Łukasiewicz Citation1957, p. 4; Malink Citation2013, pp. 81–82.

15 Cf. Crivelli Citation2004, p. 162 n. 39. Cf. also de Rijk Citation2002, p. 177; Mignucci Citation2007, pp. 135–136.

16 At APr. 1.9 30a21–3, Aristotle plainly admits that a term is a member of the plurality associated with a term (cf. Malink Citation2013, p. 52).

17 Cf. Malink Citation2013, p. 74.

18 Cf. Barnes Citation2007, p. 412.

19 Cf. Malink Citation2013, pp. 65–66, 68.

20 Cf. Morison Citation2008, p. 214; Morison Citation2015, pp. 138, 142; Vlasits Citation2019, p. 5.

21 Cf. Liddell, Scott, and Jones Citation1940 s.v. διορίζϵιν.

22 Cf. Crivelli Citation2012, p. 143 n. 47.

23 Cf. Crivelli Citation2015a, p. 131.

24 Notice that here ${\mathsf{T}}_1$ is not an axiomatic theory in the sense of a formal axiomatic theory, since it is not formulated in a formal language. This, however, can be easily done by considering a first-order language L with countably many variables $u,v,w,\dots$ and one binary predicate constant R. The set of formulas Fm of L is built up in the standard way, using the propositional connectives and quantifiers. Then, ${\mathsf{T}}_1$ is a formal axiomatic theory in L resulting from adding on top of any sound and complete axiom system for classical first-order logic the axiom $\mathrm{\forall }u\mathrm{\forall }v(uRv\equiv \mathrm{\forall }w(vRw\supset uRw\left)\right)$.

25 A sketch of the proof is in Malink Citation2009, p. 117.

26 Cf. Malink Citation2013, pp. 67–68, 81–82. Cf. also Crivelli Citation2012, pp. 121–122.

27 τὸ δὲ ἐν ὅλῳ ϵἶναι ἕτϵρον ἑτέρῳ καὶ τὸ κατὰ παντὸς κατηγορϵῖσθαι θατέρου θάτϵρον ταὐτόν ἐστιν (Arist. APr. 1.1, 24b26–28).

28 Aristotle commits himself to this equivalence many times: cf. APr. 1.4, 25b32–35; 27, 43b11–14; 32, 47a13; 2.1, 53a21–22, 53a23; 2, 53b30, 53b4–5, 54b25–6, 54b28–29, 55a6, 55a36–37; 3, 55b27–28, 55b35–36, 56a26, 56a29–30, 56a33–34, 56b1; 4, 56b37–38, 57a13–14, 57a19, 57a20–21; 6, 58b28; 20, 66b15–16; 21, 67a33–34; 22, 68a16; 68a21–22; 23, 68b21; APo. 1.15, 79a36–37, 79a38, 79a39–40, 79b2, 79b5, 79b9–10, 79b12, 79b14–15, 79b17; 16, 80a40–b1, 80b4; 17, 80b27–28, 80b37–38. Cf. also Striker Citation1996, pp. 216–217; Mignucci Citation2000, pp. 3–4; Barnes Citation2007, p. 388; Corkum Citation2015, p. 798.

29 τί τὸ ἐν ὅλῳ ϵἶναι ἢ μὴ ϵἶναι τόδϵ τ῀ωῷδϵ (Arist. APr. 1.1, 24a13–14).

30 Cf. Crivelli Citation2004, pp. 254–256; Crivelli Citation2012, p. 116.

31 Cf., e.g., Arist. APr. 1.1, 24a16–22; 4, 25a39–b3; 20, 39a15–16; Bäck Citation2000, pp. 176, 249. Cf. also Mignucci Citation2000, p. 3; Malink Citation2013, p. 84; Corkum Citation2015, p. 811 n. 12; Corkum Citation2018, p. 211; Vlasits Citation2019, pp. 2–3.

