INFORMAL LOGIC XIV. 1 , Winter 1992 Rules for Plausible Reasoning DOUGLAS WALTON University oj Winnipeg Key Words: Plausible reasoning; confidence fac- tors; presumption; argument; linked arguments; convergent arguments; argumentation. Abstract: This article evaluates whether Rescher's rules for plausible reasoning or other rules used in artificial intelligence for "confidence factors" can be extended to deal with arguments where the linked-convergent distinction is important. Many of those working in the field of argumentation now accept the idea that there is a third type of reasoning distinctive from deductive and inductive reasoning called plausible reasoning, a kind of reasoning based on tentative, prima jacie, defeasible weights of presumption which can be as- signed to the propositions in an argument.' Some theorists have now even offered sets of rules (calculi) for plausible reasoning. The set of rules presented by Rescher (1976) is perhaps the best known to those of us working in informal logic and argu- mentation. But within the field of artificial intelligence, where presumptive reasoning based on "confidence factors" is very important, e.g. in applying expert systems of technology, various proposals for rules of this type have been advanced. This paper evaluates Rescher's rules, and one set of rules from AI (Intelliware, 1986) with a view to seeing whether or to what extent such accounts of plausible rea- soning could be useful for, or adapted to, the needs of informal logic. Taking into account the vital distinction between linked and convergent arguments, new, more general rules for plausible reasoning are proposed which would be useful for evaluating argumentation in a critical dis- cussion, in the sense of van Eemeren and Grootendorst (1984) and Walton (1992). 1. Systems of Plausible Reasoning Rescher's system of plausible reason- ing follows a conservative way of evaluat- ing an argument. The least plausible proposition in a set is the weakest link in the chain of argumentation, because it represents the greatest possibility of going wrong or getting into trouble. Hence Rescher's accounts of plausible inference are generally based on the weakest link idea. It is easy to appreciate how this idea fits the context of a critical discussion. The respondent has the obligation or function of asking critical questions in response to an argument advanced by a proponent in a critical discussion. Natu- rally, a critical respondent is trying to resist being persuaded by his partner's argument. He has the job of seeking out the weakest premises, and attempting to challenge or question these premises especially. This has two consequences. One is that the pro- ponent always tries to boost up these weak premises, or potential avenues of escape (loopholes) for the respondent. The pro- ponent always tries to have all premises as potentially being able to be backed up so that they are more plausible than the con- clusion the respondent doubts or resists. But second, the respondent is always drawn towards these weakest links (loop- holes) in his adversary's line of argument. So the conclusion he is supposedly being pushed towards conceding can never be rationally rated as more plausible, for him, than that weakest premise. Another important context of applica- tion of plausible reasoning is that of deciding on a course of action based on the 34 Douglas Walton advice gathered from the solicited opinion of an expert authority on a question (Rescher, 1976, p. 6). The user interface of an expert system is designed for a very similar use. For it is the user of the system who must draw conclusions from a set of facts and rules in a knowledge base which represents the systematization of an expert's knowledge in a given domain of expertise. In using an expert system, it must be rec- ognized that exceptions to accepted rules may exist, and therefore an approach to reasoning which assigns confidence fac- tors (CF's) as rough guides to reliability of advice has proved most successfuL The wayan expert reasons, however, in arriving at a conclusion in her field of expertise, is quite different from the way a (nonexpert) user reasons in drawing con- clusions from what the expert says. The user is typically engaged in deliberating on what to do, and quite often the context is that of a critical discussion concerning the pro and contra points of view on a possible course of action being considered.2 For example, in judging the alleged fal- laciousness of an argumentum ad verecun- diam, the problem is typically to evaluate how an appeal to expert opinion was used in a critical discussion between two par- ties. 3 The expert is a third party whose opinion was appealed to as a move made by one of the participants in the critical discussion. In such a case, the rules of plausible reasoning need to be formulated in the context of the critical discussion. Although plausible reasoning involves a qualitative judgment of relative compari- son of propositions, as opposed to a quan- titative-numerical calculus, formalized systematization of general rules for plausible reasoning have been