Dialectical School

2. The beginnings of propositional logic

All we know about the so-called founder of the so-called DialecticalSchool, Clinomachus of Thurii, is that he was the first who wroteabout predicates (katêgorêmata) and propositions(axiômata) and the like (DL 2.112). Our evidence forDiodorus and Philo confirms that they, too, conceived of logic as alogic of propositions. It was the Stoics who systematically developeda logic of propositions and devised subtle classifications ofpredicates. (By contrast, Aristotle’s logic had been a logic ofterms, where subject and predicate terms were in principleinterchangeable.) The logic of the Dialectical School thus was aprecursor of Stoic logic.

At the time of Diodorus and Philo, a distinction between simple andnon-simple propositions was in circulation among the minor GreekSocratic philosophical sects (Sextus Empiricus [SE]Against theLogicians 2.93–95). There is some dispute over whether thisreference is to the Dialectical School, or merely to dialecticians ingeneral (Ebert 1991; Barnes 1993). Following the Dialecticians, theStoics divided simple propositions into affirmations and negations (DL7.68; SE,Against the Logicians 2.93–95). Amongnon-simple propositions, which were thought to be composed of simpleones, conditionals, disjunctions and conjunctions were distinguished.Disjunctions or conditionals featured as premises in many of thelogical paradoxes and sophisms which members of the Dialectical Schooldiscussed. It is likely that Diodorus and Philo examined thetruth-conditions of all three kinds of non-simple propositions;however, we know their views only for the case of the conditional (seebelowSection 3).

Their treatment of conditionals and modalities implies that —like most Hellenistic philosophers — Diodorus and Philo workedwith a concept of proposition that differs from ours in that it allowstruth-values to change over time. For instance, the standard examplefor a simple proposition, ‘It is day’, does not contain acovert fixed date or definite time determination; rather, it rangesover times, changing its truth-value twice daily, all the whileremaining the same proposition. Thus we may think of propositions asthey were understood by Hellenistic philosophers as functions of time.Diodorus Cronus can perhaps also be credited with the beginnings oftemporal logic, as both his theory of the conditional and his accountsof the modalities are built on logically relevant temporal propertiesof propositions (see below sects.3.2 and4.2). Moreover, he distinguished between propositions in the present tenselike ‘Helen has three husbands’ and ‘These men aremarrying’ and propositions in a tense of completion (theaorist), ‘Helen had three husbands’ and ‘These menmarried’, and observed that it is possible for propositions likethe latter two to be true, without there ever having been a time atwhich a corresponding one of the former type was true (SE,Againstthe Physicists 2.97–8). See Crivelli 1994; Duncombe2023.

Apart from the various logical puzzles and sophisms, there are onlytwo topics on which we can be sure of a positive contribution to logicby members of the Dialectical School. These are the positions ofDiodorus and Philo on the theory of conditionals and on modal logic.Both topics involve notorious difficulties and were extensively andintensely discussed by Hellenistic logicians; so much so that thedisputes became part of the general knowledge of the intelligentsia ofthe time (SE,Against the Grammarians 309–10).Moreover, the theory of modalities was believed to have far-reachingresults for other areas of philosophy.

3. Conditionals

In the debate about the conditional (sunêmmenon) thepoint of disagreement concerned the right truth-conditions of aconditional (Cicero,Academics II 143). This controversy wasplayed out against the background of a common acceptance of whatcounts as a conditional, and what its function is. Conditionals wereunderstood to be non-simple propositions containing one proposition asantecedent and one as consequent. The antecedent has the particle‘if’ prefixed to it; the standard form is ‘Ifp,q’. A conditional serves to manifest therelation of consequence (akolouthia): it announces that itsconsequent follows from (akolouthein) its antecedent (SE,Against the Logicians 2.110–112). This relation ofconsequence was also commonly taken to manifest the relation betweenpremises and conclusion in a valid argument.

