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Stanford Encyclopedia of Philosophy

Democritus

First published Sun Aug 15, 2004; substantive revision Sat Jan 7, 2023

Democritus, known in antiquity as the ‘laughingphilosopher’ because of his emphasis on the value of‘cheerfulness,’ was one of the two founders of ancientatomist theory. He elaborated a system originated by his teacherLeucippus into a materialist account of the natural world. Theatomists held that there are smallest indivisible bodies from whicheverything else is composed, and that these move about in an infinitevoid. Of the ancient materialist accounts of the natural world whichdid not rely on some kind of teleology or purpose to account for theapparent order and regularity found in the world, atomism was the mostinfluential. Even its chief critic, Aristotle, praised Democritus forarguing from sound considerations appropriate to naturalphilosophy.

1. Life and Works

According to ancient reports, Democritus was born about 460 BCE (thus,he was a younger contemporary of Socrates) and was a citizen ofAbdera, although some reports mention Miletus. As well as hisassociate or teacher Leucippus, Democritus is said to have knownAnaxagoras, and to have been forty years younger than the latter (DK68A1). A number of anecdotes concern his life, but their authenticityis uncertain.

The work of Democritus has survived only in secondhand reports,sometimes unreliable or conflicting: the reasoning behind thepositions taken often needs to be reconstructed. Much of the bestevidence is that reported by Aristotle, who regarded him as animportant rival in natural philosophy. Aristotle wrote a monograph onDemocritus, of which only a few passages quoted in other sources havesurvived. Democritus seems to have taken over and systematized theviews of Leucippus, of whom little is known. Although it is possibleto distinguish some contributions as those of Leucippus, theoverwhelming majority of reports refer either to both figures, or toDemocritus alone; the developed atomist system is often regarded asessentially Democritus’.

Diogenes Laertius lists a large number of works by Democritus on manyfields, including ethics, physics, mathematics, music and cosmology.Two works, theGreat World System and theLittle WorldSystem (see the entry ondoxography of ancient philosophy), are sometimes ascribed to Democritus, although Theophrastus reportsthat the former is by Leucippus (DK 68A33). There is more uncertaintyconcerning the authenticity of the reports of Democritus’ ethicalsayings. Two collections of sayings are recorded in the fifth-centuryanthology of Stobaeus, one ascribed to Democritus and another ascribedto an otherwise unknown philosopher ‘Democrates’. DKaccepts both as relating to Democritus, but the authenticity ofsayings in both collections is a matter of scholarly discussion, as isthe relationship between Democritus’ atomism and his ethics.

2. Atomist Doctrine

Ancient sources describe atomism as one of a number of attempts byearly Greek natural philosophers to respond to the challenge offeredby Parmenides. Despite occasional challenges (Osborne 2004), this ishow its motivation is generally interpreted by scholars today.Although the exact interpretation of Parmenides is disputed, he wastaken to have argued that change is merely illusory because of someabsurdities inherent in the idea of ‘what is not’. Inresponse, Leucippus and Democritus, along with other Presocraticpluralists such as Empedocles and Anaxagoras, developed systems thatclarified how change does not require that something should come to befrom nothing. These responses to Parmenides suppose that there aremultiple unchanging material principles, which persist and merelyrearrange themselves to form the changing world of appearances. In theatomist version, these unchanging material principles are indivisibleparticles, the atoms. The idea that there is a lower limit todivisibility is sometimes taken as an answer to Zeno’s paradoxes aboutthe impossibility of traversing infinitely divisible magnitudes(Hasper 2006). Reconstructions offered by Wardy (1988) and Sedley(2008) argue, instead, that atomism was developed as a response toParmenidean arguments.

The atomists held that there are two fundamentally different kinds ofrealities composing the natural world, atoms and void. Atoms, from theGreek adjectiveatomos oratomon,‘indivisible,’ are infinite in number and various in sizeand shape, and perfectly solid, with no internal gaps. They move aboutin an infinite void, repelling one another when they collide orcombining into clusters by means of tiny hooks and barbs on theirsurfaces, which become entangled. Other than changing place, they areunchangeable, ungenerated and indestructible. All changes in thevisible objects of the world of appearance are brought about byrelocations of these atoms: in Aristotelian terms, the atomists reduceall change to change of place. Macroscopic objects in the world thatwe experience are really clusters of these atoms; changes in theobjects we see—qualitative changes or growth, say—arecaused by rearrangements or additions to the atoms composing them.While the atoms are eternal, the objects compounded out of them arenot. Clusters of atoms moving in the infinite void come to formkosmoi or worlds as a result of a circular motion thatgathers atoms up into a whirl, creating clusters within it (DK68B167); thesekosmoi are impermanent. Our world and thespecies within it have arisen from the collision of atoms moving aboutin such a whirl, and will likewise disintegrate in time.

