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Dialectic (Greek: διαλεκτική, dialektikḗ; German: Dialektik), also known as the dialectical method, refers originally to dialogue between people holding different points of view about a subject but wishing to arrive at the truth through reasoned argumentation. Dialectic resembles debate, but the concept excludes subjective elements such as emotional appeal and rhetoric.[1] It has its origins in ancient philosophy and continued to be developed in the Middle Ages.

Hegelianism refigured "dialectic" to no longer refer to a literal dialogue. Instead, the term takes on the specialized meaning of development by way of overcoming internal contradictions. Dialectical materialism, a theory advanced by Karl Marx and Friedrich Engels, adapted the Hegelian dialectic into a materialist theory of history. The legacy of Hegelian and Marxian dialectics has been criticized by philosophers such as Karl Popper and Mario Bunge, who considered it unscientific.

Dialectic implies a developmental process and so does not naturally fit within classical logic. Nevertheless, some twentieth-century logicians have attempted to formalize it.


There are a variety of meanings of dialectic or dialectics within Western philosophy.

Classical philosophy[edit]

In classical philosophy, dialectic (διαλεκτική) is a form of reasoning based upon dialogue of arguments and counter-arguments, advocating propositions (theses) and counter-propositions (antitheses). The outcome of such a dialectic might be the refutation of a relevant proposition, or of a synthesis, or a combination of the opposing assertions, or a qualitative improvement of the dialogue.[2][3]

The term "dialectic" owes much of its prestige to its role in the philosophies of Socrates and Plato, in the Greek Classical period (5th to 4th centuries BC). Aristotle said that it was the pre-Socratic philosopher Zeno of Elea who invented dialectic, of which the dialogues of Plato are examples of the Socratic dialectical method.[4]

Socratic method[edit]

The Socratic dialogues are a particular form of dialectic known as the method of elenchus (literally, "refutation, scrutiny"[5]) whereby a series of questions clarifies a more precise statement of a vague belief, logical consequences of that statement are explored, and a contradiction is discovered. The method is largely destructive, in that false belief is exposed and only constructive in that this exposure may lead to further search for truth.[6] The detection of error does not amount to a proof of the antithesis. For example, a contradiction in the consequences of a definition of piety does not provide a correct definition. The principal aim of Socratic activity may be to improve the soul of the interlocutors, by freeing them from unrecognized errors, or indeed, by teaching them the spirit of inquiry.

In common cases, Socrates uses enthymemes as the foundation of his argument.[citation needed]

For example, in the Euthyphro, Socrates asks Euthyphro to provide a definition of piety. Euthyphro replies that the pious is that which is loved by the gods. But, Socrates also has Euthyphro agreeing that the gods are quarrelsome and their quarrels, like human quarrels, concern objects of love or hatred. Therefore, Socrates reasons, at least one thing exists that certain gods love but other gods hate. Again, Euthyphro agrees. Socrates concludes that if Euthyphro's definition of piety is acceptable, then there must exist at least one thing that is both pious and impious (as it is both loved and hated by the gods)—which Euthyphro admits is absurd. Thus, Euthyphro is brought to a realization by this dialectical method that his definition of piety is not sufficiently meaningful.

In another example, in Plato's Gorgias, dialectic occurs between Socrates, the Sophist Gorgias, and two men, Polus and Callicles. Because Socrates' ultimate goal was to reach true knowledge, he was even willing to change his own views in order to arrive at the truth. The fundamental goal of dialectic, in this instance, was to establish a precise definition of the subject (in this case, rhetoric) and with the use of argumentation and questioning, make the subject even more precise. In the Gorgias, Socrates reaches the truth by asking a series of questions and in return, receiving short, clear answers.


