Download Chapter 1 Error February 09

Chapter 1 Error
Jan.’10
“The philosophy of reasoning, to be complete, ought to comprise the
theory of bad as well as of good reasoning.”
John Stuart Mill
“There is no such thing as a classification of the ways in which men may
arrive at an error; it is much to be doubted whether ever can be.”
Augustus DeMorgan
1.1 Logic and error
Mistakes of reasoning are part of a rich assortment of blunders, bloopers and
miscues. There are mechanical errors, perceptual errors, errors of memory, errors of
misinformation, strategic errors, tactical errors, and errors of judgement. Some errors
matter and others don’t. There are silly mistakes, catastrophic mistakes, harmless
mistakes, costly mistakes, inconsequential mistakes, lucky mistakes, stupid mistakes,
instructive mistakes, insightful mistakes, brilliant mistakes, and so on.1 As traditionally
understood, logic investigates the principles of right reasoning. Since error is what right
reasoning is supposed to avoid, it would appear that a theory of reasoning must also
develop an account of bad reasoning. Although error is caught up in logic’s ambit, its
typical treatment plays down the connection. The error in which mainstream logicians
have the largest stake is invalidity. But deductive logic is a theory of validity. Invalidity
trails along as a byproduct. Invalidity is the absence of validity; and validity wears the
trousers in deductive logic. Seen this way, errors of reasoning are parasitic on the rules of
right reasoning – rules, so to speak, for the regulation of the mind. There is a clear
procedural suggestion in this for the would-be theorist: Get the rules of right reasoning
sorted and errors of reasoning will fall out automatically.
Augustus De Morgan is one of many logicians to have favoured this idea. Since
there are mumberless ways in which reasoning can go wrong, De Morgan thought that a
logic of error could only be a logic of right reasoning, the number of whose principles are
comparatively few.2 One might say with De Morgan – and with the rule-violation
community generally – that there are no stand-alone or non-parasitic theories of
deductive error in logic. We could think of a stand-alone theory of error as one which has
something to say, among other things, about how errors are both inapparent enough to
We note the E.R.R.O.R. convenors’ description of the scope of a conference of the same name held at
Virginia Tech in June, 2006. Seen their way, errors encompass mistakes of inference, flawed methods,
statistical errors, misspecified models, anomalous results, erroneous verdicts, deceptions in nature, biases
and fallacies. Proceedings of the conference, under the guest editorship of Kent Staley, Jean Miller and
Deborah Mayo, appear in Synthese, 163 (2008), 299-442.
2
De Morgan (1847), chapter 13: “As to mere inference, the main object of this work [Formal Logic], it is
reducible to rules; these rules being all obeyed, an inference, as inference is good; consequently a bad
inference is a breach of one or more of these rules.”
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escape detection and yet sufficiently recognizable to permit correction, and about the
factors implicated in this conversion of error-concealment to error-detection.
Traditionally minded logicians have long been accustomed to regarding the rules
of right reasoning as formal constraints, that is, as constraints that hold independently of
propositional content. This gives rise to a further difficulty. James Oliver and Gerald
Massey have argued  independently  that a non-parasitic approach to invalidity is not
possible for formal logic.3 Given that natural language arguments instantiating invalid
forms can themselves be informally valid, these critics disavow the capacity of
mathematical logic to generate a theory of errors of reasoning in natural languages.
Although any natural language argument instantiating a valid form is valid, natural
language arguments instantiating valid forms also instantiate invalid forms.4 This
produces an asymmetry, in which a formal theory of validity is thought to be possible,
but not one of invalidity. This tells us something interesting and important about a standalone logic of deductive error. Not all of it can be squeezed into the formal logics of
deduction.
The old idea that an error reasoning is reasoning that violates the rules for right
reasoning also echoes in logic’s traditional approach to inductive error, in which the
property of inductive strength is the principal target. In the logician’s technical sense, a
piece of reasoning is inductively strong just in case it exhibits a conditional probability of
requisitely high value linked to an appropriately qualified set of data. We might say that
whereas deductively valid reasoning is truth-preserving, inductively strong reasoning is
probabilistically clinching. Then reasoning is inductively defective when it is not
inductively strong. Here, too, rightness rather than wrongness is the focal issue. Inductive
strength wears the trousers in inductive logic. But here, too, if it could be shown that
there are significant ranges of cases in which good but invalid reasoning is not, in this
technical sense, inductively strong, this would be something a theory of error should take
proper note of.
We may put it, then, that a significant part of logic’s traditional understanding of
error can be set out in the following biconditional:
Proposition 1.1
RR-RULE-VIOLATION: E is an error of reasoning if and only if there exists a
truth-preserving rule R of right reasoning or a probabilistically clinching rule of
right reasoning R and E violates R or R.
Let us note in passing some of the ways in which Proposition 1.1 could go wrong. It
might, for one thing, be too narrow in its focus. That is, there might be pieces of
reasoning which, though neither valid nor probabilistically clinching are nevertheless
good. A second possibility that is it embodies an equivocation about error. That is, it
might strike us upon reflection that reasoning that is erroneous in a given way might not
be erroneous in some further way, and that there are situations in which this non3
Oliver (1967), Massey, (1975a, 1975b, 1981).
Consider, for example, the argument Harry’s shirt is red, Harry’s shirt is coloured. Although
semantically valid, it instantiates the invalid form α, β. This is what some writers call “material validity”.
See, for example, Brandom (2000).
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2
erroneousness trumps erroneousness. There are lots of cases in which the inductive
strength of an invalid argument clearly trumps its validity. A further possibility is the
Proposition 1.1 is felled by a defective presupposition. For example, if a carefully crafted
stand-alone account of error were to embody a version of logical particularism, then we
would have reason to suppose that reasoning rightly would not be a matter of following
rules, hence that reasoning badly is not a matter of not following them. As we proceed,
there will be occasion to give favourable consideration to these possibilities.
Suppose the RR-Rule violation conception of errors of reasoning were correct.
Then it would be possible to have deep and detailed accounts of trouser-wearing concepts
such as – on the deductive side – validity, entailment, logical truth, provability, deduction
and consistency and – on the inductive side – probability, inverse probability, likelihood,
conditionalization, confirmation, and so on. We could say that theories of this sort render
“thick” accounts of their target concepts. Correspondingly, concepts dealt with as the
mere complements of concepts themselves thickly provided for would be thinly deal
with. The logic of error portended by Proposition 1.1 is a thin theory of error. What’s
wanted is something thicker.
1.2 The mainstream
It would be wrong to leave the impression that logicians have never been
attracted by prospects for a thick theory of errors of reasoning. Nicholas Rescher’s Error
is a recent attempt in that direction.5 The same was true of Richard Whateley, though less
recently, and also of J.S. Mill (as the first of this chapter’s epigraphs makes plain).6, 7
Similarly, in saying that this or that is or has been true of “traditionally minded” or
“mainstream” logicians, or that these things are or have been true of them “for the most
part”, the quoted qualifications are essential for accuracy. Logic is an ancient discipline
with a vast literature. Even since taking the mathematical turn at the comparatively recent
mid-point of the century just past, logic has produced a sprawling pluralism, and in many
respects a rivalrous one. The mainstream in logic is not a precisely defined idea, and “by
and large” denotes trend-lines, not strict invariability. We may think of first order
quantification theory as the paradigm case. But even as modern logic was evolving in that
direction, there were rival developments, many of them highly regarded and influential,
and some of them of sufficient staying power to have taken on a certain venerability of
their own. We can think of first order logic as the metropolitan centre of the discipline,
with (say) modal and intuitionist logics a near and prosperous suburb, and many-valued,
relevant, and paraconsistent logics hunkering down nearby. Perhaps situation, gametheoretic, computer and dialogue logics are a longer commute, but still within striking
distance of downtown. Informal logic is further out; some would say beyond the pale.
Inductive logic, also a suburb, is almost a city in its own right; and it has in turn its own
city centre (probability theory and confirmation theory) and suburbs (nonmonotonic and
5
Rescher (2007).
Whately (1836), Mill (1961), bk. 5.
7
Schmitt (2007) observes that Rescher omits to mention the three modern philosophers who have given the
most attention to error: Bacon (1960), esp. bk. I, Malebranche (1980), esp. bks. 1 and 2, and Hume (1978),
esp. bk. I, pts. 3 and 4. It would be something of a stretch, however, to classify this trio as logicians. With
the exception to which we now turn, logicians aren’t much concerned with positive theories of error.
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default logics, logics of plausibility, and so on). Still, however flexibly configured, there
is a mainstream in logic, and the things we have said to be true of it are indeed true of it
in the main. But the mainstream is an evolving enterprise. Its configuration in the past
forty years have lent it a more presumptive character than it formerly had.8 By my lights,
this is not the time to place artificial constraints on further innovation. Better that new
work be judged after it has been done, rather than pre-judged and not done at all.
1.3 The Gang of Eighteen
Notwithstanding logic’s generally thin approach to errors of reasoning, there is a
category of error coincident with the founding of logic itself, and off and on a part of
logic’s research programme ever since, which has attracted special and somewhat thicker
attention. These are the fallacies, whose traditional list is a loose assemblage of which the
following is not untypical: ad baculum, ad hominem, ad populum, ad verecundiam, ad
ignorantiam, ad misericordiam, affirming the consequent, and denying the antecedent,
begging the question, many questions, hasty generalization, equivocation, biased
statistics, gambler’s, post hoc, ergo propter hoc, composition and division (of which
secundum quid is a special case), faulty analogy, and ignoratio elenchi (of which straw
man is a special case). If we don’t count secundum quid and straw man separately, this
makes for a list of eighteen, known – with a certain light heartedness  as the Gang of
Eighteen.9 Some of these, especially question-begging and many questions, are seen by
some investigators as dialectical fallacies rather than errors of inference. But the
traditional view is that most, if not all, of the items on this list instantiate the ruleviolation conception of error as we have formulated it here.
