Synthese (2010) 175:327349DOI 10.1007/s11229-009-9512-1

Petitio principii and circular argumentation as seenfrom a theory of dialectical structures

Gregor Betz

Received: 9 August 2008 / Accepted: 22 March 2009 / Published online: 5 April 2009 Springer Science+Business Media B.V. 2009

Abstract This paper investigates in how far a theory of dialectical structures shedsnew light on the old problem of giving a satisfying account of the fallacy of petitioprincipii, or begging the question. It defends that (i) circular argumentation on theone hand and petitio principii on the other hand are two distinct features of com-plex argumentation, and that (ii) it is impossible to make general statements about thedefectiveness of an argumentation that exhibits these features. Such an argumentation,in contrast, has to be evaluated on a case-by-case basis. Petitio principii, this paperthence suggests, is one name for, in fact, a multitude of different and quite complexdialectical situations which require specific analysis and evaluation.

Keywords Petitio principii Begging the question Fallacy Dialectical structure Dialectics Argumentation

1 Introduction

Petitio principii, or the fallacy of begging the question, is an important concept of oureveryday argumentative practice. Despite the fact that it has been studied extensivelyin argumentation theory,1 no consensus view on how to analyze that fallacy has yetemerged. In particular, it is still controversial as to whether begging the question isbest analyzed in epistemic or in dialectical terms. This paper, by embedding the prob-lem within the normative framework of a theory of dialectical structures,2 attempts todevelop a new proposal which is not committed to one of these views.

1 Walton (2006) gives a review of current research.2

Theory of dialectical structures refers to an approach to argument reconstruction and evaluation(Betz cf. 2005, 2008, forthcoming). Key concepts of that approach will be introduced later.

G. Betz (B)Universitt Stuttgart, Stuttgart, Germanye-mail: gregor.betz@philo.uni-stuttgart.de

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Petitio principii, or begging the question, is usually associated with circularargumentation.3 Moreover, such kind of argumentation is, at least traditionally, sup-posed to be fallacious. This paper challenges both views, defending:

(i) Not every circular argumentation begs the question, and not every petitio prin-cipii is based on a circular argument.

(ii) It is impossible to make general statements about the defectiveness of a complexargumentation solely on the basis that it realizes a circular dialectical pattern.Such an argumentation has to be evaluated on a case-by-case basis.

It should be noted, though, that the first conjunct of claim (i) has already beendefended in the literature; accordingly, Walton (2005, p. 89) gives a persuasive exam-ple of a non-fallacious circular reasoning. As regards claim (ii), Wilson (1993) seemsto reach a similar conclusion.4 Moreover, it goes without saying that those claims aredefended relative to the specific approach to argument reconstruction and evaluationthis paper pursues.

Our reasoning takes off from the analysis of a simple and well-known exampletheGod and the Bible case (Sect. 2). Based on this analysis, we will subsequently distin-guish three dialectical patterns which may characterize petitio principii (Sect. 3). Anargumentation, generally, can be evaluated against the discursive aims, as identified bythe theory of dialectical structures, which rational proponents are supposed to pursue(Sect. 4). A case-by-case consideration in Sect. 5 then yields that for every aim andevery dialectical pattern there exist both a fallacious and a perfectly correct argumen-tation. A second, more complex example illustrates how a case-by-case evaluationcan be implemented (Sect. 6). Finally, the conclusion attempts to locate the notion ofpetitio principii in the normative framework this paper sets up. As a preliminarystep, however, the remainder of this introduction reviews the key concepts of a theoryof dialectical structures which will be used later.

The theory of dialectical structures is an approach for reconstructing and evalu-ating complex argumentation and debate.5 These are analyzed as so-called bipolarargumentation frameworks6 consisting in deductively valid arguments (premiss-con-clusion structures) and theses. The set of reconstructed arguments and theses shallbe labeled T . An argument a1 T supports (attacks) an argument a2 T if andonly if the conclusion of a1 is equivalent to (contradicts) a premiss of a2. The sup-port- and attack-relation, U and A respectively, that are thus defined on T makeup the dialectical structure of the debate = T, A, U . A two-colored, directedgraph is an appropriate mathematical model of this structure, and a helpful visualiza-tion, too. Accordingly, arguments are the graphs nodes; a green (red) arrow betweentwo arguments signifies that one supports (attacks) the other. I will use line-styleto express these relations conveniently, i.e. A(a, b) a b and U (a, b)

3 E.g. Walton (2006, p. 278).4

I believe that the problem of analyzing question-begging argument, generally, cannot be satisfactorilycharacterized syntactically; the fallacy is not a formal fallacy. Better prospects for a plausible account liein investigating the pragmatics of argument Wilson (1993, p. 328).5 For a detailed exposition of this approach see references in footnote 2. Sample argument maps whichillustrate this method can be found at http://www.argunet.org.6 See Cayrol and Lagasquie-Schiex (2005a, b) who extend the framework of Dung (1995).

