Microsoft Word - Yu Zenker - 363-387.doc © Shiyang Yu and Frank Zenker. Informal Logic, Vol. 42, No. 2 (2022), pp. 363–387. Identifying Linked and Convergent Argument Structures: A Problem Unsolved SHIYANG YU College of Philosophy Nankai University Tianjin P.R. China FRANK ZENKER Faculty of Administration and Social Sciences, ICFO Warsaw University of Technology 00-661 Warsaw, Poland frank.zenker@pw.edu.pl Abstract: To analyze the argument structure, the linked vs convergent distinction is crucial. In applying this distinction, argumentation scholars test for variations of argument strength under premise revision. A relevance-based test assesses whether an argument’s premises are individu- ally relevant to its conclusion, while a support-based test assesses whether premises support the conclusion independently. Both criteria presup- pose that evaluating an argument’s strength is methodologically prior to identifying its structure. Yet, if ‘argument structure’ is a concept of analysis, then a structural analysis would precede evaluating an argu- ment’s strength. We problematize that state-of-the-art methods to identify structures fail, because they rely on evaluative judgments, and so “put the cart before the horse.” Résumé: Dans cet article, j'adopte un cadre pluraliste sur l'argumentation, où les normes qui dirigent la construction et l’évaluation de l’argumentation dé- pendent du but de notre engagement dans cette pratique. Un domaine d'argu- mentation spécifiquement épistémique est distingué, et je soutiens, sur la base de découvertes récentes en épistémologie modale, que ce domaine est dirigé par la norme modale de sécurité, selon laquelle une croyance est sûre juste au cas où elle serait produite par une méthode qui ne produirait pas facilement une fausse croyance. Bien que ce critère soit bien connu et non controversé en épistémolo- gie, il n'a jusqu'à présent pas été appliqué aux théories épistémiques de l'argumen- tation. Je montre la fécondité d'introduire cette norme modale dans notre théorie de l'argumentation en soutenant que cela permet une perspective nouvelle et supérieure sur la pertinence de l'interlo- cuteur persistant dans la théorie de l'argumentation, et plus généralement sur la relation entre les normes dialectiques et épistémiques. Keywords: argumentation, convergent, dependence, linked, relevance, structure 364 Yu & Zenker © Shiyang Yu and Frank Zenker. Informal Logic, Vol. 42, No. 2 (2022), pp. 363–387. 1. Introduction As the three central research questions for the study of argument structure, A. Francisca Snoeck Henkemans (2001, pp. 101f.) iden- tifies the following: (1) Definition: how to define argument struc- ture and its types? (2) Analysis: how to identify the structure of a specific argument? (3) Intellectual history: how did the concept ‘argument structure’ and its typologies develop? Other than by briefly reviewing scholarly approaches (Sect. 2), here we mostly neglect (3). Since a reasoned view on (2) grounds in a reasoned answer to (1), we treat (1) as the most fundamental question. We claim that the distinction between linked and convergent argu- ment structures remains unclear today because (1) is yet to be answered satisfactorily. The concept of ‘argument structure’ is complementary to the concept of ‘argument scheme.’ Whereas an argument structure “characterizes the ‘external organization’ of the argumentation” (van Eemeren et al. 2014, p. 21), an argument scheme “defines […] how the ‘internal organization’ of the argumentation is to be judged” (2014, p. 19). Argumentation scholars who theorize such structures on a logical approach tend to highlight the goal of “de- termin[ing] whether the premises constitute good reasons for accepting the conclusion, good in the sense of transferring the acceptability of the [accepted] premises […] to the conclusion” (Freeman 2011, p. 109). The focus thus rests on the structure of the argument-as-product, itself a simplified static representation of the dialectical process of arguing. Scholars who pursue a dialecti- cal approach, by contrast, highlight the functions that argumenta- tion structures fulfill in the process of argumentation (Snoeck Henkemans 2001, p. 101). Here, “the focus of interest concerns how well a critical discussion has come to a reasoned resolution of some disputed question” (Freeman 2011, p. 109).1 1 Use of the term ‘reason’ occurs on the understanding that pragma- dialecticians use ‘argument,’ as opposed to ‘standpoint,’ in ways that roughly correspond to how other scholarly traditions use ‘reason’ or ‘premise,’ as opposed to ‘conclusion.’ Identifying Linked and Convergent Argument Structures 365 © Shiyang Yu and Frank Zenker. Informal Logic, Vol. 42, No. 2 (2022), pp. 363–387. As argumentation scholars today seek “to provide theoretical instruments for analyzing, evaluating and producing argumentative discourse in an adequate way” (van Eemeren 2018, p. 5), the main theoretical approaches use different labels to denote argument structures (Fig. 1). Informal logicians, for instance, distinguish serial, linked, and convergent structures, while pragma- dialecticians speak of subordinative, coordinative, and multiple structures. These terms, however, fail to entail a substantial difference (Snoeck Henkemans 2001, p. 101). We adopt the former terminology. Fig. 1 Three argument structures, with the top-most node representing the conclusion, and the other nodes the premises. The central distinction is that between a convergent and a linked structure. Drawing this distinction is what Geoff Goddu rightly calls “the problem of structure” (2007a, p. 11; his italics). Multiple premises of a convergent structure (i.e., convergent prem- ises) are interpreted logically as alternative lines of support, or dialectically as alternative lines of defense, for the standpoint. In a linked structure, by contrast, multiple premises provide only a single line of support/defense. In a serial structure, finally, a single 366 Yu & Zenker © Shiyang Yu and Frank Zenker. Informal Logic, Vol. 42, No. 2 (2022), pp. 363–387. premise provides a single line of support/defense for a conclusion, where this premise is itself a conclusion supported by another line of support/defense. The serial structure thus amounts to a hierar- chical arrangement of a single line of support/defense that involves an intermediate conclusion. Virtually all structural analyses recur to the mutual in- /dependence or ir-/relevance of the premises. Roughly, if an argu- ment’s premises are independent of other premises, or are irrele- vant to the conclusion, then the argument instantiates a convergent structure, otherwise a linked structure. Our main claim is that judgements of premise-dependence or -relevance that inform a structural analysis depend on evaluating the argument’s compara- tive ability to