32 Cf. Arist. Cat. 5, 3a29–32; Ph. 1.1, 184a25–26; 4.3, 210a14–15, 17–18; GC 1.3, 317b5–7; Metaph. Δ 25, 1023b17–19; 26, 1023b26–32; Mignucci Citation1997, pp. 69–70; Malink Citation2013, p. 84.

33 Cf. Crivelli Citation2012, p. 124.

34 Cf. Mignucci Citation1997, p. 66; cf. also Mignucci Citation1996, p. 4; Mignucci Citation2000, p. 13. The translation of 24b26–30 proposed by Tricot Citation1983, p. 5 reflects this interpretation. Mignucci's mereological interpretation of Aristotle's syllogistic will be discussed in another paper.

35 Cf. Mignucci Citation1997, p. 66.

36 Cf. Vlasits Citation2019, p. 5.

37 Cf. Vlasits Citation2019, pp. 6–8, where the class of models $〈D,⊑〉$ such that $⊑$ is a preorder on D is indicated with $\mathbb{P}$. Corcoran's natural deduction calclus, called RD, is given in Corcoran Citation1972.

38 Cf. Corkum Citation2015.

39 Cf. Corkum Citation2015, pp. 800–809. On the minimal axiomatization of the relation of parthood, cf. Simons Citation1987, pp. 28, 31, 362; Casati and Varzi Citation1999, p. 39; Cotnoir and Varzi Citation2021, p. 114. Since in the presence of weak supplementation anti-symmetry is redundant, there is no need to take a weakly supplemented partial order in order to minimally characterize the relation of parthood (cf. Cotnoir and Varzi Citation2021, pp. 118–119). However, Corkum Citation2015, p. 805 stresses that ‘the question of the reflexivity of the relation is not germane to issue whether Aristotle's part-whole talk is genuinely mereological’.

40 Suppose that there was a ‘null’ individual, that is, a z such that for every x, z is part of x. Suppose that there is an individual y distinct from z. Then, z is part of y. By the principle of weak supplementation, there is another part of y which is disjoint from z. But since z is part of everything, nothing is disjoint from it.

41 Cf. Corkum Citation2018, pp. 215–216.

42 Cf. Corkum Citation2015, pp. 801–802, 805; Corkum Citation2018, pp. 213–214.

43 λέγω δὲ καθόλου μὲν ὃ ἐπὶ πλϵιόνων πέφυκϵ κατηγορϵῖσθαι (Arist. Int. 7, 17a39–40). Cf. also Arist. Cat. 5, 3b17–18.

44 Cf. Crivelli Citation2009, pp. 20–22.

45 Cf. Crivelli Citation2015b, pp. 23–24. Cf. also Mignucci Citation1965, p. 69.

46 ϵἰ δὲ ἐκ στοιχϵίου, δῆλον ὅτι οὐχ ἑνὸς ἀλλὰ πλϵιόνων (Arist. Metaph. Z 17, 1041b22–23). Cf. Cotnoir and Varzi Citation2021, p. 117.

47 Cf. Bostock Citation1994, pp. 245–246; Burnyeat Citation2001, pp. 60–64.

48 Castelli Citation2018, p. 39 has assembled evidence from the Metaphysics that Aristotle's account of universals is incompatible with the principle of weak supplementation.

49 Vlasits Citation2019, p. 2.

50 Cf. Vlasits Citation2019, p. 11: ‘the language of parthood is the closest that Aristotle could have come describing preorders without modern technical terminology’.

51 Vlasits Citation2019, pp. 10–11 argues that Aristotle rejected weak supplementation, but again, his claim that ‘just because Aristotle conceptualized the predication relation in mereological terms, we do not thereby need to think that he was committed to weak supplementation for that relation’ is simply denied by modern classical mereology.

