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Draw me an argument! What for? Graphs, maps, and diagrams in the study of argumentation

Michel Dufour
Cet article est une traduction de :
Dessine-moi un argument ! Pour quoi faire ? Schémas, cartes et diagrammes dans l’étude de l’argumentation [fr]

Résumé

Like most scientific or technical fields, the field of argumentation uses methods of presentation and cartographic representation of its objects. These illustrative processes are said to have cognitive or methodological advantages. Today, there are two main reasons for using them: pedagogical and heuristic – both of which are supposedly served by using these tools to describe or evaluate an argument, an argumentative exchange, or even a discourse. After a discussion of the notions of map and cartography, the article examines their contribution to argumentation, on the one hand by questioning their demarcation, particularly in relation to written representations, and on the other by placing them in the wake of a graphic tradition common to logic and mathematics. In contrast to a spontaneously empirical approach, which only sees data on a map, this study asks and discusses again: What is to be mapped or seen mapped, and for what purposes? What ingredients are considered worthy of appearing in a map legend? Finally, what are the expected pedagogical outcomes?

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Texte intégral

It is not so much the making of plans, the use of symbolic thought, as the externalizing and communication of those plans, transactions in symbolism, that are the marks of man.

J. Goody, The Domestication of the Savage Mind (1977: 159)

Introduction

  • 1 On the organisation of the lists and their variations, see also Sève (2010).

1It would be a mistake to rely on the French title of Jack Goody’s (1977/1979) book – La raison graphique – to reflect on the relationship between graphics and reason. The literal translation of the otherwise evocative English original – The Domestication of the Savage Mind – appears only as a subtitle in the French version. But this translation artifact has its merits since the book is about the invention, not of graphics, but of writing and its methodological consequences. Above all, it suggests the idea, which is the guiding principle here, of greater proximity between writing and cartography than between writing and speaking. There is no need to detour into ideograms or countless alphabetical fonts to recall that writing is a form of graphics. This blurring of the boundaries between writing and cartography is one of the interests of Goody’s book, particularly when he insists on the two-dimensional modes of the written word1. Faced with a list or an accountant’s spreadsheet, are we dealing with writing or cartography, the latter now understood in a generic sense to include maps, diagrams, figures, drawings, sketches, and schematics? We are still waiting for a clear answer accompanied by a decisive argument. The question remains when the mapped object is a logos. Can it seriously be said to be “outside the text”?

2From this theoretical perspective, we will consider the use of cartographies in argumentation and the benefits that we should be able to draw from them. The first section will deal with the concept of map, which is part of the very notion of cartography, and in particular with the descriptive or representational function generally ascribed to it. We will insist, however, on the illusions that can be created by an overly empirical and encyclopedic conception of a map, forgetting the choices made prior to its elaboration. We will also recall the two main functions of a map – synthetic and analytical – that recur in its typical uses.

3The second part will build on the previous points to consider the use of cartography in argumentation. First, we will defend (and illustrate) the idea of continuity, or at least visual proximity, between the written and graphic presentation of a syllogism or an argument, with the choice of one or the other mode depending on the objective pursued, namely, the identification or evaluation of the object in question. This is an opportunity to revisit a long tradition of graphic representations designed to assess the accuracy of a conclusion, a line of reasoning, or an entire argument.

4Finally, a third section discusses the results of studies on the pedagogical merits of using cartographies in argumentation. Overall, the results are mixed, although there is a trend towards greater competence among students who have benefited from exposure to these tools. In fact, the results vary according to the type of cartography and the intended use, which, in our opinion, confirms the vagueness of the concept of cartography and the difficulty of overcoming its entanglement with other cognitive resources, especially the written word.

1. Maps, charts, diagrams, graphs and schematics

5We will limit ourselves here to an understanding of the commonplace nature of the concept of cartography and its cognitive issues, restricting it to its current association with a single sense, visual perception.

6The etymology confirms this: cartography means map, and a map, whether figurative or not, suggests a solid and (relatively) durable support, with the privilege of “cardboard” and its relatives, despite fluid maps drawn on sand. Today, this privilege seems to be threatened by screens, which are nonetheless solid: the archetype remains flat or planar, rather rigid, and unfolds in two dimensions, despite “relief maps” and “3D” cartographies. Hence the first question: how can we accommodate argumentation within the invisible order of logos and the essentially spatial principle of cartography? Indeed, the voice fills the three dimensions of space, but the primitive form of the spoken word is primarily inscribed in the linear axis of time, an order of succession, not of simultaneity, as space is.

7The crucial innovation that opened a second dimension to discourse was writing: cartography” was joined by “graphy,” and with it, the hand, generally equipped with a technical extension – a stylus or paintbrush for the oldest. We can rightly speak here of the deployment of discourse and even of explanation in its original sense of unfolding, akin to a distinctively cartographic mode of intelligibility. “A good diagram is worth more than a long speech”, said Napoleon. Spread over two geometric dimensions, the diagram – if it’s good – would be more concise and “speaking” than a discourse carried by a single axis, audio-temporal for oral, visual-spatial for standard linear writing.

