J. Vuillemin, Necessity and Contingency

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index.
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alk.
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History.
3.
Diodorus Cronus, 4th cent. B.C. 4. Philosophy, Ancient. I. Title. II. Series.
1996
1983
International, and Xerox PARC to
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monographs, working papers,
1 The Master
irrevocability. Signification of the first premise. 7
1.4
Prior's
interpretation:
It
supposes
two
supplementary
premises,
it supposes
the first
premise ambiguous.
Caelo,
2.2 The principle of the conservation of modal status. 16
2.3 The principle of the possible realization of the possible
interpreted
2.4 The principle of possible realization of the possible as
principle of synchronic contraction of the possible and
diachronic
2.5 The
2.7 The
2.10 Sketch
of a
formal reconstruction
of the
2.11 Sketch of a formal reconstruction of the Master
a rgument .
3 8
Diodorus
Solution.
43
3 .2 Tw o possible inte rp reta tion s as regards the object of the
Diodorean modalities: nominalism
3 .4
4
an d secundum rem: Ockham's conception on Prior's
hypothetical reconstruction. Modality
4.2 Inadequacy of Ockham 's solution. In crimination of the
principle of
and the
r e t u rn .
97
Chrysippus.
105
5.1
th e
C O N T E N T S / vii
5.2 Chrysippus' do ubt about th e interdefinabili ty of the
modalities. From th e non-possibility of an event's
occurrence
Chrysippus. 115
temporal
5.5 A Philonian doubt about th e second premise? 126
Part III Systems of Contingency: The
Lyceum,
6 Towards Rehabilitating
6.1 De Interpretatione,
and the law
a
34-18
6
17
a
22-19
6
6.6 A ristotle 's general conception confirms th e De
Interpretatione solution; the
truth-values. 154
6.8 Second interpre tative hypothesis: propositions w itho ut a
determ inate truth-value. 157
6.9 Th ird in terp reta tive hypoth esis: probability. 161
7 Epicurus and
excluded middle: the
7.2
Second logical in terp retat ion of the Epicurean negation of
th e
excluded middle:
7.3 A re the
hypoth eses and rejection of the excluded m iddle. 182
7.5 Epicureanism and the Master Argument . 185
7.6
of
8.2 A ristotle's dogm atic definition of truth called into
question (De Fato, XIV ). 210
8.3 Carneades and the M aster A rgu m en t (De Fato, IX). 212
8.4 From Carnead es to the logics of
"fictive"
names:
8.5
interpretation of t ru th. 219
9 Platonism and Conditional Necessity. 225
9.1 Platon ism and the princ iple of con ditional necessity. 225
9.2
Consequences
modality, causality and
conditional necessity and the substantiality of the
sensible w orld for the Platon ic and Platonistic theories of
modality, causality and
10.2
Diodorus' solution.
and the
/ ix
Bibliography
267
8/8/2019 J. Vuillemin, Necessity and Contingency
Introduction
The Master A rgu m en t , recorded by Epictetus, indicates that Diodorus
had deduced a contradiction from the con joint assertion of three propo-
sit ions. Epictetu s adds that three solutions
of the aporia had
the other two.
The a rgument , w hich has to do w i th necessity and con tingen cy and
therefore
all. In any case
at
logical terms, without excessive worry about historical plausibility
and w i th the foregone conclusion that it was sophistic since it directly
imperiled
our
of
freedom.
On
both of these coun ts I have taken exception to recen t trad ition .
The success of the arg um ent w ith the A ncients , and w ith A ncients w ho
were
no mean logicians, seemed reason for presuming that the Master
Argumen t
is not
real
one. On the other side, I looked for a classical text containing
the propositions stated by Epictetus and which could have furnished
Diodorus
with the material for his argument. I believe to have found
such a text in Aristotle 's De Caelo.
In order
in the
propositions thus
restored, I had in my t u rn to translate them into logical terms. It is
unlikely
the
t rans-
lation I have proposed tries to remain faithful to its models as they
have been handed down to us, it inev itably gives them a precision they
did not have in themselves.
This
historical
There
be
no ticed. The proposition s f iguring in the M aster A rgu m en t are in-
OR
C O N T I N G E N C Y :
THE
M A S T E R A R G U M E N T
terpreted in terms of temporal modal logic where both the modalit ies
and the statem en ts the y govern have chronological indices. This means
that the
from
our experience of t ime.
To bring to its complete end the research undertaken here, I w ould
have
and to
solutions
it is susceptible of, an explicit axiomatic system form alized according
to a set of rules. For want of competence, of stamina, of time, I have
been content
the
Master Argument to Greek philosophy at large. In this way it is seen
that principles are
mentioned
in the Epictetus passage but which must have played their
role
is
legit imate or not. As one well imagines, the debate on the issue was
continued by the philosophers of the Middle Ages and the Moderns.
This was quite generally done in ignorance of the Master Argument
itself;
specify
I N T R O D U C T I O N / xiii
Acknowledgements
This book builds on earlier studies (Vuillemin, 1979, 1983) and espe-
cially
on
Necessite
ou
contingence,
I'aporie
de
Diodore
Paris, 1984). I considerably shortened this last work and eliminated
the
Argument .
I corrected it on the occasion of an objection raised by
M.H. Angstl (his letter and my reply to it were published in the All-
gemeine Zeitschrift fur Philosophic, XI 3, 1986, pp. 79-87). Finally I
added
translated into English this new French version.
My thanks go to Professor J. Moravcsik for his criticism, Bill Gra-
h a m , w ho revised the text, M rs. L. von Kam pen , w ho transform ed a
manuscript into a book, and to Patrick Suppes, my good friend.
Shortcomings of Some Past
Interpretations. Conditions to be
Fulfilled by any Acceptable
1.1 The text of Epictetus.
Ep ictetus gives the following account of the a r g u m e n t .
Here, it seems to me, are the points upon which the Master Argu-
men t w as posed: there is, for these three proposit ions, a conflict
between any two of them taken together and the third: 'Every true
proposition
logically follow from
used
the
plausibility
prove
the
following: 'Noth-
in g is possible which is not presently true and is not to be so in the
future ' .