3.1 Philonian conditionals

Philo’s criterion for the truth of a conditional istruth-functional. Later in antiquity, it was generally accepted as aminimal condition for the truth of a conditional. Philo maintainedthat a conditional is false when and only when its antecedent is trueand its consequent false, and true in the three remaining cases:whenever the antecedent is false, and when both antecedent andconsequent are true (SE,Against the Logicians2.113–114). Thus, the Philonian conditional is false only incases like ‘if it is day, it is night’, since when it isday, ‘it is day is true, but ‘it is night’ is false(SEAgainst the logicians 2.115).Thus this notion of aconditional comes very close to that of modern material implication(Hurst 1935, 485; Heal 2022). (It is not the same, since truth forHellenistic philosophers is relativized to times.) Philo’ssuggestion is remarkable in that it deviates noticeably from theordinary language understanding of conditional sentences and requiresabstraction on the basis of a concept of truth-functionality.

Remarkable as it is, Philo’s view has the following twodrawbacks: First, as in the case of material implication, for thetruth of the conditional no connection of content between antecedentand consequent is required. Thus a Philonian conditional will be truewhenever the consequent is true, for example, during the day ‘Ifvirtue benefits, it is day’ is Philonian true. Likewise, aPhilonian conditional will be true whenever the antecedent is false,for example, during the day ‘if it is night, it is day’ isPhilonian true. This introduces a variant of the so-called‘paradoxes of material implication’ (Relevance Logic,Conditionals 2.3; Lemmon 59–60, 82). The presentation ofPhilo’s view in our sources shows that the ancients were awareof this problem (SE, ibid. 113–117). Second, due to thetime-dependency of Hellenistic propositions, Philo’s criterionimplies that conditionals can change their truth-value over time: forinstance, ‘If it is day, it is night’ is true at night,but false during the day. This is counter-intuitive as regards theordinary use of if-sentences. Moreover, as the concept of aconditional was also meant to provide for logical consequence betweenpremises and conclusion, this leads to the problematic result thatarguments can in principle change from being valid to being invalidand vice versa.

3.2 Diodorean Conditionals

For Diodorus, a conditional proposition is true if it neither was noris possible that its antecedent is true and its consequent false (SE,Against the Logicians 2.115–117). The reference to timein this account (‘was … is possible’) suggests thatthe possibility of a truth-value change left open by Philo’struth-condition was one of the factors to be improved on.

We do not know whether Diodorus had his own modal notions in mind whentalking about possibility in his criterion, or just somepre-technical, general concept of possibility, or whether he perhapsintended to cover both (Denyer 1981, 39–41; Sedley 1977,101–2). (The verb used here for being possible,endechesthai differs from the word used for possibility inDiodorus’ modal theory, which isdunaton.) If oneassumes that he had his own modal notions in view when giving thisaccount (seeSection 4.2 below), his truth-criterion for the conditional stands in thefollowing relation to Philo’s: a conditional is Diodorean truenow if and only if it is Philonian true at all times. Diodorus has, asit were, quantified the Philonian criterion over time. The conditional‘If I walk, I move’ is now true because at no time is theantecedent true and the consequent false. Thus for Diodorus, aconditional cannot change its truth-value. If it is true (false) atone time, it is true (false) at all times. If on the other hand oneassumes that Diodorus had some unspecified general notion ofpossibility in mind when producing his account, the criterion will becorrespondingly less specific. However, it would presumably still be aminimal requirement that it is never the case that the antecedent istrue and the consequent false.

