Bayesian Informal Logic and Fallacy 

KEVINKORB Monash University 

Abstract: Bayesian reasoning has been 
applied formally to statistical inference, 
machine learning and analysing scientific 
method. Here I apply it informally to more 
common forms of inference, namely natural 
language arguments. I analyse a variety of 
traditional fallacies, deductive, inductive and 
causal, and find more merit in them than is 
generally acknowledged. Bayesian 
principles provide a framework for 
understanding ordinary arguments which is 
well worth developing. 

Resume: On a applique la theorie de 
probabilite de Bayes aux raisonnements 
statistiques, aux sciences informatiques, et it 
I'analyse de la methode scientifique. J'elargis 
son champ d' application it I 'analyse 
d'arguments courants qu'on trouve dans Ie 
langage ordinaire, et des sophismes 
traditionnels, deductifs et inductifs. J e trouve 
en certains de ces sophismes un bien-fonde 
qui n' est generalement pas reconnu. La theorie 
de Bayes nous donne une far;:on de 
com prendre des arguments ordinaires, et une 
telle application merite d'etre developpee 
davantage. 

Keywords: Bayesian reasoning, logical fallacies, statistical fallacies, causal fallacies, teaching 
probabilistic reasoning. 

1. Introduction 

Sophists, and the exercise of their profession-sophistry, have received very bad 
press, ever since Plato, who, together with his mentor Socrates, was an implacable 
foe of the sophists' ideas and methods. Indeed, the pair were so successful that 
labeling one a "sophist" has been a favored ad hominem attack on an antagonist 
ever since. l But the press, as often happens, have got hold of a half-truth and 
represented it as the whole. The sophists engaged in a kind of applied philosophy, 
for money-a practice which has only recently re-emerged in philosophical "therapy" 
and bioethical and environmental consulting. In particular, the sophists instructed 
people how to argue persuasively, a matter of considerable interest in the world's 
first democracies. Despite this focus on persuasion, the sophists were never so 
uninterested in the truth as Socrates and Plato would have us believe, nor were 
their sophistical arguments ever so unlikely to lead to the truth. 

My interest here, however, is not in rehabilitating the reputations of sophists 
and their sophistries per se. Rather, I shall attempt to rehabilitate the reputations of 
certain fallacies, modes of reasoning which have commonly been taken to lead 

©Informal Logic Vol. 23, No.2 (2003): pp.41-70 



42 Kevin Korb l 
from truth to falsity, or, at any 'rate, to provide no guidance towards the truth. 2 

Some of this work has been done previously, since the status of the various fallacies ! 
has always been a central and controversial issue for informal logic, the study of 
arguments in their natural setting (natural language) rather than in the formal setting 
ofmathematicallogic. J 

I shall treat three varieties of reasoning: deductive, inductive or statistical, and 
causal. All three are host to a large variety offallacies. And all three have given rise 
to substantial literatures within cognitive psychology, as psychologists have 
attempted to identify the fallacies to which people are prone and to understand the 
reasons why we are susceptible in those ways (e.g., for work on deduction, 
induction and causality see respectively Wason,1960; Kahneman, Siovic and 
Tversky, 1982; and Sperber, Premack and Premack, 1995). It is perhaps worth 
noting that my treatment of such a broad range of types of reasoning is intended to 
be indicative rather than comprehensive; the extension of Bayesian analysis beyond 
the select cases I examine should, however, frequently be straightforward. 

Some efforts to explain the results of the cognitive psychology of human error 
have aimed at defending ordinary modes of reasoning using an evolutionary line of 
thought. Essentially, the thought is that human modes of reasoning must be 
efficacious in arriving at the truth, since arrival at the truth is a precondition of 
survival and reproduction in many actual scenarios. Therefore, despite the claims 
of psychologists that many of these forms of reasoning fail to match our best 
normative standards, including Bayesian reasoning principles, and so are fallacious, 
they are to be endorsed as good, if imperfect, discoverers ofthe truth. Arguments 
along these lines are due to LJ. Cohen (1981) and Chase, Hertwig and Gigerenzer 
(1998). 

My approach is quite different. Rather than supposing that there is something 
wrong with the normative standards that the psychologists appeal to when describing 
some human forms of reasoning as illusory, I shall suppose that there is something 
right with those standards. In particular, I suppose that Bayesian principles are 
indeed normative, prescribing how we ought to reason. In the case of some 
"fallacies," however, what is illusory is the supposedly Bayesian standard: the 
Bayesian principles have either not been applied or they have been misapplied. 
When applied, Bayesian reasoning fully or partially endorses the "fallacy." This 
kind of point is perfectly compatible with the possibility that less than fully normative 
heuristics provide evolutionarily useful guidance in decision making. It is a continuing 
difficulty for Bayesian theory that its principles are very easy to misunderstand 
and misapply, as has been repeatedly demonstrated in the philosophical, statistical 
and psychological literatures. I hope that my treatment of the fallacies may help in 
this regard. 



Bayesian Informal Logic and Fallacy 43 

2. What is a Good Argument? 

Let us first consider what makes a good argument. Charles Hamblin, in his Fallacies 
(1970), launched an influential attack on the standard alethic conception of a good 
argument as a sound argument-one with only true premises and valid inferences. 
The first objection, and a telling one, is that such a criterion may be accidentally 
fulfilled, our premises may be true even though we have no good reason to believe 
that they are. In such a case, we wouldn't want to accept that the argument was 
good; the concept of an accidentally good argument appears to be empty. The 
goodness of our arguments is an achievement that we are responsible for, and so 
if our premises in fact need to be true, that truth must be something we can 
account for, rather than luck into. 

Such considerations lead quite naturally to an epistemic criterion for good 
argument: an argument is good just in case its premises are known to be true and 
its inferential force apparent. Against both this epistemic conception of argument 
and its alethic predecessor, Hamblin argues that they ignore the argumentative 
context and, in particular, the relation between the argument and its intended 
audience. Instead, Hamblin encourages us to adopt a dialectical view: a good 
argument is one that persuades us of the truth of its conclusion-and so has 
accepted premises and compelling inferences (1970, p. 242). "One of the purposes 
of argument, whether we like it or not, is to convince" (p. 241). The classic view 
of argument, whether alethic or epistemic, aims exclusively at identifying the 
normative character of good arguments. 

Ralph Johnson (1996, chapter 9) expresses a number of reservations about 
involving dialectical criteria. Knowledge of the intended audience is required in 
order to determine what premises might be acceptable as a background to 
argumentation. Johnson wonders whether we are in fact stymied without having 
an audience in mind; in practice, people seem to produce good arguments without 
anyone in particular in mind. But I think it should not be too hard to accept that in 
such cases there is an implicit audience in mind; in any case, such arguments are 
clearly produced within a cultural context that settles for practical purposes the 
unargued acceptability or not of a large range of propositions. Nevertheless, Johnson 
is surely right that dialectical criteria on their own will not suffice (p. 178): "Suppose 
that I discover that my audience accepts a proposition which I know-or strongly 
believe-to be false, but which would, if accepted, provide strong support for the 
conclusion. According to dialectical criteria, it seems that I not only may but 
should use that proposition." The relevance of dialectical standards for good 
arguments does not dispel the relevance also of normative standards. 

I suggest that the best view of good argument is a fusion of Hamblin's point 
that good arguments are persuasive and the classic view that they are normatively 
correct. An argument can fail to be a good one by employing identifiably false 
premises; it can fail to be good by employing weak inferences; and even when 



44 Kevin Korb 

employing exclusively true, known and understood steps and premises, it can fail 
to be good by failing to persuade, by being directed at the wrong audience. As 
Ralph Johnson remarks, "[the] more fundamental purpose of argument is rational 
persuasion" (1996, p. 173). And indeed that purpose can only be served by 
arguments that are jointly rational (normatively grounded) and persuasive. 4 

3. Bayesian Reasoning 

My focus here will be on the normative. More specifically it will be on understanding 
arguments from a Bayesian perspective, rather than from the more common point 
of view, that the logic of arguments can best be understood as an application of the 
logic of mathematics, or formal logic. Although this is the less common perspective, 

it is not exactly new-dating from before the 18th century and Bishop Butler's 
pronouncement that "probability is the very guide to life" (Butler, 1736). Regardless 
of its heritage, and despite considerable activity in developing a Bayesian account 
of scientific method, the Bayesian perspective on good argument has yet to be 
articulated. 5 In order to develop a Bayesian perspective I first need to describe the 
normative standard that I intend to apply; so, I briefly describe Bayesian reasoning 
and its conditions for application. 6 

Bayes' Theorem (Bayes, 1763) reports the relation between the probability of 
a hypothesis (conclusion) given some evidence-P(hle)-and its probability prior 
to any evidence P(h) and its likelihood P(elh), the probability that the hypothesis 
implies for the given evidence. In particular, 

P(hle) = (P(elh) X P(h)) 
(P(e)) 

(1) 

The theorem is just that and is not controversial. As we will see below, Bayesian 
evaluation theory is controversial: it asserts that the proper way to evaluate 
hypotheses just is to adopt the probabilities conditional upon the available evidence-
as supplied by Bayes' theorem-as our new posterior probabilities; that is, we can 
abandon our prior distribution P(.) in favor of our posterior distribution 
P'(.)=P(.le).7 This move to a posterior distribution is called Bayesian 
conditionalization. 