1.1. Map expectations

  • 2 They are countless: business cards, credit cards, greeting cards, member cards...

8According to the most naïve empirical view, a map represents, by analogy, a land, a view based on the dream of an exhaustive reproduction according to two orders, spatial and ontological. In addition to the flat support of the playing card (carte) and its relatives2, “card” (carte) evokes such a descriptive function, often considered primary, by presenting a land, a term itself already laden with soil and earth. Let’s bet that a survey would confirm that this term, when it does not immediately evoke the object map, is mainly associated with geography, even if this primordial descriptive function is also found in other disciplines of space such as astronomy or geology. A map shows a set of data in a typically spatial order but with a change of scale, which is also a spatial operation.

9The ideal would be that “everything is here” and “in good order,” at the cost of infinite resolution: zooming in on an area would constantly reveal, to the point of exhaustion, details that were initially imperceptible. This dream map would be perfectly isomorphic to the land it is supposed to miniaturize. Like Noah’s Ark, it would be an integral microcosm of reduced dimensions. Our pixelated world knows that spatial resolution can be “high”, even very high, but hopes for infinity remain illusory. Below the finest grain, we are off the map; completeness remains a distant horizon.

10In practice, therefore, a map remains finite and its richness finite. What is more, its creation is governed by choices, goals, and objectives. This is why different types of cartography often coexist within the same discipline, including argumentation studies. The dissolution of the dream of exhaustiveness, of the “total”, is partly compensated by cartography’s opposite advantage of not getting lost in an infinite number of details. Ancient sea charts, relatively detailed for coastal contours but coarse or even empty inland, were already an invitation to sobriety. Today, contrary to the dream of encyclopedic reproduction, a map is usually “only” a road or river map or only shows the relief, the greedy having to be satisfied with a multitude of specialized layers. By restricting ourselves to a limited ontology and a finite grain, announced by a limited set of symbols – legend and scale – we renounce the seductions of naive empiricism without abandoning the project of description.

11It is a short step from geography to geometry but a big gap. We know them as sister sciences, as “cadastral” sciences, whose etymology refers to the census, and therefore also to lists, objects where the boundary between the written word and the icon is blurred. What they have in common are questions such as: Where are the borders? Where are the limits? But behind the apparent platitude of his famous slogan, “a map is not the territory,” Alfred Korzybski (1933-2015), father of a semantics presented as general, recalls two antagonistic tendencies in the use of cartographies, which we may find in the divergence of destinies between geography and geometry. In cartography, the geographical tendency is characterized by the predominance of a referential descriptive function, whereas the fate of geometry has been a story of distance. Today, from abstraction to abstraction, this science and its possible cartographies have only a distant relationship with the matter of the territory, despite the persistence of an archaic generic affinity: the continuous and its parts. Above all, contrary to the concern for reality and truth seen as correspondence, typical of the empiricism of geography, geometry opts for abstraction and generality, playing with standard objects whose “real” existence is more enigmatic than that of Mont Blanc or the Atlantic, attested by a geographical map. The circle and the equilateral triangle, drawn freehand on the blackboard by the geometry teacher, claim to represent the Circle and the Equilateral triangle, despite the imperfection of the gesture that produced them, just as the canonical transcription of an argument claims to represent the speaker’s point of view. What does such a cartography, especially an argumentative one, represent? Concrete or abstract, already there, an imagined possibility, or both? Behind this question lie two millennia of puzzlement and philosophical debate about the reality of the ideal. The point is to remain wary of a priori univocity that sees cartography as all empirical or all theoretical.

  • 3 Ironically, the modern atom was quickly “broken,” its fission distinguishing it from the ancient on (...)
  • 4 Time-honored expressions in chemistry.

12Let’s return to the ambivalence between writing and graphics. Long before it reached relative agreement on the ancient dispute over the ultimate material constituents, ultimate because unbreakable – the famous “atoms”3 – chemistry had a deep affinity, but equally controversial, not only with the “elements” presumed to make up the world but also with the manipulation of symbols. The famous chemical “formulas” make use of one of the few alphabetic “languages” in which letters work as nouns. Above all, they allow us to shift almost seamlessly from writing to graphics, from the alphabet to ideograms. Like Napoleon’s slogan, let’s hope that a simple pictorial example using the “raw formula” and “developed formula” of a molecule4, will be better than a long speech at blurring the boundary between writing and cartography.

