Content uploaded by Michiel Van Lambalgen
Author content
All content in this area was uploaded by Michiel Van Lambalgen on Mar 09, 2016
Content may be subject to copyright.
The logical response to a noisy world
Keith Stenning
Human Communication Research Centre, Edinburgh University
Michiel van Lambalgen
Department of Philosophy, Amsterdam University
January 5, 2012
To appear in: The Psychology of Conditionals edited by
Mike Oaksford. OUP.
1 Introduction
We live in a noisy world and our premises are subject to doubt. There is a thriv-
ing school of opinion in the psychology of reasoning that this means that it is
both desirable, and inevitable for realism’s sake, that theories of human reason-
ing should be framed in probability theory rather than mental process theories in
logical frameworks; a prominent recent example is Oaksford and Chater [15]. A
forceful earlier statement of this position is
. .. it is, in fact, rational by the highest standards to take proper account
of the probability of one’s premises in deductive reasoning . . .
performing inferences from statements treated as absolutely certain is
uncommon in ordinary reasoning. We are mainly interested in what
subjects will infer from statements in ordinary discourse that they may
not believe with certainty and may even have serious doubts about
. . . [26, p. 615 ]
This approach assumes that reasoners come with their interpretations of materials
already fixed. How otherwise could they have any ideas about the probabilities
1
of component propositions? When subjects hear, to take a famous example we
will return to, “If she has an essay, she is in the library” how could they have any
idea about a probability without knowing anything about who ‘she’ might be or
anything else about the situation? As Cummins [6] has shown, there is a strong
correlation between subjects’ willingness to draw for example a modus ponens
inference from such conditionals, and the ease with which they can retrieve ‘ad-
ditional’ and ‘alternative’ conditionals, such as “If the library is shut, she isn’t in
the library”, or “If she has a reference book to read, she’s in the library”. Does
this mean that subjects come to such experiments with an estimate of the prob-
ability of all such propositions which they merely need to retrieve? Or does it
mean that there is a process of discourse interpretation which can explain how
discourses composed of these conditionals give rise to interpretations on the basis
of which subjects are willing to hazard likelihoods? A process of interpretation,
moreover, which is endlessly sensitive to subtelties of the kind of discourse sub-
jects believe they should engage in? Does Cummins’ result suggest, rather than
an archive of probabilities, that subjects estimate likelihoods from the ease of re-
trieving schematic knowledge about alternatives and defeaters? At least in the sit-
uations where they construe the task as interpreting a kind of fiction about typical
students and typical libraries? Our most basic claim is that what subjects cannot
do is to engage in no process of interpretation. We cannot study some ‘central’
process of inference, without studying its ‘front-end’ process of interpretation.
So, in contrast to the probabilists, we believe that the loudness of the world is one
important reason why human reasoning is a two component process [25]. We im-
pose interpretations on discourse before we can reason from those interpretations,
and we generally start by reasoning to interpretations as if their assumptions were
perfectly certain within those interpretations. That is, our guiding question as dis-
course comprehenders is “What would it be like for the speaker’s discourse to be
absolutely true?” Only when we have constructed an intended interpretation, can
we derive any consequences from the interpretation, or decide how it relates to the
world. Needless to say, we believe the twin theories of reasoning to interpreta-
tions and reasoning from them must be couched in logic, though in very different
logical frameworks than have hitherto figured in the psychology of reasoning. Our
aim in this paper is to motivate such a logical account, by showing that probabil-
ity cannot provide the kind of defeasible framework within which interpretations
can be established. Our positive logical account has appeared elsewhere [23, 25];
here we concentrate on the shortcomings of probabilistic analyses of defeasible
reasoning. We start by discussing a number of faulty assumptions about logic that
have hindered its application to actual human reasoning.
2
1.1 Dusting off logic
The probabilistic critique of ‘logicist’ accounts of the psychology of reasoning
trades on several assumptions about logic that may perhaps be reasonably applied
to logic’s applications in some psychological theories (mental logics, mental mod-
els, . . .), but are quite foreign to a modern logical approach to human reasoning
and discourse. First there is an assumption that logic is to be identified with classi-
cal logic whose logical forms can be read directly from natural language sentence
forms. Second, the interpretative component of logic is ignored at the expense of
total concentration on the derivational component. Third, it is assumed that using
probability theory as a basis for rational analysis does not involve using a logic,
and so avoids the criticism levelled at ‘logicist’ cognitive science.
The first assumption, that logic must be identified with classical logic, is unfortu-
nately still rather common. On the formal side, it overlooks the great wealth of
logical systems currently explored. More to the point, it unjustifiably takes classi-
cal logic to be the only logic with normative force [25, Chapters 2, 11]. Once one
drops this assumption, we must assume instead that the process of interpretation
of natural language sentences is a substantial one, and is best viewed as involv-
ing the setting of parameters not only for the syntactic form of natural language
sentences, but for their semantics, and for the concept of validity inherent in the
reasoning task at hand. The end result of parameter setting is to fix on what we
call a logical form [25, Chapters 2].. Here we will concentrate on the contrast
between defeasible logics of interpretation and classical logics of derivation, but
many other logics are often involved. A notable further example is deontic logic
as involved in some selection task variants [22].
The second assumption was that logic is to be identified with its formal, deriva-
tional component. The probabilistic approach identifies ‘logicist’ approaches with
certainty of conclusions given premises and sees the central advantage of proba-
bilistic approaches as their capacity to handle uncertainty. But even traditional
logic always assumed that uncertainty was to be accommodated in its interpre-
tative component. Traditional logic (that is, logic as it was practised and taught
before the era of formalisation) had no formal account of interpretation but, as we
shall see, it shares this property with current probabilistic approaches, and besides
traditional logic had a substantial informal account of the criteria for coherent
interpretation. This problem perhaps arises from identifying logic with its appli-
cations in the foundations of mathematics, a late 19C employment. Traditional
logic was developed as a theory of the discourse of argumentation – a subject
matter as prone to uncertainty as it is possible to find.
On the third assumption, that the relevant properties of logic do not apply to prob-
3
abilistic approaches, one must counter that probability theory simply is a (family
of) logic(s) – probability logics – and as such qualifies as a target of all the crit-
icisms levelled at ‘logicist’ approaches. In fact probability logic is a deductive
theory which vastly extends the range of certainty of inference. For example,
classical probability theory assures us that if the probabilities of Xand Yare
both .5 and are independent, then the probability of X∧Yis .25—exactly. Even
when probability theories only licence inferences of probability intervals rather
than point probabilities, they often give exact bounds on the confidence intervals
of our conclusions.1Everywhere uncertainty retreats before a tidal wave of ex-
actitude. We agree entirely that handling uncertainty is of great importance, but
we feel that this militates strongly against probabilistic frameworks. We agree
entirely that we must give accounts of how content affects reasoning, but for pre-
cisely this reason we propose that a ‘defeasible logic of interpretation’ is required
to explain reasoning over long-term knowledge to an interpretation.