52 Different translations of ἄλλον δὲ τρόπον τὰ καταμϵτροῦντα τῶν τοιούτων μόνον (1023b15–16) have been proposed. Some translators take Aristotle to use καταμϵτρϵῖν + genitive (τῶν τοιούτων ). Thus, the translation of 1023b15–16 would be something like the following: ‘in another way, only those which are the measure of such things [i.e. quantities] [are parts]’ (cf. Taylor Citation1801, p. 137; Viano Citation1974, p. 335). Other translators take τῶν τοιούτων to express a partitive genitive, introducing a whole (the use of parts pertinent to the first definition), a part of which is denoted by τὰ καταμϵτροῦντα. Thus, the translation of the passage would be something like the following: ‘in another way, of the parts of the first way, only those which measure [a quantity] [are said to be parts]’ (cf. Schwegler Citation1847–1848, p. i. 96; Rolfes Citation1921, p. 115; Ross Citation1924, p. i. 339; Carlini Citation1928, p. 180; Tredennick Citation1933, p. 279; Warrington Citation1956, p. 40; Apostle Citation1966, p. 97; Russo Citation1973, p. 164; Seidl Citation1984, p. 239; Tricot Citation1991, p. i. 213; Kirwan Citation1993, p. 60; Duminil and Jaulin Citation2008, p. 216; Bodéüs and Stevens Citation2014, p. 69). Other translations are ambiguous (cf. Hope Citation1952, p. 117), or omit τῶν τοιούτων (cf. Reale Citation2004, p. 253). We may notice that at Ph. 6.7, 237b28–29, Aristotle constructs καταμϵτρϵῖν with the accusative, which tells against the first translation. We follow this guidance and adopt the second translation.

53 Μέρος λέγϵται ἕνα μὲν τρόπον ϵἰς ὃ διαιρϵθϵίη ᜂν τὸ ποσὸν ὁπωσοῦν (ἀϵὶ γὰρ τὸ ἀφαιρούμϵνον τοῦ ποσοῦ ᾕ ποσὸν μέρος λέγϵται ἐκϵίνου, οἷον τῶν τριῶν τὰ δύο μέρος λέγϵταί πως), ἄλλον δὲ τρόπον τὰ καταμϵτροῦντα τῶν τοιούτων μόνον· διὸ τὰ δύο τῶν τριῶν ἔστι μὲν ὡς λέγϵται μέρος, ἔστι δ᾽ ὡς οὔ (Arist. Metaph. Δ 25, 1023b12–17). Cf. also Metaph. Z 10, 1034a32–33.

54 Cf. also Alexander in Metaph. 423.36–39 Hayduck; Asclepius in Metaph. 349.29–31 Hayduck; Aquinas in Metaph. 1093 Cathala/Spiazzi; Bonitz Citation1848–1849, p. ii. 272; Pfeiffer Citation2018, p. 62.

55 Cf. Arist. Cat. 6, 4b33; Ph. 4.12, 221a2; 6.2, 233b3, 233b5, 233b9; 6.7, 237b28, 238a7, 238a12, 238a14; 6.10, 241a13; 8.10, 266b23; Cael. 1.6, 273a32. Bonitz Citation1870, pp. 371b61–372a2 refers to some of the occurrences in the Physics as being synonymous with ἀναμϵτρϵῖν, but gives no further explanation.

56 Cf. Ross Citation1924, p. i. 339. Other commentators who are silent or uninformative are: Rolfes Citation1921, p. 115; Russo Citation1973, p. 164; Seidl Citation1984, pp. 203, 407.

57 Cf. Aquinas (see in Metaph. 1093 C./S.; moreover, Aquinas takes the dividing part and the divided quantity in both definitions to be quantitas minor and quantitas maior, respectively. Cf. also (possibly) Carlini Citation1928, p. 180 n. 4.

58 Cf. Maurus Citation1668, p. vi. 459b; cf. also Schwegler Citation1847–1848, p. iii. 237; Reale Citation2004, p. 959; Castelli Citation2018, p. 161.