Diodorus’ criterion bears some resemblance to the modern conceptof strict implication, although the exact relation between Diodoreanand strict conditionals is controversial (see Hurst 1935; Mates 1949;Hájek 2009). What is clear is that Diodorean conditionals sharesome disadvantages of strict conditionals, in that we encounter aparallel to the ‘paradoxes of strict implication’ (Lemmon1965, 153–4). As in Philo’s case, no connection of contentis required between antecedent and consequent. This time, whenevereither the antecedent is impossible or the consequent necessary, theconditional will be true, regardless of whether there is any relevantconnection between the two constituent propositions. So for instance‘If the earth flies, Axiothea philosophizes’ would be truefor Diodorus, since the antecedent was considered impossible (DL7.75), while ‘If Axiothena philosophizes, virtue isbeneficial’ would be a true Diodorean conditional, since theconsequent was considered necessary (DL 7.75) One example given of aDiodorean-true conditional, ‘If it is not the case that thereare indivisible elements of things, then there are indivisibleelements of things’ (SE,Outlines of Pyrrhonism 2.111),suggests that there was some awareness of these paradoxes inantiquity. In any event, the Stoic Chrysippus developed a system ofrelevance logic in which neither the ‘paradoxes of materialimplication’ nor the ‘paradoxes of strictimplication’ occur.

4. Modal logic

Modal logic is the second topic where we have evidence about thepositions of Philo and Diodorus and their influence on the Stoics(Kneale and Kneale 1962, 117–28; Bobzien 1993; de Harven 2016).Although the modalities were generally discussed under the heading of‘On things possible’, the Hellenistic modal systems wereeach built on a set of four modalities: possibility, impossibility,necessity and non-necessity. The matter of dispute was which systemwas the right one; that is, which one adequately described themodalities inherent in the world. In connection with this, anextra-logical concern provided additional fuel to the debate: thebelief that if propositions stating future events that will not happenall turn out to be impossible, the freedom and choices of individualswould be curtailed. This is a variation on the problem of logicaldeterminism which is known from Aristotle’sOnInterpretation 9. Several of the arguments devised and discussedby the Dialectical School touch upon this issue — the ReaperArgument, the Lazy Argument (Bobzien 1998, 78–81,180–233), and the Master Argument.

For Diodorus and Philo, as for the Stoics the modalities wereprimarily considered asproperties of propositions or statesof affairs. There is no discussion of modal propositions, i.e. ofpropositions that contain modal operators, such as ‘It ispossible that it is day’ or ‘It is necessary that virtuebenefits’, nor of iterated modalities.

4.1 Philonian modalities

Philo’s view on possibility has survived in several othersources (Alexander,On Aristotle’s Prior Analytics 184;Philoponus,On Aristotle’s Prior Analytics 169) butonly in one are his accounts of all four modal notions reported:

Possible is that which is capable of being true by theproposition’s own nature … necessary is that which istrue, and which, as far as it is in itself, is not capable of beingwrong. Non-necessary is that which, as far as it is in itself, iscapable of being false, and impossible is that which by its own natureis not capable of being true. (Boethius,On Aristotle’s OnInterpretation 2, II 234)

So according to Boethius the basic feature of Philonian modalities issome intrinsic capability of the propositions to be or not to be trueor false. That this feature is intrinsic is plain from the phrases‘by its own nature’ and ‘in itself’. In onesource, both phrases are used to characterize Philonian possibility(Simplicius,On Aristotle’s Categories 195); hence bothphrases may have originally applied to all four accounts.

Philo’s notion of possibility is often thought to be a broadnotion of logical possibility (de Harven 2016, 3–4) with theonly constraint on a proposition being possible being that it is amatter of logical consistency. Sometimes our sources seem to groundthis consistency in the intrinsic features of the subject of thatproposition. For example, according to Philo ‘this piece of woodburns at the bottom of the ocean’ is possible, since theintrinsic features of the wood give it the power to burn (AlexanderOn Aristotle’s Prior Analytics 184.7–9;SimpliciusOn Aristotle’s Categories 195,30–196,3), and, presumably, there is no logical contradiction in saying‘this piece of wood burns at the bottom of the ocean’. Itis just that the extrinsic circumstances of being in the ocean preventthe wood from burning (Bobzien 1993, 77). However, it is unclearwhether this explanation is wholly adequate, since‘preventing’ is itself a modal notion, so this explanationof the evidence risks circularity.

