This essay originally appeared in Argumentation, 8 (1994) 241-50. Copyright © 1994, Peter Suber.

Under A Non-Foundational Model of Argument
Peter Suber, Philosophy Department, Earlham College


I find (as others have found) that question-begging is formally valid but rationally unpersuasive. More precisely, it ought to be unpersuasive, although it can often persuade. Despite its formal validity, question-begging fails to establish its conclusion; in this sense it fails under a classical or foundationalist model of argument. But it does link its conclusion to its premises by means of acceptable rules of inference; in this sense it succeeds under a non-classical, non-foundationalist model of argument which is spelled out in the essay. However, even for the latter model question-begging fails to link the conclusion to premises that the unconvinced would find more acceptable than the conclusion. The essay includes reflections on the conditions under which the circularity of mutually supporting claims can avoid question-begging and legitimately be persuasive.

This paper is an attempt to describe what is objectionable and unobjectionable about begging the question, viewed from the standpoint of a non-classical model of argument.

Argument as connection

Let us call the view that arguments establish conclusions the classical model of argument. In this section I sketch a non-classical model in which arguments merely connect propositions to one another by means of rules of inference. The result is an ensemble of linked propositions. Ensembles do not establish conclusions; they only show the other propositions with which (roughly speaking) conclusions must stand or fall.

If arguments 'establish' conclusions it is only relative to premises and rules of inference. This is not a new or original claim, nor is it even denied by the classical model; but it is underemphasized by the classical model and forgotten by many educated people in practice. In practice many proponents of the classical model speak of arguments as proofs in an ancient sense of the term —as sufficient reasons for accepting their conclusions. But on the non-classical model, arguments are only maps that show us the logical neighbors of the conclusion, its presuppositions and consequences.

As shorthand I will refer to the classical model of reasoning as vertical, and the non-classical model as horizontal. In general, vertical reasoning is foundational, and horizontal reasoning is non-foundational or coherentist. In vertical reasoning, we (hope to) establish conclusions; in horizontal reasoning we (hope to) establish a linkage among propositions.

There are as many kinds of linkage as rules of inference. For example, modus ponens creates a typical kind of linkage.

  1. If p, then q.
  2. p.
  3. Therefore, q.

In this example, 1, 2, and 3 belong to the same ensemble, which may have many other members. Propositions 1 and 2 are linked to 3 by modus ponens. Propositions 1 and 2 are not linked to each other at all; or if we say (by courtesy) that they are linked by conjunction, then at least it remains true that they are not linked by a rule of inference. By a 'link' between propositions, I will mean a link created by a rule of inference under which one proposition follows from the other.

There are only two ways in which a proposition can become a member of an ensemble: stipulation and implication from premises already in the ensemble by means of a rule of inference. In the example above, 1 and 2 became members by stipulation, 3 by implication from 1 and 2 under the rule of modus ponens.

Rules of inference are the sole authorities for the validity of inferences, hence for the existence of links that create new members of ensembles. So when we say that some members are linked to others by means of a set of rules of inference, that is shorthand for saying they are linked by virtue of 'valid' inferences under that set of rules.

Rules of inference are stipulated. In practice, we are not always sure whether we use certain rules, and could not enumerate the rules we do use. But in formalizing our theories, we limit valid inference to those valid under explicitly stipulated rules.

Informally, an ensemble of propositions is meant to capture (more or less) the idea of a philosophical position or worldview. As a formal structure, an ensemble is closest to what some logicians call a theory, which can be recursively defined as a set of propositions containing all the consequences of the set (when the consequences are wffs of the same language, and follow from members under stipulated rules of inference). An ensemble has a seed, or subset of stipulated members. The others members are determined by the rules of inference and future stipulations.[Note 1]

In this way an ensemble is always consistent if its seed is consistent, its rules transmit consistency, and its future stipulations are limited to consistent additions. In what follows, I will speak only of consistent ensembles.

Ensembles are coherent in weak and strong ways. They are weakly coherent in that all co-members of the same ensemble are consistent with one another. They are strongly coherent in that some co-members (not necessarily all) have the relation of 'implication under the rules' to some others.

It would be absurd to claim that the horizontal model of reasoning is new or original. On the contrary, I think it is as old as the classical, vertical model. It is important because it has not always been disentangled from the vertical model, because it explains what argument is and ought to be better than the vertical model,[Note 2] and because it explains question-begging differently from the vertical model.

Deductive strength

If p q[Note 3] is a valid inference under some rules of inference, and q p is not, then we will say that p is stronger than q. If we know that p q, but nothing about q p, then we can say that p is at least as strong as q. If both p q and q p, then p and q possess exactly the same strength.

