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A working memory model of relations between
interpretation and reasoning∗
Keith Stenning and Michiel van Lambalgen
February 25, 2005
To appear in Cognitive Science, 2005.
Abstract
Interpretation is the process whereby a hearer reasons to an in-
terpretation of a speaker’s discourse. The hearer normally adopts a
credulous attitude to the discourse, at least for the purposes of inter-
preting it. That is to say the hearer tries to accommodate the truth of
all the speaker’s utterances in deriving an intended model. We present
a nonmonotonic logical model of this process which defines unique min-
imal preferred models and efficiently simulates a kind of closed-world
reasoning of particular interest for human cognition.
Byrne’s ‘suppression’ data [5] are used to illustrate how variants
on this logic can capture and motivate subtly different interpretative
stances which different subjects adopt, thus indicating where more fine-
grained empirical data are required to understand what subjects are
doing in this task.
We then show that this logical competence model can be imple-
mented in spreading activation network models. A one pass process
interprets the textual input by constructing a network which then com-
putes minimal preferred models for (3-valued) valuations of the set of
propositions of the text. The neural implementation distinguishes easy
forward reasoning from more complex backward reasoning in a way
that may be useful in explaining directionality in human reasoning.
1 Introduction and outline of the paper
This paper proposes a new engagement between logic and the study of the
cognitive processes of deductive reasoning. Most concretely we propose a
default logical model of some data from Byrne’s ‘suppression task’ [5]. This
model reveals how results from the logic programming literature yield the
∗Comments from two anonymous referees have helped us to improve the presentation
of the paper. The second author is grateful to the Netherlands Organization for Scientific
Research (NWO) for support under grant 360-80-000. This paper was inspired by a visit
to Waltter Schaeken and Kristien Dieussaert (cf. [11]).
1
implementability of the model in spreading activation networks. More gen-
erally, we show how this logical model changes our construal of the cognitive
issues. This proposal connects several seemingly unconnected topics and we
owe the reader an outline of their new relations:
(1) We distinguish between two main kinds of logical reasoning: reason-
ing from a fixed interpretation of the logical and nonlogical terms in the
premisses, and reasoning toward an interpretation of those terms.
(2) We illustrate the distinction by means of Byrne’s ‘suppression task’ [5].
Byrne took her data (a ‘failure’ to apply classical logic) to be evidence
against rule-based accounts of logical reasoning and favouring ‘mental mod-
els’; but this follows only if experimental subjects in her task reason from
a fixed interpretation. If what they are doing is reasoning to a consistent
interpretation of the experimental materials, their answers can be shown to
make perfect logical sense, albeit in a different logic.
(3) We then show that reasoning toward an interpretation has an impor-
tant feature in common with planning, namely model-construction, and can
be treated by the same logical tools, for which we take logic programming
with negation as failure. This is a particular form of nonmonotonic logic
which has be shown to be extremely useful in discourse interpretation in
van Lambalgen and Hamm [46].
(4) We show how the deviation from classical logical reasoning found in
Byrne’s data and those of others (e.g. Dieussaert et al. [11]) can be ac-
counted for in the proposed formalism.
(5) We then turn to a possible neural implementation of the proposed formal-
ism for reasoning toward an interpretation (or, alternatively, for planning).
Here the semantics of the formalism, Kleene’s strong three-valued logic, be-
comes prominent, since it allows a slick implementation the logic in terms
of isomorphic neural nets coupled by inhibitory connections. We view this
implementation as a contribution toward the study of the involvement of the
memory systems in reasoning. Also the observed difference in performance
between ‘forward’ reasoning patterns (e.g. modus ponens) and ‘backward’
reasoning patterns (e.g. modus tollens) is easily explained in this setup.
(6) The discussion section considers the much-debated issue of the com-
putational complexity of nonmonotonic reasoning, and relates the material
in the body of the paper to so-called ‘dual process’ theories of reasoning (for
which see e.g. Pollock [28], Stanovich [37] and Evans [13]).
The psychology of reasoning literature alludes to the implication of defeasi-
2
ble processes in interpretation at a number of points, and there are several
different logical and computational frameworks for modelling defeasible rea-
soning in AI. However, it should be clear that we propose here nothing less
than a fundamental change in the standards of the field, and the consequent
goals and treatments of evidence. Many psychologists would have us accept
that logic is a quite separate endeavour from psychology and that logical
models are somehow merely ‘technical’. But it was Johnson-Laird [18] him-
self who proposed that the psychology of reasoning requires competence
models as well as performance models. With this much even the mental
logicians can agree. Mental models theory contains within itself various dif-
ferent proposals about competence models for various tasks, but these are
rarely clearly separated from the performance modelling, nor, crucially here,
even distinguished from each other.
Formal logic is a discipline which studies competence models of reason-
ing, and which provides a body of systems and mathematical techniques for
creating others. The inter-relations between systems are well understood
from a century’s worth of study. Psychologists’ failure to exploit this knowl-
edge, the existing systems, and the available techniques for creating new
competence models, has led them, with few exceptions, to fail to make basic
psychological distinctions (e.g. between defeasible interpretation and mono-
tonic derivation here) and to fail to make simple predictions from the logical
distinctions they have made (e.g. that deontic and descriptive conditionals
pose quite different problems in the selection task (see Stenning and van
Lambalgen[42]). Indeed a substantial proportion of psychological effort has
gone into showing that logic cannot provide insight into the basis of human
reasoning, with disastrous consequences for the psychology of reasoning. We
maintain, on the contrary, that using these logical systems as competence
models opens up the possibility of much richer psychological accounts be-
cause the variety of systems and their relations allow modelling of the many
different things that subjects are doing within and between experiments.
This paper uses formal-logical machinery extensively, and some may
think excessively. But for those who feel that the logical technicalities are
not needed for a psychological understanding, we offer the following anal-
ogy: in the same way as optics is the geometry of light rays, logic is the
mathematics of reasoning systems. No one would maintain that the tech-
nicalities of optics are irrelevant to how the visual system infers, say, form
from motion. Likewise, no consideration of actual human reasoning pro-
cesses can proceed without careful attention to mathematical constraints on
these processes..
1.1 Two kinds of reasoning
Our most general aim in this paper is to differentiate interpretation tasks
from derivation tasks with which they are sometimes confused, and in doing
3
so, to establish a productive relation between logical and psychological stud-
ies of human reasoning and memory. So we should start by saying what we
include under the term interpretation. We intend to use the term broadly
for all the structures and processes which connect language to the specifics
of the current context.
All men are mortal. Socrates is a man. So he is mortal. are two pre-
misses and a conclusion. Without interpretation they have no truth values.
Interpreted in the currently relevant range of contexts, both premisses may
be true, or one or other or both false e.g. because Socrates is immortal in
his appearance in the works of Plato, or where Socrates is a dog. Even if
both are true, the ‘he’ in the conclusion may refer to someone other than
Socrates. Agreeing about what is going to count as being a man or mor-
tal, who Socrates is in the current argument, and the antecedent of ‘he’ are
all parts of the process of interpretation. Under one interpretation anyone
who has died is definitely not immortal, whereas under another this may
not be the case. Women may or may not be intended to be included in the
denotation of ‘men’, and so on.
Our claim in this paper is that central logical concepts such as logical
category, truth, falsity and validity can also vary across interpretations; we
will therefore refer to these concepts as ‘parameters’. For instance, we pro-
pose to apply a logic which makes ‘if ... then’ into something other than a
sentential connective, and the notion of validity into ‘truth in all preferred
models’. If we are right that the logic proposed is an appropriate model for
certain kinds of discourse then the parameters characterizing this logic will
have to be set in the process of assigning an interpretation, with the psycho-
logical consequence that they may also be the subject of misunderstanding
when parties make different settings.
Such misunderstandings are, we claim, endemic in the communication
between psychologists and their subjects in reasoning tasks. Elsewhere we
have argued that Wason’s selection task can be understood as largely in-
voking interpretative processes in subjects (Stenning and van Lambalgen
[41], [42]). Stenning and Cox [39] have shown that syllogistic reasoning has,
for the typical undergraduate subject, significant interpretative components,
and that these interpretative processes play an important role in determining
subsequent reasoning. Here we extend this approach to Byrne’s suppression
task[5] – a more obviously interpretative task than either the selection task
or syllogisms. The suppression task is a case of what we will call credulous
text intepretation, where, by definition, listeners engage in credulous text
interpretation when they attempt to guess the speaker’s intended model (i.e.
a model which makes the speakers’ utterances true and which the speaker
intends the hearer to arrive at). A credulous attitude to interpretation con-
trasts with a sceptical attitude under which a hearer seeks countermodels
of the speaker’s statements – models which make all the speaker’s premisses
true but their conclusion false.
4
This contrast between credulous and sceptical attitudes underlies one of
the most pervasive logical contrasts between classical monotonic and non-
classical nonmonotonic concepts of validity. Thus we wish to put forward a
picture of cognition which takes logical categories seriously – one in which
participants engaging in discourse are thereby constructing their own inter-
pretations appropriate to their purposes, and this requires the invocation of
multiple logical systems.
The history of the confusion between interpretation and derivation pro-
cesses in psychology is important to understanding the point of the present
paper. Logic has always assumed that the process of interpretation of a frag-
ment of language (an argument) into a formal representation is a substantial
process, and even traditional logic at least provided a characterisation of
what constituted a complete interpretation (and the range of possible such
interpretations).
In traditional, pre-symbolic. logic, it was assumed that logic could say
little about the process whereby assumptions were adopted, rejected, or
modified, or how alternative interpretations of natural languages into arti-
ficial ones was achieved. Traditional logical theory narrowly construed was
pretty much defined by the limits of what could be said about reasoning from
interpretations wheresoever those interpretations came from. Nevertheless,
interpretation was always assumed to be a substantial part of the process
of human reasoning, and much of traditional logical education focussed on
learning to detect illicit shifts of interpretation within arguments – equivoca-
tions. It also trained students to distinguish ‘fallacies’ which can be viewed
as patterns of assumption easily mistaken for valid patterns of conclusion,
thus confusing interpretative with derivational processes.1Traditional logic
may have had no formal theory of interpretation but it had very prominent
place-holders for such an apparatus, and considerable informal understand-
ing of it. It was obvious to all concerned that interpretation was the process
whereby content entered into the determination of form. We shall see be-
low, in section 2.1, that interpretation plays a much more explicit role in
the modern conception of logic.
1.2 The suppression task and its role in the psychology of
reasoning
Suppose one presents a subject with the following innocuous premisses:
1As we shall see, classical derivational fallacies are often valid in nonmonotonic logics
of interpretation. This is a cognitively important insight. Rather than having to model
fallacies as arbitrarily introduced rules of inference (or reject rules altogether), or for ex-
ample speculate about ‘real world conditionals often being biconditionals’, seeing fallacies
as valid patterns in contrasting logics provides scope for an explanation of why a subject
with a different construal of the task may have a different notion of validity and so draw
different inferences.
5
(1) If she has an essay to write she will study late in the library.
She has an essay to write. In this case roughly 90% of subjects draw
the conclusion ‘She will study late in the library’. Next suppose one
adds the premiss
(2) If the library is open, she will study late in the library.
In this case, only 60% concludes ‘She will study late in the library’.
However, if instead of (2) the premiss
(3) If she has a textbook to read, she will study late in the library.
is added, then the percentage of ‘She will study late in the library’ – con-
clusions is comparable to that in (1).
These observations are originally due to Ruth Byrne [5], and they were
used by her to argue against a rule-based account of logical reasoning such
as found in, e.g., Rips [32]. For if valid arguments can be suppressed, then
surely logical inference cannot be a matter of blindly applying rules; and
furthermore the fact that suppression depends on the content of the added
premiss is taken to be an argument against the role of logical form in rea-
soning. We will question the soundness of the argument, which we believe to
rest on two confusions, one concerning the notion of logical form, the other
concerning the aforementioned distinction between reasoning from and rea-
soning to an interpretation. But we agree that the data are both suggestive
and important. We will give a brief overview of how these data have been
treated in the literature following on from Byrne’s paper, and will provide
some reasons to think that these treatments may not be the last word.
Readers not interested in the survey can move on to section 1.2.4 without
loss.
