Dialectic Method, Definition, and Value in Plato’s Sophist

Método Dialético, Definição e Valor em O Sofista de Platão

Abstract

The aim of this paper is to examine the different passages of the Sophist that, although not dealing directly with the Good, can, however, hint at some aspects of Plato’s conception of value. In this regard, I mainly concentrate on the dialectic as practised in this dialogue and on the definitions offered of the figure of the sophist. In contrast to some Platonist scholars that have held that the Stranger’s method, unlike the Socratic practice of philosophy, does not discriminate the better from the worse, but only like from like, and, accordingly, that the dialectic method, as used in the Sophist, is a method of a neutral, aseptic character, I try to show that the dialectic and the definitions of the sophist have a notable evaluative charge.

Keywords:
Dialectic Method; Definition; Value; Sophist; Plato.

Resumo

O objetivo deste artigo é examinar as diferentes passagens do O Sofista que, embora não tratem diretamente do bem, podem, no entanto, sugerir alguns aspectos da concepção de valor em Platão. Nesse sentido, concentro-me principalmente na dialética praticada neste diálogo e nas definições oferecidas da figura do sofista. Em contraste com alguns estudiosos que sustentam que o método do estrangeiro, ao contrário da prática socrática da filosofia, não discrimina o melhor do pior, mas apenas o semelhante do semelhante e, portanto, que o método dialético, tal como utilizado na obra O Sofista, é de caráter neutro e asséptico, procuro mostrar que a dialética e as definições do sofista possuem uma notável carga avaliativa.

Palavras-chave
Método Dialético; Definição; Valor; O Sofista; Platão.

Introduction

There is probably no subject more important and more constantly present in Plato’s work than that of the Good. In an early period we find Plato devoted to the investigation of the nature of good life in the form of a criticism of the meaning of certain words, putting in Socrates’ mouth questions about the content of certain expressions and about the value that must be granted to them. In a further stage in his work, Plato continues his pursuit asking about the nature of virtue and how it can be acquired. In the Republic-a work of his full maturity, as is well known, which represents the most accomplished synthesis of all the fundamental themes of Platonism-the philosopher assigns a central position to this topic when he makes of the Form of the Good the ultimate aim of the intellectual and moral education of the guardians of the “beautiful city” (Resp. 527 c)¹. In a final contemplation that will carry to perfection the education received, the guardians “must raise the eye of the soul to the universal light which lightens all things, and behold the absolute Good; for that is the pattern according to which they are to order the State and the lives of individuals, and the remainder of their own lives also” (540 b)². And this ethico-political role of the Form of the Good is indissociable-as is taught in the sixth book, the locus classicus for the discussion on the supreme Form-from a very problematic double role: its epistemological function as the condition of possibility of the knowledge of the Forms, and its metaphysical function as the cause of being and of the perseverance in being³. Another dialogue, the Philebus, is entirely consecrated to discussing the nature of the Good. And in the Timaeus, the philosopher finds the necessary ultimate reason of the existence of the universe in the goodness of the demiurge, a postulate that would have such profound impact on the imagination of the European continent throughout the following centuries⁴.

This constant presence of the concern with goodness-in the background of which the long shadow of Socrates and his art of the care of the soul (θɛραπɛία τῆς ψυχῆς) are glimpsed⁵- throughout the dialogues, justifies our interest in the question of the Good in a dialogue like the Sophist⁶, in which, however, the Good is never expressly dealt with-and that is most probably the reason why the subject of this paper has hardly been discussed by the scholars. This is even more odd and disconcerting since the reader would expect the criticism of the figure of the sophist to be carried out on the basis of the appeal to the knowledge of the Good on the part of the philosopher, in accordance with the moral point of view that prevails not only in the Republic, but also in the Gorgias and the Phaedrus, the two dialogues consecrated to the criticism of rhetoric-especially bearing in mind that the second section of the Sophist is the place where Plato develops his most detailed research on the supreme genera. Or at least we would expect to be reminded of the role played by the Form of the Good as an ultimate goal in those passages devoted to reflecting on the nature of the dialectic method, a method that, as was held in the Republic, would not make sense unless it were illuminated by the supreme Form⁷.