proposed by Rescher (1976) and other systems of rules are in use in AI programs. Among the six formal rules for plausible reasoning given by Rescher (1976, p. 15), perhaps the most fundamen- tal and characteristic rule is the conse- quence condition. This condition requires that when a group of mutually consistent propositions entails a particular proposi- tion, then the latter proposition cannot be less plausible than the least plausible pro- position in the original group. This rule is also called the least plausible premise rule, and it defines the essential characteristic of plausible reasoning as a kind of logical inference, in Rescher's calculus. In artificial intelligence, a variety of sets of different types of rules have been given, for example, in expert systems research, to provide the "inference engine" for deriving conclusions in a data base where the facts and rules lead, at best, to tentative conclusions based on degrees of confidence. In the language of AI, a rule is a condition that may have several ante- cedents (premises) where the collection of antecedents is treated as a conjunction of simple propositions (facts). In one leading approach, outlined by Intelliware (1986), the rule for calculating confidence factors (CF's) for and takes the minimum plausi- bility value (confidence factor). Formally, p/aus(A 1\ B) == min (p/aus A, p/aus B) Then to calculate the plausibility of a con- clusion based on a set of premises, we multiply the plausibility value of the rule with the plausibility value obtained from the premises (by the conjunction rule above, where there is more than one premise). Formally, p/aus( conclusion) == p/aus(premises) x p/aus(ruJe» This approach (hereafter called the product rule) is quite different from Rescher's in several important respects, most notably perhaps in allowing a plausibility value for the inference itself. And then, of course, the product rule is itself basically different from Rescher's in the specific formula of calculation used. The basic formal rules of plausible rea- soning are given by Rescher (1976, p. 15), and comparable rules for inexact inference for expert systems are given by Intelliware (1986), Main Menu, Inexact Inference, pp. 3-9). However, recent developments in the area of argumentation indicate two important kinds of exceptions to these rules. Accordingly, these rules need to be modified, extended and developed in new directions. The first exception concerns the distinction between two kinds of condi- tionals. 4 In a must-conditional, 'If A then B' means that B is true in every instance in which A is true, with no exceptions. In a might-conditional, 'If A then B' means that B may be expected (presumed) to be true in a preponderance of typical instances in which A is true. But the linkage between A and B is a matter of typical or customary expectation, which can admit of excep- tions. The plausibility value of a must-con- ditional is always equal to 1 (certainty), whereas the plausibility value of a might- conditional, v, can range between 0 (of no value as a plausible presumption) and 1 (maximally plausible): 0 ~ v ~ 1. The set of rules in Rescher (1976, p. 15) is defined only for must-condi- tionals, but recent developments in artifi- cial intelligence-see Forsyth (1984), Bratko (1986) and Intelliware (1986)- show a clear practical need for considera- tion of rules of inference where "confi- dence factors" (certainty factors) need to be taken into account, by using inference rules with values of less than one for might -condi donals. It has already been noted above that in Intelliware (1986) a rule (conditional prop- osition) can be assigned a confidence fac- tor of less than one as a value. When inferring a conclusion from a set of pre- mises, the way to calculate the value of the conclusion is to mUltiply the value of the rule (conditional) by the value of the least plausible (lowest confidence factor) pre- mise. In InteIliware (1986, Main Menu, Inexact Inference, p. 6), the following example of calculating CF's for a single rule with a value of .60 is given. The aster- isk (*) stands for multiplication (product). Rule 1: CF 0.60 Stock 12 is volatile Rules for Plausible Reasoning 35 IF Stock 12 is high tech AND Stock 12 is in demand Evaluate Rule I: CF == 0.90 CF = 0.60 CF(Rule I) = Min (0.90. 0.60) * 0.60 = 0.36 This type of rule allows us to derive conclusions using a might-conditional, or as it is called in AI, a rule that is assigned a confidence factor of less than one (CF < 1). Rescher (1977, p. 6) introduces a pro- visoed assertion relation, AlB, meaning that A ordinarily obtains provided that B obtains, other things being equal, which he insists (p. 7) is not to be identified with implication. However, "for simplicity" (p. 8), he supposes that moves in dialogue of the form AlB are "always correct," mean- ing that disputants can never make errone- ous or incorrect claims about them. Rescher's comment (p. 8) is that this assumption "eliminates various complica- tions" that do not matter for his present purposes. But this assumption also removes the possibility of dealing with