We start with Byrne’s explanation in [5] of how mental models theory
explains the data. ‘Mental models’ assumes that the reasoning process con-
sists of the following stages:
(i) first the premisses are understood in the sense that a model is constructed
on the basis of general knowledge and the specific premisses
(ii) an informative conclusion is read off from the model
(iii) this conclusion is checked against possible alternative models of the sit-
uation.
In this particular case the model for the premisses p→q, r →qthat is
constructed depends on the content of pand r, and the general knowledge
activated by those contents. If rrepresents an alternative, the mental model
constructed is one appropriate to the formula p∨r→q, and the conclusion q
can be read off from this model. If however rrepresents an additional condi-
tion, that model is appropriate to the formula p∧r→q, and no informative
conclusion follows.
6
There are several problems with this explanation and Byrne’s use of the
suppression effect in the ‘rules versus models’ debate. The first is that ‘rules’
and ‘models’ may not be as mutually exclusive as they are often presented,
and in particular there may be both ‘rules’ and ‘models’ explanations of
what is going on [40, 43]. The ‘mental rules’ account of logical reasoning
is falsified only on a somewhat simplistic view of the logical form of the
experimental materials. As will be seen below, an account of logical form
more in line with current logical theorizing shows that in the paradigm case
(2), modus ponens is simply not applicable.
The second problem is that the ‘models’ explanation seems more a redescrip-
tion of the data then a theory of the processing that is going on in the
subject’s mind when she arrives at the model appropriate for p∧r→q:
exactly how does general knowledge lead to the construction of this model?
It is a virtue of Byrne’s proposed explanation that it generates this ques-
tion, but we do not think it has been solved by mental modellists. It is
precisely the purpose of this paper to propose a solution at several levels,
ranging from the logico-symbolic to the neural. We do so because we believe
that the suppression task encapsulates important aspects of the process of
natural language interpretation, which have wider implications as well (for
some of these see van Lambalgen and Hamm [46]; also see the paper [47] on
the relation between reasoning patterns and executive defects in autism).
In the remainder of this review we discuss some work on the suppression
task related to interpretative processes, and we indicate areas where we be-
lieve further work (such as that presented here) needs to be done. In order
to facilitate the discussion of other contributions it is useful to make some
preliminary distinctions.
The benefit of seeking formal accounts of interpretations is that it forces
one to be clear about the task as conceived by the subject/experimenter.
Unfortunately, few authors have been entirely clear about what they con-
sider to be the (range of) interpretation of the materials which subjects
should make, let alone what proportions of subjects actually adopt which
interpretations. For example, although a considerable number of Byrne’s
subjects (about 35%) withdraw the modus ponens inference after the sec-
ond conditional premiss is presented, many more (about 65%) continue to
draw the inference. What interpretation of the materials do these subjects
have? If it is the same, then why do they not withdraw the inference too?
And if it is different, then how can it be accommodated within the semantic
framework that underpins the theory of reasoning? Does failure to suppress
mean that these subjects have mental logics with inference rules (as Byrne
would presumably would have interpreted the data if no subject had sup-
pressed)? The psychological data is full of variation, but the psychological
conclusions have been rather monolithic.
Lechler [22] used informal interviewing techniques to elicit subjects’ likely
interpretations of similar materials and showed that a wide range of inter-
7
pretations is made.
The psychological literature has discussed some varieties of interpreta-
tion, and in particular it has distinguished between a suppositional and a
probabilistic interpretation. It is worth our while to discuss these briefly,
since the task becomes determinate only when the interpretations are. In
the end we shall query how radically these interpretations really differ, but
for now the distinction is useful.
1.2.1 Suppositional interpretations
Subjects may choose to interpret the task ‘suppositionally’ in the sense of
seeking a interpretation of the materials in which all the conditionals are
assumed to be absolutely true, and proceeding from there by deduction.
Implicitly, Byrne [5] assumes that this is what subjects do, while assuming
furthermore that subjects adopt the material implication as their model of
the conditional. As we will see below, the combination of the two assump-
tions has been criticized (rightly, in our view), but it should be pointed out
that one may formulate the suppositional interpretation also as: ‘seeking an
interpretation of the materials in which all conditionals are assumed to be
true as eventually interpreted. The rub is in the italicized phrase: we need
not assume that the prima facie interpretation of a discourse is also defini-
tive. In particular the logical form assigned to a statement on first reading
(or hearing) may be substituted for another one upon consideration. Only
if the proviso expressed by the italicized phrase is disregarded, can the sup-
pression effect be taken to falsify ‘rule’ theories. The reader may be forgiven
if at this stage it is unclear how the phrase ‘as eventually interpreted’ can
be given formal content; explaining this will be task of the technical part of
the paper.
It is of some importance to note that when using the expression ‘deduc-
tion’ as above, it is not implied that the mental process involved is derivation
in classical logic. What we mean is that the processes are such as to yield an
interpretation (or re-interpretation) of the premisses which we can entertain
as true (what we call a credulous interpretation), from which one can then
proceed to deduce consequences by whatever logic one finds appropriate.
There are however circumstances in which we would be well-advised
to adopt a non-credulous interpretation. Consider the possiblity that the
succeeding conditionals are asserted by different speakers than the initial
ones, and suppose it is clear that they are intended to voice disagreements:
Speaker A: “If she has an essay, she’s in the library – she’s a very
diligent student you know”
Speaker B: “Nonsense! If her boyfriend has called she’ll be at the
cinema – she’s an airhead”
Now it would be utterly inappropriate to credulously accomodate. The
dialogue must be represented as involving a contradiction and a different,
8
non-credulous logic is required to do that. Classical logic is useful here, be-
cause it allows one to declare a rule to be false if there is a counterexample.
By contrast, credulous logics such as the nonmonotonic logic we will be using
here do not allow the possibility for falsifying a rule, since putative coun-
terexamples are re-interpreted as exceptions. The following terminology is
intended to capture these distinctions. An absolute-suppositional interpreta-
tion is one in which the discourse-material (here the conditionals) is taken at
face-value, without the need or indeed the possibility for re-interpretation.
Asuppositional interpretation per se is one in which re-interpretation is
allowed.
1.2.2 Probabilistic interpretations
The literature on the suppression task contrasts suppositional readings, as
explained above, with ‘probabilistic’ readings in which the strength of asso-
ciation between antecedent and consequent of the conditionals is variable.
The intuition behind this is that a second conditional premiss may decrease
that strength, in which case a subject is no longer willing to draw a conclu-
sion. One way to formalize this idea is to express the conditional ‘if Athen
B’ as a conditional probability P(B|A) = x. The task is then interpreted
as judging how the arrival of subsequent premisses affects the conditional
probabilities of the truth of the candidate conclusions. However, there are
are other ways of conceiving ‘strength of association’; see for example the
discussion of Chan and Chua [7] below.
There is a clear difference, both theoretically and empirically, between
absolute-suppositional and probabilistic readings; but one outcome of the
analysis will be that the probabilistic interpretation can do the work assigned
to it only if the subject may re-interpret the materials as postulated in the
suppositional interpretation. Armed with these distinctions, we now turn to
a brief inventory of some important previous work on the suppression task,
in chronological order.
1.2.3 Review of work on the suppression task
A number of authors believe that the main problem with the ‘mental mod-
els’ account of the suppression effect lies in its assumption of an absolute
connection between antecedent and consequent. Thus, Chan and Chua [7]
upbraid both the ‘rules’ and ‘models’ camps for failure ‘to give a principled
account of the interpretative component involved in reasoning’. Chan and
Chua propose a ‘salience’ theory of the suppression task, according to which
antecedents are more or less strongly connected with their consequents –
‘operationalised as ratings of perceived importance [of the antecedent in the
second conditional premiss] as additional requirements for the occurrence of
the consequent [7, p. 222].’ Thus Chan and Chua adopt a ‘weak regular-
9
ity’ interpretation of the conditional instead of the material implication (i.e.
more like a conditional probability), and they assume their subjects do too.
They correctly argue that both the ‘rules’ and the ‘models’ theories do not
fit this interpretation, because
[W]e believe that in the suppression experiments, the way subjects cog-
nise the premiss ‘If Rthen Q’ may be more fine-grained than merely
understanding the antecedent as an additional requirement or a possi-
ble alternative.
Their main experimental manipulation accordingly varies the strengths of
the connections between antecedents and consequents, and shows that the
magnitude of the suppression effect depends upon such variations.
George [15] presents strong evidence that subjects vary as to whether
they adopt an absolute-suppositional or a statistical-tendency interpreta-
tion of the materials, and that the proportion of interpretations is strongly
affected by the response modes offered. By getting subjects to first rate
their beliefs in the conditionals used, he showed that about half his subjects
adopted absolute-suppositional interpretations in which modus ponens was
applied regardless of the subjects’ belief in the conditional, whereas for the
other half of subjects there was a strong correlation between their degree of
belief in, and their willingness to draw conclusions from, conditionals. He
further shows that the higher incidence of ‘suppression’ of the conditional in
his population relative to Byrne’s is substantially due to offering response
categories graded in likelihood. George was the first to observe that instead
of being seen as contributing to a debate between mental models and men-
tal logics, Byrne’s results point to some ‘non-standard forms of reasoning
which are outside the scope of both theories’. As far as formal theories of
these non-standard patterns goes, George mentions Collins [8] and Collins
and Michalski [9].
Stevenson and Over [44] present an account of the suppression effect
which is based on a probabilistic interpretation, along with four experi-
ments designed to support the idea that subjects at least may interpret the
materials of Byrne’s experiment in terms of conditional probabilities. Their
methods of encouraging a probabilistic interpretation are interesting. Apart
from the fact that subjects were asked to make judgements of (qualitative)
probabilities of conclusions, in their Experiment 2 the instructions were to
imagine that the premisses were part of a conversation between three people
(our italics). Presumably, having three separate sources was intended to di-
minish the likelihood of subjects focussing on the sources’ intentions about
the relations between the statements ie. to prevent subjects interpreting the
statements as a discourse.
While accepting that some subjects do interpret the materials supposi-
tionally, Stevenson and Over believe that this is somehow unnatural and that
the probability-based interpretations are more natural. Two quotes give a
10
flavour of their position. In motivating the naturalness of a probabilistic
interpretation they argue that
. . . it is, in fact, rational by the highest standards to take proper ac-
count of the probability of one’s premisses in deductive reasoning [44,
p. 615].
. . . 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 [44,
p. 621].
In our opinion, at least two issues should be distinguished here – the
issue what knowledge and belief plays a role in arriving at an interpretation,
and the issue whether the propositions which enter into those interpretations
are absolute or probabilistic. We agree that subjects frequently entertain
interpretations with propositions which they believe to be less than certain.
We agree that subjects naturally and reasonably bring to bear their general
knowledge and belief in constructing interpretations. And we agree that
basing action, judgement or even belief on reasoning requires consideration
of how some discourse relates to the world. But we also believe that sub-
jects can only arrive at less than absolute interpretations (anymore than at
absolute interpretations) by some process of constructing an interpretation.
Stevenson and Over need there to be a kind of discourse in which statements
are already interpreted on some range of models known to both experimenter
and subject within which it is meaningful to assign likelihoods. Most basi-
cally, hearers need to decide what domain of interpretation speakers intend
before they can possibly assign likelihoods.
So how are we to interpret claims that subjects’ interpretations are ‘prob-
abilistic’? First there is a technical point to be made. Strictly speaking, of
course, one is concerned not with the probability of a premiss but with a pre-
sumably true statement about conditional probability. In that sense the sup-
positional and probabilistic interpretation are in the same boat: the authors
cannot mean that subjects may entertain serious doubts about their own
assignments of conditional probabilities. In fact, in this and other papers on
probabilistic approaches to reasoning there is some ambiguity about what is
intended. At one point we read about ‘probability of one’s premisses’, only
to be reminded later that we should not take this as an espousal of the cogni-
tive reality of probability theory: ‘Even relatively simple derivations in that
system are surely too technical for ordinary people [44, p. 638].’ So there is
a real question is to how we are to interpret occurrences of the phrase ‘con-
ditional probability’. Incidentally, Oaksford and Chater [25] present models
of probabilistic interpretations of the suppression task and show that these
models can fit substantial parts of the data from typical suppression exper-
iments. What is particularly relevant in the current context is that these
11
authors, like Stevenson and Over, do not present the probability calculus as
a plausible processing model, but merely as computational-level model in
Marr’s sense – that is a model of what the subjects’ mental processes are
‘designed’ to perform. In contrast, the nonmonotonic logic presented here
as a computational model for the suppression effect is also intended as a
processing model, via the neural implementation given in section 5.