Another ,
keep, will maintain
these two: 'There is a possible which neither is presently true nor
will be so; the impossible does not logically
follow
about
the
past
489)
w i th w hom A nt ipater is general ly
in
agreement (SVF III A n t . 30). Oth ers (namely Chrysippus, SVF
II
nor w ill be so; every tru e proposition about
the past is nece ssary'; but t hen the impossible
follow s
logically from
the possible. But there is no way to maintain the three
propositions
4 / N E C E S S I T Y OR C O N T I N G E N C Y : THE M A S T E R A R G U M E N T
at
1
To this account Epictetus adds an ironic com m entary on the virtue s
of
erudi t ion. To one who boasted of hav ing read A ntip ater 's treatise
on the M aster A rgu m en t he exclaims: "W hat m ore do you have,
you,
for
quest ion?
You might jus t as well speak to us of Helen, of Priam and of that isle
of
wil l" .
2
The irony of Epictetus is aim ed ne ither at the do ctrine nor at the
moral consequences that could
but
only
at the vanity of logical quibbles. W e can no t speculate the n on any
opposition
of
had to the
Such is the only explicit text on the Master Argument.
M ost interp retation s and reco nstructions of the argu m en t that have
been given
all
agree,
implicitly at least , on d en ou n cin g i ts disastrous m oral consequen ces
and on flushing out
freedom called into question
in the first
tw o premises. He played, in the first premise, either on two possible
senses of the w ay in w hich a prop osition can be concern ed w ith the
past
or on two senses of the w ord 'necessary'. Or finally, he played on a m ore
general ambiguity hidden
which
possible would result from
impossible cannot result
from a possible. Thus nothing is possible which nei ther is nor
will
be".
The minor premise illustrates Diodorus' second proposition. The con-
ditional major premise has Diodorus ' third proposition as antecedent
Epictetus, 1916, II, 19 1-4; Dor ing , 1972, p. 131; the parentheses refer to the von
Arnim col lection,
translates axoXoutfetv
correctly by
Z E L L E R ' S
I N T E R P R E T A T I O N / 5
and the neg ation of his second as con sequen t. The correlation of the
two must evidently express the argument's first proposition. "The hy-
pothetical major premise
past
wh e n
of two m utu ally exclusive cases one has turned out to be realized,
the possibility of the other finds itself cancelled out, since that which
has
happened
case,
then,
is
now impossible; had it been possible beforehand, then, in the opinion
of Diodorus,
a
possible".
m e n t
does in-
deed have th e temporal sense of 'follow in succession', whereas, in the
minor pre m ise, on the co n trary, it has the logical sense of
'follow
log-
ically'. In the major premise, from the realization of a possible can
and must result th e de facto impossibility of another possible, since
the
But the
minor premise signifies tha t no th ing is possible of w hich the realization
would
result
in
in itself.
According to P.M. Schuhl , it is possible to restore a purely logi-
cal meaning to the condit ional major premise, thereby validating the
argument in rendering its elements homogeneous. It is not that the
impossibility
of an event simply succeeds its possibility: it is rather
a
contrary.
4
To
il lustrate w ith an example from K ur t von Fritz,
5
the condit ional major
premise would mean "If the Carthaginians win today at Cannes, it fol-
lows
answered that
impossibility, which is not actually far removed from the impossibility
due to temporal succession, as understood by Zeller.
M r. G .H. von Wr ight has given an elegant and logically im pecca-
ble, formally modified version of the intuit ion contained in this type of
in te rpre ta t ion .
6
He begins by suppo sing p to be possible. A s a con-
sequence of
t,
consequence
74-75.
5
6 / N E C E S S I T Y OR C O N T I N G E N C Y : TH E M A S T E R A R G U M E N T
later than t, it will be impossible that p at t'. Thus the supposition
of the no n-re alization of the possible leads to an im po ssibility, if one
interprets the second Diodorean premise as
follows:
impossible at some
m o m e n t of t ime.
In inte rpre ting the second Diodorean premise as a proh ibit ion of the
proposition p's
brings
to
light
the
metaphysical,
as
opposed
first
type of interpretation is obliged to suppose. The difficulty is that the
retrospective impossibility of p , which, according to the first premise,
follows
upon
the
non-realization
of
p
at
t
begins
t',
later
t han t, n o m at ter w hat hypothesis might b e made as to the cont inui ty o f
t ime.
be
i f
t' coincided
w i th t, and the second premise is then supposed to be meant to assure
the validity of the syn the t ic judgm en t by w hich the conservation of the
modali ty of p from t to t' is guaranteed.
In all these cases, the co m m en tators reject the validity of the M aster
A r g u m e n t in accusing Diodorus of confusing a de
facto
necessity
w ith a
necessity of principle; and in so doing they are immediately in
conflict
Epictetus
these interpretat ions from
'follow
logically'. In this w ay the trad ition of the A nc ients is respected, tho ug h
w i t h o u t , of course, an a priori refusal to examine whether the not ions
of
possible
by
'follow
[logically]'
(aKoXovtieli
1
) is
that
w hich the Megaro-Stoics u se for designating w hat
w e
succession.
8
For
Diodorus,
' < ? ' follows
from
'p'
if at no t ime do we have both 'p' t rue
and 'q '
false. Moreover,
the
prin-
ciple
' the impossible does not follow from th e possible' is false in the
Diodorean system if interpreted in the sense of temporal succession. A
sta tement of the type
'it
is possible that p' can be t rue at present and
false in the fu ture" .
9
Tho ugh a temp oral sense of the term "follow"
is
tion between the antecedent and the consequent and the consequent
7
For
8
On this point, cf. ibid., pp. 128-138, especially, p. 132.
9
Boudot, 1973, p. 445. As the author points out, Rescher and Urquhar t , 1971,
p. 192, agree with
A M B I G U I T Y IN THE F I R S T P R E M I S E / 7
themselves express temporal modalities connected by a simple material
implication.
irrevocability.
Signification
premise.