13A different sequence could be chosen, for example, moving H2O one notch to the right, to stress the progress of the unfolding from the horizontal of the writing towards its perpendicular on the flat sheet. The development of the formula thus concisely integrates additional information on the relative arrangement of the atoms (angles, distances), with the last representation even adding one information on the relative masses of the atoms, with oxygen being bigger than hydrogen. It should also be noted that while some entities and parameters are “observable” (but with an electronic microscope), others can only be measured by indirect methods. We thus remain only partially within the primitive paradigm of an empirical map, in the sense of recording a reality that is “directly” observable. On the other hand, an eventual mapping of atomic or subatomic particles will no longer have much in common with an immediate empirical map since it will necessarily require mediation, typically that of measuring instruments.

14Let’s emphasize two points. The illusory empirical evidence associated with the ordinary notion of a map is likely to numb the critical mind or keep it asleep. Like all representations, a map can be said to be fictitious in several senses, either in its entirety (imaginary maps) or in part, mixing existing and non-existing objects, observable or non-observable, independently of other types of error. Correspondingly, the proportion of observable (as opposed to non-observable) objects among the symbolized ones can vary significantly according to the kind of cartography considered. Thus, the abstract character generally ascribed to mathematical objects easily suggests that cartographies in this field are devoid of empirical anchorage and, as such, can be considered theoretical, imaginary, constructed, agreed upon, etc. Nevertheless, even if it mobilizes less the eye of the body than the eye of the mind, cartography remains, at least metaphorically, visual and spatial.

1.2. Two major functions

15Following Goody, we can say that a map is generally intended to be shared and to have more than one use. However, it has two general types of function: synthetic and analytical-didactic.

16The synthetic function refers more to the activity of the map’s author or producer than to that of its user. In fact, a map not only sorts and selects but also groups and coordinates a plurality of symbols unified in this way. We will probably not speak of a map, but rather of a mere drawing or picture, to mean something that is an illustration of a whole without distinguishable parts or elements. In this sense, a map, a diagram, or a scheme distinguishes and connects, creating a synthesis that elaborates a product of an art.

17The analytical-didactic function, on the other hand, is more concerned with the user’s activity. Like writing, cartography has the notorious advantage of being a lasting device: by keeping track, it can act as an aide-memoire. But it would be doubly wrong to claim that a map can only contain what has been deliberately incorporated into it: beyond its potential function as a memory aid, we can sometimes learn something by analyzing it.

18Despite the wear and tear of time, the creator of a map may not have forgotten what he put on it, and an experienced user may use it to revise a familiar route: neither will learn much from it. On the other hand, a novice explorer will expect something new, perhaps just the answer to one of his questions. This confirms the possibility of the immediate empirical value typically associated with a map. Even if it is true that all perception involves inferential processes, on a map, we “see” a situation, a global formalism, before possibly inferring the answer to the question asked (Where am I? How do I get there? etc.). In this way, sharing a map assures a didactic function that it would not, in principle, have for its author if his memory were infallible. However, it can also have unexpected heuristic value prompted by observation. For example, by raising the question of the number of possible routes between two cities, just as the observation of an octagon can lead to ask the value of the interior angle between two adjacent sides.

19It would be a mistake to think that even the map’s creator is omniscient. In this field, as in others, the richness of the work can exceed that of the artist. As the example of the octagon suggests, a study can reveal regularities, structural elements such as symmetries, or analogies that escaped the creator at the design stage. In this way, the informational resources of a cartography can exceed those originally intended.

2. Mapping arguments

20Given the commonplace use of cartographies and other iconic representations in most scientific disciplines, why should we be surprised to find them in the field of argumentation studies? Indeed, the foregoing points lead us to expect a plurality of “argumentative” cartographies.

21Even if the practice of argumentation is lost in the mists of prehistoric times, its study is sometimes considered recent, despite a renewed interest in the theories and practices of antiquity that motivates numerous borrowings. The revival of the field in 1958, with the two seminal works by Toulmin (The Uses of Argument) and by the Perelman-Olbrechts-Tyteca tandem (The New Rhetoric. A Treatise of Argumentation), helps explain a renewed interest in argumentative cartographies. Indeed, with the proliferation of new forms of graphical representation since then, especially in the wake of Toulmin’s famous diagram and mainly conveyed through English-language textbooks inspired by the Critical Thinking and Informal Logic movements, this (mainly didactic) use may have given an impression of novelty, perhaps less in principle than in content.

22A large proportion of people involved in contemporary argumentation studies are English-speaking philosophers, often from the “analytic” tradition, whose interest in logic is well known. It is hardly surprising, then, that the ontology of most of the new argumentation mappings, even after distancing themselves from some formal approaches, are still based on the two fundamental pillars of logic: proposition and inference. Indeed, informal logic maintains a central role in the concept of argument, understood as a premise-conclusion system. Thus, in the cartographies favored by this approach, it is not surprising to find two symbolic figures, respectively associated with the proposition and the inference.

2.1. Traffic between writing and mapping an argument

23What does it mean to map an argument? To draw a logos, to re-present a discourse. But re-presenting a discourse by playing with words can be an acceptable definition of writing, especially if it involves transcribing a thought or ideas, as we commonly say.