8
Here is a small sample of the rehabilitation of agency and context by logics more or less within striking
distance of the mainstream: For epistemic logic see Hintikka (1962), Fagin et al. (1995), Meyer and van der
Hoek (1995), Gochet and Gribomont (2006); deontic logic von Wright (1951), Lemmon (1957), (Hilpinen
(1981), Horty (1994); temporal logic Prior (1957), Gabbay et al. (1994), Øhstrøm and Hasle (1995), ter
Meulen (1997), Gabbay et al. (2000); dynamic logics Harel (1979) and van Benthem (1996); situation
semantics Barwise and Perry (1980) and Barwise and Seligman (1997); dialogue and interrogative logic
Lorenzen and Lorenz (1978), Barth and Krabbe (1982) and Hintikka (1989); relevant logic Sperber and
Wilson (1986/1995) and especially Gabbay and Woods (2003); nonmonotonic logic (McCarthy (1959),
McCarthy and Hayes (1969), Krauts et al. (1990), Makinson (1994), Schlecta (2004, 2007), Makinson
(2005) and Bochman (2008) ; game-theoretic logic Hintikka (1968a) and Hintikka and Sandu (1997);
default logic Reiter (1980), McDermott and Doyle (1980), Bach (1984), Krauts (1986), Shoman (1988),
Lifshitz (1994), Rescher (2006), Antoniou and Wang (2007); logic programming Kowalski (1979) and
Alferes and Pereia (1996); probabilistic and abductive logics Magnani (2001), Williamson (2002), Gabbay
and Woods (2005, 2006), Aliseda (2006); logics of belief dynamics Alchourrón et al. (1985), Gabbay et al.
(2000, 2001); decision theoretic logic Cooper (2001), Pollock (2006); evolutionary logic Cooper (2001);
domain-specific logic Toulmin (1958); informal logic Walton (1998), Johnson (1996), Johnson and Blair
(2002), Freeman (2005), Finocchiaro (2005), Hitchcock (2006b).
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I first invoked the name of the Gang of Eighteen at the ISSA conference of 1990. See my “Who cares
about the fallacies?” in van Eemeren et al. (1992) and reprinted with changes as chapter 1 of Woods
(2004). Compare this list with another recently completed one: slippery slope, appeal to precedent, many
questions, vagueness, begging the question, ad hominem, ad populum, ad verecundiam, ad ignorantiam, ad
baculum, ad misericordiam, ignoratio elenchi, straw man, red herring, equivocation, affirming the
consequent, denying the antecedent, biased statistics, hasty generalization, non-cause, post hoc ergo
propter hoc, consequent, illicit major, illicit minor, secundum quid, two wrongs, gambler’s and false
analogy (Tindale (2007)). Tindale’s is a Gang of Twenty-Eight. Other lists reflect similar differences, but
all retain a substantial common core, for which the Gang of Eighteen is reasonably representative.
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The Gang of Eighteen throws up an interesting question for a theory of error.
Some logicians cleave to the idea that all that need be said about them as reasoning errors
is that they violate a RR-rule. If so, all a logic of the fallacies need disclose is the various
RR-rules that the eighteen respectively violate. On this approach it would be the RR-rules
that wear the trousers, and the ensuing theory would be an indirect or parasitic theory of
error. On the other hand, the fallacies literature is replete with attempts to make more
detailed diagnoses of the conditions that make the fallacies errors, as well as of the
conditions which disguise their erroneousness. To the extent that this is so, fallacy
theories trend towards the thick.
In 1970, the Australian logician Charles Hamblin issued a harsh indictment of
logic’s failure to handle the fallacies in theoretically deep ways.10 He admonished
logicians to repair this omission. It was Hamblin’s contention that “we have no theory of
fallacy at all, in the sense in which we have theories of correct reasoning or inference.”11
Much in the spirit of Mill, Hamblin was calling upon logicians to produce a stand-alone
theory of errors of reasoning. Responses to this challenge include early papers with
Douglas Walton, constituting what came to be known as the Woods-Walton Approach,12
as well as my solo work The Death of Argument: Fallacies in Agent-Based Reasoning
(2004) and more recently Maurice Finocchiaro’s Arguments About Argument (2005). A
rather different, though also influential, reaction to Hamblin are Frans van Eemeren and
Rob Grootendorst’s Argumentation, Communication and Fallacies: a PragmaDialectical Perspective (1992) and Walton’s A Pragmatic Theory of Fallacy (1995). A
good collection of readings is Fallacies: Classical and Contemporary Readings, edited
by Hans Hansen and Robert Pinto (1995). A principal difference between the WoodsWalton Approach and the approach developed by van Eemeren and Grootendorst, and
extended by Walton’s more recent work, concerns the dialectial character of the fallacies.
According to van Eemeren and Grootendorst, fallacies are inherently dialectical
infelicities, that is to say, errors for which it is necessary and sufficient that they offend
against the procedural rules of disputatious argument. On the Woods-Walton Approach,
not only are the fallacies not dialectical as such, but hardly any of the items in the
traditional lists have any kind of dialectical significance. This is also my present view. I
shall return to this point in chapter 3.
From the earliest beginnings of the Woods-Walton Approach, it has been
generally appreciated that virtually none of the candidates that show up on the traditional
lists is inherently fallacious. So conceived of, fallacy-making is a highly contextualized
affair. If F is the form of argument named by a purported fallacy, then according to the
Woods-Walton Approach, there are instantiations of F which aren’t at all fallacious. It
may be, for example, that a good many instances of the ad hominem form of argument
are fallacious, but not all are.13 In an Afterward to The Death of Argument, I entertained a
10
Hamblin (1970).
Hamblin (1970), p. 11. What Hamblin here intends by “theories of correct reasoning or inference” are the
mainstream deductive and inductive logics.
12
See, for example, Woods and Walton (1972, 1974, 1975, 1982), among others. These papers are
collected in Woods and Walton (1989), reissued as Woods and Walton (2007).
13
Consider a case: You and your opponent disagree about his thesis τ. At one point in the argument you
make the ad hominem remark, “But what you’ve just now said contradicts τ!” Suppose this remark is true.
Where, then, is the fallacy? Notwithstanding these non-erroneous exceptions, the Woods11
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more radical possibility. It is, in effect, that a suitably reflective logic of error might lend
its support to one or all of the following three claims, concerning which I will be the first
to say that, on their face, they may strike the informed reader as dismissible out of hand.
My purpose in announcing them here, so early in the proceedings, is not in hopes of their
ready acceptance but rather a willingness to give them some consideration. The burden of
verifying them is distributed throughout the book  introduce then now in the spirit of
cards on the table.
Proposition 1.3a
THE RARITY THESIS:
The traditional fallacies are rarely committed.
Proposition 1.3b
THE NEGATIVE THESIS
This being so, they aren’t fallacies.
Proposition 1.3c
THE BENIGNITY THESIS
To a non-trivial extentt, the traditional fallacies are cognitively benign strategies
of reasoning.
One of the questions triggered by Hamblin’s attack on the sorry state of the fallacies
programme in logic is why its state is so sorry. Why, we might well ask, is fallacy theory
so difficult? Is it that logicians, who are smart enough for, say, inaccessible cardinals and
Bayesian nets, aren’t smart enough for fallacies? Is it that fallacies are just too hard to be
cracked even by very smart crackers? Whatever the details, a mature theory of the
fallacies would have to make good on two points. It would have to diagnose the fallacymaking features of the traditional list; and it would have to explain their appeal. The
successful discharge of this pair of tasks would go a long way towards answering the
question of why theoretical accord is so elusive an objective for fallacy theorists.
There is another way of answering this question, an altogether different way.
Suppose that Proposition 1.3a is true and that, this being so, Proposition 1.3b is also true.
This would show that
Proposition 1.3d
CONCEPT-LIST MISALIGNMENT: The items on the traditional list are not in
the extension of the traditional concept of fallacy.
If that were so, it would yield a robust explanation of why it is so difficult to provide a
deep diagnoses of the wrong-making features of those very types of reasoning. If they
aren’t fallacies in the first place, then they won’t exhibit features in virtue of which they
are. Equally, if Proposition 1.3c were true, we would have a good explanation of the
difficulties associated with the task of shedding real light on the good-appearing features
Walton Approach tended to favour the idea that arguments having the forms of the eighteen are fallacies by
default, that is, are guilty until proved innocent. It was not until 1995 that I myself saw my way to
exempting the ad baculum from any such presumption. See Woods (1995). I’ve been a slow learner.
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of these types of bad reasoning. For, again, if they aren’t types of bad reasoning, how
surprising could it be that they don’t appear to be bad? This triggers another task for a
thick theory of errors of reasoning. It should try to explain why it is that fallacy theorists
should have been attracted to the view that, contrary to Proposition 1.3d, the eighteen are
indeed traditional fallacies. In other words, a thick theory of error should try to explain
what explains that error.
To the best of my knowledge, Maurice Finocchiaro is the first logician to have
floated a rarity thesis, in a way that also suggests a version of the negative thesis.14
Finnocchiaro invites us to examine the documentary record – for example, the writings of
Galileo  which are replete with efforts to arrive at the truth of things by reasoning about
them. Where, Finocchiaro asks, do we find the eighteen? The question is rhetorical; it
answers itself. Galileo made errors of reasoning; but he didn’t make those errors.15 That
being so, then in addition to the claim that we don’t often see them in practice,
Finocchiaro is drawn to the view that the eighteen aren’t of sufficient importance to
justify serious theoretical attention. In a way, this gives a different answer to the question
“Why is fallacy theory so difficult?” It is that logicians have had unrealistic expectations
for it. As will become clear as we proceed, I agree with Finocchiaro regarding the
scarcity of the eighteen, but I part company from him on the score of their importance.16
It is an understatement that Propositions 1.3a  1.3d have a certain audacity. They
stir things up if true, and are not ideas to be lightly bandied about. As I say, I stir them
now, not in the expectation that they will win supporters on a first hearing, but rather
more in the spirit of early warning of what lies ahead. I take it as given that a logic of
reasoning badly will go nowhere fast unless it makes substantive progress with the
fallacies, unless, that is, it provides a satisfactorily thick response to Hamblin’s challenge.
The concept-list misalignment thesis says, in effect, that in their traditional approaches
logicians have fundamentally misconceived the fallacies project. This forms a large part
of my response to Hamblin. Better to give early notice of it, than to allow it to creep up
on us unawares.
