1 A common failure in psychiatric disorders: executive function

‘Executive function’ is an umbrella term for processes responsible for higher-level action monitoring and control that are necessary for maintaining a goal and achieving it in possibly adverse circumstances. The concept of executive function originated from neuropsychological study of patients with impairments in this area, and the field still suffers from lack of a comprehensive positive characterisation of just what executive function is. There is also no unanimity on how to partition executive function into meaningful subcomponents, but for the purposes of this paper we take executive function to be composed of planning, initiation, inhibition, monitoring, coordination and control of action sequences, leading toward a goal held in working memory.

In this paper we will study two psychiatric disorders with marked deficiencies in executive function, autism and attention deficit hyperactivity disorder (henceforth ADHD). Autism is accompanied by often astounding behavioural rigidity, that is, stereotyped and contextually inappropriate behaviour. In ADHD one can observe difficulties with stimulus-controlled task-switching (indicative of failures in motor inhibition), with working memory, and with flexibility and planning (for an overview see Pennington and Ozonoff (1996)). The reason for our interest in these disorders is our conviction that there is a strong connection between logic and executive function: at a logical level, the operation of executive function can be described as conditional reasoning and aberrations thereof. This may seem surprising, especially given the prevalent conception of reasoning as a conscious and somewhat laborious activity starting from explicitly given premisses according to the canons of classical logic. How then can fast and largely automatic executive function be profitably described by a logic? It will turn out, however, that the logics most useful in this context, variants of closed world reasoning, do allow fast and automatic processing.

Our objectives in this paper are to promote the benefits of logical analysis for the cognitive understanding of diverse behaviour, especially psychiatric disorders, as well as to shake up some assumptions about logical analysis itself. We see these benefits as uncovering abstract similarities between apparently unrelated behaviours and differences between apparently similar patterns. At the micro-level this involves relations between laboratory tasks, but can also involve relations between different psychiatric categories, as well as between normal and abnormal development. Autism and ADHD will be our specific examples. The paper therefore starts with a theoretical exposition of the connections between executive function and logic. We draw on accounts proposed in Stenning and van Lambalgen (2005) and summarised and extended in Stenning and van Lambalgen (2007) and provide references as we go. We then go on to apply the insights obtained to autism and ADHD. We will derive predictions from the theoretical account concerning aspects of both non-linguistic tasks and discourse processing in both autism and ADHD. The discourse prediction for autism has been verified, while the one for ADHD is currently being tested.

2 Logic and executive function

In our discussion we adopt as organizing principle David Marr’s three levels of inquiry into a cognitive phenomenon (Marr 1982, Chapter 1):

  1. (1)

    identification of the information processing task

  2. (2)

    specification of an algorithm which computes that function

  3. (3)

    neural implementation of the algorithm specified

Recall that executive function is composed of planning, initiation, inhibition, coordination and control of action sequences leading toward a goal held in working memory. In the following we abstract from the coordination and control component, and concentrate on goal maintenance, planning and (contextually determined) inhibition.

By definition, planning consists in the construction of a sequence of actions 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. The relevant properties include stable causal relationships obtaining in the world, and also what might be termed ‘inertia’, in analogy with Newton’s first law. If a property has been caused to hold by the occurrence of an event, we expect that the property persists until it is terminated by another event. This is the inertial aspect of causality: a property does not cease to hold (or come to hold) spontaneously, without identifiable cause. Such inertia is a prerequisite for successful action in the world; and we will have to find a formal way to express it. It does however not suffice for successful planning.

The problem is that in the definition of planning, ‘will achieve’ definitely cannot mean: ‘provably achieves in classical logic’, 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, but classical logic forces one to consider all models of the premisses, including those that contain farfetched possibilities. Therefore the question arises: how to characterize formally what makes a good plan?

A reasonable informal suggestion is: the plan works to the best of one’s present knowledge. More formally, this idea can be reformulated semantically as: the plan achieves the goal in a ‘minimal model’ of reality; where a minimal model is characterized by the property that, 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, and only explicitly mentioned causal influences are represented in the model. This makes planning a form of non-monotonic reasoning.

We thus postulate that the logical idea underlying planning is closed world reasoning: the principle which says that every proposition which is not forced to hold by the data available can be assumed to be false. This can apply to propositions about occurrences of events as well as those expressing causal relationships. One may identify a number of areas to which closed world reasoning is applicable, each time in slightly different form:

  1. (1)

    lists: train schedules, airline databases, ...

  2. (2)

    diagnostic reasoning and abduction

  3. (3)

    unknown preconditions

  4. (4)

    causal and counterfactual reasoning

  5. (5)

    attribution of beliefs and intentions Footnote 1

  6. (6)

    construction of discourse models, in particular event structures from verb tenses Footnote 2

It is of some interest that several psychiatric disorders come with disturbances in one or more forms of reasoning from this list. As we will see, children with ADHD tend to have difficulties with ordering events in a narrative. Autists have difficulties with at least 3, 4, and 5. They also have a special relationship with lists, in the sense that they feel lost without lists, such as timetables to organise daily activities; they have great difficulty accommodating unexpected changes to the timetable, and try to avoid situations such as holidays in which rigid schedules are not applicable. One may view this as an extreme version of closed world reasoning, sometimes even applied in inappropriate circumstances. But before one concludes from this that autists are good at closed world reasoning to the point of over-applying it, one must carefully distinguish several components of closed world reasoning. On the one hand, there is the inference from given premisses to a conclusion. In (Stenning and van Lambalgen 2007, Chapter 8) it is shown that such inferences can be executed fast by suitable neural networks. In a wider sense, non-monotonic reasoning also involves ‘pre-processing’ the given situation or discourse, that is, encoding the law-like features of a situation in a particular type of premisses. Laws and regularities always allow exceptions, and skill at ‘exception handling’ is required—which involves identifying and encoding the relevant exceptions, and knowing when ‘enough is enough’. The heading of the next section, ‘Coping with non-monotonicity’, refers to this last aspect, at which autists appear to do worse than normals, although they behave normally with respect to the non-monotonic inferences themselves.

We have thus identified closed world reasoning as (a component of) the top level in a Marr-type analysis of executive function. A good formal representation of closed world reasoning as relevant to planning is the event calculus as formulated in logic programming (van Lambalgen and Hamm 2004). An informal description suffices for our purposes. Formally, this involves a goal, a description of the current situation, a description of causal effects and preconditions of actions. The program then derives a sequence of actions which will achieve the goal if no unforeseen events occur, but execution must be stopped and plan recomputed if relevant change of context occurs.

This puts constraints on the implementation of the algorithm, Marr’s third level, in the sense that the following components are required: working memory needs to hold the goal and the current world model, semantic memory is necessary for the storage of causal properties of actions, and of a general theory of causality; working memory again computes the sequence of actions comprising the plan. Working memory deficits may thus lead to problems with goal maintenance, and we shall see an instance of this in our discussion of ADHD. In the logical model of executive function proposed here inhibition is represented through the special logical form of causal properties of actions, where the link from action to effect is mediated by a slot labelled \({\neg ab}\):

$$ A \wedge \neg ab \rightarrow E $$

This conditional is read as ‘if A and nothing abnormal is the case, then E’, where the expression ‘nothing abnormal is the case’ is governed by closed world reasoning. For instance, if there is nothing known about a possible abnormality, i.e., if the causal system is closed, one concludes \({\neg ab}\), hence from A it follows that E. If however there is information of the form Cab, i.e., if there is a context C which constitutes an abnormality, and C is the case, then the link from A to E is inhibited. In the neural model of closed world reasoning proposed in (Stenning and van Lambalgen 2007, Chapter 8) ab corresponds to an artificial neuron situated between the neurons for A and E, such that C is connected to ab via an inhibitory link; and this is the general way of incorporating contextual influences. Defects in the inhibitory neurons would thus lead to deficient context processing, as we see in autism. In (Stenning and van Lambalgen 2007, Chapter 8) we present some recent evidence indicating that in the brain of autists inhibition is compromised at the neurological level, among other reasons because inhibitory interneurons are underdeveloped.