In Platonism and Neoplatonism, dialectic assumed an ontological and metaphysical role in that it became the process whereby the intellect passes from sensibles to intelligibles, rising from idea to idea until it finally grasps the supreme idea, the first principle which is the origin of all. The philosopher is consequently a "dialectician".[7] In this sense, dialectic is a process of inquiry that does away with hypotheses up to the first principle.[8] It slowly embraces multiplicity in unity. The philosopher Simon Blackburn wrote that the dialectic in this sense is used to understand "the total process of enlightenment, whereby the philosopher is educated so as to achieve knowledge of the supreme good, the Form of the Good".[9]

Medieval philosophy[edit]

Logic, which could be considered to include dialectic, was one of the three liberal arts taught in medieval universities as part of the trivium; the other elements were rhetoric and grammar.[10][11][12][13]

Based mainly on Aristotle, the first medieval philosopher to work on dialectics was Boethius (480–524).[14] After him, many scholastic philosophers also made use of dialectics in their works, such as Abelard,[15] William of Sherwood,[16] Garlandus Compotista,[17] Walter Burley, Roger Swyneshed, William of Ockham,[18] and Thomas Aquinas.[19]

This dialectic (a quaestio disputata) was formed as follows:

  1. The question to be determined ("It is asked whether...");
  2. A provisory answer to the question ("And it seems that...");
  3. The principal arguments in favor of the provisory answer;
  4. An argument against the provisory answer, traditionally a single argument from authority ("On the contrary...");
  5. The determination of the question after weighing the evidence ("I answer that...");
  6. The replies to each of the initial objections. ("To the first, to the second etc., I answer that...")

Modern philosophy[edit]

The concept of dialectics was given new life at the start of the 19th century by Georg Wilhelm Friedrich Hegel, whose dialectical model of nature and of history made dialectics a fundamental aspect of reality, instead of regarding the contradictions into which dialectics leads as evidence of the limits of pure reason, as Immanuel Kant had argued.[20][21] Hegel was influenced by Johann Gottlieb Fichte's conception of synthesis, although Hegel didn't adopt Fichte's "thesis–antithesis–synthesis" language except to describe Kant's philosophy: rather, Hegel argued that such language was "a lifeless schema" imposed on various contents, whereas he saw his own dialectic as flowing out of "the inner life and self-movement" of the content itself.[22]

In the mid-19th century, Hegelian dialectic was appropriated by Karl Marx and Friedrich Engels and retooled in what they considered to be a nonidealistic manner. It would also become a crucial part of later representations of Marxism as a philosophy of dialectical materialism. These representations often contrasted dramatically and led to vigorous debate among different Marxist groups.[23]

Hegelian dialectic[edit]

The Hegelian dialectic describes changes in the forms of thought through their own internal contradictions into concrete forms that overcome previous oppositions.[24]

This dialectic is sometimes presented in a threefold manner, as first stated by Heinrich Moritz Chalybäus, as comprising three dialectical stages of development: a thesis, giving rise to its reaction; an antithesis, which contradicts or negates the thesis; and the tension between the two being resolved by means of a synthesis.[25][26]

By contrast, the terms abstract, negative, and concrete suggest a flaw or an incompleteness in any initial thesis. For Hegel, the concrete must always pass through the phase of the negative, that is, mediation. This is the essence of what is popularly called Hegelian dialectics.[27]

To describe the activity of overcoming the negative, Hegel often used the term Aufhebung, variously translated into English as "sublation" or "overcoming", to conceive of the working of the dialectic. Roughly, the term indicates preserving the true portion of an idea, thing, society, and so forth, while moving beyond its limitations. What is sublated, on the one hand, is overcome, but, on the other hand, is preserved and maintained.[28]

As in the Socratic dialectic, Hegel claimed to proceed by making implicit contradictions explicit: each stage of the process is the product of contradictions inherent or implicit in the preceding stage. On his view, the purpose of dialectics is "to study things in their own being and movement and thus to demonstrate the finitude of the partial categories of understanding".[29]

For Hegel, even history can be reconstructed as a unified dialectic, the major stages of which chart a progression from self-alienation as servitude to self-unification and realization as the rational constitutional state of free and equal citizens.

