The Petitio: Aristotle's Five Ways

Canadian Journal of Philosophy

The Petitio: Aristotle's Five WaysAuthor(s): John Woods and Douglas WaltonSource: Canadian Journal of Philosophy, Vol. 12, No. 1 (Mar., 1982), pp. 77-100Published by: Canadian Journal of PhilosophyStable URL: http://www.jstor.org/stable/40231241 .

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CANADIAN JOURNAL OF PHILOSOPHY Volume XII, Number 7, March 1982

The Petitio: Aristotle's Five Ways

JOHN WOODS and DOUGLAS WALTON, University of Lethbridge, University of Winnipeg*

If one looks to the current textbook lore for reliable taxonomic and

analytical information about the petitio principii, one is met with con-

ceptual disarray and much too much nonsense. The present writers have recently attempted to furnish the beginnings of a theoretical reconstruction of this fallacy which is at once faithful to its formidable

complexity yet useful as guide for its detection and avoidance.1 The fact is that the petitio has had a lengthy and interesting history, and in this

paper we shall want to explore certain features of its development, such as it may have been. The principal origins of the concept of circular

argument are to be found in Aristotle. The Aristotelian doctrine recurs with variations in the sophismata literature of the middle ages and in

logic texts and manuals right up to the present day. A particularly significant deviation from Aristotle's approach makes its appearance in astute

* Research for this paper was supported in part by a grant from the Social Sciences and Humanities Research Council of Canada, for which the authors express their gratitude.

1 See Hamblin (1970) and Woods and Walton (1975).

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John Woods and Douglas Walton

and challenging remarks by De Morgan in his Formal Logic. This issue posed by De Morgan's treatment is also reflected in J.S. Mill's famous dictum that all deductive (syllogistic) logic is circular.

The complete and careful history of the petitio remains to be done, as a study of importance in its own right. Though we do not attempt the task here, we shall look to Aristotle as the major exponent of the view that circular argument is an identifiable species of incorrect argument worthy of serious investigation. For present purposes we are not so deeply interested in the historico-exegetical examination of the textual evidence of what Aristotle really wrote, or really meant, as we are with achieving some insight and guidance that might enable us to reconstruct a coherent conceptual background of the fallacy.

1 . Pattern of the Standard Treatment

Many traditional and contemporary logic texts and manuals describe the fallacy of begging the question as the blunder of illicitly assuming what is to be proved. The conclusion of an argument, we are told, may not itself be taken as a part of the evidentiary basis of a premiss of the argument.

The fallacy has various traditional names. Hamblin (1970, 32) states that the petitio principii might be explained by noting that 'principium petere' is the vulgate translation of Aristotle's original Greek to zB oqxy\ aheiÔai, which means something rather like, 'beg for that which is the question-at-issue' - whence the familiar name, 'begging the question.' In a disputation of the Greek pattern, if one person proposes an argument to another, he may ask to be granted premisses on which to build. Thus, according to Hamblin (1970, 33), 'the fallacy consists in asking to be granted the question-at-issue, which one has set out to prove.'

In Woods and Walton (1975) it is suggested that two broad conceptions of the petitio dominate, but that in both cases the key notion involved is unclear and problematic. It is now not so clear whether this suggestion is sound; but we will briefly develop it here, and then consider a potentially more promising way of drawing the original distinction. The original distinction was between two conceptions of the petitio, the dependency conception, and the equivalence conception. According to the DEPENDENCY CONCEPTION, an argument is circular if some premiss actually depends on the conclusion as part of the evidential backing of that premiss. The metaphor of the circle can easily be appreciated. For normally the direction of an argument is from the premisses towards the conclusion;

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however If the premisses somehow depend on the conclusion, the argument must also be directed the other way,

and we are driven back in a circle. A more exact geometrical analogy would be that of two points on the circumference of a close figure. Any route from one to the other means that there is also a route back to the first. Exponents of the dependency conception very often cast their

description in epistemic terms - an argument is said to be noncircular

only if one may know that each premiss is true without knowing that the conclusion is true. The suggestion is that there must be available some

epistemic route to the premisses, with no compulsory detour through the conclusion. One must be able to establish that the premisses are true without having to infer (any of) them from the conclusion, or from some proposition that can be known only by inference from the conclusion.

According to the EQUIVALENCE CONCEPTION, an argument is circular where the conclusion is tacitly or explicitly assumed as one of the

premisses, i.e. where the conclusion is equivalent to, or even a repeti- tion of, a premiss. Thus Copi (1972, 4th ed.):

If one assumes as a premiss for his argument the very conclusion he intends to

prove, the fallacy committed is that of petitio principii, or begging the question. If the proposition to be established is formulated in exactly the same

words both as premiss and as conclusion, the mistake would be so glaring as to

deceive no one. Often, however, two formulations can be sufficiently different

to obscure the fact that one and the same proposition occurs both as premiss and conclusion.

Here the problem is that the required notion of 'equivalence' bears a non-trivial epistemic load. Just how 'close' do two propositions have to

be to cause an argument in which they occur as premiss and conclusion to beg the question? Just as in the first conception the basic notion of

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dependency is not defined yet clearly exceeds ordinary logical implication, so too with the equivalence conception: more than ordinary logical equivalence would seem to be involved. But again, an explication of the appropriate relation of equivalential propinquity is left un- done.

