There is no one logic to model human reasoning. The case for interpretation

Full text

!
"!
There%is%no%one!logic&to&model&human&reasoning:&the&case&from&interpretation.!
Alexandra Varga, Keith Stenning, Laura Martignon
Universities of Giessen, Edinburgh, Ludwigsburg
!
1. Introduction
We are interested in computational models for human reasoning at the performance, or process level.
Cognitive modelling amounts to the use of some formalism in order to provide a productive description of
particular cognitive phenomena. “Productive” has an explanatorily-oriented, twofold meaning: on the one
hand, the description helps a better understanding of the phenomena, and second, it can be used to generate
empirical predictions aiming to refine the theory that backs the model. By ‘performance model’ we imply
that the formalism is actually used by real human agents in real reasoning contexts, wittingly or not. The
reasoning process at the psychological level is an instantiation of the formal model. The model is thus not
merely a theoretical formal description of reasoning. The ‘wittingly or not’ specification points to the need
of the modelling enterprise to include those forms of reasoning which are merely implicit, or below-
awareness. A model of such reasoning processes involved in, e.g., the low-level planning to mount on the
bike and start riding to University, or understanding an utterance in one’s native language, amounts to
expressing these unwitting processes and subsequent behaviors ‘as if’ they were the result of computations
expressed in a formal language.
We propose that the highest level of explanatory productivity, or information gain, can be achieved by a
multiple-logics approach to cognitive modelling. In brief, this is so because of the complex differences
between different kinds of reasoning which cannot be adequately captured by the formal properties of a
single system. A multiple-logics approach is mandated because an all-purpose logic of human reasoning
conflicts with the many things that humans may use reasoning for (Achourioti et al. 2014), e.g., to prove
beyond reasonable doubt that the accused is guilty of the crime, to prove beyond all doubt Fermat’s
theorem, to make the child understand the moral behind the story of the Ants and the Grasshopper. This
would remain so even if all of the many formal candidates could be reconstructed in a single highly
expressive logical system, because its use in human reasoning would be too resource-demanding; in other
words, computational efficiency is an opportunity cost of expressive power. Performance models should at
all points keep the balance.
Cognitive modelling from a multiple-logics perspective is also sanctioned by the history of
psychological research. Suffices to mention here the observation that classical logic does not fit with the
experimental evidence of actual human reasoning; some examples are the results in Wason’s selection task
(1968), the ‘biases’ marring syllogistic reasoning (Evans et al., 1983), or the suppression effect first
observed by Byrne (1989). For instance, human flexibility in reasoning, e.g., the withdrawal of previously
validly derived conclusions when new information is added to the premise set, does not afford description
in terms of a monotonic formalism such as classical logic. Everyday reasoning is most often non-
monotonic. However monotonicity can be triggered by, e.g., by task instructions that create a dispute
setting (Achourioti et al., 2014). The bottom line is that different forms of reasoning, meant to achieve

!
#!
different goals, should be modelled in a formalism that bears the context-dependent properties of the
inferences.
The main purpose of the current paper is to review the ‘bridging potential’ of a multiple-logics
approach; we thus summarise some results of the approach, draw some interim conclusions, and propose
avenues for future research. The roadmap is as follows. We start in Section 2 by introducing the distinction
between two kinds of reasoning, interpretation and further reasoning from that interpretation. We introduce
the working example of a formalism, namely Logic Programming, and emphasise its application to
interpretative processes. The remainder of the paper develops the argument based on taking interpretation
seriously. In Section 3 we develop the change of direction from the use of classical logic as a monolithic
standard of human reasoning. Section 4 describes in detail a case of pre-linguistic, implicit reasoning and
summarises the modelling work in Varga (2013). It shows how the previously described logical and
psychological aspects can be integrated. Section 5 exemplifies the multiple-logics approach by describing
the use of Logic Programming and fast and frugal heuristics for better understanding subjects’ reasoning
processes; we emphasize the consequential methodological advantage of theoretical unification of the
fields of reasoning and of judgement and decision-making. We end with some suggestions for further
development of the multiple-logics approach, based on collaborative modelling among different systems.