Given this view, the simplest way of understanding the concept of the 
confirmation or support offered by some evidence is as the difference between the 
prior and posterior probabilities of a hypothesis; that is e supports h just in case 
S(hle)=d!(hle)-P(h»O (cf. Howson and Urbach, 1993, p. 117). A second measure 
of support, the ratio oflikelihoods e given hover e given not- h is equally defensible: 8 

A (elh) = (P(elh)) 
df (P(el-,h)) 



Bayesian Informal Logic and Fallacy 45 

It is a simple theorem that the likelihood ratio is greater than one if and only if 
S(hle) is greater than zero. A (elh), (or, simply, A) can be understood as a degree 
of support most directly by observing its role in the odds-likelihood version of 
Bayes'theorem: 

O(hle) = A O(h) 
This simply asserts that the conditional odds on h given e should equal the prior 
odds adjusted by the likelihood ratio. Since odds and probabilities are interconvertible 
(O(h)=d/(h)IP(-.h)), support defined in terms of changes in normative odds 
measures changes in normative probabilities quite as well as S(hle). However, A is 
simpler to calculate. Indeed, since a likelihood is just the probability of evidence 
given a hypothesis, and since hypotheses often describe how a causal system 
functions given some initial condition, finding the probability of the evidence 
assuming h is often a straightforward computation. What a likelihood ratio reports 
is the normative impact of the evidence on the posterior probability, rather than the 
posterior probability itself(i.e., the other necessary ingredient for finding the posterior 
is the prior probability of h). However, confirmation theory is primarily concerned 
with accounting just for rational changes of belief, and so A turns out to be a useful 
tool for understanding confirmation. 

There are two other "tools" that a Bayesian analysis of argumentation delivers 
(or requires): 

Priors. Some critics of Bayesianism (e.g., Gigerenzer and Murray, 1987) take the 
manifest relevance in many cases of considerations about prior probabilities, 
together with the silence of pure Bayesian principles on the subject of how to 
choose priors, to imply the inadequacy of Bayesian theory. But this has matters 
exactly backwards. Unadulterated (unextended) Bayesian theory indeed does 
not specify how to find priors. But it is precisely Bayes' Theorem which makes 
plain the relevance of the prior probability of a hypothesis for posterior belief, 
and so the inadequacy of methods ignoring priors. Considerations about rational 
priors do come into Bayesian analysis, as will be seen below, by way of possible 
extensions to unadulterated Bayesian theory. 

Probabilistic Relevance (Conditional Dependence). One proposition is 
probabilistically relevant to another just in case conditioning on one changes the 

probability of the other; i.e., a and bare probabilistically relevant if and only if 
P(alb):;t:P(a) (or, equivalently, P(bla):;t:P(b)).9 Probabilistic relevance may vanish 
(or appear) between two propositions when the truth value of some further 
proposition(s) becomes known. For example, if John and Jane both ate the 
same soup at a restaurant, then one coming down with food poisoning will 
change the probability that the other will; however, if the common causal factor-
that bacterial state of the soup-becomes known independently, then the health 
of John and Jane are rendered independent of each other (see Figure 1). That 
is, their states of health are conditionally independent given the state of the 
soup. In the philosophical literature, this is known as "screening off," following 



46 Kevin Korb 

Hans Reichenbach (1956). Conversely, two independent propositions may 
become dependent given the value of a third proposition. For example, in 
judging whether to respond to a burglary alarm, a security company may discount 
the probability of a burglary upon learning that a small earthquake has struck, 
even though burglaries and earthquakes, without an alarm, are (approximately) 
independent. The importance of probabilistic relevance to understanding some 
fallacies will also be seen below. 

/ 
G 

Bacteria in soup 

Figure 1: Screening off 

4. The Justification and Limits of Bayesian Reasoning 

4.1 Probabilistic Coherence 

The key justification for Bayesian principles has been, and remains, the "Dutch 
book" arguments due to Frank Ramsey (1931). These arguments demonstrate an 
inability by a bettor violating the probability axioms to avoid a guaranteed loss 
from a finite sequence of fair bets, regardless of the outcomes of the individual 
bets. The assumptions required to obtain the Dutch book result are minimal: the 
bettor's subjective degrees of belief must violate one of the probability axioms; the 
bets offered are individually fair (the odds are got by the ordinary ratio of probabilities 
given by the bettor); the bettor must be willing to take either side of a fair bet; the 

. bettor must be willing to take any number of such fair bets. 

Some have found the arguments insufficient reason to take the probability 
axioms as a normative guide for SUbjective degrees of belief. For example, various 
commentators have suggested that a willingness to take any finite number of bets, 
even when they are fair, is itself irrational, and have naively claimed that therefore 
the Dutch book is defective (e.g., Bacchus et aI., 1990; Chihara and Kennedy, 
1979). However, the Dutch book does not presuppose that making such a series 
of bets is always rational. The point is rather that the subsequent guarantee of 
losses appears to be attributable only to the initial incoherence and not the willingness 
to take fair bets. It matters not that incoherent bettors can manage to avoid 
guaranteed losses by refraining from betting; the point is that, should they bet, 
their subjective beliefs are advising them wrongly. 



Bayesian Informal Logic and Fallacy 47 

4.2 Conditionalization 

Matters are less clear-cut for the justification of conditionalization. David Lewis 
(reported in Teller, 1973) produced a diachronic Dutch book, intending to show 
that violating conditionalization will lead to guaranteed losses on combinations of 
fair conditional bets. The argument turns out to fail in a variety of contexts, in 
particular when the new evidence reveals more about the weakness of one's 
conditional probability structure than about the empirical world. 

As Colin Howson (2000) has pointed out, conditionalization is the invalid 
inference rule: 

P(BIA)=r, P'(A)=1 

thus, P' (B)=r (2) 

This inference will be valid when and only when a third premise is adopted: 

P'(BIA)=P(BIA) (3) 

Some might take the view that (3) is wrong and (2) is therefore a fallacy, and so 
suppose themselves justified in ignoring Bayesian conditionalization universally. 
An extremist Bayesian, perhaps in reference to the diachronic Dutch book, might 
take (2) to be universally valid and, hence, (3) to be universally true. But it is clear, 
as Ramsey (1931) himself pointed out, that there are many circumstances where 
the acquisition of new evidence will rationally upset our conditional probability 
structure. The simplest examples are suggested by Howson (2000), where 
adherence to (3) implies a logical inconsistency. But even before Howson produced 
such examples, it should have been obvious that the extremist Bayesian stance is 
hugely implausible, for it implies that a rational agent must in its very beginning 
adopt a complete conditional probability structure which can in the future never be 
rationally revised and only slavishly followed. 

The normative standing of conditionalization arises from the fact that the retention 
of conditional probabilities, although not justifiable universally, is clearly defensible 
across a wide range of cases. This is most prominent in understanding scientific 
method: it would be an unusual case for the measurement e of an experimental 
outcome to lead us to revise its probability of discovery conditional upon any 
hypothesis h , that is, the likelihood P(elh). This is in part because likelihoods are 
just the kinds of things that our hypotheses generally supply to us. For example, if 
we assume a coin-tossing exercise to be a binomial process h , then the probability 
of any reported binary sequence under that process is readily calculable, and is 
P( elh). There is ordinarily no room, simply upon discovering the particular outcome 
e, to argue for a revision of P(elh).10 In many non-scientific cases, although the 
probabilities involved may be more obscure or qualitatively understood, the same 
point applies: ifthe conditional probability structure is well thought-out in advance-
possibly being given by a clear causal theory-then observing an experimental 
outcome will not disturb it, and (3) applies, with conditionalization being justified. 

It is plain that such reasoning is not itself Bayesian. Bayesian principles obligate 
conditionalization given (3), but are silent on the whys and wherefores of adopting 



48 Kevin Korb 

a prior probability distribution, and therefore of adopting a conditional probability 
structure (which is a function of the joint prior). The normative principles of 
Bayesian theory, in order to gain any useful leverage, must be supplemented by a 
theory of rational change of conditional belief structure-when and why and how 
(3) is violated. 

I shall not here attempt any such supplementary theory. By developing and 
applying Bayesian theory to a range of cases in informal logic, however, it will 
become clear that such a theory must allow for the correctness of (3) across 
many scientific and non-scientific domains of understanding, providing a large, if 
circumscribed, range for Bayesian reasoning. 

4.3 Priors 

The single most influential criticism of Bayesianism has been and remains: 
where do prior probabilities come from? There is a broad range of answers supplied 
by Bayesians; indeed, the different flavors of Bayesianism are largely determined 
by how one answers this question. Extreme objectivists (such as Carnap, 1950, 
and maximum entropists like Jaynes, 1989) find prior probabilities in the structure 
of language; extreme subjectivists (such as de Finetti, 1937) find priors in the 
wind. Moderates, such as Lewis (1980) and Good (1983), find them where they 
can get them. 

In the face of such variety, and controversy, over legitimate sources of prior 
probabilities, many prefer to opt out of the game entirely, and seek some method 
which doesn't require a subjective choice of one probability over another, nor 
obscure reasons for one variety of prior over another. Orthodox statistical inferences, 
such as significance tests and confidence interval estimations, do not require prior 
probability distributions. A confusing factor is that any specific such inference 
can be recast as a Bayesian inference, and, working Bayes' theorem in reverse, the 
corresponding prior probability can be inferred. I I So, a sequence of such non-
Bayesian inferences will determine a prior probability distribution-or else be 
demonstrably incoherent. Although many inferential methods do not require an 
explicit prior commitment to probabilities, they nevertheless, in application, require 
an implicit commitment to either probabilities or incoherence. This makes for an 
amusing game for Bayesian critics of those alternatives: finding cases where the 
prior probabilities implied by an inferential performance are manifestly absurd. 
The other side of the game is to define cases where the prior probabilities are 
irresistible. The operation of simple gambling apparatus provides indefinitely many 
examples. And in such cases the Bayesian analysis is generally accepted. 