In the next section, we will provide a brief sketch of our own positive logical ap-
proach through its account of the suppression task in the next section [5, 23, 25].
The last reference gives an overview of the reanalysis of several psychology of rea-
soning tasks in this interpretative framework: the selection and suppression tasks,
categorial syllogisms and some non-linguistic executive function tasks. In the
middle section we will examine the Oaksford & Chater probabilistic account of
the suppression task and show, first that they too require an additional initial inter-
pretative component which cannot be couched in probability theory, and second,
that even when this interpretative component is added, the resulting probabilistic
treatment cannot cover all the argument patterns which people readily employ.
2 A logic of discourse interpretation
Whilst traditional logic had a substantial but informal understanding of interpre-
tation it had no formal account of the reasoning involved. Its formal accounts,
such as they were, were of derivations within an interpretation – examples of
which are known and loved by every introductory student. Interpretation figured
mostly in the injunction that it must be held constant throughout an argument –
no equivocation (surreptitious shifting of interpretation) was to be allowed. Since,
with the advent of symbolic logic, teachers were often content to teach the formal
manipulation of the symbols within systems, the process of interpretation was
backgrounded in introductory courses, often to the point of disappearence.
1There are important differences among probabilistic approaches (for example, [18] and [15]
differ significantly), but they share the property of assuming prior interpretation.
4
Since the 1970s, artificial intelligence researchers have provided formal accounts
of defeasible logics of interpretation [13, 20]. Defeasibility, the capacity to with-
draw earlier conclusions on the arrival of new evidence, is clearly a criterial re-
quirement for any process of interpretation, even though it is equally obviously
anathema to derivational processes. Unfortunately, despite their invention by
those interested in computer implementation, the motivating belief that these ‘nat-
ural’ logics must be easier for people to use turned out to be disappointed. These
logics are provably seriously computationally intractable – more so than classical
logic which was already known to be highly intractable. Out of the frying pan into
the fire. This impasse had a baleful effect on the application of defeasible logics
in psychology.
Fortunately, in the eighties and early nineties, the more tractable alternative logics
of interpretation which we will use here became generally available . It is of some
interest that they originated obliquely from the very practical research program of
adapting logic as a computer programming language, issuing in languages such as
PROLOG. Whilst the practitioners saw themselves as using fragments of classical
logic to achieve efficient computations, later logical study of the resulting systems
revealed that they could more perspicuously be conceptualised as computing in
defeasible logics. The particular logic we will introduce here is known as ‘logic
programming with negation as failure’.
Needless to say, the tractability of these new defeasible logics is purchased at
a certain price, in this case, as one should expect, in expressiveness of the lan-
guages concerned. Logic programming [7] restricts conditionals to ones of the
form L1∧. . . ∧Ln→A, where Ais atomic, and the Liare either atoms or
negated atoms.2However, we believe this loss of expressiveness is well worth
the price. The fact that the resulting languages can be used in programming is a
good indication that they are expressive enough to get lots done, and some of the
restrictions (such as that against some syntactic iteration of the implication) also
arguably apply to natural language conditionals. We think this is a good compro-
mise balance between expressiveness and tractability. This is a stark contrast to
the intractability of probability theory which is acknowledged also by those ad-
vocating the probabilistic approach. This acknowledgment forces them back to
a position where probability is a distant competence theory whose relation to the
data we will argue is hard to understand, and whose implementation they have
little to say about. Our defeasible logic can be shown to have a direct neural
implementation and so brings competence and performance dramatically closer
together [23], [25, Chapter 8].
2The restrictions on the antecedent can be liberalised somewhat, but it is not allowed to have
conditionals in the antecedent.
5
These defeasible logics specify valid patterns of reasoning over databases of de-
fault conditional rules. We conceive of these rules as representing long term gen-
eral knowledge of environmental regularities, and of connections between the
agent’s beliefs and his actions.3Since we are concerned with modelling dis-
courses, we conceive of the new statements of the discourse arriving sequentially
and being incorporated into the ‘hearer’s’ working memory (WM) which is con-
ceived of as the activated part of long term memory, rather in the style in which
ACT-R would model this relation (Anderson and Lebiere [2]).
We model conditionals whose surface form is ‘if pthen q’ as default rules, which
are defined as logic programming clauses of the form: p∧ ¬ab →qread as
“If p and nothing is abnormal, then q”.4The abnormality (and possibly other
propositions as well) is governed by closed world reasoning (CWR), which says
that if a proposition is not known to be true, it may be assumed false. Thus, if
there is no information that an abnormality will occur, we may assume that none
will occur, and the above conditional reduces to p→q. CWR is the logic that
one often applies in planning and prediction. There is an unlimited number of
events – abnormalties – which could interfere with our goal to be in London on
August 28, 2007 (ranging from major natural or man-made disasters to strikes, to
waking up too late and missing the plane). The vast majority of these events are
not accounted for in our plan to go to London at said date – that is, they are treated
as if they will not occur. CWR leads to a non-monotonic logic. New information
may become available which forces us to consider an event as an abnormality
which was not previously classified as such.
With this much logical background we can introduce the main reasoning task to
be studied in this chapter, the suppression task (Byrne [5]). When subjects are
presented with the modus ponens material “If she has an essay she studies late
in the library. She has an essay.” they almost universally draw the conclusion
she studies late. When instead they are presented with the same premises plus
the premise: “If the library is open she studies late in the library”, about half
of them withdraw the inference. Byrne concludes that inference rules cannot be
used to explain the former performance since they evidently cannot explain the
latter non-inference. Rules, if they are to be invoked must be invoked universally
and uniformly. So much the worse for mental logics – roll on mental models,
concludes Byrne.
3Stenning and van Lambalgen [24] discusses the role of default conditionals in ‘executive
function’, an umbrella term for processes involving planning of actions to achieve a given goal.
4Strictly speaking we would have to write p∧ ¬ab(p, q)→q, a notation that emphasises
that the abnormality is specific to the conditional. We ask the reader to remember that this is the
intended interpretation and we will continue using the less cumbersome notation.
6
Stenning and van Lambalgen [23] analyse this task in terms of CWR. In the con-
dition of the suppression task where we are given that ‘she has an essay’ and ‘if
she has an essay she’s in the library’, the argument proceeds as follows. The un-
derlying logical form of the premise set is {p, p ∧ ¬ab →q}. By itself this does
not justify the modus ponens inference, but now we invoke CWR, in the guise of
the assumption that since no abnormalties are mentioned, to conclude that ¬ab is
in fact true; whence qfollows.