Knowing well that no theme can be more fundamental for Plato than that of the Good, the aim of this paper is, therefore, to examine the different passages of the Sophist that, although not dealing directly with the Good, can, however, suggest something about it that sheds some light on the conception held by Plato, and we will also link them-even if with long stitches-with other passages of the Platonic corpus that might be pertinent to the subject.

The first indications about the importance of value

We find the first indications about the importance of value in the opening lines of the dialogue, when Theodorus says of the Stranger who is going to lead the research that he, being a philosopher, is therefore “divine” (θɛῖος) (Soph. 216 b)⁸. It is important to point out this feature of the philosopher, I think, because “divine” is the chief qualifier in Plato’s ethics and aesthetics. In fact, goodness, beauty and truth are closely connected in Plato, since they find support in each other, and together constitute the sphere labeled as that of “the divine”. In Plato’s conception of life, each of these three Forms is meaningful if the other two are on its side, if it can refer to them and live on them and for them. In the Philebus-to quote just one of the many passages in Plato’s work that can be adduced to illustrate this point-it is stated that the Good is the conjunction of three forms: beauty, symmetry, and truth (Phil. 65 a)⁹.

Taking this into consideration, it can be said that, despite the fact that the Form of the Good-as I previously pointed out-is never mentioned in the Sophist, the dialogue does deal with the Good as far as the issue at stake is the possibility of enunciating true judgments. The core of the Sophist deals with the question of the necessity of the false judgment, and its main result is recognizing to non-being a certain kind of existence-and, let it be said in passing, Plato gives thus a logico-metaphysical status to that well-known observation about the indestructible nature of evil that had appeared in the Thaetetus and that would have such a great impact on Neoplatonism¹⁰. But if Plato is led to that conclusion, it is because what he longs for is to prove that it is possible to speak in a legitimate sense about truth, and that is the reason why the object of the dialogue is beautiful (καλόν, Soph. 259 c): because thanks to it we will not find ourselves “deprived of philosophy” (260 a).

In the dialogue, the “divine” character of the philosopher appears once more when, in the course of the search for the figure of the sophist, that of the philosopher is discovered. Whereas the figure of the sophist is difficult to discern due to the gloom of the place he inhabits-the darkness of non-being-the difficulty in finding the philosopher lies in the very brightness of the realm his soul inhabits, that is to say, in the lightness or clarity of the form of being or of the divine (254 a-b), a theme that reminds us of the metaphor of the light by which the Form of the Good is thought of in the Republic, and also of that in the Seventh Letter (340 b) of the man kindled with the fire of philosophy.

The dialectic method in the Sophist and value

On the path of pursuing the sixth definition of the Sophist¹¹ the art of discriminating (διακριτικὴ τέχνη, Soph. 226 c) is divided intο the art that separates the better from the worse and the art that separates like from like. The purificative species of the art of discriminating is, in turn, subdivided into two: the purification of the body and that of the soul. When recalling the many kinds of bodily purifications that exist, and that are too “trivial” to be worth mentioning, the dialogue arrives at the idea that dialectic is a formal method. The passage where the method is described reads as follows:

The dialectical art never considers whether the benefit to be derived from the purge is greater or less than that to be derived from the sponge, and has not more interest in the one than in the other; her endeavour is to know what is and is not kindred in all arts, with a view to the acquisition of intelligence; and having this in view, she honours them all alike, and when she makes comparisons, she counts one of them not a whit more ridiculous than another; nor does she esteem him who adduces as his example of hunting, the general’s art, at all more decorous than another who cites that of the vermin-destroyer, but only as the greater pretender of the two. (227 a-b; Jowett’s translation.)¹²

Some Platonist scholars have held that the Stranger’s diairetic method belongs to the right species of the art of discriminating. It has been maintained, with the support of the passage just quoted, that the Stranger’s method, unlike the Socratic practice of philosophy, does not discriminate the better from the worse, but only like from like, and, accordingly, it has been understood that the dialectic method, as used in the Sophist, is a method of a neutral, aseptic character¹³. However, I think that this interpretation is problematic due to the following reasons.