Bayesian Informal Logic and Fallacy 49 

4.4 Formalism 

It seems plausible that one ofthe reasons that Bayesian analysis outside the simple 
gambling cases has been widely resisted is that Bayesian analysis is founded on the 
application of a formal calculus: the simple cases are clearly and non-controversially 
formalizable, whereas the difficult cases all have non-formal aspects. 

Consider the analogy of formal deductive logic (FDL). FDL, also, for a long 
time was considered a major candidate for formalizing human inferences, including 
even scientific inferences in the hypothetico-deductive methods of Nagel (1961) 
and Hempel (1965) and in the falsificationist method of Popper (1934). It has 
been typical of logic textbooks to claim, at least implicitly in the examples if not 
also explicitly, that FDL provides the normative standard for analysing natural 
language arguments. That is, an argument is to be considered good (sound) if and 
only if both its premises are true and its translation into FDL produces a formally 
valid inference (e.g., the popular texts of Mates, 1972, and Copi and Cohen, 1990). 
However, this model for understanding natural language arguments has largely 
come unglued since 1970. Instruction in logic was widely assumed to provide 
benefits to students' ability to reason and argue. However, the empirical evidence 
suggests otherwise (van Gelder, 2000), and since around 1970 the related movements 
ofinformallogic and critical thinking have made substantial imoads in that supposed 
use offormallogic (e.g., such texts as Kahane, 1971, and Scriven, 1976). 

One substantial difficulty for FDL is that there is no univocal translation of 
typical natural language arguments into the language offormallogic: dealing with 
the ambiguities, complexities and hidden assumptions of ordinary arguments turns 
out to be more complex than anything to be found in deductive logic per se. But 
a more telling difficulty is that natural language arguments clearly and non-
controversially come in degrees of strength, whereas the tools of FDL yield in all 
cases a simple two-valued verdict: valid or invalid. Precisely that same difficulty 
arises in all deductivist attempts to understand scientific inferences: in the end, for 
example, all Hempel's theory could say of any theory was that it was confirmed or 
notY 

So, if the formal methods of deductive logic are relatively useless for 
understanding natural language arguments or scientific reasoning, why should we 
believe that the formal methods of the probability calculus can do any better? 
Well, of course, regarding the last difficulty, the probability calculus was specifically 
designed to cope with degrees of strength for propositions, \3 so at least Bayesian 
analysis is not so obviously the wrong tool. Nevertheless, it is certainly correct 
that Bayesian analysis, at least in the first instance, is formal and human reasoning 
is not, so it may seem that the tool and the problem remain quite unmatched. What 
I aim to show is that the formal, quantitative tools of Bayesian analysis support 
counterparts that are qualitative and semi-formal; these can shed light on what is 
normative and productive in analysing arguments, without providing a complete 



50 Kevin Korb 

solution to the problems raised by human argumentation. Bayesian philosophers 
have already done much to make such a case in the study of scientific method 
specifically. 

4.5 Successes in the Bayesian Analysis of Scientific Method 

The concept ofthe likelihood ratio provides a simple but effective tool for analysing 
the impact of evidence on conclusions (hypotheses). For example, it makes clear 
why Karl Popper's (1959) insistence that scientific hypotheses be subjected to 
severe tests makes sense. Intuitively, a severe test is one in which the hypothesis, 
iffalse, is unlikely to survive; that is, whereas the hypothesis predicts some outcome 
e, its competitors do not. Since the hypothesis predicts e, P(elh) must be high; 
since its competitors do not, P(el-.h) must be low. These jointly imply that the 
likelihood ratio is very high. Therefore, a severe test will be highly confirmatory if 
passed and highly disconfirmatory if failed-and so provides the most efficient 
approach to testing a hypothesis. 

A second example is the preference which experimental scientists exhibit ceteris 
paribus, when confronted by two possible tests of a theory, for that test which is 
most different from one that the theory has previously passed. 14 For example, 
Eddington was faced with two approaches to testing Einstein's general theory of 
relativity (GTR) in 1919, either repeating the analysis of Einstein himself of the 
precession of Mercury's perihelion or checking the predictions which GTR made 
of a "bending" of starlight by the mass of the sun, observable during a total eclipse. 
Despite the fact that astronomical observations of the motion of Mercury are 
cheaper and simpler, Eddington famously chose to observe the starlight during the 
eclipse over the Atlantic. Why? 

Intuitively, we can say that this is because testing a new experimental 
consequence, as opposed to a repeated experiment, offers a more severe test of 
the theory: the predicted new outcome is less likely to come about than the repeated 
outcome, and so when it does come about it offers greater confirmation of the 
theory under evaluation. A more formal Bayesian analysis directly supports this 
intuition. Where d is the prior experimental result, e its repetition and f some new 
variety of test result, the ceteris paribus clause above implies P(dI,h)=P(elh)= 
P(flh) . . Since e is closer to d than is j, P( eld) > P(~d). It follows directly from the 
Bayesian understanding of confirmation that/then has greater confirmatory power 
than does e , after conditioning on d.' IS 

Since 

P jhlf)= (P j/lh)P jh)) > 
(P j/)) 

(P jelh)P jh)) 
(P j/)) = P jhle) 

P j/) < Pje) 
by assumption above and 



Bayesian Informal Logic and Fallacy 51 

because (what is typical, at any rate) we can take the probability of the evidence/, 
given that we know whether or not is true, to be independent of the probability of 
the evidence e (h screens offffrom e). More applications of such Bayesian analysis 
to scientific method can be found in Howson and Urbach (1993) and Korb (1992). 

5. Logical Fallacies 

So, Bayesian analysis has a clear and unique justification in dealing with situations 
of uncertainty, in Ramsey's Dutch book treatment. Furthermore, it has had notable 
success in aiding our understanding of scientific inference. Clearly, Bayesian 
reasoning is at least a plausible candidate for assisting with our understanding of 
human inferences as manifested in ordinary argument. 

Many of the forms of reasoning classically identified as fallacious have long 
been held not to be fallacious under various circumstances by various commentators. 
What has been missing, however, is a unifying theory supporting them which 
applies to more than a handful of cases. I believe Bayesian analysis, and, in particular, 
the examination of plausible prior probabilities and the relevant likelihood ratios, 
offers the skeleton of such a unifying theory. I will present Bayesian analyses of a 
few commonly discussed fallacies, illustrating in turn the application of likelihood 
ratios, probabilistic relevance and considerations about prior probabilities. The 
more general applicability of these types of analysis elsewhere should be clear. 

5.1 Affirming the Consequent: Reasoning with the Likelihood Ratio 

Affirming the consequent is perhaps the most blatant of fallacies. It takes an 
assumed conditional and, instead of applying to it the asserted truth of its antecedent 
as in modus ponens, it attempts to reverse the process, as in: 

All humans are mortal. 

Socrates is dead. 

Therefore, Socrates is human. 

Despite being the most blatant of fallacies, and disparaged in popular texts on 
logic, this form of argument is pervasive in the sciences: 

If evolutionary theory is true, we would expect to find antibiotic-resistant 
strains of bacterial diseases appearing. 

We do find that! 

So, indeed, evolutionary theory is true. 

The only reason we do not see versions of this particular argument with any 
frequency is that evolutionary theory by now has been so well-confirmed that 
such argument is utterly unnecessary.16 

This kind of reasoning is so wide-spread that, in addition to being a named 
fallacy, it has a name as an accepted form of reasoning, namely, "inference to the 
best explanation" (IBE). Probabilistically, what is going on is an inference of the 



52 Kevin Korb 

following form: 

P( elh) is high 
e is observed 

Therefore, P'(h) is high 

Comparing this with the Bayesian requirement for confirmation, that the likelihood 
ratio be high, we see that there is a suppressed premise, that the alternative 
hypothesis should fail to support the evidence, i.e., that P(el-,h) is 10wY In the 
case of Socrates, there are a great many explanations of Socrates' death alternative 
to Socrates being a human, as many as there are species, and so IBE fails. IS In the 
case of evolution, there is no serious alternative explanation to the development of 
antibiotic resistance-the alternatives to some form of evolutionary theory, plausibly 
construed, make such a development a miracle. Hence, the Bayesian criterion for 
high confirmation of evolution theory is satisfied. 

This analysis not only endorses the value of "affirming the consequent" in 
some circumstances, as others have done before, but also clarifies the difference 
between such circumstances and those where affirming the consequent really is 
dubious. Affirming the consequent is emphatically not a fallacy. It is a central 
feature of scientific inference, in fact, quite properly deserving its special title as 
"inference to the best explanation." But neither is it an unexceptionable form of 
inference. It is only in Bayesian considerations oflikelihood and prior that a principled 
distinction between good and bad applications of this form of reasoning has ever 
been made. 

5.2 The Appeal to Popularity: Probabilistic Relevance 

In some circles (or circumstances) it is popularly believed that there are witches; 
in others, it is believed that hairless aliens walk the planet. If a bald appeal to the 
popularity of a belief were enough to establish its acceptability, then reasonable 
beliefs and the arguments for them would ebb and flow with the tides of fashion. 