transfer the acceptability of the premises to its con- clusion. This is what creates the problem of distinguishing linked from convergent structures in the first place. Also known as argument strength, the transfer of acceptability is modelled as a function of what scholars variously call the argu- ment’s justificatory force, its weight, or the degree of support that premises lend to the conclusion. Since the identification of argu- ment structure thus relies on evaluating premise in-/dependence and ir-/relevance, analyzing the support that premises lend to a conclusion requires evaluating the contents of premises and con- clusion, and how these contents relate to each other, as well as to additional, possibly contravening information. A structural analy- sis, however, is thought to occur before evaluating argument strength: argument analysis is preparatory to argument evaluation (Freeman 2011, p. 141; van Eemeren and Grootendorst 1992, p. 95f.; Walton 1996, p. 79, p. 81). But to settle a structural question, analysts in fact recur (implicitly) to evaluative judgments, and so engage in argument evaluation (see Goddu 2007a, p. 19). This, we argue, defines ‘argument structure’ in the wrong way. This definition necessarily leads to problems in analysis, because matters of analysis depend on matters of definition. At any rate, the definition challenges the idea that ‘argument struc- ture’ is an analytic rather than an evaluative concept. We purpose- fully ignore the question whether the linked-convergent distinction Identifying Linked and Convergent Argument Structures 367 © Shiyang Yu and Frank Zenker. Informal Logic, Vol. 42, No. 2 (2022), pp. 363–387. is at all valuable.2 Our goal is rather to demonstrate that state-of-the- art methods by which to draw this distinction are ineffective because the definition of argument structure and its types is defective. We start by introducing a static logical and a dynamic dialec- tical approach to argument structure (Sect. 2). As the dynamic aspect of the dialectical approach is far from obvious, we hold that the current state of this approach amounts to yet another static approach (Sect. 3). We then turn to two ways of distinguishing convergent from linked structures using support-based and rele- vance-based tests (Sect. 4). Since both ways entail testing for variations of argument strength under premise revision, we claim that this has things backwards (Sect. 5). Our conclusions are in Sect. 6. 2. Logical and dialectical approaches to argument structure 2.1. The logical approach A logical approach pays attention “only to the structur[al] aspects of argument structure as they manifest themselves in the product of the reasoning process” (Snoeck Henkemans 2001, p. 101). Following Monroe Beardsley’s (1950) distinction between con- vergent, divergent, and serial structures, the relevant terminology nevertheless cites the term ‘support.’ This particularly invokes an evaluative aspect regarding the support that an argument’s premis- es lend to its conclusion. A convergent argument is defined as one where “several inde- pendent reasons support the same conclusion,” while in a diver- gent argument “the same reason supports several conclusions,” whereas a serial argument “contains a statement that is both a conclusion and a reason for a further conclusion” (Beardsley 1950, p. 19). What proves crucial here is that the support for the same conclusion is said to arise from reasons, or premises (Adler 2008), that are in a relevant sense independent. The first to distinguish a convergent from a linked structure seems to have been Stephen Thomas (1997 [11973]) (Snoeck Henkemans 2001, p. 108). For the linked structure, Thomas ob- 2 For a negative answer, see Goddu (2007b). 368 Yu & Zenker © Shiyang Yu and Frank Zenker. Informal Logic, Vol. 42, No. 2 (2022), pp. 363–387. serves that reasons provide inter-dependent support to a conclu- sion. “When a step of reasoning involves the logical combination of two or more reasons, they are diagrammed as linked” (Thomas 1997, p. 50; his italics). By contrast, “[w]hen two or more reasons do not support a conclusion in a united or combined way”—by which Thomas means that “each reason supports the conclusion completely [sic] separately and independently of the other”—then “the reasoning is convergent” (Thomas 1997, p. 52; his italics). Irving Copi and Carl Cohen similarly distinguish a convergent from a linked structure such that, in the former, “each of the […] premises supports the conclusion independently. Each supplies some warrant for accepting the conclusion and would do so even in the absence of the other premiss” (1990, p. 19; italics in origi- nal). In a linked argument, by contrast, the “premisses must work together to support their conclusion,” which is to say that premises “work cooperatively” (1990, p. 20). Robert Pinto and Tony Blair (1993) likewise distinguish “be- tween a ‘group’ of premises that together form one inference and ‘independent’ groups of premises which can be seen as parallel inferences to arrive at the same conclusion” (Snoeck Henkemans 2001, p. 112). Here, ‘dependence’ expresses that “the premises work in combination to support the conclusion,” and ‘independ- ence’ expresses that “the premises of each group are able to pro- vide their support without any help from premises in any other group make them independent of each other” (Pinto and Blair 1993, p. 77; see Snoeck Henkemans 2001, p. 112). Adopting this idea, Leo Groarke, Christopher Tindale and Linda Fisher (1997) improve on its formulation: “[l]inked premis- es work together. Taken independently, they do not support the argument’s conclusion. Convergent premises do not require each other, for they support the conclusion independently of the argu- ment’s other premises” (Groarke, Tindale and Fisher 1997, p. 35; italics added; see Snoeck Henkemans 2001, p. 114). Without denying the crucial role of premise in-/dependence for an analysis of argument structure, other scholars recur to the concept of ‘relevance’ as a supplementary factor to ‘dependence’. In discussing tree-diagrams as a means of mapping an argument’s logical structure, for instance, Ralph Johnson and Tony Blair Identifying Linked and Convergent Argument Structures 369 © Shiyang Yu and Frank Zenker. Informal Logic, Vol. 42, No. 2 (2022), pp. 363–387. observe for linked arguments that “two or more premises are relevant in combination,” whereas in convergent arguments there are “two or more distinct, independent grounds for a conclusion” (1994, pp. 36-38; italics added). Independence and irrelevance are thus associated with a convergent structure, while dependence and relevance are associated with a linked structure. Like Johnson and Blair (1994), Trudy Govier’s (2010) version of this distinction places relevance next to independence: “Linked premises can support the conclusion in the argument only when they are taken together; no single premise will give any support to the conclusion without the others. […] When the sup- port is of the convergent type, each premise states a separate rea- son that the arguer thinks is relevant to the conclusion. In these cases, premises are not linked and are not interdependent [i.e., in- dependent] in the sense that each one could support the conclusion without the others.” (Govier 2010, pp. 37f.; italics added) In sum, when distinguishing a convergent from a linked argu- ment structure—with the concept of relevance added, or not—the in- /dependence of the premises is central. We return to this in Sect. 4.1 and subsequently argue that premise-dependence fails as a useful criterion (Sect. 5.1). Let us first turn to the dialectical approach. 