1
Stamatis D. Gerogiorgakis
A CONTRIBUTION TO THE INFLUENCES OF THE DIODOREAN
MODALITIES FROM THE MIDDLE AGES TO THE PRESENT
My main aim is to show the importance of certain conceptions of modalities
related to the Diodorean1 modalities (named thus after the Megarian Diodorus Cronus,
a younger contemporary of Aristotle’s). First, I want to draw the attention on some
ancient conceptions of modalities, which have been partially forgotten today. Then I
will focus on the influences of these conceptions from the Middle Ages to the 20th
century. More specifically, I want to review Diodorus’s Byzantine critics and to show,
that the positive treatment, which Diodorean modalities received by Abaelard,
Thomas Aquinas and Leibniz, was a western reply to the austere, “orthodox”
Aristotelianism of the East. Finally, I will comment on the employment of Leibnizian
(and ipso facto Diodorean) modalities by Kurt Gödel in his “Ontological Proof”, a
dimension which has been, as far as I know, neglected in the Gödel-bibliography.
To understand what kind of modalities are employed by a Diodorean model, let us
take an example, which was coined in reality by Aristotle (but is likely to have been
Aristotle’s reply to Diodorus):
For “p” meaning: “a sea battle takes place in t1”, we get:
p in t0
(1)
p in t1
(2)
p in t2
(3)
According to a very plausible intuition of modalities in time, if (2) holds, then (1)
and (3) hold too. This is something, in which Aristotle and Diodorus (except for
differences in the primitive notion of modalities) would tend to agree. Given, now, the
iteration pp (cf. Prior Analytics 22a12-17), then (1) and (2΄) are compatible. I.e.
if (1) holds, then ‘¬p in t1’ (2΄) is, at least, probable. But this probability implies:
(pp), so that (2΄) would become incompatible with (3). A completely different
notion of possibility now, which Aristotle speaks of (ibid, 22b11-14), postulates:
pp. This would make (1) fit with (3) but, again, not with (2΄). Aristotle
maintained this ambiguity of the possibility notion and tried to resolve it by pointing
out, that p being necessary or not has to do with the time aspect. This is just what
Diodorus had denied! Consequently, Diodorus had identified being possible in t0 with
being actual in t1 and being necessary in t2 – but, again, the latter means to him:
having been necessary already in t0. I.e., (2) being true has to lead me to believe that
‘p’ was necessary from the very beginning and, analogously, (2΄) being true has to
lead me to believe that ‘p’ was impossible from the very beginning. In Diodorus’s
words: given (2), (3) and the non-sequitur of impossibility out of possibility leads to
the conclusion that there are no possibilities which are not actualities (i.e. there are no
possibilities which are not necessarily to happen at some time!). The Diodorean view
on possibility is expressed in what the history of logic calls “the Master Argument”.
According to this, everything which is possible, is either happening now, or going to
happen later (cf. Epictetus, Dissertationes ab Arriano digestae, lib. II, 19, 1; also
Alexander of Aphrodisias, In Aristotelis Analyticorum Priorum librum I
commentarium, 184). Therefore, Diodorus could accept no kind of non-necessary
events to have been possible, unless they really do not happen – but no one actually
1
To refer to the modalities of Diodorus Cronus, I use the adjective introduced by Benson Mates (1953):
“Diodorean” instead of Arthur Prior’s (1955): “Diodoran”. Also Lemmon (1977) followed Mates’s
terminology.
2
could call “event” something, that did not occur after all. Given the Diodorean
iteration: “everything possible is necessarily to occur at some time”, which for
Aristotle would be the contingent proposition pp, is after Diodorus a strict
implication: p  p.2
The most prominent accounts on the Master Argument are Prior’s (1955) and
Becker’s (1960) papers. Interpretations of the relations between Aristotle’s SeaBattle-Argument and the Master Argument were provided by Pierre-Maxime Schuhl
(1960), Dorothea Frede (1970) and Kurt von Fritz (1978). Nico Strobach (1998)
formalized the Sea-Battle-Argument as a model of temporal-modal logic, as Prior did
before him with the Master Argument. All these people agree at a very basic point:
Aristotle gave in Peri hermeneias the basic transformation rules to relate predicates of
the kind: “it is possible” and “it is necessary”, in a way opposite to this of Diodorus.