Nevertheless, there seems to be some merit to the appeal to popular belief. 
Johnson (1996) points out a direct relevance between popular belief and 
(propositions concerning) the outcome of democratic elections! But even when 
there is no direct (or indirect) causal chain leading from popular belief to the truth 
of a proposition, there may well be a common cause that relates the two, making 
the popularity of a belief a (possibly minor) reason to shift one's belief in a 
conclusion. Presumably, a world in which no one believes in witches would support 
a moderately smaller rational degree of belief in them than one in which many do, 
at least prior to the development of science. In general, open mindedness is at least 
partially characterized by a willingness to be persuaded that one's beliefs may be 
wrong, and perhaps in the first instance simply by the sheer number of people of 
different mindedness. The Bayesian point of view accommodates these observations, 
and again supports the demarcation of telling and pointless appeals to popularity 
via the Bayesian standard of probabilistic relevance. 



Bayesian Informal Logic and Fallacy 53 

As A (and the related measures of Bayesian confirmation) comes in degrees, 
another immediate consequence of the Bayesian analysis is that relevance also 
comes in degrees. In our world the popularity of belief in witches, for example, 
can be understood to be relevant to there being witches, but only in a minor way. 
In particular, the support it gives to belief in witches can be swamped by (or 
screened off by) other, more strongly relevant, information. That is, our situation 
may well be describable (crudely) via Figure 2. Popularity of a belief may well in 
general be associated with the truth of what is believed; so, lacking any clear 
scientific judgment (say, during the Dark Ages), common belief in the efficacy of 
witchcraft may well rationally lift our own belief, if only slightly. Nevertheless, 
given an improved understanding of natural phenomena and the fallibility of human 
beliefformation (perhaps some time in the future!), the popular belief is no longer 
relevant for deciding whether witches exist or not: science accounts for both the 
belief in witches and their unreality. 

Science of belief &I: 
nature 

/ 
e 

Figure 2: Belief in witches screened off 

5.3 Ad Hominem: Probabilistic Relevance and Priors 

Argument ad hominem is directing criticism at the presenter of some original 
argument rather than at the original argument itself. Its characterization as a fallacy 
implies that there is an attempt to deflect attention from the original argument-
that is, that the ad hominem is a form of red herring, irrelevant to the original 
question. Walton (1989, p. 151) gives a nice example: 

[An] instance of[an] ad hominem imputation of bias occurred 
during a debate on abortion in the Canadian House of Commons: "It is 
really impossible for the man, for whom it is impossible to be in this 

situation, to really see it from the woman's point of view." 

As Walton notes, it is of course correct that a man cannot be in the woman's 
situation, but the suggestion that this makes it impossible for the man to "see" the 
situation from the woman's point of view fails to address anything substantive the 
man may have said. 



54 Kevin Korb 

Despite its being a "classic" fallacy, many have expressed strong doubts that 
such argument should generally be considered fallacious. Joseph (1906) pointed 
out that it is standard practice in court to consider the reliability of witnesses, 
which would not be defensible were argument ad hominem fallacious without 
reservation. Or again, if someone can be shown to have a strong motive to dissemble, 
then it would be foolish to simply ignore that. The real question is whether or not, 
in the particular case at hand, the question raised about the human arguer is relevant 
to the original issue. If a is the ad hominem attack and h the original hypothesis 
under contention, what we would like to know is whether or not the truth of a 
would rationally influence our belief in h; i.e., whether P(hla)IP(h). 

If a liar is exposed in court, then surely that is relevant. If an anti-abortionist is 
exposed as male, the relevance is, at best, obscure. In these cases, the relevance 
(or irrelevance) is based upon the origin of the testimony. Although reference to 
the origin of an argument has been denounced as the "genetic fallacy," so long as 
the plausibility or acceptability of any of the premises in an argument relies to any 
degree upon the believability of the one putting them forward, argument ad hominem 
and the "genetic fallacy" will be pointed and relevant. The impulse to discount 
these forms of argument as fallacy appears to stem from an over-idealized view of 
argumentation: one where the merits of every argument can be assessed in 
abstraction from all context. In the real world, considerations of time and 
circumstance are inescapable, and ignoring the reliability of witnesses is simply 
irresponsible. 

5.4 Epistemic Direct Inference: Probabilistic Relevance and Priors 

This leads us straight to the issue of direct inference (or, the "statistical syllogism") 
since it is a primary way by which we assess probabilities relative to sources, and 
so a plausible source of some of the prior probabilities needed to operate Bayes' 
theorem. Formally stated, direct inference is the inference rule: 

Direct Inference (DI): If the frequency of objects of type A within the 
reference class of objects of type B is r, that is, F(AIB)=r, then if object 
x is selected from B with a physically uniform randomizing device (and 
the agent knows that), then P(xEA)=r. 

This is what endorses, for example, the assertion that the probability of drawing a 
red ball from an urn containing equal number of red and blue balls (and the result 
being from a known physically uniform selection device) is 112. This is not a 
controversial rule. However, since it requires a physically random selection device, 
its preconditions are rarely satisfied. 

A more widely applicable rule generalizes direct inference to cases where the 
properties of the selection device are unknown: 

Epistemic Direct Inference (EDI):Suppose F(AIB)=r and the agent has 
no reason to believe that the selection process for XEB is biased with 



Bayesian Informal Logic and Fallacy 55 

respect to membership in A (this is a kind of total evidence condition). 
In particular, the agent does not know of any proper subset CcB, with 
a known frequency F(AIc), both that F(AIC)tF(AIB) and that XE C. 
Then it can defeasibly be inferred that P(xEA)=r. 

ED! is of course a fallible, inductive rule. As such, it has been subjected to 
attack, for example by the notable Bayesian Isaac Levi (1980). It is easy to develop 
cases where it will lead astray. And it is quite common for the preconditions of 
ED! to be satisfied by multiple, competing reference classes. Thus, for example, 
we may know that 10% of the English are Catholics and that 5% of academics are 
the same, without knowing anything useful about the intersection set, English 
academics. This is the "problem of the reference class," which has yet to find a 
generally compelling solution. Regardless, ED! is a rule which humans and other 
animals use widely and successfully, at any rate to all appearances. And compelling 
or not, there are proposals which are probably already a satisfactory first step 
towards modestly successful autonomous agents, such as that of Reichenbach 
(1949), to employ the smallest reference class for which we have reliable statistics. 
It is in fact hard to imagine how a species in a complex environmental niche could 
survive for very long if it did not use ED! or some heuristic rule that mimicked it. 
The burden is surely on the critics to supply an alternative rule that is as useful as 
ED!. 

Given EDI, we can make immediate sense of arguments ad hominem, as either 
irrelevant or relevant. A successful ad hominem shows that the presenter of the 
argument comes from a reference class other than the one presumed, and indeed 
a reference class that biases matters, whether favorably or unfavorably. Thus, an 
appeal to expert authority identifies a favorable bias for a prior probability to be 
attached to an assertion attributed to the authority. The appeal is reasonable and 
relevant if the authority's expertise is pertinent to the assertion at issue. Similarly, 
an unfavorable biasing factor is introduced when a witness is discovered to be a 
liar. Even then, the argument ad hominem may fail to be relevant if the original 
argument in fact does not actually rely upon the credibility of its presenter-if, for 
example, all of its premises are common knowledge or previously accepted. In 
such a case, an ad hominem argument cannot but be irrelevant. 

5.5 The Value of Saying 

An interesting related phenomenon is the "value of saying." When a proposition is 
stated, as opposed to being left implicit in anargumentative exchange, there are at 
least two distinct effects on the hearer: 

• An attentional effect-the hearer's cognitive apparatus will be differently deployed, 
in order to interpret the statement. Semantic and probabilistic connections with 
the statement will be followed, or followed further, resulting in some shift in the 
degrees of belief in associated propositions. 



56 Kevin Korb 

• A probability shift-the hearer's degree of belief in what is said may shift. In 
particular, the fact that the interlocutor has made the utterance will tend to shift 
the hearer's belief in the statement up or down, depending upon the credibility 
of the speaker, in an application of ED I. That is, if the speaker has a credibility 
lower than the hearer's initial belief in the statement, the belief in the statement 
may diminish, and vice versa. The degree of shift will depend both upon the 
discrepancy between the speaker's credibility and that initial belief and upon the 
conviction with which the hearer holds that initial belief, as reflected in the 
hearer's conditional probabilities. 

5.6 Logical Fallacies in Sum 

It should be fairly clear already that the Bayesian tools for argument analysis-
prior probabilities, likelihood ratios and relevance--cannot reasonably be expected 
to resolve all issues arising in informal logic. For example, any semantic analysis 
of arguments is simply outside the scope of this kind of Bayesian analysis. So, the 
classic fallacy "A gray elephant is a gray animal, so a small elephant is a small 
animal" depends upon an equivocation in the predicate "is small." As such, no 
Bayesian analysis per se will spot the problem. 

The prerequisites for Bayesian conditionalization to support a hypothesis-
stable conditional probabilities, some significant prior probability, and a likelihood 
ratio greater than one-provide tools for analysing the merits of various kinds of 
argument, as I have shown. The analysis can clearly be extended to additional 
varieties of argument, providing a common and principled theme to this work of 
informal logic. Semantics, the origin and revision of conditional probability 
structures, and the origin of prior probabilities are matters that lie beyond any 
ordinary Bayesian analysis. So, too, does the story of how normative arguments 
mayor may not persuade the target audience, to which rhetoric and psychology 
contribute. 