2.2. The dialectical approach Whereas a logical approach to argument structure focuses on the argument-as-product (as an abstract inferential object where rea- sons support a conclusion), the dialectical approach connects the concept of ‘argument structure’ with that of the ‘dialectical situa- tion.’ Here, it “depends on the antagonist’s doubts and the way the arguer [i.e., the proponent] attempts to deal with these doubts what the resulting structure of [the] argument will be” (Snoeck Henke- mans 2001, p. 119; see van Eemeren and Grootendorst 1984; 1992; 2004). Specifically, multiple (i.e., convergent) and coordina- tive (i.e., linked) structures are treated “as resulting from different types of defensive moves aimed at removing different forms of criticism” (Snoeck Henkemans 2001, p. 121; italics added; see Snoeck Henkemans 1992). 370 Yu & Zenker © Shiyang Yu and Frank Zenker. Informal Logic, Vol. 42, No. 2 (2022), pp. 363–387. Coordinative argumentation is a response to a criticism of sufficiency and can be neutralized in one of two ways. In a direct or cumulative defense, the protagonist adds at least one new rea- son; in an indirect or complementary defense, the protagonist refutes the antagonist’s counter-reason. In both cases, old and new reasons must be somehow combined, “because the arguer can only convince the opponent of the acceptability of the standpoint if [s]he succeeds in removing the opponent’s doubt or criticism regarding the sufficiency of the [entire] argumentation” (Snoeck Henkemans 2001, p. 121). In multiple argumentation, “the only connection between the arguments [or reasons] is that they are all advanced as a [separate] defence of the same standpoint” (Snoeck Henkemans 2001, p. 121). As before, there are two ways of offering a defense. The protagonist may “withdraw his [original] argument and undertake a new attempt to defend the standpoint” (pp. 121f.; italics add- ed); or “in anticipation of a possible non-acceptance of his argu- ment, the protagonist may advance a new argument […] moti- vated by the (potential) failure of a previous attempt” (p. 122; italics added). James Freeman similarly identifies argument structure with respect to the dialectical situation. In a convergent structure, “two or more premises are each independently relevant to the conclu- sion,” and each premise is “given to answer the question—Can you give me an additional reason?” (1991, p. 94). In a linked structure, “two (or more) premises must be taken together or are intended to be taken together to see why we have one relevant reason for the conclusion,” such that “at least one of [the] linked premises [must be] offered to answer the question—Why is that (the remaining premise or premises) relevant?” (1991, p. 94). This serves to distinguish two types of premise combination: “premises involving relevance combination are linked, while premises in- volving modal combination are convergent” (Freeman 2011, viii). In agreement with Snoeck Henkemans (1992), Freeman holds that “the modality qualifies the standpoint” by “express[ing] […] different levels of commitment to the proposition advanced by the standpoint” (2011, p. 120). For instance, ‘Socrates is certainly guilty of corrupting the youth’ and ‘Socrates is possibly guilty of Identifying Linked and Convergent Argument Structures 371 © Shiyang Yu and Frank Zenker. Informal Logic, Vol. 42, No. 2 (2022), pp. 363–387. corrupting the youth’ both “express the same proposition, but the standpoints taken with respect to the proposition are different in each case, since each involves a different degree of commitment to the proposition” (Freeman 2011, p. 120; see Snoeck Henkemans 1992, p. 110). With a modal combination, then, “[e]ach premise may give some reason for the conclusion, but their combined weight constitutes a stronger case” (Freeman 2011, vii). In sum, the logical approach targets the argument-as-product resulting from a process of reasoning or argumentation. Different structures are determined via the support-relation among premises and conclusion, i.e., whether reasons support the conclusion indi- vidually or jointly. The dialectical approach, by contrast, which targets the argumentative process, determines argument structures according to whether the reasons defending a standpoint against an antagonist’s doubts do so individually or jointly. 3. Argument dynamics We saw that a logical approach focuses on structures that manifest themselves in a product of reasoning or argumentation, whereas a dialectical approach focuses on structures that arise in the process of defending a standpoint against an opponent’s doubt or criticism. The term ‘argument[ation] structure’ thus refers either to the spe- cific arrangement of conclusion-supporting reasons in a static product, or to the constellation of defensive moves in a dynamic process that unfolds under an opponent’s critical pressure. In the pragma-dialectical theory, for instance, the purpose of identifying argument structure is to elaborate how such defensive moves contribute to resolving a difference of opinion on the merits (van Eemeren and Grootendorst 2004). Whereas the distinction between supporting a conclusion and defending a standpoint merely reflects a preferred theoretical perspective regarding the goal of offering reasons, the distinction between a static argument-as-product and a dynamic argumenta- tion-as-process is substantial, because extracting only the premises and the conclusion—as is typical for an argument-as-product— entails neglecting, indeed deleting, material that is constitutive of the argumentation-as-process. But the dialectical approach to 372 Yu & Zenker © Shiyang Yu and Frank Zenker. Informal Logic, Vol. 42, No. 2 (2022), pp. 363–387. argument structure fails to accomplish the task of representing the dynamic process of argumentation. To see this, consider the argu- mentation A-1 by Thomas (1986), as cited in Freeman (2011, p. viii). On the dialectical approach, an analyst would