might-conditionals by showing how to derive conclusions from them in combina- tion with premises in plausible reasoning. What is needed is a more realistic or prac- tical concept of frame-based conditionals (provisoed assertion relations) that are suitable to the needs of persuasion dialogue. Might-conditionals are frame-based conditionals to the effect that if one propo- sition A is plausible, and another set of presumptions S are plausible in the com- mitment set of a respondent, then another proposition may be presumed to have a certain weight of plausibility. For example, consider the two propositions below. A: Jones is less than five feet tall. B: Jones is an All-Star forward on the NBA Los Angeles Lakers. If A is taken as a proposition in a commit- ment set of a participant in argument, then given what we all know about basketball (viz. it is practically necessary for a basket- 36 Douglas Walton ball player to be fairly tall, we would nor- mally expect, in order to be successful as an All-Star forward on the NBA Los Angeles Lakers), then B would not be plausible as a proposition in that participant's commit- ment set. Similarly, if B were taken as a plausible presumption, by a might-condi- tional, it would follow that A would not be a plausible presumption in that same set. In fact, from the point of view of plausible argumentation, A and B are "opposites" of each other (assuming they are in the same commitment set, which also contains the set S of plausible presumptions about suc- cessful players in the NBA). In short, there is a clash or opposition between A and B. Not a logical inconsis- tency, but a pragmatic inconsistency which reflects a tug of opposing plausibility weightings. 2. Linkage of Premises in a Critical Discussion The second type of exception to con- ventional systems of plausible reasoning concerns a requirement on the linkages between pairs of premises in an argument advanced by a proponent in a critical dis- cussion. The additional requirement needed here is that the premise-set as a whole must be taken to be plausible by the respondent to whom an argument in per- suasion dialogue is directed. Otherwise, the least plausible premise rule (reflecting the conservative point of view) might fail. This requirement of linkage of a set of premises in a useful argument in a critical discussion reflects the importance that should be placed on consistency (coher- ence) in a commitment set to be used as a set of premises to convince someone of a conclusion. Indeed, the primary way that interactive reasoning functions to produce maieutic insight is through the criticism of inconsistencies in an arguer's position. By dealing with the presumptive inconsisten- cies found by a critic, a participant in inter- active reasoning can come to a deeper understanding of his own position (com- mitment set). When discussing the rules of plausible inference, we start with a set of proposi- tions, A, B, ... , each of which can be assigned a plausibility value. For example, the plausibility value of the proposition A is written as plaus(A). For any proposition A, the value of A is subject to the condi- tion: 0 S plaus(A) S 1. In other words, a maximal plausibility (totally reliable) proposition can be assigned a value of 1, and a proposition that would not count as plausible, one of no useful value to per- suade a respondent of a conclusion, can be given a value of 0. 5 The basic axiom of plausible inference is the consequence condition (Rescher, 1976, p. 15): when a set of mutually con- sistent propositions A I , ... , An implies some other proposition B by valid deduc- tive argument, then the plausibility of B cannot be less than the plausibility value of the least plausible proposition among the set AI , ... , An. In short, If AI, ... , An imply B, then plaus(B) 2 MIN plaus(AI , ... , An) This consequence condition settles how conjunction is to be defined in plausible in- ference. The following plausibility rule for conjunction gives this definition. See Intelli- ware (Main Menu, Inexact Inference, p. 3). plausCA 1\ B) == MIN (plaus(A), plausCB» That is, the plausibility of the conjunction A 1\ B always reduces to the plausibility value of the lesser of the two propositions, A, B. How the consequence rule determines the conjunction rule above has been shown by Rescher (1976, p. 16, theorem 3). First, recall that the following three forms of inference are deductively valid. (II) A 1\ B A (12) A 1\ B B According to the consequence condition, the plausibility of the conclusion of a deductively valid argument must be as great as the plausibility of the least plausi- ble premise. Since A A B is the only premise of (11), it follows that the plausi- bility of A must be at least as great as that of A A B. Similarly for (12), the plausibility of B must be at least as great as that of A A B. In other words, (TI) plaus(A) ~ plaus(A A B); plalls(B) ;:0: plaus(A A B) Hence whichever of A or B has the lesser plausibility, it still must have a value at least as great as that of A A B. In other words, (T2) MIN(plaus(A), plaus(B» ~ plaus(A A B) But now, looking at (13), we