We saw above that the geometrical analogy of the dependency idea is that of a pair of points on a closed space. The idea behind the equivalence conception is that an adequate understanding of what is stated by a premiss takes us back to, and comprehends, the very conclusion itself. The idea of a circle or loop closed figure is agains the basis of the trouble: if we start at some point, then, the figure being closed, we must eventually come back to that same point.

o The root notion of the closed loop suggests that both the dependency and the equivalence conceptions have something in common, and analysis may eventually establish that, at bottom, the two conceptions specify one and the same fallacy.

A second dichotomy that permeates the traditional literature on the petitio is that between (a) an epistemic, as opposed to (b) a games- theoretic idiom of analysis. This second fundamental dualism can best be understood by looking to Aristotle's original account of fallacy.

2. Second Thoughts about the Dependency and Equivalence Concep- tions.2

We used to think of the Dependency and Equivalence Conceptions of the petitio as two different and competing notions of a logical depravity. Moreover, we supposed that each contending idea suffered from conceptual unclarities peculiarly its own. In the one case, there were the

2 For the invitation to which, and some useful guidance, our thanks to a referee of this Journal.

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semantical uncertainties pertaining to 'dependency*; in the other, to 'equivalence'. But this last proposal, that each conception has its own peculiar conceptual challenges, is almost certainly wrong. It is quite straightforward, for example, to formulate the Dependency Concep- tion, so as to involve a reliance upon the notion of equivalence, thus: 'According to the Dependency Conception, an argument is circular if some premiss depends for some of its evidential backing on another premiss equivalent to the conclusion/ And similarly, the Equivalence Conception can be formulated so as to involve the idea of dependency: 'According to the Equivalence Conception, an argument is circular if the conclusion depends for some of its evidential backing on itself.

What the foregoing would seem to suggest is that the distinction between what we have been calling the Dependency and Equivalence Conceptions of the petitio is not so much between two different senses or concepts of circularity (the one dominated by the concept of

dependency, the other by the concept of equivalence), but rather between two different kinds of syntactic structure that can give rise to the

petitio, now considered as we said a few lines back, as a solitary logical depravity. The one syntactic structure can be (crudely) represented this

way:

A /. A

The other (crudely) as follows:

A :. B

B :. A

The first represents an argument, the second an argument chain. And as we shall see in the sequel, this is a distinction of considerable importance for petitio-theory, never mind whether it was what we were im-

perfectly trying to get at with the now questionable distinction between the Equivalance and Dependency Conceptions.

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3. The Five Ways of Question-Begging

According to Hamblin (1970, 74) Aristotle gives two quite different accounts of question begging as between the Prior Analytics and the Topics and De Sophisticis Elenchis. In the latter two works, the context of the question-and-answer debate predominates, and the expression 'beg the question' is therefore appropriate. In the Prior Analytics, however, Aristotle was In the process of developing a context-free Logic of pure form' - so says Hamblin (1970, 74) - and it is understandable that the word 'assume' should so frequently occur there as an alternative to 'beg.' Hamblin finds that the doctrine of the Topics makes little sense when transported into the Prior Analytics. What is at issue in the latter work is the formal validity of argument forms; and in this context the petitio tends to be a rather inconsequential matter. The mutual deducibility of two statements need not be in any interesting way routinely 'fallacious.' We momentarily defer from the difficulties of unravelling the account of the Prior Analytics, and turn now to the more readily intelligible account of the Topics. In Topics 162b34, Aristotle lists five ways of begging the question.

(1) The first occurs were 'any one begs the actual point requiring to be shown' (162b35). Aristotle adds that this version of the fallacy is more apt to escape detection if different terms are used that have the same meaning. What Aristotle seems to have in mind is what we are calling the equivalence conception of the petitio; for, he says, if the premiss and conclusion are in some sense 'close in meaning7 or 'obviously the same statemenf the fallacy itself will be obvious, but if the premiss and conclusion are sufficiently different 'in formulation/ though not in meaning, the fallacy could be difficult to spot. The key question of course is what is meant by 'proximity7 or 'distance' of equivalence- seeming.

(2) The second way 'occurs' whenever one begs universally something which he has to demonstrate in a particular case' (163a1). This form of the petitio, called a universal argument by Woods and Walton (1975), is a commonly cited type of the fallacy. If Bob is asked by Bill to show that something a has property F and Bob replies that everything (in some domain to which a belongs) has F, he may be accused of begging the question unless it is clear that some basis for the universal proposition is available that does not evidentially rest on the singular statement to be proved.

(3) 'A third way is if any one were to beg in particular cases what he undertakes to show universally ...' (163a5). To contemporary ears, this is a less familiar diagnosis and it is difficult to be quite clear as to what Aristotle might have had in mind. Presumably (3) is the converse of (2). Still, it is natural to wonder where is the fallacy in attempting to prove that

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The Petitlo: Aristotle's Five Ways

everything in some domain has property F by successively showing that for individuals a, b, c, ... in the domain, a has F, and b has F, and c has F, and so forth. Arguing for a universal statement on the basis of its instances has well-know inductive shortcomings, but it is hard to see exactly why the petitio must be one of them. Perhaps, as with case (2), the answer is to be sought within the dialectical context of the argument. What was critical in (2) was whether the arguer had available in his pool of admissible evidence some basis for the universal premiss that did not require dependency on the particular conclusion to be proved. So too in (3), it is possible that the wrong that it comments on is that the evidence for each of the particular propositions that is put forward as a premiss is somehow dependent on the universal conclusion to be demonstrated.