The deductive strength of a proposition is always relative to rules of inference. Once our rules are stipulated, the strength of a given proposition is always relative to that of other propositions. In standard logics, all propositions follow from a contradiction under the rules; hence, contradictions are maximally or absolutely strong in such logics. Similarly, logical truths (tautologies in propositional logic) are minimally strong because they are implied by every proposition.

Convertible premises and conclusions

In non-foundational philosophies, propositions support one another rather than trace their support back to a given, self-evident, or self-supporting foundation. Ensembles are designed to capture this circle of support formally. It may be tempting to say that in horizontal reasoning, all propositions in the same ensemble are reciprocally premises and conclusions. There is an important sense in which this is true, and this is what makes the reasoning horizontal rather than vertical.

However there is also a sense in which it is false. In the example above, 1 and 2 do not support one another; nor does their conjunction stand in that kind of reciprocal relationship to 3. The distinction between premises and conclusions is preserved in horizontal reasoning. There are at least two ways in which we can distinguish premises from conclusions in an ensemble.

  1. By deductive strength. If p is stronger than q under our rules of inference, then p q is a valid inference under those rules and q p is not. Hence p is a premise for q in a way that q can never be a premise for p. The horizontal model of reasoning does not dispense with the concept of deductive strength, because it does not dispense with rules of inference, and so must recognize this way of distinguishing premises and conclusions.

    However, note that q conjoined with other statements may yield a conjunction that is stronger than p, and so may serve in a set of premises for p. In that sense, q is not a 'permanent conclusion' in relation to p, and p not a 'permanent premise' in relation to q. Taking all the propositions in the ensemble together, almost any member can serve in a set of premises for any other member. If we allow premise-sets to contain superfluous members,[Note 4] then this form of premise-conclusion convertibility is always the case.

  2. By psychological acceptance. If I already accept p (for whatever reason), but have doubts or no settled opinion about q, then for me p is a premise for q in a way in which q is not a premise for p.

If we say that 'proof' in an informal sense is to link the less certain to the more certain, or the novel to the familiar, then ensembles provide proofs in this sense; and when they do, premises and conclusions are not (psychologically) convertible.

Apart from superfluous premises and psychological convertibility, whether premises and conclusions are convertible in an ensemble is a function of the stipulated rules of inference. In an ensemble with just two rules, when one rule allows A B, and the other allows B A, then premises and conclusions would be perfectly convertible. Under standard rules, however, and their likely variations, convertibility will fail much of the time, and this will limit the freedom to reason in a circle.

Validity and soundness

Begging the question is objectionable, but not because it is a formal fallacy. The inference,

p p

is formally valid under standard rules of inference. Admitting the validity of this inference clashes with a key assumption of the vertical model, viz. that premises provide good reasons to accept conclusions. We want to say that p is not a good reason to accept p. If we had doubts about p and asked why we should believe it, the reply 'because p is true' would not answer our question or address our doubts.

On the horizontal model, admitting the validity of p p is entirely harmless; it means only that p is always linked to itself, or a member of the same ensembles as itself. This means in turn that a rule of inference validates the inference from a proposition to itself. The rule that does so is present in every standard logic and not at all exotic.

On the vertical model, premises are or ought to be more certain or fundamental than their conclusions; this makes p p hard to take. On the horizontal model, premises (with their conclusions) are simply nodes in a network of inferential relations, and it is not remarkable that p would bear the relation of 'implication under the rules' to itself. The vertical model can accept the rule of inference that validates p p, but it cannot accept p p as a specimen of a good argument, where the 'good' in 'good argument' goes beyond formal validity to the norms of argument provided by the vertical model of reasoning. In fact, if the formal validity of p p is not merely admitted on technical grounds, but welcomed on philosophical ones (even if other dimensions of it bother us), that is probably a sign that we already accept the horizontal model of argument.

In p p we assume the conclusion as a premise. We can also assume an equivalent of the conclusion:

(p q)[Note 5], p q

But this is also formally valid under standard rules. The problem with these arguments is not invalidity; it is that they would not persuade the unconvinced of the truth of their conclusions. Anyone who did not accept the conclusion and needed persuading would for that very reason reject at least one premise in each argument. For them, these arguments would be valid but unsound.

We are not surprised that valid arguments often fail to persuade. Under both models of reasoning, readers who reject a premise or rule need not accept the conclusion; and the validity of the inference is not an obstacle to a reader's rejection of a premise or rule. Whether a reader will reject a premise or rule is contingent on that reader's individual psychology and prior convictions, and no amount of care in insuring the formal validity of the reasoning will make these contingencies irrelevant. I describe the conditions of unpersuasiveness in psychological terms in order to emphasize, not their relativity to individuals, but their independence from the formal properties of the argument studied by logicians. Unpersuaded readers of valid arguments may be perfectly 'logical'.