First, it is necessary not to overlook that the Stranger says that he has no name for the right species of the dichotomy (226 d), and this would not have been put in his mouth if it were his own philosophical method. Secondly, it should be taken into account that the definitions of the sophist that are offered, are quite distant from not implying evaluation. These definitions are not a model of neutrality; rather, they are full of pejorative nuances, since the “hunting” of the sophist is carried out under the most adverse light possible. In fact, as it happens, in the very context of the sixth definition where the description of the method appears, at the end of it, Plato expresses clearly his reservations when it comes to putting the activity of his master and that of the sophist on the same level. Although the art of divisions has been unable to separate Socrates from the sophists, the Stranger does not speak in formal terms of the difference between the former and the latter: “I am afraid to say [that the ministers of the art of purifications are] the sophists. Lest we should assign to them too high an honour (γέρας)” (231 a).

Thirdly, I think it will be useful to bring to mind something, by way of general consideration about Plato’s philosophy, in order to interpret this passage correctly. If dialectic is a science, as Plato held it was, if it is an ἐπιστήμη, then it cannot detach itself from the consideration of the Good. The object of dialectic is, of course, the Forms, but in Plato the knowledge of the Forms is never just the understanding of truth, the mere grasping of the connections existing among essences, but also the intuition of the moral and aesthetic realities that are inherent to them. In this sense, Plato’s Forms are always values. In all the dialogues where the question arises, the science of dialectic is no less the knowledge of good and evil, of beauty and goodness, than it is of truth. It is for this reason that in dialogues like the Gorgias and the Phaedrus Socrates defends that rhetoric is not a τέχνη¹⁴. Art is ἐπιστήμη, which is based on the knowledge of the Good. Therefore it cannot be independent of ends and an instrument in the service of values morally opposed, as Gorgias defends. Dialectic, in so far as it is able to become an art, cannot limit itself to classifying existence in terms of likeness and unlikeness in an “aseptic” way. In the Sophist, once it has been proved that genera participate in the mixture, making thus language and, therefore, philosophy possible, Plato describes his method practically in the same terms used in the Phaedo and the Republic, as the path that leads to a correct (ὀρθή) vision of the world, since it teaches what genera harmonize (συμφωνɛῖν) with others and what are incompatible with each other (253 b-c). That vision is correct both in a logical and in an ethical sense, since-it is said a few lines further below-the person who sees the world in this way is “pure” (καθαρός) and “just” (δίκαιος) (253 e).

Fourth, we must remember that in the Statesman-a dialogue with which the Sophist bears a very close connection-Plato makes an important remark of a general character about the procedure of διαίρησις. He says that, just as letters are not taught to children in order to enable them to read some particular words, but to recognize any word and solve all kinds of grammatical problems, so the purpose of that dialogue is not to define the figure of the statesman, but to make the reader a better dialectician in all kinds of questions (Pol. 285 c-d). Keeping in sight that the main purpose of the philosophical dialogue has an educational character, it is easy to understand that what Plato means in the passage of the Sophist that I am commenting, is not that dialectic has lost its normative role, but only that, in order to become able dialecticians, it makes no difference to search for the genus of the sophist, or for that to which a pumpkin belongs-just to remember the comedian Epicrates’ well-known joke.

The precise meaning of our passage seems to be the following. In the Republic Plato talks of dialectic doing away with hypotheses (τὰς ὑποθέσɛις ἀναιροῦσα, Resp. 533 c). The principles postulated in those sciences that make use of them when reasoning-mathematics, for instance-must be given an ultimate justification in which the dialectic process of giving an account reaches its conclusion. This justification of hypotheses is achieved thanks to the knowledge of the Good. From a practical point of view, this “doing away with hypotheses” that dialectic carries out, means that it is the one science that can understand the social usefulness of all other knowledge and that it is, therefore, the foundation on which the art of politics must be grounded. Rhetoric, we are told, so far as it governs without adopting the point of view of the Good, is an irrational form of ruling the State (534 d).

Therefore, and bearing in mind that dialectic reaches the Form of the Good and that, as I have just said, this means, among other things, that the dialectician knows in what sense all things concerning human life are useful, what Plato says in the passage of the Sophist that I comment is not that dialectic does not evaluate the Forms and their imperfect copies in the sensible world, but, quite the opposite, he is simply pointing out-in passing-that dialectic is the science that enables the philosopher to recognize the usefulness (ὠφέλɛια, a term that for every Socratic philosopher always has a moral meaning) that any art might have, whether it is a noble art like medicine or one regarded as more modest-but useful-like the art of cleansing with a sponge.