(4) The fourth way to beg the question is 'piecemeal/ i.e. to propose an argument for a conjunctive conclusion by claiming each conjunct, as a single premiss. The simplest form of this version of the fallacy is the argument form known as 'conjunction': rP, Q, therefore P A Q.1

(5) The fifth form of the fallacy is to 'beg the one or the other of a pair of statements that necessarily involve one [another7 (163a1 1). We find this specification obscure on two counts. First, is case (5) distinct from case (1), assuming equivalence to be mutual implication? Second, just how do we identify dialectical contexts in which it can be supposed to be fallacious to prove that P implies Q by assuming that Q implies P? True enough, if we assume that P implies Q on the assumption of its converse, then we have in effect established equivalence of P and Q, but where is the fallacy in the 'second half of a proof of equivalence? Surely such is a common enough - and impeccable - procedure in ordinary mathematical proofs. Conceivably Aristotle has in mind a situa- tion in which proving either side of a co-implication is a relatively trivial matter when the other side is already given; so the problem would be to prove the one side without assuming the other. Aristotle's example suggests such a possibility: 'if he had to show that the diagonal is incom- mensurable with the side, and were to beg that the side is incommen- surable with the diagonal' (163a13). Thus the fallacy consists in trying to prove that a certain relation obtains (in this case not implication but in- commensurability) by asking it to be granted that its converse obtains. If the relation is - by previous context or agreement - a symmetrical one, then the proof will be trivial in a way reminiscent of (4). However, the point remains that a trivial proof need not always commit the fallacy of begging the question, and so we are in the end brought back to the same question posed by (1): when is an equivalence so 'obvious' or its terms so sementically 'near' to guarantee that granting one side of it in a proof of the other is fallacious in given dialectical contexts?

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The five ways are Aristotle's attempt to show how the petitio is a 'logical' fallacy, and they do indeed sound a logical chord in modern ears if only because they are addressed to conjunctions and quantifiers. Still, the five ways provide no definitive answer to the question of when the petitio may be said to be a logical fallacy. On the contrary, they merely sharpen our intitial question of how to analyse the notice of equivalence that underlies the equivalence conception of the petitio. Could this concept of equivalence be essentially epistemic, thus pulling the analysis of the petitio towards epistemology and away from the purely truth-theoretic framework of ordinary logic?3 And, likewise, will a deeper analysis of the errors of the five ways require an epistemic con- strual of dependency and near-equivalence?

4. The Epistemology of the Petitio

At the outset (Pr. An. 64b30) Aristotle sets the epistemic tone of the discussion when he reminds his readers that demonstration proceeds from what is more certain and prior. To beg or assume the original question' is only one amongst various ways that a demonstration can fail. One way that a demonstration can fall short is when the demonstrator argues from premisses that are less known or equally unknown in relation to the conclusion that is to be proved. This point bears emphasis. Aristotle is distinguishing between the petitio and the case in which the premisses fall short of the epistemic requirement that the premisses must be at least as well or better known than the conclusion. This latter failure is simple lack of evidence. Begging the question is certainly a sort of demonstrative failure, (Pr. An. 64b28); in fact it is a special case of demonstrative failure by way of epistemic evidential bereftness.

Aristotle formulates his account of begging the question in terms of his well-known doctrine that we know some things through themselves - that is, some propositions are 'self-justified' - whereas other things are known through something else. He (Pr. An. 64b37) characterizes begging the question as follows: '. . . whenever a man tries to prove

3 In Woods and Walton ( 1 975), we argue for the interpretation of the petitio as an

epistemic shortcoming. In Woods and Walton (1979) we set out a theory of circular demonstration based on the von Wright-Geach notion of entailment, an

epistemic notion, Sanford (1977) also favours an epistemic approach whereas Barker (1976) has argued against this approach.

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The Petitlo: Aristotle's Five Ways

what is not self-evident by means of itself, then he begs the original question/ This is a point of some importance, for it shows Aristotle as holding that cyclic justification need not always be question-begging.4 Whereas certain propositions may legitimately find an epistemic basis of

justification in themselves, in other cases self-grounded ness is fallacious. Aristotle makes a third point of considerable interest. It is that an

arguer might at the outset assume what is in question, for example if he were to attempt to prove a statement A by the very assumption of A. We could represent this simple inference as "A-Â.1 It is also possible to

prove A by means of itself indirectly through a sequence of other evidential steps. The example Aristotle gives (65a1) contains three

terms, and has this form, where the arrow again indicates the direction of each step of evidential justification: 'A-^C-^B-Â1. If we suppose that an arguer tries to prove A through B, but then prove B through C, and finally C through A,

' ... it turns out that those who reason thus are

proving A by means of itself (Pr. An. 65a3). The sequence of demonstration begins at A and terminates at A, and can be represented as a cycle of a directed graph.5

A

A C B

4 Rescher (1977) in a chapter on cognitive methodology sketches a theory of the

cyclical evolutionary justification of knowledge. Such evolutionary cycles would seem to be ampliative in nature, yielding new information by an aufgehoben- like process of refinement.

Incidentally, we do not hold that acceptance of Aristotle's doctrine of cir-

cularity, relying as it does upon the concept of self-evidence, commits one to

believing that, as Aristotle himself did, that there are indeed self-evident propositions. It is possible to hold to his account of the fallacy, while simply rejecting the exceptions that Aristotle wished to make. We owe this instructive point to a referee of this Journal.

5 We are thinking of the -* as a binary relation, or if you like, as an arc of a directed graph. Thus A -* B represents the inferential step from A to B. A major question here is: what is an inferential step in epistemological terms? This question is treated in Walton (1980).