A special case of unpersuasiveness arises when the reader who rejects a given conclusion ought also to reject a premise or rule. The force of this ought comes from loyalty to the ideal of consistency and nothing more. If a reader rejects a conclusion, then to be consistent with that rejection, she must reject a premise or rule equivalent to the rejected conclusion.[Note 6] This sort of case arises whenever a premise or rule restates the rejected conclusion —in short, in cases of begging the question. Such arguments may be formally valid, but they should not persuade the unconvinced.

If it helps clarify matters, we may distinguish begging the question of truth from begging the question of certainty. We beg the question of truth when we presuppose the truth of the conclusion (or its equivalent) in the premises. We beg the question of certainty when we offer premises at least as uncertain or implausible as the conclusion. Ultimately, however, I want to collapse this distinction, and assert that what is objectionable about the former is that it is a species of the latter.

Let us say that arguments that should be unpersuasive to the class of readers who already reject the conclusion, because a premise or rule restates the rejected conclusion, are intrinsically unpersuasive arguments.

Now in fact question-begging arguments may persuade anyone, since people are astonishing creatures; but they offer no reason to believe the conclusion other than the conclusion itself or its surrogate. Some good psychological research could be done on when intrinsically unpersuasive arguments are persuasive to actual readers anyway. The reader may simply be imperceptive or inattentive, of course; but there are more interesting cases. These forms of dullness might be induced by the argument itself, the way a magician's patter can distract one's attention, thereby making the trick surprising, and thereby making the patter credible. Or the reader's rejection of the conclusion might be reconsidered by reading its paraphrase in the premises. Or the circle in the argument might be very large or well-disguised.[Note 7]

In short, we cannot say that intrinsically unpersuasive arguments (such as begged questions) are never persuasive, for that is false. Nor can we say that they are never persuasive for logical readers, since that would abuse many logical readers and miss the point; these arguments are (when valid) as logical as any others, at least when 'logical' is taken formally. We can only say that intrinsically unpersuasive arguments ought not to persuade anyone. Question-begging, then, is not pointless because it fails to persuade; it is pointless because it ought to fail to persuade.[Note 8]

One reading of the etymology of the phrase 'to beg the question' is compatible with this account. Since argument only establishes conclusions relative to premises, every argument begins with undemonstrated premises. The premises may be demonstrable, but only in other arguments with their own undemonstrated premises. So in order to appeal to readers to accept our argued conclusions, we must first appeal to them to accept our unargued premises. We 'beg' their indulgence of our point of departure. In question-begging arguments we beg their indulgence of the whole issue, or 'question', including the conclusion.[Note 9]

Irreflexive arguments that do not beg the question can also be valid but (for some readers) unsound. However, when they fall into the hands of readers who are unconvinced by the conclusion, they offer non-circular or "straightforward" reasons for accepting the conclusion. When the reasons offered in the premises are at least as certain (for the reader) as the conclusion, then the entire argument may still be unpersuasive, but it is not intrinsically unpersuasive. To persist in denying the conclusion, the reader must deny at least one premise or rule of inference, which can be a non-trivial job requiring much inquiry, ingenuity, or self-deception.

Logical and social tasks of argument

Linking a conclusion to premises by means of rules of inference is, let us say, a logical task. Persuading readers to accept a conclusion is a social task. On the horizontal model, the social task should succeed when the conclusion is linked to premises that the reader already accepts, by means of rules that the reader already accepts.

No logical properties of the argument can form necessary or sufficient conditions of success at the social task, because persuasion can be irrational. This is why we shift from descriptive to normative language. The reader who accepts the premises and rules ought to accept the conclusions linked to those premises by those rules, but may not in fact do so.

Question-begging arguments accomplish the logical task as well as irreflexive arguments. In fact they do so more self-evidently. But they fail at the social task more often than irreflexive arguments. In fact, they are intrinsic failures at the social task; they ought to fail in every case even if they occasionally succeed.

An argument that ought to fail at the social task commits a kind of fallacy, even if it is impeccable at the logical task. Since the argument can be formally valid, the fallacy must informal. The fallacy does not consist in reasoning badly, but in reasoning pointlessly. It consists in perversely offering means (argument) that fail to advance the end (persuasion), like spanking a baby to make it stop crying. We can call this the fallacy of imprudence, since it reveals a failure in the rationality by which we select means to advance our ends (which rationality Kant calls prudence). As a fallacy of imprudence, using a question-begging argument to win a reader's belief is a fallacy in the same class as using an argument to win a reader's love.