In supposing that void exists, the atomists deliberately embraced anapparent contradiction, claiming that ‘what is not’exists. Apparently addressing an argument by Melissus, a follower ofParmenides, the atomists paired the term for ‘nothing’with what it negates, ‘thing,’ and claimed that—in aphrase typical of the atomists—the one ‘no more’exists than the other (DK 67A6). Schofield (2002) argues that thisparticular phrase originated with Democritus and not his teacherLeucippus. By putting the full (or solid) and the void ontologicallyon a par, the atomists were apparently denying the impossibility ofvoid. Void they considered to be a necessary condition for localmotion: if there were no unoccupied places, where could bodies moveinto? Melissus had argued from the impossibility of void to theimpossibility of motion; the atomists apparently reasoned in reverse,arguing from the fact that motion exists to the necessity for voidspace to exist (DK 67A7). It has been suggested that Democritus’conception of void is that of the (temporarily) unfilled regionsbetween atoms rather than a concept of absolute space (Sedley 1982).Void does not impede the motion of atoms because its essential qualityis that of ‘yielding,’ in contrast to the mutualresistance of atoms. Later atomist accounts attest that this‘yielding’ explains the tendency of bodies to drift intoemptier spaces, driven out by collision from more densely packedregions (LucretiusDRN 6.906–1089).

Some controversy surrounds the properties of the atoms. They vary insize: one report—which some scholars question—suggeststhat atoms could, in principle, be as large as a cosmos, although atleast in this cosmos they all seem to be too small to perceive (DK68A47). They can take on an infinite variety of shapes: there arereports of an argument that there is ‘no more’ reason forthe atoms to be one shape than another. Many kinds of atoms caninterlock with one another because of their irregular shapes and hooksat their surface, accounting for the cohesiveness of some compounds.It is not clear whether the early atomists regarded atoms asconceptually indivisible or merely physically indivisible (Furley1967). The idea that there is a smallest possible magnitude seems tosuggest that this is the lower limit of size for atoms, althoughnotions like being in contact or having shape seem to entail that eventhe smallest atoms have parts in some sense, if only mathematically orconceptually.

There are conflicting reports on whether atoms move in a particulardirection as a result of their weight: a number of scholars have triedto reconcile these by supposing that weight is not intrinsic to theatoms, but is a result of the centripetal tendencies set up in thecosmic whirl (cf. O’Brien 1981; Furley 1989, pp. 91–102). Atomsmay have an inherent tendency to a kind of vibratory motion, althoughthe evidence for this is uncertain (McDiarmid 1958). However, theirprimary movement seems to result from collision with other atoms,wherein their mutual resistance orantitupia causes them tomove away from one another when struck. Democritus is criticized byAristotle for supposing that the sequence of colliding atoms has nobeginning, and thus for not offering an explanation of the existenceof atomic motionper se, even though the prior collision withanother atom can account for the direction of each individual atomicmotion (see O’Keefe 1996). Although the ancient atomists are oftencompared to modern ‘mechanistic’ theories, Balme warned ofthe danger of assuming that the atomists share modern ideas about thenature of atomic motion, particularly the idea that motion is inertial(Balme 1941).

According to different reports, Democritus ascribed the causes ofthings to necessity, and also to chance. Probably the latter termshould be understood as ‘absence of purpose’ rather than adenial of necessity (Barnes 1982, pp. 423–6). Democritusapparently recognized a need to account for the fact that thedisorderly motion of individual distinct atoms could produce anorderly cosmos in which atoms are not just randomly scattered, butcluster to form masses of distinct types. He is reported to haverelied on a tendency of ‘like to like’ which exists innature: just as animals of a kind cluster together, so atoms ofsimilar kinds cluster by size and shape. He compares this to thewinnowing of grains in a sieve, or the sorting of pebbles riffled bythe tide: it isas if there were a kind of attraction of liketo like (DK 68B164). Although this claim has been interpreteddifferently (e.g. Taylor 1999b p. 188), it seems to be an attempt toshow how an apparently ordered arrangement can arise automatically, asa byproduct of the random collisions of bodies in motion (Furley 1989,p. 79). No attractive forces or purposes need be introduced to explainthe sorting by the tide or in the sieve: it is probable that this isan attempt to show how apparently orderly effects can be producedwithout goal-directioned forces or purpose.