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Aristotle puts the same point in the language of conditionals. We are saying If A then C, and if C then B, and if B then A/ consequently, we are saying, in effect 'If A then A/ Those who reason thus merely say a particular thing is, if it is . . / (Pr. An. 65a7). The basic point is that the in- direct version of assuming the question at issue via the intermediary of an intervening sequence of steps, ultimately reduces to the same principle of the simpler case, in which one initially assumes the question itself. In both cases we have a self-terminating (closed-loop) sequence.

What is wrong with the inference rA-*A1? Nothing whatever if the arrow gives formal entailment. But Aristotle shows by reductio what is wrong if we take the arrow to represent epistemic justification; for if we allow ^-A1 for any A then 'everything will be self-evidenf (65a8). But this, says Aristotle, is impossible. Allowing that 'self-inference' is entailment-theoretically unexceptionable, it would be ruinous for the theory of epistemic inference. The fallacy, then, is a matter of epistemic misinference.

5. The Blurred Edges

The remainder of the remarks in the Prior Analytics are very obscure. Hamblin (1970, 74) also finds these passages puzzling, and devotes some effort to their clarification. Though Hamblin's remarks (pp. 74-7) are helpful, we are not entirely convinced that they adequately explain all that is deeply bothersome about Aristotle's text.

So as to indicate the flavour of these problems, we quote the kernel passage (64b10 to 64b18), also quoted (minus the last two lines) by Hamblin (p. 75).

If then it is uncertain whether A belongs to C, and also whether A belongs to B, and if one should assume that A does belong to B, it is not yet clear whether he

begs the original question, but it is evident that he is not demonstrating: for what is as uncertain as the question to be answered cannot be a principle of a demonstration. If however B is so related to C that they are identical, or if they are plainly convertible, or the one belongs to the other, the original question is

begged. For one might equally well prove that A belongs to B through those terms if they are convertible. But if they are not convertible, it is the fact that

they are not that prevents such a demonstration, not the method of

demonstrating. But if one were to make the conversion, then he would be do-

ing what we have described and effecting a reciprocal proof with three propositions.

What is wrong, says Hamblin, is that 'because of the trivial satisfaction of one of the premisses, the other premiss and the conclusion are each as

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good or as bad as the other, so that the argument from one to the other is nugatory; and that this is so is shown, among other things, by the fact the premiss will, in this case, be as uncertain as the conclusion' (Hamblin [1970] 75). Hamblin believes that this account of the matter will only work if A and B are identical. If A and B are merely convertible (coextensive but different in meaning, like 'man' and 'featherless biped') a faultless, non-circular syllogism would seem to result. But the wor- risome question is: exactly why is the argument,

A belongs to C B is identical to C Therefore, A belongs to B

a petitio? Now, as Hamblin points out, if the first premiss and the conclusion are equally uncertain, then the argument does nothing to increase the epistemic weight of the conclusion. From this epistemic perspective, a 'nugatory argument indeed. But that does not make it a

petitio, either by Aristotle's own account of the petitio or by conven- tional contemporary intuitions.

Hamblin nonetheless finds the argument to be circular because the second premiss is an identity rather than say mutual convertibility. Why we might ask, should such an identity make for circularity? We can think of only one justifying explanation of the ostensible circularity of the above argument. It is that the identity, insofar as it expresses a

logical' equivalence relation, generates the strong and recognizable semantic equivalence of the first premiss and the conclusion by way of the identity expressed by the second premiss. By these lights, the first

premiss and conclusion are recognizably semantically equivalent and therefore the argument is circular according to the first of Aristotle's the five ways of the petitio. A central idea of this interpretation is that if B and C are identical, then their identity is 'obvious' or 'transparent enough to make it clear that the first premiss makes the same statement as (that is, semantically equivalent to) the conclusion. Hence, an

equivalence petitio. However, Hamblin's interpretation of the case is quite different (p.

77):

If it is uncertain whether all B's are As, and equally uncertain whether all Cs are

As, we cannot use one to prove the other, since premisses must always be

more certain - more immediately known - than their conclusions. If Bs and

Cs are the same things whether because the concepts are identical or merely because the terms are convertible, "All Cs are As" seems to be inferable from

"All Bs are As," but also vice versa' but there cannot be genuine inferences both

ways, or there could be argument in a circle. Hence the apparent inference is

really fallacious.

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Hamblin now seems to be saying, in apparent contrast with his previous interpretation, that the argument can be circular irrespective of whether the second premiss expresses an identity or some weaker relation, such as mutual convertibility.6 Hamblin is now saying that if someone were to infer 'A belongs to B' from 'A belongs to C and also vice versa, then the argument would be circular. But this is a claim about what might or could happen in certain dialectical circumstances, and not a reason as to why any argument in the form in which Aristotle actually gives it actually commits the petitio as it stands. In other words, Hamblin shows how a petitio might arise from Aristotle's case, but we do not think that his analysis establishes that such an argument must represent a petitio as it stands. In particular, we wonder whether Hamblin has said quite enough to warrant his claim that the first premiss 'seems to be inferrable' from the conclusion. It is worth noting that even if the argument unfolds in just one 'direction,' say from 'A belongs to C to 'A belongs to B,' Aristotle still concludes that a petitio is committed, provided that B and C are bound by the appropriate relation of semantic propinquity.