Rescher rightly observes that the “most extensive and longstanding discussion of
error in philosophy has revolved around fallacies”.17 But he goes too far in adding that
this discussion “has flourished since the days of Aristotle’s Sophistical Refutations”.18 In
Fallacies, Hamblin’s complaint was that fallacy theory had not flourished, that it had
become the sick man of logic.19 Hamblin was right.
Historically fallacy theory has been the province of logicians, just as in that same
tradition logic is the province of philosophers. There is reason to think that these
customary alliances are at risk of petering out. There is these days easily as much cuttingedge logic done outside departments of philosophy as within them, and psychologists, to
whom mainline logicians pay little heed, have been investing in fallacies research in
14
Finocchiaro (1981/2005).
Finocchiaro (1980).
16
Although Finocchiaro (personal correspondence) advises me that he is presently more open than he used
to be to this view.
17
Rescher (2007), p. 15.
18
Rescher (1970), p. 16; emphasis added. Could it be that by a “flourishing” discussion that there has been
rather a lot of it? If so, I withdraw my demurral.
19
Considered as a contribution to fallacy theory, On Sophistical Refutations is a noble mess. See here
Woods (2008).
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rather impressive ways. Not nearly enough logicians are aware of, or would care about
were it otherwise, the considerable investment by psychologists in fallacious reasoning.20
It is wide-ranging work, at times overlapping with AI, statistics and machine learning. It
includes “predictive modeling theory, multiple regression formulae, neural networks,
naïve Bayes classifiers, Markov Chains, Monte Carlo algorithms, decision tree models
and support vector machines, [as well as] the well-known heuristics and biases program
.”21 The psychologists’ list is also quite substantial, although with not much in
common with the fallacies that have traditionally occupied logicians. Prominent are the
base-rate neglect fallacy, the conjunction fallacy, covariation illusions, the interview
effect, hindsight bias, the prosecutor’s fallacy, the Lake Wobegon Effect, and the
overconfidence bias.22 Still, some psychologists do direct their attention to the logician’s
traditional fallacies, particularly to the conditions affecting people’s ability to recognize
them. The psychological approaches roughly subdivide into those that retain the
traditional idea that the fallacies really are fallacious (for example Neuman (2003),
Neuman, Weinstock and Glasser (2006), Neuman and Weitzman (2003), Ricco (2003),
and those that hold that whether or not an argument in the form of a traditional fallacy is
actually fallacious will vary with contextual and, in some approaches, Bayesian factors.23
These psychological researches greatly enlarge the fallacies component of any
logic of error that aims to meet Hamblin’s challenge. In some ways this is unfortunate.
Not only does it present the logician of error with a heavier than usual workload, not only
does it require him to work in areas that traditionally lie beyond his own theoretical
reach, but it also makes for a significantly more complex theory than anything logicians
have yet to produce. It would be folly to think that this book could pronounce definitively
on the psychological literature. But there are aspects of these contributions that are of
importance for my developing account, and I shall make some effort to give them their
due. A point of interest is that psychologists do a better job overall in locating their work
on the rationality of human performance within a network of assumptions, if not always a
worked out theory, of the psychological make-up of reasoning agents. This is an
advantage (and another complication) that a logic of error should be ready to follow up
on. See Proposition 1.7 to come.
20
An important exception is L.J. Cohen, who takes on the heuristics and bias crowd in psychology,
especially as relates to the conjunction fallacy. See, for example, Cohen, (1981, 1982). See also Gigerenzer
and Selten (2001).
21
Bishop and Trout (2005), p. 12.
22
For a (small) sample of the psychological literature on error see Griliches (1974), Edwards (1968),
Wason (1966), Kahneman et al, (1982), Kahneman and Tversky (1996), Cheng and Van Ness (1999),
Kahneman and (Tversky 2000), Nisbett and Ross (1980), Gigerenzer and Selten (2001), Gigerenzer (2005),
Mellers, Hertwig and Kahneman (2001), Dunn (2004), Fillenbaum (1977), Sweetser (1990). Psychological
studies to date have concentrated on deductive and probabilistic and inductive reasoning, with somewhat
less attention given to decisional and causal reasoning and very little to argument as such. There is no
simple paradigm at present; in fact, there are at least four main approaches that are currently in contention.
These are the mental models account e.g., Johnson-Laird and Byrne (1991), mental logics e.g., Rips,
(1994), rational analysis and information gain e.g., Chater and Oaksford (1999), Oaksford, Chater,
Grainger and Larking (1997), domain specific reasoning schemas e.g., Evans and Over (1996), and dualprocess models, e.g. Evans (2007). Nisbett and Ross (1980) calls upon philosophers to involve themselves
in sorting out normative issues that arise in psychology (pp. 13-14). Bishop and Trout (2005) is an
interesting recent response to this invitation.
23
See for example Rips (2002), Oaksford and Hahn (2004), Hahn and Oaksford (2006a, 2006b, 2007a,
2007b), Hahn, Oaksford and Bayindir (2005).
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1.4 Mishandling error
An especially egregious favouritism is the one lavished by logicians upon
deduction:
There was a long tradition in philosophy according to which good reasoning had
to be deductively valid. However, that tradition began to be questioned in the
1960s, and is now thoroughly discredited (Pollock (2008), p. 451).
One of my tasks in this book is to show that logic’s traditional contributions to a theory
of error are unsatisfactory, and to explain why this is so. I will say that logic’s
involvement with error is compromised by unjustified assumptions about or indifference
to reasoning as it is actually practised by real-life individual agents, and that this has
produced some regrettable setbacks and omissions. Not only has logic not done a good
job in analyzing fallacious reasoning, but notwithstanding good accounts of validity and
inductive strength, these are rarely standards that bind the transactions of the ordinary
reasoner on the ground. Accordingly, as suggested above, let’s make it official that
Proposition 1.4a
MISHANDLING ERROR: In their concentration on the properties of invalidity
and inductive weakness, logicians distort the concept of errors of reasoning and
significantly underdetermine what is required for a realistic treatment of the
fallacies.
Corollary: The same fault is widely present in the psychological literature on
norms of reasoning.
Proposition 1.4a is offered in the spirit of some emphatic words from Marvin Minsky:
There have been serious attempts as far back as Aristotle, to represent common
sense reasoning by a “logistic” system . No one has been able successfully to
confront such a system with a realistically large set of propositions. I think that
such attempts will continue to fail, because of the character of logistic in general
rather than from defects of particular formalisms (Minsky (1975)).
It is one thing to lodge a complaint and another to show that there is something to
it. Its full defence is the complete book. But for the present, let us briefly consider two
cases, beginning with validity. Suppose it were a requirement that no argument is good
unless its premisses deductively implied its conclusion. If this were so, any reason for the
negation of that conclusion would be inconsistent with the original premisses. Another
way of saying the same thing is that, on the present assumption, nothing consistent with
the original premisses can be a reason for anything inconsistent with the original
conclusion. If we now equate “being a reason for” with “deductively implies”, then there
can be no reason against a proposition for which there exists a reason except where the
reasons are themselves inconsistent with one another. It is a hefty exclusion. It suppresses
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large ranges of cases in which there are reasons, different but pairwise compatible
reasons, for and against a proposition. In a case in which it is known that a murder could
only have been committed by a single person, it completely rules it out that there is
evidence that Harry did it and different but compatible evidence that Sarah did it. It is a
clearly excessive unrealistic constraint on being a reason for.
Apart from that, the validity requirement makes all conflicting arguments about a
common subject reciprocally question-begging. Suppose that the disputed proposition is
α. Harry’s argument is β1  βn, α. Sarah’s is γ1, , γ, α. If, by the present standard,
Harry’s argument is good, its premisses beg the question against Sarah’s conclusion. If
instead Sarah’s argument were good, its premisses would beg the question against
Harry’s conclusion. But, as logicians traditionally have supposed, question-begging
arguments are bad arguments. So any valid argument for a conclusion that is in dispute is
a bad argument.
Suppose, even so, that validity were the standard you preferred to persist with. It’s
a preference that would cost you. It will cost you the value of right reasoning. If good
reasoning is always deductively good reasoning, then, as a matter of fact, most of the
reasoning that is actually done is bad reasoning. But the reasoning that is actually done
plays an undeniable role in our doing rather well overall. We survive, we prosper, and
occasionally we build great civilizations. These would be impossible in the absence of
reasoning as it actually occurs. If it is for the most part bad reasoning, then
Proposition 1.4b
THE DEVALUATION OF GOOD REASONING: If it were true that reasoning
is bad when it is not deductively good, then in the actual circumstances of life,
reasoning well is a matter of little value for beings like us.
Consider now the inductive strength requirement. A friend cuts open a lemon and
squeezes its juice into a glass. “Try it”, he says. You have had nicely sweetened
lemonade before, and you used to be partial to lemon pie. So you give the raw juice a try.
It is awful, much too bitter for your taste. This is your first exposure to raw lemon juice
and you resolve to make it your last. You ground this resolve in the belief that lemon
juice is too bitter for you, that is, that all drinks of raw lemon juice would be awful for
you in the way that this one has been. By traditional logical lights, the belief that grounds
your resolve is one that your one-shot citrussy encounter failed to justify. Your belief was
a generalization from a single case. It commited the fallacy of hasty generalization. It was
bad inductive reasoning.
The lemon juice case raises two different questions. One is whether it is
reasonable for you to avoid raw lemon juice in future. The other is whether it is
reasonable of you to generalize its awfulness (for you). I am concerned here only with the
second question. Suppose that we were to side with the tradition. Then the belief that you
derived from that sample is a product of bad reasoning. This, in turn, gives us two options
to consider. One is that, given the badness of the reasoning that generated it, your belief
should have been resisted, that you should not have yielded to the belief until after dutiful
implementation of Mill’s Methods or the theorems of confirmation theory, or some such
10
thing.24 The other is that, bad reasoning or not, what counts here that the belief is true,
and it redounded to your cognitive advantage not to have resisted it. The first option
appropriately generalized makes us inductive misfits on a very large scale, and renders
spurious much of what we take ourselves as having learned from experience. The other
restores all that was lost in the first option, and downgrades the imperatives of inductive
strength in the domain of ampliative knowledge. Here, too, there are costs and benefits.