24As noted above, Goody emphasized the prototypical character of the list in the invention of writing, mainly through its development in the two dimensions of a plane. Now, an argument with several premises has long been presented, rewritten, or mapped in the form of a list, tolerating several variations and a possible continuum towards a form that can be considered more indisputably cartographic, as already suggested by the variations on the theme of H2O borrowed from chemistry. For example:

.

25Faced with these different versions of the same argument, the principle of a clear distinction between writing and cartography may prove difficult to justify. What makes version V, for example, more (or less) graphic than IV or VI?

26The use of specific non-alphabetic symbols is often taken as one of the distinguishing criteria of “symbolic” or “formal” logic. But even here, the simple case of the arrow (→) can produce confusion. First, its use is not typical of logic or mathematics, unlike a symbol such as the universal quantifier. In general, we know how to interpret this kind of arrow: as the expression goes, it invites us to follow it. In standard logic, the script “pq” reads as “If p, then q,” which is an expression established for the type of proposition known as “conditional” or “hypothetical.” It is also sometimes used as a sign of inference or of “implication” in a sense that will not be defined here, given the numerous debates around this notion since antiquity. If the arrow is interpreted as a symbol of inference, we are already very close to its meaning in most contemporary cartography.

27In order to move from the register of writing to that of mapping, is it sufficient to move from horizontal writing, such as p → q, to a vertical one as:

p

q?

28This grey area between everyday writing, symbolic (or formal) writing, and cartography raises, above all, the tricky question of the modalities of our inferential activity in the face of these different modes of representation. Is it comparable or even the same when stimulated by mapped reasoning, read reasoning or heard reasoning, all following the same course and leading to the same conclusion? The question arises with sharpness in relation to the nature and status of what might be called a “logical scheme”, a notion assumed to be general because it can be exemplified by various kinds of cartography as well as by the vertical stacking of lines that represent it in writing. This is not a new question, especially when it comes to the presentation of a single argument or the whole chain of arguments that make up a reasoning that is supposed to induce conviction, which in turn is supposed to go hand in hand with its evidential value, as in mathematics or mathematical physics (Brown 1996; Aberdein and Dove 2012).

29Identifying the presence of premise(s) and conclusion(s) can be considered a necessary or minimal condition for argumentation (Dufour 2024; Siegel 2023, 2024), although an argument is rarely explicitly expressed in a reduced premise-conclusion form. Moreover, an argument can be said complex or compound to mean that at least some of its premises are justified and thus have at the same time the status of conclusions of “sub-arguments” whose premises can be said to be of “second-order.” In antiquity, the overall structure resulting from this regression of reasons was one of the possible meanings of the term epicheireme (Plantin 2018). As with a map, we can imagine stepping back, “zooming out,” and thus not limiting cartography to the contours of an elementary argument but visualizing an epicheireme or a set of arguments that share the same conclusion, in other words, what I mean by a “developed argument”. As can already be seen in many textbooks, the scope of an argumentative cartography can thus go beyond an elementary argument by depicting the different levels, stages, or steps of a developed argument. The following diagram gives an example where, in addition to propositions and conclusions, specific graphic elements are introduced to delimit not only the main argument (in the rectangle) but also possible sub-arguments (in the ellipses) and even the whole developed argument (in the peripheral hexagon).

30Of course, there is no obligation to include these demarcations. Several of the papers in this issue do so, using a variety of graphic representations to make certain stages of the argument more obvious.

  • 5 See Jacquette (2011) for other modes of symbolization and visualization.

31As we have seen it when discussing the concept of map, certain symbols can be integrated (or discarded) to introduce (or eliminate) other information. For example, in the previous figure, nothing indicates the relationship between premises of the same level, such as s and t. But we can choose to include symbols to stress it. This is a familiar choice in geography, where cartographers prioritize information on the map by using or not using a particular symbol. In our example, if s and t are “linked” rather than “independent” premises, we can indicate this by adding the symbol “&” between them, its absence or a simple comma meaning that they are independent5.

32We can go further and try to map, for example, all the arguments made by a speaker during an hour-long exchange with several interlocutors on different topics, some of them argumentative. In this way, the developed arguments of the speaker’s entire discourse could consist of several isolated hexagons or rectangles, in line with the previous model.

33Another extension is to map an argument on the model of ancient dialectics, which requires a further change of scale since it operates on a broader level than that of a single developed argument. In this way, we can consider the processual dimension of a dialectical exchange by distinguishing the argumentative contributions of the different interlocutors or parties involved (Yoshimi 2004; Harrell 2022). To this end, it is possible to play with the overall layout and/or enrich the legend, for example, by introducing notation to symbolize interactions between arguments, such as negations, doubts, or refutations. As with a geographical map, certain aspects can be privileged over others, depending on the objectives chosen a priori and, of course, without deciding about unforeseen aspects that may be discovered thanks to such cartographies.