Though Hamblin's explanations are not, therefore, very satisfying, our own suggestion is not entirely adequate either. Aristotle seems to

say that his specimen argument begs the question whether the relation of the second premiss is identity or even the weaker relation of mutual convertibility, or mutual inclusion, as genus to species. As we have said, our own best interpretation is that these relations, presumably equivalence relations, transparently establish the first premiss as a sufficiently near-equivalent to the conclusion to make the argument a petitio of Aristotle's type (1). However, we might direct to our interpretation the same cavil that we offered to Hamblin's; namely, that we have shown only that the argument could be circular under certain percep- tions of equivalence specific to the circumstances of some particular arguer, and relative to his own logical acuity.

We have no doubt that Hamblin is on the right track in regarding these problems from an epistemological perspective. As he says, 'the clue is Aristotle's concern with how we come to know things' (p. 76). Well and good; but problems and obscurities arise when we try to apply general epistemological concepts to specific instances of syllogistic argumentation. How are we to deal with these?

6 We do not however complain of an inconsistency as between the interpretation of pages 75 and 77. The remarks of p. 75 pertained to the question of whether the argument is circular, whereas the point of p. 77 concerns the possibility of there being a circle.

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6. Some Properties of Epistemic Inferences

The remarks in the Posterior Analytics 72b33 to 73a5 recall the pungent clarity of the earlier passage, Prior Analytics 65a7. We remember that in 65a7, Aristotle had made the point that a looping configuration of inferences of the form rA-*O*B-*A1 reduces ultimately to the form rA-*A\ Here we let the -stand for the binary relation that represents inference. We think of it as a binary epistemic relation, and in these passages we see that Aristotle is working towards the specification of some of its properties. For example, one way to view his proof of Pr. An. 65a7 is to assume the transitivity of-*, and then the proof is straightforward. For given rA-*C and rC-*B1, we may infer by transitivity that rA-*B\ But we also have it that 'B-Â1 and therefore again by transitivity of-* we have it that rA-*A\ But did Aristotle in fact wish to claim transitivity for -»?

It is clear that in the argument of the Post. An. to which we now turn that he did wish to explore the consequences of transitivity. In Post. An. 72b33-73a5 Aristotle shows that if we assume the transitivity of inference, and subjoin to it symmetry of inference, then reflexivity follows, as one would have expected.

The advocates of circular demonstration are not only faced with the difficulty we have just stated: in addition their theory reduces to the mere statement that if a thing exists, then it does exist - an easy way of proving anything. That this is so can be clearly shown by taking three terms, for to constitute the circle it makes no difference whether many terms or few or even only two are taken. Thus by direct proof, if A is, B must be; if B is, C must be; therefore if A is, C must be. Since then - by the circular proof - if A is, B must be, and if B is, A must be, A may be substituted for C above. Then "if B is, A must be" = "if B is, C must be", which above gave the conclusion "if A is, C must be": but C and A have been identified. Consequently the upholders of circular demonstration are in the position of saying that if A is, A must be - a simple way of proving anything.

The proof has the following form.

(1) If A-*B, then if B- C then A- C. (Transitivity)

(2) If A-B, then B-A. (Symmetry)

(3) If A-*A. (Reflexivity)

The proof is independent of specific properties of -except that-*must be binary. Putting A in for C in (1) we get: if rA-^B1, then if 'B-Â1 then rA-*A\ But by (2) we know that if we have rA-*B\ we

89




always have rB-*A1. Thus in general, if we have rA-*B\ we must also have rA-*A\ Hence the transitivity of inference gives the circularity of all inference.

This proof is extremely important for the epistemic analysis of the petitio. The absurdity of always being able to prove a statement by its own self-evidence is avoided, Aristotle is telling us, by requiring that epistemic inference be nonsymmetrical if transitive. In fact, the essential non-symmetry of the concept of demonstration at issue is independent- ly argued for by Aristotle at Post. An. 72b25: ' . . . demonstration must be based on premisses prior and better known than the conclusion; and the same things cannot simultaneously be both prior and posterior to one another . . .' Thus if B is demonstrated from A it follows that A is prior to B and B is posterior to A. Since A and B cannot be concurrently prior and posterior we have it that B is not prior to A. Hence B is not prior to A, and A is not demonstrated from B. The relation of 'being demonstrated from' is non-symmetrical.

These remarks have special relevance for the distinction between the equivalence and dependency notions of the petitio. At times Aristo- tle seems to us very close to drawing this distinction, and the two proofs of Post. An. 72b33 and Pr. An. 65a7 suggest that the equivalence conception, suggested by the reflexivity of-*, and the dependency conception, suggested by the symmetry of-*, are logically related to each other. Aristotle well might be suggesting Pr. An. 65a7 that a dependency chain of inferences resembles in its epistemic upshot a simple equivalence petitio of the form rA-Â1 (which could be diagrammed in graph- theorectic terms as a loop),

o A

because even a many-termed sequence of inferences, if it exhibits the circular dependency pattern,

90




A2 A,

A3«T /*~"\ A3«T >A6

A4 A5

comes back at some stage to the same point Alr and so differs from the simple loop from A to A only in its inessential complexity. In neither case is any epistemic advance made. Although immediate circularity and mediate circularity are distinct, the transitivity of -* collapses important features of the distinction.

Moreover, at Post. An. 72b33, we seem to have it that given the trans-

itivity of -+ for as many terms A^ as you care to choose, a dependency petitio always implies an equivalence petitio, and so again there is less difference between the two conceptions than one might initially have

supposed. Given the transitivity of demonstration, mutual demonstration is just as bad as blatant self-demonstration.