Marxist dialectic[edit]

Marxist dialectic is a form of Hegelian dialectic which applies to the study of historical materialism. Marxist dialectic is thus a method by which one can examine social and economic behaviors. It is the foundation of the philosophy of dialectical materialism, which forms the basis of historical materialism.

In the Marxist tradition, "dialectic" refers to regular and mutual relationships, interactions, and processes in nature, society, and human thought.[30]: 257 

A dialectical relationship is a relationship in which two phenomena or ideas mutually impact each other, leading to development and negation.[30]: 257  Development refers to the change and motion of phenomena and ideas from less advanced to more advanced or from less complete to more complete.[30]: 257  Dialectical negation refers to a stage of development in which a contradiction between two previous subjects gives rise to a new subject.[30]: 257  In the Marxist view, dialectical negation is never an endpoint, but instead creates new conditions for further development and negation.[30]: 257 

Karl Marx and Friedrich Engels, writing several decades after Hegel's death, proposed that Hegel's dialectic is too abstract.[31] Against this, Marx presented his own dialectic method, which he claimed to be "direct opposite" of Hegel's method.[32]

Marxist dialectics is exemplified in Das Kapital. As Marx explained dialectical materialism,

it includes in its comprehension an affirmative recognition of the existing state of things, at the same time, also, the recognition of the negation of that state, of its inevitable breaking up; because it regards every historically developed social form as in fluid movement, and therefore takes into account its transient nature not less than its momentary existence; because it lets nothing impose upon it, and is in its essence critical and revolutionary.[33]

Class struggle is the primary contradiction to be resolved by Marxist dialectics because of its central role in the social and political lives of a society. Nonetheless, Marx and Marxists developed the concept of class struggle to comprehend the dialectical contradictions between mental and manual labor and between town and country. Hence, philosophic contradiction is central to the development of dialectics: the progress from quantity to quality, the acceleration of gradual social change; the negation of the initial development of the status quo; the negation of that negation; and the high-level recurrence of features of the original status quo.

Friedrich Engels further proposed that nature itself is dialectical, and that this is "a very simple process, which is taking place everywhere and every day".[34]

In Marxism, the dialectical method of historical study is intertwined with historical materialism, the school of thought exemplified by the works of Marx, Engels, and Vladimir Lenin.

For Lenin, the primary feature of Marx's "dialectical materialism" (Lenin's term) is its application of materialist philosophy to history and social sciences. Lenin's main contribution to the philosophy of dialectical materialism is his theory of reflection, which presents human consciousness as a dynamic reflection of the objective material world that fully shapes its contents and structure.

Later, Stalin's works on the subject established a rigid and formalistic division of Marxist–Leninist theory into dialectical materialism and historical materialism. While the first was supposed to be the key method and theory of the philosophy of nature, the second was the Soviet version of the philosophy of history.

Dialectical naturalism[edit]

Dialectical naturalism is a term coined by American philosopher Murray Bookchin to describe the philosophical underpinnings of the political program of social ecology. Dialectical naturalism explores the complex interrelationship between social problems, and the direct consequences they have on the ecological impact of human society. Bookchin offered dialectical naturalism as a contrast to what he saw as the "empyrean, basically antinaturalistic dialectical idealism" of Hegel, and "the wooden, often scientistic dialectical materialism of orthodox Marxists".[35]

Theological dialectics[edit]

Neo-orthodoxy, in Europe also known as theology of crisis and dialectical theology,[36][37] is an approach to theology in Protestantism that was developed in the aftermath of the First World War (1914–1918). It is characterized as a reaction against doctrines of 19th-century liberal theology and a more positive reevaluation of the teachings of the Reformation, much of which had been in decline (especially in western Europe) since the late 18th century.[38] It is primarily associated with two Swiss professors and pastors, Karl Barth[39] (1886–1968) and Emil Brunner (1899–1966),[36][37] even though Barth himself expressed his unease in the use of the term.[40]

