1 This characterization of logical determinism is broad enough to include even Aristotle: he apparently held that all propositions ‘about’ the past and present are necessary. Rhet. Ill, 17, 1418a3-5; De Int. 9, 19a23-25; Eth. Nic. 2, 1139b7-9; and De Caelo I, 12, 283b13 are cited by Hintikka, Jaakko “Aristotle and the ‘Master Argument’ of Diodorus,” American Philosophical Quarterly, I (April 1964), pp. 101-2.Google Scholar However, Aristotle apparently wished to deny that 'future contingent’ propositions are necessary. In order to do so, he at least toyed with the idea of withholding a truth value from such propositions. Vid. De Int. 9.

2 Hintikka, op. cit., pp. 101-14.

3 Prior, Arthur “Diodoran Modalities,” Philosophical Quarterly (St. Andrews), 5 (July 1955), pp. 205-13.CrossRefGoogle Scholar Time and Modality (Oxford, 1957). “Diodorus and Modal Logic,” Philosophical Wuarterly (St. Andrews), 8 (July 1958), pp. 226-30. Past, Present and Future (Oxford, 1967). See also: Mates, Benson Stoic Logic (Berkeley, 1973).Google Scholar Rescher, Nicholas “A Version of the ‘Master Argument’ of Diodorus,” Journal of Philosophy, 63 (August 11 1966), pp. 438-45.CrossRefGoogle Scholar Michael, Frederick Seymour ‘What is the Master Argument of Diodorus Cronos?“ American Philosophical Quarterly, Vol I (July 1976), pp. 229-35.Google Scholar Sutula, John “Diodorus and the ‘Master Argument,“’ Southern Journal of Philosophy, 14 (Fall 1976), pp. 323-43.CrossRefGoogle Scholar

4 For a brief English summary of the historical connections of the Megarians and Stoics, see Mates, op. cit., pp. 5-10. Euclides (ca. 430-360 B.C.) was the founder of the Megarian school, which is numbered among the ‘minor Socratic’ schools. The Megarian Diodorus Cronos was a somewhat -younger contemporary of Aristotle (384-322). Zeno (ca. 350-260), the founder of the Old Stoa, was a pupil of the Megarian Stilpo and a friend of Diodorus’ pupil, the Megarian Philo.

5 1046b29ff.

6 See the example in Empiricus, Sextus Adversus Dogmaticos, II, 115-17 (Cambridge, Mass. and London [Loeb], 1957), p. 298.Google Scholar

7 Ibid.

8 These modal propositions are not to be read in such a way that the time variable is included in the scope of the modal term, e.g., “Socrates will die at t.” Rather, they are to be interpreted so that the ‘whole’ temporally indeterminate modal proposition, e.g., “it will be the case that p” or “it is necessary that p,” is true or false at each time t.

9 The concept of ‘basic modal logic’ is that of Jan Lukasiewicz, Aristotle's Syllogistic (Oxford, 1957), pp. 135-37. Lukasiewicz attributes thesis (2) to Aristotle principally on the basis of a passage from the Prior Analytics (i., 16, 36a15) in which Aristotle indicates that ˜ϕ entails M ˜ϕ. Alexander of Aphrodisisas generalizes this claim. (op. cit., p. 136) However, see also De Int. 13, 23a7ff.: τὸ γαρ δυνατὸν οὐχ ἁπƛῶζ ƛέγϵτσι, ἀƛƛαὰ τὸ μὲν ὃτι ἀƛηθὲζ ὡζ ἐνϵρϵία ὂν, οἷον ὁυνατὸν βαὁίζϵιν οτι βαὁίζωι, χαι ὅƛqζ ὁυναψὸν ϵἶναι ὅτι ἤὁη ἔστι χατ᾿ ἐνέργϵιαν ὁ ƛέγϵται ϵἰναι ὁυνατόν. The issue is perhaps complicated by passages such as that at Meta. K, 9,

1 066a4-5, where Aristotle says that “When a house exists, it is no longer buildable” (ἀƛƛ´ ὅταν οἰχƛα ἦ, οὐχέτι οἰχοὁομητόν). See also the parallel discussion at Physics Ill, i., 201b9ff. These passages suggest that the actualization of a potentiality means the ‘loss’ of that potentiality. Perhaps the proper conclusion is that the relation between the concepts of ‘being possible/being the case’ and ‘potentiality/actuality’ needs to be more closely investigated.