Aristotle gave an account there, or rather an interpretation, for two different meanings
of the word “possible” (the first meaning of “possible”, which Aristotle dealt with,
would correspond e.g. to “able”, the second to “inclined” – cf. Prior Analytics 23a714
), whereas it was Diodorus’s claim to keep the possibility notion unambiguous. All
modern commentators agree, as well, that the Sea-Battle-Argument and the Diodorean
modalities were formed in some kind of interdependence with each other. Not all, but
a good deal of the interprets, again, agree that Aristotle developed his argument only
later, after the Diodorean modalities were already broadly discussed. 3 To form what
he thought to be a sustainable modal logic, Aristotle presented in Peri hermeneias (18
b) a model for a temporal-modal system, which takes both meanings of “possible”
into account and still vindicates the notion of the futura contingentia. Aristotle holds
there that the shift of modalities within temporal contexts results from modalities’
having to do with our ‘perspectivist’ holding ‘p’ or ‘p’ to be true, not with p in the
end turning out to be true or not. Clearly then, whether ‘p’ or ‘p’ holds, is – unlike
‘p’ being or not being true! – dependent upon the circumstances we say so. Time is
only one of these circumstances. So, Aristotle thinks that futura contingentia
‘become’, so to say, necessary only post festum for the time after the fact, but not
retrospectively for the time before the fact. I.e. no contingency becomes necessary,
unless it is already past. And, what is more, he thinks that this aspect of temporalmodal logic is independent of what conception one has of modalities.
As Prior (1955) and Becker (1960: 258-60) pointed out, the Diodorean account of
modalities can only be confronted by not accepting its premises (or, less radically: by
not accepting the interpretation, which Diodorus gives to its premises). Nevertheless,
given the premises and their Diodorean interpretation, its validity is invulnerable.
Aristotle remarked in the Metaphysics 1046 b29-33 that the Megarian (and, effectively,
Diodorean) view on modalities is counter-intuitive (“atopon”) – NOT that it is invalid
Cf. also Lemmon (1977: 4-5), who calls indeed strict implication: “Diodorean”.
Prior (1962:138) offers no evidence to support his view, that the arguments were set the other way
around, to the effect that “[t]he aim of the Master Argument was to refute the Aristotelian view […]”.
Same holds for P.-M. Schuhl (1960). Also Oskar Becker (1960: 258) tried to make sense of the view,
that it was Diodorus who answered to Aristotle and V.A. Kyrkos (1981: 358-9) pointed out, that
Diodorus was too young to have been the first to express his views in the controversy with Aristotle.
But Kurt von Fritz (1978: 112-3) was persuasive enough offering the reasons to disregard this. Aristotle
had explicitly disqualified the modalities of “the Megarians” in the Metaphysics 1046 b29-33. It was only
natural then, to specify his view on modalities later, in Peri hermeneias. The shift from the too general
critique in the Metaphysics to the more specific argument in Peri hermeneias seems to presuppose
Aristotle’s getting familiar with a more detailed Megarian theory in the meantime – and Diodorus is the
only Megarian known to us, who did offer a more detailed view on modalities. Therefore, I hold the
fact that Diodorus was younger than Aristotle as secondary to the real events about the theory shift in
ancient modal logic.
2
3
3
(“akyron”): In Peri hermeneias, Aristotle obviously wanted to impose, that the
inference of necessity out of possibility in the Master Argument is a non sequitur, just
if one takes another interpretation of “being possible”. The counter-example to the
Master Argument, which Aristotle employs in the Sea-Battle-Argument makes indeed
a non-Diodorean interpretation of “being possible”, and a very intuitive one maybe,
but still does not show, that the Master Argument is invalid.
Just like in the Sea-Battle-Argument of Peri hermeneias, Aristotle’s views of
necessity in the Poetics (1452 a) show that he could not but believe that the Master
Argument is counter-intuitive (but still, there is no evidence that he thought, that it
was invalid). Aristotle recalled there the story of Mitys, an honourable man of Argos,
who was killed during some uprising. His fellow-citizens must have honoured him
exceptionally, as they raised his statue. Some time after his death, something
sensational happened: Mitys’ statue fell from the pedestal to kill someone who was
passing by. Not anyone though, because the dead was the very person who was known
as Mitys’ murderer. The incident was thought of as Mitys’ revenge and as such it was
seen as resulting from a higher decision. For Aristotle, however, only a poet without
wit could ever choose this story – a trivial one, as he thinks – to be his scenario.
Because necessity put forth by a divine decision, is here not the case, argues Aristotle.
He makes the point that not everything that is admirable (“thaumaston”), is deliberate
(“epitedes”). The falling of Mitys’s statue, so it seems, had to be an accident
according to Aristotle. The Mitys-case has not been reviewed so far for its modal
implications, which is a pity, as there are indeed noteworthy ones. An event remains,
says Aristotle, a contingency from any point of view, even if from our real point of
view it has already occurred – i.e. although it became necessary from the point of
view after it occurred, as we know from Peri hermeneias (loc. cit.). Aristotle favors
here again the conception of futura contingentia and presupposes an intuitive meaning
of ‘being possible’ – e.g. the one, which he introduced in the Prior Analytics.