3 Coping with non-monotonicity in autism

Autism is a clinical syndrome first described by Leo Kanner in the 1940s, often first diagnosed in children around 2–3 years of age as a deficit in their affective relations and communication. The autistic child typically refuses eye-contact, is indifferent or hostile to demonstrations of affection, and exhibits delayed or abnormal communication, repetitive movements (often self-harming) and is rather indifferent to pain. Autistic children do not engage spontaneously in make-believe play and show little interest in the competitive social hierarchy, and in personal possessions. Autism comes in all severities—from complete lack of language and severe retardation, to mild forms continuous with the ‘normal’ range of personalities and IQs. Autism is sometimes distinguished from Asperger’s syndrome—‘autism without language impairment’—but Asperger’s is probably just the mild end of the autistic spectrum. Autistic children share many symptoms shown by deaf and by blind infants, possibly because of the social isolation imposed by these conditions. There are known biochemical abnormalities associated with autism. There is some evidence of a probably complex genetic basis. Psychological analyses of autistic functioning are not inconsistent with or exclusive of such biochemical or genetic level analyses.

3.1 Theory of mind and reasoning

A famous experiment, the ‘false belief’ task (Perner et al. 1987), investigates how autistic subjects reason about other people’s belief. The standard design of the experiment is as follows. A child and a doll (Maxi) are in a room together with the experimenter. Maxi and child witness a bar of chocolate being placed in a box. Then Maxi is brought out of the room. The child sees the experimenter move the chocolate from the box to a drawer. Maxi is brought back in. The experimenter asks the child: ‘Where does Maxi think the chocolate is?’ The answers to this question reveal an interesting cut-off point, and a difference between autists and normally developing children. Before the age of about 4 years, the normally developing child responds where the child knows the chocolate to be (i.e., the drawer); after that age, the child responds where Maxi must falsely believe the chocolate to be (i.e., the box). By contrast, autists go on answering ‘in the drawer’ for a long time.

The outcomes of these experiments have been argued to support the ‘theory of mind deficit’ hypothesis on the cause of autism. Proposed by Leslie (1987), it holds that human beings have evolved a special ‘module’ devoted specifically to reasoning about other people’s minds. As such, this module would provide a cognitive underpinning for empathy. In normals the module would constitute the difference between humans and their ancestors—indeed, chimpanzees seem to be able to do much less in the way of mind-reading. In autists, this module would be delayed or impaired, thus explaining abnormalities in communication and also in the acquisition of language, if it is indeed true that the development of joint attention is crucial to language learning (as claimed for instance by Tomasello (2003)).

This seems a very elegant explanation for an intractable phenomenon, and it has justly captured the public imagination. Upon closer examination the question arises whether it is really an explanation rather than a description of one class of symptoms. For instance, the notion of a ‘module’ is notoriously hazy. In this context it is obviously meant to be a piece of dedicated neural circuitry. In this way, it can do the double duty of differentiating us from our ancestors and being capable of being damaged in isolation. But it is precisely this isolation, or ‘encapsulation’ as Fodor called it, that is doubtful. One reason is that evolution does not generally proceed by adding new modules, but instead by tweaking old ones, and another is that much of the problem of functionally characterising human reasoning about minds is about interactions between modules. ‘Theory of mind’ requires language to formulate beliefs in and it also entails a considerable involvement of working memory, as can be seen in ‘nested’ forms of theory of mind, as in Dunbar’s example

Shakespeare intended us to realise that Othello believes that Iago knows that Desdemona is in love with Cassio.

However, as soon as one realises that a ‘module’ never operates in isolation, then the ‘theory of mind deficit’ hypothesis begins to lose its hold. We are now invited to look at the (possibly defective) interactions of the ‘module’ with other cognitive functions (language, working memory, ...), which leads to the possibility that defects in these functions may play a role in autism. And there is of course also the problem of what the ‘module’ would have to contain, given that for instance reasoning about other people’s desires (as opposed to beliefs) is possible for both autists and non-human primates.

Apart from these theoretical problems, it is experimentally controversial at what stage ‘theory of mind’ abilities emerge. False-belief tasks were initially proposed as diagnosing a lack of these abilities in normal 3-year-olds and their presence in normal 4-year-olds (Leslie 1987). Others have proposed that irrelevant linguistic demands of these tasks deceptively depress 3-year-olds’ performance. For example, in the ‘Maxi’ task, the child sees the doll see the chocolate placed in a box, and then the child but not the doll sees the chocolate moved to the drawer. Now if the child is asked ‘Where will the doll look for the chocolate first?’ (instead of ‘Where will the doll look for the chocolate?’) then children as young as two can sometimes solve the problem (Siegal and Beattie 1991). Onishi and Baillargeon (2005) use data from non-verbal expectation of looking in infants of only 15 months to argue for the beginnings of effective reasoning about false beliefs at this age. Although this data can, as the authors point out, alternatively be interpreted in terms of more superficial strategies of looking for things where they were last seen, even this requires the child to preserve distinctions between who last saw the object and where. Nevertheless, all these arguments push reasoning about mental states earlier in ontogeny.

3.2 Reasoning in the false belief task

We will now analyse attribution of belief as it occurs in the false-belief task as a species of closed world reasoning. For convenience we first give an informal argument, before proceeding to a rigorous logical analysis.

An agent solving the task correctly first of all needs to have an awareness of the causal relation between perception and belief, which can be stated in the form: ‘if φ is true in scene S, and agent a sees S, then a comes to believe φ’. Applied to the situation at hand, this means that Maxi comes to believe that the chocolate is in the box. An application of the principle of inertia now yields that Maxi’s belief concerning the location of the chocolate persists unless an event occurs which causes him to have a new belief, incompatible with the former. The story does not mention such an event, whence it is reasonable to assume—using closed world reasoning—that Maxi still believes that the chocolate is in the box when he returns to the experimenter’s room. An explanation for performance in the false belief also needs to account for the incorrect answers given by children younger than 4 and autists. These subjects almost always answer ‘in the drawer’, when asked where Maxi believes the chocolate to be. To model this, we borrow a notion from executive dysfunction theory, and hypothesise that the ‘prepotent response’ is always for the child to answer where it knows the chocolate to be. In some children, this response can be inhibited, in other children it cannot, for various reasons which we shall explore below. Now for the formal analysis, which has to combine reasoning about information and about the world.

3.2.1 Formal language and principles

We need a language comprising at least the following predicates Footnote 3:

  • R a (p): agent a reports her belief that p

  • see a (p): agent a sees that p

  • told a (p): agent a is told that p

  • ded a (p): agent a deduces that p

  • B a (p): agent a has the belief that p

  • q(i,t): the chocolate is at location i at time t

  • ab a : an exception which obstructs agent a’s inferences

  • clipped(s, i,t): at some time between s and t, the chocolate ceases to be at location i.

The agent’s information state B a (p) satisfies properties such as the following:

$$see_a(q(i,t)) \rightarrow B_a(q(i,t)) $$
$$told_a(q(i,t)) \rightarrow B_a(q(i,t))$$
$$ded_a(q(i,t)) \rightarrow B_a(q(i,t))$$

We furthermore postulate that a given clause of the form φ→ ψ may be replaced by B a (φ) → B a (ψ).