In all sources the concept of possibility stands out, and so it seemslikely that Philo built his set of modal notions on a concept ofinternal consistency, as given in his account of possibility.Philo’s modal concepts are thus defined by resort to another,perhaps more basic, modal concept. As to the kind of logicalconsistency Philo had in mind, we learn nothing more. Notwithstandingthis, there can be little doubt that Philo’s modal conceptssatisfy a number of basic requirements which normal systems of modernmodal logic tend to satisfy, too. (We assume here that Philo acceptedthe principle of bivalence, as we have no contrary information.) Theserequirements are:

1. Every necessary proposition is true and every true propositionpossible; every impossible proposition is false and every falseproposition non-necessary.

According to Philo’s accounts, a proposition that is not capableof falsehood must be true; one that is true must be capable of beingtrue; etc.

2. The accounts of possibility and impossibility and those ofnecessity and non-necessity are contradictory to each other.

This can be directly read off the definitions.

3. Necessity and possibility are interdefinable in the sense that aproposition is necessary precisely if its contradictory is notpossible.

This holds for Philo’s accounts, if one neglects the differencein the two phrases ‘in itself’ and ‘by its ownnature’ or assumes that originally both were part of all thedefinitions. Then a proposition is not capable of being falseprecisely if its contradictory is not capable of being true, etc.

4. Every proposition is either necessary or impossible or bothpossible and non-necessary, i.e. contingent.

In Philo’s system this amounts to the fact that everyproposition is either incapable of falsehood, or incapable of truth,or capable of both. The fact that Philo’s modal accounts —and those of Diodorus and the Stoics — satisfy these fourrequirements is of course no proof that the ancients consciouslyreflected upon all of them or regarded them all as principles withwhich their modal theories had to comply.

4.2 Diodorean modalities

We know a little more about Diodorus’ modal theory (in additionto the passages cited for Philo, see Epictetus,Dissertations2.19, Cicero,On Fate 12, 13, 17; Plutarch,StoicRefutations 1055e-f). Still, again only one text reports all fourdefinitions of Diodorus’ modal notions:

Possible is that which either is or will be <true>; impossiblethat which is false and will not be true; necessary that which is trueand will not be false; non-necessary that which either is falsealready or will be false. (Boethius,On Aristotle’s OnInterpretation 2.II.234–235)

Two of these modal accounts are disjunctions, the other two areconjunctions. Provided that Diodorus accepted the principle ofbivalence, these definitions, too, satisfy the modal requirements (i)to (iv) above, as can be checked easily.

Apart from that, Diodorus’ modalities are of a very differentkind from Philo’s. There is no modal expression hidden anywherein his accounts. Instead, which Diodorean modality a proposition hasdepends wholly on the range of truth-values it has at present and inthe future. For example, if a proposition is always true from now on,it is now both necessary and possible; if it is, from now on,sometimes true but not always, it is possible, but not necessary.Hence it is not the case — as is sometimes assumed — thatfor Diodorus, every proposition is either necessary (and possible) orimpossible (and non-necessary). There are propositions that arecontingent in the sense of being both possible and non-necessary,namely all those which will change their truth-value at some futuretime. The proposition ‘It is day’ is such a case.

The dependency of Diodorus’ modal notions on thepropositions’ truth-values-at-times implies that somepropositions can change their modality from possible to impossible andfrom non-necessary to necessary. For instance, if we assume that theproposition ‘Artemisia is five years old’ is now true,then that proposition is now possible; but it will presumably beimpossible at some future time, namely once Artemisia has reached theage of six, since from then on it will never be true again (assuminglinearity of time).