The first premise of the M aster A rg um en t has seemed ambiguo us, and
even doubly
is past and
t rue (itav K a p e X r ) A u 9 6 < ;
d X r ) 6 e < ; )
is
necessary.
It
true,
or
past
event
is
necessary if tru e. Since w e can give a gram m atically past form to
every
'it
days',
escaped
Aris-
totle, Epicurus and Chrysippus or that they would have preferred in-
curring the risk of
from
word
'necessary'. It migh t be said that a past event is simply irrevo-
cable. When a thrown dice has come up six, the throw doesn ' t
cease
to have been aleatory. The proposition saying that the dice turned up
six could in no way be necessary, although it is about an irrevocable
event.
Such an objection, which is entirely justified, obliges us to recon-
sider th e sense of the Mas te r Argument ' s first premise and to ask our-
selves what could be the meaning of the necessity accorded to t r ue
propositions about
the past.
of
necessity,
by virtue of w hich that w hich is necessary is a
fortiori
existent
oportere
ad esse valet consequentia) an d that which is existent is
a fortiori
This axiom
seems logically
fortiori
possible. The logical sense of the modalities, as such, excludes time.
Naturally,
OR
THE
it w ould necessarily en tail, ipso facto, the logical possibility having
that
a
certain thing should have happened, i t is a fortiori possible
that that
th ing
modalities
are
applied
to
temporal events, they are generally understood, and r ight ly so, in a
different
in a
logical sense.
Irrevocability, which is a factual kind of necessity, applies to any event
whatsoever, even
it has
to pass. It follows
then tha t the real modality itself must be assigned a temporal index
dist inct from the one affecting the event to which the modality applies.
A t
present it is irrevocable, or necessary in the factual sense, that the
battle of Salamis took place. Factual possibility, the m odal c ou n terp art
of this fac tual necessity,
will
likewise be assigned a temporal index of
its ow n. But it is no table th at the re is no way of ge tting from the
past
conceived
or, at
m ost, a presen t eve nt, to the exclusion of any even t hav ing taken place.
W e
shall see
for Aristotle this privileged temporal direction of the
possible con sti tutes the e ntire content of the M aster A rg um en t 's first
premise.
not be
or tem-
poral necessity w ith irrevocability. N either the one nor the other of
these
confusions
will,
will
be
shown
to
having to resort to arguments borrowed from modal logic. Therefore
there will be no risk of subreption betw een the logical and the factual
meanings of
the first
premise ambiguous.
A . N. Prior has proposed a formal reconstruction of the Master Argu-
m e n t giving a purely logical sense to the word
'follow',
deeming
it
useful
vocability of the past.
Diodorus ' first two premises, A and B, joined to two
others,
C.
What
he
proposes
is
roughly
the
following.
10
A. If it has been tr u e that som ethin g is the case th en it is not possible
that it has never been true
that
follows
if
possible.
D. Of whatever is the case it has never been true
that
E. Of
whatever neither
is nor
ever will
be the
(at some moment) that it will never be the case.
Consider
the
propositions
premise
p in B be
the proposition ' something is the case' and the consequence q ' this
something has never been never going to be the case'. W e are justified
in saying th at if it is necessary that w hatever is the case has never been
never going
case,
case.
antecedent
of
that
p);
Fp: it will be the case that p (it wi l l be
true
E.
Prior
as u n d e f i n e d ,
defines M as ~ L ~, and adds to propostional ca lcu lus only the one new ru le
I- a — » h La and the two axioms: Lp 3 p and L (p 3 q) 3 (Lp 3
Lq).
(Contraposition)
2.
axiom,
Mod.Pan.)
4. L (p 3 q) 3 (L ~ q 3 L ~ p) (3, second added axiom, Mod.Pan.)
5.
[L(p
Mp)]
Let
1.
L(p
3~
2. L(p
w i th substitutions p/p,
~ P ~
Fp/q)
3.
(1,2,
Mod.Pan.)
5. P
(4,5
Mod.Pon.)
7.
w e
possible that something
has never been never going to be the case, then it is not possible that
it
case'.
On
the other hand, given E and substi tuting 'something will never
be the
rule
of
write:
Of w hatever ne ither is nor ever will be the case it is no t possible t h at
it has
to be the
have dem on strated
(B and D ) that if it is not possible that som ething has never been never
going to be the case then it is not possible for that something to be the
case.
will
is im possible.
This is the denial of the M aster A rgu m en t 's premise C according to
which there is a
Once
of D
But in one case it is the Megarians, in the
other the Stoics, to w hich this principle of retrogradation is at tr ibu ted .
Aristotle refuses it imp licitly.
A s for the premise E, it means that "... if a statem en t is false and
will always rem ain so in the fu ture, then there has been a past moment
at which it was true
that
be false".
14
W hat m akes this prem ise seem plausible is th at "... if
p
at
least
just
past
it wasn ' t t rue , that p would be t rue again".
15
non-dense,
16
as
Diodorus
holds,
17
13
Potest factum quicquam igitur esse, quod non verum fuerit futurum esse?
(Ci-
14
Boudot,
16
Boudot, 1973, pp. 447-448 (for an intuitive resume); Prior, 1967, pp. 49-50.
17
render motion impossible. According
himself on
o f
For
that
time
suffices
for the arg um en t, dense. Then betw een
the presen t moment ,
the antecedent
past m o m e n t , t\,
chosen arbitrarily near so as to verify th e consequen t 'p has been never
going to be the case', there will inevitably be an infinity of instants
and so at least on e ins tan t , £
2
t
0
which
affords no condi t ion on the status at t ime t of the fu ture case
in
false
t ru th -value
is satisfactory only
and
to , which implies that t\ is the immediate predecessor of to and that
time is thus discrete.
With
the c onclusion established, it can be show n tha t D iodorus'
propositions
-viz.
E-
expressing
possible.
18
On the o ther hand , i f we begin wi th that definition, the
premises used, linking
temporal
determinations,
are demo nstrable. Looked at in this way, the M aster A rg um en t reveals
such
a perfection in its alliance of coherence and simplicity that it be-
comes easy to unders tand the reputation it established for its au thor
and the respect i t inspired among the Ancients".