Diodorus and the Megarians are not mentioned in the Poetics, that much is true. But
anyway, these were not mentioned in the Sea-Battle-Argument in Peri hermeneias
either.
The few Greek logicians of the late antiquity and the Middle Ages who dealt with
the Diodorean modalities, tended to maintain, what they thought to be the “orthodox”
Aristotelian position, and allowed, intuitively enough, accidental states of affairs, by
condemning the Master Argument as a fallacy.
Thus, Alexander of Aphrodisias (2nd century PD) insisted in his commentary on
the Prior Analytics (18330 – 1846) that Diodorean modalities are ambiguous.
According to Diodorus, says Alexander, everything that is possible either is or will be.
The only original argument, which Alexander employs against Diodorus, is that there
is a strong habit (“hexis”) to call “possible” everything which is (more or less)
probable (“hos epi to pleiston kai tou aoristou kai tou ep’ elatton”). But even this is no
more than a variant of the Aristotelian view, that the possibility notion of the
Megarians is not intuitive.
John Philoponus (6th century PD) considers the use of the predicate: “possible” in
Diodorus as mistaken too. He says that according to Diodorus “possible” (“dynaton”)
has at the same time the meaning: “existent, occurred” (“hyparchon”, “ekbebekos”)
and “possible to occur but still not occurred” (“dynamenon ekbenai mepo de
ekbebekos”). I.e. Diodorus changes the meaning of “possible” in order for the term to
fit in his analysis (“alla tina tou dynatou semainomena einai phesi” – cf. In Aristotelis
Analytica Priora Commentaria, Vol 13,2, p. 16917-9). If anything, then, the Master
Argument is question begging. But John Philoponus thinks that something else is the
case, as he presents the Master Argument in terms of a quaternio terminorum:
4
Everything which is either occurring or has not occurred yet, is possible
Everything that is possible, [is said, from the point of view after its realization,
to have occurred necessarily; therefore it] has to occur.
 Everything which has not occurred yet, has to occur.
Then “possible” has clearly another sense in the major premise than in the minor
premise.
It is only natural, then, that, Michael Psellos (*1018-†1078), perhaps the most
profound of Byzantine literates, makes in his Theologica, 130-1, the point, that the
Master Argument is a fallacy (“paralogismos”) – which is something more than a
lectio difficilior of the Diodorean modalities; in fact, Psellos’s opinion is biased.
Because strong modalities are not fallacious. If anything, they are counter-intuitive.
As an aftermath, Byzantine philosophy showed a clear preference for certain
modal systems, which excluded necessity implied out of possibility, as the case with
the Diodorean modalities is. Aristotelian “orthodoxy”, turned out to be a guideline for
religious orthodoxy, as the latter insisted on modalities, which would allow chance as
an ingredient of the world-order.
Like the Greeks, Westerners too reviewed the Diodorean modalities already in the
very beginnings of the Middle Ages (cf. Boethius, Commentaria in librum Aristotelis
Peri hermeneias, ed. Meiser, Leipzig, 1880, p. 234). But unlike Byzantine
commentators and historians of philosophy, Western scholasticism considered modal
“heresies” of the Diodorean type more open-mindedly.
One generation after Michael Psellos and in the spiritual atmosphere of the schism
between Greek Orthodoxy and Roman Catholicism (1054), Peter Abaelard (*1079†1142) reviewed in his Theologia ‘Scholarium’, III, §§ 106-13 the question if God
could have created more or less than he did. Peter Damian (*1007-†1072), had
formerly put the question, if something that already happened, could be made undone
due to some action of God’s. Damian, who most possibly had not considered the
modal connotations of the question, had left it unanswered and banned it to the realm
of blasphemy. But Abaelard dismissed Peter Damian’s agnosticist verdict by means of
a modal syllogism4: he argued that due to God’s nature it is impossible to Him to act
otherwise than He acts or acted or has to act. Clearly, then, what God is about to do,
will happen (some time) necessarily. Nothing, which is reasonable for God to do, is
doomed to fail; and: everything God did, He did since it was the only reasonable thing
to do. Abaelard’s position was condemned by the council of Sens, but his attempt to
introduce God into a modal frame paid back. In Aquinas’s Summa Theologiae (liber I,
quaestio XXV, IV) we find an argument to the effect that God cannot make undone,
what is already done in the past (cf. (3) above). It is perhaps more noteworthy, that
Thomas Aquinas based his proof of God’s evidence in Summa Theologiae on a system
which permits the passing from possibility to necessity (cf. Γερογιωργάκης (2002)).