When, in the second condition of the experiment we get the extra premise that ‘if
the library is open she is in the library’, it is conceived of as triggering the rele-
vance of an abnormality – namely that the library may be closed (along with the
relevance of other general knowledge conditionals such as ‘if the library is closed,
readers are not inside’ etc. etc.). Now we do have information that something may
be abnormal, and from p∧ ¬ab →qand ab we can no longer conclude q, though
we, along with lots of subjects, may be happy to make the weaker conclusion that
‘if she has an essay and the library is open she’ll be in the library’.
Because of closed world reasoning over abnormalities, some patterns of reasoning
are valid in this logic which are invalid in classical logic, notably affirmation of
the consequent and denial of the antecedent. So from ‘If she has an essay she is in
the library’ and ‘She doesn’t have an essay’, it is valid to conclude that ‘She isn’t
in the library’, but again only by closed world reasoning. This is a valid move if
this is all there is in the database concerning essays or libraries, because we can
conclude that this is the only conditional with the relevant consequent. If we then
add the ‘alternative’ premise ‘If she has a textbook to read she’s in the library’
then this defeats the conclusion because the world is now large enough to contain
another explanation for her presence in the library. Similarly, affirmation of the
consequent is valid in this logic in the right context. From ‘If she has an essay
then she’s in the library’, and ‘She is in the library’ we can conclude that ‘she
does have an essay’, but only providing the world is closed against abnormalities
such as the library being closed. For a fuller exposition of the suppression task
treated in this logic see [23].
As an aside which we return to below, it is worth raising the question “What
is worth modelling in the suppression task?" Stenning and van Lambalgen [23]
show that each of the four conditional inference forms can be valid in the pro-
posed logic, depending on context, and therefore why nonmonotonic revisions
of conclusions should or should not occur with the arrival of different kinds of
premises. It is not unreasonable to fit this to group data such as Byrne’s, showing
that indeed the changes in proportion of subjects making or retracting conclusions
fits well with the logical model. But it is important we don’t forget that this is
a very weak sort of data. What we would really like is longitudinal data on the
7
same subject’s revisions of belief in each of the conditions studied – the model is
about belief revision, not about the development of group belief. Of course there
are problems of interference between conditions measures which forced the orig-
inal design. Our point is that other methods of investigation need to be brought
to bear. Lechler [11] used socractic dialogue to show that the range of interpreta-
tions subjects adopted was much wider than allowed for in Byrne’s analysis, and
that the classification of premises as ‘alternative’ or ‘additional’ was far from re-
liable.5So the logical model as it stands only models a particular small range of
the interpretations people adopt, but it is also clear, even from Byrne’s data, that
different subjects are doing different things and it makes little sense to model the
details of their average mental processes.
We mention here in passing that there is a highly significant difference in perfor-
mance between autists and normal controls, in that the former do not engage in
closed world reasoning with respect to abnormalities, although they are capable
of other forms of CWR. Thus autists suppress MP and MT much less, although
they are indistinguishable from normals with respect to AC and DA. See Stenning
and van Lambalgen [24] for more details.
Defeasibility is achieved in this logic by closed world reasoning, and this is central
to why the logic is tractable. In fact the semantics assigned to the logic guarantees
that a unique minimal ‘intended’ model (valuation of all the atomic propositions)
for the discourse is derivable at each step of incorporation of a new discourse
sentence. The semantics is three-valued (true, false, so-far-undecided) but not
truth functional and the concept of validity is not the classical ‘truth of conclu-
sion in all models of the premises’ but rather ‘truth of conclusion in the intended
model of the premises’. The non-truth functionality of the system is clear for all
to see. The defaults in long term memory cannot be shown to be false because of
their robustness to exceptions invoked as abnormality conditions. Genuine coun-
terexamples would require relearning – repair of the database which is a process
outside this logic. None of this is surprising as long as we keep in mind that this
is a logic of credulous discourse processing in which our goal as hearer is to make
the speaker’s statements true. that is, to find a model that makes them true. The
three-valued semantics is required to allow negation-as-failure (¬pis adopted as
true if we fail to prove p).
Interestingly, this three-valued semantics allows a neural network implementation
of the logic in which a feedforward spreading activation network computes the
intended model of a discourse in linear time [25, Chapter 8]. It should be obvious
that this is not a logic of cogitation – slow deliberate derivation of conclusions. In
psychological terms it is rather a logic of System 1 processes [8] – fast automatic
5Stenning and van Lambalgen [25, Chapter 7] features extracts of Lechler’s data.
8
processes of inference operating over general knowledge, often well outside of
conscious control, or indeed awareness – exactly appropriate for the processes of
discourse understanding for simple well-wrought discourse.
2.1 So why is discourse processing still hard?
Computational linguistics tells us that discourse processing is hard because dis-
courses are massively ambiguous and require continual backtracking in their pro-
cessing. How can a defeasible logic deliver a unique minimal intended model
at each stage? We do not pretend to have solved the computational linguistic
problem. What we are claiming is that the human problem is solved by the mo-
bilisation of general knowledge by the human processor. Human processors do
not normally experience these ambiguities because they can apply commonsense
knowledge to avoid them. The logical theory shows how this can happen in ex-
ample cases but does not explain how LTM can be organised on a global scale to
achieve this in a computer. Baggio, van Lambalgen and Hagoort [3] show how
ERP data can provide some evidence that the invocation of abnormalities in the
logic leaves traces in brain activity in psycholinguistic experiments.
Let us take an example that is used by Oaksford and Chater to argue against the
applicability of defeasible logics:
Birds fly.
One-minute-old birds don’t fly.
Can the bird Tweety at one-minute-old fly?
Here there is a conflict between defeasible rules. Our response would be that de-
feasible logical databases allow the mobilisation of general knowledge ‘theories’
to resolve such impasses and this gives a plausible explanation as to how humans
resolve them. The database also contains conditionals such as Birds hatch from
eggs,Helpless chicks take weeks to mature, and lots of other linking materials
. . .which are sufficient to arrive at the conclusion So Tweety can’t fly, yet.Con-
tra Oaksford and Chater, this tractable defeasible logic really does solve some
versions of the frame problem (Shanahan [21]).
This problem of conflict between two applicable rules is only one kind of im-
passe. The other kind of impasse the credulous discourse processor may hit is
where no models can be found i.e. where the speaker’s utterances appear to con-
tradict each other. Socrates is mortal. Socrates lives forever in the dialogues of
Plato. Given a small amount of inference about living for ever and immortality,
9
we have a contradiction. No model is available. Here, we propose, the credu-
lous processor interrupts its activity while classical reasoning attempts to repair:
“Something is wrong. There must be equivocation in the interpretation. We must
find an interpretation in which these two statements are consistent.” The intended
(unique) model of the discourse thus far, now has to be inflated into a set of clas-
sical models representing various possibilities. The single truth valuation of the
atoms of the discourse fragment define a set of all possible valuations. Now we
have to find some modification of this interpretation which will allow an intended
model in which the rogue statements are both true. Is the first Socrates perhaps the
Brazilian football player? Or are there two senses of immortality involved? The
reasoning is only classical in the sense that we may have to explore all possible
classical models in order to repair the impasse by finding some one that will do.