The most important point we want to make about probabilistic inter-
pretations is this: probabilistic and suppositional interpretations share an
important characteristic in that they necessarily require the same interpre-
tative mechanisms. For instance, as designers of expert systems well know,
it is notoriously difficult to come up with assignments of conditional prob-
abilities which are consistent in the sense that they can be derived from a
joint probability distribution. This already shows that the probabilistic in-
terpretation may face the same problems as the suppositional interpretation:
both encounter the need to assure consistency.
Now consider what goes into manipulating conditional probabilities of
rules, assuming that subjects are indeed able to assign probabilities.2Recall
that an advantage of the probabilistic interpretation is supposed to be that
conditional probabilities can change under the influence of new information,
such as additional conditionals. What makes this possible?
Let the variable Yrepresent the proposition ‘she studies late in the li-
brary’, Xthe proposition ‘she has an essay’, and Z: ‘the library is open’.
Probabilistic modus ponens then asks for the probability P(Y|X= 1). It is
clear that if the first conditional were to be modelled as a conditional prob-
ability table of the form P(Y|X), and the second as P(Y|Z), the setup
trivializes, for then the second conditional cannot influence the conditional
probability we are interested in.
What presumably happens instead is that the premisses are first inte-
grated into a Bayesian network which represents causal (in)dependencies,
and which determines the conditional probabilities that have to be esti-
mated. For instance, the processing of an additional premiss as the second
conditional is represented by the move from a Bayesian network of the form
depicted in figure 1, to one of the form as depicted in figure 2. This struc-
ture would make the subject realize that what she can estimate is the con-
ditional probability table for P(Y|X, Z), rather than those for P(Y|X)
and P(Y|Z). The additional character of the premiss is reflected in the
entry in the table which says P(Y= 1 |X= 1, Z = 0) = 0. As a conse-
quence we have P(Y= 1 |X= 1) = P(Y= 1 |X= 1, Y = 1)P(Z= 1),
i.e. probabilistic modus ponens will be suppressed in the absence of further
information about P(Z).
2We are skeptical, but also think probabilistic approaches to reasoning only make sense
under this assumption. In the following we assume subjects are able to construct Bayesian
networks as a form of qualitative probabilistic models.
12
X Y-
Figure 1: Bayesian network for probabilistic modus ponens
X
Y
HHHHHH
Hj
Z
Figure 2: Bayesian network for an additional conditional premiss
Now let us see what happens in the case of an alternative premiss, where
Zis ‘she has a textbook to read’. The specific alternative character can only
be represented by a different Bayesian network, which now involves an OR
variable (defined by the standard truth table), as in figure 3.
More complicated cases can be imagined, involving conditional premisses
which are not clearly either additional or alternative. The upshot of this
discussion of Stevenson and Over [44] is that moving to a probabilistic in-
terpretation of the premisses does not obviate the need for an interpretative
process which (a) constructs a model for the premisses, and (b) conducts
a computation on the basis of that model. This is necessary whether the
premisses are interpreted probabilistically or logically, although doubtless
much more challenging to model in the probabilistic case.
We close with a discussion of a more recent restatement of Byrne’s own
position in [6]. Byrne provides a slightly more detailed explanation of the
suppression effect in the ‘mental models’ framework, pointing to its emphasis
X
OR
HHHHHH
Hj
Z
?
Y
Figure 3: Bayesian network for an alternative conditional premiss
13
on counterexamples
According to the model theory of deduction, people make inferences
according to the semantic principle that a conclusion is valid if there
are no counterexamples to it [6, p. 350]3
and going on to argue against Stevenson and Over that
Our counterexample availability explanation of the suppression effect
. . . suggests that people disbelieve not the truth of the premisses, but
the validity of the conclusion, because they can readily think of a
counterexample to it. On our account, people do not doubt the truth
of either of the conditional premisses: A conditional that has an an-
tecedent that is insufficient but necessary for the consequent is no more
uncertain or doubted than a conditional which has an antecedent that
is sufficient but not necessary [6, p. 353].
The explanation thus attempts to stay with a classical concept of validity
and, in a roundabout way, to retain classical logic’s interpretation of the
conditional as material implication as the ‘basic’ meaning of conditionals.
An additional premiss is taken to make subjects aware that the antecedent
of the main conditional is not sufficient for its consequent, and subjects
are supposed, as a result, to invert the conditional, as evidenced by the
non-suppression of the fallacies denial of the antecedent and affirmation of
the consequent [6, p. 364]. Indeed, going one step further and adding
both an additional and an alternative premiss then suppresses all inferences
(Experiment 4, [6, p. 364ff]). The interpretation of this final experiment is
interesting:
The suppression of valid inferences and fallacies indicates that people
can interpret a conditional within a coherent set of conditionals as
supporting none of the standard inferences whatsoever. A conditional
can be interpreted as a ‘non-conditional,’ that is, as containing an
antecedent that is neither sufficient nor necessary for its consequent
[ibidem]
If one now inquires how this ‘non-conditional’ is defined, one is referred to
Table 2 [6, p. 349] where one finds that it is determined by the truth table
of a tautology. This we take to be a reductio ad absurdum of the attempt to
model conditionals using tools from classical logic only, an attempt already
made inauspicious by the weight of linguistic evidence against it. As we
will see below, nonmonotonic accounts of the conditional allow a more fine-
grained study of when and why the four inference patterns are (in)applicable,
3In principle, a counterexample is different from an exception: the former falsifies a
rule, the latter does not. It seems however that Byrne et al. reinterpret counterexamples
as exceptions, as the next quotations indicate. This terminological confusion is revealing
of the lack of clarity whether mental models is to be interpreted as a classical or as a non-
monotonic logic, and of the model-theoretic rhetoric inducing talk of ‘examples’ where
what is involved are generic rules about boundary conditions.
14
which go much beyond saying that none applies. It therefore comes as
something of a surprise to see that the penultimate section of [6] is entitled
‘Suppression and the non-monotonicity or defeasibility of inferences’, where
it is claimed that ‘the model theory attempts to provide an account of one
sort of non-monotonic reasoning, undoing default assumptions . . . ’ [6, p.
370]. We have been unable to give formal substance to this claim, but if
true the ‘mental models’ solution and the one given here should be mutually
translatable, and we look forward to seeing such a proof.
1.2.4 Conclusions and outlook
A common thread through the articles reviewed here is that their authors try
to explain the suppression effect by assuming a link between the antecedent
and consequent of a rule which is different from material implication. This
is certainly the correct way to go, but it leaves several questions unan-
swered. Human reasoning may well mostly be about propositions which are
less than certain, and known to be less than certain, but our processes of
understanding which meaning is intended have to work initially by credu-
lously interpreting the discourse which describes the situation, and in doing
so accommodating apparent inconsistencies, repairs and revisions. For in-
stance, only after we have some specification of the domain can we start
consistently estimating probabilities in the light of our beliefs about what
we have been asked to suppose. Of course, our judgements about the likely
intended interpretation are as uncertain as all our other judgements, but we
judge how to accomodate utterances in a way that makes them absolutely
consistent.
The good news about the ‘mental models’ approach is that it draws
attention to this interpretative process, but it simultaneously fails to distin-
guish between reasoning to and reasoning from an interpretation, and it is
sorely lacking in detail. The present paper shows that a much more explicit
approach to interpretation is possible once one accepts the lessons from the
authors discussed above (and from linguistic discussions about the meaning
of conditionals; see for example [41, 42, 45, 1]), that subjects generally allow
rules to have exceptions. It will be seen that there is in fact an intimate
connection between exception-handling and interpretation of discourse. As
a first step in that direction, we will now look at an area where efficient
exception-handling is of prime importance, namely planning.
1.3 Logic, working memory and planning
In some ways classical logic is the nemesis of working memory. Consider
what is involved in checking semantically whether an argument of the form
ϕ1, ϕ2/ψ is classically valid. One has to construct a model M, then check
whether M |=ϕ1, ϕ2; if not, discard M; otherwise, proceed to check whether
15
M |=ψ, and repeat until all models have been checked. This procedure puts
heavy demands on working memory, because the models which have to be
constructed are generally not saliently different, so are hard to tell apart.
By the same token, it is not easy to check whether one has looked at all
relevant models. The fact that classical logic does not fit harmoniously
with the operation of working memory already suggests that classical logic
is not the most prominent logic for interpreting discourse. Classical logic
may indeed be an acquired trick as is sometimes maintained, because it
requires overcoming the tyranny of working memory. There may however
be other logics which are very much easier on working memory, for instance
because the number of models to be considered is much lower, or because
these models exhibit salient differences; and we claim that planning provides
a good source for such logics.
By definition, planning consists in the construction of a sequence of ac-
tions which will achieve a given goal, taking into account properties of the
world and the agent, and also events that might occur in the world. Both
humans and nonhuman primates engage in planning. It has even been at-
tested in monkeys. In recent experiments with squirrel monkeys by Mc-
Gonigle, Chalmers and Dickinson [24], a monkey has to touch all shapes
appearing on a computer screen, where the shapes are reshuffled randomly
after each trial. The shapes come in different colours, and the interesting
fact is that, after extensive training, the monkey comes up with the plan of
touching all shapes of a particular colour, and doing this for each colour.
This example clearly shows the hierarchical nature of planning: a goal is to
be achieved by means of actions which are themselves composed of actions.
It is precisely the hierarchical, ‘recursive’ nature of planning which has led
some researchers to surmise that planning has been co-opted by the lan-
guage faculty, especially syntax (Greenfield [17]; Steedman [38]). There is
also a route from planning to language that goes via semantics. There is a
live possibility that a distinguishing feature of human language vis `a vis ape
language is the ability to engage in discourse. Chimpanzees can produce
single sentences, which when read charitably show some signs of syntax.
But stringing sentences together into a discourse, with all the anaphoric
and temporal relations that this entails, seems to be beyond the linguistic
capabilities of apes. One can make a good case, however, that constructing
a temporal ordering of events out of a discourse involves an appeal to the
planning faculty (see van Lambalgen and Hamm [46]).
We defined planning as setting a goal and devising a sequence of actions
that will achieve that goal, taking into account events in, and properties
of the world and the agent. In this definition, ‘will achieve’ cannot mean:
‘provably achieves’, because of the notorious frame problem: it is impossible
to take into account all eventualities whose occurrence might be relevant to
the success of the plan. Therefore the question arises: what makes a good
plan? A reasonable suggestion is: the plan works to the best of one’s present
16
knowledge. Viewed in terms of models, this means that the plan achieves
the goal in a ‘minimal model’ of reality, where, very roughly speaking, every
proposition is false which you have no reason to assume to be true. In
particular, in the minimal model no events occur which are not forced to
occur by the data. This makes planning a form of nonmonotonic reasoning:
the fact that
‘goal Gcan be achieved in circumstances C’
does not imply
‘goal Gcan be achieved in circumstances C+D’
The book van Lambalgen and Hamm [46] formalizes the computations per-
formed by the planning faculty by means of a temporal reasoner (the Event
Calculus) as formulated in a particular type of nonmonotonic logic, namely
first-order constraint logic programming with negation as failure.4Syn-
tactically, logic programming is a good formalism for planning because its
derivations are built on backward chaining (regression) from a given goal.
Semantically, it corresponds to the intuition that planning consists in part
of constructing minimal models of the world. The purpose of [46] is to show
that the semantics of tense and aspect in natural language can be formally
explained on the assumption that temporal notions are encoded in such a
way as to subserve planning. For our present purposes we may abstract
from the temporal component of planning, and concentrate on the skeleton
of the inference engine required for planning, namely propositional logic pro-
gramming. Planning proceeds with respect to a model of the world and it
is hypothesised that the automatic process of constructing a minimal model
which underlies planning also subserves discourse integration. We present
our analysis of the suppression task as evidence for this hypothesis.
Nonmonotonic logics abound, of course,5but logic programming is at-
tractive because it is both relatively expressive and computationally effi-
cient.6Below we shall see that logic programming also has an appealing
implementation in neural nets, and that it may thus shed some light on
the operation of working memory. Taken together, the proven merits of
logic programming in discourse processing [46] and its straightforward im-
plementation in neural networks suggests to us that it is relevant to cognitive
modelling. We are not aware of any other nonmonotonic logic which has this
4This system is originally due to Kowalski and Sergot [20]), with improvements by
Shanahan [34, 35, 33, 36], and the authors of [46].