1 9
ceived
directly
at least, to de t e rmin i sm: a statement true sometimes, but not always,
is true without being necessary.
Tw o
The first
the second in the hypothesis of an ambiguity in the
interpretation
of
which
an equivalent is supposed
Diodorus there is a relation between these u n ities and the
defini t ion
of the possible:
since no body
inf ini te
n u m b e r of parts, w hy call such a
division
division
is
indefinite,
not reaching
infinity. It is to him that our explicit sources trace back th e affirmation of inf ini te
divisibili ty
(no te
80, pp.
111-112).
18
That
is to say: the possible is whatever is or will be and only
that.
19
Boudot,
OR
THE
to be
to an
increasing degree
of
'neig hbo urh oo d', the three n otions of succession, contiguity and con-
tinuity. "The A ristotelian co n tinu um
is
defined
in a
certain sense w ell-linked (Ephexes an d Echomenon ) , of parts poten-
tially separated by lim iting points" .
20
con t inuum th us involves its den sity. So
D iodorus'
Prior's
reconstruct ion, would fail to touch Aristotle in the least. Not only
would
it not touch him in fact, but the introduction of clause E of the
discontinuity
of
be
to a
conscious admission
o f
defeat. If w e t h i n k of Diodorus as having distinguished himself
from
the "ancient" M egarians in saving the m odal distinctions, it w ould have
to be added that as soon as conf ronted w i th a dense time-and we can
presume
that on this point the Aristotelian representation had rallied
the assent of the scientific com m un ity-those distinction s collapse. The
Mas te r Argument is thus demoted to the rank of a particular school's
a rgument .
Prior's
this way,
exploited an a m bigu ity of langu age. But this is to fly in the face of
the tradition which has it that, contradicted on this point by Chrysip-
p u s ,
equivocation.
22
Surely,
one might argue, this premise does not in itself produce necessity, for
Diodorus dis t inguishes
from
'pseudo-dated'
20
Granger, 1976, p. 306; for the ph ilosop hical analysis of this A ristotelian contin-
uum cf. Vu illem in, 1962, pp. 185-198.
21
Boudot, 1973, p . 470; Mrs. Kneale denounces th e confusion, 1962, p . 121.
2 2
A ul i usG e l l i u s , Nodes Atticae,
Bk. XI I ,
p. 128.
Ockhamist
suspect
as we
is no am-
biguity in Diodorus ' first premise, it would no longer be possible to
avoid necessitarianism
in a
case to
l imit
ourselves to authentically past events , where there is no room for for-
m al confusion, to
the
move, for it would no longer be s imply a grammatical mistake
that
the
univer-
sal
favor
of its solidity; dismiss the supposition that c r ud e
ambiguities
events
three premises without having either
to
postulate
the
discreteness
of
tion. Above all, in order to avoid introducing principles inappropriate
to the
by w ay
of comparison, to shed light on the objective sense of the latter.
23
Boudot,
t ru th
to the ne-
to the
si t t ing
does
follow,
is
equivalent
to
'It
tru e yesterday tha t Socrates
will be sitting the day after tomorrow' , w hich is itself necessary because past".
This
status
wi th the formal expression of the tense of the statemen ts
in
question
assures
the originality of Diodorus with respect to the 'ancient ' Megar-
ians (Blanche, 1965, pp. 133-149). It also limits the importance of th at originality.
Sedley (1977,
to the
Megarian 'school'
(pp. 74-78), m aki ng him rathe r a represen tative of the Dialectical 'school',
which
affinity
from
th e point of view both of the the ory of mo tion
and of that of the modalities, for us to still consider Diodorus to be a 'Megarian'
philosopher.
This constitutes th e essentials of Boudot's elegant solution (1973).
OR
THE
There is such a text . It is tha t of an a rgument of Aristotle 's con-
cerning demonstra t ions
2 6
26
With the exception of Cherniss (1962) and Hint ikka (1973), though even here ,
none of Hintikka's references to this passage (p. 94, p. 152, p. 164, p. 183) actually
analyzes Aristotle 's manner
exists
in
suppose this capacity
/14/
only of the present and the future . The same goes too if
that
/15/
the
possible:
supposed
to
o f
of the
eternal , t ho ug h sensible , substances which are not subject to generation and decay
from the
16 / N E C E S S I T Y OR
THE M A S T E R
A R G U M E N T
The passage
In the first,
Aris tot le
shows that noth ing that has been generated can last sempiternally,
in
the
of
an
affirmation,
will
serve to co nsti tute the M aster A rg um en t as w ell . Since the f irst
of these last three principles has the form of a logical co n jun ction , i t
will
be analyzed , for clarity's sake, as tw o ind epe nd en t principles. The
complete list then will consist of
1)
modalities,
principle of conditional necessity, 3) the principle of the impossibility
of
past,
subsistence of a possible
and these alone, are explicitly used in the De
Caelo
text. A ll but the
first , and these alon e, are to be explicit ly m ention ed in Epicte tus ' text
on
the
2.2
The
principle
status.
De
Caelo to which the Master Argu-
m e n t will have no recourse. The modalities, according to the Stagirite,
are not simple operators governing statements or propositions. They
are the characteristic properties of substances. Ther e are t h u s tw o sorts
of
is
to
and
endowed
w i t h pe rpe tual mot ion , b) contingent substances, now existing, now
not existing, subject to full materiality, to generation and decay. It is
an axiom of Aristotle 's philosophy that a substance cannot change it s
modal status. Whether necessary or cont ingent , it is so by na ture . It
will retain its modal status then and will never be seen to change it.
The con sequen ce of this p rin cip le is used in lines 7-9 of the tex t.
Suppose
that
something
that
or
should have come, to be. It thus belongs to the category of cont ingen t
things .
By
modal s tatus
it will the n retain later the capacity of not existing.
This principle and its consequence would merit an investigation on
their own; but since they play no role in the Master Argument and
OF THE
STATUS
/ 17
since
its
validity
on the
other hand,
four
for the sake of clarity, are the very ones operative in the Master
A r g u m e n t .