Suppose b is an agent thinking about the behaviour of agent a. We model b’s responses as the result of a competition between two rules, both of which are instantiations of the general response rule schema

$$B_b(\varphi) \wedge \neg ab_b \rightarrow R_b(\varphi)$$

where ab b indicates a possible circumstance which prevents b from reporting his belief. Footnote 4 This response schema says that if b believes φ, his prepotent response is to report that φ; but the rule is inhibitable in that the occurrence of ab b may prevent the report. Obviously this response is itself an instance of the more general schema 1 which we took as the basis of our treatment of executive function in Sect. 2.

The first substitution instance of rule 5 says that if b knows what the location of the chocolate is, he will report that location, barring exceptional circumstances:

$$B_b(q(i,t)) \wedge \neg ab_b \rightarrow R_b(q(i,t))$$

which arises from rule 5 by the substitution φ: = q(i,t).

The second rule says that b will report a’s information state, again

$$B_b(B_a(q(i,t)))\wedge\neg ab_b \rightarrow R_b(B_a(q(i,t)))$$

It arises from rule 5 by the substitution φ: = B a (q(i,t)).

In the case of a false belief, these rules are in competition, and we have to ensure that only one is operative; i.e., the rules must inhibit each other mutually. This is achieved by means of the abnormalities. The inhibitory effect of 6 on 7 is modelled by the clause

$$R_b(q(j,t)) \rightarrow ab_b$$

which combined with 7 expresses that b’s own response interferes if j ≠ i.

The inhibitory effect of 7 on 6 is modelled by a clause which expresses task understanding: b should report a’s beliefs, even if he knows them to be wrong:

$$B_b(B_a(q(i,t))) \rightarrow ab_b$$

This formula expresses that b’s prepotent response 6 is inhibited if agent b has information that agent a has (possibly incorrect) information about the location of the chocolate.

It is essential to note that the false-belief task not only involves fairly modest reasoning about beliefs, but more importantly also reasoning about the world. The interaction between the inertial properties of the world and the information of an agent a is given by:

$$B_a(q(i,s)) \wedge s < t \wedge \neg B_a(clipped(s, i,t)) \rightarrow ded_a(q(i,t))$$

with clipped governed by clauses such as

$$q(i,s) \wedge chocolate\hbox{-}moved(r) \wedge s < r < t \rightarrow clipped(s,i,t)$$
$$q(i,s) \wedge chocolate\hbox{-}melted(r) \wedge s < r < t \rightarrow clipped(s,i,t)$$

3.2.2 Closed world reasoning in the normal child older than 4 years

Let the location variable i range over {1,2}, where 1 = box, 2 = drawer. Also let a be Maxi, b the child, t 0 the time at which Maxi leaves the room, and tt 0 the time at which b must answer the experimenter’s question, i.e., report Maxi’s belief state. We will represent b’s report as a statement of the form R b (B a (p)). We assume that b believes that q(2,t), and we omit the derivation of this fact. We must explain why the normal child responds with R b (B a (q(1,t))), and not with R b (q(2,t)). As mentioned above, the explanation assumes that of the competing conditionals 6 and 7 the first is inhibited by the second through a condition on ab b reflecting the child’s understanding of the task:

$$B_b(B_a(q(i,t))) \rightarrow ab_b$$

We first show that in these conditions b will not respond with his own knowledge of the whereabouts of the chocolate. The response R b (q(2,t)) would require \({\neg ab_b}\), i.e., that ab b cannot be derived. Now ab b reduces to B b (B a (q(i,t))) by Eq. (9): if the latter can be proved, so can the former. But as we will prove next, one has B b (B a (q(i,t))), whence also ab b . Thus the prepotent response 6 is inhibited and R b (q(2,t)) is not a possible response for b. This will help in showing that the antecedent of rule 7 is satisfied.

The second part of the proof shows that b will report a’s beliefs. We know q(1,s) ∧ see a (q(1,s)) for some st 0t. It follows that B a (q(1,s)) by Eq. (2). Intuitively, because nothing happens between s and t 0, and Maxi leaves after t 0, one may conclude \({\neg B_a(clipped(s,i,t))}\), i.e.,

$$ ?B_a(clipped(s,i,t))\mathsf{fails}$$

Formally, one can show a query like

$$?B_a(chocolate\hbox{-}moved(r) \wedge s < r < t)$$

fails, even though the chocolate is actually moved. Indeed, applying Eq. (2)

$$?B_a(chocolate\hbox{-}moved(r) \wedge s < r < t)$$

reduces to queries such as the following, which all fail:

$$?see_a(chocolate\hbox{-}moved(r) \wedge s < r < t)$$

It thus follows that


By inertia (i.e., Eq. (10)), we then have ded a (q(1,t)), whence B a (q(1,t)) by 4. Since b can perform the preceding deduction, also B b (B a (q(1,t))). We have already seen in the first part of the argument that therefore \({\neg R_b(q(2,t))}\), and it follows by clause 8 that \({\neg ab_b}\). We must therefore have R b (B a (q(1,t))) by rule 7.

3.2.3 Attribution of beliefs and closed world reasoning in the younger or autistic child

As mentioned in the introduction to this Sect. (3.2), in this case b’s response rule is effectively of the form

$$B_b(q(i,t)) \rightarrow R_b(q(i,t))$$

i.e., rule 6 without the inhibiting condition. In the Maxi task we thus get the response R b (q(2,t)). This response cannot be inhibited: the form of the rule does not even allow b to consider a’s information sources. But the effect of the response R b (q(2,t)) is that rule 7 is inhibited, whence R b (B a (q(1,t))) is not a possible response.

The rule 14 may be primitive, or a consequence of failed task understanding. In the latter case, the child has not yet incorporated rule 9, so that closed world reasoning leads to \({\neg ab}\). In this case cognitive development may be viewed as relating the variable for an exception in 6 to internal theories about mental processing and about the world. The child may also substitute the rule 9 with the following theory relating world and information

$$q(i,t)\rightarrow B_a(q(i,t))$$

introducing a short circuit which bypasses the set of principles 2–4, 10, 11 .... The rules 14 and 7 are then no longer in competition, but yield the same answer.

As mentioned in Sect. 1, there is an interesting neural aspect to this discussion of the rule 14, which we can only touch upon very briefly here. In the artificial neural network model of closed world reasoning proposed in (Stenning and van Lambalgen 2007, Chapter 8) ab b corresponds to a neuron situated between the neurons for B b (q(i,t)) and R b (q(i,t)), such that the condition B b (B a (q(i,t))) in rule 9 is connected to ab b via an inhibitory link. Interestingly, there is recent mounting evidence that autists have difficulties with inhibition at the neural level, for which see (Stenning and van Lambalgen 2007, Chapter 9). For instance, it appears that in autists’ brains far more excitatory than inhibitory synapses remain after pruning; furthermore, because autists’ brains undergo a growth spurt in the first 2 years followed by rapid decrease of growth, slowly maturing neurons like the inhibitory interneurons remain underdeveloped. Clearly these connections can be no more than suggestive, since it remains to be shown that the artificial neural networks proposed in (Stenning and van Lambalgen 2007, Chapter 8) can actually be found in the brain; but it remains interesting that a logical analysis can potentially be connected to neurophysiological investigations.

3.3 What this analysis of the child’s reasoning tells us

Let us first explain some of our design decisions. The reader might wonder why a task called the ‘false belief task’ is not analysed logically in terms of one of the logics of belief that logic has developed. And indeed the predicate B a applied to propositions acts somewhat like a modal belief operator B a as used in multi-agent epistemic logics (since we need iterated belief operators, for instance in the clause reflecting task understanding, 9, and in rule 7). The trouble with such an analysis is that, on the one hand, standard axioms for belief such as positive and negative introspection appear to play no role in the derivation of the responses of the two categories of subjects, and on the other hand, the real work in the derivation is done by assumptions concerning the relation of belief and sensory information, concerning persistence of belief over time, and concerning belief reports. In other words, the reasoning is mostly about belief formation, not so much belief manipulation. For the same reason it does not appear to be very useful to analyse the false-belief task in terms of a possible world semantics for epistemic logic, since such a semantics is concerned with how to get from one belief state to another, which is not the main issue in the false-belief task.