5See Pelletier and Elio [26] for a plea for more extensive investigations into the psycho-
logical reality of nonmonotonic logics. The present paper proposes that the suppression
effect may be used as a testbed.
6This is because it does not involve the consistency checks necessary for other non-
monotonic logics, such as Reiter’s default logic [31]. This point is further elaborated in
section 6 below.
17
range of features, but definitely do not claim that there cannot be any such
logic.7’8
2 The suppression effect
The suppression task sets subjects the problem of finding an interpretation
which accommodates premises which at least superficially may conflict. It
is therefore provides a good illustration of how default logic can be used to
model human interpretation and reasoning. It is one of the benefits of for-
malisation that it reveals many aspects of the data which are in need of clar-
ification and suggests how finer-grained data might be found, as well as how
the formalism may be modified to account for richer data. As the literature
review above revealed, there are several prominent interpretations subjects
may adopt, and there are certainly more than was mentioned there. All
that is attempted here is to provide an illustrative model of one important
interpretation subjects adopt. We believe this is a reasonable reconstruction
of how Byrne appears to believe her subjects are interpreting the materials.
As mentioned in the introduction, if one presents a subject with the
following premisses:
(4) a. If she has an essay to write she will study late in the library.
b. She has an essay to write.
roughly 90% of subjects9draw the conclusion ‘She will study late in the
library’ (we will later discuss what the remaining 10% may be thinking).
Next suppose one adds the premiss
(5) If the library is open, she will study late in the library.
and one asks again: what follows? In this case, only 60% concludes ‘She
will study late in the library’.
However, if instead of the above, the premiss
(6) If she has a textbook to read, she will study late in the library
7Consider for instance the defeasible planner OSCAR developed by Pollock (see e.g.
[29]). This planner is built on top of a theorem prover for classical predicate logic, and
is thus more expressive than logic programming. But the gain in expressiveness is paid
for by less pleasing computational properties, since such a system cannot be interpreted
semantically by iterations of a fixed point operator, a prerequisite for an efficient neural
implementation.
8An anonymous referee asked whether it might not be possible to model the suppres-
sion effect using assumption-based systems (see e.g. Poole [30]). The short answer is
‘yes’, because the assumption-based framework is so broad that any nonmonotonic logic,
including logic programming, can be modelled in it (see Bondarenko et al. [4]). But the
translation does not necessarily increase perspicuity.
9The figures we use come from the experiment reported in Dieussaert et al. [11].
18
is added, then the percentage of ‘She will study late in the library’–conclusions
is around 95%.
In this type of experiment one investigates not only modus ponens (MP),
but also modus tollens (MT), and the ‘fallacies’ affirmation of the consequent
(AC), and denial of the antecedent (DA), with respect to both types of added
premisses, (5) and (6). In Table 1 we tabulate the relevant data, following
Dieussaert et al. [11], since the experiments reported in this study have
more statistical power than those of Byrne [5].
19
2.1 Logical form
The conclusion that Byrne draws from the experimental results is that
. . . in order to explain how people reason, we need to explain how pre-
misses of the same apparent logical form can be interpreted in quite
different ways. The process of interpretation has been relatively ne-
glected in the inferential machinery proposed by current theories based
on formal rules. It plays a more central part, however, in theories based
on mental models [5, p. 83].
Byrne thus sees the main problem as explaining ‘how premisses of the same
apparent logical form can be interpreted in quite different ways’. We would
question the accuracy of this formulation, and instead prefer to formulate
the main issue as follows: it is the job of the interpretative process to assign
logical form, which cannot simply be read off from the given material. In
other words, the difference between Byrne and ourselves appears to be this:
whereas she takes logical form to be more or less given, we view it as the
end result of a (possibly laborious) interpretative process. This difference
is connected to a different view of what logical form is. Psychologists have
often taken this to mean the formal expression which results from translating
the surface structure of the given sentence into a chosen formal language;
this is apparently how Byrne conceives of logical form. However, from a
logician’s point of view much more is involved. Let Nbe (a fragment of)
natural language. A more complete list10 of what is involved in assigning
logical form to expressions in Nis given by:
1. La formal language into which Nis translated
2. the expression in Lwhich translates an expression in N
3. the semantics Sfor L
4. the definition of validity of arguments ψ1, . . . , ψn/ϕ, with premisses ψi
and conclusion ϕ.
We can see from this list that assigning logical form is a matter of set-
ting parameters. For each item on the list, there are many possibilities for
variation. Take just one example, the choice of a formal language. One
possibility here is the ordinary recursive definition, which has clauses like
‘if A,Bare formulas, then so is A→B’, thus allowing for iteration of the
conditional. However, another possibility, and one which we shall choose, is
where formation of A→Bis restricted to A, B which do not themselves con-
tain a conditional. Furthermore, these parameters are independent: if one
has decided to translate Ninto a propositional language with connectives
¬,∨,∧,→, one is still at liberty to choose a semantics for these connectives;
10See [42, section 1.1] for discussion.
20
and, perhaps more surprisingly, one is also at liberty to choose a definition
of validity. The classical definition of validity: ‘an argument is valid if the
conclusion is true in all models of the premisses’, is but one possibility; the
general form of a nonmonotonic notion of validity: ‘an argument is valid if
the conclusion is valid in all preferred models of the premisses’ is another.
In fact, applying the classical definition of validity means that one must
leave out of consideration all information we happen to have, beyond the
premisses. This is typically almost impossible to achieve for those without
logical training.11
In the remainder of the paper we will reanalyse the data regarding the
suppression effect with the above distinctions in mind, and show their rele-
vance for the process of interpretation and its relation to working memory.
One outcome of the analysis will be that there is no ‘suppression effect’ in
the sense of suppression of formal reasoning – indeed the observed reason-
ing patterns, ‘fallacies’ included, conform to well-known logical forms. We
will illustrate the main logical ideas involved using (propositional) logic pro-
gramming with negation as failure, to which we give a brief introduction in
section 3.12 Before we do so, we briefly discuss some aspects of the meaning
of the natural language conditional.
2.2 A logical form for the conditional
We have seen above that several authors have tried to explain the suppres-
sion effect by assuming that the conditional expresses a regularity rather
than a material implication, as in classical logic. Furthermore, the litera-
ture on the semantics of natural language provides a wealth of data showing
that the identification of the conditional with the material implication is
not generally warranted. It is impossible to do justice to that vast litera-
ture here, so we content ourselves with references to two books containing
important review articles: [45] and [1].
The meaning of the conditional that we shall focus on is that of a law-
like relationship between antecedent and consequent. An example of a law
expressed by means of a conditional is
(7) If a glass is dropped on a hard surface, it will break.
or
(8) If a body is dropped, its velocity will increase as gt2.
What is common to both examples is that the antecedent hides an endless
number of unstated assumptions: in the first case, e.g. that the glass is not
caught before it falls, etc., in the second case, e.g. that there are no other
11Byrne replaced all subjects in her sample who had taken a course in logic.
12For a fuller introduction, consult [12].
21
forces at work on the body, etc..13 We will therefore give the general logical
form of lawlike conditionals ‘if Athen B’ as
(9) If A,and nothing abnormal is the case, then B.
where what is abnormal is provided by the context; we will shortly see ex-
amples of this in the suppression task. The preceding formulation, however,
still explains ‘if . . . then’ in terms of ‘if . . . then’, so we must now inquire
seriously into the meaning of the conditional. We contend that the condi-
tional is often not so much a truth functional connective, as a license for
certain inferences.14 One reason is the role of putative counterexamples, i.e.
situations where Aand not-B; especially in the case of lawlike conditionals,
such a counterexample is not used to discard the conditional, but to look
for an abnormality; it is thus more appropriate to describe it as an excep-
tion. Thus, one use of the conditional is where it is taken as given, not as
a statement which can be false, and we claim that this is the proper way
to view the conditionals occurring in the suppression task, which are after
all supplied by the experimenter.15 Stenning and van Lambalgen [41, 42]
use these same observations about natural language conditionals to explain
many apparently unrelated phenomena in Wason’s selection task.
Having posited that, in the present context, the conditional is rather
a license for inferences than a connective, we must determine what these
inferences are. One inference is, of course, modus ponens: the premisses A
and ‘if Athen B’ license the inference that B. The second type of inference
licensed by the conditional may be dubbed ‘closed world reasoning’: it says
that if it is impossible to derive a proposition Bfrom the given premisses
13In this respect, the conditional provides an interesting contrast to the universal quan-
tifier, with which it is often aligned. To slightly adapt an example due to Nelson Goodman:
one can say
(i) All the coins in my pocket are copper.
in order to express a contingent generalisation, but one would not so readily, with the
same intent, say
(ii) If a coin is in my pocket, it is copper.
precisely because it is hard to imagine the law-like connection between antecedent and
consequent which the conditional suggests. The attempt to interpret this tends to conjure
scenarios of reverse alchemy, in which gold coins just moved into the pocket turn to copper.
14A connective differs from a license for inference in that a connective, in addition to
licensing inferences, also comes with rules for inferring a formula containing that connective
as main logical operator.
15The preceding considerations imply that the conditional cannot be iterated. Natural
language conditionals are notoriously hard (although not impossible) to iterate, especially
when a conditional occurs in the antecedent of another conditional – ‘If, if conditionals
are iterated, then they aren’t meaningful, then they aren’t material’ is an example; one
more reason why ‘if ... then’ is not simply material implication.
22
by repeated application of modus ponens, then one may assume Bis false.
This kind of reasoning is routinely applied in daily life: if the timetable says
there is no train leaving Edinburgh for London between 5 pm and 5.10 pm,
one assumes there is no such train scheduled. Closed world reasoning is
what allows humans to circumvent the notorious frame problem (at least to
some extent): my reaching for a glass of water may be unsuccessful for any
number of reasons, for instance because the earth’ gravitational field changes
suddenly, but since I have no positive information that this will happen, I
assume it will not. We hypothesise that closed world reasoning plays an
important part in reasoning with conditionals; in particular, the suppression
effect will be seen to be due to a special form of a closed world assumption.
We now recast the preceding considerations as a formal definition in logic
programming. The next section is inevitably somewhat technical.
3 Logic programming
A good framework for a conditional with the properties outlined above is
logic programming, a fragment of propositional (or predicate) logic with a
special, nonclassical, semantics. For the sake of clarity we start with a
fragment of logic programming in which negation is not allowed; but note
that the proposed notion of conditionals does require negation.
Definition 1 A(positive) clause is a formula of the form p1, . . . pn→q,
where the q,piare propositional variables; the antecedent may be empty.
In this formula, qis called the head, and p1, . . . pnthe body of the clause.
A(positive) program is a finite set of positive clauses.
Until further notice, we assume that propositions are either true (1) or false
(0), but the semantics is nonetheless nonclassical. The only models to be
considered are those of the following form
Definition 2 Let Pbe a positive program on a finite set Lof proposition
letters. An assignment Mof truthvalues {0,1}to L(i.e. a function M:
L−→ {0.1}) is a model of Pif for q∈L,
1. M(q) = 1 if there is a clause p1, . . . pn→qin Psuch that for all i,
M(pi) = 1
2. M(q)=0if for all clauses p1, . . . pn→qin Pthere is some pifor
which M(pi) = 0.
The definition entails that for any qnot occurring as the head of a clause,
M(q) = 0. More generally, the model Mis minimal in the sense that a
proposition not forced to be true by the program is false in M; this is our
first (though not final) formulation of the closed world assumption.
We will next liberalize the preceding definitions and allow negation in
the body of a clause.
23
Definition 3 A(definite) clause is a formula of the form (¬)p1∧. . . ∧
(¬)pn→q, where the piare either propositional variables, >or ⊥16 , and
qis a propositional variable. Facts are clauses of the form > → q, which
will usually be abbreviated to q. Empty antecedents are no longer allowed.
Adefinite logic program is a finite conjunction of definite clauses.