In using the phrase 'advocates of circular demonstration/ Aristotle

opens up the possibility that some might care to argue for the non- fallaciousness of circular inferences. Aristotle's own view is that some

propositions are self-evident and known through themselves (Pr. An. 64b34). Here we have the suggestion that sometimes cyclic inference

sequences are epistemically permissible and sometimes not. In recent times, coherence theorists, Hegelians, Marxists, and idealists have shown some friendliness for cyclic patterns of epistemic justification. For Aristotle, however, it is clear that his basic notion of the 'epistemic priority' of premisses in a demonstration categorically rules out cyclic se-

quences. The Aristotelian programme for the study of petitio must, ob-

viously enough, be a special topic within the study of epistemic demonstration and it needs to be determined whether an adequate notion of epistemic inference will tolerate some form of cyclical demonstration. Hamblin is giving us the right orientation by attempting to place Aristotle's contribution to the petitio at the interface of his logic and his theory of knowledge.

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7. Is All Deductive Logic Circular?

Any theory of the petitio must confront Mill's notorious charge in his System of Logic that all deductive logic commits this fallacy. Mill's argument is not new - it may also be found in Sextus Empiricus - but it raises many general questions about the role of deductive formal logic in the analysis of the concept of argument, a concept of argument, by the way, that is broader than that implicit in the Aristotelian treatment.

The thrust of Mill's argument is quite straightforward. Consider a traditional example of a deductively valid argument,

All men are mortal. Plato is a man. Therefore, Plato is mortal.

Mill (1843, Ihlll) argued that the major premiss depends on the conclusion in the sense that we cannot be assured that all men are mortal 'unless we are already certain of the mortality of every individual man.' If it is doubtful that Plato (or anyone) is mortal, then it is at least as doubtful that all men are mortal. Thus Mill seemed to be thinking of circularity in terms of the epistemic dependency conception.

Many critics, Cohen and Nagel notably among them (1934, 177ff.), argue that Mill sets his standards too high, and that it is not necessary, in order to know that a universal statement is true, to know that each and every instance of it is true. This draconian requirement is, they find, hardly consonant with the practice of experimental science. Poor Mill has been virtually fusilladed by critics for committing the howler of thinking of a general statement as a simple enumeration of finite instances. Perhaps, his critics concede, the above argument would be circular if we though of general statements in this way. Indeed if the major premiss were a finite list of all the mortal men, Plato would occur in it and the argument would commit an equivalence petitio. However, insofar as there is little temptation to think of universal quantifications as finite conjunctions, Mill's arguments, we are assured, may safely be brushed aside.

It is not to the point here to attempt to settle the substantive issues involved in this controversy, (these are more fully outlined in Walton [1977]), but it is interesting to note that, however one feels about quantifiers, the option is still open to a proponent of Mill's argument to reconstrue the dependency relation so that Mill's argument can still go through. Mill might argue (and perhaps did) that if one knows that the conclusion is false, then one knows that the major premiss is false. And the point still holds if the major is more than a finite conjunction, the

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question now raised is whether what is appealed to is a correct account of the dependency relation as it pertains quite generally to the petitio.

Turning aside such difficult problems, De Morgan in his Formal Logic (1847, 254f.) directed a criticism against Mill's type of argument that is so pointed and so clever that we like to call it De Morgan's Deadly Retort: The whole objection tacitly assumes the superfluity of the minor; that is, tacitly assumes we know Plato to be a man, as soon as we know him to be Plato/ We are assuming, for example, that Plato is not a dog or a ship. De Morgan is entirely correct in noticing that Mill's argument overlooks the second premiss.

De Morgan was able to generalize his Deadly Retort into an analysis of the petitio that stands in direct opposition to Mill's treatment. Where Mill argued that all deductively valid arguments beg the question, De

Morgan holds that no deductively valid arguments (with certain

qualifications) ever beg the question. To begin with, De Morgan shrewdly observed that, correctly speaking, the most favourable account of the petitio pertains to what is assumed in one premiss: The most fallacious pair of premisses, though expressly constructed to form a certain conclusion, without the least reference to their truth, would not be assuming the question, or an equivalent7 (De Morgan (1847) 257). thus the petitio is a function of the number of premisses. This view too, like Mill's own, had its predecessors. The Stoics disputed over whether

one-premissed arguments are really genuine arguments. Antipater of Tarsus (fl. 150 B.C.) led the group that stood for the existence of

arguments such as It is day, therefore it is night.' The opposition main- tained that this should be filled out to read: If it is day, it is night. It is

day. Therefore it is night.7 The issue raises pointed questions about what constitutes an adequate argumentive context for the petitio and its

analysis. We remember that Aristotle in Posterior Analytics (73a10) required

that an argument should have at least two premisses. In fact, the reader will recall that a syllogism is required to be constructed in such a way that each of its two premisses is logically independent of the other. Thus what we call De Morgan's Thesis (Woods and Walton, 1977) could be stated succinctly within traditional logic: No syllogism begs the question. However, the deeper interest of De Morgan's remarks lie in their ap- plicability beyond the archaic syllogistic, and it is possible to extend them in this direction if we introduce the notion of a superfluous premiss. A premiss may be said to be superfluous in a valid argument if,

7 See Benson Mates, Stoic Logic (Berkeley and Los Angeles: University of Califor- nia Press 1961).

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and only if, the argument is invalid if the premiss is deleted. Then our up-to-date version of De Morgan's Thesis becomes: No valid argument with more than one premiss, and no superfluous premiss, begs the question. The reformulated thesis is extremely interesting because it gives us what amounts to a partial decision method for the determination of circularity. It does not tell us what to do with one-premissed arguments, assuming there are such things, but it does tell us that every non- superfluous multi-premissed valid argument is non-circular. We say 'partial' decision method for obvious reasons. Superfluity is defined for valid arguments, and, by Church's Theorem, the decision problem of first order logic is unsolvable. However, Church himself has provided ten classes of special case in which first-order validity is solvable, and Ackermann has dealt with special case solutions even more com- prehensively.8 It would appear, then, that where premiss-deletion of a first-order valid argument, which is in fact special-case decidable, does not generate an argument which is undecidable, superfluity is effective- ly decidable; and so, also, would non-circularity be decidable.