can see that according to the consequence condition, the plausibility of A A B must be at least as great as the plausibility of whichever of A or B has the lesser value. In other words, (T3) plaus(A A B) ~ MIN (plaus(A), plaus(B» Putting (Tl) and (T2) together yields the plausibility rule (T3) for conjunction given above. It has been shown then that the con- junction rule follows from the conse- quence condition. So conceived, the rules for plausible inference are parallel to the rules for deductive inference. Just as conjunction was defined as a logical constant in the theory of deductive reasoning, so too con- junction will have a rule (T3) that defines it as a constant in the theory of plausible reasoning. So conceived, also, the theory of plausible reasoning presupposes the concept of deductive logical consequence that is defined in the theory of deductive reasoning. By these lights, plausible reasoning has a formal aspect which appears to make a calculus with formal rules of inference. This parallel begins to break down, however, when certain kinds of cases of Rules for Plausible Reasoning 37 plausible reasoning enter the picture. These examples undermine the plausibility rule for conjunction, and with it, the funda- mental least plausible premise rule. The latter rule states that, in a deductively valid argument (where the premises are logically consistent) the conclusion must be at least as plausible as the least plausible premise. But consider the following argument. Case 0: (PI) Jones is less than five feet tall. (P2) Jones is an All-Star forward on the NBA Los Angeles Lakers. (C) Jones is a less than five-Foot tall All-American forward on the NBA Los Angeles Lakers. In this case, there may be evidence that makes (P I) highly plausible, and also other evidence that suggests that (P2) is highly plausible. But although the form of argu- ment in case 0 is deductively valid, and the premises are logically consistent with each other, the conclusion is not highly plausi- ble. In fact, it is implausible. And since case 0 is of the form (I3), the plausibility rule for conjunction also fails in case O. Case 0 is a linked argument, in the sense that both premises (PI) and (P2) are required to derive (C) by a deductively valid argument form. If either of (P I) or (P2) is omitted, the argument ceases to be valid. But in some other sense perhaps, case 0 may not appear to be a linked argu- ment, in that it would seem to be somehow characteristic of this type of argument that the line of evidence for (PI) should be separate from, or distinct from, the line of evidence for (P2) and vice versa. But it does not seem obvious what "separate from" means in this context. This is a pro- blem we return to below. One might wonder how plausible rea- soning compares to probable reasoning in this type of case. In case 0 above, part of the problem appears to be that the premises are probabilistically dependent on each 38 Douglas Walton other so that the conditional probability of either on the other is less than its uncondi- tional probability alone. But the problem does not disappear by attempting to restrict the rules to sets of premises that are proba- bilistically independent of each other. Case 1: (P I) The first flip of this coin will be heads. (P2) The second flip of this coin will be heads. (C) Both the first and second flip of this coin will be heads. In this case, like the one above, the pro- bability (or plausibility) is less than the probability (or plausibility) of the least probable (plausible) premise. Plausibility seems parallel to probability in this type of case. But, at any rate, plausibility does not follow the least plausible premise rule. And this failure is instantiated in its basic failure to follow the plausibility rule for conjunction in these cases. Possibly to deal with this kind of exception, Rescher (1976, p. 15) adds the requirement of the compatibility condition: all propositions in a plausibility evaluation set must be "logically compatible and materially consonant with one another." To be materially consonant (footnote, p. 15) is meant "logical compatibility with certain suitable 'fundamental' stipulations of extra-logical fact." But what are these "fundamental stipulations of extra-logical fact"? Rescher does not tell us, and the resulting gap makes it hard to apply the least plausible premise rule, and to know where it is applicable to argumentation and where not. For clearly the exceptional cases above indicate that the rule is not applicable in some instances. The third exception to the conventional rules of plausible reasoning arises through the distinction between linked and conver- gent arguments, now commonly used in informal logic. The exception noted in the present section arises because, in linked arguments, the premises must be con- nected together in such a way as to provide a plausible commitment set or position from which the respondent can be persuaded to accept a particular conclu- sion. In the next section, another exception arises through the fact that not all argu- ments advanced in persuasion dialogue are linked arguments. In a linked argument, a bundle of premises is taken together as a fixed set representing the commitment set of a respondent at one move in dialogue. How- ever, in dynamic interactive reasoning, "new knowledge" may be added to the com- mitment store of a participant in dialogue. 