2
The prin ciple of the conservation of modal status is a consequence of the Aris-
totelian
a
30-
1069*2): a) the sensible corruptible substances, b) the sensible incorruptible sub-
stances, c) the im m ateria l and imm obiles substances. The th ree kinds are such
tha t ,
means being
passage
To
allow perishable substances to transgress their essence and accede to immorta l i ty
and ind estr uct ibil i ty w ould be to destroy the very l imits that define the nature of
t h ings and to
make contrary capacities
generable
and
the perishable are not so by chance and by fortune, it is seen that they are so
by na ture . Indeed, al l that there is is e i ther by n a tu r e or by chance-for w e leave
aside here that which
the
capacities that are proper to them . If they do n't keep the m an d they change into
other dispositions, their modification wil l be e ither the result of chance-in w hich
case there would be but one and the same disposition for both natural things and
fo r those resulting from chance and fortune-or the mo dification will come about,
even
for
in
t u r n wil l take place either by nature or as a result of chance. This is why natural
capacit ies must have l imits. But if these capacities have limits, then that also,
which has a birth and is subject to death,
wil l
be so by nature before that death.
I t can the n be seen that w hat happens n ow to the nature and the matter thus
subject to
th ings
and the i r existence, on the other their privation and their inexistence, has a limit
beyond
which
it changes no more. That is why it is necessary that that which is
engendered
should
is why
death too wil l come to it, in its time. Likew ise, w e do no t surprise that which is
always subject to death transg ressing its proper natu re. Otherw ise, since it will
have persisted
fo r
some time
of its
na ture ,
it will
equally persist
in the
disposition th at renders it im mo rtal and, k eeping i ts nature , i t
will
which
it was changed. That is w hy there must then be several
capacities at once for an inf ini te tim e, and since w e have established th at the action
of this capacity persists, then what w e have established will be false, which can no
more be the case than the rest w e have spoken of so often." (Them istius,
De Caelo
6
OR
THE
2.3 The principle of the possible realization of the
possible
interpreted
modal logic.
3
The reasoning, or rather the tw o parallel lines of reasoning that com-
pose our text are among those
reductio
ad
absudum
to singular propositions
possible cannot
of
reasoning
to
singular propositions of existence. W hat the re is actually question of
is a de te rminate individual, the Heavens; but the reasoning has to do
wi th
not
to
exist
(1.7-8). The reasoning is seen to be transcen den tal and un iversal, since
the
only 'predicate' u n d e r which such indeterminate individuals
are
o f
th e syllogistic form of science and particularly o f the modal syllogistic.
The directing principle of this m ode of reasoning has g enerally been
expressed in terms of pure modal logic. It is this conception that is
to be set out f irst . But, on subsequently examining the Aristotelian
text, this
possible comes down to none other than the diachronic and
synchronic
modalities.
From the possible as such, nothing can be concluded, for a modal
conclusion
is legitimate only if it proceeds a fortiori, that is to say,
from the necessary to the actual or from the actual to the possible.
Start ing from the consideration of a possible th en , the on ly thin g
w e can do is to analyze the consequences that would result from its
realization.
If
selves have
th e
realization turn out to be impossible, that realization is impossible in
t u rn . By a second contraposition it can fur ther be shown
that,
It ,
4
which
one must
be able to realize a possible is called "ecthesis by demodalization" by G. Granger
(1976, p.
THE PRINCIPLE OF THE POSSIBLE REALIZATION OF THE POSSIBLE / 19
Consider
as
existent
and examine th e consequence q that follows necessarily from it. If q
is contradictory, i.e. not realizable in any possible world, we have the
right to conclude that p is
also
impossible. This is the import of the
M aster A rg um en t 's second premise according to w hich the impossible
does
possible.
5
A s a typical application of the principle Aristotle gives that of the
incommensurabili ty
of the diagonal of the squ are w ith its side: "All who
establish
an
is
false
and prove the hypo thetical conclusion w hen som ething impossible
results
from
the
supposition
diagonal of the square has no common measure with the
side,
since if
one supposes it to have a common measure then the odd numbers are
equal
hypothetically
the incom men surabil i ty
of the diagonal, since a falsehood results from it s negat ion" .
6
Thus if p
signifies tha t there is a comm on measure betw een the side and diagonal
of
the
once both even
and odd , then given that if p is t r ue q is necessarily true as well,
from
th e impossibility of q th e impossibility of p is inferred, i.e. th e
incommensurabi l i ty of the
side
square.
From this principle of the possible realization of the possible one
can draw yet another principle whose intuitive significance is
perhaps
clearer. Namely, it is equivalent to say of a th ing tha t it is possible and
that
the conjunction of the possibility of that thing with its realization
is
possible.
7
A t
this point th ere is need for a no te of cautio n. It hap pen s, bo th
in
th e case of Aristotle 's reasoning and in that of the Master Argu-
men t
tha t
possible under consideration produces
a contradictory consequence only by vir tue of an as yet non-explicit
hypothesis. It is
tion justifies the
the
realization of the possible and the hypothesis but in no w ay does it jus-
5
weak
6
An.Pr., I, 23, 41"23-30. This example is used by A ristotle in the De Caelo, I 11
281°7.
7
The principle in question here is: Mp = M(p.Mp). Since T contains L (p
D
q)
D
(L p
3L q )
and the ru le of nec essitation , it is n or m al. Hence given
p
3
normal,
Mp = M(p • Mp) ( N o te of B. G r a h a m ) .
tify the conclusion that there is incompatibility between the possible
in
hypothesis.
8
This
latter
the
pre-
mise.
It w ould resu lt in a sophism in the d istribution of the modali-
ties. Consequently, whenever a reasoning by realization is employed to
demonst ra te
rather
the
logical incompatibility between the possibility of one th ing and the re-
ality of ano the r, simp le application of the principle that the impossible
does not logically follow from the possible is not enough. What must in
fact be demonstrated
the conjunction of the realization of the possible
and the hypothesis, but from the co nj un ction of simply the possible
and the
is
required.