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a b. An argument is said to be -valid, in contrast to deductively valid, if and onlyif it is not attacked by other -valid arguments.7 Premisses which are neither equiv-alent nor contrary to conclusions of -valid arguments are -free, or simply freeas we shall say for the sake of simplicity in this paper. If a sentence p can be deducedfrom other sentences Q by using but the inferential relations explicit in the dialecticalstructure , p is -deducible from Q. Eventually, different proponents might adhereto different arguments in a complex debate. The stance-attribution maps each pro-ponent onto a subset of the debates arguments and theses, thereby specifying for eachproponent which arguments she approves. In order to be a fully specified, rationallyadoptable position, such a subdebate is required to be in equilibrium (i.e., roughly,must not contain contrary claims), closed (i.e. must comprise all arguments in thedebate whose premisses have already been accepted in the subdebate), and complete(i.e. must contain the thesis p or the thesis p if both sentences occur in the debate).8

2 A first example: the god and the bible case

The God and the Bible case is probably one of the most common illustrations of petitioprincipii and circular argumentation.9 Already Descartes, in his letter of dedication inthe Meditations, noted that

[...] although it is quite true that the existence of God is to be believed since it istaught in the sacred Scriptures, and that, on the other hand, the sacred Scripturesare to be believed because they come from God (for since faith is a gift of God, thesame Being who bestows grace to enable us to believe other things, can likewiseimpart of it to enable us to believe his own existence), nevertheless, this cannotbe submitted to infidels, who would consider that the reasoning proceeded in acircle.10

The argumentation is supposed to prove the existence of God by referring to theBible which says that god exists; the trustworthiness of the Bible, in turn, is thencederived from its being the word of God. This informal argumentation can be recon-structed in different ways: we shall consider two versions in the following.

The first version consists of two arguments:Argument A: Bible as testimony

(A1) The Bible represents testimony of a source presumed to know aboutthe existence of God.

(A2) The Bible says that God exists.

7 Thus, a recursive status assignment is defined on the argumentation framework. Compare the review byPrakken and Vreeswijk (2001) for alternative approaches.8 Whereas only the requirement of being in equilibrium will figure prominently in this papers argumenta-tion.9 See also Walton (2005, pp. 9799).10 Descartes (1996, p. 3).

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(A3) If x represents testimony of a source who is presumed to be in a positionto know and x says that p, then p is the case.

(A4) Thus: God exists.Argument B: God omniscient

(B1) God is omniscient.(B2) If God is omniscient and does exist, then the Bible represents testimony

of a source presumed to know about the existence of God.

(B3) God exists.(B4) Thus: The Bible represents testimony of a source presumed to know

about the existence of God.As premiss (A1) is equivalent to conclusion (B4), argument B supports argument

A, in short B A. Yet, as (A4) is also equivalent to (B3), A supports B, too, and wehave a directed green circle B A B.

Frankly, argument B is not really a very sophisticated reconstruction and its plaincircularity does not really seem to do justice to the informal argumentation. Thus thefollowing second version:

Argument C: God omniscient(C1) The Bible is the word of God. (The Bible represents testimony of God.)(C2) God is omniscient. (God is presumed to know about everything.)(C3) Thus: The Bible represents testimony of a source presumed to know

about the existence of God.In contrast to B, C requires but two premisses to warrant (A1). Also, the argumenta-

tions circularity in C A is not as apparent as in B A. C A does in particularnot give rise to a directed green circle. Walton, whose reconstruction I tried to capturewith arguments A and C, refers to the notions of equivalency and dependency circular-ity in order to distinguish between such cases.11 Accordingly, B A is an instanceof equivalency circularity, whereas in C A the conclusion (A4) and the crucialpremiss (C1) are not equivalent, but one depends on the other, in a certain sense.12The following analysis will give us a clearer idea about what kinds of dependenciesare involved in circular argumentation.

In C A, the crucial premiss (C1) is not equivalent to the central conclusion(A4), yet it somehow presupposes its truth, or, more specifically, (A4) is a necessarycondition for (C1). This means: (C1)(A4) is analytic.13 So the conclusion which

11 See Walton (2005) and Woods and Walton (1975).12 Cited from Walton (2005, p. 87). Yet, the notions of equivalency and dependency circularity are notcoherently defined in the literature. In Walton (2006), for instance, dependency circularity is said to holdiff the line of reasoning used to support the premise requires the conclusion as a part of the reasoning(p. 281). In that case, however, dependency circularity would simply reduce to equivalency circularity if thesupporting argumentation were explicitly reconstructed. Ritola (2001), though, defines: The dependencyconception means, roughly, that an argument begs the question because at least one of the premises of theargument is dependent on the conclusion, i.e. cannot be true, reasonably believed or accepted unless theconclusion is also true. (p. 295) We shall see that this later interpretation hints at the distinction betweenmaximal redundant and circular argumentation I introduce below.13 (C1) is only true if its names do refer, and if the name God refers then God exists.