intuitively identify A-1 as a convergent structure, because the proponent seems to defend the standpoint against doubt or criticism using three independent reasons, R1-R3. A-1 An argumentation with a convergent structure (Freeman 2011, p. viii) [R1] His swimming suit is wet. [R2] His hair is plastered down. [R3] He is wearing swimming goggles. Therefore [Conclusion] He’s been swimming. What kind of dialectical situation might be associated to A-1? As per Freeman’s method of reconstructing the dialectical situa- tion, to connect any two convergent premises one may imagine the antagonist intermediately asking: “Can you give me an additional reason?” One can thus transform argumentation A-1 into the dia- logue D-1 between, say, Nancy and Tony. D-1 A dialogue reconstructed from A-1 Tony: “You see, he’s been swimming [conclusion], because his swimming suit is wet [R1].” Nancy: “Can you give me an additional reason?” Tony: “His hair is plastered down [R2].” Nancy: “Well, can you give me an additional reason?” Tony: “All right, he is wearing swimming goggles [R3].” In D-1, Tony initially forwards only R1, whereas R2 and R3 arise in response to critical pressure by Nancy. Were the dialogue D-1 to unfold as described, then two assumptions would normally hold. First, Nancy was insufficiently convinced by R1, and receiv- ing R2 did not change this. Otherwise, why would she continue to ask for an additional reason? Second, compared to R2 and R3, R1 deserves identification as Tony’s original premise, because R2 and Identifying Linked and Convergent Argument Structures 373 © Shiyang Yu and Frank Zenker. Informal Logic, Vol. 42, No. 2 (2022), pp. 363–387. R3 are forwarded only in response to critical pressure by Nancy. Compared to R3, moreover, R2 seems to enjoy a priority status because Tony offered R2 before R3. Neither assumption, however, can be readily justified, because the transformation of A-1 into D-1 requires information that is absent from A-1. Can one nevertheless readily identify the struc- ture of a dialogue? It turns out that one rather cannot, or so D-2 shows, where R3 is withdrawn under critical pressure. D-2 A dialogue where reason R3 is withdrawn (1) Tony: “You see, he’s been swimming [conclusion], be- cause his swimming suit is wet [R1], and swimming makes one’s suit wet [R2]. Moreover, his hair is plastered down [R3].” (2) Nancy: “But, actually, I saw his hair was dry.” (3) Tony: “Well… all right, my guess [R3 withdrawn].” Given Tony’s original utterance in (1), R1 and R2 instantiate a linked sub-structure, which, like R3, feature separate premises, whereas R1, R2, and R3 together instantiate a convergent struc- ture. So, if R3 is withdrawn in (3), then the structure constituted by R1 and R2 turns out to be linked. Now, is the structure in D-2 linked or convergent? The best answer, apparently, is that the structure has changed from a convergent to a linked structure. Notice that this issue is general, pertaining to any transformation of an argumentation into a dialogue. Other than withdrawing a reason, of course, arguers can make other changes, e.g., revising a reason or even a conclusion. To our best knowledge of the literature, the dialectical approach to argu- ment structure has so far failed to discuss these changes, leaving it unclear whether a dialectical analysis can adequately deal with argument dynamics. So, without denying the theoretical and ter- minological differences between ‘supporting a conclusion’ and ‘defending a standpoint against doubt or criticism,’ respectively between a static and a dynamic perspective, the current state of the dialectical approach to argument structure is virtually indistin- guishable from the logical approach. Both target a static argument- as-product. 374 Yu & Zenker © Shiyang Yu and Frank Zenker. Informal Logic, Vol. 42, No. 2 (2022), pp. 363–387. We now turn to ways of testing whether arguments-as- products feature linked or convergent structures. 4. Testing for linked and convergent structures 4.1. Support-based tests To distinguish a linked from a convergent structure, Douglas Walton (1996, pp. 119f.) lists five tests (T1-T5; here listed in a modified order).3 Generally, if the truth of an argument’s anteced- ent (comprising the premises) leads to the truth of its consequent (the conclusion), then the argument passes the test, resulting in a positive test-result; otherwise the test-result negative. A positive test-result indicates that the argument’s structure is linked, whereas a negative test-result indicates that it is convergent. In each case, the test-criterion is the effect exerted upon the support-relation if a single premise is considered false or is suspended (i.e., neither known to be true nor false). In this case, the conclusion receives insufficient support or no support at all. T1-T4 are binary tests, while T5 reports the test-result in comparative, yet vague terms (ordinal measurement level). Except for T-2, which Walton develops in analogy to T-1, T-3, and T-4, the other four tests draw on previous literature. T1 Falsity/no support If one premise is false, then the conclusion no longer re- ceives any support. T2 Falsity/insufficient support If one premise is false, then the conclusion receives insuffi- cient support. 3 To identify the structure of an argument, Walton’s pragmatic theory of argu- ment structure distinguishes four types of evidence: the argument type, textual evidence such as indicator words, contextual evidence on the purpose of the argument, and the test-result (Walton 1996, p. 152). Here we focus on the tests themselves. For a critique of the relations between the four evidence types, see Goddu (2007a, pp. 13f.). Identifying Linked and Convergent Argument Structures 375 © Shiyang Yu and Frank Zenker. Informal Logic, Vol. 42, No. 2 (2022), pp. 363–387. T3 Suspension/no support If one premise is suspended, then the conclusion receives no support. T4 Suspension/insufficient support If one premise is suspended, then the conclusion receives in- sufficient support. T5 Degree of support If the joint strength of the argumentation is much greater than if each premise is considered separately, then the argu- ment has a linked structure. T1, called the Copi-Cohen test (Copi and Cohen 1990, p. 20), shall not only indicate “whether the premises ‘work cooperatively’ or ‘independently’” (Walton 1996, pp. 109f.), but also “whether each [premise] is absolutely needed for the other [premise(s)] to provide any support at all to the conclusion” (1996, p. 111). By weakening the antecedent-condition—from a premise being false to being suspended—one obtains T3, called the Freeman test (Freeman 1988, p. 178). T3 tests for non-zero support by asking whether, “if we suspend the one premise, does the other give any reason at all to support the conclusion?” (Walton 1996, p. 113; italics