Failure to do so will turn the discourse from credulous cooperative to classical
adversarial, at least until some mutually acceptable repair is found.
Our claim is that we have provided a logical framework for interpretation whose
known properties make it plausible that these problems can be solved and will
scale up. Much of the empirical psychology remains to be done. Note that in
this account of ‘suppression’, nothing is suppressed, and the account is com-
pletely neutral as to whether the reasoning is done over syntactic rules or over
models. Both proof-theoretic and equivalent model-theoretic expositions can be
given. The task cannot provide any evidence whatsoever for resolving debates
about rules versus models.
This concludes our survey of what we believe is the proper setting of the sup-
pression task: defeasible reasoning on discourse interpretations. We now turn to a
critical examination of a very different probabilistic analysis of the same task, that
of Oaksford and Chater [14] (reprinted in [15]). Our purpose here is general how-
ever: we believe the potential of probability theory to model defeasible reasoning
has been overestimated, and the suppression task allows us to show why.
3 A probabilistic model for the suppression task
Oaksford and Chater [14] attempt to show that a probabilistic model can account
for the observed suppression effects in conditional reasoning. We will carefully
analyse the assumptions underlying the model, because they are fairly typical of
applications of probability to human reasoning. These assumptions can be divided
into four broad categories:
10
I. philosophical assumptions forcing a close connection between uncertainty
and probability (while denying a role for logic)
II. largely implicit assumptions about what is and what is not to be included in
the model
III. coordinating definitions linking the probabilistic model to the task at hand
IV. assumptions concerning what counts as validation of the model.
In our discussion below we indicate the category of assumptions at issue by the
corresponding Roman numeral.
3.1 Probability and logic
The following quote from [14] provides much material for reflection on I.:
[M]uch of our reasoning with conditionals is uncertain, and may be
overturned by future information; that is, they are non-monotonic.
But logic based approaches to inference are typically monotonic, and
hence are unable to deal with this uncertainty. Moreover, to the ex-
tent that formal logical approaches embrace non-monotonicity, they
appear to be unable to cope with the fact that it is the content of the
rules, rather than their logical form, which appears to determine the
inferences that people draw. We now argue that perhaps by encod-
ing more of the content of people’s knowledge, by probability theory,
we may more adequately capture the nature of everyday human in-
ference. This seems to make intuitive sense, because the problems
that we have identified concern how uncertainty is handled in human
inference, and probability is the calculus of uncertainty [14, p. 100].
The argument sketched here is that logic is not useful as a model of reasoning,
because it is either monotonic, in which case it flies in the face of data such as
suppression, or non-monotonic, in which case it is closer to the data, but still far
off, because it cannot account for the difference between additional and alterna-
tive conditionals, which is supposed to be one of content not form. Probability
fares much better in this respect, because it can incorporate changes in content as
changes in the values of probabilities. Furthermore, and most importantly, there
are a priori arguments to show that probability theory is the normatively justified
calculus of uncertainty. We will discuss this issue first.
11
3.1.1 Probability and uncertainty
The standard justification for probability – interpreted subjectively – as represen-
tation of uncertainty proceeds via so-called ‘Dutch Book arguments’, a simplified
version of which goes as follows.6Assume given a sample space Xand a set of
events Aon X, which has a Boolean structure. For ease of exposition we take A
to be the powerset of X. What is important to note, however, is that the Boolean
structure of the set of events is given not derived; in other words, one must as-
sume that the events satisfy classical logic. This severely restricts the kinds of
uncertainty to which probability theory can be applied.
In the next step it is assumed that one’s degree of belief in the occurrence of an
event Ein Acan be determined via betting. In very rough outline, the agent’s
degree of belief in Eequals the price of a promise to pay 1if E occurs, and 0
otherwise. The phrase ‘a promise to pay’ refers to a subtlety in the wager: the
bookmaker decides after you have set the price which side of the bet will be the
agent’s. Thus, the agent knows that the bookmaker will either buy such a promise
from him at the price set, or will require the agent to buy such a promise, again at
the price set by the agent. The price set by the agent is the subjective probability
assigned to E. A Dutch Book is a betting scheme which guarantees a gain for the
bookmaker. A famous foundational result in subjective probability then says that
the following are equivalent
1. no Dutch Book can be made against the agent
2. the agent’s degrees of belief satisfy the axioms of probability
Since it is supposedly irrational to engage in a bet in which you are sure to lose, the
result is often glossed as: a rational assignment of degrees of belief is one which
satisfies the axioms of probability theory. This is a so-called synchronic Dutch
Book theorem; the diachronic Dutch Book theorem is concerned with Bayesian
updates of beliefs and justifies the rule of Bayesian conditionalisation
(BaCo) The absolute subjective probability of event Dgiven that ev-
idence Ehas been observed is equal to the conditional probability
P(D|E). Here Eshould encompass all available evidence.
(BaCo) will be important in constructing probabilistic analogues for the standard
propositional inference patterns MP, MT, AC, DA.
6For an excellent exposition, see Paris [17].
12
An objection against justifications of this type is that the agent is all-knowing and
‘all-willing’ as well: he must be willing to bet on any event in Aand he must
fix a unique price for the promise, instead of, say, an upper and a lower bound.
While these idealisations may be alright in some philosophical contexts, they are
definitely out of order when it comes to cognition. It is not excluded that agents
reason with subjective probabilities in some situations, but there is no guarantee
that they assign probabilities to every event of interest.
There exists a way out of these problems, at least technically, by considering finite
approximations to the space of all events. To fix ideas, suppose the space of all
events corresponds to the denumerable set of propositions p0, p1, p2, p3, . . .. An
agent is aware of only finitely many propositions at any one time, but over time
this set may increase. For the sake of argument we may suppose that the above
betting procedure assigns a joint distribution Pnover An={p0, . . . , pn}, for all
n≥1. In principle, however, the Pn(pi)can be different for all n, and hence there
is no guarantee that there exists a limn→∞ Pnwhich defines a joint distribution
over the whole space {p0, p1, p2, p3, . . .}. The reader may wish to consider the
special case where the Pnare actually truth value assignments to the p0, . . . , pn;
in this case limn→∞ Pn(pi)exists only if the truth value of pichanges finitely
many times. That is, incoming information about propositions not considered so
far may non-monotonically change the truth value of pi, but only for finitely many
propositions – the last change must be monotonic.