This said, there remain intuitive considerations on the false-belief task which suggest that some sort of modal principle of positive introspection is operative after all. An experiment by Clements and Perner (1994) shows that normal 3 year olds may give the wrong answer in the false belief task, while simultaneously looking at the right place. The interesting paper ‘Knowing about knowing’ (Hauser 2003) glosses this result by saying that these 3 year olds have (implicit) knowledge about the right response, but no knowledge of their knowledge, i.e., no explicit knowledge. This distinction can be represented by a slight change in the set-up. We keep the predicate R a (p) for ‘agent a (verbally) reports her belief that p’, but introduce a new predicate A a (p) with the intended meaning ‘agent a acts out her belief that p’, for example by looking. We then get two general response schemata instead of the one given as 5, namely

$$B_b(\varphi) \wedge \neg ab_b \rightarrow A_b(\varphi)$$


$$B_b(B_b(\varphi)) \wedge \neg ab_b \rightarrow R_b(\varphi)$$

Only positive introspection leads to congruent answers here. That is, the argument given above for the normal child older than 4 now applies to ‘acting out’ only, i.e., with R b replaced by A b everywhere; positive introspection is needed to give the corresponding verbal response.

On the assumption that the analysis captures the processing that is going on in the child’s mind, we can isolate the executive and ‘theory’ components in the false belief task. To start with the latter, the ‘theory’ component for the normally developing child consists of the rules 2, ..., 10, 11 ..., and the response rule 6. If we define executive control generally as concerned with maintaining a goal and planning, coordination, inhibition and control of action sequences leading toward that goal, then in this particular task executive control has to maintain the goal ‘Find out what Maxi’s belief state is, and report truthfully’, and to set up a sequence of steps leading to this goal. This first involves keeping the linking rule 9 active in working memory, and also the goal ?B a (q(i,t)). Second, a computation has to be started, regressing from the activated goal; this is essentially the closed world argument given above. Given the connection between closed world reasoning and planning, one can see from this analysis that executive function is essentially involved in solving the false belief task. Inhibiting the prepotent response 6 is only one, relatively minor, facet of this involvement.

These two components together explain competent verbal performance in the false belief task. If any of these is lacking, one may expect wrong answers. If the child does not answer that Maxi believes the chocolate to be in the box, this may thus be due to a failure to maintain the goal or rule 9 in working memory, a failure to apply closed world reasoning, for instance due to rule 15, a failure to grasp inertia (principle 10 plus closed world reasoning), a failure to grasp the connection between sensory input and information state (principles like 2 together with closed world reasoning), or simply having a different primitive response rule replacing 6, one which cannot be inhibited, as in 14. We know that children below the cut-off age overwhelmingly answer that Maxi believes the chocolate to be in the drawer, whereas a possible answer could also be ‘anywhere’. This reduces the possible causes of the failure to those which generate a response rule which is essentially of the form 14. As shown in Sect. 3.2.3 this still leaves several possibilities, from deficient neural encoding of rules to a failure in closed world reasoning about information sources. That is, the defect could be located at the level of neurotransmitters, at the level of executive function, or at the level of theory of mind; at the moment one cannot say which.

What the analysis also shows is the implausibility of a theory of mind module, in the sense of an encapsulated piece of neural tissue dedicated exclusively to theory of mind. The force of the set of rules 9, 2, ..., 10, 11 ... comes from combining notions which have to do with mental representation, and notions useful in understanding (and acting in) the natural world, by means of a powerful inference engine operated by executive function. Theory of mind must be viewed as a superstructure built on top of a theory of causality in the world, powered by a theorem prover in executive function. Although this superstructure is a quite considerable extension, and so could be differentially affected, some researchers have claimed that in autism the foundation itself, the theory of causality, is compromised. This will be discussed next, in Sect. 3.4. We then turn to a more detailed discussion of executive dysfunction in Sect. 3.5, and will provide a formalisation of one of its important experimental paradigms, the box task. The formalisation of the false belief task reveals formal similarities with the suppression task, a reasoning paradigm much studied in the adult reasoning literature. The box task exhibits even stronger similarities to the suppression task, and this has led us to try the latter task on a population of autistic subjects. The results (see Sect. 4), while not statistically significant due to the small sample size, are highly suggestive of a very specific deficit in autism.

3.4 Counterfactual reasoning

A counterfactual version of the false belief task due to Riggs and Peterson (2000) was designed with the purpose of showing that (in normally developing children) failures in counterfactual reasoning lie at the root of unsuccessful performance in the false belief task. Briefly, the set-up is as follows. In each condition, a false belief task and a corresponding physical state task based on the same ingredients were constructed. For instance, the analogue of the Maxi task was the following. A child, a mother-doll, and an experimenter are in a kitchen. The child sees that there is a chocolate in the fridge. The mother-doll now bakes a chocolate cake, in the process of which the chocolate moves from fridge to cupboard. The experimenter now asks the child:

  • (1) Where would the chocolate be if mother hadn’t baked a cake?

In two of the three experiments, the pattern of answers is highly correlated with that on the false belief task. Below the cut-off age of 4, the child answers: ‘in the cupboard’; afterwards, it answers ‘in the fridge’. Apparently there is no theory of mind involved in answering (1) correctly; instead one needs insight into the ‘inertia’ of the world: ‘things only change places when there is an explicit cause’. It is interesting to inquire what prompts the younger child to answer ‘in the cupboard’; a simple failure to apply inertial reasoning might as well lead to the answer ‘it could be anywhere’, say because of the events that could have happened in this alternative world. Answers such as this would be a consequence of applying causal reasoning without closed world reasoning for occurrences of events. The answer ‘in the cupboard’ more likely reflects a failure to apply causal reasoning altogether, reverting instead to the prepotent response. It must be noted here that in one out of Riggs and Peterson’s three experiments, the false belief task was considerably more difficult for the children than the counterfactual task, which is what one would expect given the analysis of Sect. 3.2. We will return to this issue below, in connection with autists’ behaviour on this task.

We will now restate the previous considerations in terms of the formal machinery introduced for the false belief task; this will allow a more precise comparison. The main difference with the argument in Sect. 3.2 is that the child now compares two belief states of herself: what she knows to be true and what she is asked to assume. To economise on notation we will not introduce a separate operator for knowledge, and will continue to use B to mark beliefs which are not necessarily true.

The general form of the response rule is now

$$\varphi \wedge \neg ab_{ch} \rightarrow R_{ch}(\varphi)$$

where the subscript ch indicates that the rule is applied by the child herself. The substitution instance of interest to us is

$$q(i,t) \wedge \neg ab_{ch}\rightarrow R_{ch}(q(i,t))$$

The analogue of the substitution instance 7 is now the simpler

$$B_{ch}(q(i,t))\wedge\neg ab_{ch}\rightarrow R_{ch}(q(i,t))$$

which comes together with a clause inhibiting 19, where j ≠  i

$$R_{ch}(q(j,t)) \rightarrow ab_{ch}$$

Let t = 0 denote the time of the initial situation, t = 1 the time of moving the chocolate, t = 2 the time of asking question (1). The child is asked to imagine that chocolate-moved(1) did not occur. Understanding this task amounts to the adoption of the rule

$$\varphi\wedge B_{ch}(\neg \varphi) \rightarrow ab_{ch}$$

which inhibits the child’s report of the true situation φ if she pretends to believe \({\neg\varphi}\).