In dialectical theology the difference and opposition between God and human beings is stressed in such a way that all human attempts at overcoming this opposition through moral, religious or philosophical idealism must be characterized as 'sin'. In the death of Christ humanity is negated and overcome, but this judgment also points forwards to the resurrection in which humanity is reestablished in Christ. For Barth this meant that only through God's 'no' to everything human can his 'yes' be perceived. Applied to traditional themes of Protestant theology, such as double predestination, this means that election and reprobation cannot be viewed as a quantitative limitation of God's action. Rather it must be seen as its "qualitative definition".[41] As Christ bore the rejection as well as the election of God for all humanity, every person is subject to both aspects of God's double predestination.

Dialectic prominently figured in Bernard Lonergan's philosophy, in his books Insight and Method in Theology. Michael Shute wrote about Lonergan's use of dialectic in The Origins of Lonergan's Notion of the Dialectic of History. For Lonergan, dialectic is both individual and operative in community. Simply described, it is a dynamic process that results in something new:

For the sake of greater precision, let us say that a dialectic is a concrete unfolding of linked but opposed principles of change. Thus there will be a dialectic if (1) there is an aggregate of events of a determinate character, (2) the events may be traced to either or both of two principles, (3) the principles are opposed yet bound together, and (4) they are modified by the changes that successively result from them.[42]

Dialectic is one of the eight functional specialties Lonergan envisaged for theology to bring this discipline into the modern world. Lonergan believed that the lack of an agreed method among scholars had inhibited substantive agreement from being reached and progress from being made compared to the natural sciences. Karl Rahner, S.J., however, criticized Lonergan's theological method in a short article entitled "Some Critical Thoughts on 'Functional Specialties in Theology'" where he stated: "Lonergan's theological methodology seems to me to be so generic that it really fits every science, and hence is not the methodology of theology as such, but only a very general methodology of science."[43]


Friedrich Nietzsche viewed dialectic as a method that imposes artificial boundaries and suppresses the richness and diversity of reality. He rejected the notion that truth can be fully grasped through dialectical reasoning and offered a critique of dialectic, challenging its traditional framework and emphasizing the limitations of its approach to understanding reality.[44] He expressed skepticism towards its methodology and implications in his work Twilight of the Idols: "I mistrust all systematizers and I avoid them. The will to a system is a lack of integrity".[45]: 42  In the same book, Nietzsche criticized Socrates' dialectics because he believed it prioritized reason over instinct, resulting in the suppression of individual passions and the imposition of an artificial morality.[45]: 47 

Karl Popper attacked the dialectic repeatedly. In 1937, he wrote and delivered a paper entitled "What Is Dialectic?" in which he criticized the dialectics of Hegel, Marx, and Engels for their willingness "to put up with contradictions".[46] Popper concluded the essay with these words: "The whole development of dialectic should be a warning against the dangers inherent in philosophical system-building. It should remind us that philosophy should not be made a basis for any sort of scientific system and that philosophers should be much more modest in their claims. One task which they can fulfill quite usefully is the study of the critical methods of science".[47] Seventy years later, Nicholas Rescher responded that "Popper's critique touches only a hyperbolic version of dialectic", and he quipped: "Ironically, there is something decidedly dialectical about Popper's critique of dialectics."[48]

The philosopher of science and physicist Mario Bunge repeatedly criticized Hegelian and Marxian dialectics, calling them "fuzzy and remote from science"[49] and a "disastrous legacy".[50] He concluded: "The so-called laws of dialectics, such as formulated by Engels (1940, 1954) and Lenin (1947, 1981), are false insofar as they are intelligible."[50] Poe Yu-ze Wan, reviewing Bunge's criticisms of dialectics, found Bunge's arguments to be important and sensible, but he thought that dialectics could still serve some heuristic purposes for scientists.[51]