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is supposed to be warranted by the argumentation follows from one premiss, namely(C1), plus an analytic truth alone, as the following argument makes explicit:

Argument D: Short-circuit(D1) The Bible is the word of God. (The Bible represents testimony of God.)(D2) If the Bible is the word of God, God exists. (analytic)(D3) Thus: God exists.

The original arguments A and C, argument D which pinpoints the dependency, plustwo theses that state (A4) and (C1) make up the following dialectical structure 1:

This diagram shows that arguing for (A4) via C and A is redundant in the sensethat there is a shorter inferential path that leads to (A4), namely via D. Is this a generalcharacteristic of an argumentation we use to brand as petitio? Is not circularity in thesense of a directed green circle but the fact that an argumentation can be short-circuitedby adding but analytic premisses what defines petitio? As a first step to sort this out,the next section systematizes the dialectical patterns at issue.

3 Three types of redundant dialectical patterns

First, a word on the notion of a dialectical pattern. Dialectical patterns are realizedby dialectical structures. A green directed circle or a vicious circle14 are examplesof dialectical patterns. Types of argumentation can be distinguished by the dialecti-cal patterns their corresponding dialectical structures realize. Thus, a directed greencircle is, according to its first reconstruction, a feature of the God and the Bibleargumentation.

In the previous section we have rather unspecifically diagnosed a certain redun-dancy in the second reconstruction of the God and the Bible case. Let us now definea redundant argumentation as follows: A green tree (i.e. a dialectical structure withgreen edges, no circles and one sink) is called a redundant argumentation if and onlyif its central conclusion follows from a proper subset of the structures free, that isunsupported, premisses plus, possibly, additional analytic sentences.

An apparently interesting special case of a redundant argumentation occurs whenthe central conclusion follows from a single free premiss alone plus, possibly, addi-tional analytic sentences. We shall call such an argumentation maximal redundant. If,moreover, the single free premiss that warrants the central conclusion is equivalent to

14 Compare Betz (2005).

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that conclusion, the argumentation is called circular. The following diagrams depictthe three dialectical patterns.

Redundant argumentation ((p r) q analytic), 2:

Maximal redundant argumentation (p q analytic), 3:

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Circular argumentation, in an analogous diagram, 4:

Note that this last diagram illustrates that a circular argumentation is a special caseof redundant argumentation, on the one hand, and involves a directed green circle, onthe other hand.

The second reconstruction of the God and the Bible case as depicted in 1 is anexample of maximal redundancy (3) whereas the first reconstruction (arguments Aand B) exemplifies a circular argumentation (4).

In the remainder of this section I would like to comment on some peculiarities ofthese dialectical patterns.

First of all, it might have occurred to the reader that we have so far been dealing withgreen graphs only. Does circularity, however, not arise in debates where proponentsdo support as well as attack other arguments, too? This is certainly the case and thatis the reason why it is interesting to note that the pattern depicted in 3 is equivalenttoi.e. the debate can equally be reconstructed asthe following structure 3:

or 3 :

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which include red edges besides green ones.The further analysis of maximal redundant argumentation in the next section applies

thus to dialectical structures that realize patterns 3 and 3 as well. Our discussion ishence not limited to green graphs only.

As a next point, consider circular argumentation. Green circles are exempted froma theorem on dialectical structures15which stated roughly that in a given dialecti-cal structure a sentence can be -deduced from the free premisses if and only if itis the conclusion of a -valid argumentfor reasons that relate to the proof-methodapplied. Reconsidering green directed circles we see that the theorem is actually falsefor these structures: Although all arguments in a (isolated) green circle, for not beingattacked, are -valid, the truth of the free premisses doesnt warrant the truth of anyconclusion. In a circular argumentation, the premisses truth guarantees only that thearguments conclusions are equivalent. In fact, it is a common proof-strategy for equiv-alence claims in mathematics to construct such a circular argumentation. As this iswell-known, scholars have wondered how to distinguish equivalence proofs whosevalidity is beyond any doubt on one side, and circular argumentation which seem tobe, to put it mildly, at least questionable, on the other side. I will briefly indicate howthese cases can be dealt with within a theory of dialectical structures. Assume that thecentral theses in a debate are ck and cl (k = l and 1 k, l n) and that we wantto distinguish a proponent who argues in a circular way for ck and a proponent who,on the same grounds, just holds that ck and cl are equivalent. What helps is that thereasoning can be reconstructed in two different ways, namely as arguments a1 . . . anor as argument an+1:

15 Which is proven in Betz (forthcoming).

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where an+1 has the following form:Argument E: Circular equivalence proof

1) ck 2) pk+1

.

.

.

i) pli+1) Thus (from 1 i): cli+2) Thus (from 1, i+1): ck cli+3) cl i+4) pl+1

.

.