Such a principle indeed has been clearly stated by Leibniz. To An-
toine who in his Dialogue with Laurent Valla
sets forth
"the philoso-
contradic-
tories are both possible; can they also both exist?" H e thus denounces
the sophism of distribution w hich would have it
that
th e real world,
in so far as it excludes the possible worlds, would also render them
impossible by the same token .
10
8
Having
recourse
position is that p be compat ib le wi th
Mp.
Suppose
In
conjunction
is
equivalent
to :
hypothesis,
~ Mp.M(p.Mp)
conjunction
~ Mp.Mp.MMp.
9
Leibniz , Gerhard t , 1978, p. 359; Jalabert, 1962, p. 371. It is important to insist
here
on the word 'may ' which has as consequence th e impossibility o f applying
th e
impossibility
th e
philosophers ' rule
P R I N C I P L E
OF THE P O S S I B L E R E A L I Z A T I O N OF THE P O S S I B L E / 21
That
10).
It
will
be seen below ,
however, that the modalities figuring in this latter principle, instead of
belonging to pure modal logic, are affected by temporal indices. If one
were then to continue
inconsistency w ould
creep into the Aristotel ian passage, for the second principle of the De
Caelo
text w ould the n have to be expressed as the con jun ction of no n-
homogeneous terms,
the first
the second temporally indexed modal expressions.
2.4 The principle of possible realization of the
possible as principle of synchronic contraction
of the
is it
principle of
10
that
this capacity cannot be actualized whi le the th ing in quest ion
is in actuality
by
past.
bility
of not exist ing Aristot le speaks of then is not a simple logical or
atemporal possibility.
will
persis t th rou gh ou t al l the
f u t u r e
bu t
year and in the past. Such doubly indexed modali-
t ies are hom ogeneo us w ith the prin ciple of con dit ional necessi ty, as w e
shall see,
an d
the
principle
of
conservation
(for
w hat sense w ould i t m ake to speak of the
conservation
given
m om ent ( th at w e shall f ix as now for perspicuity 's sake) that som ething
( 2 ) ~ M(p. ~p) D ~ (Mp.M ~p)
and by virtue of non-contradiction and
modus
ponens:
(3 )
should
be
realized
or
actualized
at
either in the past or in
the f utu re (possibly coincident w ith the now ) such
that
be
realized
at
ti.
The
contracts a possibility posited diachronically over the interval onto an
instant of the interval.
By contraposition, if at every instant, ti, of an interval \N — 1\ < e ,
where ei ther
that
p at t.
It might be objected that this proposed formulation is en tirely in -
adequate
for
expressing
possible.
To
synchronically contract a possible by saying that it is possible at t\
that p at ti is surely not to realize that possible or to say tha t p at t\.
But we
did not
that
possibility
is
of the
interpreta t ion will have been satisfied. The aim of the following section
will
11
A ristotle exem plifies his principle now by cases w ith no relation to
t ime ,
281
b
10-20
the
possibility
of s tanding and si t t ing. For a given agent, it is impossible at a given
momen t to be at any t ime both sitting and standing. A nd it is clearly
having in mind such doubly indexed modalities that he says in the
chapter of Metaphysics w here he refutes the Megarics (0,3,1047
a
23-
2 5 ) : "A th ing is capable of doing something if there will be no th ing
impossible
the
contraction
(~
w e to
De
Caelo text in terms of doub ly temporally indexed mod ali ties , m ay
be
expressed
(t){~
L
N
~p
t
• ( « < *i < NV N < ti < t ) ] } .
W h a t is the relat ion between the law of modal logic
(a ) L(p 3 q)
O F T H E
POSSIBLE R E ALI Z AT I O N
O F T H E POSSIBLE / 23
Then
it
would
be
true
t\
would obtain, destroying the
Let us show on what conditions B can be deduced from
(a):
2. h L{
(a).
4.
(
.
(3«i)M[p
t l
• (t <
ti
t l
( l
p
MNP I, 10)
There are fou r statements in this deduction that merit justification: 1, 5, 7 and 9.
The validity of the first
statement stems
t inc lusive ly .
The fifth statement is a characteristic axiom of the logic of the temporal modal-
ities.
then
p at
t, since the synchronic possibility puts added strictures on the logical possibility.
Statement
formula.
That
formula
is
legiti-
mate when the variable is interpreted substitutionally. And in fact a substitutional
interpretation
is
The definition 9 is that of synchronic possibility. To say
that
N,
is to say that it is logically possible
that p be the case at t\ with t\ a function of t and of
./V
potentiality of
will
be
impossible
to
contract
the
possible.
A n objection could be derived from this last case. "Suppo se, it
would be said, that I can n ow do something at a given t ime tomorrow.
It may happen that this possibility will be maintained du ring some
t ime
< ^ i <
an
at
( 2
an
event
be
produced which makes impossible that patti. Therefore it may be now
possible
that
p
at
t\
and this possibility may persist un t i l t^ w i t hou t
that p
Metaphysics
16-21), Aristot le requires
that having a potency does not h appen w i tho ut given cond i t ions th at
distinguish
a genuine potency from a rh etorical one: "To add, he says,
the q ualification 'if n othin g external preve nts it' is no fur the r necessary;
for it has the potency on the terms on w hich this is a potency of acting,
and it is not in all
ci rcumstances
but on
cer ta in condi t ions, among
which will be the exclusion of externa l h indrances , fo r they are barred
by
some
line
10 Aristot le uses, wi thout formu-
lating it, the principle of con ditional necessity. Which are the cases, he
asks, in wh ich the consequence d raw n
from
t ion of
to the
negative possible)
that
a thing which has not been but then comes to exist
will
not
is
not
by the
pr inciple
of non
contradiction applied to temporal things, since it is impossible for one
and the
same
thing to be and not to be at the same time. The second
exclusion is stronger. It prevents a negative capacity's coexisting, in
so far as the time of the event governed by that capacity is concerned,
wi t h an
affirmative actuali ty; just
would prevent
an affirmative capacity's coexist ing w ith a negative actuality. Thus the
actuality of
not-p
of
not-p excludes th e capacity of p whi le no t-p.