2. The proposed view of reasoning and an example of formal implementation.
We view human reasoning with broad teleological lenses, i.e., from the perspective of goals broadly
construed1. We are mostly concerned with everyday reasoning, i.e., the processes involved in habitual
activities such as conversations, disputes, stories, demonstrations, etc.
Stenning and van Lambalgen (2008) set forth two kinds of processes: reasoning to an interpretation of
the context, and reasoning from that interpretation. From a goal-centered perspective, we now propose to
extend the analysis by viewing interpretation as a sub-goal of further reasoning from that interpretation2.
As a sub-goal, it is necessary for accomplishing the purpose that calls for reasoning. Hence models of
different kinds of reasoning should not neglect the ubiquitous interpretative component.
Language processing is perhaps the clearest instantiation of the two reasoning stages. When speakers
ask their interlocutors a question, they must first process the string of words in the context (linguistic and
extra-linguistic) and produce an interpretation or model of it; in order to achieve the default purpose of
communication fast and efficiently, these computations are aimed at the one model intended by the
speakers. Because of this assumption that the right interpretation is in terms of what “(s)he must have
meant to ask”, the interpretative process is a paradigmatic case of credulous or cooperative reasoning. Of
course, credulity is only the beginning of the story. Should the first interpretation be unsatisfactory, e.g., all
of a sudden being asked by one’s life-time partner the question “How old are you?”, hearers might resort to
compensatory mechanisms, e.g., taking into account metaphorical meanings, asking for clarifications. Once
the model is available the interlocutors can start to compute what they believe to be the contextually
appropriate answer – this is reasoning from the interpretation. The reasoning path is not linear – suffices to
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
"!$%%!&'()!*+(,-((+).!)/!0-'1+2'%3%'!1%'%)')45!+.!$%,1+).!67!!
#!8%!9+(:!1)!%0;:&(+<%!1:&1!1:%=%!+(!.)!,).1=&*+,1+).!>%19%%.!>%+.4!&!(->24)&'!)/!/-=1:%=!=%&().+.4?!&!(->24)&'!)/!)1:%=!
4)&'(!@%747?!-.*%=(1&.*+.4A?!&.*!>%+.4!&!4)&'!B+.!+1(!)9.!=+4:1C7!!

!
D!
think about discourse understanding, where additional utterances usually require model updates or re-
computations of the initial model.
The focus of cooperative interpretation on constructing a minimal contextual model can be described as
the use of closed-world assumptions to frame the inferential scope (van Lambalgen and Hamm, 2004,
Stenning and van Lambalgen, 2008). The basic format is the assumption for reasoning about abnormalities
(CWA), which prescribes that, if there is no positive information that a given event must occur, one may
assume it does not occur. In practice, these ‘given events’ are abnormalities with respect to the smooth,
habitual running of a process; for example, a metaphorical interpretation is abnormal with respect to the
literal one, and thus disregarded in minimal model construction. A conditional abnormality list is attached
to each conditional; the list should be viewed as at the back of reasoners’ minds (Varga, 2013). That is,
abnormalities are reasoned about only when evidence arrives (otherwise the assumption would be self-
defeating). CWAs require construction of a minimal interpretation based only on what that is derivable
from explicitly mentioned information. This is why they can be said to restrict or to ‘frame’3 reasoning to
manageable dimensions. Interpretation with CWAs is therefore a plausible candidate to model the
reasoning of real agents’ (i.e., agents with limited memory and computational resources) in real-time. It can
provide accurate descriptions of quasi-automatic interpretations, which are computed fast and effortlessly.
The CWA is captured by all three parameters of Logic Programming – LP (syntactic, semantic, and
definition of validity), a computational logic designed by AI researchers for automated planning
(Kowalski, 1988); it is the formal system that we use to instantiate our proposal. We view the utilization of
such a formalism to model human inferences as a contribution to the bridge that this workshop seeks to
build.
Whereas an extensional formal approach deals with sets of items and with relations between those, an
intensional one deals with characteristics and constitutive properties of the items in these classes.
Relatedly, Logic Programming is an intensional formalism because its semantics is given in terms of
computations of completion, and is therefore not directly truth-functional. We adopt the formal description
of the logic set forth in van Lambalgen and Hamm, 2004 (applied to constructing a formal semantics of
time) and Stenning and van Lambalgen, 2008 (applied to explanations of human data in a series of
psychological experiments, e.g., Wason’s task, the suppression task, the false belief task).