Even some Marxists are critical of the term "dialectics". For instance, Michael Heinrich wrote, "More often than not, the grandiose rhetoric about dialectics is reducible to the simple fact that everything is dependent upon everything else and is in a state of interaction and that it's all rather complicated—which is true in most cases, but doesn't really say anything."[52]


Since the late 20th century, European and American logicians have attempted to provide mathematical foundations for dialectic through formalisation,[53]: 201–372  although logic has been related to dialectic since ancient times.[53]: 51–140  There have been pre-formal and partially-formal treatises on argument and dialectic, from authors such as Stephen Toulmin (The Uses of Argument, 1958),[54][55][53]: 203–256  Nicholas Rescher (Dialectics: A Controversy-Oriented Approach to the Theory of Knowledge, 1977),[56][57][53]: 330–336  and Frans H. van Eemeren and Rob Grootendorst (pragma-dialectics, 1980s).[53]: 517–614  One can include works of the communities of informal logic and paraconsistent logic.[53]: 373–424 


Building on theories of defeasible reasoning (see John L. Pollock), systems have been built that define well-formedness of arguments, rules governing the process of introducing arguments based on fixed assumptions, and rules for shifting burden.[53]: 615–675  Many of these logics appear in the special area of artificial intelligence and law, though the computer scientists' interest in formalizing dialectic originates in a desire to build decision support and computer-supported collaborative work systems.[58]

Dialog games[edit]

Dialectic itself can be formalised as moves in a game, where an advocate for the truth of a proposition and an opponent argue.[53]: 301–372  Such games can provide a semantics of logic, one that is very general in applicability.[53]: 314 


Mathematician William Lawvere interpreted dialectics in the setting of categorical logic in terms of adjunctions between idempotent monads.[59] This perspective may be useful in the context of theoretical computer science where the duality between syntax and semantics can be interpreted as a dialectic in this sense. For example, the Curry-Howard equivalence is such an adjunction or more generally the duality between closed monoidal categories and their internal logic.[60]

See also[edit]