10 Lewis, David Counterfactuals (Cambridge, Mass., 1973), p. 8.Google Scholar

11 Meta. θ,3,1047a15-16.

12 One place where Aristotle appears to accept thesis (4) is at the beginning of Meta. θ, 4, 1047b3-4: ψανϵρὸν ὅτι οὐχ ἐνὁέχϵται ἀƛηθὲζ ϵἶναι τὸ ϵἰπϵῖν ὁυνατὸν τοὁί οὐχ ἔσται.

13 See note 1.

14 Lukasiewicz offers an alternative interpretation of this passage, one that seems to to me to be somwhat less plausible in view of the context. He suggests that Aristotle may have in mind simply a sort of necessitas consequentiae, which applies not to syllogisms but to “singular propositions about events“: L(ϕ ⊃ ϕ). (op. cit., pp. 151-53)

15 The hypothesis that this argument is directed against early (pre-Diodorean) Megarian doctrine is that of Hintikka. (op. cit., pp. 1 08-9).

16 Ibid., pp. 109-10. Boethius’ report (Commentarii in Librum Aristotelis [In De Interpretatione], 2nd ed., Meiser, 234) of Diodorus’ account of the alethic modalities may thus be interpreted as follows: at t, p is possible iff (p(t) v fp(t)); at t, p is impossible iff (˜ p(t) ˜ “'Fp(t)); at t, p is necessary iff (t') 9t<t') ⊃ p(t')); at t, pis non necessary iff (∃t') ((t<t') p(t')).

17 Hintikka, op. cit., p. 110.

18 Perhaps Aristotle's view is that, with respect to the indexical present, there are no forks or branches to the ‘left’ (the past), but perhaps branches to the ‘right' (the future). As the ‘present’ ‘moves’ from left to right, the parallel ‘untraversed' or unactualized branches are ‘erased’ in such a way that there always is but one past and present.

An interesting discussion of Aristotle's own treatment of future contingent propositions in De lnterpretatione 9, “Time, Truth, and Necessity,” by Professor Robert H. Kane, was presented to a University of Texas at Austin ‘Backward E' logic symposium during the spring of 1977. It is hoped that Prof. Kane's work will soon appear in print.

19 A temporally determinate proposition can always be transformed into a temporally indeterminate proposition by adding a time variable. Thus, “Socrates dies at 399 B.C.” becomes something like ‘At t, it is the case that Socrates dies at 399 B.C.’ This propositional function is understood to be true for all values of t.

20 See notes 1 and 18.

21 18b31-32

22 Aristotle's somwhat perplexing disjunction of present and future throughout the following argument calls to mind Diodorus’ definition of possibility. I have therefore attempted reconstructions of the argument substituting the modality of possibility for each occurrence of the ‘present or future’ disjunction, but without any degree of success. The reconstruction I present in this paper is the best I have been able to devise.

23 Cicero, De Fato, vii., 13 (Cambridge, Mass. and London [Loeb], 1960), p. 206.

24 Mates (op. cit., p. 39) indicates that although Diodorus usually attributes the modalities to “what are in effect propositional variables,” the first premise of his 'Master’ argument (τῷ πᾶν παρϵƛηƛυθὸζ ἀƛηθϵζ αναγχαῖον ϵἶναι - “everything past that is true is necessary“) seems to predicate necessity of “propositions“ (i.e., temporally determinate propositions). However, with the aid of our ‘simple past’ operator ‘P', we can devise an interpretation of the first premise that seems true and also analyzes the modality of necessity as applying to temporally indeterminate propositions: Pp(t) ╟ LPp(t).

25 On Interpretation, 9 (Cambridge, Mass. and London [Loeb], 1962), p. 134:. “Eτι ϵἰ ἔστιν ƛϵυχὸν νῦν ἀƛηθὲζ ἠν ϵἰπϵῖν πρότϵρον ὅτι ἔσται ƛϵυχόν.

26 Prior, in fact, uses a similar but stronger premise, p(t) ╟ Hfp(t) (“from a thing's being the case it necessarily follows that it has always been going to be the case“), in his reconstruction of the ‘Master’ argument. See Past, Present and Future, p. 33.