The cost of the first is the devaluation of ampliative knowledge. The cost of the second is
the devaluation of the technician’s notion of inductive strength. For anyone pledged to
the value of knowledge thesis, the course is clear. So, again, let’s make it official:
Corollary: If it were true that reasoning is bad when it is not inductively strong,
then in the actual circumstances of life, reasoning well is a matter of little value
for beings like us.
1.5 Bayesian interlude
Our present example is but one of a large class of cases in which an agent is
trying to reason his way to a conclusion on the basis of such data as he is able to muster
for it. Traditionally-minded logicians see this as a case in which the agent is seeking to
build an evidence-based justification for the conclusion in question. Since justification is
a matter of degree, let us say (for now) that degrees could be represented by numbers 
say the real numbers in the unit interval. In the probability calculus, probabilities are also
numbers in this interval. In what is by a large margin the dominant way of analyzing
degrees of justification, the way of Bayesian epistemology,25 degrees of justification are
modelled as probabilities and are subjected to the rigours of the probability calculus.
Suppose, then, that there is some proposition β whose justification you seek by way of
supporting evidence. Suppose that your evidence-searches are going rather well. Not only
does β1 lend evidentiary support to α, but so do β2, , βn. It is clear intuitively that the
greater the amount of aggregated positive evidence, the more secure you can be in the
justification of α. In fact, however, the Bayesian apparatus reverses this expectation.
Thanks to the conjunction axiom of the probability calculus, increases in the evidence for
a proposition lessen its justification. That is to say, the more evidence there is for a
proposition α, the less it is justified to accept α on that evidence.
There is a standard response to this difficulty. It is to rule that the conjunction
principle is not probabilistically valid, that is, does not transmit to a conclusion at least
the same degree of justification as provided by its premisses (Kyburg (1970)). This raises
I simplify. In one sense, beliefs are not the sorts of things to be turned off and on at will. In another – the
acceptance sense – some discretion is permitted, but upon reflection not as much as one might think. Let α
be a proposition you believe to be false and yet which you accept for cognitively virtuous reasons. For
example, let α be the proposition that populations are infinitely large. And suppose your reasoning for
accepting it is its indispensability for the correct account of how natural selection goes in actual
populations. We might be right to think that α nevertheless is something whose acceptance could be
rejected, but not in any very practical sense by serious-minded population biologists. For more on the
cognitive virtue of accepting propositions known to be false, see Woods and Rosales, (2010).
25
So named in recognition of Bayes’ theorem, which is the probability calculus’s main theorem about the
impact of new evidence on an agent’s current beliefs. A recent and accessible primer is Bovens and
Hartmann (2003).
24
11
the obvious question. Which rules of, say, the propositional calculus are probabilistically
valid? The answer is that none is. It is true that any deductive argument proceeding from
a single premiss has a conclusion at least as justified as its premiss.26 But in every other
case, in which the premisses are essentially (i.e., non-redundantly) multiple, probabilistic
validity is lost. This is not to say that deductive justification is never possible, at least in
principle. Let ⌐β1, , βn ⊦ α¬ be any such rule. Then whether to ascribe a degree of
justification to β which is at least the equal of the degree of justification of the βi, depends
on how the probability of the conclusion in the fact of those premisses actually computes.
It bears on this that the computations in question are not determined by the logical form
of premiss and conclusion. The probability of ⌐α  β¬ is typically a material fact about α
and β. This means that the required computations must proceed instantiation-byinstantiation, depending on the particular contents of the propositions in question. But
this is an intractable problem, requiring much too much computation for the human
system to bear. Harman (1973) furnishes a striking example of the difficulty. Suppose
your belief-set contains 300 beliefs, and that your task is to compute the probabilities of
individual beliefs conditional on belief-conjunctions. It is a hefty number, 2300, several
orders greater than the number of elementary particles in the universe. Far too many to
compute; and far too many to store as (innately secured) primitive probabilities.
Premissed-based justification of beliefs is impossible. These and other difficulties
afflicting Bayesian epistemology is a principal motivation for theories of defeasible
reasoning. We follow this up in chapter 6.
A further, and related part of our criticism of logic’s traditional approach to error
is its failure to give due weight to the question of what a theory of error should look like.
The mantra, “an error of reasoning is anything that violates a rule of logic”, leaves it
open that, as logic has been dominantly conceived of since 1879, a rule of logic has
nothing directly to do with the rightness or wrongness of reasoning. We will shortly say
why we take this to be so. If this proves to be right, we will need to build a theory of error
from a different starting point, from some minimally adequate understanding of what
reasoning is for, how it is transacted, and what assets it calls into play. This will be our
orientation here. We take it, knowing that when all is said and done it may be more
trouble than it’s worth. When they talk about invalidity and inductive strength, logicians
are in a class by themselves. They know what they are talking about, and they are better
at it than anyone else. In taking a non-traditional approach to the logic of error we lose
the second advantage outright, concomitantly compromising the first.
1.6 The Koch Principle
Edward Koch was mayor of New York City from 1977 to 1989. New York is an
unruly place at the best of times, and those were not the best of times. Koch was famous
for his walk-abouts. He and an aide would pop up unannounced in every nook and cranny
of the city – bars, ball games, union picnics, subways, street corners, parks – and would
mingle with the people they met there. “Hi, I’m Ed Koch, your mayor! How am I doing?”
Nearly always, Koch would get an earful from his fellow inhabitants, and often he heard
26
Which raises the question of whether such justifications are question-begging. For more on begging the
question, see chapter 10.
12
what he might have wished not to hear. Even so, what he did hear was invaluable to him.
Koch realized that being a good mayor is virtually impossible in the absence of serious
reflection on what it takes to be a good mayor. There is an obvious parallel to the work of
theory-building. Doing it well requires consideration of what it takes to do it well. Koch’s
walk-abouts was a profitable operationalization of this maxim. The feedback he received
gave him fruitful occasion to reflect on what it takes to be a good mayor. In writing this
book, I have tried to follow the Koch Principle. In Kochean moments, I have sought at
least some basic understanding of what it would take to produce a thick logic of errors of
reasoning. This is a methodological question. There are also first-order substantive
questions. This is a book heavy on the substantive side, but I want to leave room for
methodological reflection. I am not less interested in knowing how to go about producing
a logic of error than in knowing what error actually is.
1.7 Naturalizing logic
The defence of Proposition 1.4, the mishandling error thesis, will draw upon the
resources of a naturalized logic. Naturalized logic stands to logic as naturalized
epistemology stands to epistemology. In each case, there is a principled openness to
empirical factors, especially the lawlike disclosures of the relevant branches of natural
science. So a naturalized logic is an empirically sensitive logic. Naturalized logic
currently occupies a still more distant suburb than the others we have mentioned. One of
my hopes here is to move it closer to the centre. Since naturalized logic is a part of the
natural science of human reasoning, it is an approach to reasoning that pays attention to
what people are like, to how they are put together and what they get up to when they
reason.
The naturalization of logic is far from a new idea, having been floated explicitly
in Quine’s “Two Dogmas of Empiricism” and prefigured in Dewey’s experimental
logic.27 It is also present in the early writings of Toulmin:
Logic  may have to become less an apriori subject than it has recently been .
Not only will logic have to become more empirical; it will inevitably tend to be
more historical. (Toulmin (1958), p. 257).
Putnam also advances the idea28, as does Finocchiaro under the heading “empirical logic”
and Scriven under the heading of “probative logic”.29 Finochiaro writes (rhetorically):
[I]s logic an abstract science that studies entailment, truth functions, the calculus
of propositions, predicates, relations, identity, etc.; or is it a social science that
studies the mental activities of reasoning and argument? If the former is the case,
how does logic relate to mathematics? Is it just a branch of mathematics? Or does
it provide the foundations of mathematics? If logic is a social science, how does it
relate to experimental cognitive psychology? In any case, aside from the issue of
the meaning to be attached to “logic”, there is a thriving enterprise called formal
27
Quine (1951), Dewey (1916).
Putnam (1968).
29
Scriven (1987, 2009).
28
13
or symbolic logic, and there exists an important human activity consisting of
reasoning and argumentation; so we may ask, what is or ought to be the
relationship between formal or symbolic logic and reasoning or argument?
(Finocchiaro (2005), pp. 6-7).
Scriven is similarly minded:
I take the term ‘probative’ from jurisprudence, where it means ‘having the quality
or function of proving or demonstrating’ a conclusion  . I use it by contrast with
two other established processes for doing the same task, notably deduction, and
induction in a special sense of the latter. The special sense is intended to preclude
using ‘inductive’ to cover all non-deductive reasoning, and refers to the type of
induction for which we have some kind of logical analysis, namely statistical
inference from samples to populations. I make no claim about the relation of
‘probative’ to several other terms that have been coined to refer to legitimate
inference types other than deduction . These include logics called defeasible,
non-monotonic, default, autoepistemic, paraconsistent and relevant (hope springs
eternal!): reasoning called plausible, prima facie, presumptive, , conductive ,
and of course Toulmin’s rebuttable, Perelman’s argumentative, and Peirce’s
abductive reasoning . But I do think that they were all coined from the same
sense of need to identify a category of legitimate inference other than the classic
duo. (Scriven, 2009; p. 2 of pre-publication text.)30
Empirically sensitive logic exhibits important and interesting differences from its
mathematical cousin. Mathematical logic defines its properties on linguistic structures or
by way of relations between linguistic structures and set theoretic abstracta. Championed
by its four principal branches – set theory, model theory, proof theory and recursion
theory – mathematical logic has scaled the upper reaches of intellectual attainment. It has
made important gains in both the foundations and methodology of mathematics. The
foundational contribution was largely of philosophical interest. Frege and Russell sought
a basis for logicism, for the reduction of mathematics to logic. Others, Boole among
them, favoured the same idea in reverse – the reduction of logic to mathematics. The
methodological contribution was also of philosophical significance, but it threw its net
more widely, fostering the idea that the rigour of expression and standards of proof
exemplified by mathematical logic might also serve as a methodological guide for other
branches of learning.
The success of mathematical logic is tied in no small way to its freedom from the
drag of context, and from the semantic distractions of sentences that brim with
propositional content. In the new paradigm, sentences don’t say anything; and they are
treated without regard to when they were uttered, by whom or in response to what. The
nub of this observation is that there are no people in mainstream mathematical logic. A
good many logicians think that the discovery of people-free, context-independent,
symbolic languages was the linch-pin of the success of the mathematical turn. Whence
the iconic importance of Frege’s notational breakthrough in the Begriffsschrift of 1879
30
See again footnote 8.