3. Linked and Convergent Arguments Revisited The third exception concerns the dis- tinction between two kinds of argument techniques represented in argument dia- gramming, namely linked and convergent arguments. Since the reader conversant with informal logic is already familiar with these techniques of argument diagram- ming, no further, more elaborate examples need to be presented here. It is enough to note that convergent and linked arguments can be combined into larger networks of argument structures, by means of serial connections joining subarguments together. The basic rule of plausible reasoning in the Rescher framework, as noted, is the least plausible premise rule, which states that in a deductively valid argument, the conclusion must be as plausible as the least plausible premise. This rule works well in critical discussion for linked arguments, but not for convergent arguments. Typi- cally, in a convergent argument, a conclu- sion is based on some existing evidence, but then some new and independent evi- dence comes along. If this new evidence is stronger than the old evidence, there should be an upgrade of the plausibility value of the conclusion, based on the value of the new premises. In such a case, if there is one "old" premise and one "new" premise, for example, the value of the con- clusion should be set at the value of the most plausible premise-in this instance, the value of the "new" premise. It follows that the least plausible premise rule is not universal for plausible reasoning. It fails in convergent arguments. It also fails where the linkage between premises and conclusion is that of a might -conditional. The distinction between convergent and linked argumentation is not modelled in classical logic where, for example, we have valid forms of inference like 'A A B, there- fore A'; and the deduction theorem allows us to treat separate premises as a grouped conjunction of propositions in a single premise. But in a critical discussion the distinction between uses of these two types of argument is fundamental because each of them has to be defended against criti- cisms in a fundamentally different way. In a linked argument, the respondent, who is inclined to be resistant to being convinced of the proponent's conclusion, will try to reject the premises if the argu- ment is otherwise convincing. And he will seek out the weakest of the premises, for if one premise alone fails, the whole argu- ment fails to persuade successfully. But in a convergent argument, each premise is a separate line of argument. So if one fails, the proponent can rely on the other. This funda- mental difference is basic to the structure of using inference in critical discussion. In figure 0, there are two premises A and B, used as a basis to support a con- clusion C. LINKED CONVERGENT A\/A\/C o 0 C C Figure 0 In the linked argument, both premises A and B are needed to prove C. In the conver- Rules for Plausible Reasoning 39 gent argument each of A and B is inde- pendent of the other.6 What this means, in dialectical terms, is that the use of each type of argument has a distinctive prag- matic rationale. This duality of pragmatic rationale was recognized and clearly stated by Windes and Hastings (1965), in their discussion of how to organize a proof when your goal is to construct a convincing case in order to persuade an audience to accept a particular proposition. Within such a context of per- suasion dialogue, Windes and Hastings postulate (1965, p. 215) that there will be an "over-all argument" that states the issues (the global level of argumentation), and subarguments that are local con- tentions supporting these global issues. Serial argumentation connects some sub- arguments to other subarguments, result- ing in extended chains of argumentation in a proof. What is especially interesting here is that Windes and Hastings clearly distin- guish between linked and convergent argu- ments, and articulate a basic principle of plausible inference governing each type of argument. First, they describe linked argu- mentation, and express what is, in effect, a statement of the weakest link principle as applicable to linked argumentation. In con- vincing an audience of a particular propo- sition, they wrote, there may be several issues, and the principle of argumentation is: "Each one of the issues must be estab- lished for the proposition to be estab- lished." (1965, p. 216) In other words, as they put it: "If any issue is not proved, then the proposition is not proved." (p. 216) They recognize, as well, that this principle of reasoning is typically embedded in a larger process of a chain of arguments that may be quite long. This statement of Windes and Hastings expresses the basic pragmatic rationale behind linked argumentation in the context of persuasion dialogue. It expresses the idea that a