27 18b 11-12: ὥστϵ ἀϵὶ ἀƛηθὲζ ἦν ϵἰπϵῖν ὁτισῦν τῶν ƛϵνομένων ὄτι ἔστιν ἠ ἔσται.

28 The past operator here operates on a disjunction of propositional function to produce a new propositional function; there is really only one time variable in the resulting function despite the appearance of my somewhat clumsy notation.

29 18b13-15: .

30 18b 15-16:

31 Op. cit., pp. 109-10.

32 De Fato, vii., 13: at hoc, Chrysippe, minime vis, maximeque tibi de hoc ipso cum Diodoro certamen est. llle enim id solum fieri posse dicit quod aut sit verum aut futurm sit verum, et quidquid futurum sit id dicit fieri necesse esse et quidquid non sit futurum id negat fieri posse.

33 Ibid., vi., 12: si Fabius oriente Canicula natus est, Fabius mari non morietur.

34 Sextus Empiricus, Adversus Dogmaticos, II, 115.

35 Op. cit., pp. 45-6.

36 If (11) is true, there is no (present or future) time at which ϕ(t) is true but ψ(t) false. Suppose that ϕ(t) is necessary (in the Diodorean sense); it is thus true at all present or future times. Consequently, if there is no time at which ϕ(t) is true but ψ(t) false, ψ(t) also must be true at all present or future times and, hence, necessary in the Diodorean sense.

37 Chryssipus, but not Cleanthes, also held this view according to Cicero, De Fato, vii., 14: omnia enim vera in praeteritis necessaria sunt, ut Chrysippi placet dissentienti a magistro Cleanthe, quia sunt immutabilia nee in falsum e vero praeterita possunt convertere.

38 I certainly do not mean to suggest that only the two assumptions I consider in the text would verify thesis (5) within the framework of a Diodorean semantics. The two views I consider, however, seem to have some historical importance.

39 In order to verify (5) in this manner, it is not necessary to assume that the truth values of all temporally indeterminate propositions become ‘static’ at the same (present or future) time; it is only necessary that there be some such time at which the truth value of each proposition becomes ‘frozen.'

40 Principal aspects of the doctrine are 1) the existence of a world order that is eventually destroyed in a conflagration (ϵχπυρωσιζ), 2) the eternal recurrence of this process, and 3) the identity of objects and sameness of detail in each successive world order. With respect to aspect 3), however, at least one fragment suggests that what ‘recur’ are something like ‘counterparts’ ('something that does not deviate from Socrates’). See). Arnim, von Stoicorum Veterum Fragmenta, Vol. II, Sec. 626Google Scholar (Lipsiae, 1903), p. 190: (Origenes contra Celsum) ἵνα μὴ Σωχράτηζ πάƛιν γένϵται, ἀƛƛ’ ἀπαράƛƛαχτόζ τιζ τῷ Σωχράτϵι, γαμήσων ἀπαράƛƛαχτού τινα Ξανθίππη, χαὶ χατηγορ ηθησόμϵνοζ ὑπὸ ἀπαράƛƛαχτων ‘ Aνύτῳ χαὶ Μϵƛητῳ.

41 See the discussion in Mates, op. cit., pp. 5-8.

42 Servius ad Vergil Aen. iii., 376, reproduced in the Loeb edition of De Fato as Fragment 2, p. 246: Fatum est conexio rerum per aeternitatem se invicem tenens, quae suo ordine et lege sua variatur, ita tamen ut ipsa varietas habeat aeternitatem.

43 It would be rash indeed to suggest that the doctrine of eternal recurrence was 'developed’ by the Stoics with the explicit purpose of meeting logical exigencies pertaining to fatalism. There are any number of plausible sources for the view, e.g., Greek myth, the Pythagoreans, Empedocles, and, perhaps, Heraclitus. See Hahm, David E. The Origins of Stoic Cosmology (Columbus, Ohio, 1977),Google Scholar especially Ch. VI. ‘The Cosmic Cycle.’ My suggestion is that perhaps the problem of the necessity of future events, analyzed within a Megarian modal framework, can help us better understand the relation of the doctrine of recurrence to the concept of fate.

44 It is characterized in the fragment as a “definitio fati secundum Tullium.“