added). T4 does the same for non-zero but insufficient support (van Eemeren and Grootendorst 1984, p. 91; Windes and Hastings 1965, p. 216). Specifically, T4 tests for insufficient sup- port that is typical for multiple argumentation, where among “‘a series of separate and individual arguments [read: reasons]’ for a conclusion […] it does not matter [with respect to supporting the conclusion] which [reason] is chosen” (Walton 1996, pp. 114f.; italics original; see van Eemeren and Grootendorst 1984, p. 91). Apparently, T2 is designed in analogy to T1, T3 and T4. T1-T4 are binary tests, whereas T5 addresses a comparative notion of support. Inspired by Thomas (1981, p. 52) and Malcolm Acock (1985, p. 83)—wherefore Walton calls it “Thomas-Acock test”—T5 tests “how well the conclusion was supported before [a] premise was removed versus how well [the conclusion] is support- ed once the premise is taken away” (Walton 1996, p. 121). Thom- 376 Yu & Zenker © Shiyang Yu and Frank Zenker. Informal Logic, Vol. 42, No. 2 (2022), pp. 363–387. as’ and Acock’s tests, however, do test for distinct states of affairs. For Thomas, “[t]he test for a linked argument is: if one premise is taken away, the conclusion is more weakly supported than it was when that premise was in the argument” (Walton 1996, p. 125). According to Thomas’s test, then, the argument ‘(i) his swimming suit is wet; (ii) his hair is plastered down; therefore, he’s been swimming’ instantiates a linked structure, because if (i) or (ii) is suspended, the conclusion is less supported than otherwise. This test-result is counter-intuitive; scholars would normally consider the argument to instantiate a convergent structure. According to Acock’s test, it holds for a linked argument that “the sum of the amount of support given independently [by the premises, to the conclusion] is less than the amount of support [the premises] give to the conclusion when taken together” (Acock 1985, p. 83; see Walton 1996, p. 125). This means Acock’s test compares the joint support that the entire premise set lends to the conclusion to the sum of the support that each premise lends indi- vidually. The test thus differs from Thomas’s in that Acock’s test must first (somehow) determine the support each premise lends individually and form the sum, and then compare that sum to the joint support lent by the premise set. By contrast, Thomas’s test determines the joint support, and then establishes whether sus- pending a premise does result in reduced support, without sum- ming the support that each premise lends individually. Robert Yanal (1988, p. 42; Walton 1996, p. 127; c.f. Yanal 1991; 2003) has refined Thomas’s test by altering the change in support from ‘is greater/less than’ to ‘is much greater/less than,’ yielding T5.4 As Walton observes, what T5 “literally says is that, when we remove the premise (or component argument [sic]) in question, the level of support for the conclusion drops considera- bly,” which entails that “the argument is no longer strong enough to meet the [contextually determined] level of burden of proof […] to make the conclusion acceptable. So construed, [T5] amounts to the same finding as [T4]” (1996, p. 166). 4 Goddu (2003) provides a more detailed account of Yanal’s test (aka the ordinary summing test). He contends that this test, as well as two related ver- sions of it, fail because the tests cannot identify convergent structures in all relevant cases. Identifying Linked and Convergent Argument Structures 377 © Shiyang Yu and Frank Zenker. Informal Logic, Vol. 42, No. 2 (2022), pp. 363–387. Yet, T5 does not require that the support changes from suffi- cient to insufficient. T5 merely requires that, in a linked structure, the degree of joint support is much greater than the support that is generated if the premises are considered individually. It therefore remains possible that, even after a very large increase in strength, the argument nevertheless cannot meet the sufficiency requirement. Hence, T4 and T5 are distinct tests, that use distinct test-criteria. In sum, all tests rely on the variation in strength/support to ad- dress the mutual dependence of the premises. This makes consid- ering the variation of strength/support inherent in the concept of ‘dependence,’ and hence in the concept of ‘argument structure.’ Yet, defining argument structures based on strength variation does not always succeed. As we explain in Sect. 5.1, T1-T5 therefore fail to be absolute tests. First, we turn to yet another way of distinguishing linked from convergent structures, which we criticize in Sect. 5.2. 4.2. Relevance-based test We saw in Sect. 2 that argument structure types can be defined by using relevance as a supplementary factor to dependence. Freeman holds that an argument instantiates a convergent structure if “two or more premises are each independently relevant to the conclu- sion,” which means that “[e]ach gives a separate piece of evidence for the conclusion” (2011, p. 94; italics added). Particularly prem- ises that involve modal combinations are said to be convergent because “each premise may give some reason for the conclusion, whereas their combined weight [i.e., support] constitutes a strong- er case” (2011, p. vii). By contrast, Freeman considers premises involving relevance combinations as linked, such that “premises which taken individually do not constitute even relevant reasons for a conclusion [may,] when taken in combination [, …] consti- tute one obviously relevant reason” (2011, p. viii). This implies that Freeman’s test is not exclusively based on relevance, but also on dependence, because the test involves considerations of strength variation. The relevance relation between any two statements, P and Q, Freeman submits, is best thought of “as a ternary relation between P, Q, and a set of inference rules I” (Freeman 2011, p. 130; see 378 Yu & Zenker © Shiyang Yu and Frank Zenker. Informal Logic, Vol. 42, No. 2 (2022), pp. 363–387. Freeman 1992). Here, I may contain formal deductive or inductive rules, as well as material inference rules (e.g., Toulmin’s war- rants). For instance, ‘Harry was born in Bermuda’ is relevant to ‘Harry is a British subject,’ because I contains the rule ‘from x is born in Bermuda, infer that x is a British subject’ (1992, pp. 131f.). This consideration is based on Charles Peirce’s (1955, p. 130) inference habit which “convey[s] us from one judgment to anoth- er” (Freeman 2011, p. 130). It is in virtue of an inference habit, then, that one can “perceive or intuit relevance” (2011, p. 130). This ternary-relation of relevance Freeman defines as follows: “A statement P is relevant to a statement Q if there is some infer- ence rule in the canonical set C licensing the move from P to Q. Similarly, a set of statements P1, P2[,] …, Pn is relevant to a statement Q if there is some n-premised inference rule in C licens- ing the inferential move from P1, P2[,] …, Pn to Q.” (Freeman 2011, p. 131) Though Walton (1996, p. 113), who characterizes T3 as ‘the Freeman test’, claims that this test uses only premise dependence based on strength variation (see our Sect. 4.1), Freeman here distinguishes linked from convergent argument structures by considering relevance besides dependence. Relevance-based tests do thus seem to be more developed than support-based test. Yet, as we show in Sect. 5.2, Freeman’s test must likewise be grounded in support/strength, and so is also problematic. 