.

j) pkj+1) Thus (from i+3 j): ckj+2) Thus (from i+3, j+1): cl ckj+3) Thus (from i+2, j+2): ck cl

In the first case, circularity is a feature of the dialectical structure whereas in thesecond case, the circular reasoning occurs in the inference pattern of the argument(sharp brackets indicate assumptions for the sake of the argument). A proponent whomerely claims that ck and cl are equivalent asserts but the argument an+1 whereasa proponent who additionally maintains that ck and cl are true would also stick toarguments a1 . . . an . Thus, the different reconstructions of circular reasonings as aninference pattern or as a dialectical pattern plus the stance attribution over a debatedistinguish the two cases at hand.

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The importance of being able to distinguish between these cases stems from ourintuition that one circular reasoning (the mathematicians) is unproblematic whereasthe other one isnt. But what exactly is the problem with circularity or, more generally,redundancy? This is the question to be dealt with in the next two sections.

4 Evaluating redundant argumentation

Assume that once we have reconstructed the arguments put forward in a debate wesee that the dialectical structure realizes a circular argumentation. What does this tellus about the debate? Did any of the proponents make a mistake? Not necessarily. Infact, the bare dialectical structure doesnt allow for any such inference: Dialecticalstructures are evaluatively underdetermined. The thesis of evaluative underdetermin-ation can be substantiated by a simple example. Reconsider structure 4 and assumethat Anna argues with Bianca and Chris. Anna holds both q and p2 whereas Biancabelieves but q while Chris thinks that p2 is true and q isnt. Now, in order to convinceBianca (that p2) Anna introduces argument a1 and in order to convince Chris (that q)she establishes argument a2. Annas argumentation is perfectly fine, yet it gives riseto a green circle. Dialectical structures alone dont tell us whether something goeswrong in a debate.

In addition to (i) the dialectical structure, (ii) the stance attribution, that is whoasserts which arguments, as well as (iii) the purpose of the respective arguments, or,as I would prefer to put it, the discursive aims the proponents are pursuing, have to betaken into account when evaluating redundant argumentation. This is not new. Waltonrecently reaffirmed this insight of argumentation theory: It is assumed that puttingforward an argument in a dialogue has a purpose, and that the success or failure of theargument can be judged in relation to the fulfillment of that purpose.16 What seemsto be, in contrast, controversial is the nature of the purpose arguments are supposed tofulfill: Are, in particular, epistemic purposes necessarily involved in an evaluation ordo they even have priority over other purposes, e.g. conversational ones?17 I will notcontribute to that particular dispute. Rather, I want to discuss the success or failureof redundant argumentation in the light of the discursive aims proponents can ratio-nally pursue according to the theory of dialectical structures. All these aims can beequally interpreted as epistemic in a broad sense, or as dialectical. What is interestingand new is that, according to the theory of dialectical structures, there is not merelyone epistemic (or dialectical) purpose of argumentation, namely to get the other partyto accept a proposition that s/he doubts,18 but a plentitude of purposes. Table 1 is asystematic and slightly simplified exposition of these discursive aims.

These different discursive aims are not on an equal footing. Categories I and IIappear to be rather technical compared to III and IV. Moreover, I and II are hardlyideas proponents have consciously in mind when debating. This might suggest the fol-lowing grading: Whereas aims I and II guide our interpretation of a debate and serve to

16 Walton (2005, p. 87)17 See Wilson (1988), Ritola (2001).18 Walton (2005, p. 91)

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Table 1 The system of discursive aims (cf. Betz, forthcoming)

(a) Defensive (b) Offensive

I Completeness I assert exactly one of a pair ofcontradictory sentences occurringin the debate.

II Closedness There is no argument in the debate Ido not assert but which I had toassert given the sentences I havealready accepted.

III Equilibrium The sentencesa which occur in thesubdebate I assert are consistent.

The sentencesa which occur in thesubdebates of my opponents areinconsistent.

IV Burden of proof The central theses I want to maintainare -deducible from the freepremisses in my subdebate. (1storder burden of proof)

The central theses I want to maintainare -deducible from the freepremisses in my opponentssubdebates. (2nd order burden ofproof)

a More precisely: the free premisses and the conclusions of -valid arguments

determine the different proponents stances in the first place, i.e. represent standardsof adequate reconstruction which are thence fulfilled qua rational reconstruction ofthe debate, this does not hold for aims III and IV: Even in an as rational reconstructeddebate, they are not necessarily attained. The situation is similar to the evaluation ofsingle arguments. As we reconstruct arguments as deductively valid, formal fallaciesare excluded qua reconstruction and deductive validity is not an aim proponents canfail to meet in an as rational reconstructed debate.19

With categories III and IV left, there are altogether still four different epistemicaims proponents pursue when introducing new arguments into a debate. The next sec-tion will investigate in how far the pursuit of these aims might bring about maximalredundant or even circular argumentation on one hand and whether the fact that thedebate realizes one of these dialectical patterns might undermine the achievement ofthese discursive aims on the other hand.