14
and of the subsequent improvements wrought by Peano.31 It is an achievement that
prompted Quine to remark in the preface to Methods of Logic that logic is an ancient
discipline, but since 1879 it has been a great one.32
In Aristotle’s hands, the logic of syllogisms was also the study of linguistic
structures.33 But, as the Topics and On Sophistical Refutations make clear, the theory of
syllogisms was designed to be the theoretical heart of a wholly general theory of real-life
interpersonal argument. The theory of argument is a larger logic, one that abounds in
people, the real-life purveyors of challenge and defence, of utterance and counterutterance, and it is a logic heavy with social context. It is an empirically sensitive logic.
This distinction can be marked by saying that the logic of syllogisms is logic in
“in the narrow sense” and that the more general theory of argument, of which syllogisms
are the theoretical core, is logic “in the broad sense”. I will say something further about
Aristotle’s syllogisms in chapter 3. For now let us simply observe that we ourselves could
apply a similar distinction.34 Mainstream mathematical logic is logic in the narrow sense
and the logic of how real people actually reason is logic in the broad sense. Everyone,
Aristotle included, appreciates that next to logic in the narrow sense, logic in the broad
sense is messy.
There are plenty of systems of logic, calling themselves in one way or another
“agent-centred” and “goal-oriented”, in which representations of agents appear; that is,
symbols for agents are a dedicated part of syntax. But theories with terms for agents are
not agent-centred in the sense that I intend, unless they take into account the resources an
agent has at his (or its) disposal for the discharge of that class of tasks – chiefly beliefrevisional and decision theoretic – in which logic has historically shown an interest.
Related considerations are those affecting the cognitive constitution of reasoning agents,
what they are like as knowers. Most logics take up the task of saying what reasoning is
like without bothering to say what reasoners are like. The common assumption appears
to be that in getting the reasoning part right the reasoner part drops into place virtually
automatically. For if good reasoning is reasoning in such-and-such ways, what could a
good reasoner be if not a being (or device) that reasons in those ways? Against this, logic
empiricized urges the converse dependency. Without independent consideration of what
reasoners are like, an account of right reasoning is one that seeks leverage without a
fulcrum. Accordingly,
Proposition 1.7
31
Frege (1879/1967), Peano (1908).
Quine (1982).
33
Aristotle defined a syllogism as a sequence of propositions whose terminal member (the conclusion) is
necessitated by the others (the premisses), whose conclusion doesn’t repeat a premiss and whose premisses
are free of redundancy (Soph. Ref. 165a 1-3 ). So the property of syllogisity, as we might call it, is
instantiated by valid noncircular arguments, and is lost if any premiss is removed or a new one added. The
premiss-irredundancy condition makes for a relevant and nonmotonic logic. If, as many commentators
suppose (see here Shoesmith and Smiley (1978)), there is a further condition that bans multiple
conclusions, then Aristotle’s is also an intuitionistic logic.
34
A distinction between arguments in the narrow sense and arguments in the broad sense is also present in
Ralph Johnson’s distinction between the “illiative core” of an argument and its “dialectical tier. See
Johnson (2000).
32
15
REASONERS: An account of reasoning requires independent engagement with
an account of reasoners.35
1.8 Naturalized logic as practical
An empirically sensitive naturalized logic is a logic centred on the practical. It is a
logic that takes “the practical turn”. There are different interpretations of the word
“practical”, both in its application to reasoning and to the logics that study it. Among
philosophers a practical logic is a logic of decision and action.36 Others identify a goaloriented practical agent as one who transacts his reasoning with comparatively scant
cognitive resources and subject to comparatively modest performance standards, and
correspondingly a practical logic as the study of the reasoning of practical agents in this
sense.37 On a still further interpretation, a practical logic is a generalization of C.W.
Morris’ notion of the pragmatics of language-use to that of the pragmatics of agent-based
reasoning.38 This, too, is not a new idea. Aristotle’s logic of refutation is in this sense a
practical logic. Closer to home, Richard Montague’s treatment of natural languages
embeds a nicely developed mathematical pragmatics.39 Similarly, when Jaakko Hintikka
admitted agents into the epistemic adaptation of the modal system S4, he pragmaticized
his semantics. He included among the system’s logical truths sentences whose negations
would be self-defeating for an agent to utter, never mind that they are not sentences that
à la Tarski, have a model in every interpretation.
There is yet another sense of the practical influencing our reflections. It provides
that practical reasoning is reasoning about practical matters and that practical matters are
matters pertaining to “the necessaries of life”, or what the French call ce qu’il faut pour
vivre. Although giving a precise meaning to the necessaries of life is no easy task, the
main idea is readily exemplified. For some people, the question of whether sets are more
35
Some slight headway on this question has been made by computer scientists. Wooldridge and Jennings
(1995) distinguish between a weak sense of agency, which is generally accepted in the computer science
community, and a stronger sense that has attracted less of a consensus. According to the weak conception
(p. 116), agents are autonomous; they exercise control of their inner states and actions. They are social;
they interact linguistically; they are reactive; they read their environment and respond to changes in it.
They are proactive; they initiate goal-directed actions. The stronger conception provides (p. 117) that
agents are mobile; they move about physical and/or electronic networks. They are honest; they don’t
knowingly communicate misinformation. They are cooperative; agents will do what is asked, and will
avoid having conflicting goals. Finally, agents are rational; they act to achieve their goals in line with their
beliefs. The Wooldridge-Jennings list is indifferent to the distinction between institutional and individual
(concerning which, see below). As they themselves acknowledge, some elements of their characterization
are more plausible than others. There is something right about the autonomy criterion, but, as we shall see
in due course, it requires qualification. Similarly, we might better think of honesty and cooperation as
default features of agency, rather than as necessary conditions of it. Perhaps this most interesting item on
their list is rationality, which is nothing more or less than using one’s “reason” in trying to achieve one’s
goals.
36
See, for example, Millgram (2008), p. 732: “ ‘Practical reasoning’ is philosopherspeak for figuring out
what to do, and is contrasted with theoretical reasoning, that is, figuring out what to believe, or what the
facts are.” See also Pollock (2006).
37
Gabbay and Woods (2003, 2005), Bruza et al. (2006), d’Avila Garcez et al. (2007) and Magnani
(forthcoming).
38
Morris (1971).
39
Montague (1974).
16
NBG-like than ZF-like is of considerable theoretical interest, but no one seriously
imagines that it has much bearing on the necessaries of life. On the other hand, whether
this fungus is edible is unmistakably – in Peirce’s words  a Vital Affair. Practicality in
the present sense induces a loose rank-ordering of an agent’s interests. Roughly speaking,
practical matters take precedence. Whatever else one does in life, necessaries are to be
dealt with first, and the others, as may be, come next, depending on interest and
circumstance.
Logic naturalized has a practical orientation in all four senses of the word.40 It is
concerned with the connection between thinking and doing. It ties its accounts of
reasoning to the resources available to the reasoning agent on the ground. It analyzes
reasoning with due regard for the respective positions of utters and hearers and of the
asymmetry between the first and third person. It takes into account the priority of the
necessaries of life. Important as their differences might be, more important still are the
points on which they agree. At bottom, a theory of practical reasoning in any of these
senses or all is a theory that gives load-bearing significance to the factors of agency and
context. A theory of reasoning gives load-bearing significance to the factors of agency
and context to the extent to which its theorems are relativized to those factors. I intend
that mine be a logic of practical reasoning in this basic sense.
Even so, I would not want to minimize the preference shown by philosophers to
the decision/action sense of the practical. Lest my embrace of a more resource-bound
conception should appear too dismissive, we should pause with it awhile before moving
on. My view is that the link to action is already present in an even more basic way in any
theory in which agents play a load-bearing role. What, after all, is an agent? An agent is a
being (or device) that does things. We may take it, then, that the reasonings undertaken
by an agent A, no matter how abstract or recherché, are those that have some bearing on
A’s conduct as an agent. His reasoning about breakfast may prompt him to reach for the
Rice Krispies rather than the Coco-Puffs; his reasoning about the Continuum Hypothesis
may prompt him to try to undermine the independence proofs or merely to reflect further
on the cost to set theory of making do without it. These we might say are “cognitive
actions”. Perhaps there are those who find profit in denying the status of real actions to
cognitive actions. If so, I don’t count myself among them. If this is right, then all agentbased reasoning is practical in the present sense of practical. Up to a point we may say
what we like; there are better things to do than squabble about nomenclature. Still I think
that there is something to recommend the following way of seeing how cognitive agents
pursue their various tasks:

Some cognitive tasks require comparatively large numbers (quantities, amounts)
of cognitive resources, such as information, time and computational capacity. Call
these, for short, big tasks. Call the others small tasks. Bigness and smallness are
comparative notions, and as used here neither is intended evaluatively. ‘Big’ and
‘small’ are technical terms. To classify a task as big is not to praise it; and to
classify a task as small is not to judge it unimportant or subpar.
40
The practical side of empirical logic is discussed in Gabbay and Woods (2003, 2005) under the heading
of “a practical logic of cognitive systems”, and in Gabbay and Woods (2001) under the title “The new
logic”. Hanna (2006), to much the same effect, speaks of “cognitivist logics”.
17

There is a critical distinction between institutional agents, such as NASA or the
Government of Finland, and individual agents, such as any reasonably wellaccoutered human being. One way of marking this distinction is by saying that it
is typical of institutional agents that they execute big tasks and of individual
agents that they execute small tasks.

Accordingly, it is typical of institutional agents that they transact their cognitive
agendas with greater numbers of cognitive assets than are typically available to
individual agents in the transaction of their cognitive agendas.

In the usage currently in view, an agent is reasoning “practically” when the task
he is launched upon is small.
In the interests of nomenclatural harmony, I emphasize that this particular usage is
stipulative. This raises an obvious question. How likely is it that the stipulated concept
will “find a market”. On the evidence available since the idea’s launch in 200141 the
answer would appear to be “Not much”. In what follows, I’ll adopt the sensible practice
of calling the logic of this book by the name that more readily fits the logic’s features.
