Plato‘s Quincunxes

Alexandre Losev

Abstract. The Five Greatest Kinds discussed in Plato‘s Sophist are taken to be just one instance of a fivefold structure found in various related texts. Contemporary linguistic theories are a source for ideas about its functioning.

1504326172_tekken7_SMALL   PDF    Keywords: Plato, categories, semiotic triangle, logical square

The Sophist is undoubtedly one of the most remarkable Platonic dialogues. The title anounces its main content but there is a wealth of tangential topics in the the text. If the sophist and the philosopher are presented in the contrast of darkness and light (254a), they are somehow correlated to non-being and being, a pair which attracts even more interest. The sophisticated and fascinating discussion may easily lead to forgetting that Plato wrote at a time when neither logic nor grammar[1] were conceived as self-standing disciplines. Much of the dialogue revolves around linguistic issues and actually its methodological side might be considered as even more important than the substance. It was much later that ,sophist‘ became a disparaging qualification, just as being became an issue pertaining to religious ideologies that developed in post-Hellenistic times[2]. It was the success of Aristotle‘s categories that spurred Platonists to scrutinize their own canon with the Sophist becoming a „proof“ that Plato already had worked out the topic. Porphyry‘s rewriting of the four Aristotelian predicables as five visibly follows the schema of the Five Greatest kinds and this in turn would be a further argument to revisit Plato‘s original texts in such a perspective.

Besides generic studies of Plato‘s thought, the extent of comments on even this single dialogue discourages attempts at survey and reviewing the items just for this secondary topic is no mean feat[3]. A modest attempt will be made here to reconsider the fivefold schema from the Sophist and to trace some links and continuations in other texts e.g. the Philebus, Timaeus[4], and elsewhere. Seeing a pattern and establishing connections hopefully would help readings all related texts. Plato unambiguously states that understanding of which kinds and/or things (can) mix is crucial for activities like his own, pertaining to philosophy (253d). Just before launching the discussion about the five greatest kinds, his main protagonist, The Stranger, comments thatif we cannot grasp being and not-being with perfect clearness, we shall at any rate not fail to reason fully about them, so far as the method of our present inquiry permits“(254c). So, just understanding, what a set of notions such as the one from the Sophist allows to achieve, is the modest aim of the present inquiry.

The Quincunx

The Five Greatest kinds (μεγίστα γένη) are introduced in two steps, starting with the remark that among them are surely being itself, rest and change (τό τε ὂν αὐτὸ καὶ στάσις καὶ κίνησις). Next are added the same and the different (τό τε ταὐτὸν καὶ θάτερον) (254d-55b). Obviously there are two pairs of closely related terms and another one, single, which seems to be, so to say, pivotal:


Presenting and describing them in this way emphasizes that the exact words do not matter as much as the structure that they form and which will be called conveniently a ‚quincunx‘.

The naive explanation here amounts to a narrative and one may be even tempted to say ‚myth‘: the Same and the Different somehow mix, combine or blend, producing Being which is from two distinct types, either Rest or Change. Inverting the perspective Plato rather hints that there are two main principles (archai) that govern being which consists from a mix of sameness and difference. His exposition begins with the brief assertion that rest and change do not mix (252d) and the Stranger explains in detail that the change is „identical and different” (256b), “is not different and is different” and it is “with being and non-being” (256d). Disregarding nuances of language, it seems rather natural to object that the different and the identical are strong abstractions or some second order derivatives from forms of relation and their logical nature makes them antinomically distinct. Quite the opposite, rest and change are relative, especially when thinking about mechanical motion with our usual post-Galilean understanding.

This conundrum will disappear if one assumes the structure to be more important than the words in it, a move recalling the Saussurean solution which redrew the langue / parole contrast.