Democritus regards the properties of atoms in combination assufficient to account for the multitude of differences among theobjects in the world that appears to us. Aristotle cites an analogy tothe letters of the alphabet, which can produce a multitude ofdifferent words from a few elements in combinations; the differencesall stem from the shape (schêma) of the letters, as Adiffers from N; by their arrangement (taxis), as AN differsfrom NA; and by their positional orientation (thesis), as Ndiffers from Z (DK 67A6). These terms are Aristotle’s interpretationof Democritus’ own terminology, which has a more dynamic sense(Mourelatos 2004). This passage omits differences of size, perhapsbecause it is focused on the analogy to letters of the alphabet: it isquite clear from other texts that Democritus thinks that atoms alsodiffer in size.

He famously denies that perceptible qualities other than shape andsize (and, perhaps, weight) really exist in the atoms themselves: onedirect quotation surviving from Democritus claims that ‘byconvention sweet and by convention bitter, by convention hot, byconvention cold, by convention color; but in reality atoms andvoid’ (DK 68B9, trans. Taylor 1999a). There are differentaccounts of this distinction. Furley argues that the translation‘convention’ should not be taken to suggest that there isanythingarbitrary about the perception of certain colors,say: the same configuration of atoms may be regularly associated witha given color (Furley 1993; cf. Barnes 1982, pp. 370–7). WhatDemocritus rejects with the label ‘merely conventional’is, perhaps, the imputation of the qualities in question to the atoms,or perhaps even to macroscopic bodies. Mourelatos (2005) drawsthe contrast as that between intrinsic and relational properties.

While several reports of Democritus’ view, apparently directquotations, mention exclusively sensible qualities as being unreal, areport of Plutarch includes in the list of things that exist only byconvention the notion of ‘combination’ orsunkrisis. If this report is genuinely Democritean, it wouldbroaden the scope of the claim considerably: the idea that anycombination—by which he presumably means any cluster ofatoms—is ‘unreal’ or merely‘conventional’ suggests that Democritus is drawing a moreradical distinction than that between sensible and nonsensiblequalities. The implication would be that anything perceived, becauseit is a perception of combinations of atoms and not atoms themselves,would be suspect, not merely thequalia experienced by meansof individual sense organs. One report indeed attributes to Democritusa denial that two things could become one, or vice versa (DK 68A42),thus suggesting that combinations are regarded as conventional.

Commentators differ as to the authenticity of Plutarch’s report. Asthe wordsunkrisis does not occur in other reports, Furley(following Sandbach) suggests that it is most likely an error forpikron, ‘bitter’ which occurs instead in anotherreport. However, Furley concedes thatPlutarch at leastunderstands the earliest atomists to be committed to the view that allcombinations of atoms, as much as sensible qualities, should beunderstood as conventional rather than real (Furley 1993 pp.76–7n7). This would suggest that everything at the macroscopiclevel—or, strictly, everything available to perception—isregarded as unreal. The ontological status of arrangement orcombination of atoms for Democritus is a vexed question, that affectsour understanding of his metaphysics, his historical relationship toMelissus, and the similarity of his views to the modernprimary-secondary quality distinction (Wardy 1988; Curd 1998; Lee2005; Mourelatos 2005; Pasnau 2007). If we take the‘conventionality’ thesis to be restricted to sensiblequalities, there is still an open question about Democritus’ reasonfor denying their ‘reality’ (Wardy 1988; O’Keefe 1997;Ganson 1999).