linked argument is only as strong as its weakest premise. For if any 40 Douglas Walton premise (issue) is not proved, in a linked argument, then the conclusion is not proved. In a linked argument, the premises are interdependent, and if the audience doubts one premise, or finds it weak and unconvincing, then the audience will not be persuaded by the argument to accept its conclusion. Windes and Hastings went on (p. 216) to recognize a second type of argumenta- tion where there are "independent lines of reasoning" that "lead to the same conclu- sion," i.e. what we have called convergent argumentation. They cite the following case, where "three reasoning processes" are used to support the conclusion, 'The corn crop of Dullnia is failing.' Case 2: I. Dullnia is buying corn on the world market. (Reasoning from effect to cause.) 2. The testimony of an agricultural expert who visited Dullnia. (Testimonial evidence.) 3. The presence of drought and poor growing conditions this year. (Cause to effect.) In describing the pragmatic rationale of this type of (convergent) argument in persua- sion dialogue, Windes and Hastings claim that both the number and the plausibility of the component arguments can be important (p. 217). Two other pieces of advice they offer the advocate generally-whether the argument is linked or convergent-are to use as many different lines of argument as possible, "giving precedence to the strong- est proofs." (p. 218) This significant remark suggests another pragmatic ration- ale that (in the present author's opinion) is especially and distinctively applicable to convergent argumentation. This is the rationale, from the point of view of the advocate of a convergent argument in a persuasion dialogue, of giving precedence to the strongest line of argument, where more than one (independent) line of sup- port for your conclusion is available. These pragmatic rationales for linked and convergent arguments both have a dual nature, reflecting the character of persua- sion dialogue. From the point of view of the proponent, or advocate of an argument, his function is to persuade the respondent by finding premises that will meet the bur- den of proof for that respondent. From the point of view of the respondent, his func- tion is to critically question the premises of the proponent's arguments, finding a way to resist being persuaded, if he can. This framework leads to the following characteristic general formulations of a pragmatic rationale and a plausibility rule for both of these types of argumentation in persuasion dialogue. PRAGMATIC RATIONALE FOR LINKED ARGUMENTATION: If the respondent succes~Jully questions one premise, the whole argument fails to meets its burden of proof So the respon- dent can choose 10 attack one or the other. PRAGMATIC RATIONALE FOR CON- VERGENT ARGUMENTATION: If the respondent questions one premise, the other can be brought to bear to back up the conclusion. So the respondent needs 10 attack both, to refute the argument. Match- ing each of these pragmatic rationales is a corresponding rule for plausible reasoning. PLAUSIBILITY RULE FOR LINKED ARGUMENTS: C has the value of the least plausibility value of the pair (A, B). PLAUSIBILITY RULE FOR CONVER- GENT ARGUMENTS: C has the value of the greater plausibility value of(A, B). From the point of view of the critical questioning of linked and convergent argu- ments, each type of argument has its own characteristic type of strategy as well. STRATEGY FOR QUESTIONING A LINKED ARGUMENT: Generally attack the weaker (weakest) premise (other things being equal). STRATEGY FOR QUESTIONING A CONVERGENT ARGUMENT: There is no point in starting by attacking the weaker premise. You might as well attack the stronger premise right away. These differences have fundamental impli- cations for the project of formulating rules of plausible reasoning for use in a critical discussion. 4. New Rules for Convergent and Linked Arguments The basic idea of plausible reasoning has, to this point, been typified by the least plausible premise rule. This rule, it will be recalled, states that the conclusion of a deductively valid argument is at least as plausible as the least plausible premise of the argument. Now we have distinguished between linked arguments and other kinds of arguments like convergent, divergent and serial arguments. However, some important exceptions to the least plausible premise rule need to be explained. For while the least plausible premise rule holds generally for valid linked arguments at the local level, it is superseded by other rules of plausible inference in convergent argu- ments, and in some serial arguments. The least plausible premise rule derives its justification from the character- istics of the critical discussion as a context of use. Generally, an argument in a critical discussion is a kind of interchange where the proponent of an argument is trying to persuade the recipient (respondent) of the argument to accept the conclusion. How- ever, generally speaking, it is a feature of this kind of dialogue that the recipient does not accept the conclusion of the argument, at