5 A case distinction of redundant argumentation

According to the previous section, proponents can pursue different aims in a debate.Since an argumentation must be evaluated in the light of the purpose it shall serve, thereare two questions we have to address for each dialectical pattern we are interested in:(i) Can the effective pursuit of the respective discursive aim bring about the dialecticalpattern? (ii) Can the achievement of the respective aim be undermined because theargumentation realizes the dialectical pattern? This section discusses these two ques-tions for 8 aim-pattern pairs, mainly by constructing examples of dialectical structuresthat warrant a positive answer. The results of this discussion are presented in Table 2,where + and are indicating the affirmation of questions (i) and (ii) respectively.

19 I thank Christian Voigt for reminding me of this analogy.

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Table 2 The evaluation ofdiscursive patterns in the light ofthe discursive aims according tothe theory of dialecticalstructures, see text forexplanation

Circularity Maximal redundancy

(IIIa) +/ +/(IIIb) +/ +/(IVa) +/ +/(IVb) +/ +/

Accordingly, +/ symbolizes that the dialectical pattern can both result from thepursuit of as well as prevent the achievement of the discursive aim. The fact that forevery aim-pattern pair both questions are answered positively is an indication of thedialectical patterns ambivalence: Redundancy and circularity have to be judged ona case-by-case basis because there are both situations where they facilitate or hinderthe achievement of the discursive aimswhere the argumentation which brings themabout is thus rational or not. The systematic investigation spares redundant argumen-tation in order not to inflate the distinction of cases, leaving its discussion to a separateparagraph at the end of the section.

(IIIa)-C: Equilibrium, defensive, circularity To see whether a circular argumentationcan arise because proponents pursue aim (IIIa), consider the following consecutivestates of a dialectical structure that is asserted by a single proponent.

The green circle emerges because of the introduction of arguments a6 and a7 whosepremisses are assumed to be well confirmed in the debate. As a consequence, the pre-viously isolated argument a5 is now firmly embedded in the inter-connected maingraph. Moreover, assume that a6s conclusion is p1 and that a4 and a7 have equiva-lent conclusions. This implies that on the background of the free premisses in state(2), q is equivalent to a1s conclusion. And so is p1. Thus, while p1 and q could

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have contradicted argument a1 in state (1) of the debate, this is impossible in state(2). There is another way to put this idea. Both in (1) and (2) the following condi-tional equilibrium-claim holds: If the set of free premisses is logically consistent, thenthe set of free premisses plus the conclusions does not contain contrary sentences(i.e. the dialectical structure is in equilibrium). Because the green circle, however,reduces the set of free premisses significantly and thereby weakens the antecedentconditions, that equilibrium-claim is stronger in (2) than in (1). In sum, the introduc-tion of the green circle supports the proponents goal of asserting a dialectical structurewhich is in equilibrium.

Is it equally possible that a circular argumentation obstructs the achievement ofconsistency? Consider the following two states:

where a2 and a6 have equivalent conclusions. The introduction of a5 and a6 doesntincrease the inferential density of the dialectical structure since the equivalence ofthe arguments conclusions is already established in state (1). While (2) is thereforenot less prone to inconsistency than (1), it introduces in turn arguments with newpremisses which might conflict with other assertions and thence increases the chancesthat the entire debate the proponent asserts is not in equilibrium anymore. Here, the(circular) argumentation has a tendency to obstruct equilibrium.

(IIIa)-Mr: Equilibrium, defensive, maximal redundancy A maximal redundant argu-mentation can promote equilibrium of a proponents subdebate for reasons similar tothose given above, as the following state illustrates (we assume throughout this sectionthat p q is analytic):

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If a2 and a3both resting on supported premissesare simply introduced in orderto integrate a1 into the main debate, this reduces the set of free premisses and thereforewarrants a stronger conditional equilibrium-claim.

Assume, however, that a1, being isolated, is introduced into a debate, yielding thefollowing dialectical structure:

Here a1 doesnt increase the inferential density and causes a premiss-proliferationwhich might in turn trigger new inconsistencies.

(IIIb)-C: Equilibrium, offensive, circularity This is an example where a circular argu-mentation emerges because a proponent attempts to disturb the equilibrium of heropponents subdebate:

Consider state (1), first. There are two proponents, Anna and Bianca. Anna believesp and q, as well as a1 and a2. Bianca believes q, but, strongly rejecting the conclusionof a1, denies p which she attacks with argument a3. With a view to demonstrating

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that Biancas subdebate is not in equilibrium, Anna introduces argument a4 that restson premisses Bianca has acknowledged elsewhere, including q. That is how the circlearises in state (2).

The failure, however, to disturb the equilibrium of an opponents subdebate canalso be linked to circular argumentation.

Here, Chris attacks p with a1. Dave, in an attempt to expose Chris inconsistency,introduces in state (2) arguments a2 . . . a4 whose premisses which are not supported inthe circle are conclusions of other arguments Chris approves. Yet Dave failed to showthat Chris subdebate is not in equilibrium: Rejecting the newly introduced argumentsconclusions, and thereby a premiss of each, he can resist accepting them as -valid.

(IIIb)-Mr: Equilibrium, offensive, maximal redundancy To see how a maximal redun-dant argumentation can arise when proponents try to disturb their opponents subde-bates equilibria, consider the following dialectical structure.