3. Theory of Perception

Democritus’ theory of perception depends on the claim thateidôla or images, thin layers of atoms, are constantlysloughed off from the surfaces of macroscopic bodies and carriedthrough the air. Later atomists cite as evidence for this the gradualerosion of bodies over time. These films of atoms shrink and expand;only those that shrink sufficiently can enter the eye. It is theimpact of these on our sense organs that enables us to perceive.Visible properties of macroscopic objects, like their size and shape,are conveyed to us by these films, which tend to be distorted as theypass through greater distances in the air, since they are subject tomore collisions with air atoms. A different or complementary accountclaims that the object seen impresses the air by theeidôla, and the compacted air thus conveys the image tothe eye (DK 68A135; Baldes 1975). The properties perceived by othersenses are also conveyed by contact of some kind. Democritus’ theoryof taste, for example, shows how different taste sensations areregularly produced by contact with different shapes of atoms.

Theophrastus, who gives us the most thorough report of Democritus’theory, criticizes it for raising the expectation that the same kindsof atoms would always cause similar appearances. However, it may bethat most explanations are directed towards thenormal caseof a typical observer, and that a different account is given as to theperceptions of anontypical observer, such as someone who isill. Democritus’ account why honey sometimes tastes bitter to peoplewho are ill depends on two factors, neither of which undercut thenotion that certain atomic shapes regularly affect us in a given way.One is that a given substance like honey is not quite homogeneous, butcontains atoms of different shapes. While it takes its normalcharacter from the predominant type of atom present, there are otheratom-types present within. The other is that our sense-organs need tobe suitably harmonized to admit a given atom-type, and the dispositionof our passageways can be affected by illness or other conditions.Thus someone who is ill may become unusually receptive to an atom-typethat is only a small part of honey’s overall constitution.

Other observed effects, however, require a theory whereby the sameatoms can produce different effects without supposing that theobserver has changed. The change must then occur in the object seen.The explanation of color seems to be of this variety: Aristotlereports that things acquire their color by ‘turning,’tropê (GC 1.2, 315b34). This is theDemocritean term that Aristotle had translated as‘position,’thesis, i.e. one of the threefundamental ways in which atoms can alter and thus appear differentlyto us. Aristotle gives this as the reason why color is not ascribed tothe atoms themselves. Lucretius’ account of why color cannot belong toatoms may help clarify the point here. We are told that if the sea’satoms were really blue, they could not undergo some change and lookwhite (DRN 2.774–5), as when we observe the sea’ssurface changing from blue to white. This seems to assume that, whilean appearance of a property P can be produced by something that isneither P nor not-P, nonetheless something P cannot appear not-P.Since atoms do not change their intrinsic properties, it seems thatchange in a relational property, such as the relative position ofatoms, is most likely to be the cause of differing perceptions. In theshifting surface of the sea or the flutter of the pigeon with itsirridescent neck, it is evident that the parts of the object aremoving and shifting in their positional relations.

By ascribing the causes of sensible qualities to relational propertiesof atoms, Democritus forfeits theprima facie plausibility ofclaiming that thingsseem P because theyare P. Muchof Theophrastus’ report seems to focus on the need to make itplausible that a composite can produce an appearance of properties itdoes not intrinsically possess. Democritus is flying in the face of atleast one strand of commonsense when he claims that textures producethe appearance of hot or cold, impacts cause colour sensations. Thelists of examples offered, drawing on commonsense associations oranecdotal experience, are attempts to make such claims persuasive.Heat is said to be caused by spherical atoms, because these movefreely: the commonsense association of quick movement with heating maybe employed here. Betegh (2020) suggests that larger void spaces aredirectly associated with heating, rather than that rarefactionindirectly causes heat by allowing freer and more frequent atomicmotion.

Aristotle sometimes criticizes Democritus for claiming that visible,audible, olfactory and gustatory sensations are all caused by touch(DK 68A119). Quite how this affects the account of perception is notclear, as the sources tells us little about how touch is thought towork. Democritus does not, however, seem to distinguish between touchand contact, and may take it to be unproblematic that bodiescommunicate their size, shape and surface texture by physicalimpact.

4. The Soul and the Nature of Living Things

In common with other early ancient theories of living things,Democritus seems to have used the termpsychê to referto that distinctive feature of living things that accounts for theirability to perform their life-functions. According to Aristotle,Democritus regarded the soul as composed of one kind of atom, inparticular fire atoms. This seems to have been because of theassociation of life with heat, and because spherical fire atoms arereadily mobile, and the soul is regarded as causing motion. Democritusseems to have considered thought to be caused by physical movements ofatoms also. This is sometimes taken as evidence that Democritus deniedthe survival of a personal soul after death, although the reports arenot univocal on this.