The general case in which the Pnare probability distributions is a difficult mathe-
matical problem (Rao [19]), much studied in the Bayesian literature, for example
in connection with ‘hierarchies of beliefs’ (see Brandenburger and Dekel [4] for
an interesting example). In order for the limit limn→∞ Pnto exist for all events,
a complicated coherence condition must be satisfied. A special case of this con-
dition will be of interest later when discussing the suppression task. Suppose P2
is the joint distribution over p, q and P3the joint distribution over p, q, r. The
coherence condition then implies the following relation among conditional prob-
abilities:
P2(q|p) = P3(q|p∧r)P(r|p) + P3(q|p∧ ¬r)P(¬r|p).(1)
This property is the probabilistic analogue of monotonicity.
Summing up the discussion so far: we have seen that many assumptions are nec-
essary to conclude that probability theory is the uniquely designated formalism
to deal with every form of uncertainty. In fact the assumptions can be glossed
by saying that the application of probability assumes quite a bit of certainty: the
structure of events must be of a very specific type described by classical logic, the
13
agent must be certain about the price he wants to set for any bet, and the agent
must also be certain that his future probabilities conform to the coherence con-
dition. This is too much to ask for a real cognitive agent. Of course, we do not
deny that the brain uses frequency information about the environment – e.g. the
frequency with which a given word occurs – in processing. We just doubt that,
where frequency information is lacking, subjective probability can always do duty
for frequencies.
It is therefore unclear why subjective probability can claim the status of a com-
putational model (in the sense of Marr) for dealing with uncertainty [14, p. 103].
Note that Marr did not intend his computational level to be an idealised model
only. This level specifies the inputs and outputs – and their relation – of a given
cognitive function, and thus functions as a declarative specification of the algo-
rithm at the next level. Its character as specification of an algorithm means that
the computational level is not just idealisation and is connected in a lawlike man-
ner to the algorithm.
3.2 Setting up the model
Of course, for the purposes of the argument we grant Oaksford and Chater the
assumption that subjects do indeed assign degrees of belief to the events of interest
in the suppression task. This assumption (of type III.) can be divided into two
components:
1. reasoning with a conditional p→qis de facto reasoning with the condi-
tional probability7P(q|p)– which is assumed to be known exactly
2. subjects assign precise probabilities to atomic propositions
Three parameters then suffice to determine a joint probability on {p, q}:a:=
P(p), b := P(q), := P(¬q|p). This last parameter is the ‘exception parame-
ter’, which models the defeasibility of the conditional p→q. We will see below
that suppression of MP and MT in the presence of an additional conditional is
explained via modulation of , but we first discuss the two premise inference pat-
terns. MP is viewed as (BaCo) applied to P(q|p). That is, if pis given as
the categorical premise, it is assumed this premise constitutes all the available
evidence, so that (BaCo) can be applied. This way of justifying probabilistic
MP gives it a nice non-monotonic flavour, for if in addition to pthe categorical
7As a consequence, some reasoning with iterated conditionals cannot be represented, as is also
the case in closed world reasoning.
14
premise ris given, (BaCo) can no longer be applied to P(q|p). Thus, it would
seem that a probabilistic analysis is in principle suitable to model the suppression
task.8In a similar way, MT is viewed as (BaCo) applied to P(¬p| ¬q), AC as
(BaC0) applied to P(p|q), and lastly DA as (BaCo) applied to P(¬q| ¬p).
The three parameters a, b, suffice to generate the required conditional probabil-
ities via the probabilities of conjunctions. For example, P(p∧ ¬q)is computed
as P(p)P(¬q|p) = a,P(p∧q) = P(p)P(q|p) = a(1 −),P(¬p∧q) =
P(q)−P(p∧q) = b−a(1 −). It follows that e.g. P(p|q) = P(p∧q)
P(q)=a(1−)
b,
and so on for the other conditional probabilities.
3.2.1 Validating the model for two premises
Why is it plausible to assume that subjects reason according to such a probabilistic
model? Oaksford and Chater attempt to show this by fitting the model to charac-
teristic data on two premise inferences (here taken from Byrne [5]). The fitting9
proceeds as follows: was set to .1, and values of a, b were sampled10 in the
interval [.1, .9].11 The values of the four conditional probabilities were averaged
over the a, b in the sample. Just to give an example, the value of the average for
P(q|p)obtained in this way was 90.00. As a number, this is not dramatically
different from the 97.36% rate of endorsement in Byrne’s data, which are also av-
eraged over subjects. This is a validation of sorts,12 but it would be much stronger
if the model were fitted to individual subjects – after all it’s they who are sup-
posed to reason with subjective probabilities. Further principled objections to this
procedure will be discussed in section 3.3.3.
8This is not the justification given in [14], where it is simply observed that subjects, when given
p, seem to rate the probability of qas proportional to P(q|p). The present exposition makes it
clearer why a probabilistic analysis of the suppression task is prima facie plausible.
9The assumptions used here are of type IV.
10If the probabilities a, b were frequencies, there would be no need to sample, because they
would be uniquely determined by the environment.
11Actually the sampling space was also restricted by the requirement that a<b. This is an
assumption which should also count as a parameter of the model, so that we now have four param-
eters and four data points. The requirement is somewhat peculiar in a context where P(q|p)<1.
Note also that the rarity assumption prominent in Oaksford and Chater [16] is now dropped, since
a, b > 0.1, because ‘[conditional] inferences are specific to context’ [14, p. 103].
12If the problem indicated in footnote 11 can be solved.
15
3.3 Incorporating additional and alternative conditionals
There are two motivations for using probability theory as a model of reasoning
in the suppression task. One is that Bayesian conditionalisation is itself a non-
monotonic principle: an extension of the evidence pmay invalidate a previous
posterior probability for qderived by (BaCo) applied to P(q|p).13 The second
motivation is that the conditional probability P(q|p∧r)may differ from P(q|p),
so that strengthening the antecedent of a conditional has a real effect. Thus there is
face validity in the attempt to model suppression via change in conditional prob-
ability. As we will see, however, it is not straightforward to give a probabilistic
account of the processes involved in changing the conditional probabilities.
There is also a weak probabilistic explanation of the suppression task, which is
content with observing that because (BaCo) is a non-monotonic principle, ex-
pansion of the original evidence pwith an additional conditional invalidates the
probability assignment to q, so that suppression of MP is expected. Such an ex-
planation would need to show how (the antecedents of) additional conditionals
differ from (the antecedents of) alternative conditionals, and in any case requires
an account of how a conditional, i.e. a conditional probability, can be incorporated
into the evidence. Since Oaksford and Chater do not take this road, we shall not
explore it either.
3.3.1 Additional conditionals
It is worth quoting Oaksford and Chater in full on suppression of MP:
Additional antecedents (or exceptions), for example that there is petrol
in the tank with respect to the rule if you turn the key, the car starts,
concern the probability of the car’s not starting even though the key
has been turned – that is, they concern . If you do not know there
is petrol in the tank, you cannot unequivocally infer that the car will
start (MP). Moreover, bringing to mind additional factors that need to
be in place to infer that the car starts – for example the battery must
be charged – will increase this probability [i.e. ]. [. . . ] [I]f there are
many additional antecedents, that is, is high, the probability that the
MP inference will be drawn is low [14, p. 104].