Strong Diodorean modalities, which were found so appalling in the Greek
commentaries between the 2nd and the 11th centuries, were restored by the scholastics.
Leibniz (Theodicy, §§ 170-1) noticed that the point in the third book of Abaelard’s
Introductio ad theologiam5 is an instance of Diodorus’s Master Argument: Leibniz is
In Abaelard’s passage above, the discussion is on an argument about God’s actions being necessary,
which was first posed by Boethius, an early reviewer of Diodorus’s modalities (cf. loc. cit.). Peter
Damian, who had realised the problem without trying to resolve it, is not mentioned in the third book of
Theologia ‘Scholarium’. But characteristically enough, Abaelard discussed Aristotle’s See-BattleArgument in Peri hermeneias just before the passage, to which I refer here.
5
This is no other work but Theologia ‘Scholarium’ itself. The title, which Leibniz mentions for this
work of Abaelard’s: ‘Introductio ad theologiam’, was coined in 1616 and stayed in use until 1935,
when the new title was introduced by H. Ostlender. Cf. Petri Abaelardi, Opera Theologica, vol. III, ed.
by E.M. Buytaert & C.J. Mews, Turnholti, 1987: 203, footnote.
4
5
right on this: Abaelard’s argument holds for systems, in which, whatever is possible in
t0, not only becomes necessary in t2, but has had to be held as necessary
retrospectively since t0. So was the case in the Master Argument too. Leibniz
explicitly says, that, what we find in Abaelard’s theory are the Diodorean modalities,
which Leibniz himself employed, to form his own ontological proof (Theodicy, § 184;
Monadology, §§ 44-5). The passing in Leibniz’s ontological proof is again one from
possibility to necessity. Still, Leibnizian modalities are stronger: what is possible
there, becomes necessary in eternity, due to God’s power (which is the same as His
inability to do something which is not necessary). And, finally, it was under the
influence of his reading of Leibniz just before and during the 2nd World War that Kurt
Gödel wrote his “Ontological Proof”, which proves the existence of the entity which
incorporates all positive properties but none negative.6
Stamatis D. Gerogiorgakis
University of Patras
REFERENCES (SELECTIVE)
Becker, O., “Zur Rekonstruktion des Kyrieuon Logos des Diodoros”, in: Derbolav J.
and Nicolin F. (eds.), Erkenntnis und Verantwortung, Düsseldorf, 1960, 250-63.
Γερογιωργάκης, Σ., Αναγκαίο ον και νομοτέλεια, Αθήνα, 2002.
Feferman, S. et. al. (ed.), Kurt Gödel. Collected Works, Vol. III, New York/Oxford,
1995.
Frede, D., Aristoteles und die Seeschlacht, Göttingen, 1970.
Fritz, K. von, Schriften zur griechischen Logik, Bd. 2, Stuttgart/Bad Cannstatt, 1978.
Kneale, W. & Kneale, M., The Development of Logic, Oxford, 1962.
Κύρκος, Β.Α., “Ο Αριστοτέλης και οι μεγαρικοί”, Φιλοσοφία 10-11 (1980-’81), 34662.
Lemmon, E.J. & Scott, D., An Introduction to Modal Logic, Oxford 1977.
Mates, B., Stoic Logic, Berkeley/Ca., 1953.
Prior, A.N., “Diodoran [sic] modalities”, Philosophical Quarterly 5 (1955), 205-12.
_________ , Time and Modality, Oxford, 1957.
_________ , ‘Tense Logic and the Continuity of Time’. Studia Logica, 13 (1962),
133-48.
Schuhl, P.M., Le dominateur et les possibles, Paris, 1960
Sobel, J.H., “Gödels Ontological Proof”, in: Thomson J.J. (ed.), On Being and Saying.
Essays for Richard Cartwright, Cambridge, Mass., 1987, 241-61.
Strobach, N., “Logik für die Seeschlacht - mögliche Spielzüge”, Zeitschrift für
philosophische Forschung 52 / 1 (1998), 105-119.
Gödel’s most complete manuscript of this proof from February 1970 is to be found in: Sobel (1987:
256-7) and in Feferman (1995: 403-4). Kurt Gödel dictated a fuller version to Dana Scott at exactly the
same time (in: Sobel (1987: 257-8)) – or Scott knew exactly how to make a better sense of Gödel’s
ideas. There are Gödel’s own manuscripts with outlines of the same proof from, which are dated back
to 1941 (= Feferman (1995: 428)). For the debts of Gödel’s ontological proof to Leibniz, cf. Sobel
(1987: 241) as well as the “Introductory Note to *1970”, written by R.M. Adams, in Feferman (1995:
388 ff.).
6