One difficulty faced by materialist theories of living things is toaccount for the existence and regular reproduction of functionallyadapted forms in the natural world. Although the atomists haveconsiderable success in making it plausible that a simple ontology ofatoms and void, with the minimal properties of the former, can accountfor a wide variety of differences in the objects in the perceptibleworld, and also that a number of apparently orderly effects can beproduced as a byproduct of disorderly atomic collisions, the kind offunctional organization found in organisms is much harder toexplain.

Democritus seems to have developed a view of reproduction according towhich all parts of the body contribute to the seed from which the newanimal grows, and that both parents contribute seed (DK 68A141; 143).The theory seems to presuppose that the presence of some material fromeach organ in the seed accounts for the development of that organ inthe new organism. Parental characteristics are inherited when thecontribution of one or other parent predominates in supplying theappropriate part. The offspring is male or female according to whichof the two seeds predominates in contributing material from thegenitals. In an atomist cosmos, the existence of particular species isnot considered to be eternal. Like some other early materialistaccounts, Democritus held that human beings arose from the earth (DK68A139), although the reports give little detail.

5. Theory of Knowledge

One report credits Democritus and Leucippus with the view that thoughtas well as sensation are caused by images impinging on the body fromoutside, and that thought as much as perception depends on images (DK67A30). Thought as well as perception are described as changes in thebody. Democritus apparently recognized that his view gives rise to anepistemological problem: it takes our knowledge of the world to bederived from our sense experience, but the senses themselves not to bein direct contact with the nature of things, thus leaving room foromission or error. A famous fragment may be responding to such askeptical line of thought by accusing the mind of overthrowing thesenses, though those are its only access to the truth (DK68B125).Other passages talk of a gap between what we can perceive and whatreally exists (DK 68B6–10; 117). But the fact that atoms are notperceptible means that our knowledge of their properties is alwaysbased on analogy from the things of the visible world. Moreover, thesenses report properties that the atoms don’t really possess, likecolors and tastes. Thus the potential for doubt about our knowledge ofthe external world looms large.

Later philosophers adapted a Democritean phraseou mallon or‘no more’ in the argument that something that seems both Pand not-P is ‘no more’ P than not-P. Arguments of thisform were used for sceptical purposes, citing the conflicting evidenceof the senses in order to raise concern about our knowledge of theworld (de Lacy 1958). Democritus does not seem to be pursuing aconsistently skeptical program, although he does express concern aboutthe basis for our knowledge.

The idea that our knowledge is based on the reception of images fromoutside us is employed in Democritus’ discussion of the gods, whereinit is clear that our knowledge of the gods comes fromeidôla or giant films of atoms with the characteristicswe attribute to the gods, although Democritus denies that they areimmortal. Some scholars take this to be a deflationary attack ontraditional theology as based on mere images (Barnes 1982, pp.456–61), but others suppose that the theory posits that theseeidôla are really living beings (Taylor 1999a, pp.211–6). Although atomism is often identified as an atheistdoctrine in later times, it is not clear whether this is reallyDemocritus’ view.

6. Indivisibility and Mathematics

The reasons for supposing that there are indivisible magnitudesapparently stem from Zeno of Elea’s account of paradoxes that arise ifextension is understood to be infinitely divisible, i.e. composed ofan infinite number of parts. The atomists may have sought to avoidthese paradoxes by supposing that there is a limit todivisibility.

It is not clear, however, in what sense the atoms are said to beindivisible, and how the need for smallest magnitudes is related tothe claim that atoms are indivisible. Furley suggests that theatomists may not have distinguished between physical and theoreticalindivisibility of the atoms (Furley 1967, p. 94). The physicalindivisibility of the atoms seems to be independent of the argumentfor indivisible magnitudes, since the solidity of atoms—the factthat there is no void within them—is said to be the reason whythey cannot be split. The existence of void spacebetweenatoms is cited as the reason why they can be separated: one latesource, Philoponus, even suggests that atoms could never actuallytouch, lest they fuse (DK 67A7). Whether or not Democritus himself sawthis consequence, it seems that atoms are taken to be indivisiblewhatever their size. Presumably, though, there is a smallest size ofatom, and this is thought to be enough to avoid the paradoxes ofinfinite divisibility.