The child may derive by closed world reasoning that \({\neg clipped(0,fridge,2)}\), whence by equation 10 ded ch (q(fridge,2)). It follows that B ch (q(fridge,2)), whence the child will report q(fridge,2) by rule 20, if in addition \({\neg ab_{ch}}\). To establish this latter formula, we have to show (using 21) that \({\neg R_{ch}(q(cupboard,2))}\).

That the response R ch (q(cupboard,2)), triggered by the rule 18, is indeed inhibited, follows from the fact that q(fridge,2) and q(cupboard,2) are incompatible, so that \({B_{ch}(\neg q(cupboard,2))}\), whence ab ch by 22, which inhibits the rule 18.

The younger child is again hypothesised not to use 18, but to use the uninhibitable prepotent response (18 stripped of \({\neg ab_{ch}}\)) instead, for any of the reasons outlined in Sect. 3.2 (failed task understanding, deficient neural encoding ...).

If we now compare the two tasks, we see that the reasoning involved is very similar, but that the false belief task requires a more extensive set of principles. Thus, failure on the counterfactual task may be expected to lead to failure on the false belief task, because in both cases it is the prepotent response that is assumed to be operative, perhaps as a derivative effect. Success on the counterfactual task by itself does not imply success on the false belief task, because the calculations for the latter involve combining reasoning about information sources, inertial properties, and closed world reasoning. In this sense false belief reports are a proper subspecies of counterfactuals, and it would be interesting if they could be shown to be harder for some populations.

There is some experimental evidence which bears on this issue. Peterson and Bowler (2000) have compared performance on false belief tasks and counterfactual tasks in populations of typically developing children, children with severe learning problems, and autistic children. The normal children showed high correlation on these tasks, but a dissociation became apparent in the other two groups. In all groups, those who failed the counterfactual task overwhelmingly failed the false belief task, suggesting that the kind of reasoning going on in the former is a necessary ingredient of the latter. About 75% of the typically developing children who pass the counterfactual task also pass the false belief task, but these percentages get lower in the other groups: 60% in children with learning difficulties, 44% in autistic children. The authors speculate on the additional factors going into a false belief computation, and suggest that one factor is the necessity to ‘generate’ Maxi’s false belief, whereas in the counterfactual task the false statement is given. They then go on to relate this feature to other supposed failures of generativity in autism, such as the difficulty of spontaneous recall compared to cued recall. While we would not put it in precisely this way, there is something to this distinction, in that in the false belief task the child has to see the relevance of Maxi’s not witnessing the crucial event, for the ensuing computation. In the counterfactual task all the ingredients are given, and only an inertial computation is necessary.

3.5 Executive dysfunction and the box task

Russell’s executive function deficit theory (Russell 1997) takes as basic the observation that autists often exhibit severe perseveration. They go on carrying out some routine when the routine is no longer appropriate, and exhibit great difficulty in switching tasks when the context calls for this (that is, when switching is not governed by an explicit rule). This perseveration, also observed in certain kinds of patients with frontal cortex damage, would give rise to many of the symptoms of autism: obsessiveness, insensitivity to context, inappropriateness of behaviour, literalness of carrying out instructions. Task-switching is the brief of executive function, a process (or processes) responsible for high-level action control such as planning, initiation, co-ordination, inhibition and control of action sequences. Executive function is hypothesised to be necessary for mentally maintaining a goal, and pursuing it in the real world under possibly adverse circumstances.

Executive function is called upon when a plan has to be redesigned by the occurrence of unexpected events which make the original plan unfeasible. Autists indeed tend to suffer from rather inflexible planning. Here are two examples, furnished by a single (Asperger) patient, to illustrate the phenomenon.

(1) If she wants to go to the supermarket, she must make a shopping list. If she finds items in the supermarket that she needs, but which do not figure on her list, she has to go home, append the needed item to the list, and return to the supermarket. Occasionally she has to go through this loop several times.

(2) She has a fixed route from home to work. If a detour is necessary because of construction work on the road, she does not know what to do, because she has only one plan, whose execution is now thwarted.

The second example above is an instance of the inability to inhibit the prepotent response to a stimulus, even when it is known that the response is inappropriate. This phenomenon is illustrated in a paradigmatic experiment designed by Hughes and Russell (1993), the ‘box task’ (see Fig. 1).

Fig. 1
figure 1

Russell’s box task

The task is to get the marble which is lying on the platform (the truncated pyramid) inside the box. However, when the subject puts her hand through the opening, a trapdoor in the platform opens and the marble drops out of reach. This is because there is an infrared light-beam behind the opening, which, when interrupted, activates the trapdoor mechanism. The switch on the left side of the box deactivates the whole mechanism, so that to get the marble you have to flip the switch first. In the standard set-up, the subject is shown how manipulating the switch allows one to retrieve the marble after she has first been tricked by the trapdoor mechanism.

The pattern of results is strikingly similar to that exhibited in the false belief task: normally developing children master this task by about age 4, and before this age they keep reaching for the marble, even when the marble drops out of reach all the time. Autistic children go on failing this task for a long time. The performance on this task is conceptualised as follows. The natural, ‘prepotent’, plan is to reach directly for the marble, but this plan fails. The child then has to re-plan, taking into account the information about the switch. After age 4 the normally developing child can indeed integrate this information, that is, inhibit the prepotent response and come up with a new plan. It is hypothesised that autists cannot inhibit this prepotent response because of a failure in executive function. But to add precision to this diagnosis we have to dig deeper.

It is important to note here that the ability to plan and re-plan when the need arises due to changed context is fundamental to human cognition, no less fundamental than ‘theory of mind’ abilities. Remember executive function was first conceived as the capacity for dealing flexibly with novel situations. Human beings (and other animals too) act, not on the basis of stimulus-response chains, but on the basis of (possibly distant) goals which they have set themselves. A goal together with a world-model lead to a plan which suffices to reach the goal in the assumed circumstances. But it is impossible to enumerate a priori all events which might possibly form an obstacle in reaching the goal. It is therefore generally wise to keep open the possibility that one has overlooked a precondition, while at the same time not allowing this uncertainty to inhibit one’s actions. It is perhaps this flexibility that autists are lacking. This point can be reformulated in logical terms. The autist’s concept of a rule is one in which the consequent invariably follows the antecedent. By contrast, a normal subject’s rule is more likely to be of the exception-tolerant variety. Indeed, Russell writes (following a suggestion by Donald Peterson)

[T]aking what one might call a ‘defeasibility stance’ towards rules is an innate human endowment – and thus one that might be innately lacking ... [H]umans appear to possess a capacity – whatever that is – for abandoning one relatively entrenched rule for some novel ad hoc procedure. The claim can be made, therefore, that this capacity is lacking in autism, and it is this that gives rise to failures on ‘frontal’ tasks – not to mention the behavioural rigidity that individuals with the disorder show outside the laboratory (Russell 2002, p. 318).

Russell goes on to say that one way this theory might be tested is through the implication that “children with autism will fail to perform on tasks which require an appreciation of the defeasibility of rules such as ‘sparrows can fly’.” This is what we shall do; but to get started we first need a logical description of what goes on in the box task.

3.5.1 Closed world reasoning in the box task

For the formalisation we borrow some self-explanatory notation from the situation calculus. Let c be a variable over contexts, then the primitives are

  • the predicate do(a,c), meaning ‘perform action a in context c

  • the function result(a,c), which gives the new context after a has been performed in c.

The actions we need are g (‘grab’), u (‘switch up’), d (‘switch down’). We furthermore need the following context-dependent properties:

  • possess(c): the child possesses the marble in c

  • up(c): the switch is up in c (= correct position)

  • down(c): the switch is down in c (= wrong position).