The CWA provides the notion of valid inference in LP, as truth preserving inferences in minimal
models where nothing abnormal is the case. Relatedly, the LP conditional is represented as p & ~ab
!
q,
which reads as “If p and nothing abnormal is the case, then q”. Closed-world reasoning manifests itself in
that, unless positive evidence (i.e., either explicit mentioning, or facts inferable from the database with the
LP syntactic rules), the negation of the abnormality conjunct holds true. The syntactic expression of closed-
world reasoning is the derivation rule of negation-as-failure – NAF. If a fact can be represented as the
consequence of falsum ⊥, thus it cannot be derived by backwards reasoning from program clauses, its
negation is assumed true and the fact is thereby eliminated from the derivation. When resolving the query q
given a program with clauses p & ~ab
!
q and ⊥!ab, q reduces to p & ~ab, from which p is derived by
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
D!8%!-(%!1:%!1%=0!9+1:!=%/%=%.,%!1)!1:%!/=&0%!;=)>'%0?!%747?!E,F&=1:5!G!H&5%(?!"IJI7!
K!HL''*)>'%=!&.*!M%.,&.&!N&0'+!@#OOIA!,=+1+,+(%*!1:%!M'%%.%!(%0&.1+,(!-(%*!>5!$1%..+.4!&.*!3&.!P&0>&'4%.!@#OOQA!>5!

!
6!
means of NAF. Use of negation-as-failure in derivations means that goal resolution checks if a goal clause
can be made true in a minimal model of the program. A minimal model is a semantic interpretation of the
information available in the logic program. It is a ‘closed world’ in the sense that facts not forced to occur
by inferences over the program clauses using the LP syntactic rules are assumed not to occur. The system’s
three-valued Kleene semantics (procedural in nature) warrants the construction of a unique minimal model,
which is the only interpretation of concern of the current input for reasoning5. Minimal models are
provided by a semantic restriction of logic program clauses, called completion. It is obtained by
introducing disjunction between all the bodies (antecedents) with the same head (consequent) in a program,
and substituting implication with equivalence between the disjunctive body and the head.
The use of CWAs in interpretation is only the beginning of the intensional, or meaning-directed part of
reasoning. Computations of a minimal preferred interpretation have been described at the psychological
level by Stenning and van Lambalgen (2008) as an interaction between the knowledge base of long-term
memory and incoming input (e.g., new discourse statements, or new observations), in search for relevant
information. Novel input may override the assumption and lead to subsequent model extensions by
inclusion of the encountered abnormalities. This is a constitutively difficult task because at any give point,
the vast majority of the long-term memory knowledge base is irrelevant. The Kleene semantics models
this phenomenon by setting propositions to value U (undecided), which can develop to either T or F as a
result of further inferences. The extensions of minimal models are also minimal. LP reasoning is thus
inherently non-monotonic. Because of this it aligns with both the efficiency and the flexibility of everyday
reasoning, evidence of which were the main reasons for breaking out of the monolithic classical logic
approach.
Let us relate this to the empirical sciences of human reasoning. What is most missing in the literature is
detailed consideration of a positive account of the mental processes of interpretation, and of the interplay of
the two forms of reasoning. In psychological experiments, when subjects are presented with the premises
of a syllogism, they must first make sense of the information presented in order to be able to perform the
inferences they are asked for. Reasoning to an interpretation must be acknowledged at face value by
cognitive scientists when operationalizing theories into testable hypothesis, when deciding on the standards
for response evaluation, when interpreting the empirical data, and obviously, when setting forth
computational models for better understanding the cognitive phenomena. Despite a long period of utter
neglect7, recent work in the psychology of reasoning has started to acknowledge the role of interpretation,
e.g., Bonnefon and Hilton, 2002, Hertwig et al., 2008, Politzer, 2004. We believe this is a salutary new
direction which calls for development of its consequences in modelling; consequently we argue that
intensional formalisms are a necessary (though certainly not sufficient) ingredient of models for reasoning.