Demanding, or expecting, Bayesian principles either to deliver the complete 
story of argumentation or else to stand exposed as inadequate sets an impossible 
standard for any formal system. Bayesian analysis clearly can deliver much of 
what is important. But the point of this Bayesian analysis completed will be simply 
to provide a useful framework for identifying and dealing with inferential relations 
between statements, one which supplements and must be supplemented by 
considerable additional analytic resources. 

6. Statistical Fallacies 

The psychological literature on statistical fallacies has been developed largely by 
Bayesian psychologists, that is, by psychologists who accept Bayesian principles 
as their normative standard. Hence, we should expect to find that the fallacies 
which they identify are indeed fallacies from the Bayesian perspective. However, it 
ain't necessarily so. Consider Kahneman and Tversky's best known heuristic: the 



Bayesian Informal Logic and Fallacy 57 

Representativeness Heuristic. This was introduced in order to explain why many 
people reason in a way obviously violating probabilistic rules. According to Tversky 
and Kahneman (1973), with this heuristic "an event is judged probable to the 
extent that it represents the essential features of its parent population or generating 
process." Thus, in one notorious experiment, a hypothetical Linda, described as 
an active leftist when a student, is judged more likely to have subsequently become 
a feminist bank tel1er than ... a bank tel1er!-a conclusion violating the rules of 
probability. Tversky and Kahneman's extremely plausible suggestion is that people 
find the concept of Linda more stereotypical of the subset offeminist bank tel1ers 
than of bank tellers generally, and then they substitute this statement of 
stereotypicality for the requested response of which vocational outcome is more 
likely. 

The moral that Tversky and Kahneman would like their readers to draw, of 
course, is that this latter substitution of stereotypicality for probability is a mistake, 
that we humans have got it fundamental1y wrong. For example, in summarizing 
their work on representativeness, they write "In his evaluation of evidence, man is 
apparently not a conservative Bayesian: he is not Bayesian at al1" (Kahneman and 
Tversky, 1972). I would like to suggest that this is quite a wrong interpretation 
(and not just because of the sexist implication that women are superior to men). 

It has long been known that strict Bayesian inference is computational1y 
complex; that is the main reason why computerized expert systems, until quite 
recently, never used proper probability computations (cf. Neapolitan, 1990, Chap. 
4). Indeed, Bayesian inference has been proven to be NP-hard (Cooper, 1990), 
which means in practice that complete algorithms to perform such inference require 
exponential increases in time as the problem to be dealt with increases (e.g., as the 
number of variables increases). In short, it is too much to expect computers to 
perform strict Bayesian inference; it would then be absurd to expect humans to do 
so, when relative to computers we are so much worse at arithmetical computations. 
If computers require heuristic short-cuts to approximate ful1 Bayesian inference, 
then so much more do humans require heuristic short cuts. 

So what might such heuristic short cuts look like-rules that approximately 
fol1ow the normative standard, at least in common cases, even if they break down 
under less usual circumstances? Given the performance pressures on humans and 
their ancestors in evolution, in addition to yielding true or "good enough" answers 
in run-of-the-mi11 circumstances, it would undoubtedly also be advantageous if 
such heuristics were quick and easy to compute. We can cal1 such rules/as! and 
frugal heuristics. So, what are they? Well, that is a moot point, but one wel1 worth 
investigating in the cognitive psychology of human inference. A very promising 
candidate for one though is the representativeness heuristic. I suggest that it has al1 
of these characteristics: (1) It is relatively fast. For obvious reasons humans have 
evolved substantial capacities to recognize and categorize objects and events. The 
use of stereotypes is clearly a part of this set of abilities. And there is compelling 
evidence, for example, that humans are better and faster at categorizing stereotypical 



58 Kevin Korb 

members of classes than non-stereotypical members (Rosch, et al., 1976; Rips, 
1975). (2) Given all of the mental capacity already dedicated to recognition and 
categorization, and its relation to stereotypicality, frugality follows. (3) It is also 
clear that in a very broad range of cases, the stereotypical answer is also the right 
one. This is clear on evolutionary considerations, since we (and other animals) 
demonstrably employ such reasoning tendencies and so were they commonly 
seriously suboptimal we (and such other animals) would no longer exist. But also 
on statistical grounds, use of the stereotype (mode) of a class to infer properties of 
an unknown member of the class must in a very large range of cases yield a 
practically useful answer (see, for example, Holte, 1993, and Korb and Thompson, 
1994, for statistical studies showing the effectiveness. of classifiers drawing upon 
such extremely simple classification models). You can and should judge a book by 
its cover, if you haven't the time to examine its contents. 19 

Many other heuristics investigated by Bayesian psychologists, such as availability 
and anchoring (see Kahneman, Slovic and Tversky, 1982), have the same properties. 
Indeed, the Bayesian analysis of the logical fallacies can be viewed in the same 
way: many of the fallacies are fast and frugal heuristic techniques for inference, 
which nevertheless can lead us astray. Considering how we might improve our 
reasoning abilities, a question at least underlying much of the work of cognitive 
psychology, it would be more useful to examine the circumstances in which our 
heuristics fail us, and what might flag such circumstances, rather than simply 
decry our inability to reasoning cogently. Labelling these heuristics as fallacies 
tends to act as a substitute for critical reasoning about them, rather than an 
encouragement to reason critically. 

7. Causal Fallacies 

7.1 Post Hoc 

The classic "causal fallacy", dreaded by many generations of logicians, starting 
with the first one, Aristotle, is post hoc ergo propter hoc-that is, the argument 
that "because B happens after A, it happens because of A" (Aristotle, RhetoriC). 
As Pinto (1995) points out, this characterization is ambiguous. Many recent writers 
have interpreted it as an inference concerning particular token events: because A in 
fact occurred before B, it caused B (e.g., Rottenberg, 1991; Copi and Cohen, 
1990). No doubt a more useful interpretation is available in terms of event types: 
that because events of type A are followed by events of type B, As cause Bs. This 
is a more useful interpretation because in token form the fallacy is strictly 
counterfactual-it is one which people do not in fact commit; from the mere fact 
that a particular event preceded another, practically no one concludes a causal 
relationship. How often is it said that John Wilkes Booth caused the death of 
Kennedy, or the invention of the piano initiated World War II? The denunciation of 
post hoc in these terms is simply fatuous. 2o What might ground a singular causal 



Bayesian Informal Logic and Fallacy 59 

claim is the acceptance of the corresponding claim relating event types. Ifwe may 
be warranted in claiming that consuming anti-oxidents prevents some cancer in 
general, then we may be warranted in asserting the same in particular cases as 
well. 

7.2 Correlation and Causation 

The denunciation of post hoc, in type language, is closely allied to the dictum of 
many statisticians: "correlation does not imply causation." Indeed, Walton (1989) 
simply identifies the two: post hoc occurs "when it is concluded that A causes B 
simply because one or more occurrences of A are correlated with one or more 
occurrences of B." This identification of Walton's is mistaken because a defining 
characteristic of post hoc is the known temporal precedence of cause to effect, 
whereas correlation is a symmetrical relation. Of course, those who oppose the 
statisticians' dictum, including me, do not propose that we infer causal relations 
which do not respect temporal ordering; rather, we propose that causal relations 
may be inferred even lacking explicit temporal knowledge. In artificial intelligence 
methods to do this are known as causal discovery algorithms. If these methods 
can be demonstrated to be successful, then the statisticians' principle will have 
been demonstrated to be false or misleading. 

The denial of the basis for causal discovery-learning causal structure from 
correlational information--<:ontinues in the statistical literature (see the recent debate 
in McKim and Turner, 1997);21 however, since the mid-1980s, the case against it 
has thinned considerably. A clear and compelling philosophical argument for causal 
discovery was already put by Glymour et al. (1987, part I). But more compelling 
for some will be the many and varied successes of causal discovery algorithms in 
practice, which have been reported over the last decade in the artificial intelligence 
literature. Every volume of the annual conference Uncertainty in AI since 1990 
contains multiple reports of new and successful applications of causal discovery 
algorithms. The technique has been remarkably successful for something founded 
on a mere fallacy! Despite this, the nay-sayers continue to be unimpressed; for 
example, Humphreys and Freeman (1996) suggest that it is all a Baconian 
(Cartesian?) dream of mechanizing thought, which will come crashing down in a 
heap. Of course, a Humean concern might give any inductive program pause, but 
no reason has yet been produced for thinking causal discovery is in that respect 
any more vulnerable than any other scientific activity.22 

What causal discovery does is precisely to distinguish those cases of correlation 
between A and B which are best explained by a causal structure relating them from 
those which are best explained otherwise. How can that be done? The basic insight 
was codified already by Hans Reichenbach (1956). In his "Principle of the Common 
Cause" he asserted (1956, p. 157): "If an improbable coincidence has occurred, 
there must exist a common cause." By this formulation Reichenbach did not mean 
to be ruling out the possibility that, given coincident phenomena, one might cause 



60 Kevin Korb 

the other, directly or indirectly. ,Rather, this formulation becomes active just in 
case we can rule out such a connection-so, the Common Cause Principle in 
effect is: reliable, reproducible coincidences do not occur-magic does not exist. 
The statisticians' denial of the rationality of inferring causation from correlation 
simply leaves the reliable, persistent correlation inexplicable, which is tantamount 
to endorsing magic. 