The following equations give the rules appropriate for the box task

$$down(c)\wedge do(u,c)\wedge \neg ab'(c)\rightarrow up(result(u,c))$$
$$do(g,c) \wedge \neg ab(c)\rightarrow possess(result(g,c))$$

We first model the reasoning of the normal child >4 years. Initially, closed world reasoning for ab(c) gives \({\neg ab(c)}\), reducing the rule 24 to

$$do(g,c) \rightarrow possess(result(g,c))$$

which prompts the child to reach for the marble without further ado. After repeated failure, she reverts to the initial rule 24, and concludes that after all ab(c). After the demonstration of the role of the switch, she forms the condition

$$down(c)\rightarrow ab(c)$$

She then applies closed world reasoning for ab to 26, to get

$$down(c)\leftrightarrow ab(c)$$

which transforms rule 24 to

$$do(g,c) \wedge up(c)\rightarrow possess(result(g,c))$$

Define context c 0 by putting cresult(u,c 0) and apply closed world reasoning to rule 23, in the sense that ab′(c) is set to \({\bot}\) due to lack of further information, and → is replaced by ↔. Finally, we obtain the updated rule, which constitutes a new plan for action

$$down(c_0)\wedge do(u,c_0) \wedge c = result(u,c_0)\wedge do(g,c)\rightarrow possess(result(g,c))$$

As in the previous tasks, both the normal child younger than 4, and the autistic child are assumed to operate effectively with a rule of the form

$$do(g,c)\rightarrow possess(result(g,c))$$

which cannot be updated, only replaced in toto by a new rule such as 29.

It is tempting to speculate on the computational complexities of both these procedures. The preceding considerations suggest that what Russell called in the quote above ‘abandoning one relatively entrenched rule’ may indeed be costly, but that normal humans get around this by representing the rule in such a way that it can be easily updated. It is instructive to look at the computation that the normal child older than 4 is hypothesised to be performing. The only costly step appears to be the synthesis of the rule 26; the rest is straightforward logic programming which, as we have argued in (Stenning and van Lambalgen 2007, Chapter 8), can proceed automatically. The rule 24 is never abandoned; a new rule is derived without having to ditch 24 first.

To close this discussion, we compare the false belief task to the box task from the point of view of the formal analysis. The tasks are similar in that for successful solution one must start from rules of the form \({p\wedge \neg ab\rightarrow q}\), identify conditions which constitute an abnormality, and apply closed world reasoning; and also that in both cases a failure of ab to exercise its inhibitory function leads to the inability to inhibit the prepotent response. A difference is that in the false belief task, one needs a ‘theory’ relating ab to sensory, or inferred, information, whereas it suffices to operate with rules for actions in the box task.

3.6 The suppression task as a formal analogue of the box task

When considered formally, all tasks mentioned have a logical structure in common, besides showing undeniable differences. The common core is closed world reasoning applied to possible exceptions. It is therefore an interesting challenge to try to devise a task which captures precisely this common core. Surprisingly, a task with the required properties has been around for a long time in the adult reasoning literature, although it was not treated as such: Byrne’s ‘suppression task’ (1989).

If one presents a subject with the following premisses:

  • (2) 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 subjects Footnote 5 draw the conclusion ‘She will study late in the library’. Next suppose one adds the premiss

  • (3) 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’. This known as the ‘suppression’ of modus ponens.

However, if instead of the above, the premiss

  • (4) 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’—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, (3) and (4). The results are that MT is suppressed in the presence of a premiss of the form (3) (but not (4)), and that both AC and DA are suppressed in the presence of a premiss of the form (4) (but not (3)).

Byrne interpreted her data in terms of the ‘mental rules’–‘mental models’ debate, viewing the results as support for the latter. In Stenning and van Lambalgen (2005), we gave a very different interpretation of the suppression phenomenon as an instance of closed world reasoning. Given the formal analogy between the box task and the suppression task, we are led to expect that autists have a very specific difficulty with closed world reasoning about exceptions. This should show up in a refusal to suppress the inferences MP and MT in case the second conditional premise is of the additional type. To show that the problem is really specific to exceptions, and not a problem about integrating new information, or with closed world reasoning generally, one may compare autists’ reasoning with AC and DA, in which case suppression is independent of exception-handling. Here one would expect behaviour which is comparable to normals. One must thus distinguish two forms of closed world reasoning that play a role here. On the one hand there is closed world reasoning applied to abnormalities or exceptions, which takes the form: ‘assume only those exceptions occur which are explicitly listed’. On the other hand there is closed world reasoning applied to rules, which takes the form of diagnostic reasoning: ‘if B has occurred and the only known rules with B as consequent are A 1B, ..., A n B, then assume one of A 1,...,A n has occurred’. These forms of closed world reasoning are in principle independent, and in our autist population we indeed observed a dissociation between the two.

4 Autists’ performance in the suppression task

In order to test these hypotheses, formulated generally as

  • (5) Autists can apply closed world reasoning, but have a decreased ability in handling exceptions to rules

Smid (2005) conducted an experiment on a population of six autists (young adults) with normal intelligence (IQ > 85) and language abilities from a psychiatric hospital in Vught (Netherlands). The tests administered to the subjects involved a false belief task (the ‘Smarties’ task Footnote 6), propositional reasoning with 2 premisses (MP, MT etc.), the Wason selection task, the suppression task, reasoning with prototypes, and analogical reasoning. The method consisted in having tutorial interviews with the subjects, which were taped and transcribed (including annotation for pauses and emphases). This data has since been augmented from a comparable population at Nijmegen University Medical Hospital. Table 1 presents the combined data as relevant to the suppression task.

Table 1 Results on suppression task in autists (n = 20)

As predicted, suppression of fallacies (DA and AC) with an alternative premiss does occur and the percentages we find are roughly the same as those found in research with normal subjects (cf. Table 1). Suppression of MP, by contrast, is much rarer in our subjects than in normals. In the dialogues subjects often ignored the additional premiss completely in their overt reasoning. With regard to suppression of MT the results are less dramatic, and harder to interpret, in particular because the rate of endorsement for MT with an alternative premiss is somewhat higher than that for the base case. Nevertheless, the percentage of MT conclusions drawn to problems with an additional premiss is higher than for the normal subjects—autists are not suppressing.

These observations lend some support to the hypothesis (5) that it is specifically processing exceptions that creates difficulties for autistic subjects. DA and AC showed the pattern familiar from normals, suggesting that this type of closed world reasoning, where exceptions do not figure, presents no atypical difficulties. The behaviour in MP and MT conditions (especially the former), where implicit exceptions need to be acknowledged to achieve suppression, was different from normals, showing much less suppression, and moreover non-suppression for different reasons: a total disregard of the additional premiss. Perhaps (5) is not the ultimate formulation of the hypothesis, but that there is something very distinct about autists’ handling of defeasible rules seems certain.

5 Executive function and verb tenses

We have analysed the executive function component in several tasks in which autists are known to experience difficulties, and we have seen that the behaviour of autists in these tasks can be viewed formally as a very specific deficit in executive function, namely the inability to inhibit a prepotent response. There are several aspects to executive function, however, of which inhibition is only one, and in this section we extend our methods to encompass a disorder in which goal maintenance may be impaired, namely ADHD. This disorder affects about 6% of children (mainly boys) and is characterised by persistent and developmentally inappropriate levels of inattention, impulsivity, and hyperactivity. It has been hypothesised to be an executive function disorder, and indeed children with ADHD score significantly lower on a number of standard tests measuring components of executive function, such as planning, inhibition, and self-monitoring. The precise pattern of executive deficits in ADHD is not yet known, and it is not yet determined whether there is a single executive deficit that explains most of the symptoms. Below we will investigate consequences of the hypothesis that goal maintenance is affected in ADHD, evidence for which can be found in Shue and Douglas (1992); Pennington and Ozonoff (1996). We will be particularly interested in the effect of deficient goal maintenance on language processing, and we therefore briefly introduce some of the relevant neuropsychological data.