In order to give a semantics for negation, the closed world assumption is
internalised to what is known as ‘negation as failure’: ¬ϕis true if the
attempt to derive ϕfrom the program Pfails. The proper semantics for
definite programs requires a move from two-valued logic to Kleene’s strong
three-valued logic (introduced in [19, p. 332ff]), which has the truth values
undecided (u), false (0) and true (1). The meaning of undecided is that the
truth value can evolve toward either true or false (but not conversely).17
This semantics will be introduced in greater detail below. It is an important
fact, however, that the models of interest can be captured by means of the
following construction
Definition 4 (a) The completion of a program Pis given by the following
procedure:
1. take all clauses ϕi→qwhose head is qand form the expression
Wiϕi→q18
2. replace the →’s by ↔’s (here, ↔has a classical interpretation given by:
ψ↔ϕis true if ψ, ϕ have the same truth value, and false otherwise).
3. this gives the completion of P, which will be denoted by comp(P).
(b) If Pis a logic program, define the nonmonotonic consequence relation
|≈ by
P|≈ ϕiff comp(P)|=ϕ.
If P|≈ ϕ, we say that ϕfollows from Pby negation as failure, or by closed
world reasoning. The process of completion is also referred to as minimisa-
tion.19
16We use >for an arbitrary tautology, and ⊥for an arbitrary contradiction
17This is therefore different from saying that the truth value is fuzzy, which requires a
linear order of truth values, with between 0 and 1.
18In the customary definitions it is assumed that if there is no such ϕi, then the expres-
sion ⊥ → qis added. This case cannot occur here.
19Readers who recall McCarthy’s use of an abnormality predicate ab may wonder why
we do not use circumscription [23] instead of logic programming, since circumscription
also proceeds by minimising the extension of ab. There is a technical reason for this:
circumscription cannot be easily used to explain the fallacies; but the main reason is
that logic programming unlike circumscription allows incremental computation of minimal
models, and this computation will be seen to be related to the convergence toward a
stable state of a neural network. This shows why model-construction can here proceed
automatically.
24
Using the terminology introduced above, our main hypothesis in explaining
Byrne’s data as conforming to the logical competence model of closed world
reasoning can then be stated as
(a) the conditionals used in the suppression task, far from being
material implications, can be captured much more adequately
by logic programming clauses of the form p∧ ¬ab →q, where
ab is a proposition letter indicating that something abnormal is
the case;
(b) when making interpretations, subjects usually do not con-
sider all models of the premisses, but only minimal models, de-
fined by a suitable completion.
The readers who wish to see some applications of these notions to the sup-
pression task before delving into further technicalities, may now jump ahead
to section 4, in particular the explanations with regard to the forward infer-
ences MP and DA. The backward inferences MT and AC require a slightly
different application of logic programming, which will be introduced in the
next subsection. The final subsection will look at the construction of models,
which is necessary for the connection with neural networks.
3.1 A strengthening of the closed world assumption: in-
tegrity constraints
So far we have applied the closed world assumption to atomic formulas (for
instance ab) and their negations: if ab is not in the database, we may assume
it is false. We now extend the closed world assumption to cover program
clauses as well: if ϕ1→q, . . . , ϕn→qare all the clauses with nontrivial
body which have qas head, then we (defeasibly) conclude that qcan only
be the case because one of ϕ1, . . . , ϕnis the case. We therefore do not
consider the possibility that qis the case because some other state of affairs
ψobtains, where ψis independent of the ϕ1, . . . , ϕnand such that ψ→q.
This kind of reasoning might be called ‘diagnostic’.
This notion of closed world is not quite expressed by the completion of a
program. For in the simple case where we have only a clause p→qand a fact
qis added, the completion becomes (p∨ >)↔q, from which nothing can be
derived about p. What we need instead is a principled way of adding qsuch
that the database or model is updated with p. The proper technical way of
achieving this is by means of so-called integrity constraints. To clarify this
notion, we need a small excursion into database theory, taking an example
from Kowalski [21, p. 232].
An integrity constraint in a database expresses obligations and prohi-
bitions that the states of the database must satisfy if they fulfill a certain
condition. For instance, the ‘obligation’ to carry an umbrella when it is rain-
ing may be formalized (using a self-explanatory language for talking about
25
actions and their effects) by the integrity constraint
Holds(rain, t)→Holds(carry −umbrella, t).(1)
The crucial point here is the meaning of →. The formula 1 cannot be an
ordinary program clause, for in that case the addition of Holds(rain, t) would
trigger the consequence Holds(carry −umbrella, t) which may well be false,
and in any case does not express an obligation.
A better way to think of an integrity constraint is to view the consequent
as a constraint that the database must satisfy if the antecedent holds. This
entails in general that the database has to be updated with a true statement
about the world. First a piece of terminology. A formula ϕis used as a query,
and denoted ?ϕ, if one tries to determine whether ϕfollows from a program
P. The ‘success’ or ‘failure’ of a query is usually defined syntactically, but
we will not introduce a derivational apparatus here, and provide only a
semantic characterization: a query ?ϕsucceeds with respect to a program
P, if comp(P)|≈ ϕ, i.e. if ϕis entailed by the completion of P. Likewise, a
query ?ϕfails with respect to a program P, if comp(P)|≈ ¬ϕ.
To return to our example, there will be an action take −umbrella, linked
to the rest of the database by
Initiates(take −umbrella,carry −umbrella, t).
Suppose the database contains Holds(rain,now ), then the integrity con-
straint requires us to update the database in such a way that the query
?Holds(carry −umbrella,now ),
succeeds. The appropriate way to do so is, of course, to take an umbrella
and inform the database that one has done so.
It is also possible to have an integrity constraint without a condition,
such as Holds(rain, t) in the above example. An entry in someone’s diary
like ‘appointment in Utrecht, Friday at 9.00’ expresses an unconditional
obligation to satisfy HoldsAt(be-in-Utrecht,Friday-at-9.00), and presented
with this integrity constraint, the internal database comes up with a plan to
satisfy the constraint. Such unconditional integrity constraints are especially
useful for the kind of examples discussed here. Readers who wish to see
applications to the suppression task may now jump ahead to the second
part of section 4, which treats the backward inferences MT and AC. The
next (and final) subsection is only essential for the neural implementation.
3.2 Constructing models
In this last subsection, we explain how minimal models can be efficiently
computed given a definite logic program. As above we start with a simpler
case. Recall that a positive logic program has clauses of the form p1∧. . . ∧
26
pn→q, where the pi, q are proposition letters and the antecedent (also
called the body of the clause) may be empty. Models of a positive logic
program Pare given by the fixed points of a monotone20 operator:
Definition 5 The operator TPassociated to Ptransforms a model M(viewed
as a function M:L−→ {0,1}, where Lis the set of proposition letters) into
a model TP(M)according to the following stipulations: if vis a proposition
letter,
1. TP(M)(v)=1if there exists a set of proposition letters C, true on
M, such that VC→v∈P
2. TP(M)(v) = 0 otherwise.
Definition 6 An ordering ⊆on models is given by: M ⊆ N if all proposi-
tion letters true in Mare true in N.
Lemma 1 If Pis a positive logic program, TPis monotone in the sense
that M ⊆ N implies TP(M)⊆TP(N).
This form of monotonicity would fail if a body of a clause in Pcontains a
negated atom ¬qand also a clause ¬q→s: one can then set up things in
such a way that sis true at first, and becomes false later. Hence we will
have to complicate matters a bit when considering negation, but this simple
case illustrates the use of monotone operators. Monotonicity is important
because it implies the existence of so called fixed points of the operator TP.
Definition 7 A fixed point of TPis a model Msuch that TP(M) = M.
Lemma 2 If TPis monotone, it has a least and a greatest fixed point. The
least fixed point will also be called the minimal model.
Monotonicity is also important because it allows incremental computation
of the minimal model. As noted above, by itself circumscription does not
give this computational information. Computability is a consequence of the
syntactic restrictions on logic programs.
So far we have only modelled positive programs, but the logic programs
that we need must allow negation in the body of a clause, since we model
the conditional ‘pimplies q’ by the clause p∧ ¬ab →q. As observed above,
extending the definition of the operator TPwith the classical definition of
negation would destroy its monotonicity, necessary for the incremental ap-
proach to the least fixed point. One solution is to replace the classical
two-valued logic by a particular form of three-valued logic, Kleene’s strong
three-valued logic, designed for modelling the process whereby reasoning al-
gorithms take us from uncertainty to definitive values [19, p. 332ff]. This
20Monotonicity in this sense is also called continuity.
27
p¬p
1 0
0 1
u u
p q p ∧q
1 1 1
0 0 0
u u u
1 0 0
1u u
0 1 0
0u0
u1u
u0 0
p q p ∨q
1 1 1
0 0 0
u u u
1 0 1
1u1
0 1 1
0u u
u1 1
u0u
Figure 4: Three-valued connectives
logic has truth values {u, 0,1}with the partial order u≤0 and u≤1. Here,
uis not a degree of truth, but rather means that the truth value is, so far,
undecided. The chosen ordering reflects the intuition that ucan ‘evolve’
toward 0 or 1 as a result of computation. The truth tables given in figure
4 (taken from [19, p. 334]) are then immediate, as readers should satisfy
themselves.
In addition we define an equivalence ↔by assigning 1 to ϕ↔ψif ϕ,ψ
have the same truth value (in {u, 0,1}) , and 0 otherwise.
We show how to construct models for such programs, as fixed points of
a three-valued consequence operator T3
P. We will drop the superscript when
there is no danger of confusion with its two-valued relative defined above.
Definition 8 A three-valued model is an assignment of the truth values
u, 0,1to the set of proposition letters. If the assignment does not use the
value u, the model is called two-valued. If M,Nare models, the relation
M ≤ N means that the truth value of a proposition letter pin Mis less
than or equal to the truth value of pin Nin the canonical ordering on u, 0,1.
Definition 9 Let Pbe a program.
a. The operator TPapplied to formulas constructed using only ¬,∧and
∨is determined by the above truth tables.
b. Given a three-valued model M,TP(M)is the model determined by
(a) TP(M)(q) = 1 iff there is a clause ϕ→qsuch that M |=ϕ
(b) TP(M)(q)=0iff there is a clause ϕ→qin Pand for all such
clauses, M |=¬ϕ
The preceding definition ensures that unrestricted negation as failure ap-
plies only to proposition letters qwhich occur in a formula ⊥ → q; other
28
proposition letters about which there is no information at all may remain
undecided.21 This will be useful later, when we will sometimes want to re-
strict negations as failure to ab. Once a literal has been assigned value 0 or
1 by TP, it retains that value at all stages of the construction; if it has been
assigned value u, that value may mutate into 0 or 1 at a later stage.
Lemma 3 If Pis a definite logic program, TPis monotone in the sense
that M ≤ N implies TP(M)≤TP(N).
Here are three essential results, which will turn out to be responsible for
the efficient implementability in neural networks.
Lemma 4 Let Pbe a program.
1. Mis a model of the comp(P)iff it is a fixed point of TP.
2. The least fixed point of TPis reached in finitely many steps (n+ 1 if
the program consists of nclauses).
Lemma 5 If Pis a definite logic program, TPis monotone in the sense
that M ≤ N implies TP(M)≤TP(N).
Lemma 6 1. The operator T3
Phas a least fixed point. The least fixed
point of T3
Pwill be called the minimal model of P.
2. All models Mof comp(P)are fixed points of T3
P, and every fixed point
is a model.
In this context, the nonmonotonic consequence relation P|≈ ϕ(see defini-
tion 4) is given by ‘comp(P)|=3ϕ’, or in words: all (three-valued) models
of comp(P) satisfy ϕ. Observe that the relation |≈ is completely determined
by what happens on the least fixed point. Larger fixed points differ in that
some values uin the least fixed point have been changed to 0 or 1 in the
larger fixed point; but by the monotonicity property (with respect to truth
values) of Kleene’s logic this has no effect on the output unit pairs, in the
sense that an output value 1 cannot be changed into 0 (or conversely).
4 How this explains nonclassical answers in the
suppression task
The explanation of Byrne’s data will be presented in two stages, correspond-
ing to the forward inferences (MP and DA) and the backward inferences (MT
and AC).
21This parallels a similar proviso in the definition of the completion.
29
Before we present the explanation we want to make a remark of a
methodological nature. Once one acknowledges that there are many alter-
native logical models besides classical logic, logic takes on a more interesting
combination of normative and descriptive roles in empirical investigations.