5. Problems 5.1. Evaluating T1-T5 Prima facie, the fundamental difference between linked and con- vergent argument structures is whether the premises support a conclusion jointly, or in a combined or united way (Thomas 1997, p. 52), i.e., whether the premises “work together to support con- clusion” (Copi and Cohen 1990, p. 20). This, however, does not yield a well-specified criterion to distinguish a linked from a con- vergent structure. For the linked structure, ‘working together’ presumably means that the premises jointly increase the conclu- Identifying Linked and Convergent Argument Structures 379 © Shiyang Yu and Frank Zenker. Informal Logic, Vol. 42, No. 2 (2022), pp. 363–387. sion’s acceptability. But this also holds for a convergent argument, because convergent premises lend comparatively “more support to the conclusion collectively than each [convergent premise] would individually” (Walton 1996, pp. 111f.), that is, “convergent prem- ises together [make] a stronger case for the conclusion than either [premise] by itself” (Freeman 2011, p. ix). So, convergent premis- es do likewise work together to increase the conclusion’s accepta- bility. In brief, premises work together in a convergent and a linked structure. A relevant difference between both structures might arise if ‘working together’ meant that the premises in a linked structure work together necessarily, whereas this need not hold for a con- vergent structure. The presumably most straightforward way of interpreting necessity is to specify it as ‘the degree of support each premise is required to lend to the conclusion.’ On this interpreta- tion, if the premises of an argument with a linked structure support the conclusion independently, then the support that each premise lends to it cannot meet this required degree. In a convergent struc- ture, by contrast, the support that each premise lends to the conclu- sion must meet this required degree. But this interpretation— which corresponds to the mainstream view today—precisely creates the problem of clearly distinguishing linked from conver- gent structures, because it places argument evaluation before argument analysis. If the degree of support is specified as the degree that is re- quired for sufficient support, what could explain why linked prem- ises must work together necessarily is that each linked premise lends insufficient support to the conclusion (although premises jointly lend sufficient support). Convergent premises, by contrast, need not work together, because each convergent premise already lends sufficient support to the conclusion. The specification ‘suffi- cient support’ would work for T2 and T4. Yet for T1 and T3, the required degree of support would have to be specified as ‘any sup- port,’ such that in a linked argument, and absent a given premise, the conclusion receives no support. And although the formulation of T5 does not imply a clear specification of the required degree of support in terms of sufficient or any support, it can be interpreted as 380 Yu & Zenker © Shiyang Yu and Frank Zenker. Informal Logic, Vol. 42, No. 2 (2022), pp. 363–387. a significant increase in the degree of support that results from considering the linked premises jointly as opposed to individually. All five tests thus assume that the necessity of linked premises to work jointly is based on a variation between two states of sup- port. The first state amounts to all linked premises jointly support- ing the conclusion. The second state amounts to only one premise supporting the conclusion, namely that premise which is not sus- pended or considered false. This at least holds for arguments with two premises. Arguments with more than two premises face a distinct complexity problem, addressed below. To see how this plays out using an example where the re- quired degree of support is specified as ‘sufficient support’, we can contrast the argumentation A-2, which would normally be analyzed as a convergent structure, with the argumentation A-3, which would normally be analyzed as a linked structure: A-2 A convergent argumentation (Freeman, 2011, p. viii) [R1] His swimming suit is wet. [R2] His hair is plastered down. Therefore [C] He’s been swimming. A-3 A linked argumentation [R1] His swimming suit is wet. [R1-C] A wet swimming suit implies one has been swim- ming. Therefore [C] He’s been swimming. The following problems now arise: The inconsistency problem On the assumption that A-3 instantiates a linked structure, both of its premises are individually necessary and do jointly provide sufficient support to the conclusion. In this case, if R1-C were suspended, then C would be left with insufficient support from R1. By contrast, on the assumption that A-2 instantiates a convergent structure, each of R1 and R2 can support C sufficiently. Both assumptions together, however, do result in an inconsistency: R1 Identifying Linked and Convergent Argument Structures 381 © Shiyang Yu and Frank Zenker. Informal Logic, Vol. 42, No. 2 (2022), pp. 363–387. by itself can, but at the same time cannot, support C sufficiently. Let’s call this the inconsistency problem. One way of resolving the inconsistency problem would be to stipulate that, before judging the sufficiency of support in a con- vergent structure, all implicit premises must be made explicit. This may seem to explain that, in A-2, C is not independently supported by R1, whereas C is so-supported once the implicit inference rule R1-C is added, as in A-3. But this way of addressing the incon- sistency problem provides only a seeming solution. For once the linked structure is treated in the same way—such that a recon- struction of implicit elements is allowed (even required) before evaluating the dependence relation—R1 and R1-C in A-3 would cease to be dependent, because in reconstructing the argument ‘R1 therefore C,’ one may supply R1-C as an implicit premise. Simi- larly, if R1 is suspended, then, after a reconstruction, R1-C alone can support C sufficiently. The linked argumentation ‘R1, R1-C, so C’ would thus change into a convergent argument, because its premises individually support the conclusion sufficiently. The presupposition problem As a yet more serious problem, when maintaining