Accordingly, “practical” is out and “resource-bound” is in.
Let me say again that it would be a mistake to misjudge the bigness of
institutional cognitive effort. It is true that, given the resources we individuals are able to
draw upon, institutional agents are able to take on tasks whose execution-standards
exceed in rigour (and cost) the typical requirements for an individual’s cognitive success.
In some respects, the abilities of institutional agents certainly exceed those of individuals.
But we must not infer from this that in their respective domains of operation institutional
agents are typically better knowers than individual agents typically are. Consider the
sprawling collective entity picked out by the name “modern science”. Modern science
takes on tasks of impressive – in some cases, mind-boggling – bigness. Getting things
right in science is immensely difficult, and the standards of success are very tough. John
Ionnidis and his colleagues have argued that most published scientific research is wrong.
(See the online journal, Public Library of Science (PloS) Medicine.) Suppose Ionnides is
right. Then it could easily be the case that a typical individual has more success with his
cognitive targets than modern science does with its. In that case, and in that sense, any
old Harry would be a better knower than modern science.42
Suppose now – contrary to the position we are advancing here – that cognitive
success were reserved for outcomes that met the standards of what we are calling big
tasks. If true, then individual knowers would in the performance of the tasks that are
typical of them be the subjects of a massive scepticism, in which beings like us hardly
ever know anything. If that in turn were also right, the consequences for a theory of error
would be as catastrophic as they are direct. For we would have it that with regard to the
41
Woods et al. (2002).
I note in passing the not wholly disreputable idea advanced by some philosophers – notably Richard
Jeffrey – that for all its successes, it is neither the role nor the destiny of science to acquire knowledge.
(Jeffrey, 1992; see also Gillies, 1993). In other words, some philosophers have given up on the value of
knowledge, which is an obvious extension of the value of good reasoning thesis. This seems to me to be
perverse.
42
18
tasks that are typical of him the individual is nearly always in error. My complaint, so far,
against the standard approaches taken by logicians to reasoning is that, if sound, the
catastrophe theory would be the right account of the reasoning of human individuals.
1.9 Agency and patiency
In this book I want to concentrate on individual agents, operating alone or in quite
restricted n-person interactions. This is a considerable constraint, which we impose in the
interests of space. There is a good deal of the story of human error, including nearly all of
what can be said of collective error, that lies on the other side of this self-imposed
limitation. It is a story for another occasion. Given the stipulations at hand, my focus on
individuals is also a focus on resource-bound reasoning, that is, reasoning about small
things, in our present technical sense of that term. So this would be a good juncture, in
the spirit of Proposition 1.7, to make a further point about practical agency. As we saw,
the tie to action is furnished by the concept of agent. An agent is a being who does things.
On the other hand, a patient (in the Latin sense of “patior”) is a being who has things
done to him. Human beings are a duality of the active and the passive; they are both
agents and patients. It is easy to see that a patient’s patiency plays a large role in the
successful discharge of his agency. There are lots of cases – indeed virtually all – in
which an agent cannot do unless he is done to. Human beings are organisms, denizens of
the causal order. Causality bears on the present contrast in two ways. A person’s patiency
is a matter of being on the receiving end of causal inputs. His agency is a matter of
dishing them out. Agency is making things happen. But you can’t make things happen
without the requisite things happening to you.
As we proceed, I will propose a number of theses about the individual agent. I
will concentrate our attention of the individual agent’s reasoning behaviour. I do this
because I want to know whether the traditional approach to fallacies of reasoning can be
made to stand. So I am interested in what can be said of the good and the bad of what
occurs when an agent reasons. One might think that this focus is restricted to the
reasoning behaviour of an individual when he is functioning as an agent, that is, as a
maker of things to happen. Since every agent is also a patient, the word “agent” serves as
an expository convenience. We should not want to rule it out in advance that a successful
account of an individual’s reasoning might have to take heed not only of his wherewithal
for doing but also his wherewithal for being done to. We should not want to ignore the
possibility – indeed the great likelihood – that an agent’s success in producing a bit of
reasoning is conditioned in a non-trivial way by the limits on how he is able to be acted
on.
It is a commonplace among philosophers that beings like us want to know things.
We want to know what to believe and want to know what to do. But, as often will be the
case, the individual agent will find that he has a belief unpreceded by any desire to have
it, and that he is in the throes of an action wholly unanticipated. For at least large
percentages of such cases, it is more plausible to attribute the beliefs and the actions to
patiency than to agency. Thus Harry was “made” to believe, rather than “elected” to
believe; and his action was triggered rather than chosen. It may be that upon reflection
there will be good reasons to restrict the account of reasoning to what happens when a
19
person is functioning as an agent, independently of considerations that impinge on his
patiency. But we should not uncritically assume so going in.
1.10 Psychologism
An empirically sensitive logic of errors of reasoning is agent-centred, goaloriented and resource-bound. Investigators who make room for context and agency are
drawn to a form of what used to be called the Laws of Thought approach43 and, needless
to say, are committed to an element of psychologism in logic. This psychologism is not
inadvertent.44 Since human agents come with psychologies as standard equipment, once
you re-admit them to logic, you admit their psychological make-ups as well, warts and
all. I said earlier that ours is a logic naturalized on the model of the naturalization of
epistemology. The expression “epistemology naturalized” originated with Quine as the
title of a celebrated paper.45 The original subtitle of that paper was “The case for
psychologism.”46 Psychologism is once again an open question in the research
programme of logical theory. Its re-emergence should not be prejudged. Better to wait
and see how, once it is up and running, a psychologically real agent-based logic fares as a
theory of reasoning. In this I cast my lot with John Macnamara: There is no need for a
“border dispute” between logic and psychology.47
There is a kind of psychologism that logic empiricized rightly eschews. It is the
psychologism which asserts the strict reducibility of logic to empirical psychology. The
psychologicism that the logic embraces is of a gentler providence. It is one that asserts
“an essential connection between the logical and the psychological”.48 Our view of the
connection resembles the one attributed to Peirce by Pietarinen, especially as regards its
third disjunct:
[Psychologism] is the view according to which logical laws are subordinate to, or
rest on, or at the very least are not entirely autonomous from, facts of human
psychology.49
All this gives rise to a nice problem. It is a problem thanks to which psychologism is an
open question in logic. It challenges us to determine in what, if not reducibility, the
essential tie between naturalized logic and psychology consists. I don’t have the means of
43
See, for example, Ellis (1979), p.v.
The hostility towards psychologism in logic is ably assessed in Jacquette, (2003). See also Pelletier and
Elio (2005), Gabbay and Woods (2003), chapter 2, Gabbay and Woods (2007b), and a special issue of
Studia Logica, 88 (2008). Brockhaus (1991) and Godden (2005) are historically interesting.
45
Quine (1969).
46
See Willard (1989).
47
Macnamara (1986).
48
Hanna (2006), p. 27.
49
Pietarinen (2006), p. 411. Pietarinen continues the passage as follows: Like Kant, Peirce is leery about
“psychological concepts in logic  What remains indispensable to his work, though, is mental vocabulary,
involving as it did the notions of the interpreting mind exemplified in thought. This was essential to his
broad conception of logic ”. These words suggest that Peirce’s view might have been something like this:
That although a broad logic – i.e., the general theory of signs – requires the use of a mentalistic vocabulary,
no theorem of logic can be secured by the methodology of psychology alone. I myself am not so sure.
Again my advice is to wait and see. But see section 1.10 just below.
44
20
completely closing that question here. Perhaps we will make some progress with it as we
proceed. As a modest first step, we could agree that a good naturalized logic should make
itself aware of what psychologists and neurobiologists have to say about reasoning.
Logicians and psychologists should start reading one another.
The logic I want for the analysis of errors of reasoning has a number of features.
It is a naturalized logic, hence sensitive to empirical considerations. It is a psychologistic
logic; hence some of the empiricist considerations to which it is sensitive will be drawn
from psychology. It is an agent-centred and resource-bound logic. It is a logic that takes
seriously premiss-conclusion relations other than the classical pair of deductive and
statistico-experimental consequence. It is also a logic some of whose target properties
exceed the reach of formal expression.
1.11 Conservative resistance
Once you admit people to logic, you open the door to the tangle of complexity
and the blur of inexactitude. The history of logic is dominated by a dislike of these
orientations. It is hard enough to get logic in the narrow sense right. Getting the logic in
the broad sense right is a challenge of a different order. This has engendered both
resistance and turf-wars. The methods and precepts that have taken logic in the narrow
sense to the heights of theoretical achievement don’t “pay the rent” for logic in the broad
sense. So why bother with logic in the broad sense? Why give up the freedom and the
theoretical power that comes with the suppression of agency and context? (Resistance).
Why recognize it as logic at all? Why do we not see that logic which is, in my sense,
practical stands to logic as fools’ gold stands to gold? Why not send it over to rhetoric or
to psychology? (Territoriality).
Our openness to psychological inputs is not carelessly ventured. It is made in
recognition of the fact that logic is an ancient one and that psychology, in comparison, is
a Johnny-come-lately. It is made in recognition that logic won its spurs before
psychology was ever thought of as a stand-alone science. And it is made in recognition of
the fact that it is only fairly recently that psychology could claim the status of mature and
deep scientific respectability. I myself am of the view that, by any fair measure, the
psychology of perceptual representation is a mature and deep science, and that, although
not yet as well-advanced, the psychology of propositional representation is on track for
the same kind of success. Of course, cognitive psychology has no lessons to pass on to
pure first order quantification theory, which is as near to mined-out as a great theory gets.
Practical logic is another matter. It is built for psychological inputs, and it is nowhere
close to the stages of development that the better parts of psychology have now attained.
So the last thing that our openness to psychology is, is a patronizing one. Let us say, then,
that:
Proposition 1.11
EMPIRICAL SENSITIVITY: A practical logic of errors of reasoning should take
note of, when they are available, the lawlike pronouncements of the relevant parts
of descriptively adequate psychological theories. Any case in which a provision of
the logic deviates from provisions of the psychological account, it should explain
the discrepancy and assess its significance.
21
Neither is our openness to psychology a cringing one. Logicians have something
to learn from psychology, but not everything that psychology pitches is worth catching.