2.2. Descriptive synthesis vs. evaluative analysis

34A remarkable feature of the argumentative cartographies considered so far is their descriptive character, post hoc, except for the implicit premises that may be introduced as part of a logical “reconstruction”. These cartographies just graphically re-produce a previous discourse, whether it is limited to an argument or extends to the scale of an exchange, especially a controversy. In this way, we remain within the perspective that we have called synthetic. Nothing is said or shown about the value, in particular the logical value, of the mapped arguments: if these cartographies are to be strictly empirical, they mainly note that such and such a point of view is claimed to be supported by such and such a premise. However, informal logic, which is concerned with the actual practice of argumentation, usually distinguishes between the identification and evaluation of an argument, even if this distinction is not always clear. From this point of view, the cartographic variants above are typically based on the perspective of a survey, but a survey that presupposes the identification of arguments. However, different cartographies can be put at the service of different functions, not necessarily or exclusively descriptive. This is the case with formal logic, which is essentially normative, where the formal or symbolic character of the graphic authorizes and conditions the evaluation of the argument. It is not just the presentation of an argument or set of arguments that is expected, but an answer to the question: “What are they worth?” Here we are in a domain that particularly defies the boundary between writing and cartography, and we “see”, literally and figuratively, what a well chosen mode of formatting shows (and demonstrates), to such an extent that a machine can exploit this cartography and decide “blindly” on the correctness of a conclusion supposed to “visibly” impose itself. This kind of conventional metaphor, even if banal, has long been evidence of the kinship between visibility and the recognition of the correctness of a line of reasoning.

35The demonstrative power of certain cartographies can be illustrated by the most straightforward example: “I like it because I like it.” Such a circular argument has two remarkable properties. The first is that it has little epistemic value: it does not teach anyone anything except, sometimes, in a derivative way, by drawing on resources left implicit. But it is correct from a logical point of view: it is (deductively) valid since its conclusion cannot be false if its premise is true. When the arrow is used as a derivation symbol, it can be “formally” written or mapped as p → p, where p represents the proposition “I like it.” Here is a similar example, where the symbol V symbolizes the disjunction “or”: p → (p V q). By simple inspection (and some prior knowledge of the disjunction) we can evaluate the argument. This type of representation is at the heart of the “tree method”, a graphical evaluation procedure familiar to logicians (Vernant 2001; Lepage 2001). It should be remembered, however, that the mere possibility of symbolizing, formalizing, or mapping an argument does not say anything about its a priori value: it is not enough, even if it can sometimes contribute to its determination.

2.3. A few historical antecedents

36Contemporary argumentation studies have developed cartographic practices, particularly from a pedagogical perspective. In recent decades, studies of argument mining have flourished, i.e. the automatic search for or reconstruction of arguments from natural language discourses. These works exploit cartographies for various purposes (Lawrence and Reed 2019; Reed, Walton and Macagno 2007; Sbarsky et al. 2008). While some aspects are innovative, others are part of a long tradition.

37Recent historical research into graphical representations in argumentation quickly led to Monroe Beardsley’s Practical Logic (1950), then to The Principles of Judicial Proof (1913), where John Henry Wigmore attempted to reconstruct the progression to the final decision of all arguments presented in a court case (Goodwin 2000). But in history, as elsewhere, the direction and outcome of a search generally depends on what we are looking for, on the initial problem. And now, two already discussed aspects of cartography become important. Firstly, the resolution and extension of the map: are we interested in a single argument, a developed argument, or even a whole debate without any particular limits of space and time? Secondly, in terms of the distinction between identification and evaluation, is the goal simply to describe or reconstruct a particular statement, passage, or exchange, or does it go so far as an evaluation from a logical or other point of view? You will probably grant that, in practice, an evaluation presupposes at least a provisional identification, even if we must return to it to decide on implicit elements that escape a cartography intended to be strictly empirical by sticking to the manifest discourse. Thus, the empirical approach of the historian or linguist, who wishes to record all the argumentative moments of an exchange, can stand alongside the evaluative approach of the lawyer or logician, who wishes to distinguish the decisive elements from those which, in his eyes, have no demonstrative value. Their cartographies may, therefore, diverge.

38In the case of Wigmore, we can guess the aim of this late 19th-century American academic and jurist. Goodwin (2000) confirms that his graphic method had an evaluative purpose. Based on the description of the articulation of all the arguments exchanged, it was intended “to help lawyers analyze the evidence in a trial.” (p. 223) The scope and purpose of the cartography are thus well defined without prejudging the actual outcome. In fact, unlike other mappings, particularly logical ones, Wigmore’s does not claim to be able to reach a conclusion about the case under consideration simply by inspecting the cartography: the aim is to help lawyers take stock, not reach a conclusion in the judge’s place. It should be noted that this cartography fulfills at least the two valuable functions of aide-memoire and argumentative synthesis of the trial but that it can also draw attention to a neglected but crucial aspect of the problem.