In so far as dialectic is a method of a universal character-a method not limited to some part of the intelligible world, but that seeks to know the whole of reality-it does not disregard any Form. It is for this reason that in the Parmenides the old master reprimands Socrates for having succumbed to the thought of restricting the world of the Forms to noble beings. When young Socrates matures and philosophy gets a firmer grasp on him, he will not despise any Forms, not even those of things such as hair, mud, or dirt (Parm. 130 c-e). Only in this sense is dialectic a formal method.

It might be objected that it is not adequate to explain a passage of the Sophist resorting to a previous dialogue, such as the Republic, but that, on the contrary, previous dialogues should be read in the light of the subsequent ones. This possible objection is, in my opinion, hardly plausible, as in several of the dialogues posterior to the Sophist Plato maintains an ontology similar to that of the Republic. Moreover, we fortunately have the Seventh Letter-the last writing of Plato about the nature of philosophy-where we find essentially the same doctrine as the one of books VI and VII of the Republic.

The Statesman is an example that confirms what I have said. In that dialogue we are told that there is a science, politics, that is over the rest of the arts and rules them¹⁵. Whereas the other arts are each one concerned with some special action of their own, political science interweaves in the most correct way all matters affecting the city (Pol. 305 d-e). Political science, “perhaps the science most difficult to acquire and the greatest” ([ἡ ἐπιστήμη] σχɛδὸν . . . χαλɛπωτάτη καὶ μɛγίστη κτήσασθαι, 292 d), dealing with “the most important things” (τὰ μέγιστα, 302 a)-dialectic, then-fulfils its function of supervising the rest of sciences making use of the “just mean” (τὸ μέτριον)-the measure provided by the Forms-in order to discriminate good from evil and so making all their works “good and beautiful” (284 a-b)¹⁶.

The definitions of the sophist and the good

In accordance with this, all the definitions of the sophist elaborated in the dialogue are far from being objective and neutral; they are rather drawn from a certain sense of what is considered to be right and good. In the first four definitions, the idea that dominates is that the philosopher is better than the sophist because, whereas the philosopher engages in debates with no other view but to attain the knowledge of truth, the sophist does it looking for his own benefit, either in terms of honours or wealth. However, it is equally clear that in Plato’s eyes these definitions are defective, since the notion of the sophist as a merchant of spiritual nourishment-this is the common feature of the first four definitions-is clearly insufficient to define him. And the reason is that the use that the greatest sophists made of dialectic has a justification going quite beyond the profit motive which might have moved them to practise it. Indeed, if the sophists practised dialectic in an eristic form-and this is the fifth definition of sophistry at which the dialogue arrives-it was because they defended that any other alternative was impossible. To justify this claim, they resorted to several arguments bequeathed to them by the Presocratic tradition, of which the one relevant here is the well-known argument of the impossibility of falsehood, stating that all opinions must be considered as equally true, since every opinion has to be about some particular thing, and everything is real. Falsehood does not exist, and therefore we can only expect an agonistic use of dialectic aiming at the defeat of the opponent and not at the elucidation of truth, as was the Socratic-Platonic purpose. That is why, in order to be able to offer a description of what the sophist is “absolutely and really” (παντάπασιν ὄντως, Soph. 268 c), the dialogue has to embark on a metaphysical journey in search of the demonstration-without old Parmenides’ blessing-of the existence of non-being, a voyage completed with the seventh and definitive definition of the sophist as a master of appearances (φαντάσματα)¹⁷.

The sixth definition of the sophistry-that of the nobly-descended sophistry (ἡ γένɛι γɛνναία σοφιστική, 231 b)-is a description of the Socratic art of the care of the soul. The inclusion of Socrates’ mayeutics among descriptions of sophistry may be due, as Guthrie has pointed out¹⁸, to Plato’s desire to make the activity of his master stand out in vivid contrast to that of the rest of sophists, being those two kinds of activities certainly kindred, but none the less quite opposite to each other-as the dog and the wolf are (231 a).

The conception of the good implied by this sixth variety of sophistry is the following. The Stranger’s account depends on two analogies: one analogy holding between the dissension of the soul and the disease of the body, and another between the lack of knowledge and physical deformity. Accordingly, we are told that there are two kinds of evil (πονηρία). One is like disease (νόσος), the other appears as ugliness or deformity (αἶσχος). Disease is identified with dissension (στάσις), which is defined as “the corruption, originating in some disagreement (διαφορά), of that which is kindred by nature (τὸ φύσɛι συγγɛνές, 228 a)”. On the other hand, deformity is an unsightly want of measure (ἀμɛτρία).

