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A Normative Theory of Argument Strength 1
A Normative Theory of Argument Strength
ULRIKE HAHN
© Informal Logic Vol. 26, No. 1 (2006): pp. 1-24.
Keywords: argumentation, fallacies, Bayesian probability, argument from ignorance,
slippery slope arguments, circular arguments, Theophrastus rule
Résumé: Dans cet article nous soutenons
l’importance générale des théories
normatives de l’appui des raisons d’un
argument. Nous employons nos
recherches récentes sur les sophismes pour
appuyer l’idée que la probabilité
bayesienne pourrait contribuer à ces
théories. Ensuite nous discutons des
caractéristiques générales qui suggèrent les
avantages d’une approche bayesienne, et
évaluons les présumés points faibles
attribués aux probabilités bayesiennes et
avancés dans les publication sur
l’argumentation.
Abstract: In this article, we argue for the
general importance of normative theories of
argument strength. We also provide some
evidence based on our recent work on the
fallacies as to why Bayesian probability
might, in fact, be able to supply such an
account. In the remainder of the article we
discuss the general characteristics that make
a specifically Bayesian approach desirable,
and critically evaluate putative flaws of
Bayesian probability that have been raised
in the argumentation literature.
Cardiff University
Birkbeck College London MIKE OAKSFORD
Introduction
Some arguments we find convincing, others we do not. Is such an evaluation
simply a matter of arbitrary preference or can rational justifications for such choices
be given? Can there be such a thing as a normative theory of argument strength? It
is the contention of this paper that such a theory is not only highly desirable, but
also possible. Specifically, our recent work on the fallacies (Oaksford & Hahn,
2004; Hahn & Oaksford, in press; Hahn, Oaksford & Bayindir, 2005; Hahn,
Oaksford & Corner, 2005; and see also, independently, Korb, 2004), has sought to
develop a general account of the fallacies based on Bayesian probability. The results
of this, we will argue, give some hope that Bayesian probability might be able to
provide a general, normative theory of argument strength. In how far this will be
possible, of course, currently remains unclear. The question can only fully be
addressed by actual demonstration of the account’s adequacy in a wide range of
circumstances; consequently there is a wealth of further research to be done.
Here, we seek only to provide an argument for why investment in this project
seems worthwhile. This argument has four main parts. In the first part of the
paper we outline general considerations as to why a normative account of argument
2 Ulrike Hahn & Mike Oaksford
strength would be desirable. In the second part, we give a brief overview of our
results to date, in order to establish the case that a Bayesian theory looks at least in
contention. In the third, and main, section we draw out the particular properties of
Bayesian probability that we think make a specifically Bayesian account of argument
strength attractive. Finally, some arguments against a Bayesian approach that have
been voiced in the argumentation literature are addressed, and it is argued that the
seeming limitations of Bayesian probability are in fact strengths.
1. Why a normative theory of argument strength?
Two obvious answers dominate here: first, such a theory is of inherent theoretical
interest and, second, it is of obvious applied importance. Interest in standards of
rational inference and hence argument has, in one way or another, motivated research
on logic for much of its history (see Prakken & Vreeswijk, 2002 for an overview
of recent work concerned with natural language argumentation). Within the last
decades, numerous authors have come to doubt that logic could provide an
appropriate standard against by which to judge argument strength (e.g., Hamblin,
1970; Heysse, 1997 Johnson, 2000; also Boger, 2005 for further references).
Logic’s perceived failures have fuelled the rise of dialectical and rhetorical theories
(see e.g., Slob, 2002 for discussion). These theories have focussed on properties
of discourse, not the evaluation of the inherent qualities of sets of reasons and
claims. Nevertheless, proponents of such theories have frequently sought to use
discourse rules to evaluate at least some classes of arguments, in particular classic
fallacies, as good or bad (e.g., Walton, 1995; van Eemeren & Grootendorst, 1992,
2004). In this sense, a preoccupation with argument strength has remained even
here. On the theoretical side then, whether or not an adequate theory of argument
strength is possible, and what it would look like, is a longstanding theoretical
question demanding resolution.
At the same time, the applied importance of a theory of argument strength is
obvious. In complex societies such as ours argumentation plays a central role.
This has led not only to a burgeoning literature concerned with ‘critical thinking’
and its teaching (e.g., McPeck, 1981; Siegel, 1988; Bowell & Kemp, 2002; Woods,
Irvine & Walton, 2004), it has also motivated the development of domain specific
theories of argumentation, for example in law (for overviews see, Neumann, 1986;
Feteris, 1997). Argumentation in applied settings would be directly affected by the
development of a suitable normative theory of argument strength and the capacity
for rational resolution it would provide.
Somewhat less obviously, the question of a normative theory of argument
strength is relevant also to those pursuing alternative, seemingly opposed paths in
the form of consensus theories (e.g., Alexy, 1989; van Eemeren & Grootendorst,
2004 for examples). For one, relative to “right or wrong”, “consensus” is typically
only second best. The reason researchers default to consensus theories, whether
in the domain of theories of truth or in the domain of argumentation, is because the
ultimate prize, a normative theory of content, seems unattainable. Were a normative
A Normative Theory of Argument Strength 3
theory of content available, it is that theory we would look to for conflict resolution.
In fact, the relationship between a normative content theory of argument strength
and consensus theories of argumentation is more complex. Inevitably given their
focus on a non-content based outcome characteristic –consensus- these theories
tend to have a strong emphasis on procedure. The rules and norms they posit are
rules of engagement: proponents can only put forward claims they actually believe
(e.g., Alexy, 1989), proponents must justify claims when challenged (van Eemeren
& Grootendorst, 2004) and so on. Crucially, the need for procedural rules remains
even where objective standards of content evaluation exist. Even where the goal
becomes ‘truth’, ‘the best perspective’ or the ‘strongest position’ there will still be
rules of engagement that will make that outcome more or less likely to occur (see
also Goldman, 1994). Silencing opponents by force, for example, is undesirable
not just with regards to consensus, but also because the suppression of arguments
in discourse means that the potentially strongest argument might not be heard.