In Shue and Douglas (1992) it is shown that children with ADHD score significantly worse than neurotypical subjects on a family of tasks involving control of motor response. The paradigm case is the ‘Go/No Go’ task. In the version of Shue and Douglas (1992), the ‘go’ stimulus is a card showing an apple, and the ‘no go’ stimulus a card showing an ice cream; Footnote 7 the response is pressing a key. Viewed as a (rather minimal) planning problem the computation goes like this. There are two rules, which can be formalised as ‘ if apple, do(go)’ and ‘ if ice cream, do(no go)’. The goal is ‘do(x) now’, with x a variable to be instantiated. There are two possible unifications for x, go and no go, which reduce the goal to satisfying one of the preconditions, apple or ice cream. The (unique) successful unification then determines the action to be performed. The plan synthesised by this computation is then an IF THEN ELSE rule. Of course if all goes well, after a few trials performance is determined by the automatic IF THEN ELSE rule, and not by the explicit computation of the plan. But performance on this task in children with ADHD is significantly impaired in that many ‘no go’ trials lead to a ‘go’ response, and this suggests that the initial computation (unification and reduction of the goal ‘do(x) now’) is not executed correctly. Unfortunately, it is not reported in Shue and Douglas (1992) whether there are symmetrical errors with the ‘go’ trials. If the ‘go’ response is conceived as an unconditional response, it does not require any computation. It is this distinction that we shall apply to the analysis of verb tenses below.

Language ability also seems to be affected in ADHD, in particular with regard to semantics and pragmatics. This has been investigated using story telling tasks, in two forms. In Purvis and Tannock (1997) a folk tale ‘The father, his son and their donkey’ was read to the children, who then had to repeat the story. In Blankenstijn and Scheper (2003) the ‘Frog story’ paradigm was used, in which children were asked to narrate a sequence of 24 scenes in a picture storybook called Frog, where are you?, where a boy attempts to find his pet frog which has escaped from its jar. Footnote 8 The drawings depict various failed attempts, until the boy finds his frog by accident. The purpose of the experiment is to investigate what linguistic devices, in particular temporal expressions, children use to narrate the story, as a function of age and mental condition.

Both tasks seem to involve executive function, in particular planning and goal maintenance. One connection between narration and planning has been illuminated in Trabasso and Stein (1994), whose title says it all: “Using goal-plan knowledge to merge the past with the present and future in narrating events on line”. The main idea is that the events depicted in the book are naturally structured in time as a sequence of actions aimed toward achieving a goal, and hence the narration is like the unfolding of a plan. Executive difficulties are therefore likely to result in deviant narration patterns, and indeed several such have been observed. For our purposes the following two phenomena are of especial interest (Purvis and Tannock 1997, p. 136):

  1. (i)

    retelling story events out of sequence—this reflects a breakdown in overall goal-plan organisation

  2. (ii)

    ambiguous anaphoric references to events—this reflects a more local breakdown of planning, a failure to achieve discourse cohesion

In order to sketch a theoretical background for these phenomena, and derive an additional prediction, we have to make a detour through a recently proposed formal semantics for tense.

5.1 Verb tenses from planning

In the book The proper treatment of events (van Lambalgen and Hamm 2004) the main function of tense and aspect is considered to be the construction of event structures from discourses. This construction is moreover viewed as a planning problem: the same mechanism which in planning constructs a sequence of actions is responsible for the construction of event structures from discourses. In this set-up, verb tenses are represented as goals in the same sense as goals are used in planning. In both comprehension and production, the goal is to introduce the event corresponding to the tensed VP into the event structure. Such a goal must have two components:

  1. (i)

    location of the event in past, present or future

  2. (ii)

    meshing the event with events introduced previously

Thus, the computational function of verb tenses is much more than locating an event with respect to now; the real computational burden is borne by the incorporation of the event in the discourse. An example will make this clearer. Suppose a listener must comprehend the mini-discourse

‘Max fell. John pushed him.’

The goals in this case are

  1. (i)

    update discourse with past event e 1t fell(m) and fit e 1 in context

  2. (ii)

    update discourse with past event e 2push(j,m) and fit e 2 in context

It is hypothesised that it is the planning system which determines the event structure from the discourse; here it tries to determine the order of e 1, e 2. In order to do so, the planning system recruits causal knowledge as well as the principle that causes precede effects.

Applied to the case at hand, the planning system scans declarative memory for causal connections between e 1 and e 2 and finds (roughly) ‘e 2 causes e 1’. This fixes the temporal order of e 1 and e 2, with e 2 preceding e 1. Note that inferring this event order is a defeasible process; if the discourse had been a bit longer, say

Max fell. John pushed him, or rather what was left of him, over the edge.

the order of e 1 and e 2 would now be different, since the discourse conjures up a scenario like the following: John does something very nasty and bloody to Max which makes him fall, near the edge of precipice; he then shoves the body over the edge of the precipice: e 1 precedes e 2. Footnote 9

5.2 Deviant verb tenses and ADHD

We are now in a position to formulate a hypothesis on the production and comprehension of verb tenses in narrative tasks by children with ADHD. Recall the goal corresponding to a verb tense consists of two components

  1. (i)

    location of event in past, present or future

  2. (ii)

    meshing the event with events introduced previously

The first part of the goal can be executed immediately, one does not have to consider a reduction to other goals. It is the meshing that is computationally costly; why this is so can be seen if the condition ‘ fit e 2 in context ’ is spelled out. For a start, the general form of the condition must be something like ‘ fit e 2 in context {e, e′, e′′, ...}’ where {e, e′, e′′, ...} is a set of variables for events. These variables have to be unified with the events activated in the given context, which will come with an event ordering expressible in terms of <  (‘precedes’). The instruction ‘fit’ then asks for an extension of this ordering with e 2. In the case at hand, the general condition reduces to ‘ fit e 2 in context {e}’, and the instruction ‘fit’ can be viewed as asking which if any of the queries ?e 1e 2 and ?e 2e 1 can be made to succeed. (The argument given in section 5.1 shows that it must be the latter query.) Viewed in this manner, the computation involves

  1. (i)

    keeping the context active in working memory, both events and their order

  2. (ii)

    search and unification prompted by the ‘meshing’ condition

  3. (iii)

    executing in parallel the backtracking derivations that determine the place of the new event with respect to the old event order

  4. (iV)

    this step in itself involves a search for applicable causal information, and unification

This computation is vastly more complex than the simple location of the event in past, present, or future, which can be read off immediately from the verb tense. But what is most interesting for our purposes is that the apparently problematic ingredients in the ‘Go/No Go’ task, i.e. unification and reduction of the goal, are very prominent in the computation of verb tenses.

For language comprehension this means that the understanding of discourses like ‘Max fell. John pushed him.’ is predicted to be compromised: children with ADHD should less often get the reading where the event order is the reverse of the sentence order. For production one would generally expect decreased coherence of the narrative produced. This can take several forms. Consider first the use of the simple past tense. It has been observed (see for example Steedman 1997, p. 906) that the past tense requires the preliminary establishment of a reference point to which the event described by the tensed verb phrase is anchored. This requirement is captured in the above analysis by the instruction ‘ fit e 2 in context {e, e′, e′′, ...}’, which initiates a search for context. Thus, out of the blue a sentence like (6-a) is infelicitous, but with a temporal adverbial or a subordinate clause, felicity is restored, as in (6-b).

  • (6) a. \({\sharp}\)It rained.

  • b. Yesterday/When I stepped outside, it rained.