Assume that both Anna and Chris approve p and q. Morover, Anna holds that rand a1, the latter being her reason to assert q. Here, Chris disagrees, backing up his

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belief with a2. Anna exposes Chris inconsistency with argument a3 that rests on p,which Chris approves, plus other premisses he has already accepted.

In a slightly modified dialectical context, Anna would have failed to do so. Consider

In contrast to the previous debate, Chris attacks q. Then, Annas introducing a3is bound to fail: Chris wont accept p. And if he did, his subdebate would not havebeen in equilibrium anyway because of the short-circuit argument. So introducing a3fails to achieve the desired goal because the dialectical structure realizes the patternof maximal redundant argumentation.

(IVa)-C: 1st order burden of proof, circularity A circular argumentation can arisewhen several proponents successfully fulfill their 1st order burdens of proof. Consider

where Anna, Bianca and Chris are uncertain about p, q, and r respectively, whilebelieving the remaining theses firmly. They meet their first order burdens of proof byintroducing a1, . . . , a3, respectively, which, putting the arguments together, results inthe green circle.

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Circular argumentation, however, can severely obstruct the achievement of this aim,too. Assume Dave were sceptical about p and started to convince himself by puttingforward arguments a1, a2, . . . finally ending up at an argument with premiss p. In thatcase, p is not -deducible from the free premisses and Dave failed to meet his 1storder burden of proof because of circularity.

(IVa)-Mr: 1st order burden of proof, maximal redundancy Maximal redundancy canstem from the pursuit of 1st order burdens of proof in a multi-proponent debate.

Assume that Anna is not sure whether q, approving r while denying p. Chris, incontrast, approves both p and q, and wants to be certain that r . Anna and Chris fulfilltheir first order burdens of proof by introducing a1 and a2, respectively.

To see that an argumentation can be irrational with a view to 1st order burdens ofproof because it is maximal redundant, reconsider the above structure but suppose thatthere were but one proponent, Bianca, who tries to convince herself that q by introduc-ing a1 and a2. Now Bianca either believes that p or she doesnt. If she does, then thearguments a1, a2 do not contribute to satisfying her 1st order burden of proof as q fol-lows directly from p. If, however, she doesnt believe that p, the argument chain a1, a2,presupposing p, is of no use, either. Thus, the maximal redundant argumentation isnot effective.

(IVb)-C: 2nd order burden of proof, circularity In a many-proponent debate, circlesmay arise when proponents try to meet their 2nd order burdens of proof. Reconsiderthe dialectical structure given in (IVa)-C. Assume that Anna, Bianca and Chris eachapprove of exactly one of the theses, namely p, q, and r , respectively. Dave, however,holds that all of these theses are true. With a view to his 2nd order burden of proof, heeffectively puts forward a1, a2, a3.

Suppose, in contrast, that Bianca, in another context, argues with Chris alone andthat she tries to convince him that p. If she puts forward a green argument circle, thiswill hardly compell Chris to approve p as he can refute the arguments conclusionsaltogether. This can be put differently: The last argument Bianca introduces which

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contains premiss p and closes the green circle is not effective in the light of her 2ndorder burden of proof.

(IVb)-Mr: 2nd order burden of proof, maximal redundancy It is not too difficult totransfer our previous reasonings to this aim-pattern pair. Reconsider the dialecticalstructure depicted in (IVa)-Mr and assume that Anna accepts p and q but denies rand Dave approves r , yet neither p nor q. Chris holds that p, q and r , and thence putsforward a1 to fulfill his 2nd order burden of proof with respect to q and Dave, and a2to fulfill his 2nd order burden of proof with respect to r and Anna. This is effectiveand rational.

Still, assume Dave didnt approve r . Chris introducing a1, a2 would hardly changeDaves mind with regard to q. He doesnt believe p. Even if he did, Chris argument-chain would be useless as q follows from p alone. Chris argumentation would in anycase be irrational because it is maximal redundant.

Eventually, a note on redundant argumentation in general. Can a redundant argu-mentation emerge from the effective pursuit of discursive aims? Can it, on the contrary,hinder their achievement? First of all, recall that every circular and every maximalredundant argumentation is a redundant argumentation. The answer to both questionsis therefore positive. Now what about redundant argumentation that is not maximalredundant (and thence not circular)? The above investigation implies by an argumen-tum a minore that such an argumentation might arise in purposeful discussions. Yet,might it also be irrational, in the light of some discursive aims, to argue redundantly?This is at least indicated by the fact that the dialectical pattern of redundant argumen-tation (2) can be transformed to the pattern of maximal redundant argumentation (3)and vice versa: If one reconstructs the two arguments on the right hand side in 2 as oneargument with conclusion q that contains a premiss p r , the argumentation is indeedmaximal redundant because the conclusion follows from one of the premisses, i.e.p r , alone. Likewise, splitting up p into two sentences whose conjunction is equiv-alent to p, a maximal redundant argumentation might be reconstructed as a redundantone. If redundant and maximal redundant argumentation can be transformed into eachother and the latter is in some cases irrational, so is (probably) the former.