Question-begging arguments sometimes do commit this fallacy, but not always. Whether they do so or not depends on the ends of their users. When offered to persuade the unconvinced, they are imprudent. When offered to cement the approval of the convinced, or infuriate the unconvinced, they can be much more effective, hence prudent, hence non-fallacious in this sense. However, they can still be intrinsically unpersuasive, which shows that the fallacy of imprudence is a narrower and rarer kind of delict than intrinsic unpersuasiveness.

Persuasive and unpersuasive ensembles

Ensembles become persuasive in two ways.

  1. They may succeed at the social task, or provide proofs in the informal sense; they may link propositions on which we may have doubts to propositions that we already accept, by means of rules of inference that we already accept. In this way, ensembles spread the grounds of acceptability. If we trust some core of propositions in an ensemble, and if we trust the rules of inference or linkage, then we should trust the newly linked members. Whether we do or not is a function of the psychology of acceptance, which is bound to be slower than inference if not also divergent from it. But even when it lags, the fact that novel propositions are linked by means we trust to familiar propositions that we already trust is persuasive, both a reason and a cause for us to accept the novel proposition.

    The reverse happens as well. New propositions implied by the core we accept may be so doubtful or implausible to us that the linkage leads us to doubt the core we once accepted. Inferential links, therefore, can spread the grounds of acceptability, by modus ponens, or undo them, by modus tollens. This phenomenon is well-known but I think unintelligible for the classical model under which rules of inference take premises as true; hence for it inference can only establish conclusions, not undermine premises.

  2. As ensembles grow in size and range, they acquire more than the coherence of consistency and implication. By linking propositions to enough other propositions, covering a wide enough range, the entire ensemble becomes a coherent story for an interesting chunk of the universe. As the ensemble grows in size and range, it grows from a set of externally associated claims to a world-view. Its coherence and range become reasons for accepting its members. Let us call this vouching. Ensembles whose size and range are above a certain indefinite threshold vouch for their members;[Note 10] those below it merely contain their members. The threshold, needless to say, is not a sharp cut-off, cannot be ascertained with any precision, and need not be the same for every person.

    Nor is vouching conclusive. A vouching ensemble does not 'establish' conclusions; it is a reason and a cause to accept them. It can be outweighed or neutralized by other considerations. Moreover, a coherent story vouches in a way that is compatible with another coherent story giving an equally strong reason for accepting the negation of the same conclusion. Finally, of course, more than one vouching ensemble exists.

Vouching is an informal emergent property of ensembles. By contrast, Gödel-incompleteness is a formal emergent property of powerful systems of arithmetic; it is formal in the sense that we can prove that it is present. By contrast, we cannot prove that an ensemble vouches to one who denies it. But vouching itself is no more mysterious than Gödel-incompleteness; it is not an element or ingredient of the system, but it is a byproduct of the ingredients. Vouching can be demystified a bit if we remember that dictionaries can be helpful even if they circularly define all of their words with other words in the same dictionary. C.I. Lewis observed that dictionaries are commonly circular in this sense. Yet this organization is only useless when the dictionary, or circle, is small; when it is large, a dictionary, as we all know, can be very useful.[Note 11]

Or think of a theory (ensemble) that explains the orbits of the planets. We might not be impressed, since several other theories do so too. But if it also implies an explanation of the tides, then the range of the theory extends over the threshold, and we find the wonderful fit of the two explanations a good reason to accept both. The coherence of the two explanations vouches for each.

Or think of two chroniclers independently telling us what happened at the trial of Socrates. If their stories are consistent, then we are more inclined to believe both, although neither story might imply the other. If they cohere in the stronger sense that one implies the other, or they imply one another (perhaps even because they are the same story), then our sense that they both true may be further enhanced. At least the theory that both stories are true is a permissible explanation of the fact of their coherence. For this reason, the United States Supreme Court overturned a Texas statute that barred partners in crime from testifying for one another. Even though the coherence of their stories could be explained by collusion, it could also be explained by truth. The jury should be allowed to conclude that the coherence of the two stories vouches for them both.[Note 12]

Note that vouching is not an intrinsically unpersuasive form of argument. It is not an argument in which some premises are more doubtful than the conclusion. The main reason is that vouching is not even an argument with premise-conclusion structure, mediated by rules of inference. It is an epiphenomenon of an ensemble of many arguments of that type linked to one another. It is an emergent property of ensembles, not an argument (in the usual sense) for ensembles. So although it is circular, it is not question-begging. Or if one automatically calls all circular reasoning question-begging, then vouching is not an objectionable sort of question-begging.[Note 13]

Vouching is fascinating and complicated, and there is much more to be said about it. To explicate it more fully would require a (longer) detour into the logic of mutually supporting claims and the psychology of persuasion that is beyond the scope of this paper.[Note 14]


The vertical and horizontal models of reasoning understand begging the question very differently. Begging the question, say in the form of p p, does not establish the conclusion. Hence it fails by the tenets of the vertical model. But it does link the premise with the conclusion by means of an acceptable rule of inference. Hence it succeeds by the tenets of the horizontal model.