That thinking and speech are the same (διάνοια μὲν καὶ λόγος ταὐτόν, Soph. 263e; cp. Theaet. 189e, 190a; Phileb. 39a), is a commonsensical view that Plato endorses in his later works. However in the absence of formal theory, logic or grammar, the world becomes hoplessly entangled with features of language in ways which thought cannot trace. Even the simple – for us – distinction between correct, or „right“, and true[5] is problematic. Today it seems evident that the 5 megista gene owe much to a crude linguistic theory. Its exposition is found towards the end of the Sophist (261d-263d), where the combination of nouns and verbs is said to form speech (λόγος), which can be either true or false. Two preliminary examples are outlined in text: the consonant and vowel sounds forming syllables or words, which are either meaningful or meaningless, i.e. available or not in the language that the literate person knows (253a). This is repeated with the mention of high and low notes, from which a melody or harmony is formed, pleasant or not, according to the judgment of a musician (253b).

The construction is obviously similar, so there is no proper discussion pertaining to it. The words onoma, rhema just as logos are rather vague and general[6], still it is evident that here again there are two elements that are combined or mixed, and they produce or generate something else, for which a pair of opposite predicates is given:


The link between the athletic qualifications „true“, „false“ and the megista gene “same” and “different” is easily enough established. The specific terminology of the Sophist creates also a connection with the Timaeus, where the same schema is unambiguously recognized, albeit with some complications, as it is used there for the explanation of the world soul. “Same” and “different” are replaced by “indivisible” (αμέριστος; cp. Theaet. 205c) and “divisible” and they mix into the third essence (ουσία), which is that of the soul. It is here that arise either “opinions and beliefs” or “knowledge and understanding”, a description given in pairs of quasi-synonyms (37b-c).

knowledge (understanding)
opinion (belief)

A similar construction is recognized in the Philebus,  mentioning also sounds and notes as in The Sophist and with the same order of exposition which is an inversion of the original appearance of the 5 megista gene. A separate paragraph of the introduction (16f-d) sets out the general framework of the subject under consideration, and details come a few pages later. Socrates seems to recall the schema: “we said, it seems to me that a god reveals to us, now the infinite nature of things, now theirs limitation ”(23c7); after acknowledging the existence of a mixture, he proposes to his interlocutor “to think about the reason for mixing between the limited and the infinite and to add it as the 4th genus ”. And the interlocutor shrewdly asks him: “do you think you will also need a 5th genus to be the cause of separation?” (23f) Socrates at this point diverts consideration, but retains the possibility.


mixing (genesis)
separation (destruction)

The rhetoric that supports this construction is hardly convincing, asking whether “the creative agent always naturally leads, and that which is created follows after it…?” (27a). If in the naturalistic attitude the presence of causes is accepted only when they are manifested, in the repertoire of Platonism, reasons that have not yet caused anything would stand quite naturally. In the Philebus, Plato sketches something like a theodicy and this explains why the 5th component of the schema is not discussed, only hinted at, though its destructive nature is sufficiently obvious. The correspondence between the Philebus and the Sophist was noted already by Plutarch (Moralia 27), but the idea remained unpursued as it would lead to a dualistic philosophy (respectively: “heresy”). It was reconsidered again some hundred years ago by Paul Natorp but he also rejected it on the basis of content without noting the structuring.[7] Charles H. Kahn calls the passage in the Philebus “cosmology” and acknowledges that mixing and separation appear together in the Sophist (243b5), but his opinion is that an additional principle, a fifth component, is not needed.[8]


Another quincunx, perhaps more controversial, can be found in the apocryphal Seventh Letter (342a). The text demonstrates enviable awareness of Plato’s manner of thinking, so the identity of the author does not actually matter here: one could talk about “transformation” or about a “misunderstanding”, according to views on authorship.

In this passage components of knowledge are listed and because they are again a total of 5 in number, they can obviously be arranged once more in quincunx, though this time the narrative is somewhat different. The scheme is described as “3 + 2”, three factors which lead to knowledge and its object itself with a special emphasis how the last two, referred to as 4th and 5th, differ from the first three. Reconstruction in the spirit of the current proceedings would point out that the name and definition refer to the object for which adequate knowledge and inadequate image are possible (the approach here being much like Laws, X 895d):


object  (5)




There could not be much doubt that author behind this exposition is aware of the fivefold structure, even if he might be not Plato[9]. There are further grounds to suppose that such constructions are not just an inevitable coincidence in the works of Platonizing philosophers. Plato‘s quincunx can be recognized in Porphyry‘s Isagoge and this touches the intriguing question how the violence done to Aristotle’s text came to be so passively accepted.