59 Cf. Bonitz Citation1848–1849, p. ii. 272.

60 Cf. Apostle Citation1966, p. 315 n. 2; Kirwan Citation1993, p. 174; Legatt Citation1995, p. 192; Pfeiffer Citation2018, p. 199 n. 12. Warrington Citation1956, p. 40 even renders καταμϵτρϵῖν as ‘to divide exactly’ (similarly Viano Citation1974, p. 335: ‘essere un sottomultiplo’).

61 Cf. Acerbi Citation2007, p. 432 n. 410.

62 See Section 8.

63 Cf. Davey and Priestley Citation2002, p. 37.

64 Cf. Arist. Int. 7, 17b16–37; 10, 20a16–20; APr. 1.5, 27a29–31; APr. 2.15, 63b23–30.

65 As we shall see in Section 7 on page 21, this assumption is warranted by the fact that ancient mathematics denies that a unit is a number: since f is defined as assigning integers to terms, there is no reason why it should assign something that is not theoretically a number.

66 Moreover, the restriction excludes that a term is assigned to the value 1, which would immediately entail the existence of a term which is part (divisor) of all terms. While not problematic per se, this could arguably be at odds with the central tenet of classical mereology which rules out the existence of a ‘null’ individual, namely an individual that is part of everything.

67 πρότϵρον γὰρ ϵἴρηται πῶς τὸ κατὰ παντὸς λέγομϵν (Arist. APr. 1.4, 25b39–40).

68 ϵἰ ἔστι παντὸς κατϵγορϵῖσθαι τὸ ἐν ἀρχ῀ηῆ λϵχθέν (Arist. APr. 1.4, 26a24).

69 ὥρισται γὰρ καὶ τὸ κατὰ μηδϵνὸς πῶς λέγομϵν (Arist. APr. 1.4, 26a27).

70 Cf. Alexander in APr. 54.6–21, 54.31–55.7, 60.21–61.5 W.; Mignucci Citation1969, p. 216 n. 9; Crivelli Citation2012, pp. 118–120, 129–130; Malink Citation2013, pp. 37–38; Ebert Citation2015, pp. 359–360.

71 Cf. Malink Citation2008, pp. 523–524, 528, 529 n. 33, 533–535; Malink Citation2013, pp. 39–40, 88–89.

72 Cf. Malink Citation2008, p. 535; Malink Citation2013, p. 40.

73 Cf. Malink Citation2008, pp. 526, 535; Malink Citation2013, pp. 40–41.

74 Cf. Malink Citation2013, pp. 313–315, 318–319. As Malink Citation2013, p. 229 himself notes, ‘the body of Aristotle's claims of invalidity and inconcludence can be reduced to a certain subset of such claims. Aristotle himself does not engage in such a project. […] Part of the reason for this may be that performing such reductions would make the presentation of his syllogistic considerably more difficult’.

75 A terminological note on pairs of premisses (we take ‘AE.1’ as an example): the string ‘AE’ indicates that the first premiss is an A-proposition and the second one an E-proposition; the number ‘1’ indicates that the pair is in the first figure.

76 A thorough analysis of these books, commonly known as the arithmetical books, is Mueller Citation1981, pp. 59–84.

77 Cf. Knorr Citation1975, §7; Acerbi Citation2009, pp. 84–86.

78 The reference works are Einerson Citation1936a, Citation1936b; Smith Citation1978; Acerbi Citation2009.

79 Cf. Einerson Citation1936b, pp. 155–156 on Barbara, and Smith Citation1978, pp. 202–203 on Cesare and Camestres.

80 μέρος ἐστὶ μέγϵθος μϵγέθους τὸ ἔλασσον τοῦ μϵίζονος, ὅταν καταμϵτρ῀ηῆ τὸ μϵῖζον (Elem. V. Def. 1).

81 Cf. Heath Citation1956, p. 115. Notice that Euclid also defines the converse relation, being-a-multiple-of (πολλαπλάσιον  ): ‘the greater is a multiple of the less when it measured by the less’ (πολλαπλάσιον ὲ τὸ μϵῖζον τοῦ ἐλάττονος, ὅταν καταμϵτρῆται ὑπὸ τοῦ ἐλάττονος   Elem. V. Def. 2).