De Morgan does not, so far as we know, mention Mill or others by name, but he concedes that the sort of argument that we have ascribed to Mill is ingenious. The opposition between these two points of view is of more than historical interest because it poses general questions about the limits of truth-theoretic approaches in the analysis of the petitio. It is easy nowadays to appreciate the economy of alethic frameworks represented, say, by first-order logic, especially given the various obscurities and problems known to inhere in epistemic approaches. Many would suspect that bringing the concepts of knowledge or belief into the analysis of the petitio could reduce the detection of circularity to an essentially subjective or psychological enterprise. Archibishop Whately in his Elements of Logic (1840) took the position that it is not generally possible to draw a precise line between circular and non- circular argument: for one person, the premises might be more evident than the conclusion, while for another, a premiss might not be admitted except as a consequence of the conclusion.9 In short, significant economies are promised if we can avoid the Aristotelian appeal to epistemic notions. Nevertheless, the very slipperiness of the petitio to which Whately draws our attention would seem in fact to constitute a problem even for the purely alethic approach. Sanford (1972) has

8 Alonzo Church, Introduction to Mathematical Logic, I, pp. 246-257; and Wilhelm Ackermann, Solvable Cases of the Decision Problem (Amsterdam: North Holland 1954)

9 For further elaborations of this issue, see Woods and Walton (1975).

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pointed out that certain forms of argument, such as the disjunctive syllogism, are circularity-relative, depending on the epistemic background conditions in which the argument is proposed. Given any non-circular instance of the disjunctive syllogism we care to choose, there is always available a circular counterpart where the disjunctive premiss is based on the conclusion. But the disjunctive syllogism is a many-premissed form of argument, and neither premiss is superfluous. Unless we are prepared to suppress all factors whatever of epistemic background context, such cases of slippage would seem to count decisively against De Morgan's Thesis.

Is there yet a way out? One possibility for fanciers of De Morgan's Thesis is sketched in Woods and Walton (1975). The counterexample to De Morgan's thesis alleges a circular disjunctive syllogism, as follows.

^P v q

where the bent arrow indicates the dependency of the first premiss on the conclusion. But perhaps, it might be suggested, what we have here is not one argument but two:

p v q

"P and q

q P v q

In actual practice, the two are often combined in piggy-back fashion, where the first premiss of the first argument is an interim conclusion. But it is the conclusion of the second argument as well. This can be ex- hibited as a chain of argument composed of two sub-arguments linked as follows:

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q

p v q

q

But notice now that the conclusion is identical with a premiss, and this argument chain is obviously De Morgan-circular. De Morgan's Thesis would seem to be vindicated.

The strategy involved in the rebuttal of this counterexample can be generalized as follows. An argument is construed as a sequence of sub- arguments in which the conclusion of a predecessor-segment functions as a premiss in a successor-segment. In general we will have the following sort of array, for premiss-sets Pj. and conclusions Cj.

Pi

q

p2-

c2

Pi-

Ci

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The Petltio: Aristotle's Five Ways

Now if the argument really is circular, then given sufficient analysis of it, the conclusion will turn up again somewhere in the sequence as premiss.

De Morgan's Defence brings us back to the very questions that Aristotle had raised, and it collapses with its appeal sooner or later to epistemic considerations. What, after all, do the arrows the diagram represent if not some or other epistemic notion of groundedness? What is more, the linkages of the sub-argument sequences of De Morgan's Defence unavoidably give rise to questions as to the transitivity and other properties of the arrow - the very questions that preoccupied Aristotle. In seeking for what this retroductive process of 'pressing back behind the premisses' really amounts to, we are returned directly to the epistemic considerations of the Prior Analytics.

8. Conclusions

1 . Hamblin is right to emphasize that the treatment in the Topics is fundamentally different from that of the Prior Analytics. In the Topics, the fallacy of begging the question is committed by a participant in a disputation if he asks his opponent to concede the very proposition to be proved. In the Prior Analytics, the fallacy of begging (or assuming) the question is committed when one tries to prove what is not 'self-evidenf by means of itself. The former context is dialectical, the latter epistemic.

2. Each explication has its own theoretical difficulties. The first account presupposes a theory of rational disputation (dialec- tic). The second presupposes an epistemological theory of self- evidence and epistemic priority.10

3. Both accounts share a common difficulty. We see from the treatment of the Topics that the interesting cases of the petitio crop up where the proposition asked to be conceded is not or- thographically speaking, the Very proposition' to be proved, but rather one that, being semantically 'too close,' would be recognized by the participants as being virtually the same pro-

1 0 The question of what precisely could be the difference between a dialectical and

epistemic context of petitio is treated in Woods and Walton, (1978).