In the Topics Porphyry had found the 4 predicabilia (idion, genos, symbebekos, horos) and disregarding the logic of their system replaced the last item with two: „species“ and „difference“. He does not expose the canonical structure which nevertheless could be unambiguously guessed:


However there is a complete misunderstanding of the original intent. Aristotle, while developping his logic, discovered the logic squares: a pair of independant predicates generates 4 cases. Most conspicuous examples of its application are the 4 elements, 4 causes, 4 types of propositions. The 4 predicables are unmistikably arrived at through the same process e.g. cobining the pair intrinsic/extrinsic with singular/nonsingular:






Actually the invention of the logic square is a major achievement and today there is a vast literature on such constructs (and a biannual World Congress since 2010). Plato‘s musing about the philosopher knowing what does mix could be reworded as knowing independent predicates that will produce 4 cases. On the one occasion when 5 components had to be co-present Aristotle obviously knew how to proceed logically: a new predicate halved the universe into immutable and mutable realms with the 4 elements exhaustively describing this last part. History however inverted their order and so the name “quintessence“ was adopted:



Plato‘s linguistics

Ultimately, outside of Plato’s text the quincunxes do not have any particular value and they should be seen best as an idiosyncrasy of his.[10] The case about logos (Soph. 261-3). however appears to be an important exception as it indisputably captures a basic understanding of the language phenomenon. Gathering Plato’s remarks from various places allows to eventually produce an even more pertinent structure. Even if it does not occur anywhere as a whole it would eventually summarize his views, while making conspicuous the absence of a key concept, that of a linguistic sign. One would have to go through the detail of contemporary semiotics and check what plausibly Plato had already guessed, which is not really feasible here. His writings hint at a development from a naive paradigm of „naming“ to understanding propositional constructs. The relation of a name to its referent is not the same for proper and common names, so referring is not same as meaning; while meaning can be arrived at through speech there is not a guarantee for existence or being.

linguistic signs

The important point of the Sophist, the understanding of speech (logos), on the one hand, as a blending or combination and, on the other, as truth, suggests how to deal with what is called “non-being”: “we will not fully agree, but only to the extent that the affixed “no” and “not at all” reports some difference in the names in question or rather the objects in connection with which the names of the negative statement are then pronounced (απόφασις)“ (Soph. 257c). Another name is derived from the name, a negation is applied to the concept and it is understood through the objects that would be included in its scope. Disentangling the Parmenidean problem in the dialogue is seen to occur mainly through the analysis of language with the Greatest kinds only indirectly contributing. The explanation of speech with an interplay of 5 components is indeed most illuminating.

Aristotle in his On Interpretation is seen to have exposed the basics of a cognitive or semiotic conception that remains as the received view even today. Nearly a hundred years ago Ogden and Richards gave its popular graphic representation as a „triangle of reference“ or „semiotic triangle“ but they did not add anything substantial to it.  „Just as all men have not the same writing, so all men have not the same speech sounds, but the mental experiences, which these directly symbolize, are the same for all, as also are those things of which our experiences are the images (De Interpr.  I.4). Words, mental states, things (or states of affairs) appear to form a single structure even if it is readily seen to break down into three pairs of elements. Charles Morris  suggested that with respect to signs there would be three separate directions of study, known as semantics, syntax and pragmatics.[11] Language, being a historical product, is found to contain idiomatic constructions that are syntactically incorrect or even meaningless. More generally, separating semantics from syntax for natural languages turned out to be rather difficult, so ultimately it is the empirical study of language use which decides about acceptable and unacceptable expressions, between sense and non-sense. These proposition can be summarized in a schema:


Undoubtedly, the elements of modern semiotics can be seen among Plato’s considerations, and his five-element scheme will do well enough to represent its structuring.