13This assumes that the additional conditional can itself be regarded as (probabilistic) evidence
on which it is possible to conditionalise.
16
In more formal terms, the argument for the suppression of MP seems to be the
following. Initially the subject works with the conditional probability P(q|p)
derived from a joint distribution on {p, q}. The antecedent of the additional con-
ditional enlarges this space to {p, q, r}(where rstands for ‘there is petrol in the
tank’), and this leads to a new representation of P(q|p). One may write
P(q|p) = P(p∧q)
P(p)=P(p∧q∧r)
P(p)+P(p∧q∧ ¬r)
P(p)=(2)
=P(p∧q∧r)
P(p)=P(p∧q∧r)
P(p∧r)
P(p∧r)
P(p)=P(q|p∧r)P(r)
where the last equality follows under the plausible assumption that pand rare
independent.
In an orthodox Bayesian approach, P(q|p∧r)P(r)is simply a different repre-
sentation of P(q|p), but the values of these expressions must be the same, since
it is assumed that the subject had access to the joint distribution over {p, q, r}in
computing P(q|p). But this is not how Oaksford and Chater use probability.
They assume that the subject assigns a lower probability to P(q|p)in the en-
larged representation P(q|p∧r)P(r); the quote suggests that this is because
the subject lowers P(r)from 1 to a value smaller than 1 when becoming aware
of possible exceptions. This requires the kind of transition between algebras of
events that we studied in section 3.1, but now we are in great difficulties, because
such transitions must be governed by equation 1, which in the case at hand boils
down to
P2(q|p) = P3(q|p∧r)P(r).(3)
Thus, in Oaksford and Chater’s model the subject must change his probabilities
in a manner that conflicts with the Bayesian desideratum of striving toward a co-
herent probability distribution over all events. As a consequence, no Bayesian
explanation of the transition P2(q|p)and P3(q|p∧r)P(r)can be given – this
transition must remain outside the model (a type II. assumption). In order to ac-
count for the phenomena the model therefore needs to be supplemented with a
theory about non-monotonic changes in degrees of belief.
A technical aside The reader might be tempted to object that the above repre-
sentation of the increase of information is obviously not what is intended. More to
the point would be a construction in which the probability spaces remain the same,
but the probability distributions change. For our running example this would mean
17
that the probability space is in both cases {p, q, r}, but that the probability distri-
bution first assigns probability 0 to ¬r, and upon becoming aware of the additional
conditional r→q, a non-zero probability. The trouble with such a suggestion is
twofold. Firstly, from a Bayesian point of view, the transition from the a pri-
ori probability P(¬r) = 0 to the a posteriori P(¬r)>0is not allowed, since
this cannot be achieved via (BaCo): conditionalising on more evidence cannot
make a null probability positive. Secondly, one would like to have the assurance
that incoming information generates a stable probability distribution in the limit.
However, the convergence theorems that guarantee this,14 also require a null prob-
ability to remain null. Clearly it does not help to assume that the initial probability
of ¬ris very small, because this would bring us back to the situation where the
probability is essentially defined on the set of all propositions, and not on a finite
subset.
Again the conclusion must be that Bayesian probability has too much monotonic-
ity built in to account for non-monotonic belief change as witnessed in the sup-
pression task. A form of closed world reasoning with probabilities must be devel-
oped which allows an agent to set conditional probabilities to 0 by default, and to
change these to a positive value if relevant new information comes in.
3.3.2 Alternative conditionals
Let us now look at distinction between additionals and alternatives: additionals
are related to P(¬q|p), alternatives to P(q| ¬p)– note that this is determined by
a, b, . The effect of incorporation of an alternative conditional is that P(q| ¬p)
must increase, but must remain constant to account for MP with alternative con-
ditional. This could mean that bincreases – more alternative antecedents ‘means’
there are more possibilities for the antecedent to occur, and the probability of the
given antecedent adecreases.
We find the same mixture of probabilistic and non-probabilistic modelling as-
sumptions in Oaksford and Chater’s account of the suppression of DA:
Alternative antecedents, such as information that the car can also be
started by hot-wiring, with respect to the rule if you turn the key, the
car starts, concern the probability of the car starting even though the
key has not been turned; that is P(q| ¬p). If you know that a car can
be started by other means, you cannot unequivocally infer that the car
will start even though the key has not been turned. Moreover, bringing
14The so-called ‘martingale convergence theorems’, see [27, Ch. 10].
18
to mind other alternative ways of starting cars, such as bump-starting,
will increase this probability. [. .. ] It is therefore an immediate con-
sequence of our model that if there are many alternative antecedents,
that is, P(q| ¬p)is high, the probability that the DA inference should
be drawn is low [14, p. 104].
Bearing in mind the analysis we gave above of the suppression of MP, it is now
easy to see that what the Bayesian model prima facie predicts (via (BaCo)) is the
following:
if the conditional probability P(q| ¬p)increases, then the probability
of DA decreases.
The essential question is, however, whether the Bayesian model also covers the
first stage of the subject’s reasoning, in which the presence of alternative an-
tecedents increases the initial P(q| ¬p). The argument seems to be this, using
the same notation as in section 3.3.1: the subject makes a distinction between
P2(q| ¬p)and P3(q| ¬p)given by
P3(q| ¬p) = P3(q|r∧ ¬p)P3(r| ¬p) + P3(q| ¬r∧ ¬p)P3(¬r| ¬p)(4)
which, in the example discussed though not always, can be simplified to
P3(q| ¬p) = P3(q|r)P3(r| ¬p) + P3(q| ¬r∧ ¬p)P3(¬r| ¬p)(5)
since r→ ¬p. In this formula, the term P3(q|r)is large, since it represents the
conditional r→q, and the term P3(q| ¬r∧ ¬p) = P3(q| ¬(r∨p)) is much
smaller, especially when P3(¬r| ¬p)= 1, since it then represents P3(q| ¬p).
Oaksford and Chater then appear to argue as follows. In the case of one con-
ditional premise p→qonly, the subject works with the conditional probabil-
ity P2(q| ¬p)on the probability space {p, q}. This probability is then identi-
fied with P3(q| ¬p)on the probability space {p, q, r}under the assumption that
P3(¬r| ¬p)= 1. Consideration of the alternative antecedent r→qthen leads to
a decrease in P3(¬r| ¬p), so that a larger proportion of the large term P3(q|r)
contributes to P3(q| ¬p), whence P3(¬q| ¬p), i.e. the probability that ¬qis
concluded, decreases. But we can now see that this reasoning is not Bayesian,
since it requires the subject to update P3(r| ¬p)from zero to strictly positive.
This update must necessarily remain outside the model.