This means the insights of researchers developing consensus theories are unlikely
to become obsolete. Some reorientation and adjustment to specific rules of discourse
would likely be necessary were a normative theory of content to supplant the
emphasis on consensus, but normative theories of content and procedural theories
ultimately pursue complimentary goals (Goldman, 1994; Hahn & Oaksford, in
press), both of which have an important role to play.
Finally, a normative theory would provide an organizing framework for
descriptive work. This may seem counterintuitive at first, but attempts at scientific
description of human argumentative behavior within cognitive science, and
particularly within cognitive psychology would benefit hugely from a normative
theory around which to structure research.
Normative theories of behavior and thought provide standards against which
actual human performance can be compared. For one, this provides a ready set of
questions for descriptive research to address- specifically, how far do human
beings match up to these standards and where specifically do they fall short.
However, normative standards also play a vital role in interpreting and understanding
human behavior. For a cognitive scientist, complete explanation encompasses
several, mutually informative levels of description (Marr, 1982). Human cognition
as a computational process is to be understood at the hardware level that governs
how a process is actually implemented, at a representational or algorithmic level
that characterizes the procedures involved, and at the highest level, the so-called
computational level, through a general characterization of the problem the cognitive
process is seeking to address. Though these levels constrain one another, they also
exhibit some degree of independence.1 Because of this degree of independence,
the task of explaining any computational system is not complete before all three
levels have been addressed. Normative theories have an important role to play in
computational level explanation, particularly so within the so-called rational analysis
of behavior (e.g., Anderson, 1991; Chater & Oaksford, 2000; Oaksford & Chater,
1998a; Chater & Oaksford; 1999a). Rational analysis seeks to understand human
behavior as an approximation to some ideal behavior, typically as an adaptation to
4 Ulrike Hahn & Mike Oaksford
some environmentally posed problem. Rational analysis seeks to characterize a
problem faced by the cognitive system, develop what would be the ‘best’ solution,
and then to determine the extent to which system behavior can be seen to be an
approximation of that solution, even though it might fall short in certain
circumstances. Consequently, normative theories can form part of functional ‘why’
questions in the analysis of behavior which are essential to the understanding of
purposeful behavior.
These considerations are exemplified by the enormous success of the
psychological literatures on “naïve statistics”, decision-making and logical reasoning
(see e.g., Kahneman, Slovic & Tversky, 1982; Holyoak & Morrison, 2005). In all
three cases, clear normative theories have prompted obvious research questions
with regards to the extent to which human beings conform to normative
considerations. This has not only led to considerable insight into what humans can
(and cannot) readily do and informed our understanding of human rationality; by
provoking explanation of seeming deviations from normative theories it has also
led to theories about mechanism and process, that is, theories about the specific
cognitive means by which reasoning in these contexts is achieved. Though these
areas are not complete, they are, by the standards of psychology, fairly “mature”
and developed areas of research. It does not seem unreasonable to suppose that
the lack of a clear normative theory of argument strength is one of the reasons
why the wider psychological study of argumentation is underdeveloped by
comparison.
Experimental research in this area is currently fragmented across a variety of
individual specialist domains. There is, on the one hand a wide literature on
‘persuasion’ which has, by and large, examined non-strength related circumstances
in their influence -typically specifically on attitudes (see e.g., Maio & Haddock, in
press, or Johnson, Maio & Smith-McLallen, in press, for an overview). There is
also, in addition to the logical reasoning literature, a much more narrow literature
examining several kinds of inductive arguments (e.g., Osherson et al, 1990). Finally,
there are a few studies that have been informed by a broadly pragma-dialectic
perspective (Neuman, 2003; Neuman & Weizman, 2003; Neuman, Weinstock &
Glasner, in press; Weinstock, Neuman & Tabak, 2004; Rips, 1998, 2002) as well
as a body of developmental research (for a review see Felton & Kuhn, 2001).
All of these have proceeded in virtually complete isolation from one another
and there is nothing even remotely like an integrated account to be had (but see
Rips, 2001 and Oaksford & Hahn, in press, for integration of at least some of
these). A normative account of strength could serve to link these different bodies
of research, as well as generate novel testable predictions of its own.
There is reason to believe that a specifically probabilistic, normative account
would be particularly useful here. In parallel to our development of a normative
theory, we have begun a program of experimental research involving the fallacies
from a Bayesian perspective (Oaksford & Hahn, 2004; Hahn, Oaksford & Bayindir,
A Normative Theory of Argument Strength 5
2005), and this work has natural connections to the literature on belief and attitude
change under the header of the ‘subjective probability model’ which have yet to be
explored (e.g., Allen & Kellermann, 1988; Allen, Burrell & Egan, 2000; Hample,
1977, 1978, 1979; McGuire, 1960; Wyer, 1970, 1974; Wyer & Goldberg, 1970).
We detail the (normative) Bayesian reconstruction of fallacies in the next section in
order to motivate further the desirability of a Bayesian account.
2. Why might Bayes provide a theory of argument strength?
Support for the idea that Bayesian probability could provide a normative theory of
argument strength comes from the fact that it has successfully been used to explain
a considerable range of fallacies, in that it captures key intuitions about the relative
strength of arguments. In conjunction with Bayesian probability’s pedigree as a
normative framework , this suggests it might be able to supply the long-desired
formal treatment of the fallacies (Hamblin, 1970). This work can be seen as providing
a formal explication of the epistemic account of the fallacies (e.g., Siegel & Biro;
1997; Ikuenobe, 2004). Because this work is published elsewhere, we provide
only a brief overview here.
Oaksford and Hahn (2004) first sought to explain through a Bayesian account
arguments from ignorance such as
(1) Ghosts exist, because nobody has proven that they don’t.
Individual arguments are composed of a conclusion and evidence for that conclusion.
Both conclusion and evidence have associated probabilities which are viewed as
expressions of subjective degrees of belief. Bayes’ theorem provides an update
rule for the degree of belief associated with the conclusion in light of the evidence.
Argument strength, then, on this account is a function of the degree of prior
conviction, the probability of evidence, and the relationship between the claim and
the evidence -in particular how much more likely the evidence would be if the
claim were true.
A Bayesian account captures, among other things, the difference between positive
and negative evidence and allows one to capture the intuition that the positive
argument (2a) is stronger than the negative argument (2b):
(2a) Drug A is toxic because a toxic effect was observed (positive
argument).