  • 6 Although sometimes still attributed to Aristotle, its authorship should be attributed to Apuleius, (...)
  • 7 This bestseller has been regularly republished. We quote from the 1802 version available online. Th (...)

39The project of mapping for evaluative purposes is older than that of description. Indeed, Wigmore’s contribution – on the condition that we accept a more limited and, above all, non-dialectical description – can be largely predated by the graphic representation of the famous “square of oppositions” or “logical square,” schematizing the logical relations of the four propositions: “All A is B,” “Some A is B,” “No A is B,” and “Some A is not B”6. An illustration of these four elementary propositions, and of the syllogisms or pseudo syllogisms that can be constructed from them, can be found, much later, in the Lettres à une princesse d’Allemagne (Letters to a German Princess), published in French by the German mathematician Leonard Euler between 1768 and 1772 (Euler 1802)7. He himself was probably inspired by Leibniz's graphical innovations in reasoning and evaluation, notably in his essay De formae logicae comprobatione per linearum ductus, where he uses in parallel a first original two-dimensional diagram evoking both a “tree” and a “list,” and a second one using the circular forms popularized by Euler (Leibniz 1903: 292-321). In the first type, the horizontal segments represent the concepts (e.g. “to be B,” “to be C,” etc.), their length symbolizing their extension. Their logical relations (inclusion, disjunction, intersection) are represented by their vertical alignments (total, absent, partial): if all Bs are C, the vertical projection of the B segment is thus totally included in the C segment.

40The second type of Leibniz graph is the one popularized by Euler, who presents the principle as follows: “As a general notion contains an infinite number of individual objects, we may consider it as a space in which they are all contained. Thus, for the notion of man we form a space in which we conceive all men to be comprehended.” (in Euler I, 102: 396). This space is represented by a circle marked with a letter. From here, cartographies are introduced that use such circles to illustrate various elementary propositions that express relations of inclusion or intersection between concepts. Euler then illustrates syllogistic relations. For example, the famous Barbara-form (“All A is B, all B is C; therefore, all A is C” argument) is mapped as follows:

41In the following century, the English mathematician John Venn took up the “Euler circle” representation and generalized it, by standardizing the use of non-circular contours. Today it is more common to speak of “Venn diagrams” to designate this type of cartography, another variant of which was introduced by Lewis Caroll, his compatriot and contemporary. Commonly used in mathematics and logic, these diagrams have been widely adopted in argumentation textbooks (Govier 1992; Ennis 1996; Dufour 2008).

42Euler has not gone so far as to represent a developed argument, a fortiori a controversy: he works on a small scale, that of a single argument of the syllogistic type, and from an evaluative perspective: his aim is to show (in both senses of the term) whether this argument, or more generally its type, is probative, “valid” in the logician’s sense. Referring to the nineteen logical patterns he lists and illustrates, Euler writes: if the two first propositions, or the premises, are true, you may rest, confidently assured of the truth of the conclusion” (I, 105: 407). Hence the possibility of extending their use to the detection of fraud. What about the following argument borrowed from Euler? Correct or fallacious?

Some learned men are misers;

But no miser is virtuous;

Therefore, some virtuous men are not learned (I, 105: 409).

43Let’s schematize the situation with the following diagram where S symbolizes the learned (scholars), A the avaricious, V the virtuous:

44The truth of “Some learned men (scholars) are misers” is symbolized by the overlap of S and A, their intersection translating the existence of (some) learned misers, in other words, the truth of the proposition “Some learned men are misers”. The truth of “No miser is virtuous” is symbolized by the disjunction of A and V. As Euler rightly points out, the conclusion “Some virtuous men are not learned (scholars)” could be true because, according to the second premise, V could partially overlap S without contact with A. But this possibility is not required by the premises proposed. So, the diagram demonstrates that it is possible for the premises to be true but for the conclusion to be false and, thus, for the argument to be invalid.

45The value of such a diagram undoubtedly lies more in its analytical, heuristic, and even pedagogical power than in its descriptive accuracy. Is it merely a decision-making aid, as in Wigmore’s case? This is not what Euler implies, for the diagram imposes the right conclusion:

These circles, or rather these spaces, for it is of no importance what figure they are of, are extremely commodious for facilitating our reflections on this subject, and for unfolding all the boasted mysteries of logic, which that art finds it so difficult to explain; whereas, by means of these signs, the whole is rendered sensible to the eye (I, 103: 397).

46Against the objection that to understand a demonstration, one has to go through syllogistic structures, which are repulsive because they are foreign to ordinary speech, Euler argues that a logical structure is accessible in other ways, even through natural speech. In a letter to the princess, he concludes:

It is not necessary, however, that our reasonings should always be proposed in the syllogistic form, provided the fundamental principles be the same. In conversation, in discourse, and in writing, we rather make a point of avoiding syllogism (I, 105: 410).