It would be futile to try to deduce such a principle
from
THE P R I N C I P L E OF C O N D I T I O N A L NECESSITY / 25
to pointing this out in the case of reasoning by possible realization.
1 2
Suppose
th e negative possibility of an event and the hypothesis of that
even t's existence actualize the possibility. B y vir tu e of the princ iple of
contradict ion
from
that
dis tr ibution
of the mod alities to conc lude that the po ssibility of the no n-realization
of an event is incompatible with its realization. The logical incompat-
ibility is in fact
by the passage
on e of two th ings mus t be the
case.
of the actuality and the
contrary capacity
to be a specific principle.
To decide the issue let us examine the consequences of this posited
incompatibili ty.
By the reciprocal definitions of the modalities, if p is necessary then
it is not possible that
not-p.
The necessity of a thing, then, excludes
the con trary poten t ial ity of that th ing . To main tain that the ac tuality
of a thing excludes the contrary potentiality of
that
assimilate actuality and necessity in that respect. But it
will
be seen
that in Chapter IX of De Interpretations Aristotle does explicitly as-
similate
actuality
and
actuality
and
sometimes
are
not .
In
so far as, and while, they are in actuality they behave as necessary
substances. On ly tha t necessity is temp orally con ditioned by the ac tu-
ality, and ceases
differences
celestial things
and the temporal actuality of sublunar things, these differences are not
differences of degree and the actuality of a contingent thing has, for a
finite
t ime,
the same necessity as that belonging to the actuality of an
eternal or sempiternal thing.
12
The
sophism of the distribution of the modalities may be written thus:
* ~ M(3t)(pt- ~
pt)
~ pt).
The p r inc ip le of condi t iona l necessity is w r i t t e n :
~ (pt • M t ~ pt) or ~ (~ P t •
that
Does
the
principle
of
of the
modalities? It does in that, while p, one can conclude the necessity
of
p
from
detached
from
its
condition,
13
The
principle
of
actuality.
14
Socrates, who is sitting, might be standing. The contradiction disap-
pears
modalities,
principle
latter.
event w i t h which
the modality has to do. Suppose it possible at time ti (in the sense
of "real" possibility) that
take
place
at
time
< 2 - Suppose f u r t h e r that at time ti the event in question is not tak-
in g place. What the principle of conditional necessity tells us under
such
cannot be identical w i t h time t . On
the other hand, there is nothing contradictory in maintaining that it is
possible
past
chapter
Ethics, Aristotle develops
being
other
than they are". And he winds up by saying: concerning the past
there is no longer any choice. That is why no one chooses that Illium
should have been ravaged; for no one deliberates about that which has
been, but
possible (ivdexonevov);
and indeed what has been cannot not have been. This is why Agathon
13
((t){M
.( t =
can be
sequence
is d rawn in the sentence of lines 13 and 14 of the De
Caelo
past
is
irrevocable
to it, all
potential i ty
of
opposites regards the present or the fu ture, the irrevo cability of the
past
follow s by con traposition : this irrevo cability residing in the dis-
junc t ion
affirmative member
or the
This dis junct ion
in no way be
if
logically
possible th at it sho uld not have taken place. Th e logical imp oss ibil-
ity in
to do
w i t h
an event dated in the past . But once endow ed w ith i ts tw o tem po ral
indices, this necessity,
same
direction as the na tura l movement o f real po ssibility:
i.e.,
longer applying
but to
our relation to things. It is in virtue of this relation
that
will
be spoken of. Let us then give each of the modalities a
double temporal index. Let us say that it is possible, in the sense of a
'real' po ssible,
at
that
t
2
can not be anterior to t\. W h at is
specific to
De Caelo passage
add yet a
fur the r
that
there is a (con t ingent ) potent ia li ty
of th e
w ould
be led to say of the now that it is last
year,
that
w e w ere to deliberate
about the fall of Troy. Th ere w ould result the possibility both that Troy
should have
past
would have to come to be identified with the present and the present
wi th the past. In this w ay i t would be denied that the past occupies a
different
time
from
the
present.
modalities, once they are used to characterize substances. The princi-
ple of the impossibility of realizing a possible in the
past
illustrates the
constraints affecting this d ou ble index ju st as the principle of con di-
tional
It
comes down to postulating that the index o f the possible at t\,
which
m ust , by defin ition , be equal to o r anterior to the ind ex
< 2
of the event
whose possibility is in question, can, at the same time, be posterior to
it
16
16
This is attested to by the following Commen ta ry of Sim plicius: In L. De Caelo
I,
th e
propo sition saying
that a generated thin g is indestructible implies that that th in g has the capacity
of
line
is
both
generated
an d indestructible has the capacity of not being in the direction of
th e past, since it was non-existent before being, and the capacity of be ing in the
direction
not at
th e same time that it wil l have th e capacity of being and of not
being,
he
says that all potential i ty is in the direction of present or fu ture t ime . For above
all
capable
of
becoming,
will
be but are not yet. If it is no t t rue
then
as
regards anything to say now that it is last year or that it is not last year
( t h e tw o
lessons exist =
t rue
to say now
that th e t ime of last year is, nor of any event
that
but ne ither w as it tru e to say
last
past
year
which w as
generated should
continue to be indestructible for the rest of time . Indeed , since tha t which after-
wards is was first inexistent it wil l also have, once having attained to being, th e
possibility of not be ing, though not tha t of not be ing then w hen it has already
attained
to
last year
and in
th e past. This is absurd, since there is no potentiality of that which has happened,
but
only
of
moreover:
actuality:
it
existing
year
will be now. Now is in effect supposed to have the
capacity
last
THE
I R R E V O C A B I L I T Y OF THE PAST / 29
Such a derangement of time leads to a physical impossibility an d
it comes down to the same logically to say either that when t rue , the
words
that
he
concludes:
thus
be
true
does
not
exist
last
is ungenerated though destructible. But it is right to say
that
potentiality
of
that
which has happened: for the whole past is necessary (TO fa.p
irapeXri\vd6'; -nav
nor
contingent
(nal O U T E dwarbv O U T E ivbs.\6\iEvov \e~jeTca)".