Thus, a speaker has the pragmatic obligation to provide the relevant context-setting elements. This observation can be reformulated from the point of view of the listener. The processing load of the listener is least when the reference time is provided explicitly, as in (6-b). If the speaker does not provide this reference, the listener must assume the speaker intends that the reference time can be determined from the discourse context in a straightforward manner. But the processing load of the listener is larger in this case than in the previous one, whereas the situation is of course the reverse for the speaker.

Now suppose the speaker is a child with ADHD. Producing a sentence like (6-b) requires the explicit integration of context and main event. Footnote 10 If in the mind of the child with ADHD the goal corresponding to the past tense is simplified by dropping the meshing part, this should show as an increased production of ‘bare’ past tenses as in (6-a), and a decreased use of explicit context-setting elements.

A more subtle deviation is predicted to occur with the production of perfect tenses. A scene in the beginning of the ‘Frog story’ could be described thus

  • (7) The boy woke up. The frog had gone.

Here, the past tense in the first sentence sets up the reference point for the past perfect in the second sentence, locating the departure of the frog before the reference point. Generally, the goal corresponding to a perfect tense has the following structure

  1. (i)

    locate the event in the past of the reference point

  2. (ii)

    determine the reference point from the context

Unlike the case of the past tense, here the first instruction cannot be executed without the second, and one would therefore expect significantly fewer uses of the perfect tenses in children with ADHD.

These predictions are currently being tested on a large sample of ‘Frog story’ narratives. The results obtained so far point to interesting deviations in the use of tense, of which we will provide two examples.

One interesting observation was the tendency in children with ADHD to use more quotes in their narratives, significant even in this pilot sample. For example, a narrative like (7) would be given a form like (8)

  • (8) The boy woke up. [The boy said: ] “My frog is gone!”

The effect of this change is that in (8), unlike (7), sentence order is aligned with event order, so that this strategy perhaps reflects an attempt to simplify the computations.

A second interesting observation relates to the examples (6) above. The experimenters tested whether subjects’ utterances involved one event (as in (6-a)), or on the contrary more than one event (as in (6-b)). As we have seen, utterances in which the main event is described in the past tense need a subordinate clause to provide a reference time if the speaker wants to ease the processing load for the listener. Children with ADHD used such combinations of events into one utterance significantly less than the normal controls. This is consistent with the hypothesis ventured above, that difficulties with maintaining a complex goal (corresponding to the verb tense) in memory lead to concentration on the purely temporal part of the goal, at the expense of locating the event explicitly with respect to other events.

Before we summarise the results of this paper, we must draw attention to a difference between the analysis of ADHD and that of autism. In the latter case we could pursue the analysis all the way to the neural level, because of the close connection between unknown preconditions and inhibition. So far, we lack a similar analysis here. We drew on empirical research indicating that in ADHD goal-maintenance may be compromised, but we have not appealed to a neural model of what goal-maintenance is, and what factors lead to its being compromised.

6 Conclusion: what does a formalism do for you?

We started out from a global characterisation of executive function as composed of planning, initiation, inhibition, coordination and control of action sequences leading toward a goal held in working memory. We have seen that the information processing goals of executive function can be described quite well by a formalisation in logic programming, which thus provides a top-level analysis in the sense of Marr.

This top-level analysis provides several benefits, as we illustrated for both autism and ADHD. One immediate benefit is the derivation of testable predictions. After identifying a formal structure underlying several diagnostic tasks in autism, we pointed to a reasoning task with that same formal structure, and we predicted (correctly, as it turned out) specific differences between performance of autists and normal controls on that task. In ADHD the route to prediction was slightly different: given the well-attested difficulties of goal-maintenance in ADHD, and given furthermore the importance of goal-maintenance in the processing of tense, it was predicted that children with ADHD would show problems with integrating several events into a single linguistic utterance.

Another benefit of the formal analysis is the possibility to determine what is common and what different in theories presented as radical alternatives, such as ‘theory of mind deficit’ and ‘executive dysfunction’. The formal analysis showed that performance on the false belief task can to a large extent be explained by executive dysfunction, but not completely: some insight into the causal relation between perception and belief appears to be necessary for successful performance. That is, failure on the false belief task may be due either to executive dysfunction (in particular failures in inhibition) or to lack of insight into the causal relation mentioned. Footnote 11

The virtues of this analysis become even clearer when the focus shifts to the lower levels. We certainly do not claim that autism, say, is completely characterised by some peculiarities in reasoning (either explicit or implicit) vis à vis normal controls. What is of most importance is that the formal structure tells us something about underlying neural peculiarities. In the case of autism we saw that the analysis of the relevant diagnostic tasks highlighted the role of closed world reasoning about unknown preconditions, which at the level of neural implementation was seen to be intimately related to inhibition; and we could then appeal to data showing that for various reasons inhibition is compromised in autism.

A potential benefit of logical analysis is a reconceptualisation of the psychiatric disorders themselves, although this must remain speculative at this point. Psychiatric disorders as defined in DSM-IV are characterised as syndromes, more or less coherent collections of symptoms without an underlying theoretical rationale. One indication that there is something amiss with the DSM-IV classification is that the psychiatric disorders thus defined come with considerable comorbidity. For instance, autism is often accompanied by depression, and both autism and ADHD by various language disorders. In fact if one takes a category of symptoms as broad as executive dysfunction, then it appears under the headings of most of the major psychiatric categories in DSM-IV. There should be no surprise: executive functions are prominent among evolutionarily recent human cognitive innovations, and innovations perhaps tend to go wrong more often than older mechanisms. But there is a real issue here for psychiatry.

One possibility is that there is an essence to be discovered for each of these disorders. It may well be that a formal analysis such as the one sketched above may contribute to a different, theoretically motivated partitioning of symptoms which can also take account of the comorbidity patterns. However, there is another possibility that the several symptoms have different underlying causes, and that patients suffer from combinations of problems. On this view, there are no essences to be found corresponding to clinical labels. The need for single diagnostic labels is powerful, but may lead to an illusory realism about the discrete categorisation of psychiatric conditions. Such an account has been proposed under the name of the transdiagnostic approach (Harvey et al. 2004). An important methodological consequence for the cognitive analysis of psychiatric disorders is that most samples of patients with clinical conditions are filtered through diagnoses. For example, to qualify as autistic under DSM-IV one has to have a range of social and non-social cognitive deficits, and a correlation between them in any sample thus selected is inevitable. Interestingly, genetic studies are one source of data that sometimes sidesteps this problem. ‘Community studies’ in genetics sample the population on the basis of say postcode, and then screen for psychiatric conditions such as component symptoms of autism. In these samples, one can ask whether social and non-social deficits are really correlated. At least one, albeit preliminary, study suggests that they may not be (Ronald et al. 2005). One possible contribution of logical analyses is to search for functional commonalities between the apparently unrelated symptoms.

Our only claim here is that logical analysis is a powerful tool for revealing both cases in which apparently unrelated behaviours share a common abstract characterisation, and cases where the superficial similarity of behaviours can obscure commonalities. Both in autism and in ADHD (at least if our preliminary results hold up) there are non-linguistic patterns of behaviour which are logical analogues of discourse patterns.

Of course one can assign abstract noun phrases (like ‘executive function’) to achieve related abstractions, but abstract nouns do not carry with them the requirement to consistently capture indefinitely complex inference patterns across domains, nor do they provide clues as to how they might be implemented in the mind. Aversion to formalism sometimes leads to its rejection in favour of all the contentful insights that are unearthed in applying formalism. For example, in our analyses of reasoning, we found that we have to make assumptions about beliefs as causal effects of perception, impose particular analyses on rules in each task, bring in a theory of tense in narrative, and make many other assumptions. These contentful side-effects are not somehow distinct from the process of formalisation. They arise from the demands of formalisation, and without those demands can remain dormant indefinitely.