(2b) Drug A is not toxic because no toxic effects were observed (negative
argument, i.e., the argument from ignorance).
However, (2b) too can be acceptable where a legitimate test has been performed,
i.e., If drug A were toxic, it would produce toxic effects in legitimate test.
Drug A has not produced toxic effects in such tests.
Therefore, A is not toxic.
6 Ulrike Hahn & Mike Oaksford
Demonstrating the relevance of Bayesian inference for negative vs. positive
arguments involves defining the conditions for a legitimate test. Let e stand for an
experiment where a toxic effect is observed and ¬e stand for an experiment where
a toxic effect is not observed; likewise let T stand for the hypothesis that the drug
produces a toxic effect and ¬T stand for the alternative hypothesis that the drug
does not produce toxic effects. The strength of the argument from ignorance is
given by the conditional probability that the hypothesis, T, is false given that a
negative test result, ¬e, is found, P(¬T|¬e). This probability is referred to as
negative test validity. The strength of the argument we wish to compare with the
argument from ignorance is given by positive test validity, i.e., the probability that
the hypothesis, T, is true given that a positive test result, e, is found, P(T|e). These
probabilities can be calculated from the sensitivity (P(e|T)) and the selectivity
(P(¬e|¬T)) of the test and the prior belief that T is true (P(T)) using Bayes’
theorem:
)
)
(1)(|()()|(
)()|(
)|( TPTePTPTeP
TPTeP
eTP −+
=(1)(|()()|(
)()|(
)|( TPTePTPTeP
TPTeP
eTP −+
=(3)
)
()|())(1)(|(
))(1)(|(
)|( TPTePTPTeP
TPTeP
eTP +− −
=()|())(1)(|(
))(1)(|(
)|( TPTePTPTeP
TPTeP
eTP +− −
=(4)
As Oaksford and Hahn (2004) argue, sensitivity and selectivity, for a wide
variety of clinical and psychological tests are such that positive arguments are
stronger than negative arguments. The reason we consider negative evidence on
ghosts (1) to be so weak is because of the lack of sensitivity (ability to detect
ghosts) we attribute to our tests as well as our low prior belief in their existence
(see also Hahn et al, 2005 and Hahn and Oaksford, in press, for further discussion
and analysis of different kinds of arguments from ignorance). The Bayesian account
renders this textbook example as an argument that occupies the extreme lower end
of the argument strength range as a consequence of the specific probabilities
estimates involved. The argument is weak because of these aspects of its content,
not because of its logical structure or particular role in a discourse (cf., Walton,
1996) and other arguments with the same structure and discourse function can be
perfectly convincing. In other words, the Bayesian analysis tackles what has been
a longstanding problem for the fallacies, namely that most types of fallacy seem
prone to a proliferation of exceptions that seem more or less acceptable. The
Bayesian account allows one to distinguish ‘good’ arguments from ignorance from
less compelling ones, providing an explanation for why they are good or bad.
Hahn and Oaksford (in press) also provide a Bayesian treatment of slippery
slope arguments which as consequentialist arguments are captured using decision
theory (Savage, 1954; on decision theory and consequentialist argument see also
e.g., Lumer, 1997). According to Bayesian decision theory, choosing an action in
the face of an uncertain future should be based on an evaluation both of the utilities
A Normative Theory of Argument Strength 7
we assign to possible future outcomes and the probabilities with which we think
these outcomes will obtain. This normative framework can be applied to textbook
slippery slope arguments such as
(5) We should not ban private possession of automatic weapons, because
doing so will be the first step on the way to a communist state.
This argument seems weak, because the subjective probability of such a ban
indeed setting in motion a chain of events that will lead to the outcome ‘communist
state’ seems so incredibly low. Yet at the same time, there is evidence from legal
history that ‘slippery slopes’ have, in fact occurred (Lode, 1999), and, in everyday
life, the mechanisms behind them are exploited through techniques such as ‘foot
in the door’ advertising (e.g., Freedman & Fraser, 1966). In general, the more
there is a real chance of a feared outcome occurring the stronger a slippery slope
argument will be. Hence, in
(6a) Legalizing cocaine will lead to an increase in heroin consumption.
(6b) Legalizing cannabis will lead to an increase in heroin consumption.
the first argument (a) seems more compelling. That the degree to which one cares
about the outcome, its utility, also plays a role can be seen from the examples
(6c) Legalizing cannabis will lead to an increase in heroin consumption.
(6d) Legalizing cannabis will lead to an increase in listening to reggae
music.
where, assuming that both listening to reggae music and heroin consumption are
equally likely, (c) seems the far stronger argument. By varying both utility and
probability, perfectly acceptable examples of slippery slope arguments can readily
be generated (see for examples also Corner, Hahn & Oaksford, 2006).
Hahn and Oaksford (in press) also provide a Bayesian analysis of the
argumentum ad populum or “appeal to popular opinion” (on this see also, Korb,
2004) and the argumentum ad misericordiam, which uses an appeal to pity or
sympathy for argumentative support. In Hahn, Oaksford and Bayindir (2005) the
Bayesian account is extended to a treatment of circular arguments, and we say
more on these below.
This treatment of informal argument fallacies complements earlier research by
Oaksford and Chater that has argued in detail that a wide range of seeming ‘logical
errors’ in conditional and syllogistic reasoning are perfectly acceptable when viewed
from a probabilistic perspective, that is, widespread intuitions are rendered more
accurately by switching from logic to probability theory as a normative standard
(Oaksford & Chater, 1994, 1996, 1998b, in press; Chater & Oaksford, 1999b).
Finally, Korb (2004) has also argued that a Bayesian approach could explain
fallacies such as the appeal to authority and hence provides a framework for
understanding ordinary arguments that is well worth developing.
8 Ulrike Hahn & Mike Oaksford
The use of Bayesian probability to distinguish between warranted and
unwarranted conclusions in the context of these fallacies is an attempt to develop
a ‘reduction of fallacy theory’ in Lumer’s (2000) sense in that systematization and
explanation of the fallacies is derived from a general normative theory. In contrast
to past emphasis on deduction as the appropriate epistemological principle/standard,
however, the Bayesian approach leads to a rather different evaluation of individual
fallacies; the logical standard tends to lead to very ‘all or none’ evaluations whereas
the probabilistic account allows and explains graded variation among instances of
the same structure.