The interpretation of premise A in the logic of the real modalities must fulfi l l the
fo l low i n g
three conditions:
and
true
is
necessary,
b) it must express the relation between deliberation and thus possibility
ad
f u t u r e , on the other hand,
c)
it
Let
that which
is such
C NPI — DS M N P I • M N ~ pt
(D ef i n i t i on
of the
contingent object
of
deliberation).
a) In leaving aside mention of the truth, which is implicit in the formulas, the
first
condition
N
p
t
consequence:
In
N
pt/R,
L
N
~p
t
/S,
~ L N ~ pt)
= t.
The contrapositive of (6) is none other than (5). Consequently, to say that what
is
past and true is necessary (Diodorus according to Epictetus) comes down to
the
same
as
saying
that
there
past
To
realize
the
contingent
past
Argument 's
past
the
order
of
time,
that
such
a
realiza-
tion
leads to posi t ing that t ime has two opposite directions, as in the
Nicomachean Ethics.
come to min d. A ccording to the pr inci-
ple
of conservation of modal status every substance possesses an exis-
tence which is either immaterial and thus atemporal and absolutely
necessary,
or
sensible
but
sempiternal
and
some-
times is not . In virtue of the principle of conditional necessity, when
it
is, it is necessary while it is. When it is not, it is not with an equal
necessity and for the same reason. Then too, once it has been, it is no
longer
possible that it be or it is no longer possible that it not be, in
vi r tue
of the "necessity" of the past. The m od ality of the existence of
such a
subs tance
w hich
of a
is
expressed
of the
of
In
o ther w ords, the present is not etern al. A ristotle thu s rejects, explicitly in (8),
and implicit ly in (1) and (2) or in
(6 ) ,
creation is realized in the past, whereas th e
positive possible resulting from the sempiternality of the world is realized in the
future;
by symmetry , (9) expresses th e case of the Master Argument. These tw o
state m en ts are paradoxical because the tw o "pieces" of the con tingen t take oppo site
temporal directions.
They become contradictory when the two pieces of the contingent are asserted
simultaneously.
This
sempiternality
of the world results its positive possibility. It is the case in the Master Argument,
since, from the
negative
possibility.
<
TH E
S U B S I S T E N C E
O F A P O S S I B L E
T H A T
I S N O T T O B E R E A L I Z E D / 3 1
contingency, a thing being contingent precisely if sometimes it is and
sometimes
A t lines
De Caelo passage Aristot le asserts that,
be-
cause
was consequently non-existent
that
this potentiality
of not
existing will,
because of con dition al necessity, never be actualized. (Th at is w hy the
only means
past ) .
The
M aster A rgu m en t 's thi rd premise wi ll s imply sub st i tu te
for the
will
that of a positive possible which, also by hypothesis, will
never be actualized either.
This third premise deserves fur ther sc ru tin y, albeit because
of
only
a
minor
difficulty.
In
that
which is not presently true and never wil l be so, one
might
possible that becomes actualized would falsify such a clause,
which is un justif ied even as regards the possibles in utraque.
If one is tempted to accept this extreme interpretat ion, i t is
that
its negation is maintained by Diodorus when he says that that which
will never be actualized is impossible.
A s
regards the re lations of the possible to actualization -inev itably
fu ture , since th ere is no pote ntiality of the past-there are three concep-
tions that
arise
a
priori.
1) O ne can, w i t h Diodorus, treat the possibles
and the actualized futures ( taking
fu ture
includes
the present) as equivalent. This same thesis has been mistak-
enly
ascribed
to
verify and
17
Possibles
can but heat).
opposita are possibles w hich result either from rational
potencies containing a deliberation (the doctor can give a remedy or a poison to
the
patient)
or
can be
a
5) . L et me summ arize C ajetan (in St . Thomas, Frette, 187 5,
88-89; Oesterle, 1962,
217-221) .
These two sorts of possible are each de fined w ith
respect to that w hic h is susceptible of mov emen t. The possible ad unum, however,
can,
by
said possible because
it is in act. Contrarywise, for the possibles in opposita, the thing is said possible
that
the
futures
is
to be understood any idea in the mind of God. Those possibles that
are realized are realized by v irt u e of a "co nd ition al necessity" which
is itself beyond the
retrograde, since
in
Aris tot le , dis t inguish tw o sorts of
futures.
Some have virtue s th at could
not not be realized. The others are contingent possibles. Among these
lat ter, some
will
never
be realized. The range of possibles, then, is by no means exhausted
by the futures
that
no
non-realized future is po ssible. But he need not for that suppose that
his adversary holds that
fact
that A ris tot le adm its of the n on -realization of some p ossibles is already
enough .
This
is
clearly
famous
impossible things owed
of the Stoics re-
garding fate. The question w as w he the r , among the t h i ngs
which
never
have
been and will never be, there are possibles, or if all that is not, all
that never has been, al l that never will be, was
impossible".
18
possible
22, p. 89;
1046"36
sq.)
a s
imp ed imen t s
are removed. Heat necessarily heats a material which is present, once all insulation
is
removed.
To keep th e original sense of the possible ad unum, that of a vi r tual i ty which
develops
p
is to say that it is po ssible tha t
p ad
h a n d ,
Must
w e grant th e axiom of necessity fo r v i r tue , while refusing it for chance
(23 ° 15-16)? If it is necessary that a material substance perish then it is possible ad
unum
that
This
is in contrast to chance. If Socrates can be seated an d
not be
1 8
R E C O N S T R U C T I O N
OF THE D E
CAELO D E M O N S T R A T I O N
/ 33
19
he goes on to add: "I t h i n k that the Stoics took
it upon themselves
awful
and odious consequences
d r a w n from the ir dogm a of fatalism ". Le ibniz too gives reasons, and
different from Aristotle

J. Vuillemin, Necessity and Contingency

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