Aristotle, (1928) On Interpretation, transl. E. M. Edghill,  The University of Adelaide.

Bossi B., and T. M. Robinson, eds., (2013) Plato’s Sophist Revisited, Berlin/Boston: Walter de Gruyter GmbH.

Brisson L., (1974) Le même et l’autre dans la structure ontologique du Timée de Platon. Un commentaire systématique du Timée de Platon, Paris: Klincksieck, (4th edn., with an added bibliography 1998–2015, Sankt Augustin: Academia Verlag, 2015).

Burnyeat M. (w/ M. Frede), (2015), The Pseudo-Platonic Seventh Letter, Oxford UP.

Cornea A., (2009) Le Sophiste de Platon: cinq ou six «genres suprêmes»?, Semitica et Classica  vol. 2, p. 43-9.

Fronterotta F., (2019) Platon sur Οnoma, Rhema et Logos: théories du Semainein en Sophiste 261d-262e, Methodos, (2019) 19.

Gill M-L., (2020) Method and Metaphysics in Plato’s Sophist and Statesman, The Stanford Encyclopedia of Philosophy (Spring 2020 Edition), E. Zalta (ed.)

Graham D., Barney J., (2014) Γραμματική in Plato and Aristotle, Apeiron, Vol. 47: Iss. 4, p 513-25.

Hoekstra M., Scheppers F.(2003) Ονομα, ρήμα et λόγος dans le Cratyle et le Sophiste de Platon. Analyse du lexique et analyse du discours, L’antiquité classique, tome 72, 2003. pp. 55-73;

Kahn C., (2009) Essays on Being,  Oxford UP.

Kahn C., (2013) Plato and the Post Socratic dialogue: The Return to the Philosophy of Nature, Cambridge UP.

Kahn C. (2015) Review of The Pseudo-Platonic Seventh Letter, Notre Dame Philosophical Reviews. (9 Nov. 2015).

Morris C., (1971) Writings on the General Theory of Signs, The Hague: Mouton.

Natorp P., (1921) Platos Ideenlehre; transl., ed. and intro. 2004, V. Politis and J. Connolly, Plato’s Theory of Ideas, An introduction to Idealism, postscript by A.é Laks, Sankt Augustin: Academia Verlag.

Plato, (1921) Plato in Twelve Volumes, transl. H. Fowler. London: William Heinemann Ltd.

Plutarch, (1918) De Eapud Delphos, Moralia 27. XV (384e – 394c). transl. A. O. Prickard, Oxford: The Clarendon Press.

[1] Grammar (Γραμματική) in classical texts is nothing more than literacy (Graham 2014).

[2] Kahn C., Essays on Being (2009).

[3] Cf. (online): an annotated bibliography on Plato’s Sophist  Part 1,Part 2. and a few more recent references

at the end of the present text.

[4] Brisson’s contemporary classic “The Same and the Other in Timaeus‘s Ontological Structure”(1974) treats much of  the same, moving in its own direction.

[5] No such distinction exists in mathematics.

[6] Hoekstra (2003); Fronterotta (2019).

[7] Natorp (1921) Platos Ideenlehre; transl., Plato‘s Theory of Ideas (2004), 300-3.

[8] Kahn C., (2013), Plato and the Post Socratic dialogue, 167 n. 19.

[9] Brisson supports authenticity while Frede and Burnyeat have published The Pseudo-Platonic Seventh Letter (2015), a book rather drily reviewed by C. H. Kahn (2015).

[10] Lumping here for brevity his peculiar interest in the 5 regular solids from the Timaeus.

[11] That is,  signs in relationships with thoughts, other sings, the world: see Morris (1971), Writings on the General Theory of Signs. The semiotic triangle itself hints actually to cognitive science,  information theory and semantics.

Philosophia 26/2020, 200-209