Because Bayesian probability provides a formal framework for distinguishing
between warranted and unwarranted conclusions in the context of the fallacies,
we think it has considerable potential for advancing epistemic approaches to
argumentation (see e.g., Siegel & Biro, 1997; Goldman, 1997, 2003). Though we
are optimistic in this regard, we do not wish to claim that the definitive, long-
desired (see e.g., Hamblin, 1970) formal treatment of the fallacies has been provided
in detail; for this, it will be necessary to demonstrate how Bayesian probability
captures the bulk of the traditional list of fallacies (see Hahn & Oaksford, in press,
for further discussion of the key issues here). Even less do we wish to claim that
Bayesian probability has in any way been established as a sufficient theory of
argument strength. For the current context we wish to claim only that some success
in explaining the fallacies can be reported and that this success lends some credibility
to the idea that a general, normative, Bayesian theory of argument strength might
one day be forthcoming.
Clearly, this goal has not yet been achieved. What we wish to argue for in the
remainder of this paper is why the pursuit of a specifically Bayesian theory of
strength strikes us as worthwhile.
3. Why a Bayesian theory?
In this section, we discuss some of the main assets of the Bayesian approach,
which we think would make it particularly desirable as a normative theory.
Firstly, the Bayesian approach treats probabilities as expressions of subjective
degree of belief not as an objective property of statements and their relationship to
the world as is the case for frequentist interpretations of probability. This is important
in the context of argumentation, because many of the things we argue about involve
singular events—for example, whether or not Oswald killed JFK (see also Hahn &
Oaksford, in press). Assigning single event probabilities only makes sense from a
Bayesian subjective perspective; it is meaningless on a frequentist interpretation.2
The fact that probabilities are taken to be expressions of subjective degree of belief
also makes it particularly natural to interpret them as the degree of conviction
associated with a claim. That conviction can be a matter of degree, not just a
binary true or false, is fundamental for an adequate treatment of the fallacies. In
A Normative Theory of Argument Strength 9
particular the treatment of circular arguments has been hampered by binary notions:
Arguments involving self-dependent justification are frequent in science yet they
are hopelessly rendered viciously circular and hence unacceptable by any account
that conceives of statements put forward in an argument as only true or false (see
Hahn, Oaksford & Corner, 2005 for detailed discussion; and more on this below).
An all or none notion of dissent or assent to a claim has also led to an overexpansion
of the notion of the burden of proof (see Hahn & Oaksford, subm.), suggesting,
among other things, an explanation for why arguments from ignorance are poor
that is vacuous in practice (see Hahn & Oaksford, in press; Hahn & Oaksford,
subm. for detailed discussion).
Furthermore, the Bayesian formalism allows us to distinguish readily between
the ultimate conviction with regards to a claim that an argument brings about—
expressed as the posterior probability assigned to that claim in light of the evidence—
and the degree of change that a reason effects (one way of measuring the latter is
the likelihood ratio, see Hahn, Oaksford & Corner. 2005; Oaksford & Hahn, in
press). This is important because the ultimate degree of conviction brought about
in an argument is influenced, on the Bayesian account partly by the prior conviction
associated with the claim. Consequently, we want to evaluate arguments not just
with regards to how convinced they make us, but also with regards to how much
they made us change our beliefs. This notion of degree of change (or ‘force’ of an
argument) is important in the evaluation of argument strength, for example, because
it allows one to explain why direct premise restatements (“God exists, because
God exists”) make poor arguments even though they are deductively valid (see
Hahn, Oaksford & Corner, 2005)—a tension that has puzzled philosophers for a
long time. The ability to quantify change allows us to see clearly that such arguments
bring about no change in convictions whatsoever. This makes them maximally
ineffective as arguments and consequently maximally poor.3
That prior beliefs influence argument strength on the Bayesian account, of
course, introduces a degree of relativity into the evaluation of arguments. We
would argue that the degree of relativity afforded by the Bayesian approach is both
essential and just right.
The important role a Bayesian analysis assigns to prior belief is an instance of
a fundamental aspect of argumentation—the nature of the audience, which has
been assumed to be a crucial variable for any rational reconstruction of
argumentation (e.g., Perelman and Olbrechts-Tyteca, 1969; Goldman, 1997).
Audience relativity has the consequence that a fallacy for one person may not be a
fallacy for someone else because their prior beliefs differ. Ikuenobe (2004) makes
the same point using the argument that all cases of killing a living human being are
bad and abortion is a case of killing a living human being, therefore, abortion is
bad. This argument may provide adequate proof for someone who already believes
that a fetus is a living human being. However, for someone who does not believe
this proposition, this argument is weak or provides inadequate proof for the
conclusion.
10 Ulrike Hahn & Mike Oaksford
Crucially, however, the relativity introduced by Bayesian priors does not mean
that “anything goes”. For one, the subjectivity introduced by priors does not mean
that objectively true states cannot be reached. Individual Bayesian estimators can
be more accurate than their frequentist counterparts, as judged by the criteria of
the frequentist, even though they are biased estimators. For example, comparing
Bayesian and frequentist estimators for a population mean, the Bayesian posterior
mean will have the smaller error over the range of values that are realistic for that
population mean (see, e.g., Bolstad, 2004). Furthermore, the well-known
convergence properties of Bayesian updating mean that where enough suitable
data are available, posteriors will eventually converge on the appropriate values
regardless of priors. This latter result holds true, of course, only if the priors are
not already ‘certain’, that is, 0 or 1. In this case, no amount of data can bring
about further change. This allows one to capture the fact that some degree of
“openness to change” or basic willingness to consider an argument is necessary
for it to take an effect. At the same time it means that one does not lose the
possibility of certainty associated with logical necessity and proof.