Following Reichenbach, the causal inference is based upon probabilistic 
independencies that are revealed in observed correlations. Thus, considering a 
system with three variables, there are only the following possible causal structures 
(assuming the pairs <A,B> and <B,C> are directly related): 

a.A~B~C 

b.A~B~C 

c.A~B~C 

d.A~B~C 

Reichenbach called (c) a "fork open toward the future" and (d) a "fork open 
toward the past" and pointed out that they support distinct conditional independence 
structures: (c)-as well as (a) and (b)-have A and C marginally dependent, but 
independent conditioned upon B (i.e., P(AIC):;tP(A) but P(AIC,B)=P(AIB»; (d), 
exactly to the contrary, has A and C marginally independent, but conditionally 
dependent (i.e., P(AIC)=P(A) and P(AIC,B):;tP(AIB». 

All causal discovery algorithms are ultimately based upon this simple distinction. 
And although this basis may seem quite small, making only a binary distinction 
between two sets of models, the recursive application of the principle over models 
with many variables turns out to be very powerful (see Verma and Pearl, 1990). 
Since different causal models give distinct probabilities to data reflecting a given 
conditional independence structure, they have different likelihoods, and so Bayes' 
theorem yields different posterior probabilities. For example, ifsome observational 
evidence e were to support strongly the conditional dependence of (d) (that 
P(AIC,B):;tP(AIB», then the likelihood of (d) on that evidence would be much 
greater than the likelihood of any of the three other models, and on Bayesian 
grounds (d) would be strongly confirmed. Such Bayesian inference can be 
automated, and quite effectively for discovering the true model underlying the data 
(see, e.g., Korb and Nicholson, 2004). And, so, Bayesian analysis once again 
exposes the denunciation of a mode of reasoning as fallacious as itself fallacious. 

8. On Improving Our Probabilistic Reasoning 

So, we have seen that Bayesian reasoning can deliver more informed and insightful 
verdicts on the merits and demerits of arguments in informal logic, the correctness 
of forms of statistical reasoning, and even whether a causal inference is justified 
or not. If Bayesian reasoning is so valuable, it would be good to learn how to learn 



Bayesian Informal Logic and Fallacy 61 

it, or do it, because the evidence on the whole is that people find Bayesian reasoning 
very difficult to do (e.g., Nisbett, et aI., 1987). Indeed, the field of informal logic 
in general arose largely out of a concern over the ineffctiveness of logic teaching 
on critical reasoning (cf. Johnson, 1996); it would be reassuring to have evidence 
that we could put these techniques into practice. 

There are at least four different approaches that can be taken to using Bayesian 
calculation in practice. To illustrate, I present them in the context of an example of 
breast cancer diagnosis adapted from Hoffrage and Gigerenzer (1998; itself adapted 
from Eddy, 1982).21 

8.1 Bayes' Theorem 

The most direct way to do Bayesian reasoning is simply to employ Bayes' theorem. 
Suppose that we are presented with a woman appearing at a clinic whose initial 
test for breast cancer has proved positive. We are asked to estimate the chance 
that the woman indeed has breast cancer. As background, we are told that one in 
one hundred women appearing at the clinic have breast cancer and that the tests 
are positive given cancer 80% of the time and they also are positive 10% of the 
time in women without cancer. 

The most common response to this kind of scenario is to assert that the woman 
has an 80% chance of having breast cancer. It is not clear why most people 
respond this way. One possibility is just that people tend to find conditionals 
confusing and, in particular, confuse conditionals and their converses. In this 
case, asserting an 80% chance of cancer confuses P(cancerlPos[itive test]), which 
is the posterior probability that we were asked for, with P(poslcancer), which is 
the likelihood given in the scenario as 0.8. Some Bayesians would like the confusion 
between posterior and likelihood to explain the entire tradition of maximum 
likelihood-oriented orthodox statistics, which is probably putting too much of a 
burden on a simple confusion. Tversky and Kahneman (1982) dubbed this mistake 
"base-rate neglect," since it can arise by suppressing the term for the prior in 
Bayes' formula. 

In any case, the correct computation using Bayes' formula is: 

P(cancerlPos) = 

(P(poslcancer) P(cancer» 

(P(poslcancer) P(cancer) + (P(poslno cancer) P(no cancer) 

(.8 X .01) 

(.8 X .01)+(.1 X .99) 

.008 

.008 + .099 



62 Kevin Korb 

.008 

.017 

.075 

It is easy to see why few people will get this computation right without resorting 
to a calculator or at least paper and pencil! Assuming that the difference between 
estimating the probability of cancer as 80% and as 7.5% is one that means something, 
what needs to be noticed is just the need to employ a computational device (such 
as paper and pencil) to get the answer right, rather than relying upon an intuitive 
judgment that gets the answer wrong. Another option is to learn one of the following 
three alternative methods of computation, each of which is notably psychologically 
simpler than the one above. 

8.2 Frequency Formats 

Hoffrage and Gigerenzer (1998) advocate the use of "frequency formats" to make 
Bayesian reasoning more intuitive. Basically, this involves multiplying the probabilities 
in any given case by a sufficiently large number so that the entire problem can be 
worked in whole numbers rather than fractions. Thus, we can take the breast 
cancer numbers and multiply them by 1000. We also layout the problem in a 
classification tree (cf. Breiman et al., 1984), as in Figure 3. 

Figure 3: Classification tree for breast cancer 

To construct the tree, we start with 1000 women. One percent are presumed to 
have cancer (the prior probability); that means the left branch yields 10 women 
with cancer and the right branch 990 without. Of the 10 women with cancer, 8 
(80%) will test positive. Of the 990 without, 99 (10%) will test positive. Thus, 
confronted with a positive test and nothing else, we compute the probability of 



Bayesian Informal Logic and Fallacy 63 

cancer as 8/(8+99). This is clearly easier to handle without computational devices 
than Bayes' formula directly. Hoffrage and Gigerenzer (1998) report greater success 
in getting frequency formats into effective use than Bayes' theorem, which is 
hardly surprising. 

8.3 Odds-likelihood Bayes 

Another way of simplifying Bayesian reasoning is to encourage people to think in 
terms of betting odds, rather than probabilities. In that case, the odds-likelihood 
form of Bayes' theorem, which is far simpler to handle than the original form, can 
be employed exclusively. The breast cancer problem is then solved as follows. 

The prior odds of cancer are: 

O(cancer) 

The likelihood ratio is: 

(P(cancer» 

(P(no cancer» 99 

A = (P(poslcancer» .8 8 
(P(poslno cancer» .1 

We apply odds-likelihood Bayes: 

1 8 
O(cancerlPos) = A O(cancer) = 8 X - = -

99 99 

In other words. we simply take the confirmatory power of the evidence, the 
likelihood ratio (Bayes factor) of A = 8 and multiply that into the prior odds, 
yielding a posterior odds for cancer of 8:99. This is at least as simple as the use of 
frequency formats. 

Furthermore, the odds-likelihood approach focuses attention on the two major 
factors in Bayesian reasoning better than any of the other approaches, since it 
simply involves multiplying them together-that is, mUltiplying the prior odds 
(equivalent to the prior probability) and the likelihood ratio (i.e., the confirmatory 
power of the evidence). Given these strengths, it would make good sense to try 
building a tutorial program for Bayesian reasoning around odds-likelihood reasoning. 
Nevertheless, I must report that in teaching first-year university students I have 
had more success with frequency formats than odds-likelihood Bayes; presumably, 
the latter should be reserved for more advanced students. 

If in the end we are asked to produce a posterior probability, we will need to 
move from odds to probabilities through an additional conversion: 



64 Kevin Korb 

P( cancerlPos)= (O( cancerlpos)) 
(1 +O( cancerlpos)) 

8/99 

(107/99) 

8 

107 

8.4 Bayesian Networks 

:::::: .075 

From the human point of view, far and away the simplest method of solving such 
probability problems is just to let a computer do them for us. Bayesian network 
technology has been developed by AI researchers and statisticians since the 1980s 
(e.g., Pearl, 1988; Neapolitan, 1990) which allows both exact and approximate 
Bayesian inference to be performed with essentially no burden on the human user. 
In recent years, this technology has been incorporated into PC programs with 
typical windows interfaces (e.g., Netica, at http://www.norsys.com). The breast 
cancer problem then is handled by: building a two-variable Bayesian net as in 
Figure 4; clicking on the test node and setting it positive; reading off the display a 
posterior probability of cancer at 0.075. This is as simple as it gets. The use of 
such networks in decision analysis and decision support is almost certainly going 
to become widespread. 

P =0.01 

8 
j 

(;) 

+Tc51 

-TC51 

Cancer No Canc<:r 

0.80 0.10 

0.20 0.90 

Figure 4: Bayesian network for breast cancer 

One might wonder how such a tool is possible. After all, above I pointed out that 
Bayesian computations are NP-hard, that is to say, they grow exponentially complex 
and long as the problem size increases. There are two parts to the answer to this. 
First, a two-variable problem is computationally trivial (at least for the better and 
more capable non-humans); the computational complexity kicks in somewhere 
above ten variables (depending on the network structure and number of states per 
node). Second, methods of approximating exactly correct inference, including 



Bayesian lriformal Logic and Fallacy 65 

stochastic simulation over Bayesian networks, are available, and their improvement 
is an active area of research. These techniques make possible inference in complex 
problems that would otherwise be hopeless. 