19
3.3.3 Validating the model for three premises
The hypothesis to be tested is that subjects reason with subjective probabilities
in order to solve the suppression task. It is in the nature of this hypothesis that
it does not concern the exact values of the subject’s degrees of belief in concrete
conditionals and categorical propositions. The hypothesis is that whatever these
values are, they obey the Bayesian inference rules. This makes it difficult to test
the hypothesis directly.
Above we have seen that Oaksford and Chater take the revision of conditional
probabilities in going from simple MP to MP with additional premise to lie out-
side the scope of the probabilistic model. Thus the model provides no constraints
on the values of the probabilities in the simple condition versus the three premises
conditions. To fit the model it is therefore sufficient to estimate a, b, from the data
in each condition of a suppression task; Oaksford and Chater [14, p. 105] take
Byrne’s [5] for this purpose. Under the assumption that subjects apply (BaCo)
when making an inference, the data supply information15 about the conditional
probabilities P(q|p)(MP), P(¬p| ¬q)(MT), P(¬q| ¬p)(DA) and P(p|q)
(AC). Given these conditional probabilities, the parameters of interest, a, b and ,
can be computed. The exception parameter is obtained from the rate of endorse-
ment of MP (which is equal to P(q|p)). The parameter a=P(p)is obtained
from the rates of endorsement of MP, AC and DA via the following computation
x:= P(q|p)
P(p|q)=P(q)
P(p)
y:= P(q| ¬p)
P(¬p|q)=P(q)
1−P(p)
hence xP (p) = y(1 −P(p)) and so P(p) = y
x−y,
where the right hand side is supplied by the data. The computation for bis similar
and yields
P(q) = xy
x+y.
Observe that the parameters are computed using only three conditional probabil-
ities: P(¬p| ¬q)is nowhere used, and can be computed from the remaining
15It will soon become apparent why we use a deliberately vague expression here.
20
conditional probabilities. This puts a consistency requirement on the four data
points, thus allowing the experimenter to check whether subjects are coherent in
the probabilistic sense.
The result is that in every condition the three parameters can be fitted from the
four data points MP, MT, DA, AC, where one is looking for a fit at .01 signifi-
cance level. The question arises however whether the estimates of the parameters
a, b, can be interpreted as probabilities: what is the event which is supposed to
have probability a=P(p)? Recall the definition of P(p):pdoes not stand for
any specific proposition, but it refers to a particular role: the antecedent of the
main conditional, the content of which differs from one conditional to the next.
Similarly for the conditional probabilities used to estimate P(p): these are in fact
averages of the conditional probabilities hypothesised to be used by the subjects
with the various experimental materials. Now suppose that the parameters were
estimated using the data for a single subject. What would the estimated value
P(p)(or P(q|p)) mean in this situation? It would be an average over the very
different contents occurring in the antecedent (and consequent) of the main con-
ditional, not a degree of belief assigned by the subject to a particular event. What
the model fitting procedure does, therefore, is construct a joint probability distri-
bution (from a, b, ) of which we can be certain that it represents the degrees of
belief of no subject at all. This seems a weak justification for the model.
To summarise, we have not claimed that Bayesian probability plays no role at all
in subjects’ reasoning in the suppression task. Especially when the formulation
of the task explicitly introduces qualitative probabilistic expressions like ‘almost
always’ or ‘rarely’, as in Stevenson and Over’s graded suppression task [26], a
probabilistic model may be appropriate.16 We do claim however, that it cannot
be the whole story in those reasoning tasks in which non-monotonic (degree of)
belief revision plays an important part. If one does not incorporate this form of
belief revision into the model, one is in effect saying it falls outside the scope of
explanation. This is giving up too soon. If one is convinced that the last part of
subjects’ reasoning is indeed Bayesian conditionalisation, there arises the chal-
lenge of combining closed world reasoning with probabilistic reasoning.
More generally, the discussion points to the necessity of distinguishing between
‘reasoning to an interpretation’ and ‘reasoning from an interpretation’, as we have
done in [25]. Once the subject fixes the interpretation of the task at hand as ‘rea-
soning with uncertainty, more particularly probability’, and assigns probabilities
to the propositions of interest, reasoning may proceed entirely within (Bayesian)
probability theory. The reasoning process that leads up to fixing the interpretation
16We do not take performance in the graded suppression task to indicate that subjects always
reason according to a probabilistic model in a suppression task.
21
as probabilistic, and to fixing the probabilities, may well be of a very different
nature. Here we have argued that this reasoning to an interpretation is best seen
as a form of closed world reasoning, perhaps applied to probabilities. In any case
it seems more fruitful to investigate how logic and probability must interact than
to view them as rival approaches.
3.4 System P?
The reader acquainted with the literature might think that a fruitful interaction
between non-monotonic logic and probability theory already exists, in the form of
System P [10, 18]. This is a deductive logic for reasoning with exception-tolerant
conditionals written as α|∼ β, and read as ’if α, then normally β’. A highly
appealing feature of System P is that it allows probabilistic semantics, which are
of two kinds:
1. ’α|∼ βis true’ is interpreted as P(β|α)> θ, where θ < 1but 1−θis
infinitesimal [1].
2. ’α|∼ βis true’ if P(β|α)takes a value in a suitable interval [9]. The
details need not concern us here. It suffices to say that the semantics intro-
duces a new kind of object, ‘conditional events’ B|A, corresponding to
α|∼ β, allowing the possibility that the (absolute) probability of Ais 0; and
that the inference rules of System P in this semantics correspond to rules
for transforming probability intervals.
It thus seems that System P, together with the interval semantics, is ideally suited
to address the issues we raised above: it seems to allow a rational treatment of
events with probability 0, and it is not wedded to the assumption that probabilities
must be known precisely. This is not the case however, as can be seen when
we look at System P’s rules, presented as sentences of the form ’premises =⇒
conclusion’:
=⇒α|∼ αREFL EXIVITY AXIO M (6)
|=α↔β, γ |∼ α=⇒β|∼ αLEF T LOGICAL EQUIVALE NC E (7)
|=α→β, γ |∼ α=⇒γ|∼ βRIG HT WEAKENING (8)
α|∼ γ, β |∼ γ=⇒α∨β|∼ γOR(9)
α∧β|∼ γ, α |∼ β=⇒α|∼ γCUT (10)
α|∼ β, α |∼ γ=⇒α∧β|∼ γCAUT IO US MONOTONIC IT Y (11)
22
It is now immediately obvious that one rule of System P dashes all hopes of
treating the suppression task: the rule ORwhich forces one to treat additional
and alternative premises on the same footing.17 System P therefore reinforces the
point made earlier, that probability is too much tied to classical logic to model the
truly non-monotonic reasoning that occurs in the suppression task.