For many arguments in everyday life, of course, there will be neither proof nor
huge amounts of data. Here, priors will matter and it should be seen as a virtue that
they do. In the numerous argumentative contexts where there simply isn’t sufficient
mutually agreed evidence to bring people’s beliefs in to alignment “with the facts”
the Bayesian approach can reflect the diversity of opinion and legitimate disagreement
that will remain. 4
However, because Bayesian probability imposes constraints on the rational
assignment of degrees of belief, the possibility of agreement and disagreement are
constrained both within and across agents. The relative degree of conviction a set
of different reasons brings about, for example, will be the same for two agents
even where they differ in their priors with regards to the claim in question unless
they also disagree about properties of the reasons themselves. In other words,
even if we end up differentially convinced as a result of initial differences in priors,
we can agree on the relative strength of arguments; where we do not do so, we
must differ in other ways than just our priors for that disagreement to be rational.5
Moving on from the relativity (or not) afforded to argumentation by a Bayesian
approach, a further important asset is the probabilistic notion of relevance (see
also in the context of argument strength Korb; and, more generally, Pearl, 1988,
2000 for detailed discussion). ‘Conditional independence’ offers a dynamic notion
of relevance, that changes as information is added or deleted to a database and the
conditional independence axioms have been found to correspond to intuition about
informational relevance in a variety of contexts (Pearl, 1988). Whether this is
ultimately good enough, of course, remains to be seen. However, should more be
required, some entirely new, as yet unknown account of relevance will likely have
to be devised. It already seems comparatively clear that accounts of argumentative
relevance that rely purely on logical consequence are unlikely to do justice to the
concept of relevance required by a theory of informal argument. Similar arguments
A Normative Theory of Argument Strength 11
have been made with respect to relevance logics (Anderson & Belnap, 1975). The
concept of relevance goes beyond logical entailment, even relevant entailment. For
example, it is relevant to whether an animal has palpitations that it has a heart but
this is due to the causal structure of the world not the logical relations between
propositions (Oaksford & Chater, 1991; Veltman, 1986). In short, existing theories
of argumentation that transmit plausibilities via logical consequence relations seem
unlikely to capture all the ways in which relevance relations are established in
argument.
Our arguments in favor of a Bayesian approach so far have sought to identify
characteristics that seem well attuned to the needs of argumentation. We conclude
our survey of reasons why a specifically Bayesian theory of argument strength
would be desirable with two more general considerations: the well-established
normative standing of Bayesian probability and its connection with other bodies of
research.
The most obvious consideration in developing a normative theory is the extent
to which its normativity is indeed accepted or guaranteed. The normative standard
of probability theory meets this requirement. There is an intuitive rational justification
that underpins probability theory: Reasoning probabilistically is rational if one wants
to avoid making bets one is guaranteed to lose. Furthermore, there is a well-
defined formal calculus which guarantees that this rational principle is respected.
The theory can deal with a huge range of hypotheses—whether these be discrete,
multivalued, or continuous—and has been developed to deal with a wide variety of
circumstances, such as uncertain or cascaded evidence. However, Bayesian
conditioning, which is at the heart of the approach, follows directly from the three
basic axioms of probability theory and the notion of conditional probability. That
its assumptions are so minimal also lies at the heart of the finding that attempts to
develop new and different formalisms frequently turn out to be ‘probabilities in
disguise’ (on this issue see e.g., Cox, 1946; Horvitz, Heckerman & Langlotz,
1986; Heckerman, 1986; Snow, 1998; see also, Pearl, 1988 and Howson & Urbach,
1993 for further references). Consequently, we follow Pearl’s (1988, p. 20) view
on whether it is necessary to supplant probability theory, “…we find it more
comfortable to compromise an ideal theory [i.e., probability theory] that is well
understood than to search for a new surrogate theory, with only gut feeling for
guidance.”
The final benefit of the Bayesian approach then, is that it connects research on
everyday argument with a number of different bodies of research. At the level of
normative theory, our Bayesian approach trades on similar approaches to scientific
inference (e.g., Howson & Urbach, 1993; Earman, 1992), which, following other
authors in the area (Kuhn, 1993), we have suggested can be extended naturally to
informal argument. This is advantageous not only because of the potential theoretical
unification, but also because a lot of hard work has already been done. The Bayesian
approach to scientific inference has received much scrutiny and criticism (e.g.,
Miller, 1994; Sober, 2002); not all of this is equally relevant to everyday argument
12 Ulrike Hahn & Mike Oaksford
(specifically, the subjectivity inherent in the Bayesian approach might, to some, be
more palatable in non-scientific contexts), but a wealth of important issues have
already been well worked through and can be received into studies of argumentation.
One example of this is the issue of priors and ignorance that we will return to
below.
Within psychological research on human behavior, finally, the Bayesian approach
links up with the ever-increasing body of evidence that suggests that much of
human cognition is, in one way or the other, probabilistic. Theories and data range
from Bayesian accounts of vision (Knill & Whitman, 1996), through language
acquisition and processing (e.g., Bates & MacWhinney, 1989; MacDonald, 1994),
to many aspects of higher level cognition (e.g., Oaksford & Chater, 1998b). An
emerging picture, here, suggests that humans are probabilistic beings.6 A probabilistic
account of argument evaluation receives support from this ‘character’ argument.
4. Limitations of a Bayesian Approach
The intention in this paper is to highlight the merits of a Bayesian approach, not a
comparative evaluation with possible competitors, not least because that set itself
is not closed. However, it would be wrong to complete a discussion of a Bayesian
approach to argumentation without drawing attention to the fact that Bayesian
principles have been subject to criticism. Because the interest in Bayesian principles
in areas such as statistics and also the philosophy of science has been intense,
these criticisms have been well-discussed and many issues have more or less well-
developed replies. For an excellent overview the interested reader is referred to
Howson and Urbach’s volume on scientific reasoning (1993). We restrict ourselves
here to two issues that have found their way into the literature on argumentation.
First is the idea that what we know, or more importantly do not know, makes
the Bayesian formalism unnatural. Walton (2004, pg. 277) and Ennis (2004), for
example, seem to echo an oft heard claim that the assignment of numerical values
to premises is frequently impossible or unhelpful, in that it requires an exactness
that is not available in most cases.