The first kind of evil of the soul, its disease, is the dissension among its parts. In bad men-says the Stranger-opinions dissent with desires, courage with pleasures, and reason with pains, although all parts of the soul are necessarily akin (228 b). We find here, then, something similar to what Socrates had defended in the Gorgias against Polus: that excellence (ἀρɛτή) is produced by order, by the harmonious life of the parts of the soul-an idea, moreover, that is developed at length in book IV of the Republic, where virtue is defined as the harmonious cooperation among the parts of the soul.

The second kind of evil of the soul, its deformity, is ignorance (ἄγνοια). This is-says Plato-a wandering of mind (παραφροσύνη). An ignorant soul is that which, in spite of its natural attraction to truth, lacks symmetry with the object (228 c). The art that fights against the evil of ignorance is instruction (ἡ διδασκαλικὴ τέχνη); the one fighting against injustice is the corrective art (ἡ κολαστική). The part of instruction in which Plato is interested here is the one that takes care of that ignorance which gives birth to all the mistakes of human knowledge: presumptuousness. To that ignorance the dialogue reserves from this point the name ἄγνοια; education (παιδɛία) is the part of instruction that deals with it. As opposed to the traditional and ineffective exhortation (νουθέτησις), Plato sets up his master’s dialectic method, the healthy confutatory method. Thus understood, the dialectic method aims at purifying the soul of the prejudices that block the way to instruction. The Socratic therapy consists in refuting the mistaken opinions of the interlocutor and to gather them in order to show him the contradictions into which he falls, liberating him in this way from the presumptuous, blocked opinions that he has of himself. This is the highest benefit that can be made to someone, since the soul of the patient “will receive no benefit from the application of knowledge until he is refuted (ἐλɛγχόμɛνος), and from refutation learns modesty; he must be purged (καθαρός) of his prejudices first and made to think that he knows only what he knows, and no more” (230 d; Jowett’s translation). This is the best and wisest attitude of mind which a man can have, since it represents the condition of possibility of his access to the exercise of dialectic, understood now in the properly Platonic way as the science of the gathering (συναγωγή) and division (διαίρɛσις) of genera, and this gives the soul, as the dialogue shows, its freedom and happiness.

In fact, the liberation possessing a religious character is dragged by Plato to the philosophical terrain and declares that the spiritual experience involved in dialectic turns men into free beings (ἐλɛύθɛροι, 253 c) and leads them to a blessed state (ɛὐδαίμον, 230 e).

Consequently, I can say in conclusion that it must not be overlooked that the aim pursued by the Sophist is associated with interests of an ethical character. As it becomes clear from the passage of the Statesman mentioned above, the aim of the philosophical dialogue is to teach man to think in dialectical terms, developing his natural capacity to have a synoptic vision of reality and to divide into classes, and that implies a liberation, as far as possible, from every subjectivist element, every element coming from the senses, from feelings, from affections and from passions, that are an obstacle to the access to the world of essences-the access to truth, the only place where the soul of man finds peace and happiness.

Thanks to the learning of dialectic, man can correct, insofar as it is possible for a mortal being, that partial point of view that, as an individual, he has of reality-a perspective mutilated and connected with a lack of control of the νοῦς, the rational element, over the body-as far as rising to a contemplation of the world sub specie aeternitatis.

The difference, then, that Plato sees between the philosopher and the sophist is that, whereas the former, as far as he rises to the perspective of universality and objectivity, realizes that universal desire of mankind for the good about which Socrates has told so much, the sophist, on the contrary, lacking ἐπιστήμη and being dominated by the low passions, cannot reach the supreme wisdom, the wisdom that is goodness. That is why, in the context of the seventh and definitive definition of the sophist as a creator of illusions by means of the juggling of words (ἐν λόγοις τὸ θαυματοποιικόν)-a definition in accordance with all the other dialogues where the theme arises-Plato states on two occasions that sophistry is not a serious activity. He thus tries to discredit it in the harshest way possible, being the concept of seriousness (σπουδή, σπουδαῖος), a fundamental concept in Plato’s ethics, since, as we know from the Republic, serious activities are those which have in view the good of mankind¹⁹.