least to begin with, and he is inclined to doubt or even reject the conclusion. This being the case, the recipient of a valid argument will generally try to resist accep,ting the conclusion of an argument he has just been presented with, by seeking out the "weakest link" in the premises. Rules for Plausible Reasoning 41 In a linked argument, the respondent should try to attack the weakest premise, because that will bring the whole argument down, if he can attack this one premise successfully. From the proponent's point of view, he can expect the respondent to be convinced by his argument only to the strength (weight) provided by his weakest premise. Hence the appropriate strategic presumption to gain assent in persuasion is the least plausible premise rule. For example, suppose that Lester doubts that Nasir is a Christian, but Arlene advances the following argument. Case 3: Nasir went to church. If Nasir went to church then Nasir is a Christian. Therefore, Nasir is a Christian. If Lester does not dispute the first premise, and finds it relatively plausible, but he does dispute the acceptability of the second pre- mise, and finds it much less plausible, how should he respond to Arlene's argument? If he is a smart and reasonable critic, he would attack the second premise, as the "weakest link," and he would not find the conclusion any more plausible than he finds the (weak) second premise, even though he may agree that the first premise is highly plausible. And it is the second premise that Arlene needs to defend. So it can be appreciated why the least plausible premise rule is an appropriate rule of plausible reasoning in persuasion dialogue for valid linked arguments, like the one above. This argument is a linked argument because each premise fits together with the other to support the con- clusion. Both premises are required to sup- port the conclusion, and neither premise appears to render the other premise implausible for the respondent (or at least so we may presume, from what we know of the position of the respondent, on the information available to us as critics). However, now let us contrast a case of a linked argument with a case of conver- 42 Douglas Walton gent argument. In the linked argument below, the two premises go together to support the conclusion. Whereas in the convergent argument, the second premise does not depend on the first, or vice versa. Each premise is an independent item of evidence to support the conclusion. Case 4: There is smoke coming from the University. If there is smoke coming from the University, then there is a fire in the University. Therefore, there is a fire in the University. This example is a linked argument, because each premise goes along with the other to help support the conclusion. In the linked argument, if one premise is weaker, then the conclusion is only made as plausi- ble, through the argument, as this weaker premise. For example, in the linked argu- ment above, if the first premise is highly plausible, but the second premise is only weakly plausible, then the conclusion is only made weakly plausible by the argument. However, in a convergent argument, each premise is a separate line of evidence, independent of the other premises. There- fore the conclusion is made as plausible as the most plausible premise, if the argument is valid. This principle is illustrated in the following example. Case 5: Virgil said sincerely that there is a fire in the University. Vanessa said sincerely that there is a fire in the University. Therefore, there is a fire in the University. This example is a convergent argument, for each premise individually constitutes a plausible argument for the conclusion without requiring the support of the other premise. Now let us suppose that Virgil is a highly reliable source on the subject of the fire in the University, and that Vanessa is a less reliable source. Suppose, in other words, that the first premise is highly plausible, but the second premise is only slightly plausible. What plausibility value should we assign to the conclusion? Clearly, we can infer that the conclusion is highly plausible, that it is at least as plausible as the first premise. In short, the new rule is the following. PLAUSIBILITY RULE FOR CONVER- GENT ARGUMENTS: In a convergent argument, the conclusion is at least as plausible as the most plausible premise. This rule then contrasts with the case of the linked argument, where the conclusion is assigned a plausibility value at least as great as the least plausible premise. A complication is introduced through the fact that linked and convergent argu- ments can be combined, as below. Case 6: (j) A passerby reported smoke coming from the University. @ If a passerby reported smoke coming from the University, then there is a fire in the University. @ The Fire Chief reported a fire in the University. @) If the Fire Chief reported a fire in the University. there is a fire in the University. tID Therefore, there is a fire in the University. This example is a case of two linked argu- ments joined together in a convergent argument, as shown below. \! 0--.. ·0 ...... ~2---!