We have followed Hamblin in saying that logicians haven’t done nearly a good enough
job with the fallacies on their lists. The same can sometimes be said for the
psychologists’ treatment of the fallacies on their lists.
However drawn we might be to orthodox admonitions to forgo the psychological
in logic, the present-day determination to circle the wagons around mathematical logic
and refuse admittance to the empirical outliers is compromised by an historical oddity. It
is that mathematical logic has to a quite extraordinary degree retained the ancient
presumption that logic is a theory of reasoning50 – in some tellings it is about reasoning at
its most perfect.51 Since reasoning is what reasoners do, even mathematical logic retains
the pretence of a tie between its disclosures and the conditions under which reasoners
behave when they reason well.
How could logic succeed as a theory of reasoning without somehow taking
reasoners into account? The question is (in the logician’s sense) a complex one. It admits
of two answers. Hardliners, such as Quine and Harman, will say that logic isn’t at all
about reasoning. Others will allow – at least in principle – for some kind of connection
between the rules of logic and the rules of right reasoning. In a variation already
mentioned, a reasoner is a virtual entity implicit in the theory but without formal
recognition there. A reasoner is any device, actual or counterfactual, implementing the
logic’s norms. Other variations are more forthcoming. They show a degree of readiness
to re-admit agents and contexts into their logics as expressly load-bearing objects of
theory. And so it has been, with greater or less vigour, for some forty years and more.
The idea that the laws of logic are not universally applicable principles of
inference is at least as old as modern mathematical logic52, arising in problems noted by
Frege in (1918-1919), discussed without much success by Russell in an Appendix of
Principles of Mathematics (1903), and revived for expressive contexts by Geach (1960,
1965). The problem here is one of embedding. Suppose that α is asserted and ⌐α  β¬ is
asserted. Say that we denote these assertings by the Frege-sign ⊦. Then we have it that ⊦α
and that ⊦(α  β). Suppose we want to say that β is likewise asserted. What allows us to
say so? One might think that modus ponens will deliver the goods here. But it doesn’t. It
doesn’t because ⌐⊦α¬ does not occur as antecedent of the asserted conditional ⌐⊦(α  β)¬;
that is to say, it doesn’t because ⌐⊦α¬ isn’t embedded in ⌐⊦(α  β)¬. The inference may in
some sense be valid, but it is not valid by modus ponens. What is so striking about the
large bulk of agent-admitting logics is the conservatism with which they seek to preserve
Harman and Kulkarni (2007), p. 5, observe that “in the traditional view, a deductive logic is a theory of
reasoning”. (They themselves are non-believers.) See also Hahn and Oaksford (2007), p. 705: “By
extending a normative, probabilistic approach to at least some aspects of argumentation, we hope to show
that such an approach can generalize beyond the narrow confines of deduction and induction as construed
in the psychology of reasoning to the real human activity of which these modes of reasoning are but a small
part.”
51
This is a view also widely shared by psychologists. Evans (2002) calls it the “deduction paradigm”.
(Evans, too, is a non-believer). Syllogisms “have been extensively studied by psychologists in search of
‘the fundamental human deduction mechanism’ ” (Stenning (2002), p. 8).
52
Though implicit in Aristotle. The rules of his perversely named logic of immediate inference are not
universally valid in contexts of argumentative inference.
50
22
the provisions of the mainstream. While they give at least notional recognition of agents
(people), they are unable to shake off their addiction to deduction. Agent-admitting logics
are extensions or adaptations of mathematical logics, fashioned in the spirit of Quine’s
famous maximum of minimum mutilation.53 It provides that in making these
transformations the methodology of mainstream logics be trifled with as little as possible.
Thus agent-admitting logics would still be grammars of semi-interpreted formal
languages, overlain by syntactic theories of proof and set-theoretic theories of truth, each
fashioned with an eye on the metatheoretical plums of consistency, completeness and
soundness. I need a name for such things. Suppose we say that they are mathematically
virtuosic theories.
1.12 A methodological tension
In the province of logic, mathematically virtuosic theories have enormous appeal.
They are studies in technical finesse, and in some cases they provide deep and satisfying
reconstructions of target concepts. In the period since (say) 1879, systems of logic have
proliferated. There are more mathematically virtuosic systems of logic than you can
shake a stick at. As we have said, this has engineered a large and contentious pluralism in
logic. Anyone who has paid any attention to the historical unfolding of these
multiplicities will have no trouble in pointing to cases in which a system’s mathematical
virtuosity trumps its elucidatory power. Here are a pair of cases briefly to consider.
Almost no one believes that there are true contradictions. Nearly everyone thinks
that the very idea of true contradictions is absurd, an insult to reason itself. Yet there are
systems of dialetheic logic – the logic of true contradictions – meeting all the conditions
of mathematical success. They have recursively specifiable languages, formally adequate
proof rules and semantics, and they are sound and complete with respect to their
semantics. Dialetheic logicians publish their work in the best of the mainstream journals
and with leading university presses.54 While they are duds on the score of conceptual
adequacy, they do very well on the score of technical virtuosity. Similarly, the system of
relevant logic known as first-degree entailment (FDE) is sound and complete with respect
to a possible world semantics (the so-called “star” semantics) in which the negation
operator has no intuitive interpretation. FDE recalls the words of Quine about another but
related matter: “They think they are talking about negation, ‘’, ‘not’; but surely the
notation [has] ceased to be recognizable as negation ”. Yet FDE is widely studied and
well-regarded as a technical achievement.55 It is virtually impossible for a logician
acquainted with developments in relevant logic to be unaware of FDE.
“The price [of straying from the mainstream] is not quite prohibitive, but the returns had better be good.”
(Quine, 1970/1986, p. 86). Quine’s maxim has an AI-counterpart in theories of belief revision (Gärdenfors,
1984, 1988, Gärdenfors and Makinson, 1988, Willard and Yuan, 1990, and Nebel, 1991).
54
For example, Graham Priest’s dialetheic system LP is developed in a paper that appeared in the Journal
of Philosophical Logic (Priest, 1979) and his book, Beyond the Limits of Thought (Priest, 1995), is in its
second edition with the Cambridge University Press.
55
It even has its apologists on the conceptual side. See, for example, Restall (1999). It is worth mentioning,
by the way, that notwithstanding the mutilation of negation occasioned by its star semantics, a good many
sensible people are of the view that FDE gives a credible account of entailment. (For one thing, it blocks
the classical proof that a contradiction entails every statement whatever.) If this is right, we see that in order
53
23
Our embrace of an empirically sensitive orientation for logic reflects no disrespect
of logics that owe their success exclusively or mainly to mathematical virtuosity. Neither
does it imply a failure to appreciate that a mathematically virtuosic logic might or might
not also do well on the score of analytical elucidation, depending on which concept is up
for analysis. Error is a case in point. A logic that does well in elucidating (say)
entailment, is far from guaranteed a like success with error. A logic’s potential for good
is a function of what it is wanted for, and what it is good for is a matter of what it is good
at.
1.13 Having and drawing
Like the orthodox mainstream, a principal focus for a resource-bound logic are
properties of premiss-conclusion set-ups, with special attention to consequence-relations.
Aristotle’s logical writings reflect a relevant contrast. Although it is not one he explicitly
identifies, its influence has been felt from his day to ours. This is the distinction between
the consequences a set of beliefs has and the consequences that a reasoner will, may, or
should draw, hence a distinction between consequence-recognition and consequenceselection.56 There is a kindred distinction between the rules that generate the
consequences that a set of beliefs has and the rules that regulate which of them it would
be appropriate to draw. It is natural to ask whether these are the same rules.
The history of logic discloses two answers to this question, Yes and No.
According to the Yes-camp, a good reasoner will draw all consequences of anything he
believes. According to the No-camp (the Frege-Geach-Harman camp), a good reasoner
will draw only a proper subset of consequences of anything he believes, not excluding the
null set. There is, thus, a loose but discernible concurrence between the Yes-camp and the
logical mainstream, and between the No-camp and empirically sensitive logic. For the
Yes-side belief is closed under consequence. For the No-side it is not. The trouble with
the Yes-answer is that the number of consequences that a set of beliefs has is always far
too large for anyone to draw. The Yes-people offset this disadvantage by dressing up
their affirmative answer as a normative ideal, as something for the logically omniscient
reasoner. So understood, the more consequences you can manage to draw from anything
you believe the better your reasoning, never mind that you can’t manage to draw
anything remotely close to them all.
The No-camp displays a greater sense of realism about these things. It is reluctant
to attach the knock of the subpar to what we never do and couldn’t. According to the
Yes-camp we are notably and systematically deficient in our inference-making ways.
According to the No-camp, this is an indictment too heavy for serious countenance.
Beyond that, there is a point which the No-camp can use to its considerable advantage. It
is that when a proposition α is a consequence of something we believe, α is not always a
candidate for drawing. The No-camp subscribes to the view that one always draws only a
proper subset of the consequences of one’s beliefs. What we now see is that the No-view
extends to the empty set of those consequences. For nearly forty years Harman has been
to get its target concepts right (entailment, logical truth, etc.), it is not necessary for a logic to get all its
connectives right (that is, right according to their common meanings in natural speech).
56
Thus Aristotle distinguishes between statements necessitated by premisses and statements syllogistically
implied by them.
24
telling us (rightly) that when α is a consequence of β sometimes the right consequence to
draw is that β is false.57 But not-β is not a consequence of β. The fact that the
consequence which is sometimes right to draw when α is a consequence of β is neither α
nor any other consequence of it leads Harman to reject the traditional idea that a
deductive logic is a theory of reasoning.58
There is something right about this, and something not. What is right is that a
logic that tells us what the consequences are can’t in the general case tell us what
consequences to draw. What is wrong is the idea that the logic of deduction has nothing
to do with deducing. This goes too far. Deducers have a natural interest in not playing
fast and loose with inconsistency, hence with the consequences with which inconsistency
is interdefinable. Logics that are good at inconsistency-spotting have an obvious role to
play. They help reasoners to consider what consequences they should not draw. This was
Mill’s answer to the Pyhrronic objection that all valid deductions are question-begging.