This slightly misses the point of subjective probabilities: they are expressions—
through the use of non-extreme values—of the indefiniteness of one’s knowledge
and are introduced precisely because of uncertainty. The argument, by contrast,
makes it seem as if one not only has to know that one is uncertain, but also that
one has to know to a precise degree how uncertain (see also Howson & Urbach,
1993 pg. 87). While it is true that some number has to be specified, the resolution
with which that number is specified can vary according to context and sensitivity
analysis can be used to determine how much precision the problem at hand requires
(on sensitivity analysis see e.g., Gill, 2002). Exactly the same applies to the use of
real numbers in measuring physical, everyday quantities. In many contexts, it will
simply be irrelevant to the required outcome not only whether a measurement was
5.687 or 5.689cm but also whether it was 5.5 or 5.7cm, or even 5 or 6cm. In
A Normative Theory of Argument Strength 13
practice, one will always be restricting oneself to certain points of the scale, and
what these are will be determined by context.
Likewise, even though a specific point value probability has to be specified,
there is nothing to stop one, in a given context, restricting one’s use of the scale to
5 different points, for example 0.1, 0.25, 0.5, 0.75, 0.9 which one might take to
correspond to the verbal descriptors ‘very unlikely’, ‘unlikely’, ’50/50 chance’,
likely’, and ‘almost certain’—or any other such set of numbers by which to express
broad increments. Alternatively, one can define and use intervals over the scale. As
long as these intervals are small, one gets a generalization of the classic calculus
which is simply defined over interval values rather than point values (see e.g.,
Walley, 1991; for a broader overview and discussion see also Parsons, 2001). It is
also interesting to note here that peoples’ attitudes toward imprecision in the
specification of probabilities seems to contain an asymmetry. In experimental
contexts it has been found that while people prefer to express probabilities with
verbal descriptors and the crude distinctions they afford, they actually prefer to
receive numerical probabilities (e.g., Wallsten et al., 1993); so it does not actually
seem to be the case that numerical expressions of probability are inherently
‘unnatural’.
The issue of indefiniteness in one’s degree of uncertainty leads on naturally to
the case of maximal indefiniteness in the form of complete ignorance. Because the
Bayesian treatment of complete ignorance is one of the most widely cited
‘shortcomings’ of Bayesian probability, it seems useful to also discuss it here. The
Bayesian way to represent ignorance about a range of possibilities is to assign
them all equal probability according to the ‘Principle of Indifference’. This principle
states that when we have a number of possibilities, with no relevant difference
between them, they all have the same probability. As a way of representing ignorance,
indifference makes intuitive sense; however, it readily leads to seeming ‘paradox’.
As an example, one might take an urn filled with white and colored balls in unknown
proportions, but the colored balls consist of red balls and blue balls in equal number.
According to the Principle of Indifference, our present data—before we have
drawn our first ball—are neutral between its being colored and its being white.
Hence, we should expect it to be white with a probability of .5. However, if the ball
is colored, then it is either red or blue, and the data are also neutral between the
ball’s being either white or red or blue. Hence, according to the Principle of
Indifference, the probability of the ball’s being white is one third. Another example,
this time concerning a garden plot, is presented by Sober (2002). Once again there
is no unique way to translate ignorance into an assignment of priors: one gets one
answer if one applies a uniform prior to lengths of sides of plot and another if one
applies it to the area of the same plot. Examples of this kind can be generated ad
nauseam (see Howson & Urbach, 1993 for further examples and references).
What then, are their implications?
The paradoxes arise because the uniform assignment of probabilities is language-
or description relative. It is the primitives of the description that are assigned equal
14 Ulrike Hahn & Mike Oaksford
probability according to the Principle of Indifference. Terms derived from these
primitives will not themselves necessarily have equal probability. Moreover, they
will not do so for good mathematical or logical reason. If all six numbers of a dice
are equi-probable, then the outcome ‘greater than 1’ will have a 5 in 6 chance of
occurring. The seeming paradox occurs, because alternative sets of primitives
give rise to different probabilities for composite beliefs—but given our ignorance,
the choice between these sets seems arbitrary.
Some have seen these paradoxes as so compelling as to force the development
of alternative theories. Dempster-Shafer theory, for example, is based on the idea
that the natural way to represent total ignorance is to treat equally all possible
alternatives, whether they are primitive or composite. This would allow one to
have the same degree of belief both in the claim that all six numbers are equally
likely and that ‘greater than 1’ is as likely as not to occur (see Howson & Urbach,
1993).
The Bayesian response to this is that total ignorance is not possible. Specifically
it is not rationally possible to be uniformly unopinionated about everything. The
probability calculus provides a normative theory for the rational assignment of
belief. From certain beliefs other beliefs will necessarily follow, for example, by
logical consequence. A rational agent should be committed to these beliefs whether
or not they are held in actual fact or even entertained. There are considerable
constraints on what is a rational set of beliefs given no evidence. If I actively—in
ignorantiam—believe something about a hypothesis H, then I am forced to believe
something else about ¬H; likewise, if I actively—in ignorantiam—believe something
about the perimeter of a property, I have to consistently believe something about
its area. As Howson and Urbach argue, it should be seen as a virtue not a vice of
the theory that it brings out the impossibility of total ignorance so clearly.
The argument based on paradoxes seems compelling because it is so easy to be
unaware of these constraint—whether this is due to a failure of rationality, or a
reflection of the fact that one actually held no beliefs on the topic whatsoever. The
fact that different problem statements can give rise to different answers and that
this discrepancy might seem arbitrary if there is not much to choose between
them should likewise not be seen as a fault of the formalism. That different answers
can and will ensue depending on how a formalism is mapped onto a real world
problem is not a unique property of Bayesian inference, but occurs wherever
states of affairs are mapped onto formal models. This problem could be avoided
only if there was one single, unique way to represent the world. In lieu of that, the
problem will remain, and it is typically not the job of the formalism itself to determine
which of many possible or even plausible mappings is preferable. What is comforting
about Bayesian inference and the so-called ‘paradoxes’ involving ignorance is that
it need not always matter: once balls are drawn from the urn my estimate will
gradually become more and more accurate, whether my prior was 1/2 or 1/3.