9. Conclusion 

Despite the fact that probability theory applies a formal calculus and, as such, 
comes no more naturally to people than do mathematical logic or the differential 
and integral calculus, it should be clear that probability theory offers a useful guide 
to much of correct human inference. It can go a long way toward making sense 
not just of scientific inference and evaluation, but also of ordinary language 
arguments of a wide variety. Further, there is reason to believe that real progress 
can be made in developing pedagogical techniques for teaching probabilistic 
reasoning and its application to argumentation. Here I have only made a rudimentary 
start at applying Bayesian principles and methods to these tasks, but [ believe it 
makes clear the opportunity for a more systematic application of Bayesian techniques 
for informal reasoning. Perhaps, indeed, someone will even apply them to the 
rehabilitation of the standing of the sophists, removing the ad hominem invented 
by Plato so long ago. 

Acknowledgement 

[ would like to thank the London School of Economics and its Centre for the 
Philosophy of the Natural and Social Sciences for the provision of facilities during 
early work on this paper. [ also thank Charles Twardy and Noretta Koertge for 
helpful comments. 

Endnotes 

iS ee, for example, his Euthydemus. Even before Plato, Aristophanes castigated the sophists (and 
Socrates) in his play The Clouds. 

lFallacies are often defined as forms of reasoning which are invalid-meaning, in the case of 

deductive argument, those that do not necessarily preserve the truth of the premises, and, in the 
case of inductive argument, those that do not lead to probable truth. 
JFor a brief history of informal logic, see Johnson and Blair (1996). 
4In a series of papers on automated argumentation I, and my collaborators, have called such 
arguments nice rather than good-admittedly,justto give rise to the acronym NAG (Nice Argument 
Generator); e.g., Korb et al. (1997). 
lNot that there has been no work in this area. The work of Polya (1968) in particular comes to 
mind. But although his ideas are suggestive and aimed in the same general direction as my own 
thoughts, they appear to have been incomplete thoughts. 
6As an aside, I apologize now, once and once only, for presenting in this paper quite so an 
unabashedly Bayesian anlaysis of the fallacies. Some would want me to apologize more profusely-
or, what amounts to much the same thing, to revisit all of the major arguments for and against 
Bayesian principles, prior to attempting to further their application in any new analysis. Rather 



66 Kevin Korb 

than that, here I largely just assume such principles and see where they lead us in understanding 
a selection of fallacies. Opponents or agnostics of the Bayesian way can accept this as hypothetical 
reasoning-of a type which is necessary for the development of human thinking. In any case, there 
is already a huge literature devoted to defending and upending Bayesian and other views on 
statistical inference, to which my references already suffice to find an introduction. 
7Another cluster of controversies surrounds the interpretation of probability: what are these 
magnitudes designated by P(.) and P(.le)? The common interpretation amongst Bayesians is that 
they designate rational degrees of belief in propositions, whether they are idealized or realized. 
The merits or demerits of this subjective account of probability is not the issue of this paper. In 
any case, it is not strictly necessary to go along with this subjectivist interpretation while finding 
value in Bayesian reasoning; for example, Reichenbach (1949) presented a frequentist interpretation 
consistent with the use of Bayesian reasoning over scientific hypotheses. 
8Note that this is what Good called a Bayes factor (Good, 1983) and is not the same as what many 
statisticians call the likelihood ratio, which is instead a ratio of maximum likelihoods. That 
assumes that the hypothesis under consideration and its alternative are incomplete (indefinite) 
and can be parameterized so as to maximize the probability of the observed evidence. Here, 
however, I shall assume that we are dealing with definite hypotheses. 
9 Also equivalently, a and bare probabilistically related just in case their mutual information 
measure is non-zero (see Cover and Thomas, 1991). 
"'There are other cases, however, where the likelihoods are not so simply understood; see Korb 
(1994). 
"That is, such orthodox inferences issue in claims, implicit or explicit, about posterior probability, 
such as P(hle) that is almost certainly false; since likelihoods are generally not controversial, we 
can apply Bayes' theorem to obtain the denied prior probability P(h) (cf. with equation 1): 

P(h) = (P(hle)P(e» 
(P(elh» 

'2Similarly, all Popper could say of any theory was that it was falsified or not falsified, despite 
some extraordinary contortions to do with the "corroboration" of theories-which never quite 
managed to mean as much as confirmation. 
'3The probability calculus had its origin in a famous correspondence between Pascal and Fermat 
in 1654, posing and solving questions about various gambles. 
'4Franklin (1986) gave a prior, more complex, Bayesian analysis of this case. 
151 employ P j.) for notational simplicity rather than P(.Id), which is warranted by the fact that 
probabilities conditional upon some event satisfy the probability axioms. 
'6Never mind that there are those with minds closed against evolutionary theory; empirically 
based arguments will never have a significant impact if you start out with either sufficiently 
biased priors or sufficiently warped conditional probabilities. (And, if to the anti-Bayesian this 
appears to be a fatal concession-because it shows that it is possible to be an irrational Bayesian-
that is a common but mistaken conclusion. It is the demand for a Bayesian theory that answers 
all problems about rationality in all circumstances that is most irrational.) 
' 7To be sure, it is also required that the prior P(h) not be too low before asserting P'(h) is high. 
'8Interestingly, the proponents of IBE, such as Gilbert Harman (1965) and Peter Lipton (1991), 
seem not to have appreciated these additional requirements, which in fact imply that IBE taken as 
a general rule is defective. Incidentally, one reviewer pointed out, quite rightly, that my discussion 
ofIBE here is a considerable simplification of the subject. Much the same could be said about all 
of the individual topics as I present them in this paper-a certain lack of depth is inevitable in such 
a broad overview as I offer here. Nevertheless, I will stand by all that I say here. IBE is a well-
intentioned but itself oversimple account of induction. A far better foundation for such an account, 
in my opinion, can be found in Chris Wallace's (2005) minimum message length theory. 
191 am indebted to Gigerenzer and Todd (1999) for the phrase "fast and frugal heuristic," and 



Bayesian Informal Logic and Fallacy 67 

indeed they discuss some such heuristics that have the same sort of characteristics as 
representativeness, availability and other heuristics examined by Bayesian psychologists- i.e., 
accuracy in common cases, ease and speed of computation-although curiously they do not 
investigate those heuristics themselves. 
2'Nor does imposing a condition of temporal proximity save the charge of fatuousness. Although 
people commonly infer that hitting a light switch turns on a light, it is hardly just a temporal 
relation they are relying upon, since analogous inferences are withheld regarding the overwhelming 
majority of simultaneous cases of hitting a switch around the world. 
2iOne reader has suggested that the statisticians' slogan represents no such denial, that it merely 
reports a sensible aversion to inferring specifically that A causes B from a mutual correlation. This 
is a real distortion ofthe history of the debate, however. First, no one has ever endorsed the above 
inference, so under this interpretation the statisticians' dogma is directed at nothing. Furthermore, 
it fails to explain the long and enduring literature debating such a non-inference. That at least some 
statisticians genuinely believe correlational structure provides no useful support for causal inference 
of any kind can be seen in my references above and also in a statement of Sir Ronald Fisher's: "If 
we are studying a phenomenon with no prior knowledge of its causal structure, then calculations 
of total or partial correlations will not advance us one step" (Fisher, 1925). 
22My rebuttal to Humphreys and Freeman (1996) is in Korb and Wallace (1997). 
lJ Although Gerd Gigerenzer is the author of one of these methods, curiously he has spent much 
energy and time questioning Bayesian reasoning as a normative standard (e.g., Gigerenzer and 
Murray, 1987; Gigerenzer, 1991; Gigerenzer and Todd, 1999). Most recently, he has objected to 
Bayesian reasoning on the ground that there is no plausible evolutionary story to tell about how 
such a reasoning ability could evolve, given its computational complexity, citing the work of 
Tooby and Cosmides in evolutionary psychology (e.g., in Chase et al., 1998). Presumably, the 
idea is that, as a general principle, what cannot be done also cannot be held up as a normative 
standard of behavior, so Bayesian reasoning, being unavailable to us, offers us no standard. 
Perhaps the oddest twist in this story is that Gigerenzer's own technique for Bayesian reasoning 
promises to make Bayesian calculation far more accessible, using his "frequency formats" (Hoffrage 
and Gigerenzer, 1998; Gigerenzer, 1996). Gigerenzer is certainly right that for Bayesian principles 
to provide an effective standard they must be made possible to use. 

References 

Aristotle. On sophistical refutations. 

Aristotle. Rhetoric. 

Bacchus, F., H. Kyburg, and M. Thalos (1990). Against conditionalization. Synthese, 8, 
475-506. 

Bayes, T. (1763/1958). An essay towards solving a problem in the doctrine of chances. 
Philosophical Transactions of the Royal Society of London, 53, 370-418. Reprinted 
in Biometrika, 45, 296-315. 

Breiman, L., Frieqman, J. H., Olshen, R. A. and Stone, C. J. (1984). Classification and 
regression trees. New York: Chapman and Hall. 

Butler, J. (1736). The analogy of religion. London: Knapton. 

Carnap, R. (1950). Logical foundations of probability. Chicago: University of Chicago. 

Chase, V. M., Hertwig, R. and Gigerenzer. G. (1998) Visions of rationality. Trends in 
Cognitive Sciences, 2,206-214. 

Chihara, C. and Kennedy, R. (1979). The Dutch book argument: Its logical flaws, its 
SUbjective sources. Philosophical Studies, 36, 19-33. 



68 Kevin Korb 

Cohen, L. J. (1981). Can human rationality be experimentally demonstrated? Behavioral 
and Brain Sciences, 6, 248-249. 