Areductio ad absurdum argument reported by Aristotlesuggests that the atomists argued from the assumption that, if amagnitude is infinitely divisible, nothing prevents it actually havingbeen divided at every point. The atomist then asks what would remain:if the answer is some extended particles, such as dust, then thehypothesized division has not yet been completed. If the answer isnothing or points, then the question is how an extended magnitudecould be composed from what does not have extension (DK 68A48b,123).

Democritus is also said to have contributed to mathematics, and tohave posed a problem about the nature of the cone. He argues that if acone is sliced anywhere parallel to its base, the two faces thusproduced must either be the same in size or different. If they are thesame, however, the cone would seem to be a cylinder; but if they aredifferent, the cone would turn out to have step-like rather thancontinuous sides. Although it is not clear from Plutarch’s report how(or if) Democritus solved the problem, it does seem that he wasconscious of questions about the relationship between atomism as aphysical theory and the nature of mathematical objects.

7. Ethics

The reports concerning Democritus’ ethical views pose a number ofinterpretative problems, including the difficulty of deciding whichfragments are genuinely Democritean (see above, section 1). Incontrast to the evidence for his physical theories, many of theethical fragments are lists of sayings quoted without context, ratherthan critical philosophical discussions of atomist views. Many seemlike commonsense platitudes that would be consistent with quitedifferent philosophical positions. Thus, despite the large number ofethical sayings, it is difficult to construct a coherent account ofhis ethical views. Annas notes the Socratic character of a number ofthe sayings, and thinks there is a consistent theme about the role ofone’s own intellect in happiness (Annas 2002). The sayings containelements that can be seen as anticipating the more developed ethicalviews of Epicurus (Warren 2002).

It is also a matter of controversy whether any conceptual link can befound between atomist physics and the ethical commitments attributedto Democritus. Vlastos argued that a number of features of Democritus’naturalistic ethics can be traced to his materialist account of thesoul and his rejection of a supernatural grounding for ethics (Vlastos1975). Taylor is more sceptical about the closeness of the connectionbetween Democritus’ ethical views and his atomist physics (Taylor1999a, pp. 232–4).

The reports indicate that Democritus was committed to a kind ofenlightened hedonism, in which the good was held to be an internalstate of mind rather than something external to it (see Hasper 2014).The good is given many names, amongst themeuthymia orcheerfulness, as well as privative terms, e.g. for the absence offear. Some fragments suggest that moderation and mindfulness in one’spursuit of pleasures is beneficial; others focus on the need to freeoneself from dependence on fortune by moderating desire. Severalpassages focus on the human ability to act on nature by means ofteaching and art, and on a notion of balance and moderation thatsuggests that ethics is conceived as an art of caring for the soulanalogous to medicine’s care for the body (Vlastos 1975, pp.386–94). Others discuss political community, suggesting thatthere is a natural tendency to form communities.

8. Anthropology

Although the evidence is not certain, Democritus may be the originatorof an ancient theory about the historical development of humancommunities. In contrast to the Hesiodic view that the human pastincluded a golden age from which the present day is a decline, analternative tradition that may derive from Democritus suggests thathuman life was originally like that of animals; it describes thegradual development of human communities for purposes of mutual aid,the origin of language, crafts and agriculture. Although the text inquestion does not mention Democritus by name, he is the most plausiblesource (Cole 1967; Cartledge 1997).

If Democritus is the source for this theory, it suggests that he tookseriously the need to account for the origin of all aspects of theworld of our experience. Human institutions could not be assumed to bepermanent features or divine gifts. The explanations offered suggestthat human culture developed as a response to necessity and thehardships of our environment. It has been suggested that the sheerinfinite size of the atomist universe and thus the number of possiblecombinations and arrangements that would occur by chance alone areimportant in the development of an account that can show how humaninstitutions arise without assuming teleological or theologicalorigins (Cole 1967). Although here, as on other questions, theevidence is less than certain, it is plausible that Democritusdeveloped a powerful and consistent explanation of much of the naturalworld from a very few fundamentals.

For the reception and subsequent history of Democritean atomism, seethe related entry on ancient atomism.

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Other Internet Resources

Acknowledgments

I wish to thank the ancient philosophy editor John Cooper, A.P.D.Mourelatos and Tim O’Keefe for helpful comments and suggestions.

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