  1. ^ See Gorgias, 449B: "Socrates: Would you be willing then, Gorgias, to continue the discussion as we are now doing [Dialectic], by way of question and answer, and to put off to another occasion the (emotional) speeches (rhetoric) that (the sophist) Polus began?"
  2. ^ Ayer, A. J.; O'Grady, J. (1992). A Dictionary of Philosophical Quotations. Oxford, UK: Blackwell Publishers. p. 484.
  3. ^ McTaggart, J. M. E. (1964). A commentary on Hegel's logic. New York: Russell & Russell. p. 11.
  4. ^ Diogenes Laërtius, IX 25ff and VIII 57 [1].
  5. ^ "Elenchus - Wiktionary". 8 February 2021.
  6. ^ Wyss, Peter (October 2014). "Socratic Method: Aporeia, Elenchus and Dialectics (Plato: Four Dialogues, Handout 3)" (PDF). open.conted.ox.ac.uk. University of Oxford, Department for Continuing Education.
  7. ^ Reale, Giovanni (1990). History of Ancient Philosophy. Vol. 2. Translated by Catan, John R. Albany: State University of New York. p. 150.
  8. ^ Republic, VII, 533 c-d
  9. ^ Blackburn, Simon (1996). The Oxford Dictionary of Philosophy. Oxford: Oxford University Press.
  10. ^ Abelson, P. (1965). The seven liberal arts; a study in mediæval culture. New York: Russell & Russell. Page 82.
  11. ^ Hyman, A., & Walsh, J. J. (1983). Philosophy in the Middle Ages: the Christian, Islamic, and Jewish traditions. Indianapolis: Hackett Pub. Co. Page 164.
  12. ^ Adler, Mortimer Jerome (2000). "Dialectic". Routledge. Page 4. ISBN 0-415-22550-7
  13. ^ Herbermann, C. G. (1913). The Catholic encyclopedia: an international work of reference on the constitution, doctrine, and history of the Catholic church. New York: The Encyclopedia press, inc. Page 760–764.
  14. ^ From topic to tale: logic and narrativity in the Middle Ages, by Eugene Vance, p.43-45
  15. ^ "Catholic Encyclopedia: Peter Abelard". Newadvent.org. 1 March 1907. Retrieved 3 November 2011.
  16. ^ Kretzmann, Norman (January 1966). William of Sherwood's Introduction to logic. U of Minnesota Press. pp. 69–102. ISBN 9780816603954.
  17. ^ Dronke, Peter (9 July 1992). A History of Twelfth-Century Western Philosophy. Cambridge University Press. p. 198. ISBN 9780521429078.
  18. ^ Delany, Sheila (1990). Medieval literary politics: shapes of ideology. Manchester University Press. p. 11. ISBN 9780719030451.
  19. ^ "Catholic Encyclopedia: St. Thomas Aquinas". Newadvent.org. 1 March 1907. Retrieved 20 October 2015.
  20. ^ Nicholson, J. A. (1950). Philosophy of religion. New York: Ronald Press Co. p. 108.
  21. ^ Kant, I.; Guyer, P.; Wood, A. W. (2003). Critique of pure reason. Cambridge: Cambridge University Press. p. 495. ISBN 9780758339010.
  22. ^ Maybee, Julie E. (Winter 2020). "Hegel's Dialectics § 3. Why does Hegel use dialectics?". In Zalta, Edward N. (ed.). Stanford Encyclopedia of Philosophy.
  23. ^ Henri Lefebvre's "humanist" dialectical materialism (Dialectical Materialism [1940]) was composed to directly challenge Joseph Stalin's own dogmatic text on dialectical materialism.
  24. ^ Hegel, Georg Wilhelm Friedrich (2010). Encyclopedia of the Philosophical Sciences in Basic Outline: Part 1, Science of Logic. Cambridge Hegel Translations. Cambridge, UK; New York: Cambridge University Press. pp. 34–35. ISBN 9780521829144. OCLC 651153726. the necessity of the connectedness and the immanent emergence of distinctions must be found in the treatment of the fact itself, for it falls within the concept's own progressive determination. What propels the concept onward is the already mentioned negative which it possesses in itself; it is this that constitutes the truly dialectical factor. [...] It is in this dialectic as understood here, and hence in grasping opposites in their unity, or the positive in the negative, that the speculative consists.
  25. ^ Historische Entwicklung der spekulativen Philosophie von Kant bis Hegel [Historical development of speculative philosophy from Kant to Hegel] (in German) (Fourth ed.). Dresden-Leipzig. 1848 [1837]. p. 367.
  26. ^ The Accessible Hegel by Michael Allen Fox. Prometheus Books. 2005. p. 43. Also see Hegel's preface to the Phenomenology of Spirit, trans. A. V. Miller (Oxford: Clarendon Press, 1977), secs. 50, 51, pp. 29, 30.