4 Conclusions
Interpretation processes are necessary, whether one then applies probability the-
ory or some logic in reasoning from the resulting interpretations. In the case of
suppression, understood in probabilistic terms, interpretation shows up as the ne-
cessity to change one’s probabilities in ways not sanctioned by Bayesianism. We
claim that a computational level analysis in the sense of Marr must also incor-
porate the interpretation process, not only the reasoning once the interpretation
is chosen. This is not to deny the role of Bayesian probability in a character-
isation of the computational level. If a subject construes the task as involving
uncertain conditionals, in the sense of positive probability of exceptions, prin-
ciples like Bayesian conditionalisation may well form part of the computational
level. In this case competence theory is needed of how judgements of probabilities
can change in non-Bayesian ways. This we regard as one of the most interesting
technical challenges issuing from the present analysis. We do deny that the entire
computational level can be characterised in this manner, and also that the same
computational level analysis applies to all subjects engaged in this task.
Classical logic forced a single competence model upon reasoning tasks, and one of
the merits of the Bayesian approach is to have loosened the grip of classical logic.
But the danger exists that Bayesian probability is itself promoted to the absolute
standard of competence. The arguments purporting to show that probability is
the calculus of uncertainty are too weak to establish this, hence absoluteness of
the Bayesian standard cannot be defended in this way. On the empirical side it
is clear that different subjects are interpreting tasks very differently, and although
Bayesianism has some built-in mechanisms to deal with individual differences via
variation in assignments of subjective probabilities, this does not capture the full
range of differences. For instance, closed world reasoning, for which there exists
evidence in our tutorial data [25, Chapter 7], cannot be modelled in this way.
Oaksford and Chater view their work as an instance of ‘rational analysis’ and
consider that giving an account of the economics of information processing tasks
17If one drops O Rfrom System P one gets Makinson’s ’cumulative logic’ [12], which however
has no close relationship to probability.
23
should occupy centre stage. Our own work adopts a very different computational
level analysis of reasoning tasks, assimilating them to discourse understanding.
The familiar laboratory deductive reasoning tasks are interesting, important and
potentially psychologically insightful precisely because subjects assimilate them
to this overall process of discourse processing. Because the normal cues as to
how to make this assimilation are removed, subjects make it in a variety of ways,
many of which are not the classical logical one the experimenter expects. So we
absolutely agree with Oaksford and Chater that the classical logical computational
model is generally not a reasonable choice for data analysis. Where we disagree
is that we believe discourse processing has to be understood as at least a two-
component process, and the rational analyses of these two processes cannot be the
same. Trying to make them the same leads to the invocation of probability theory
as an overall framework, and that makes the computational theory distant from
the data, and inappropriate for dealing with the kinds of uncertainty that permeate
interpretation.
References
[1] E. Adams. The logic of conditionals. Reidel, Dordrecht, Netherlands, 1975.
[2] J.R. Anderson and C. Lebiere. The Atomic Components of Thought.
Lawrence Erlbaum, Mahwah, N.J., 1998.
[3] G. Baggio, M. van Lambalgen, and P. Hagoort. Language, linguistics and
cognition. In N Asher, R. Kempson, and T. Fernando, editors, Handbook of
the Philosophy of Linguistics. Elsevier, Amsterdam, To appear.
[4] A. Brandenburger and E. Dekel. Hierarchies of beliefs and common knowl-
edge. Journal of Economic Theory, 59(1):189–198, 1993.
[5] R.M.J. Byrne. Suppressing valid inferences with conditionals. Cognition,
31:61–83, 1989.
[6] D.D. Cummins. Naive theories and causal deduction. Memory and Cogni-
tion, 23(5):646–58, 1995.
[7] K. Doets. From Logic to Logic Programming. The MIT Press, Cambridge,
MA, 1994.
[8] J.St.B.T. Evans. In two minds: Dual-process accounts of reasoning. Trends
in Cognitive Sciences, 7(10):454–459, 2003.
24
[9] A. Gilio. Probabilistic reasoning under coherence in system p. Annals of
Mathematics in Artificial Intelligence, 34:5–34, 2002.
[10] S. Kraus, D. Lehman, and M. Magidor. Nonmonotonic reasoning, preferen-
tial models and cumulative logics. Artificial Intelligence, 44:167–207, 1990.
[11] A. Lechler. Interpretation of conditionals in the suppression task. Msc thesis,
HCRC, University of Edinburgh., 2004.
[12] D. Makinson. General theory of cumulative reasoning. In M. Reinfrank,
editor, Proceedings Second International Workshop on Non-Monotonic Rea-
soning, volume 346 of Lecture Notes in Computer Science. Springer, 1989.
[13] J. McCarthy. Circumscription – a form of non–monotonic reasoning. Artfi-
cial Intelligence, 13:27–39, 1980.
[14] M. Oaksford and N. Chater. Probabilities and pragmatics in conditional
inference: suppression and order effects. In D. Hardman and L. Macchi,
editors, Thinking: psychological perspectives on reasoning, judgment and
decision making, chapter 6, pages 95–122. John Wiley & Sons, Chichester,
2003.
[15] M. Oaksford and N. Chater. Bayesian rationality. Oxford University Press,
Oxford, 2007.
[16] M.R. Oaksford and N.C. Chater. A rational analysis of the selection task as
optimal data selection. Psychological Review, 101:608–631, 1994.
[17] J.B. Paris. The uncertain reasoner’s companion. Cambridge University
Press, Cambridge, 1994.
[18] N. Pfeiffer and G. Kleiter. Coherence and nonmonotonicity in human rea-
soning. Synthese, 146:93–109, 2005.
[19] M.M. Rao. Projective limits of probability spaces. Journal of Multivariate
Analysis, 1:28–57, 1971.
[20] R. Reiter. A logic for default reasoning. Artificial Intelligence, 13:81–132,
1980.
[21] Murray P. Shanahan. Solving the Frame Problem: A Mathematical Investi-
gation of the Common Sense Law of Inertia. MIT Press, 1997.
[22] K. Stenning and M. van Lambalgen. A little logic goes a long way: bas-
ing experiment on semantic theory in the cognitive science of conditional
reasoning. Cognitive Science, 28(4):481–530, 2004.
25
[23] K. Stenning and M. van Lambalgen. Semantic interpretation as reasoning
in nonmonotonic logic: the real meaning of the suppression task. Cognitive
Science, 29(6):919–960, 2005.
[24] K. Stenning and M. van Lambalgen. Logic in the study of psychiatric dis-
orders: executive function and rule-following. Topoi, 26(1):97–114, 2007.
Special issue on Logic and Cognitive Science.
[25] K. Stenning and M. van Lambalgen. Human reasoning and cognitive sci-
ence. MIT Press, Cambridge, MA., 2008.
[26] R. Stevenson and D. Over. Deduction from uncertain premisses. Quarterly
Journal of Experimental Psychology A, 48(3):613–643, 1995.
[27] D. Williams. Probability with martingales. Cambridge University Press,
Cambridge, 19954.
26