So, if subjects adopt the processing goals implicit in our proposed nonmono-
tonic logic, then ‘correct’ performance must be judged by that model. This
offers us the possibility of explaining, for example, two groups of subjects
drawing different conclusions in one of Byrne’s conditions as both conforming
to competence models – but different ones. Obviously, if there are sufficient
unconstrained competence models to fit any data, then our enterprise is not
an empirical one. But the reverse is true. We can readily seek corroborating
data by seeing whether subjects’ discourse-processing goals (their construal
of the task) accords with the strict assumptions of the proposed competence
models. In particular it seems that subjects can have two main discourse-
processing goals in this type of task: accomodating the speaker’s utterances,
or questioning them. As we have seen, classical logic is appropriate to the
latter goal, but not to the former. Below, we will only explain what the
appropriate pattern of answers is on the assumption that subjects are acco-
modating. The reader will readily supply the appropriate answers for the
other alternative22.
4.1 The forward inferences: MP and DA
We will represent the conditionals in Byrne’s experiment as definite clauses
of the form p∧ ¬ab →q, where ab is a proposition which indicates that
something abnormal is the case, i.e. a possibly disabling condition.
Definition 10 For our purposes, a program is a finite set of conditionals
of the form A1∧. . . ∧An∧ ¬ab →B, together with the clauses ⊥ → ab for
all proposition letters of the form ab occurring in the conditionals. Here, the
Aiare propositional variables or negations thereof, and Bis a propositional
variable. We also allow the Aito be >and ⊥. Empty antecedents are not
allowed.23.
In the following we will therefore represent (10-a) as (10-b)
(10) a. If she has an essay, she will study late in the library.
b. p∧ ¬ab →q
and (11-a) and (11-b) both as (11-c)
22We do not have a model which explains the exact distribution of frequencies of an-
swers. If the present analysis is correct, such a model would require variables determining
discourse-processing goals.
23As mentioned before, this definition is formulated because we do not necessarily want
to apply closed world reasoning to all proposition letters, although always to proposition
letters of the form ab.
30
(11) a. If the library is open, she will study late in the library.
b. If she has a textbook to read, she will study late in the library.
c. r∧ ¬ab0→q
It is essential that the conditionals are represented as being part of a defi-
nite logic program, so that they function as licenses for inference rather than
truthfunctional connectives. We show that on the basis of this interpreta-
tion, the forward inferences MP and DA and their ‘suppression’ correspond
to valid argument patterns. The main tool used here is the completion of
a program, as a formalisation of the closed world assumption as applied to
facts. We emphasise again that in virtue of lemmas 4 – 6, the completion
is really shorthand for a particular model. Thus, what we will be modelling
is how subjects reason toward an interpretation of the premisses by suitably
adjusting the meaning of the abnormalities. Once they have reached an
interpretation, reasoning from that interpretation is trivial.
The ‘backward’ inferences (MT and AC) require the closed world as-
sumption as applied to rules, and will be treated separately, in section 4.2.
MP for a single conditional premiss Suppose we are given a single
conditional p∧ ¬ab →qand the further information that p(i.e. > → p).
The full logic program for this situation is {p;p∧¬ab →q;⊥ → ab}. Closed
world reasoning as formalised in the completion gives the set {p;p∧ ¬ab ↔
q;⊥ ↔ ab}, which is equivalent to {p;p↔q}, from which qfollows. This
argument can be rephrased in terms of our distinction between two forms
of reasoning as follows.
Reasoning to an interpretation starts with the general form of a condi-
tional24, and the decision to apply nonmonotonic closed world reasoning.
As a consequence, one derives as the logical form of the conditional p↔q.
Reasoning from an interpretation then starts from this logical form and the
atomic premiss p, and derives q.
A ‘fallacy’: DA for a single conditional premiss Suppose we are
again given a conditional p∧ ¬ab →qand the further information ¬p.25
Reasoning to an interpretation starts from the program {¬p;p∧ ¬ab →
q;⊥ → ab}and the closed world assumption. As above, the end result of
that reasoning process is {¬p;p↔q}. Reasoning from an interpretation
then easily derives ¬q.
24There is nothing sacrosanct about this starting point. It may itself be the consequence
of a reasoning process, or a different starting point, i.e. a different interpretation of the
conditional may be chosen. The formalisation chosen is a first approximation to the idea
that natural language conditionals allow exceptions. For other purposes more complex
formalisations may be necessary.
25Strictly speaking ¬pis not an allowed clause, but we may interpret ¬pas obtained
from the allowed clause ⊥ → pby closed world reasoning.
31
MP in the presence of an additional premiss As we have seen, if the
scenario is such that nothing is said about ab, minimisation sets ab equal to
⊥and the conditional p∧ ¬ab →qreduces to p↔q. Now suppose that the
possibility of an abnormality is made salient, e.g. by adding a premiss ‘if the
library is open, she will study late in the library’ in Byrne’s example. We
propose that this results in the addition of a clause such as ¬r→ab, because
the possibility of an ab-normality is highlighted by the additional premiss.26
Although this is not essential, the situation may furthermore be taken to
be symmetric, in that the first conditional highlights a possible abnormality
relating to the second conditional. The circumstance that the library is open
is not a sufficient incentive to go and study there, one must have a purpose
for doing so.27 This means that the further condition ¬p→ab0for ab 0is
added. That is, reasoning toward an interpretation starts with the set
{p;p∧ ¬ab →q;r∧ ¬ab0→q;⊥ → ab;⊥ → ab0;¬r→ab ;¬p→ab0}.
Applying closed world reasoning in the form of the completion yields
{p; (p∧ ¬ab)∨(r∧ ¬ab 0)↔q; (⊥∨¬r)↔ab; (⊥∨¬p)↔ab0},
which reduces to {p; (p∧r)↔q}. Reasoning from an interpretation is now
stuck in the absence of information about r.
Here we see nonmonotonicity at work: the minimal model for the case
of an additional premiss is essentially different from the minimal model of a
single conditional premiss plus factual information.
DA in the presence of an additional premiss Now suppose we have
as minor premiss ¬pinstead of p. Reasoning toward to an interpretation
derives as above the set {¬p; (p∧r)↔q}. Since ¬(p∧r), reasoning from
an interpretation concludes ¬q. It follows that ‘denying the antecedent’ will
not be suppressed, as observed.
MP and DA for an alternative premiss The difference between this
case and the previous one is that, by general knowledge, the alternatives do
not highlight possible obstacles. This means that the clauses ¬r→ab;¬p→
ab0are lacking. Reasoning to an interpretation thus starts with the set
{p;p∧ ¬ab →q;r∧ ¬ab0→q;⊥ → ab;⊥ → ab0}
Closed world reasoning converts this to
{p; (p∧ ¬ab)∨r∧ ¬ab 0)↔q;⊥ ↔ ab;⊥ ↔ ab0},
26Obviously, this is one place where general knowledge of content enters into the selection
of appropriate logical form. Nothing in the form of the sentence tells us that the library
being open is a boundary condition on her studying late in the library.
27Evidence for such symmetry can be found in Experiment 1 of Byrne et al. [6].
32
which reduces to {p; (p∨r)↔q}. Reasoning from an interpretation then
easily derives q: no suppression.
Now consider ‘denial of the antecedent’. We interpret ¬pagain as ob-
tained from ⊥ → pby closed world reasoning. The completion then becomes
(p∨r↔q)∧(p↔ ⊥), and reasoning from an interpretation is stuck: suppres-
sion. Indeed, in Byrne’s study [5] DA for this type of problem was applied by
only 4% of the participants. However, in Dieussaert et al.’s study [11] 22%
applied DA in this case. This could be a consequence of applying negation
as failure to ras well, instead of only to abnormalities. The competence
model allows both choices.
4.2 The backward inferences: MT and AC
As we have seen, the forward inferences rely on the completion, that is, the
closed world assumption for facts. We propose that the backward inferences
rely in addition on the closed world assumption for rules. When we come
to discuss the neural implementation of the formalism we will see that this
explains to some extent why backward inferences are perceived to be more
difficult. This section uses the material on integrity constraints introduced
in section 3.1.
AC and MT for a single conditional premiss Suppose we have a
single conditional premiss p∧ ¬ab →qand a fact q. Closed world reasoning
about facts would yield the completion {((p∧ ¬ab)∨ >)↔q;ab ↔ ⊥}, from
which nothing can be concluded about p.
But now assume that reasoning to an interpretation sets up the problem
in such a way that AC is interpret as an integrity constraint, that is, as the
statement
if ?qsucceeds, then ?psucceeds.
In this case, closed world reasoning for rules can be applied and we may
ask what other atomic facts must hold if qholds. Since the only rule is
p∧ ¬ab →q, it follows that p∧ ¬ab must hold. For ¬ab this is guaranteed
by closed world reasoning about facts, but the truth of pmust be posited.
In this sense AC is valid.
For MT, the reasoning pattern to be established is
if ?qfails, then pfails.
One starts from the integrity constraint that the query ?qmust fail. This
can only be if at least one of pand ¬ab fails. Since in this situation we know
that ¬ab is true (by closed world reasoning for facts), we must posit that p
is false.28
28This shows that subjects may do a modus tollens inference for the ‘wrong’, i.e. non-
33
AC and MT for an additional premiss In the case of an additional
premiss, the program consists of
p∧ ¬ab →q, r ∧ ¬ab0→q, ¬p→ab0,¬r→ab.
Consider AC: here we start with the integrity constraint that the query ?q
succeeds. It follows that at least one of p∧ ¬ab,r∧ ¬ab0must be true. But
given the information about the abnormalities furnished by closed world
reasoning for facts, namely ¬r↔ab and ¬p↔ab0, in both cases this means
that pand rmust true, so that AC is supported.
Now consider MT, for which we have to start from the integrity con-
straint that ?qmust fail. The same reasoning as above shows that at least
one of p,rmust fail – but we don’t know which. We thus expect suppression
of MT.
AC and MT for an alternative premiss In the case of AC, an alter-
native premiss leads to clear suppression (from 55% to 16%). It is easy to
see why this must be so. Closed world reasoning for facts reduces the two
conditional premisses to p→qand r→q. Given that ?qmust succeed,
closed world reasoning for rules concludes that at least one of p,rmust be
true – but we don’t know which. It is interesting at this point to look at
an experiment in Dieussaert et al. [11] where subjects are allowed to give
compound answers such as p∧qor p∨q.29 For AC, 90.7% subjects then
chose the nonmonotonically correct p∨r.
Turning to MT, we do not expect suppression, and indeed, the require-
ment that ?qfails means that both pand rhave to be false. In Dieussaert et
al.’s data for MT, 96.3% of the subjects, when allowed to choose compound
answers, chose the classically correct ¬p∧ ¬r.
4.3 Competence and performance
We derived the pattern of suppression and non-suppression observed by
Byrne and others by means of a nonclassical competence model, namely
classical, reason. There may be a relation here with the interesting observation that for
children, the rate of MT seems to increase to around 74% for 10–12 year olds, only to
decrease again for older children, to the usual percentage of around 50% (see [14] for dis-
cussion) Could it be that the younger children are good at MT for the wrong reason, and
that the later decline is due to a not yet fully sufficient mastery of the classical semantics?
29These authors claim that Byrne’s experiment is flawed in that a subject is allowed to
judge only for an atomic proposition or its negation whether it follows from the premisses
supplied. This would make it impossible for the subjects that draw a conclusion which
pertains to both conditional premisses. Accordingly, they also allow answers of the form
(¬)A(∧)(∨)(¬)B, where A, B are atomic. Unfortunately, since they also require subjects
to choose only one answer among all the possibilities given, the design is flawed. This is
because there exist dependencies among the answers (consider e.g. the set {p∨q , p∧q, p, q})
and some answers are always true (e.g. p∨ ¬p). Thus, the statistics yielded by the
experiment are unfortunately not interpretable.
34
closed world reasoning with facts and rules, applied in the service of con-
structing a model of the discourse. We have thus adopted what Dieussaert
et al. [11] call an ‘integration strategy’: the premisses, both conditionals
and facts, are taken jointly when constructing a minimal model, and it is
assumed that no more premisses will be supplied.