Although Mill didn’t subscribe to the no-thick-theory position about error in general, he
did allow that the logic of deduction could not be a stand-alone theory of reasoning. He
saw it instead as a theory of consistency which elucidates the negative requirement, as he
saw it, that reasoners ought to avoid inconsistency.59
So far we have taken note of two kinds of consequence-mistake. There is the
mistake of drawing consequences that aren’t there, that is, the ancient mistake of non
sequitur. There is the mistaking a drawing a conclusion which, though it is a sequitur, is
not an appropriate consequence to draw, given one’s interests and one’s resources. But
there is a third way of consequence-mistakeness. It is the failure to draw a consequence
that is there and appropriate to draw. It is the mistake of not seeing what follows from
what. This, too, has been a preoccupation of logicians since ancient times. The
instantiation of proof designed to overcome consequence-blindness in mathematical
contexts. And the great dialecticians laid great store on the interrogative techniques by
which such blindspots could be removed. There is a useful lesson in this.
Proposition 1.13
OMISSION: Some errors of reasoning are consequences overlooked.
1.14 Proofs and premiss-searches
Proof searches are but a special class of problem-solving tasks. At a certain level
of abstraction a problem calls for the presentation of considerations that make it go away.
In the case of a proof search, the problem is that there is some proposition β that stands in
57
Harman (1970).
David Israel made much the same point ten years later, which he developed into an attack on the very
idea of a nonmonotonic logic. Expressed in our terms, Israel accused nonmonotic logicians of losing
control of the distinction between consequence-having and consequence-drawing. The latter, he agrees, is
nonmonotonic but not the business of logic. The former, he claims, is monotonic and the business of logic.
(Israel (1980)).
59
The having-drawing distinction is blurred by logistic systems that are (strongly) sound and complete. In
such logics, a sentence  is deducible from a set  of sentences if and only if  follows from or is a
semantic consequence of . If we allow, as such systems clearly do, that a sentence is drawable from  if
and only if it is deducible from it, then likewise the distinction between drawable consequences of  and
consequences of  collapses.
58
25
need of a certain kind of support. The problem, in effect, is a request for properly
supported propositions α1, ,αn that jointly entail β. Finding those propositions is a
premiss-selection task. It is easy to generalize the proof example. It holds for any premiss
search relative to any notion of premiss-conclusion consequence.60
We see, then, that the premiss-conclusion pair engages the reasoner in two quite
different ways, consequence-finding and premiss-finding. Each is a task performed on a
fixed element. In the first instance, the premisses are fixed and the question of the
consequences to be drawn from them is open – that is, is the problem to be solved. In the
second instance, the conclusion is fixed and the question of what it takes to prove it is
open. These are different situations, no doubt, but they are tied together in an essential
way. In each case, the success of the outcome pivots on whether these premisses, whether
at hand or to be found, do in fact bear to the conclusion the requisite consequence
relation.
The obvious question is whether the essential link to the consequence relation is
enough to qualify those tasks – the consequence-determining task and the premissfinding task – as falling properly into the ambit of logic. There is a large body of opinion
to the effect that premiss-finding is a psychological matter, that there is no such thing as
the logic of discovery (Reichenbach, 1938, Popper, 1934). My answer is twofold. First,
since the success of premiss-search is tied to whether the selected premisses stand in the
intended consequence relation, and since consequence relations are a central business for
logic, then logic has a stake in premiss-searches. Second, inasmuch as we have already
psychologized our logic, it should not be foreclosed à priori that, apart from the tie to at
least some features of the premiss-searches are fit subjects for logic. Think again, of the
factors of consistency and inconsistency, relevance and irrelevance61, and the plausible
and the implausible.62 So let us say that.
Proposition 1.14
GENERATION AND RECOGNITION: Subject to the indicated constraints both
premiss-generation and consequence-recognition are proper parts of a
psychologized logic of errors of reasoning.
1.15 Consistency again
Mill’s point was that consistency is a premiss-selection filter. Given the
interdeducibility of consequence and consistency, a logic of consequence-drawing is
directly tied to the logic of this filter. As we saw, in all going systems of mainstream
logic, a quite general constraint on good reasoning is that it engender and maintain the
60
Another example of premiss-searches relative to consequence relations of requisite strength are
refutation-arguments, in which the task of the premiss-searcher is to extract from his opponent concessions
which entail or lend credible support to the contradictory of the opponent’s own thesis. As Aristotle
remarks (Metaphysics, K5, 1062a 2-3), this is not proof in the full sense but proof ad hominem. What
Aristotle means that the premiss-conclusion connection noted by the refuting party reveals that the other
party’s retention of his thesis is inconsistent with his other concessions. Refutations are discussed in
greater detail in chapter 10.
61
Concerning which see Anderson and Belnap (1975, 1995) and Gabbay and Woods (2003). On the
psychological side see also Sperber and Wilson (1986/1995).
62
See for example, Rescher (1976) and Gabbay and Woods (2005), chapter 7.
26
reasoner’s set of beliefs in a condition of deductive consistency. Dialetheists apart, no
one doubts the reasonability, or the applicability, of the consistency constraint for most
comparatively small (or local) subsets of reasoners’ beliefs. But it is out of the question
for them all (i.e., globally), assuming it were possible to identify them all. Here are some
further strong words from Minsky:
I cannot state strongly enough my conviction that the preoccupation with
Consistency, so valuable for Mathematical Logic, has been incredibly destructive
to those working on models of the mind. At the popular level it has produced a
weird conception of the potential capabilities of machines in general. At the
“logical” level it has blocked efforts to represent ordinary knowledge, by
presenting an unreachable image of a corpus of context-free “truths” that can
stand separately by themselves (Minsky (1975)).
Even if we restrict the constraint to elementary truth functional consistency, the
computational load is well beyond the reach of the human individual. It is another
intractable task, embedded in a problem that is at least NP-hard.63
Furthermore, even if consequence-having were consistentist, consequencedrawing can only be paraconsistentist. So let us make it official: The base logic for a
thick theory of error is paraconsistent.64
Proposition 1.15a
PARACONSISTENCY: Consistency-maintenance is not a wholly general
requirement on inputs to the processes of reasoning. Individual reasoners operate
perforce in paraconsistent contexts and are not intrinsically compromised by
them.
Corollary
Consistency maintenance is covered by a more general policy of inconsistency
management.
I bring this chapter to a close with a brief remark about the connection between
inconsistency-detection and error-correction. To do this without taking up more space
than I can presently spare, let us draw attention to some historical cases in which theories
have been discovered to contain inconsistencies. Let us begin with the observation that
the most central objective of a theory of a given subject matter S is to achieve a
knowledge of S, that is, to determine which propositions about S are true, as opposed to
false, and false, as opposed to true. On the classical – or consistentist – approach every
sentence whatever is a consequence of an inconsistency. This, the ex falso quodlibet
“NP” denotes “nondeterministic polynomial time.” Informally, NP is the set of all decision problems for
which an affirmative answer admits of simple proofs of the fact that the answer is indeed affirmative. Such
proofs are verifiable in polynomial time by a deterministic Turing machine. “NP-hard” denotes the class of
problems at least as hard as the hardest problem in NP. The hardest problems in NP are called NP-complete
problems, for which no polynomial time algorithms are known. A relatively untechnical treatment of these
issues can be found in Cherniak (1986).
64
Concerning which see Gabbay and Hunter (1991), Gabbay and Hunter (1993), Jennings and Schotch
(1989), Brown (1999), Woods (2003), and Brown (2007).
63
27
theorem, generates a “catastrophe thesis” for theory construction. Suppose that our Stheory contains an inconsistency. Then, by ex falso quodlibet, every sentence of S is true
and false. The theory is unable to determine of any with respect to its sentences that it is
true as opposed to false. The central objective – indeed the very rationale – of the theory
is dashed. It is impossible for it to achieve a knowledge of S – not a jot of it.
In 1902, Russell communicated to Frege that his (Frege’s) masterwork, Der
Grundgesetze die Arithmetik, contained an inconsistency. By catastrophist reasoning,
there is absolutely nothing to learn about arithmetic or sets or anything else in the
Grundgszetze. Not one sentence of the Grundgesetze is true rather than false or false
rather than true. Two and a half century’s earlier, there was a similar situation in the
arithmetic of small numbers. Calculus, the theory of infinitesmals, embodied a known
inconsistency.65 The inconsistency proved resistant to removal until the introduction in
the 19th century of Wierestrasse’s limits and, in the 20th century, of Robinson’s
hyperreals. Again, the catastrophist line is that nothing about infinitesmals could be
learned from Newton or Leibniz, that for the first 200 years of its existence, there was no
knowledge of anything – still less of infinitesmals, differentiation and integration  to be
got from calculus.
There is no serious student of the history of mathematics who pays these dire
sayings the slightest heed. Working mathematicians took it as given that there was lots to
know about these things, notwithstanding their respective local inconsistencies. What this
tells us is that in their workaday professional responses to locally inconsistent
mathematical theories, mathematicians implicitly reject the catastrophe thesis, and
therefore also reject ex falso. It is not just that these mathematicians weren’t drawing
every consequence possessed by a local inconsistency, but rather that they weren’t
recognizing every sentence as a consequence of it. The moral of these historical cases is
clear. If the mathematicians were reasoning properly in generating the knowledge that
those theories gave them, then
Proposition 1.15b
DROPPING EX FALSO: It is an error of consequence-recognition to suppose
that every sentence is a consequence of a local inconsistency.
Corollary
A paraconsistent logic is needed not only for consequence-drawing but also for
comsequence-having.
Proposition 1.16a tells us, in effect, that inconsistency in reasoning-contexts is not
intrinsically intolerable. Proposition 1.16b goes a goodly part of the way in explaining
why: Inconsistency in reasoning is not catastrophic. Reasoning can flourish even if
tainted by inconsistency.
65
Perhaps I am taking some historical liberties here. Leibniz regarded infinitesmals as mere fictions which
stand in eliminably for very complicated exhaustion proofs and which can be shown to be wholly free of
inconsistency. Newton on the other hand based his calculus on the notions of uniform motion and
acceleration, a foundation that he held to carry no taint of paradox. Berkeley’s position was in effect, that
both these inconsistency-free claims were mistaken, and that the calculus was indeed paradoxical. I do not
have the means at hand to resolve this historical dispute. Accordingly, it suffices for our purposes here to
make do with the inconsistency of set theory.
28