We do not know what motivated Diodorus to introduce his modalnotions.^[1]. But we know that Hellenistic philosophers generally regardedDiodorus’ modal notions as jeopardizing freedom, since they ruleout the possibility that something that never happens, or is nevertrue, is nonetheless possible. Thus, if ‘Dio goes toCorinth’ is and will be false, then ‘Dio goes toCorinth’ is impossible, and then, or so the thought went, it isimpossible for Dio to go to Corinth. In this context it becomes plainwhy our sources stress the fact that something can be Philonianpossible without ever being true or obtaining.This is the relevantdifference from Diodorus with respect to logical determinism, thepoint being that Philo’s modalities, unlike Diodorus’, donot unduly limit the scope of contingency.

5. The Master Argument

Diodorus’ definition of that which is possible can be split intotwo distinct claims: first that everything that either is or will betrue is possible, and second, that everything that is possible eitheris or will be true. The first statement was not questioned byHellenistic philosophers. It is the second statement that was and isconsidered counterintuitive and in need of justification; it is thisclaim which Diodorus attempted to back up with his Master Argument(Alexander, On Aristotle’sPrior Analytics183.34–184.6; Epictetus,Dissertations 2.19).

Despite being widely known in antiquity, the argument has not comedown to us. All we have is one brief passage:

The Master argument seems to have been developed from the followingstarting points: There is a general conflict between the followingthree <statements>: (I) every past true <proposition> isnecessary; and (II) the impossible does not follow from the possible;and (III) something is possible which neither is true nor will betrue. Being aware of this conflict, Diodorus used the plausibility ofthe first two <statements> in order to show that (IV) nothing ispossible that neither is nor will be true. (Epictetus, Dissertations2.19.1)

This is usually understood as implying that the argument was groundedon statements (I) and (II), and had (IV), which is the contradictoryof (III), as conclusion; and this is about as far as the passages leadus. But how did the argument run? There is a large and technicalliterature on the Master Argument and this entry can only offer anorientation to it. We can divide approaches into four groups along twoaxes, an approach suggested by (Uckelman 2023, 246 [Other InternetResources]) and (Foster 2008, 21). The first axis is whether thepropositions in question are temporally definite or temporallyindefinite, particularly in premise (I). The second axis is whetherthe ‘follows’ relation is a temporal succession relationor a logical consequence relation, particularly in (II). Temporallyindefinite propositions are tensed propositions which can change theirtruth value over time, such as ‘it is day’, a propositionthat is true once and false once in each twenty-four hour period. Thetruth-value of the proposition depends on the time of the valuation.Temporally definite propositions are such that there is an explicit orimplicit indexing to a particular time, such as ‘it is day atnoon on 31st October 2022’. Such a proposition is, as it were,always true (or false): the truth-value does not depend on the time ofevaluation. In terms of ‘follows’, we might interpret (II)as a principle that the passage of time does not affect whethersomething is possible: once something is possible, it will not becomeimpossible. (Rescher 1966) and (Zeller 1882) take this line. Thealternative is to take ‘follows’ as a logical consequencerelation. In that case, (II) amounts to the claim that there is notlogical entailment from a possible proposition to an impossibleone.

Within that space of logically possible reconstructions of theargument, a viable historical reconstruction must satisfy a number ofconditions. It must make use of the principles (I) and (II) handeddown in Epictetus; in addition to these, it must make use solely ofpremises plausible to the Stoics; and it must appear valid. For weknow that different Stoic philosophers attempted to refute (I) or(II), but we do not hear of anyone questioning the truth of any otherpremise or the validity of the argument. Moreover, the reconstructionmust employ only the logical means and concepts available inantiquity; in particular the notions of proposition, consequence, andmodalities used must fit in with the logic of the time, and it must bepossible to formulate the argument in ordinary language. Finally, therestored argument should not have a complexity which precludes itspresentation at a social gathering, since it appears that peopleenjoyed discussing the Master argument over dinner (Epictetus,Dissertations. 2.19.8).