Cooper, G. (1990). The computational complexity of probabilistic inference using Bayesian 
belief networks. Artificial Intelligence, 42, 309-405. 

Copi, I. and Cohen, C. (1990). Introduction to logic, 10th ed. Upper SadIe River, NJ: 
Prentice Hall. 

Cover, T.M. and Thomas, J.A. (1991). Elements of information theory. New York: Wiley. 

de Finetti, B. (1937/80). Foresight: Its logical laws, its subjective sources. Translated by 
H. Kyburg in H. Kyburg and H. SmokIer (Eds.) Studies in subjective probability, 2nd 

ed, Huntington, NY: Krieger. pp. 55-118. 

Eddy, D.M. (1982). Probabilistic reasoning in clinical medicine: Problems and opportunities. 
In Kahneman, Slovic and Tversky (1982). pp. 249-267. 

Fisher, R.A. (1925). Statistical methods for research workers. Edinburgh: Oliver and 
Boyd. 

Franklin, A. (1986). The neglect of experiment. Cambridge: Cambridge University Press. 

Gigerenzer, G. (1991). How to make cognitive illusions disappear: Beyond "heuristics 
and biases." In W. Stroebe and M. Hewstone (Eds.) European Review of Social 
Psychology, vol 2; New York: Wiley. pp. 83-115. 

Gigerenzer, G. (1996). The psychology of good judgment: Frequency formats and simple 
algorithms. Journal of Medical Decision Making, 16, 273-280. 

Gigerenzer, G. and Murray, D. J. (1987). Cognition as intuitive statistics. Hillsdale, NJ: 
Lawrence Earlbaum. 

Gigerenzer, G. and Todd, P.M. (1999). Simple heuristics that make us smart. Oxford: 
Oxford University Press. 

Glymour, C., Scheines, R., Spirtes, P. and K. Kelly (1987). Discovering causal structure: 
Artificial intelligence, philosophy of science and statistical modeling. New York, 
NY: Academic Press. 

Good, I.J. (1983). Good thinking: The foundations of probability and its applications. 
Minneapolis: University of Minnesota. 

Hamblin, C. L. (1970). Fallacies. Newport News, V A: Vale Press. 

Harman, G. (1965). Inference to the best explanation. Philosophical Review, 74, 88-95. 

Hempel, C. (1965). Aspects of scientific explanation. New York, NY: Free Press. 

Hoffrage, U., and Gigerenzer, G. (1998). Using natural frequencies to improve diagnostic 
inferences. Academic Medicine, 73, 538-540. 

Holte, R. (1993). Very simple classification rules perform well on most commonly used 
datasets. Machine Learning, 11, 63-91. 

Howson, C. (2000). The logic of Bayesian probability. In D. Corfield and J. Williamson 
(Eds.) Foundations of Bayesianism, Dordrecht: Kluwer. pp. 137-159. 

Howson, C., and Urbach, p. (1993). Scientific reasoning: The Bayesian approach, 2nd 

ed. La Salle, IL: Open Court. 

Humphreys, P., and Freeman, D. (1996). The grand leap. British Journal for the 
Philosophy of Science, 47, 113-23. 

Jaynes, E. (1989). Papers on probability, statistics and physics, 2nd ed., edited by R.D. 
Rosencrantz. Kluwer. 



Bayesian Informal Logic and Fallacy 69 

Johnson, R.H. (1996). The rise of informal logic. Newport News, VA: Vale Press. 

Johnson, R.H., and Blair, J.A. (1996). Informal logic: Past and present. In Johnson (1996), 
pp.I-19. 

Joseph, H.W.B. (1906). An introduction to logic. Oxford: Clarendon Press. 

Kahane, H. (1971). Logic and contemporary rhetoric. Belmont, CA: Wadsworth. 

Kahneman, D., Siovic, P. and Tversky, A. (Eds.) (1982). Judgment under uncertainty: 
Heuristics and biases. Cambridge: Cambridge University Press. 

Kahneman, D. and Tversky, A. (1972). Subjective probability: A judgment of 
representativeness. Cognitive Psychology, 3, 430-454. 

Korb, K.B. (1992). A pragmatic Bayesian platform for automating scientific induction. 
Ph.D. dissertation, Indiana University. 

Korb, K.B. (1994). Infinitely many resolutions of Hempel's paradox. In R. Fagin (Ed.) 
Theoretical aspects of reasoning about knowledge V, San Fransisco: Morgan 
Kaufmann, 138-49. 

Korb, K.B. and Nicholson, A.N. (2004). Bayesian artificial intelligence. CRC Press. 

Korb, K.B. and Thompson, C. (1994). Primitive concept formation. Second Australian 
and New Zealand Conference on Intelligent Information Systems, 362-366. 

Korb, K.B. and Wallace, C.S. (1997). In search of the philosopher's stone: Remarks on 
Humphreys and Freeman's critique of causal discovery. British Journal for the 
Philosophy of Science, 48, 543-553. 

Korb, K.B., McConachy, R., and Zukerman, I. (1997). A cognitive model of argumentation. 
In Proceedings of the Nineteenth Annual Conference of the Cognitive Science 
Society, Mahwah: NJ, Lawrence Erlbaum Associates. pp. 400-405. 

Laplace, P. S. (181411951) A philosophical essay on probabilities. trans. F.W. Truscott 
and F.L. Emory. New York, NY: Dover. 

Levi, I. (1980). The enterprise of knowledge. Cambridge, MA: MIT Press. 

Lewis, D. (1980). A subjectivist's guide to objective chance. In R. Jeffrey (Ed.) Studies in 
inductive logic and probability, Berkeley: University of California, vol. II, 263-293. 

Lipton, P. (1991). Inference to the best explanation. London: Routledge. 

Mates, B. (1972). Elementary logic, 2nd ed., New York: Oxford University Press. 

McKim, V. and Turner, S.P. (Eds.) (1997). Causality in crisis? Statistical methods and 
the search for causal knowledge in the social sciences. Notre Dame, IN : University 
of Notre Dame Press. 

Nagel, E. (1961). The structure of science. New York: Harcourt, Brace and Ward. 

Neapolitan, R. (1990). Probabilistic reasoning in expert systems. New York: Wiley. 

Nisbett, R.E., Fong, G.F., Lehman, D.R. and Cheng, P.W. (1987). Teaching reasoning. 
Science, 238,625-631. 

Pearl, J. (1988). Probabilistic reasoning in intelligent systems. San Mateo: Morgan 
Kaufmann. 

Pinto, R. (1995). Post hoc ergo propter hoc. In H.V. Hansen and R.C. Pinto (Eds.) Fallacies, 
302-311. University Park: Pennsylvania State University. 

Polya, G. (1968) Mathematics and plausible reasoning, volume 2, 2nd ed. Princeton, NJ: 
Princeton University Press. 



70 Kevin Korb 

Popper, K. (1934/59). Logik der Forschung, translated as Logic of scientific discovery, 
translator Popper, K., London: Hutchison. 

Ramsey, F. P. (1931). The foundations of mathematics and other logical essays, edited 
by R. B. Braithwaite. New York, NY: Humanities Press. 

Reichenbach, H. (1949). The theory of probability, 2nd ed. Berkeley: University of 
California. 

Reichenbach, H. (1956). The direction of time. Berkeley: University of California. 

Rips, L.J. (1975). Inductive judgments about natural categories. Journal of Verbal 
Language and Verbal Behavior, 14,665-681. 

Rosch, E., Merris, C.b., Gray, W.D., Johnson, D.M. And Boyes-Braem, p. (1976). Basic 
objects in natural categories. Cognitive Psychology, 8, 382-439. 

Rottenberg, A.T. (1991). Elements of argument: A text and reader. Boston: St. Martin's 
Press. 

Scriven, M. (1976). Reasoning. New York: McGraw Hill. 

Sperber, D., Premack, D. and Premack, AJ. (1995). Causal cognition: A multidisciplinary 
debate. Oxford: Clarendon Press. 

Teller, P. (1973). Conditionalization and observation. Synthese, 26,218-258. 

Tversky, A. and Kahneman, D. (1973). Availability: A heuristic for judging frequency 
and probability. Cognitive Psychology, 4, 207-232. 

Tversky, A. and Kahneman, D. (1982). Evidential impact of base rates. In Kahneman, 
Slovic and Tversky (1982), 153-160. 

van Gelder, T. (2000). The efficacy of undergraduate critical thinking courses: A survey 
in progress. At http://www.philosophy.unimelb.edu.au/reason/critical/ 

Verma, T., and Pearl, J. (1990). Equivalence and synthesis of causal models. Proceedings 
of the Sixth Conference on Uncertainty in AI, San Fransisco: Morgan Kaufmann, 
220-227. 

Wallace, C.S. (2005). Statistical and Inductive Inference by Minimum Message Length. 
Berlin: Springer Verlag. 

Walton, D. N. (1989). Informal logic: A handbookfor critical argumenation. Cambridge: 
Cambridge University Press. 

Wason, P. C. (1960). On the failure to eliminate hypotheses in a conceptual task. Quarterly 
Journal of Experimental Psychology, 12, 129-140. 

Wason, P. C. (1966). Reasoning. In B. M. Foss (Ed.), New horizons in psychology, 1. 
Hammondsworth: Penguin, 135-51. 

Kevin B. Korb 
School of Computer Science 

and Software Engineering 
Monash University 

Clayton, Victoria 3168 AUSTRALIA 

korb@csse.monash.edu.au