47In a savory way, Euler recognizes the virtues of simple language, accessible to an innocent audience. But this simplicity also allows for fallacious argumentation, the denunciation of which requires us to subject what is “sensible to the eye” to the sieve of reflection in order to ensure “that the fundamental principles be the same.”

48Before returning to this interplay between what is “sensible to the eye” and possible illusions, let’s point out the contribution of the logician and Anglican Archbishop Richard Whately. In Appendix III of his Elements of Logic, first published in 1826, we find a modest footnote (2008: 380) presenting a diagram relevant to today’s argumentative cartographies but also to the “trees” of logic. Although no longer based on containers such as the circles of the two German mathematicians, this diagram remains in syllogistic perspective since it directly integrates elementary predicative propositions such as “Y is X,” but where Y and X can occasionally denote propositions. The set of recorded propositions is then arranged as an “inverted tree”, i.e with the root at the top, which Whately describes as the “ultimate conclusion.” While Euler remains at the level of a single syllogism, Whately’s tree allows for the representation of multiple logical levels or stages, thus allowing for the representation of a line of reasoning, an epicheirema, called an “argument chain.” However, the whole remains monological in that the tree does not, as in Wigmore’s, represent opposing arguments, including refutations. Differences in logical levels are represented on the vertical axis. If “Y is X” is found at level (or line) n, the premises or systems of premises that might justify Y (or X) are found at the same horizontal level just below (n-1), connected by a horizontal line underlining them. Conclusions are represented by ascending vertical lines (branches). With its stratified, vertical list-like appearance, this system is reminiscent of Leibniz’s method of representation by segments.

49Whately too relies on the pedagogical virtues of this tool. His diagram is preceded by the following comment: “Many students probably will find it a very clear and convenient mode of exhibiting the logical analysis of a course of argument, to draw it out in the form of a tree, or logical division.” It’s also worth noting that, unlike Euler’s circles, the textual content of the argument or line of argument is transferred to the tree. Thus, the evaluation of the logical whole is not based on visual considerations of its codified graphical structure but on the meaning of the text.

50Schopenhauer’s interest in Euler’s circles dates from 1810-1820, a few years before Whately’s book and The Art of Being Always Right, his own posthumously published opuscule on what he calls “eristic dialectics.” In particular, he takes up the idea that a diagram can show both the correctness or the weakness of a conclusion and, thus, of an entire argument. From the perspective of an eristic dialectician, Schopenhauer’s primary concern is to highlight fallacious biases. According to Lemanski (2023), this project was even systematic. Indeed, contrary to the impression of fragmentation that The Art of Always Being Right may give with its list of stratagems for having the last word, the project was based on the reductionist principle that “The whole of eristic is based on the fact that one confuses the logical perspective with the eristic perspective, resulting in false RDs [relational diagrams]” (Lemanski 2023: 79). For example, a common eristic abuse would be to occasionally substitute the quantifier “all” for the quantifier “some”, i.e. “All A’s are B” for “Some A’s are B” or, in graphical terms, to substitute the first of the following patterns for the second:

51So, taking Euler’s example as a starting point, the statement “Some misers are learned, and some learned people are virtuous,” which allows no conclusion about the virtue of misers, is replaced, deliberately or not, by “Some misers are learned, and all learned people are virtuous,” which shows that some misers are virtuous. The diagram thus makes “sensible to the eye” an exaggeration of the false argument, playing on a common ambiguity of the French article “les” in an expression like “Les avares sont savants” (misers are learned), where it can be understood as universal (without exceptions) as well as general (with exceptions considered rare enough to be negligible).

52In short, Schopenhauer may have initiated the dialectical opening of the use of “Eulerian” cartographies to arbitrate controversies, including the denunciation of illicit maneuvers. Following in the footsteps of Leibniz and Euler, his project is not limited to describing a structure but aims to evaluate the arguments involved.

3. Some of the educational challenges of cartography

53Despite the vagueness of the concept of cartography, what this term usually covers today in the field of argumentation seems to be quite limited and easy to recognize or circumscribe because of recurring features and problems. Firstly, unlike most of the historical cases presented here, whose aim was primarily evaluative, most contemporary cartographies have an initial descriptive and identification purpose, facilitated in recent decades by the use of specific software (Araucaria, MindMup, Rationale, Argument Diagramming, etc.). These are generally available online and have the advantage, especially in a didactic context, of making easier to produce, reproduce, correct, and erase than when using a map drawn by hand. These advantages resulting from the power of the machine are conducive to rapid, large-scale cartographic description. Beyond the simple argument, we can easily imagine mapping an entire argument or exchange of arguments. A correlated disadvantage is the need for increased learning time.

54It should be noted, however, that despite all these advantages, the use of machines is not an obligation since most of the pedagogical benefit of a cartographic mode of representation is perhaps to be found on a small scale, in the assimilation and recognition of elementary logical aspects, and then in the ability to translate them into the visual register (Harrell 2008; Twardy 2004).