Because of these historical constraints, most writers hold that theMaster Argument concerns temporally indefinite propositions and thatthe ‘follows’ relation is a logical consequence relation.This sort of approach was pioneered by Arthur Prior. (Prior 1955).Prior’s reflections on Diodorus’ Master Argument go backto his unpublished manuscriptThe Craft of Formal Logic(Markoska-Cubrinovska 2018). For an interesting recent comparison ofhow Prior and Mates reconstruct the Master Argument see Corpina andØhrstrøm 2018).

We will offer here an informal reconstruction along the line suggestedby Prior, which has the advantage of being historically credible andin keeping with the above constraints. The Epictetus passage suggeststhat the argument was presented in terms of propositions and theirmodalities, and so will the reconstruction. Other sources presentDiodorus’ modalities as modalities of states of affairs, andalternative versions of the Master argument can be producedaccordingly. (The difference between propositions and states ofaffairs is often blurred in ancient testimonies of logic.)

In line with Diodorus’ modal definition, the general conclusionof the argument (IV) may be reformulated as

(IV′)

If a proposition neither is nor will be true it isimpossible.

The first statement is less clear. It runs

(I)

Every past true <proposition> is necessary.

The Greek term used for ‘past’(parelêluthos) is a standard Stoic expression for pastpropositions, meaning not that the proposition itself subsisted in thepast, but that it is in some sense about the past. The principleoccurs also in Cicero, together with some explication:

All true <propositions> of the past are necessary … sincethey are unalterable, i.e. since past <propositions> cannotchange from true to false. (Cicero,On Fate 14)

From this passage we may infer that it was a peculiarity of all pasttrue propositions that they cannot change their truth-value tofalsehood; and that because of this they are necessary. This suggeststhat the past true propositions at issue do not include allpropositions in the past tense, but that they were those propositionswhich correspond to some past state or event. For instance, the truepast proposition ‘I went to Athens’ corresponds to theevent of my having gone to Athens. It can never become false, once itis true. Assume that I went to Athens last month. Then the proposition‘I went to Athens’ is not only true now, it will also betrue tomorrow, the day after, and in fact always from now on. Thetruth of the proposition is based on the fact that there has been acase of my going to Athens, and — whatever happens from now on— this cannot unhappen. (One can bring out this feature moreclearly by reformulating the proposition as ‘It has been thecase that I went to Athens’). On the other hand, the proposition‘You have not been to Athens’ does not correspond to apast state or event. Suppose that up to now you never went to Athens.Then the proposition is true now. Now suppose in addition that youwill go to Athens next week. After you have gone there, theproposition ‘You have not been to Athens’ is no longertrue. Hence it is not necessary now. (This proposition cannot becorrectly paraphrased as ‘It has been the case that you were notin Athens’). We may hence reformulate statement (I) as

(I′)

Every true proposition that corresponds to a past state or eventis necessary.

The second statement that functions as a premise in the argumentis

(II)

The impossible does not follow from the possible,

This was accepted by Aristotle and, with the exception of Chrysippus,by all Hellenistic logicians. At least by the Stoics it was understoodas

(II′)

An impossible proposition does not follow from a possibleone.

This amounts to the statement that if a proposition is impossible andfollows from some other proposition, then this other proposition isimpossible, too.

The following reconstruction assumes that, in addition to (I) and(II), the argument rests on a couple of further principles, whichmight have been generally taken to be valid and thus not worthmentioning, or else which might have been generally accepted by theStoics and for this reason omitted by Epictetus. The first additionalprinciple is

(V)

If something is the case now, then it has always been the casethat it will be the case.

For instance, if I am in Athens now, then it has always in the pastbeen the case that I would be in Athens (at some time). This principlegains historical plausibility from the fact that we find a version ofit in Aristotle (On Interpretation 9, 18b9–11), andthat another version of it was accepted by the Stoics (Cicero,OnDivination 1.125). The second additional principle is

(VI)

If something neither is nor will be true, then it has been thecase (at some time) that it will never be the case.^[2]

This statement is based on the idea that if a proposition presentlyneither is nor will be true, and you step back in time, as it were,then the formerly present ‘not being true’ turns into afuture ‘not going to be true’, so that from the point ofview of the past, the proposition will never be true, and thecorresponding state of affairs will never be the case. (This isassumed to hold at least for the past moment that immediately precedesthe present.) This principle has some plausibility to it. There issome evidence that it may have been discussed in antiquity (Becker1961, 253–5).