Thus far our theoretical investigation into whether circular and maximal redun-dant argumentation are irrational or not. The next section shall apply our results to anexample of a complex multi-proponent debate.

6 A second example: the scientific relativist

Redundant argumentation is not merely a theoretical possibilityit represents a dia-lectical pattern which does occur in real debates. Whether such an argumentation ispurposeful and thence rational has to be judged, according to the results of the pre-vious section, on a case-by-case basis. We will in the following have a closer look ata specific circular argumentation and see how such a case-specific judgment can becarried out.

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The example is taken from Larry Laudans ingenious book Science and Relativism:Some Key Controversies in the Philosophy of Science20 which is a fictitious dialoguebetween four philosophers of science, namely Quincy Rortabender (the relativist),Percy Lauwey (the pragmatist), Rudy Reichfeigl (the positivist) and Karl Selnam(the realist).21 One of the pivotal claims these philosophers discuss is the thesis ofunderdetermination, which, in its general version, states

Relativist: [] Strong relativism, in contrast, would hold that evidence is alwayspowerless to choose between any pair of [rival theories].22

After briefly touching the question whether this is an unreasonably strong versionof underdetermination, the discussion proceeds as follows:

Pragmatist: I dont think anyone will accuse you of taking up an easy brief.Perhaps we could now get to the solid arguments you promised us on behalf offull-blown relativism?

Relativist: The arguments are not only solid but they have the virtue of beingstraightforward. I begin by reminding you of Humes claim to the effect that nogenuinely universal statement (and I mean such statements when I talk abouttheories or hypotheses) can be deduced from a finite set of its positiveinstances. Because that is so, no theory can ever be proved to be true.

Positivist: Big deal.

Relativist: Moreover, and this point was made by Duhem almost a century ago,we can no more derive the falsity of a theory or hypothesis from the evidencethan we can derive its truth.

Realist: Youre referring, I suppose, to the holistic character of the testing situ-ation and the fact that we bring many assumptions to bear in the design of anytest of a single theory or hypothesis.

Relativist: Exactly. It is never single hypotheses or theories we test but entirenetworks of such assumptions. When the test fails, that is, when the predictedresult differs significantly from the observed result, all we know, at best, is thatwe have made a mistake somewhere; it cannot be further localized.

Pragmatist: I have serious doubts about this holism of yours, which I hope Iwill have the chance to voice later; but suppose we grant you, for now, both theHume and the Duhem theses about the nonderivability, and the nonrefutability,of theories from their positive, or negative, instances. How does this establishyour thesis that it is as reasonable to accept any theory as any other?

20 Laudan (1990).21 My example is the slightly modified reconstruction from a project that drew up the dialectical structureof Laudans entire book. Id therefore like to thank Friederike Greb, Bianca Drefahl, Eugen Pissarskoi,Martin Ebert and Gema Ortiz del Castillo!22 Laudan (1990, pp. 5556).

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Relativist: I would have thought the connection was more or less self-evident. Ifwe can never show that any theory is true, and we can never show that any theoryis false, then surely it is clear that whatever grounds we may have for choosingbetween rival theories must be practical rather than epistemic, or simply mattersof convention and simplicity. Since there is nothing in the evidence which forcesus to believe one thing rather than another, we can choose our beliefs in the lightof our private and personal interests.23

The relativists reasoning can be reconstructed as two consecutive arguments. Thefirst one starts from the Hume and the Duhem thesis and warrants that neither thetruth nor the falsity of a theory can be deductively derived from empirical evidence.The second one, building on this conclusion, claims that, as a consequence, empiricalevidence cannot decide between rivals.

Argument F: Hume-Duhem(F1) No theory can be deduced from a finite set of empirical observations.(F2) Our empirical evidence is, necessarily, finite.(F3) Thus: No theory can be deduced from empirical evidence. (Hume thesis)(F4) Every test of a theory assumes other theories to be true. (Holistic

character of testing situation)(F5) A theorys falsity could only be derived from empirical evidence, if

testing theories would rely on empirical evidence alone and not requireother, non-observational statements to hold.

(F6) Thus: A theorys falsity cannot be derived from empirical evidence.(Duhem thesis.)

(F7) Thus: Neither the truth nor the falsity of a theory can be deductivelyderived from empirical evidence.

(F8) If neither the truth nor the falsity of a theory can be deductively derivedfrom empirical evidence, then nothing in the evidence forces us to beliefone rather than another theory.

(F9) Thus: There is nothing in the evidence that forces us to belief one ratherthan another theory.

Argument G: Underdetermination(G1) There is nothing in the evidence that forces us to belief one rather than

another theory.

(G2) If there is nothing in the evidence that forces us to belief one rather thananother theory, empirical evidence is powerless to choose between rivaltheories.

(G3) Thus: Empirical evidence is powerless to choose between rivaltheories.