The data obtained by Dieussaert et al. [11] suggest however that some
answers may not so much reflect a competence model as processing con-
straints. Dieussaert et al. observed that in the case of MT for an additional
premiss, 35.3% of subjects, when allowed to choose compound answers, chose
¬p∧ ¬r, whereas 56.9% chose the classically correct ¬p∨ ¬r. The second
answer is readily explained in the present framework. The first answer can
be explained if subjects first set ab,ab0to ⊥, and report the result of this
intermediate computation: for if ?qfails, both p∧ ¬ab and r∧ ¬ab0must
be false. Since by assumption ¬ab and ¬ab0are true, both ¬pand ¬rmust
be true. We hypothesize that in a case such as this, some subjects simplify
their handling of the abnormalities, to reduce the complexity of the correct
derivation. This type of answer does not conform to a particular compe-
tence model, but reflects processing constraints. It is argued in [47] that
these processing constraints play a role in autism as well.
4.4 Summary
Let us retrace our steps. Byrne claimed that both valid and invalid infer-
ences can be suppressed, based on the content of supplementary material;
therefore, the form of sentences would determine only partially the conse-
quences that people draw from them. Our analysis is different. Consider
first the matter of form and content. We believe that logical form is not
simply read off from the syntactic structure of the sentences involved, but is
assigned on the basis of ‘content’ – not only that of the sentences themselves,
but also that of the context. In this case the implied context – a real-life
situation of going to a library – makes it probable that the conditionals
are not material implications but some kind of defaults. We then translate
the conditionals, in conformity with this meaning, into a formal language
containing the ab,ab’, . . . formulas; more importantly, in this language the
conditional is a special, non-iterable non-truthfunctional connective. How-
ever, translation is just the first step in imposing logical form. The second
step consists in associating a semantics and a definition of validity to the
formal language. For example, in our case the definition of validity is given
in terms of minimal models, leading to a nonmonotonic concept of validity.
Once logical form is thus fixed (but not before!), one may inquire what fol-
lows from the premisses provided. In this case the inferences observed in
the majority of Byrne’s subjects correspond to valid inferences (given the
assignment of logical form). Hence we would not say that content has beaten
form here, but rather that content contributes to the choice of a logical form
35
appropriate to content and context. There are many different ways of as-
signing logical form, and that of classical logic is not by any means the most
plausible candidate for common sense reasoning; indeed default systems can
provide several models of the variety of reasoning behaviour observed.
As regards Byrne’s positive suggestion that ‘mental models’ provides a
better account of what goes in a subject who suppresses MP, we do not dis-
pute that such subjects construct semantic representations (‘models’) which
do not support MP. The present proposal seems however to add more de-
tail as to how the construction is done, namely as a form of closed world
reasoning analogous to what is necessary for planning.
We now propose that it is possible to go further and use these systems
as a basis for proposals about cognitive architecture, in particular neural
implementation.
5 Closed world reasoning and working memory
We will now show that the computations underlying the suppression effect
can actually be performed very efficiently in suitable neural networks. The
observation that there is a strong connection between logic programming
and neural nets is not new (see d’Avila Garcez, Broda and Gabbay [2]),
but what is new here is a very straightforward modelling of closed world
reasoning (negation as failure) by means of coupled neural nets. This ex-
ploits the soundness and completeness of negation as failure with respect
to Kleene’s strong three-valued logic. What is of importance here is that
the relevant models can also be viewed as stable states of a neural network,
obtained by a feedforward computation mimicking the action of the con-
sequence operator associated to the logic program. We first present some
pertinent definitions30.
5.1 Neural nets
Definition 11 Acomputational unit, or unit for short, is a function with
the following input-output behaviour
1. inputs are delivered to the unit via links, which have weights wj∈R
2. the inputs can be both excitatory or inhibitory; let x1. . . xn∈Rbe
excitatory, and y1. . . ym∈Rinhibitory
3. if one of the yifires, i.e. yi6= 0, the unit is shut off, and outputs 0
4. otherwise, the quantity Pi=n
i=1 xiwiis computed; if this quantity is greater
than or equal to a threshold θ, the unit outputs 1, if not it outputs 0
30For expository purposes we consider only very simple neurons, whose thresholds are
numbers, instead of functions such as the sigmoid.
36
5. we assume that this computation takes one time-step.31
Definition 12
1. A spreading activation network is a directed graph on a set of units,
whose (directed) edges are called links.
2. A (feedforward) neural network is a spreading activation network with
two distinguished sets of units, I(input) and O(output), with the
added condition that there is no path from a unit32 in Oto one in I.
Neural networks and spreading activation networks differ in some respects.
A spreading activation network is typically conceived of as consisting of
units which fire continuously, subject to a decay, whereas in neural nets
one commonly considers units which fire once, when triggered. Some neural
networks are equipped with a ‘nice’ structure, in order to facilitate the action
of the backpropagation algorithm; e.g., one assumes that the network is
composed of input, output and an ordered set of hidden layers, such that
units are only connected to (all units of) the next layer. Spreading activation
networks require no such assumption. Although we talk of ‘backpropogation’
in our spreading activation networks to describe a process analogous to
the learning algorithm in neural nets, it does not have some of the ‘nasty’
neural implausibilities of the latter in this context. Our construction will
need a bit of both kinds of network; in logic programming one typically
computes models, which correspond to stable patterns of activation, but
sometimes one focusses on models which make a designated output true, as
when considering integrity constraints.
We now propose that the fixed points of the consequence operator as-
sociated to a program correspond to stable states in a spreading activation
network derived from the program. We start again with positive programs;
here the correspondence is due to d’Avila Garcez, Broda and Gabbay [2].
5.1.1 Example
Consider the language L={p, q, r, s, t}, and following example of a program
Pin L:P={p, p →q , q ∧s→r}. We start from the empty model M0=
∅, i.e. the model in which all proposition letters are false. We then get the
following computation:
1. M1is given by TP(M0)(p) = 1, TP(M0)(q) = TP(M0)(r) = TP(M0)(s)
=TP(M0)(t)=0
31This assumption is customary for spreading activation networks and recurrent neural
nets, although not for feedforward nets.
32The term ‘unit’ is unfortunate here, since strictly speaking input units do not compute
anything; they are just nodes where data is fed into the network.
37
p
q
s
r
t
AND
Figure 5: Network associated to {p, p →q, q ∧s→r}
2. M2is given by TP(M1)(p) = 1, TP(M1)(q) = 1, TP(M1)(r) =
TP(M1)(s) = TP(M1)(t) =0
The model M2is a fixed point of TPin the sense that TP(M2) = M2. It
is also the least fixed point; we may consider M2to be the minimal model
in the sense that it makes true as few proposition letters as possible. M2is
not the only fixed point, since M3={p, q, r, s}is also one. However, such
greatest fixed points are neurally less plausible; see below.
Figure 5 shows our proposal for the the spreading activation network
associated to the program P. Until further notice all links will be assumed
to have weight 1. The important design decision here is that →is not
represented as a unit, but as a link – this is the neural correlate of the earlier
observation that the conditional often does not act as a truthfunctional
connective.
In this picture, the node labelled AND is a unit computing conjunction.
The time-course of the spreading of activation in this network mimics the
action of the monotone operator TP. At time 0, no node is activated; at time
1, the data is fed into the network,33 and only pis activated, and at time
2, as the result of a computation, pand qbecome activated. The presence
of pin the program means that the input site pis continually activated,
without decay of activation; the activation of q(and the nonactivation of
r, s, t), is therefore also permanent. The most important point to emphasise
here is that the least fixed point of TP, that is, the minimal model of P,
corresponds to a stable pattern of activation of the network, starting from
a state of no activation. We thus claim that the computation of minimal
models is something that working memory is naturally equipped to do.
Interestingly, the greatest fixed point M3={p, q, r, s}cannot be ob-
tained by a simple bottom up computation, starting from an initial state of
no activation. In terms of neural nets, the greatest fixed point corresponds
rather to starting from a state where all units are activated, and where
the activation decays unless it is maintained by the input. This seems less
33Section 5.4 will contain a more detailed representation of input nodes.
38
AND+
AND-
Figure 6: Three-valued AND
plausible as a model of neural computation.
5.2 From algorithm to neural implementation: definite pro-
grams
To describe a network corresponding to a definite logic program, we need
units which compute ∧,∨and ¬with respect to Kleene’s strong three-valued
logic. The following trick is instrumental in defining these units (here we will
part company with [2]). The three truth values {u, 0,1}in Kleene’s logic
can be represented as pairs (0,0) = u, (0,1) = 0 and (1,0) = 1, ordered
lexicographically via 0 <1. The pair (1,1), which would represent a contra-
diction, is considered to be excluded. We shall refer to the first component
in the pair as the + (or ‘true’) component, and to the right component as
the −(or ‘false’) component. Interpret a 1 neurally as ‘activation’, and 0
as ‘no activation’, so that ucorresponds to no activation at all. Neural net-
works are now conceived of as consisting of two isomorphic coupled layers,
one layer doing the computations for ‘true’, the other for ‘false’, and where
the coupling is inhibitory to prevent pairs (1,1) from occurring.
With this in mind, a three-valued binary AND can then be represented
as a pair of units as in figure 6.
What we see here is two coupled neural nets, labelled + (above the sepa-
rating sheet) and −(below the sheet). Each proposition letter is represented
by a pair of units, one in the + net, and one in the −net. Each such pair
will be called a node. The thick vertical lines indicate inhibitory connec-
tions between units in the + and −nets; the horizontal arrows represent
excitatory connections. The threshold of the AND+ unit is 2, and that of
the AND- unit is 1.
As an example, suppose the two truth values (1,0) and (0,0) are fed
into the unit. The sum of the plus components is 1, hence AND+ does not
fire. The sum of the −components is 0, so AND- likewise does not fire.
The output is therefore (0,0), as it should be. There is an inhibitory link
39
+
-
+
-
Figure 7: Three-valued negation
between the + and −units belonging to the same proposition letter (or logic
gate) because we do not want the truth value (1,1), i.e. both units firing
simultaneously.
We obtain an AND-gate with ninputs for n≥2 if the threshold for
the plus-unit is set to n, and the threshold for the minus-unit is set to
1. Similarly, we obtain an OR-gate if the threshold of the plus-unit is set
to 1, and that for the minus–unit to n. The reader may check that these
conventions correspond to Kleene’s truth tables (see figure 4), reformulated
in terms of pairs (i, j). We also need a unit for negation, which is given in
the figure 7, where each unit has threshold 1.
Now consider the logic program Pwe used to explain the suppression
effect:
P={p, p ∧ ¬ab →q, r ∧ ¬ab0→q, ¬r→ab, ¬p→ab0,⊥ → ab, ⊥ → ab0}.
For the sake of readability, in figure 8 we give only the + net; the diagram
should be extended with a −net as in the diagram for AND (figure 6) In
this picture, the links of the form 0 →ab represent the + part of the link
from ⊥to the pair of units corresponding to ab. A NOT written across a
link indicates that the link passes through a node which reverses (1,0) and
(0,1), and leaves (0,0) in place. AND indicates a three-valued conjunction
as depicted above. The output node qimplicitly contains an OR gate: its
+ threshold is 1, its −threshold equals the number of incoming links. The
abnormality nodes likewise contain an implicit OR.
Before we examine how such networks could be set up in working mem-
ory, we trace the course of the computation of a stable state of this network,
showing that qis not true in the minimal model of the program. Initially all
nodes have activation (0,0). Then the input pis fed into the network, i.e.
(1,0) is fed into the pnode. This causes the ab0node to update its signal
from (0,0) to (0,1), so that ¬ab0changes its signal to (1,0). But no further
updates occur and a stable state has been reached, in which qoutputs (0,0).
If we view this state of activation as a (three-valued) model, we see that
40
A
N
D
A
N
D
p
not-ab
ab
r
not-ab’
ab’
NOT
NOT
not-p
not-r
NOT
NOT
q
0
0
Figure 8: Network for the suppression of MP
pis true, and the proposition letters rand qare undecided. Not surpris-
ingly, this model is also the least fixed point of the three-valued consequence
operator associated to the program.
5.3 Constructing the nets
We will now indicate briefly how such networks may be set up in working
memory, following the highly suggestive treatment in a series of papers by
Bienenstock and von der Malsburg [3, 49, 48]. Here we will only sketch a
hypothesis to be developed more fully elsewhere. There are two issues to
be distinguished: (1) the representation of conditionals as links, and (2) the
structure of the network as a pair of isomorphic graphs. It seems that the
work of Bienenstock and von der Malsburg is relevant to both issues.