that R1 or R2 by itself, or R1 and R1-C together, do support the conclusion suffi- ciently, one must presuppose that the premise(s) lend sufficient support to the conclusion. Let’s call this the presupposition prob- lem. But there is no obvious reason to presuppose that the relation between the premise(s) and the conclusion must be evaluated as a sufficient supporting relation. In A-2, for instance, one may doubt that R2 is a sufficient reason for C, because taking a shower (rather than swimming) may also cause one’s hair to be plastered down. Furthermore, when presupposing that the conclusion receives sufficient support, analysts incur a presupposition regarding the acceptability of the premises. A false premise, after all, can offer no support to the conclusion, let alone sufficient support. And since the tests T1 to T5 (see Sect. 4.1) identify argument structure as a function of the premises’ presupposed acceptability, these tests require evaluative judgements to satisfy the respective test- condition. A fully explicit version of T1, for instance, would read: ‘If one premise is assumed to be false, while the other premise is 382 Yu & Zenker © Shiyang Yu and Frank Zenker. Informal Logic, Vol. 42, No. 2 (2022), pp. 363–387. assumed not to be false, then the conclusion no longer receives any support.’ The presupposition for T1 thus is that the other premise is not false. Of course, applying T1 does not require that analysts know the acceptability of that premise. But to identify the structure of an actual (real-life) argument, they must nevertheless presuppose an evaluative judgement of that premise. The requirement on premise-acceptability in T1 (and T3) does not apply to T2 or T4. A fully explicit version of T2, for instance, should instead read: ‘If one premise is assumed to be false, while the other premise is assumed to be true (or very plausible), then the conclusion receives insufficient support.’ For the test-result to align with pre-theoretical intuitions, after all, the impact of the acceptability of the other premise on argument strength should be minimal. Otherwise, if the acceptability of the other premise is extremely low, even an intuitively convergent argument can be identified as linked. For instance, if R1 is assumed to be false, while R2 is not assumed to be true or very plausible (but is rather implausible), then C can only receive insufficient support. This would indicate that A-2 instantiates a linked structure. But this breaks with the intuition that A-2 is a convergent argument. There- fore, ‘the other premise is assumed to be not false’ is the minimal requirement for T1-T4, whereas in T2 or T4 the other premise must be assumed to be true (or very plausible). The presupposition problem generalized Not only does the focal problem involve a presupposition regard- ing the acceptability of some other premise, as well as of the se- mantic or pragmatic relation between it and the conclusion. The problem generalizes to both premises. For nothing about the tests T1 to T5 constrains the specific premise an analyst might set to the status false or suspended. Indeed, the specific premise having this status should be immaterial to identifying an argument’s structure. Otherwise, given the same test, an argument may have more than one structure. The premise that is set to the status false/suspended should therefore be selected randomly. Yet, if the other premise (that besides the randomly selected premise) has to fulfil some acceptability requirement, then the constraint for T1—‘If one premise is assumed to be false, while the other premise is assumed Identifying Linked and Convergent Argument Structures 383 © Shiyang Yu and Frank Zenker. Informal Logic, Vol. 42, No. 2 (2022), pp. 363–387. to be not false, then the conclusion no longer receives any sup- port’—should be reformulated as: ‘Assuming all premises are not false, if one premise is false, then the conclusion no longer re- ceives any support.’ The meaningfulness problem One consequence of this constraint is that the resulting structural identification in T1 pertains to possible states of an argument, states where all premises are not false. But this leaves other possi- ble states unidentified. And if some premises turn out to be false, then the argument structure may change. This clearly challenges the value of performing these tests, suggesting that they may be meaningless, and so damages the theoretical value of the concept ‘argument structure.’ The complexity problem Practically, given a complex argument with more than two prem- ises, or even more than one type of structure (i.e., a complex ar- gument comprising sub-arguments), how should an analyst identi- fy the premises that work separately, indicating a convergent structure, while other premises work jointly, indicating a linked structure? A more complicated method certainly seems to be required (Goddu 2003, pp. 219-225). According to Walton, how- ever, “the same [test for two premises] applies to any number of premises in an argument” (Walton 1996, p. 182). One can only imagine how complicated this would be, involving a great deal of presupposition, evaluation, and comparison before the structure of a complex argument is identified. In sum, we do neither claim that the dependence criterion is useless in identifying argument structure, nor that one should cease to specify the notion of ‘premises working together’ as ‘premises working together necessarily.’ We rather contend that it is problematic to interpret the notion of necessity—which func- tions as the sole criterion in determining whether premises are mutually dependent—as achieving some required degree of sup- port for the conclusion. 384 Yu & Zenker © Shiyang Yu and Frank Zenker. Informal Logic, Vol. 42, No. 2 (2022), pp. 363–387. We now turn to Freeman’s test, which combines considera- tions of dependence and relevance. 5.2 Evaluating Freeman’s test When applying Freeman’s ternary relation of relevance to A-2 and A-3, the above problem arises again. Assume one grants that A-2 instantiates a convergent structure and A-3 a linked structure. Following Freeman, that A-2 is convergent entails that each prem- ise—for instance, ‘his swimming suit is wet’—is individually relevant to the conclusion. On this understanding of relevance, anyone who agrees that A-2 is convergent should grant an infer- ence rule—e.g., ‘if one’s swimming suit is wet, then one has been swimming’—licensing the move from ‘his swimming suit is wet’ to ‘he has been swimming.’ This, however, contradicts Freeman’s own claim that A-3 is linked, because at least one individual prem- ise—namely: ‘his swimming suit is wet’—constitutes what in Freeman’s sense is a relevant reason for the conclusion. This is similar to the inconsistency problem for the support-based tests T1-T5 (see Sect. 5.1). Like the support-based tests, moreover, Freeman’s relevance- based test features a constraint on the argument to be analyzed. Recall that, according this method of analysis, a linked argument’s premises would fail to be individually relevant to the conclusion, yet are jointly relevant to it (see Sect. 4.2). So, in a linked argu- ment each premise by itself offers no support to the conclusion because there is no inference rule that connects each premise with the conclusion. Meanwhile, given that the premises may be false, they need not jointly offer any support to the conclusion either. In a convergent argument, by contrast, the combined premises lend a greater degree of support to the conclusion than each premise lends to it individually. So each premise must individually offer at least some support. Otherwise, given any randomly selected indi- vidual premise, the combined degree of support cannot be greater than the individual premises’ degree of support. Therefore, if both R1 and R2 are false, and if inference-rules connect R1 with C, and R2 with C, respectively, then A-2 would be identified as neither linked nor convergent. After all, the prem- ises do provide individually relevant reasons for the conclusion (so Identifying Linked and Convergent Argument Structures 385 © Shiyang Yu and Frank Zenker. Informal Logic, Vol. 42, No. 2 (2022), pp. 363–387. the argument is not linked), and the premises taken together fail to constitute a modal combination that offers stronger support (so the argument is not convergent). In both the relevance- and the sup- port-based tests, then, argument evaluation informs argument structure identification. But this once again challenges the idea that ‘argument structure’ is a concept of analysis. 6. Conclusion On a dialectical and a logical approach alike, a linked argument structure is distinguished from convergent structure according to whether the premises work together necessarily. This distinction is theoretically basic today and may even be considered intuitive. What remains problematic is how the mainstream scholarly view specifies ‘necessity’. Both dependence-based and relevance-based tests specify ‘necessity’ as the achievement of some required degree of support for a conclusion. But this entails that a structural analysis grounds in analysts’ evaluative judgements. This is incon- sistent with the idea that ‘argument structure’ is an analytical concept that is methodologically prior to argument evaluation. Argumentation scholars require a new way of specifying necessity that does not involve argument evaluation. Acknowledgements We thank the 2021 AILACT Essay Prize committee for selecting this paper. S.Y. acknowledges funding from the Fundamental Research Funds for the Central Universities (No. 63202058) and from the Nankai University Development Funds for the Humani- ties (No. ZB22BZ0314). F.Z. acknowledges funding from the Polish National Science Center (2019/35/B/HS1/03281) and the European Network for Argumentation and Public Policy Analysis (EU Cost Action CA 17132). References Acock, M. 1985. Informal logic examples and exercises. Belmont, CA: Wadsworth. 386 Yu & Zenker © Shiyang Yu and Frank Zenker. Informal Logic, Vol. 42, No. 2 (2022), pp. 363–387. Adler, J. (2008). Introduction: Philosophical foundations. In Reasoning: Studies of human inference and its foundations, eds. J. Adler and L. Rips, 1-34. Cambridge: Cambridge University Press. Beardsley, M. 1950. Practical logic. Englewood Cliffs, NJ: Prentice- Hall. Copi, I.M., and C. Cohen. 1990. Introduction to logic (8th ed.). New York: Macmillan. Eemeren, F.H. van, and R. Grootendorst. 1984. Speech acts in argumen- tative discussions. A theoretical model for the analysis of discussions directed towards solving conflicts of opinion. Berlin/Dordrecht: Wal- ter de Gruyter/Foris. Eemeren, F.H. van, and R. Grootendorst, R. 1992. Argumentation, communication, and fallacies: A pragma-dialectical perspective. Hillsdale, NJ: Lawrence Erlbaum. Eemeren, F.H. van, and R. Grootendorst. 2004. A systematic theory of argumentation. The pragma-dialectical approach. Cambridge: Cam- bridge University Press. Eemeren, F.H. van, B. Garssen, E.C.W. Krabbe, A.F Snoeck Henke- mans, B. Verheij, and J.H.M Wagemans. 2014. Handbook of argu- mentation theory. Dordrecht: Springer. Eemeren, F.H. van. 2018. Argumentation theory: A pragma-dialectical perspective. Cham: Springer International Publishing AG. Freeman, J. 1988. Thinking logically. Englewood Cliffs, NJ: Prentice- Hall. Freeman, J. 1991. Dialectics and the macrostructure of arguments. Berlin: Foris. Freeman, J. 1992. Relevance, warrants, backing, inductive support. Argumentation 6: 219-235. Freeman, J.B. 2011. Argument structure: Representation and theory. Dordrecht-New York: Springer. Goddu, G. 2003. Against the “ordinary summing” test for convergence. Informal Logic 23: 215-236. Goddu, G. 2007a. Walton on argument structure. Informal Logic 27: 5- 25. Goddu, G. 2007b. Against making the linked-convergent distinction. In Proceedings of the Sixth Conference of the International Society for the Study of Argumentation, Sic Sat, Amsterdam, eds. Frans H. van Eemeren et al. URL: Identifying Linked and Convergent Argument Structures 387 © Shiyang Yu and Frank Zenker. Informal Logic, Vol. 42, No. 2 (2022), pp. 363–387. Groarke, L., Tindale, C., and L. Fisher. 1997. Good reasoning matters! A constructive approach to critical thinking. Toronto: Oxford Uni- versity Press. Govier, T. 2010. A practical study of argument (7th ed.). Belmont: Wadsworth. Johnson, R., and J.A Blair. 1994. Logical self-defence (U.S. ed.). New York: McGraw-Hill. Peirce, C. S. 1955. What is a leading principle? In philosophical writings of Peirce, ed. J. Buchler, 129-134. New York, NY: Dover. Pinto, R.C., and J.A Blair. 1993. Reasoning: A practical guide. Eng- lewood Cliffs, NJ: Prentice Hall. Snoeck Henkemans, A.F. 1992. Analysing complex argumentation: The reconstruction of multiple and coordinatively compound argumenta- tion in a critical discussion. Amsterdam: Sic Sat. Snoeck Henkemans, A.F. 2001. Argumentation structures. In Crucial concepts in argumentation theory, ed. F.H. van Eemeren, 101-134. Amsterdam: Amsterdam University Press. Thomas, S.N. 1997/1986/1981/1973. Practical reasoning in natural language (4th, 3rd, 2nd, 1st ed.). Englewood Cliffs, NJ: Prentice- Hall. Walton, D.N. 1996. Argument structure: A pragmatic theory. Toronto: University of Toronto Press. Windes, R.R., and A. Hastings. 1965. Argumentation and advocacy. New York: Random House. Yanal, R.J. 1988. Basic logic. St Paul: West Publishing. Yanal, R.J. 1991. Dependent and independent reasons. Informal Logic 13: 137-144. Yanal, R.J. 2003. Linked and convergent reasons—again. In Informal Logic at 25: Proceedings of the Windsor Conference, eds. J. Anthony Blair et al. CD-ROM, Windsor, ON: OSSA.