Next the construction of the argument: Fallacies and sophisms weregenerally presented by means of an example which stands in for thegeneral case, and it is plausible that this was so for the MasterArgument as well, although other explanations for the name have beenoffered (Massie 2016, 281n3). Overall, the argumentation thenproceeded as follows: you assume of a chosen proposition that itneither is nor will be true; next, by employing (I), (II), (V) and(VI), you deduce that this proposition is impossible. Then yougeneralize the result to all propositions, since nothing in theargument hinges on the fact that this particular proposition wasselected. A suitable example can be found in Alexander’s passageon Diodorus’ notion of possibility and the Master Argument: theproposition ‘I am in Corinth.’ The argument then startswith the assumption that

(1)

the proposition ‘I am in Corinth’ neither is nor willever be true.

The conclusion to be demonstrated is that

(C)

the proposition ‘I am in Corinth’ is impossible.

By (VI) it follows from (1) that

(2)

it has been the case (at some time) that I will never be inCorinth.

By (I) (‘all past truths are necessary’), it follows from(2) that

(3)

the proposition ‘It has been the case (at some time) that Iwill never be in Corinth’ is necessary.

But since necessity of a proposition is equivalent to theimpossibility of its contradictory (modal requirement (iii) above),from (3) it follows that

(4)

the proposition ‘It has always been the case that I will bein Corinth (at some time)’ is impossible.^[3]

Now, according to (V), it holds that

(5)

if I am in Corinth, then it has always been the case that I willbe in Corinth (at some time).

This is equivalent to

(5′)

the proposition ‘It has always been the case that I will bein Corinth (at some time)’ follows from the (initial)proposition ‘I am in Corinth’.

Now we can apply (II) (‘the impossible follows from theimpossible’) to (4) and (5′), and obtain as a resultthat

(C)

the proposition ‘I am in Corinth’ is impossible.

And this is precisely what the Master argument was meant to show.Moreover, this argument appears to be valid.

Where does the argument go wrong? The ancients went in for criticizing(I) and (II), and one may indeed wonder whether (I) covers cases ofthe kind to which it has been applied above. But there are also acouple of things questionable with (V) and (VI). With a certaincontinuum theory of time, it turns out that (VI) does not hold forthose cases in which the proposition at issue has started to be falseonly at the present moment (Denyer 1981, 43 and 45). More importantly,(V) and its variants seem to smuggle in a deterministic assumption. Onone or another of these grounds, then, one may attempt to hold theMaster Argument unsound, and so to reject Diodorus’ own accountof the possible as what either is or will be true.

6.The Reaper Argument

Like the Master Argument, the Reaper Argument emerged from theDialectical School and appears to argue against future contingencies.Unlike the Master Argument, we have good evidence for how the argumentwas formulated. Our main source for the detail of the Reaper argumentis Ammonius’ commentary on Aristotle’sDeInterpretatione 9, in the 5th century of our era. But the Reaperargument had a long history. Zeno of Citium, the founder of Stoicism,paid good money for seven versions of the argument (DL 7.25 (part)= LS31M=SVF 1.279). Zeno learned the argument from an unnameddialectician. Usually scholars take it this to be a member of theDialectical School. DL 7.44 lists the Reaper alongside the Veiled Man,the Horned Man and the No Man, and DL 2.111 gives Diodorus as theauthor of the Veiled and Horned arguments. Aside from schoolaffiliation and the tenuous connection in DL, the content of theargument has suggested that Diodorus may have authored, or at leastdeployed, the Reaper as a companion to the Master Argument (Sedley1977, 98; Seel 1993, 317–18; Bobzien 1998, 79–86; Sorabji2014, 4–5).