82 μέρος ἐστὶν ἀριθμὸς ἀριθμοῦ ὁ ἐλάσσων τοῦ μϵίζονος, ὅταν καταμϵτρ῀ηῆ τὸν μϵίζονα (Elem. VII. Def. 3).

83 Cf. Mueller Citation1981, pp. 59–60

84 Cf. μονάς ἐστιν, καθ᾿ ἣν ἕκαστον τῶν ὄντων ἓν λέγϵται (Elem. VII. Def. 1)

85 ἀριθμὸς δὲ τὸ ἐκ μονάδων συγκϵίμϵνον πλῆθος (Elem. VII. Def. 2).

86 πρῶτος ἀριθμός ἐστιν ὁ μονάδι μόνῃ μϵτρούμϵνος (Elem. VII. Def. 11). Clearly, without assuming that a unity divides any number, there would be no prime numbers: cf. Acerbi Citation2007, p. 365.

87 πρῶτοι πρὸς ἀλλήλους ἀριθμοί ϵἰσιν οἱ μονάδι μόνῃ μϵτρούμϵνοι κοιν῀ωῆ μέτρῳ (Elem. VII. Def. 12). σύνθϵτοι δὲ πρὸς ἀλλήλους ἀριθμοί ϵἰσιν οἱ ἀριθμ῀ωῆ τινι μϵτρούμϵνοι κοιν῀ωῆ μέτρῳ (Elem. VII. Def. 13).

88 μέρη δέ, ὅταν μὴ καταμϵτρ῀ηῆ (Elem. VII. Def. 4 ).

89 Cf. Heath Citation1956, p. 280. Zeuthen's interpretation of ‘parts’ as ‘proper fraction’, largely inspired by the proof of Elem. VII.4, is discussed in detail in Mueller Citation1981, p. 112.

90 Following Ian Mueller, this can be made formally precise by assuming a ternary relation $M(a,b,m)$ that a number a bears with a number b when a measures b m times. An expression which Euclid uses for this relation is ‘a measures b according to the units in m’. Thus, given two numbers a and b such that a<b, we say that a is parts of b when $M(c,a,m)$ and $M(c,b,n)$, for some c. In our example, 2 is parts of 3 since $M(1,2,2)$ and $M(1,3,3)$.

91 ἔστωσαν οἱ δοθέντϵς δύο ἀριθμοὶ μὴ πρῶτοι πρὸς ἀλλήλους οἱ ΑΒ, ΓΔ. δϵῖ δὴ τῶν ΑΒ, ΓΔ τὸ μέγιστον κοινὸν μέτρον ϵὑρϵῖν. ϵἰ μὲν οὖν ὁ ΓΔ τὸν ΑΒ μϵτρϵῖ, μϵτρϵῖ δὲ καὶ ἑαυτόν, ὁ ΓΔ ἄρα τῶν ΓΔ, ΑΒ κοινὸν μέτρον ἐστίν. καὶ φανϵρόν, ὅτι καὶ μέγιστον: οὐδϵὶς γὰρ μϵίζων τοῦ ΓΔ τὸν ΓΔ μϵτρήσϵι (Elem. VII.2).

92 ἐπϵὶ οὖν ὁ Ε τὸν ΓΔ μϵτρϵῖ, ὁ δὲ ΓΔ τὸν ΒZ μϵτρϵῖ καὶ ὁ Ε ἄρα τὸν ΒZ μϵτρϵῖ (Elem. VII.1).

93 Cf. Malink Citation2013, pp. 74–75.

94 Anti-symmetry, just like the other principles governing |, simply follows from the definition of | in terms of · and the axioms of the latter.

95 καὶ τὰ ἐφαρμόζοντα ἐπ᾿ ἄλληλα ἴσα ἀλλήλοις ἐστίν (Elem. CN.7).