Concerning the first issue, they observed that, apart from the ‘perma-
nent’ connection strengths between nodes created during storage in declar-
ative memory, one also needs variable connection strengths, which vary on
the psychological time scale of large fractions of a second. The strength of
these so-called dynamical links increases when the nodes which a link con-
nects have the same state of activation; networks of this type are therefore
described by a modified Hopfield equation. Applied to the suppression task,
we get something like the following. Declarative memory, usually modelled
41
by some kind of spreading activation network, contains a node representing
the concept ‘library’, with links to nodes representing concepts like ‘open’ ,
‘study’, ‘essay’ and ‘book’. These links have positive weights. Upon being
presented with the conditional ‘if she has an essay, she will study late in the
library’, these links become temporarily reinforced, and the system of nodes
and links thereby becomes part of working memory, forming a network like
the ones studied above. Working memory then computes the stable state of
the network, and the state of the output node is passed on to the language
production system. Modification of the connection strengths is an auto-
matic (albeit in part probabilistic) process, and therefore the whole process,
from reading the premisses to producing an answer, proceeds automatically.
The second intriguing issue is how the two layers of neurons become isomor-
phically wired up through inhibitory interconnections. We believe this can
be achieved using Bienenstock and van der Malsburg’s algorithm, since this
is precisely concerned with establishing graph isomorphism. Unfortunately
these brief indications must suffice here.
5.4 Backward reasoning and closed world reasoning for rules
We now have to consider the computations that correspond to the inferences
AC and MT. We have analyzed these by means of integrity constraints, that
is, statements of the form ‘if query ?ϕsucceeds/fails, then query ?ψsuc-
ceeds/fails’. From a neural point of view, this is reminiscent of a form of
backpropagation, except that in our case inputs, not weights are being up-
dated. This distinction is however not absolute, and we will rephrase the
construction in terms of the updating of weights. We propose that this dif-
ference between the processes of forward and backward reasoning can play a
part in explaining various observations of the difficulty of backward reason-
ing such as MT. We will do only one example, AC for a single conditional
premiss; this suffices to illustrate the main idea. We are given the premisses
qand p∧ ¬ab →q. The corresponding network is given in figure 9.
In this network, if qbecomes activated, then, provided nothing happens
to maintain the activation, it will decay to a state of no activation. It follows
that the only state of the network which will maintain the activation at qis
one where pis (and remains) activated, and ab is not. One may rephrase this
as an abductive learning problem: given an output (namely qactive), the
inputs have to be readjusted so that they yield the output. Now usually the
input is given, and the weights in the network are readjusted. But since we
are concerned with units which fire continuously, a reformulation in terms
of the updating of weights is possible. For this purpose we replace what was
previously an input node pby the configuration of figure 10.
Here, the bottom nodes always fire: the bold 1indicates that this node
stands for the true (or >); likewise 0stands for the always false (or ⊥). Both
nodes are taken to fire continuously. The links from the bottom nodes to
42
q
AND
p
not-ab
NOT
ab
0
Figure 9: Network for AC
p+
p-
1
0
w
= 0,1
w = 0,1
Figure 10: Structure of input nodes
43
ab
ab-
1
0
w
= 1
w = 0
+
Figure 11: Structure of ab nodes
the p-nodes have weights 0 or 1. Activation of pwith (1,0) is now modelled
as the left link having weight 1, and the right link having weight 0, and
analogously for (0,1), with ‘left’ and ‘right’ interchanged. Nonactivation of
pcorresponds to both links having weight 0. In terms of this view of input
nodes, the links previously described as 0→ab now have the structure as
in figure 11, where the weights are fixed.
Consider again the network of figure 9, and think of this as the + part
of a net as in figure 6. Assume the inputs are as given by figures 10 and
11. In this network, the inputs always fire, and the output may, or may not,
fire. Reaching a stable state now corresponds to readjusting the weights to
match the observed input/output pattern. For instance, if qis activated,
simple perceptron learning will update p’s initial weight configuration (left :
0,right : 0) to (left : 1,right : 0).
In conclusion of this discussion of how the networks compute, we may
thus note that the relatively easy forward reasoning patterns such as MP and
DA correspond to simple feedforward networks, in which no learning occurs,
whereas backward reasoning patterns such as AC and MT, which subjects
typically find harder, necessitate the computational overhead of readjusting
the network by an (albeit simple) form of backpropagation.
6 Discussion
The purpose of this paper has been to show that studies of reasoning, once
taken beyond the ‘mental rules’ versus ‘mental models’ debate, may begin
to address the important issue of the cognitive capabilities underpinning
reasoning.
The first step consisted in clearing the ground, by showing that the im-
position of logical form on natural language utterances involves a process of
44
parameter-setting far more complex than, say, associating a material impli-
cation to ‘if . . . then’. This process was dubbed ‘reasoning to an interpreta-
tion’; it was postulated to underlie credulous interpretation of discourse, in
which the hearer tries to construct a model for the speaker’s discourse. We
saw that the logical form assigned to the conditional introduces a parameter
(namely ab), which plays an important role in integrating the conditionals
with knowledge already present in declarative memory.
In the second step, we identified a logic that is a natural candidate for
discourse integration: closed world reasoning, here treated formally as logic
programming with negation as failure. Closed world reasoning is much easier
on working memory than classical logic: a single model of the premisses
suffices, and this model can be generated very efficiently. We showed that
the phenomena observed in the suppression task conform to the pattern
predicted by logic programming.
In the third step we showed that logic programming, unlike classical
logic, has an appealing neural implementation. On our proposals working
memory maintains a minimal preferred model of the world under description
in a form in which at any point the model can be efficiently revised in the
light of new information. This is the kind of system required for an organism
to plan, and is the kind of system which might be developed by an organism
evolved for planning which then turned its architecture to the performance
of intentional communication.
Reflecting back on the psychology of reasoning literature, the computa-
tional properties of this logic are highly suggestive as candidates for ‘System
1 processes’ much discussed in dual process theories of reasoning (e.g. Pol-
lock [28], Stanovich [37] and Evans [13])
System 1 is . . . a form of universal cognition shared between animals
and humans. It is actually not a single system but a set of subsys-
tems that operate with some autonomy. System 1 includes instinctive
behaviours that are innately programmed and would include any in-
nate input modules of the kind proposed by Fodor . . . The System 1
processes that are most often described, however, are those that are
formed by associate learning of the kind produced by neural networks.
. . . System 1 processes are rapid, parallel and automatic in nature; only
their final product is posted in consciousness (Evans [13, p. 454]).
In these theories, logical reasoning is considered to belong to ‘System 2’
which is ‘slow and sequential in nature and makes use of the central working
memory system [13, p. 454]’. Our proposals certainly challenge the idea that
fast and automatic processes are thereby not logical processes, thus drawing
a slightly different boundary between System 1 and System 2 processes.
Our proposals are for implementation in spreading activation networks
and it is perhaps worth comparing the strengths and weaknesses of these
to connectionist networks. In some ways spreading activation networks are
45
more immediately neurally plausible than connectionist networks. If acti-
vation of a node is interpreted as representing sustained firing of neurons,
possibly with gradual decay of firing rate or intensity, then spreading acti-
vation networks more directly mimic neural activity than do connectionist
networks. For the latter, it is not so clear how individual spikes or bursts of
spikes are represented in the network simulations.
However, spreading activation networks are localist representations and
are therefore not primitively capable of pattern completion, fault tolerance,
etc. Connectionist networks are mainly designed for learning applications,
but the process which constructs the networks proposed here, on the basis of
discourse comprehension is not a conventional learning process. The process
that most closely corresponds to learning in that construction is the ‘wiring’
between the pairs of nodes representing propositions, and the setting of
thresholds to represent connectives. We have suggested how this process
can be achieved in a neurally plausible fashion.
One last issue concerns a charge frequently brought against nonmono-
tonic reasoning, namely that its high computational complexity rules it out
as a formal description of actual human reasoning. There are in fact two
issues here, one pertaining to search for possible exceptions or abnormalities
in the mental database, the other to the computational complexity of the
derivability relation of a given nonmonotonic logic. The first issue can be
illustrated by a quote from Politzer [27, p. 10]
Nonmonotonicity is highly difficult to manage by Artificial Intelligence
systems because of the necessity of looking for possible exceptions
through an entire database. What I have suggested is a some kind
of reversal of the burden of proof for human cognition: at least for
conditionals (but this could generalise) looking for exceptions is itself
an exception because conditional information comes with an implicit
guarantee of normality.
Translated to our formalism, the difficulty hinted at by Politzer concerns
tabulating all clauses of the form ϕ→ab which are present in memory.
But here we do not have a knowledge base in the sense of AI, with its
huge number of clauses which are all on an equal footing. The discussion of
the neural implementation in section 5 has hopefully made clear that what
counts is the number of links of the form ϕ→ab which are activated in
working memory by means of a mechanism such as Bienenstock and von
der Malsburg’s ‘fast functional links’. This search space will be very much
smaller. There remains the issue of how relevant information in long term
memory is recruited into working memory, though we assume this is achieved
efficiently through the organisation of long-term memory. We do not pretend
to have solved the problem, but equally we do not believe the AI experience
of its intractability is entirely relevant.
The second issue is concerned with the computational complexity of the
decision problem for the relation ‘ψis derivable from ϕ1, . . . , ϕnin nonmono-
46
tonic logic L’, where we may restrict attention to propositional logics only.
For example, one well-known nonmonotonic logic, Reiter’s default logic is
computationally more complex than classical propositional logic, which is
NP-complete, so in practice exponential (see Gottlob [16] for a sample of
results in this area). By contrast, if Lis propositional logic programming
with negation as failure, the corresponding decision problem is P-complete,
hence less complex than classical propositional logic (see Dantsin et al. [10]
and references given therein for discussion). This difference is mainly due
to a restriction in the syntactic form of the rules, which have to be of the
form ϕ→A, where ϕcan be arbitrary, but Amust be atomic. This re-
striction, whose main effect is to rule out disjunctions in the consequent, is
harmless in the case of the suppression task. We do not deny that there may
be other nonmonotonic reasoning tasks where this restriction causes prob-
lems; it should be noticed, however, that logic-programming has a certain
track record in problem solving in AI, which provides further evidence of
this logic’s expressiveness.
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50
Role Content
Conditional 1 If she has an essay to write she will study late in the library
Categorical She has an essay to write
Conclusion She will study late in the library (MP 88.3%)
Additional If the library stays open, she will study late in the library.
Conclusion She will study late in the library (MP 60.6%)
Alternative If she has a textbook to read, she will study late in the library
Conclusion She will study late in the library (MP 93.3%)
Conditional 1 If she has an essay to write she will study late in the library.
Categorical She will study late in the library
Conclusion She has an essay to write (AC 55.1%)
Additional If the library stays open then she will study late in the library.
Conclusion She has an essay to write (AC 53%)
Alternative If she has a textbook to read, she will study late in the library.
Conclusion She has an essay to write (AC 16%)
Conditional 1 If she has an essay to write she will study late in the library.
Categorical She doesn’t have an essay to write
Conclusion She will not study late in the library (DA 49.3%)
Additional If the library stays open, she will study late in the library
Conclusion She will not study late in the library (DA 49.2%)
Alternative If she has a textbook to read, she will study late in the library
Conclusion She will not study late in the library (DA 22%)
Conditional 1 If she has an essay to write she will study late in the library.
Categorical She will not study late in the library
Conclusion She does not have an essay to write (MT 69.6%)
Additional If the library stays open, she will study late in the library
Conclusion She does not have an essay to write (MT 43.9%)
Alternative If she has a textbook to read, she will study late in the library
Conclusion She does not have an essay to write (MT 69.3%)
Table 1: Percentages of Dieussaert’s subjects drawing target conclusions in
each of the four argument forms modus ponens (MP), modus tollens (MT),
denial of the antecedent (DA) and affirmation of the consequent (AC), in
two premiss and three premiss arguments. ‘Conditional 1’ is the same first
premiss in all cases. In a two-premiss argument it is combined only with the
categorical premiss shown. In a three-premiss argument, both are combined
with either an ‘alternative’ or an ‘additional’ conditional premiss.
51