• How to cite:
  RAMÓN CÁMARA, Begoña. Dialectic Method, Definition, and Value in Plato’s Sophist. Revista de Filosofia Aurora, Curitiba, v. 38, e202632970, 2026. DOI: https://doi.org/10.1590/2965-1557.038.e202632970
• 1
  1 On education in Plato, see R. L. Nettleship (1906).
• 2
  Jowett’s translation.
• 3
  For a presentation of the main statements on the Form of the Good made in the Republic, see G. Seel (2007), pp. 168-172; G. Santas (1983), pp. 232-241, and Th. A. Szlezák (2003), pp. 111-112.
• 4
  On Plato’s conception of the Good, see R. L. Nettleship (1897), pp. 237-394; C. H. Kahn (2004), pp. 627-640; G. Reale and S. Scolnicov (2002), and M. Vegetti (1993), pp. 207-229 and (1998), pp. 253-286.
• 5
  On this Socratic art in Plato’s Dialogues, see A. Biral (1997). «The Socratic background must never be lost sight of if we are to understand the tremendous importance attached by Plato to the discovery of the Form of the Good, and the ineffably exalted position which he gives it» (Guthrie (1975), p. 505).
• 6
  On the Sophist, see D. Ambuel (2007); N. Notomi (1999), and G. Movia (1991).
• 7
  On Plato’s dialectics see M. Dixsaut (2001); H. G. Gadamer (1968); H. Krämer (1989), and T. Robinson (1953).
• 8
  In the first lines of the Sophist Plato also alludes to the “harmonic” nature of the philosopher. The Stranger is more “temperate” than those who indulge in disputes (μɛτριώτɛρος τῶν πɛρὶ τὰς ἔριδας ἐσπουδακότων, 216 b). Parallel to the beginning of the Sophist is that of the Statesman-the second part of a trilogy that, in addition to the Sophist and the Statesman, was to be composed of the Philosopher, a dialogue that, however, was never written. The dialogue opens with Socrates’ reproach to Theodorus for having rated the sophist, the statesman and the philosopher at the same value (τιμή), whereas they are really separated by an interval which cannot be expressed by any ratio of that geometrical art in which Theodorus is an expert (Pol. 257 a-b).
• 9
  On Philebus, see R. G. Bury (1897); R. Hackforth (1945), and C. H. Kahn (2010), pp. 56-67.
• 10
  Plato’s words in the Thaetetus were: “Evils can never pass away; for there must always remain something which is antagonistic to good. Having no place among the gods in heaven, of necessity they hover around the mortal nature, and this earthly sphere” (Theaet. 176 a; Jowett’s translation.)
• 11
  On this definition, see W. K. C. Guthrie (1978), pp. 128-129, and A. Bernabé (2013), pp. 41-56.
• 12
  In the Statesman (266 d) Plato points out again the formal character of the dialectic method in very similar terms.
• 13
  For this kind of interpretation, see K. Dorter (2013), pp. 87-102, and (1994).
• 14
  On rhetoric in the Gorgias, see E. R. Dodds (1959) and M. McCoy (2008), pp. 85-110. On the subject in the Phaedrus, see G. R. F. Ferrari (1987). On the notion of persuasion, see F. Trabattoni (1993), pp. 100-148.
• 15
  Plato talks in particular of three of those arts: rhetoric, the science of the generals, and that of the judges.
• 16
  On the dialectic method in the Statesman, see K. M. Sayre (2006), pp. 113-138 and M. S. Lane (1998), pp. 33-98.
• 17
  I agree with the view on Plato’s conception of mimesis (μίμησις) in the Sophist as defended by L. Palumbo (2013), pp. 269-278. On the role played by the important category of mimesis in Plato’s philosophy, see L. Palumbo (2009).
• 18
  W. K. C. Guthrie (1978), p. 129.
• 19
  The two passages referred to are Soph. 234 a, where the Stranger says that sophistry is a jest (παιδιά), and Soph. 259 c. The passage of the Republic where Plato defines seriousness is 452 d-e.

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• Editores responsáveis:
  Léo Peruzzo Júnior e Jelson Oliveira.

Publication Dates

• Publication in this collection
  20 Apr 2026
• Date of issue
  2026

History

• Received
  16 May 2025
• Accepted
  28 Jan 2026