  27. ^ Maybee, Julie E. (Winter 2020). "Hegel's Dialectics". In Zalta, Edward N. (ed.). Stanford Encyclopedia of Philosophy (Winter 2020 ed.). Metaphysics Research Lab, Stanford University. Retrieved 2024-02-11.
  28. ^ Hegel, Georg Wilhelm Friedrich (1812). Hegel's Science of Logic. London: Allen & Unwin. p. §185.
  29. ^ Hegel, Georg Wilhelm Friedrich (1874). "The Logic". Encyclopaedia of the Philosophical Sciences (2nd ed.). London: Oxford University Press. p. Note to §81.
  30. ^ a b c d e Ministry of Education and Training (Vietnam) (2023). Curriculum of the Basic Principles of Marxism-Leninism. Vol. 1. Translated by Nguyen, Luna. Banyan House Publishing. ISBN 9798987931608.
  31. ^ Marx, Karl (1873) Capital Afterword to the Second German Edition, Vol. I [2]
  32. ^ Marx, Karl. "Afterword". link=Das Kapital [Capital] (in German). Vol. 1 (Second German ed.). p. 14. Retrieved 28 December 2014 – via Marxists Internet Archive.
  33. ^ Marx, Karl, (1873) Capital Vol. I, Afterword to the Second German Edition.
  34. ^ Engels, Frederick, (1877) Anti-Dühring, Part I: Philosophy, XIII. Dialectics. Negation of the Negation.
  35. ^ Biehl, Janet, ed. (1997). The Murray Bookchin reader. London; Washington, DC: Cassell. p. 209. ISBN 0304338737. OCLC 36477047.
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  39. ^ "American Heritage Dictionary (online)". Archived from the original on 2005-05-10. Retrieved 2008-07-26.
  40. ^ See Church Dogmatics III/3, xii.
  41. ^ Karl Barth, The Epistle to the Romans (1933), p. 346
  42. ^ Bernard J.F. Lonergan, Insight: A Study of Human Understanding, Collected Works vol. 3, ed. Frederick E. Crowe and Robert M. Doran (Toronto: University of Toronto, 1992, pp.217-218).
  43. ^ McShane, S.J., Philip (1972). Foundations of Theology. Notre Dame, Indiana: University of Notre Dame Press. p. 194.
  44. ^ Nietzsche, Friedrich (2001). The Gay Science. Cambridge University Press. p. 117. ISBN 9780521636452.
  45. ^ a b Nietzsche, Friedrich (1997). Twilight of the Idols or How to Philosophize with a Hammer. Hackett. ISBN 978-0872203549.
  46. ^ Karl Popper,Conjectures and Refutations: The Growth of Scientific Knowledge [New York: Basic Books, 1962], p. 316.
  47. ^ Karl Popper,Conjectures and Refutations: The Growth of Scientific Knowledge [New York: Basic Books, 1962], p. 335.
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  52. ^ Heinrich, Michael (2004). "Dialectics—A Marxist 'Rosetta Stone'?". An Introduction to the Three Volumes of Karl Marx's Capital. Translated by Alexander Locascio. New York: Monthly Review Press. pp. 36–37. ISBN 9781583672884. OCLC 768793094.
  53. ^ a b c d e f g h i Eemeren, Frans H. van; Garssen, Bart; Krabbe, Erik C. W.; Snoeck Henkemans, A. Francisca; Verheij, Bart; Wagemans, Jean H. M. (2014). Handbook of argumentation theory. New York: Springer-Verlag. doi:10.1007/978-90-481-9473-5. ISBN 9789048194728. OCLC 871004444.
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  58. ^ For surveys of work in this area see, for example: Chesñevar, Carlos Iván; Maguitman, Ana Gabriela; Loui, Ronald Prescott (December 2000). "Logical models of argument". ACM Computing Surveys. 32 (4): 337–383. CiteSeerX doi:10.1145/371578.371581. And: Prakken, Henry; Vreeswijk, Gerard (2005). "Logics for defeasible argumentation". In Gabbay, Dov M.; Guenthner, Franz (eds.). Handbook of philosophical logic. Vol. 4 (2nd ed.). Dordrecht; Boston: Kluwer Academic Publishers. pp. 219–318. CiteSeerX doi:10.1007/978-94-017-0456-4_3. ISBN 9789048158775.
  59. ^ Lawvere, F. William (1996). "Unity and identity of opposites in calculus and physics". Applied Categorical Structures. 4 (2–3): 167–174. doi:10.1007/BF00122250. S2CID 34109341.
  60. ^ Eilenberg, Samuel; Kelly, G. Max (1966). "Closed Categories". Proceedings of the Conference on Categorical Algebra: 421–562. doi:10.1007/978-3-642-99902-4_22. ISBN 978-3-642-99904-8. S2CID 251105095.

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