The second perceived limitation of the Bayesian approach that can be found in
the argumentation literature concerns the way argument strength is calculated
A Normative Theory of Argument Strength 15
from evidence. An alternative approach to argumentation views arguments as
presumptively acceptable (e.g., Walton, 1996). That is, they are defaults that can
be readily overturned by new information. Walton makes the direct link between
this style of reasoning and the development of non-monotonic logics (e.g., Reiter,
1980, 1985) in AI. In AI knowledge representation, the failure to develop tractable
non-monotonic logics that adequately capture human inferential intuitions, has
provided part of the impetus for the development of Bayesian probabilistic approaches
to uncertain reasoning. Nonetheless, some alternative approaches have been
developed that explicitly include some measure of the strength of an argument
(Gabbay, 1996; Fox & Parsons, 1998; Pollock, 2001; Prakken & Vreeswijk, 2002).
Moreover, it is this inclusion that provides these systems with their nice default
properties.
These systems explicitly eschew the idea that argument strength can be
adequately dealt with using the probability calculus. The root of this contention is
Theophrastus’ Rule: the strength that a chain of deductively linked arguments confers
on the conclusion cannot be weaker than the weakest link in the chain (Walton,
2004). This is a condition that cannot be guaranteed by the probability calculus.
Examples that seem to conform to Theophrastus’ rule but not to the probability
calculus, have persuaded Walton (2004) and, for example, Pollock (2001), that a
third form of reasoning should be countenanced in addition to deductive and
inductive/statistical reasoning, i.e., plausibilist reasoning.
We address first one of the examples for which probability has been viewed as
inadequate. We then address directly Theophrastus’ rule.
Walton’s (2004) example concerns negation within the probability calculus
according to which the probability of the negation of a hypothesis H, P(¬H), is
constrained to be 1 minus the probability of that hypothesis, P(H). Plausibilist
reasoning based on Theophrastus’ rule is necessary because this probabilistic
approach to negation is sometimes inappropriate. Specifically, Walton argues that
in the case of legal argumentation involving evidence, both a proposition and its
opposite can be highly plausible. He writes,
For example, suppose a small and weak man accuses a large and strong man
of assault. The small man argues that it is implausible that he, the weaker
man, would attack a visibly stronger man who could obviously defeat him. In
court, the visibly larger and stronger man asks whether it is plausible that he
would attack such a small man in front of witnesses when he knew full well
that he could be accused of assault. Here we have an argument with probative
weight on one side, but also an argument with probative weight on the
opposed side. The proposition that the large man commited assault is plausible
in light of the facts, but its negation is also plausible in light of the same facts.
The argumentation in this typical kind of case in law violates the negation
axiom of the probability calculus. (pg. 278)
From this it follows, according to Walton, that a different calculus is necessary,
namely one whereby a proposition can be plausible in relations to a body of evidence
16 Ulrike Hahn & Mike Oaksford
in a given case, whereas the negation of that same proposition can also be plausible
in relation to another body of evidence, and that a contextual and pragmatic notion
of probative evidence is required.
We do not think the example warrants a new form of reasoning. What is going
on here is simply a case of conflicting evidence. One first receives evidence that
would increase one’s posterior degree of belief in the claim, and then receives
evidence that would decrease it. In this sense, one has had evidence both to make
the claim more plausible, and evidence to reduce its plausibility. However, argument
evaluation must mean that the two sources of evidence are at some point integrated
into a final, overall assessment. At this stage, they can either cancel each other out,
or one can outweigh the other with the consequence that one’s final conviction is
shifted in the direction of the ‘weightier’ evidence, though less so than if the
counterevidence had not been received. One does not end up both more convinced
that the small man hit the large one and that he did not.7 Unless one views both
bodies of evidence as entirely equal, in which case nothing changes, weighing the
evidence will increase one’s belief in the one at the expense of the other. This is
exactly what the probability calculus allows one to achieve. The first set of evidence
increases one’s posterior degree of belief in the claim, the second decreases it and
Bayesian updating provides the mechanism whereby these beliefs are integrated
into an overall single judgment whether the two sets of evidence are received one
after the other or together.
It is also important here that not all legal claims involve a proposition and its
negation. They might equally involve two propositions that are mutually exclusive
but not complements. In fact, Walton’s example seems at times to oscillate between
negation and mutual exclusivity in that the small man’s claim that it would be
implausible that he would attack a larger man seems more relevant to a debate
about who threw the first punch as opposed to a debate about whether or not the
large man hit him. In arguing about the first punch, the answer could be the small
man, the large man, both men simultaneously or that nobody hit anybody at all. In
this context, the small man’s claim that the large man hit him first could become
more plausible according to the evidence as could the large man’s claim that the
small man hit him first, because only the sum total of the probabilities associated
with the four logical possibilities must equal to one. In other words, one could
become more convinced that one of them hit the other, as opposed to no fight
having taken place at all, but be none the wiser as to which one hit the other first.
Again, Bayesian probability will give this result without the need for a new form of
‘plausible reasoning’.
To conclude our discussion, we provide an example to illustrate why we think
Theophrastus’ rule is not a good idea. The example is drawn from our treatment of
the fallacies and involves circular arguments. Most circular arguments found in
practice do not involve a direct restatement of the conclusion among the premises,
rather the conclusion forms an implicit assumption, a presupposition, that underlies
the interpretation of the premise material. Hence, the inference involves self-dependent
A Normative Theory of Argument Strength 17
justification (also referred to as ‘epistemic circularity’, e.g., Goldman, 2003; see
Hahn, Oaksford & Corner, 2005 for discussion of the previous literature).
This is the case for the classic textbook example,
(7) God exists, because the Bible says so, and the Bible is the word of
God.
However, this is also the case for a huge number of scientific inferences, for
example
(8) Electrons exist, because we can see 3 cm tracks in a cloud chamber,
and 3 cm tracks in cloud chambers are signatures of electrons.
The scientific example which arises wherever scientists are dealing with entities
that cannot be directly observed seems a perfectly acceptable example of the
classic inference to the best explanation (Harman, 1965; see also, Josephson &
Josephson, 1994). This suggests that self-dependent justification is not inherently
wrong (see also Shogenji, 2000 and Brown, 1993, 1994 for scientific examples).
Our Bayesian account explains why (and for whom) the Bible example, which has
exactly the same structure, seems weak and the scientific example acceptable.