Moreover, the relativist highlights some further implications of this last conclusion.As, in the first chapter of the book, he holds that theory change is irrational and thatthere is no scientific progress, it is not far fetched to assign the following argument tothe relativist, too:

23 Laudan (1990, pp. 5657).

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Argument H: No progress(H1) Empirical evidence is powerless to choose between rival theories.(H2) If empirical evidence cannot choose between rivals, theory choice involves

arbitrary criteria.

(H3) If theory choice involves arbitrary criteria, there is no objective scientificprogress.

(H4) Thus: There is no objective scientific progress.The proponents continue discussing several objections. Among others, certain

methodological rules were proposed by the pragmatist and the realist whereupon therelativist asserts:

Relativist: [] Indeed, as far as Im concerned all the rules that Karl and Percyallude to are just conventions for doing science. They have no objective ground-ing in the facts of the matter and simply serve as convenient instruments forpromoting a certain kind of epistemic interest.24

Why does the relativist insist on this claim? Because its negation threatens his thesisthat there is no scientific progress, provided he assumes that if there were objectivelygrounded scientific rules, then there would also be scientific progress. Now there is afurther argument emerging around this very sentence, building on the conclusion ofargument H:

Argument I: No objective methodological rules(I1) There is no objective scientific progress.(I2) If there were objectively grounded scientific rules, then there would also

be scientific progress.

(I3) Thus: There are no objectively grounded scientific rules.However, (G1) seems to be just a special case of the statement that there are no

objective scientific rules; such rules, were they objective, would apparently force cer-tain beliefs upon us. Thus

Argument J: Evidence doesnt force belief upon us(J1) There are no objectively grounded scientific rules.(J2) Only the existence of objectively grounded scientific rules could

compell us to adopt certain beliefs rather than others in the light ofempirical evidence.

(J3) Thus: There is nothing in the evidence that forces us to belief one ratherthan another theory.

Taken together, arguments F-J make up the following dialectical structure:

24 Laudan (1990, p. 66).

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As the relativist, explicitly or implicitly, subscribes to arguments F-J, the questionarises as to whether he makes a mistake, whether he argues in a rational way at all.First of all, in the specific context of the debate cited above, the relativist puts for-ward only F and G in order to fulfill his 1st and 2nd order burdens of proof. H,I and J have been reconstructed in the light of other sections of the book. Thence, theargumentation is not fallacious as far as burdens of proof are concerned. Yet, is it insome sense purposeful, and with regard to which discursive aim? The relativist doesnot explicitly comment on this. Still, the green circle serves definitely some purpose:It demonstrates that the relativists claims and argumentation cohere; technically, thecircular argumentation increases the chances that the relativists subdebate is in equi-librium. Putting forward F-J is thence, for the relativist, in the wider context of theentire debate, perfectly rational.

7 Conclusion

Analyzing a simple argumentation, we have seen that the fallacy of petitio princip-ii can be characterized in different ways which led us to distinguish the dialecticalpatterns of redundant, maximal redundant and circular argumentation. The mere fact,however, that an argumentation realizes one of these dialectical patterns doesnt tellwhether something went wrong in the debate or not. Dialectical structures, we put it,are evaluatively underdetermined. This has further implications. First of all, argumen-tation has to be evaluated with regard to the purpose it is supposed to fulfill; wherethe theory of dialectical structures identifies what (kinds of) discursive aims propo-nents can pursue in a debate. Still, for every discursive aim there are debates wherethe respective dialectical patterns emerge because the proponents successfully pursuethe discursive aim as well as debates where the respective dialectical patterns are thereason why proponents fail to achieve that aim. As a consequence, secondly, the dia-lectical context, that is the rest of the dialectical structure that realizes the dialecticalpattern, as well as the stance attribution have to be taken into account when evaluatingredundant argumentation. In other words, such evaluations have to be accomplishedon a case-by-case basis and general statements about the defectiveness of redundantargumentation, including maximal redundant and circular argumentation, cannot bemade.

So let us return to our starting point. What is a petitio principii? Can we locatethis informal everyday concept in our analysis? Our investigation suggests the fol-lowing two alternative explications. One commits a petitio in a narrow sense if onefails to satisfy a 2nd order burden of proof because ones argumentation is maximalredundant, possibly circular. These are the negative examples given in Sect. 5 undercase (IVb). A petitio, in a broad sense, however, refers to every argumentation whichfails to achieve its aim, no matter what this aim, because it is maximally redundant,possibly circular. Accordingly, every negative example in Sect. 5 is a petitio in thisbroad sense.

Acknowledgements I would like to thank the -Klubs participants at the Institute of Philosophy, FreieUniversitaet Berlin, for a constructive discussion of ideas this paper unfolds.

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Petitio principii and circular argumentation as seen from a theory of dialectical structuresAbstract1 Introduction2 A first example: the god and the bible case3 Three types of redundant dialectical patterns4 Evaluating redundant argumentation5 A case distinction of redundant argumentation6 A second example: the scientific relativist7 ConclusionAcknowledgementsReferences

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