The argument is recorded in several sources, but the most fulsome andreliable account is in Ammonius’ CommentaryOnAristotle’s On Interpretation (131,20–131,33):

The more verbal one proceeds from some activity of ours, for example,from the activity of reaping, in the following way: (I) if you willreap, they say, it is not the case that perhaps you will reap andperhaps you won’t reap, rather you’ll reap anyway(pantos); and (II) if you will not reap, similarly, it is notthe case that perhaps you’ll reap and perhaps you won’treap, rather you won’t reap anyway. (III) But necessarily,either you will reap or you won’t reap. (IV) So‘perhaps’ has been destroyed, since it has no place eitherin the opposition of reaping to not reaping (one of these occurring ofnecessity), or in what follows from either of the hypotheses. (V) Butthe ‘perhaps’ was what introduced the contingent. (VI)Therefore the contingent is gone.

The overall structure of the argument is clear enough. The first stageof the argument tries to establish that there is no place for theexpression ‘perhaps’ regarding the future. The secondstage connects the use of the expression ‘perhaps’ withcontingency. The argument then concludes that there is no contingencyregarding the future. The first stage has a dilemma structure, thesecond stage could be seen as giving an argument from exhaustion:there is nowhere ‘perhaps’ could fit in; perhaps is neededfor contingency; so there is no contingency.

The literature on the Reaper argument is not large. Sedley suggeststhat the Reaper is ‘aimed at showing that it is never logicallycorrect to say “perhaps”’ (Sedley 1977, 98). ForSedley, the Reaper aims to demonstrate some fact about linguisticusage, namely that one cannot use ‘perhaps’ in a futuredirected way. Sedley goes on to explain that it seemsself-contradictory to say ‘If you will reap, perhaps you willreap and perhaps you will not reap’, but equallyself-contradictory to say ‘If you will not reap, perhaps youwill reap and perhaps you will not reap’. But bivalence commitsyou to one or other antecedent. The only way out of the bind would beto abandon the use of ‘perhaps’, at least in a sense thatexpresses future-directed uncertainty.

Sedley’s suggestion is interesting, but Ammonius is clear thatthe argument is supposed to rule out contingents, specifically, futurecontingents, rather than make a point about linguistic usage. The mostin depth discussion of the sources and argument is (Seel 1993). Seeldoes take the argument to be an attempt to demonstrate fatalism. Hetakes the conclusion of the argument to be ‘it is not the casethat it is now contingent that you will reap and now contingent thatyou will not reap’. Even if Seel’s reconstruction of theargument is correct, it is not obvious that fatalism results. OnSeel’s reading, the conclusion is equivalent to ‘it is nownecessary that you will reap or you will not reap’. But thatleaves open which of the two options I will take: ‘necessarily(p or not-p)’ does not entail ‘(necessarilyp ornecessarily not-p)’.

The Reaper Argument certainly originated from within the DialecticalSchool and argued for fatalism, but was Diodorus himself was theauthor? Seel suggests that Diodorus was (Seel 1993, 137–38). Onedifficulty with this attribution is that the notion of contingency inthe Reaper argument differs from what we know of Diodorus’modalities from other sources. For Diodorus, a contingent propositionis one that is true at some future time and false at some future time,so each contingent will be realised. But the Reaper seems to entertaincontingents that are never realised. Premises (I) and (II) of theReaper both commit to the unrealised possibility that you will dosomething. In the case of premise (I), if you will reap, then it isnot the case that perhaps you will reap and perhaps you will not: youwill reap anyway. So, the possibility that you will not reap willnever be realised. Likewise, premise (II) entertains the possibilitythat reaping will never be realised. So the Reaper argument seems tooperate with a non-Diodorean notion of contingency.