Self-dependent justification of this kind is captured through hierarchical Bayesian
inference. Three levels are involved here—the directly observed evidence, the
interpretation of the evidence, and the hypothesis. The observed evidence is the
Bible’s claim that God exists and the 3cm tracks seen in the cloud chamber. These
are relevant to the hypothesis because they are interpreted as the word of God and
the signature effects of electrons, respectively. This interpretation is itself uncertain
and dependent on the fact that the hypothesis is true. This extra level, however,
does not in anyway preclude Bayesian conditioning on the given observation (see
e.g., Pearl, 1988 for examples of hierarchical Bayesian inference), and making that
observation will increase our posterior degree of belief in the hypothesis. In the
case of the Bible, this increase will typically be slight, because there are numerous
other plausible interpretations of the Bible and its content other than that it is the
direct word of God, and priors, as always, also affect how convincing the argument
will be. The scientific example will seem stronger simply because (and as long as)
our estimates of the associated probabilities are different. A Bayesian analysis then
explains the difference between weak, textbook examples of self-dependent
justification and widespread scientific practice and renders scientific use of inference
to the best explanation acceptable.
By contrast, classical logic fails here (Hahn, Oaksford & Corner, 2005 for
fuller discussion). Because the conclusion must already be assumed as a premise,
and degrees of belief (i.e., ‘truth’) are all-or-none, no self-dependent argument
can bring about any change in conviction. Consequently, all such arguments are
necessarily rendered maximally poor.
Moreover, Theophrastus’ rule fails here as well. The interpretation of the
observation statement required for the inference (i.e., that 3cm tracks are signature
18 Ulrike Hahn & Mike Oaksford
effects of electrons; for example, random smudges on the screen) depends on the
hypothesis in question being true (i.e., trivially, if there is no such thing as an
electron the 3cm tracks can be whatever they like except an electron’s signature
effect). This dependency means that the probability of the interpretation being true
can never be greater than the probability of the hypothesis itself. This follows
directly from the fundamental axioms of probability theory, because the interpretation
statement, if true, would imply the truth of the hypothesis, and the probability of
any logical consequence of a statement must be at least as great as the probability
of that statement itself.
In other words, the degree of belief associated with the interpretation statement
will be a weaker (or as weak a) link in the premise material as the presupposed
conclusion itself. Hence our degree of belief in that conclusion should never rise
according to Theophrastus’ rule. Much of our scientific reasoning and
argumentation would be labelled ‘fallacious’ as a result, and my degree of belief in
electrons would remain the same regardless of what I observed in the cloud chamber
experiment or, in fact, whether or not I bothered to conduct the experiment at all.
This example, in our view, also serves to underscore the wider merit of sticking
with a well-established normative calculus in that trouble free inference procedures
are not that easy to derive, let alone to establish as normatively justified.
5. Conclusion
We have sought to argue here for the desirability not only of normative theories of
argument strength, but also for the desirability of a specifically Bayesian account.
Bayesian probability brings with it a whole host of inherent characteristics and
theoretical connections that make it an attractive candidate framework. Also, its
success in explaining a range of fallacies suggests, to us at least, that there is some
chance its promise might be fulfilled. It is hoped that the case for Bayesian probability
made here will motivate future research aimed at this goal.
Acknowledgments
The authors would like to thank Adam Corner and Adam Harris for helpful
discussions, as well as Christoph Lumer and three anonymous reviewers for their
useful comments on this manuscript.
Notes
1 A cash register, as a simple computing device, calculates a total sum of cost (computational level
description) as part of the social exchange network that constitutes a ‘sale’- several possible
algorithms for addition are available as procedures for the device at the representational level, and
each of these, finally, could be realized in a physical device in countless ways.
2 Propensity theory (e.g., Popper, 1959) is another suggested interpretation of the probability
calculus which conceptually allows single-event probabilities, while seeking to maintain a
frequentists basis. On its problems see e.g. Howson & Urbach, 1993, ch. 13 for detailed discussion.
A Normative Theory of Argument Strength 19
3 Note that this is by no means a property of all logically valid arguments (Hahn et al, 2005). It
does not apply, for example, to the inference p therefore ‘p or q’.
4 Ennis (2004) claims that assigning subjective probabilities to statements altogether ‘wipes out
disagreement’ among people and hence is inappropriate for argumentation, because we can agree
that our subjective views of a claim differ. However, we do not see how it follows from this that
we cannot be motivated to change others’ degree of belief to the extent that it does not correspond
to our own, so argument is still both meaningful and possible.
5 Specifically the likelihood ratio has to be different.
6 This, of course, does not imply that peoples assessments of probabilities are always entirely
accurate. It does mean, however, that sensitivity to the probabilistic nature of the environment
emerges as a central aspect of computational level description (see section 1). At the same time,
however, subsequent research has greatly modified some of the early claims regarding people’s
‘failings’ with regards to intuitive statistics (e.g., Birnbaum 2004, cf. Tversky & Kahneman,
1974).
7 If one did, one would be prone to (synchronic) Dutch books, that is bets one is guaranteed to
lose. Specifically, in the case where it is clear that either the A hit B or B hit A, becoming both more
convinced in A’s hitting and in B’s hitting would mean that the degrees of belief assigned to each
of these possibilities could exceed 1. Assume one thought, for example, that there was a .6 chance
that A hit B, and a .5 chance that it was B that hit A. Assume further one is willing to bet in line
with one’s degrees of belief, such that one will accept up to or equal to one’s ‘degree of belief’ x
$10 for a unit wager that pays $10 if the claim in question turns out to be true. One would then
be happy to pay a bookie $6 on A being the hitter, but also to pay $5 on B being the hitter (i.e.,
‘not A’). Regardless of who had actually done the hitting, one would then lose $1, having paid $11
on a combination of wagers guaranteed to pay exactly $10. By contrast, as long as degrees of belief
in a statement and its complement sum to one, bets on an event and its complement respecting
those degrees of belief will break even.
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Ulrike Hahn
School of Psychology
Cardiff University
Tower Building, Park Place
Cardiff CF10 3AT, U.K.
hahnu@cardiff.ac.uk
Mike Oaksford
School of Psychology
Birkbeck College London
Malet Street
London, WC1E 7HX
United Kingdom
m.oaksford@bbk.ac.uk