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position. Not identity, but strong and recognizable semantic equivalence. In the Prior Analytics, Aristotle focuses on those cases in which someone tries to prove what is not self-evident by means of a proposition that is epistemically so close to what he is trying to prove that no epistemic advance is made. Both accounts presuppose notions of 'equivalence' or 'near- equivalence' that seem clearly to exceed all familiar notions of logical equivalence. Rather we seem to have at hand the epistemic concept of recognized or perceived equivalence or

near-equivalence.

4. The account of the Prior Analytics points to the fundamental importance of distinguishing between two species of failure of demonstration: (a) inadequacy of evidence supplied by the premisses, i.e. the premisses fail to be 'better known' than the conclusion, and, (b) circularity. Since (b) is a case in which a premiss and the conclusion are epistemically on a par, (b), though different, would seem to be a species of (a).

5. Following on (2), the point is brought out that cyclic patterns of inference need not always be fallacious. Where a proposition is of the self-evident variety, the inference rA-Ân is allowable. One assumes of course that rA implies A1 always holds in the pure theory of formal deduction (epistemological and dialectical theory aside). Perhaps too, coherence theories of truth might adequately establish the acceptability of certain cyclic networks on inference or reasoning.11

6. The Prior Analytics brings out the usefulness of breaking down chains of arguments into individual links, and of constructing sequence-diagrams of complex chains of argumentation.

7. Aristotle shows that, given the transitivity of -* (the binary relation of epistemic inference), any sequence of the form

•"Aq-Â^ A1-Â2, ..., An-*An + 1, An + 1-Âq1 must reduce to a sequence of the form A0-*A0. Thus in order to distinguish between mediate circularity and immediate circularity, it is necessary to begin with a non-transitive notion of immediate epistemic inference. Yet the transitive relationship of the two

1 1 See Rescher (1972), (1977a) and (1977b).

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varieties of circularity discloses their underlying similarity of epistemic upshot. A circular inference always loops back' to its initial point of demonstration.

8. De Morgan showed that it is possible to take a constructive approach to the petitio without appealing to (the possibly subjective) parameters of dialectical or epistemic frameworks. Ac- cording to De Morgan, an argument does not beg the question provided that it has more than one premiss, and that no premiss is superfluous. Here, then, it is contended that we can dispense with epistemology or dialectics in the analysis of the petitio.

9. Nevertheless, in order to deal with plausible counterexamples to his thesis, De Morgan seems inevitably driven to a reliance upon concepts that are essentially epistemic or dialectical.

10. An adequate defence of De Morgan's account of the petitio leads back to the same problems of epistemic inference that were posed by Aristotle's theories. If it is to be useful, De Morgan's theory requires a notion of epistemic inference such that immediate epistemic inferences can be linked up in longer chains of argumentation by transitive closure.

1 1 . The opposition between De Morgan and Mill draws attention to the fact that any epistemological or dialectical theory of a logical' fallacy will strike many contemporary philosophers as odd. their own tendency is to locate the fallacy within the more familiar truth-theoretic framework of modern classical logic. The issue posed is whether the petitio is more a 'formal' matter than 'informal'. For our own part, we find this latter to be a nonissue.12

July 1980

12 See Woods and Walton (1979) and Woods (1979).

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REFERENCES

Aristotle, The Works of Aristotle Translated into English, ed. W.D. Ross (Oxford: Oxford University Press 1928)

John Barker, The Fallacy of Begging the Question/ Dialogue, 15 (1976) 241-55

Morris R. Cohen and Ernest Nagel, An Introduction to Logic and Scientific Method (New York: Harcourt, Brace & World 1934)

Irving Copi, Introduction to Logic 4th. edn. (New York: MacMillan 1972)

Augustus De Morgan, Formal Logic (London: Taylor and Walton 1847)

C.L. Hamblin, Fallacies (London: Methuen 1970)

John Stuart Mill, A System of Logic (London: Longmans Green 1843)

Nicholas Rescher, The Coherence Theory of Truth (Oxford: Oxford University Press 1973)

Nicholas Rescher (1977a), Dialectics (Albany, N.Y.: State University of New York Press 1977)

Nicholas Rescher (1977b), Methodological Pragmatism (Oxford: Basil Blackwell 1977)

David Sanford, 'Begging the Question/ Analysis, 32 (1972) 197-9

, The Fallacy of Begging the Question: A Reply to Barker/ Dialogue, 16 (1977)485-98

Douglas Walton, 'Mill and De Morgan on Whether the Syllogism is a Petitio/ In- ternational Logic Review, 8 (1977) 57-68

, 'Petitio Principii and Argument Analysis/ in Informal Logic, ed. R.H. Johnson and J.A. Blair (Pt. Reyes, Cal.: Edgepress 1980) 40-54

Richard Whately, Elements of Logic (New York: Sheldon & Co. 1840)

John Woods and Douglas Walton, 'Petitio Principii/ Synthese, 31 (1975) 107-27

, 'Petitio and Relevant Many-Premissed Arguments/ Logique et Analyse, 77-78(1977)97-110

, 'Arresting Circles in Formal Dialogues/ Journal of Philosophical Logic, 7 (1978) 73-90

, 'Circular Demonstration and von Wright-Geach Entailment/ Notre Dame Journal of Formal Logic (forthcoming)

, 'Question-Begging and Cumulativeness in Dialectical Games/ Nous

John Woods, What is Informal Logic?', in Informal Logic, ed. R.H. Johnson and J.A. Blair (Pt. Reyes, Cal.: Edgepress 1980)

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Documents

The Petitio: Aristotle's Five Ways