More precisely, it succeeds at the logical task of argument, for the horizontal argument, but fails at the social task. Here the horizontal model is more articulate than the vertical model. The argument p p does not merely fail to establish the conclusion p, it fails to link the conclusion to a premise that the unconvinced would find more acceptable than the conclusion.

Moreover, there is a sense in which circular reasoning can succeed at both the logical and social tasks of argument and still avoid objectionable or fallacious forms of question-begging. That is, large ensembles covering many topics may vouch for their members. This seems to be a case in which quantitative changes bring qualitative changes in their wake.


1. Eventually it is useful to distinguish the ideal ensemble, which includes all the consequences of the members under the rules, from the finite sub-ensemble actually considered and affirmed by some person. For some purposes only the latter is analogous to a philosophical position or worldview. In this paper, I will use 'ensemble' to refer to ideal ensembles. [Resume]

2. To argue this self-consistently, that is horizontally, would be to embed the thesis in an ensemble that is more attractive than any ensemble embedding its negation. This essay can do this only to a limited extent. I plan to offer a more extensive argument for the horizontal model in a companion paper in the near future. [Resume]

3. I use the "" rather than "" because I want to talk about arguments, not propositions. I avoid "" and "" because it would be false precision at this point to emphasize that the validity of the arguments is to be conceived syntactically or semantically. [Resume]

4. Proposition p is a superfluous member of the set of premises for conclusion q iff removing p from the set of premises does not invalidate the inference from the set to q. [Resume]

5. Here "" may express any of several possible kinds of equivalence. [Resume]

6. We do not say that she must reject any premise deductively stronger than the rejected conclusion, for then she would in effect refuse to hear any argument against her beliefs. If consistency is not to require closed-mindedness, it must not go that far. [Resume]

7. I will argue below that sometimes larger circles are legitimately more persuasive than smaller ones. Some kinds of large, persuasive circles, but not all, escape the definition of an intrinsically unpersuasive argument, and hence are legitimately persuasive. [Resume]

8. It is no accident that here we have had to repeat the transition made in the nineteenth century when, by moving from the descriptive to the normative, logicians left behind their earlier psychologism. [Resume]

9. This account summarizes Douglas Walton's etymological research in his Begging the Question: Circular Reasoning as a Tactic of Argumentation, Greenwood Press, 1991, p. 10. [Resume]

10. Since most ensembles are denumerable, hence equal, in cardinality, range is more important than size in affecting this threshold, as the examples below will show. [Resume]

11. C.I. Lewis, Mind and the World Order: Outline of a Theory of Knowledge, Charles Scribner's Sons, 1929, at p. 82. [Resume]

12. Washington v. Texas, 388 U.S. 14 (1967). For a short discussion of this case and related issues, see my The Paradox of Self-Amendment, Peter Lang Publishers, 1990, pp. 271-72. [Resume]

13. For the same reason, vouching solves the problem of how philosophical systems (ensembles) that aspire to completeness can argue for themselves. For if they appeal to premises from outside the system, they admit incompleteness; if they appeal to premises from inside, they seem to beg the question. While there may be other, harmless kinds of circular reasoning available to solve this problem, vouching passes through the horns of the dilemma. For more on this dilemma, see my "Logical Rudeness," in S.J. Bartlett and P. Suber (eds.), Self-Reference: Reflections on Reflexivity, Martinus Nijhoff, 1987, at p. 60. [Resume]

14. I plan to argue in a companion paper on the horizontal model of argument in general that vouching derives from our satisfaction in seeing a large, coherent linked network of antecedents to a proposition (answering the question why?) and a large, coherent linked network of consequents (answering the question so?). While this satisfaction may be contingent on individual psychology, an ensemble that answers the why? and so? questions offers a legitimate reason and compelling cause to be persuaded. [Resume]

The logic symbols in this file are GIFs. See my Notes on Logic Notation on the Web.

[Blue Ribbon] Peter Suber, Department of Philosophy, Earlham College, Richmond, Indiana, 47374, U.S.A. Copyright © 1997, Peter Suber.