  • 8 As we said about empirical illusion at the beginning of this chapter, from one territory more than (...)

55This emphasis on schematizing description as a prerequisite for synthetic reconstruction8 when an “ordinary” argumentative discourse is taken as a starting point, goes hand in hand with three main pedagogical objectives, which, especially in English literature, are themselves often subordinated to the broader and perhaps less clear-cut objective of developing “critical thinking.”

56Firstly, it is important for the student to be able to identify the argument(s) of a given discourse (typically a text) presumed to be argumentative, which presupposes the recognition of a minimum of typical elements, at least an associated reason and conclusion. But beyond this minimum, the challenge is often to identify the whole argumentative movement, the arguments and their sub-arguments, or even the whole argumentation in the case of a polyphonic text. This is often a difficult exercise, even at the level of an elementary argument, especially in the case of an authentic text “in the wild,” in the sense of not being designed for the pedagogical occasion. Such a text is likely to pose additional challenges, for example, because it is confusing or does not respect syntactic rules.

57A second objective, linked to the previous one, is a little more general and abstract since it consists of becoming familiar with recurring argumentative features and thus developing expertise in recognizing a general structure, which goes hand in hand with the ability to distinguish between types of argument.

58Finally, a third objective is to mobilize these skills beyond the analysis of specific cases, in particular when it comes to producing new argumentative discourses. Starting from an argumentative map without text (except propositional symbols), a possible test consists of asking students to write an argumentative discourse based on it. In this way, cartography can be exploited by moving from the discourse in natural language to the map or vice versa.

59These objectives benefit from the use of cartography. However, the specific contribution of a type of representation is not clear from the studies carried out to date, even if they highlight its contribution to pedagogical success. In fact, most of the recent empirical studies have focused on the general cognitive benefits. Despite the popularity of the clarification or economy of thought promoted by a well-conceived graphic presentation, a large number of parameters interfere in the evaluation of the pedagogical use of cartographies. It is, therefore, difficult to distinguish the specific contribution made by one form of presentation from another.

60However, there is evidence of a relative increase in the skills of students who have benefited from the use of common argumentative cartographies, both in terms of their ability to analyze and critically evaluate texts (Twardy 2004; Harrell 2011, 2012; Cullen et al. 2018) and even in terms of writing argumentative texts (Harrell 2013). Some of these studies, mostly statistical, compare the test performance of a group trained in specific cartographic techniques with a control group that has not been trained. Others compare the results of instruction using cartography with instruction limited to text-based resources (Pinkwart et al. 2008). These studies highlight the impact of other parameters that may themselves be interdependent, such as the population concerned, the initial general level of education (maps are said to benefit the least advanced students), the assessment criteria used, and the subject area concerned. Questions are raised about the impact of the type of cartography used, especially its symbolic resources, and about the capacity of a particular mode of representation to cover the full range of argumentative registers, especially those related to meta-arguments or specific disciplinary fields (Rathkopf 2024). However, the advantage of cartographic pluralism, or at least of an enrichment of the iconographic repertoire, seems obvious.

61All in all, the use of cartographies seems to be generally advantageous in pedagogical exercise and can provide non-negligible punctual or specific contributions. However, it is still difficult to assess the cognitive effect of mapping an argument or a set of arguments, especially as the notion of mapping remains unclear, despite the persistent belief in a clear contrast between the discursive written register and the iconographic register, which are often intertwined in argumentative cartographies. In terms of argumentation pedagogy, this reflection invites us to use the various cartographic resources available, in conjunction with others, in order to avoid the banal illusion of transparency or immediacy of the meaning of the image.

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Notes

1 On the organisation of the lists and their variations, see also Sève (2010).

2 They are countless: business cards, credit cards, greeting cards, member cards...

3 Ironically, the modern atom was quickly “broken,” its fission distinguishing it from the ancient one. The result was nuclear energy and the revival of the ancient debate, beyond the equivocation of “atom.”

4 Time-honored expressions in chemistry.

5 See Jacquette (2011) for other modes of symbolization and visualization.

6 Although sometimes still attributed to Aristotle, its authorship should be attributed to Apuleius, a philosopher, rhetorician, and writer of the second century AD. Hence, the name ‘Square of Apuleius’.

7 This bestseller has been regularly republished. We quote from the 1802 version available online. The letters in question (Part I, letters 103-105) deal with the perfection of languages and then with the syllogism.

8 As we said about empirical illusion at the beginning of this chapter, from one territory more than one map is in principle possible.

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Michel Dufour, « Draw me an argument! What for? Graphs, maps, and diagrams in the study of argumentation »Argumentation et Analyse du Discours [En ligne], 34 | 2025, mis en ligne le 10 avril 2025, consulté le 21 avril 2025. URL : http://journals.openedition.org/aad/9276 ; DOI : https://doi.org/10.4000/13q0g

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Michel Dufour

Sorbonne Nouvelle University (France)

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