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
K!HL''*)>'%=!&.*!M%.,&.&!N&0'+!@#OOIA!,=+1+,+(%*!1:%!M'%%.%!(%0&.1+,(!-(%*!>5!$1%..+.4!&.*!3&.!P&0>&'4%.!@#OOQA!>5!
=%/%=%.,%!1)!0)*%''+.4!1:%!(-;;=%((+).!1&(R!@S5=.%?!"IQIAT!1:%(%!&-1:)=(!;=);)(%!-(+.4!1:%!P-R&(+%9+,<!(%0&.1+,(!
+.(1%&*7!U!/-''%=!1%,:.+,&'!=%V)+.*%=!+(!&3&+'&>'%!+.!1:%!U;;%.*+W7!H%=%!9%!9+(:!1)!%0;:&(+<%!1:&1!S5=.%C(!1&(R!,&''(!/)=!&!
,));%=&1+3%!+.1%=;=%1&1+).!)/!1:%!%W;%=+0%.1&'!0&1%=+&'7!!X:%!(5.1&,1+,!=%(1=+,1+).(!).!PY!,).*+1+).&'(!).!1:%!)1:%=!:&.*?!
%747?!.).2+1%=&>+'+15?!&'')9!,)0;'%1+).!1)!(-,,%%*!+.!;=)3+*+.4!&!0+.+0&'!0)*%'!&(!&!;=%2/+W%*!;)+.1!+.!&!,));%=&1+3%!
,).1%W1?!9:%=%!%;+(1%0+,!1=-(1!+(!V-(1+/+%*7!!!
Z!U!.)1&>'%!%W,%;1+).!:%=%!+(!H%.'%!@"IJ#A7!!

!
K!
3. Making use of formalized interpretations in the psychology lab.
Stenning and van Lambalgen (2008) summarise an extended program of using computational logic to
model human reasoning, notably reasoning to interpretations. This work is a sustained meditation on the
gap between human and automated reasoning, providing one kind of answer about the nature of the
relation, and several directions for bridging the gap. If a formalism such as Logic Programming can be
shown to be a good model of some of the things that humans do in reasoning, e.g., communicating, then the
gap may be very narrow. Especially so, because LP as a model is very close to actual reasoning processes.
If one can see the ERP spikes that correspond to the firing of abnormality conditions in text processing, as
proposed by Pijnacker et al. (2010), then there may be no gap at all. The optimistic view is that the brain
may directly implement the neural LP-net model on which the predictions are based (cf. Chapter 9 in
Stenning and van Lambalgen, 2008).
We use this view here as an example, in order to draw some programmatic warnings of how easy it is,
nevertheless, to fall down this non-existent paradoxical gap. We choose this forum for this argument in an
attempt to be helpful to those approaching the gap from the automated reasoning side.
As mentioned in the Introduction, it was routinely assumed, at least until the turn of this century, that
the main deductive tasks should be interpreted by experimental subjects according to classical logical
standards. The oldest task was categorical syllogistic reasoning (Störring, 1908) and the one with the most
papers was Wason’s (1968) selection task; both were meant by experimenters to be interpreted classically
by subjects. Stenning and van Lambalgen (2004) initiated a research program presenting strong evidence
that people often actually interpret them nonmonotonically (for a review, see Baggio et al., 2014).
Psychological data is not interpretable unless the experimenter has understood the interpretation the subject
has made of the task – its instructions, as well as its material (e.g., the premises of a syllogism). It is all
very well saying “Well, the subjects should have interpreted the task as requiring CL reasoning”. If we are
making educational judgments of a certain sort, we might agree. However, given the necessity of
interpretative processes during reasoning, psychologists must first answer the question of what
interpretation subjects actually do make that governs the reasoning that leads to their observed responses.
Experimenters in reasoning have much less ‘stimulus control’ than their colleagues in perception: it is more
a process of negotiation of mutual interpretation with their subjects through the instructions. Merely asking
them to “Reason logically” is far from sufficient because of the various possible ways of understanding the
instruction.
LP is immediately relevant to laboratory tasks because it is a logic whose formal properties, e.g., non-
monotonicity, are good approximations of reasoning to cooperative interpretations8. Deriving a preferred
model for a story is a task subjects readily adopt with the most meagre of cues.
But what about classical logic (CL)? We believe that subjects do actually have some conceptual grasp of
it, given certain contexts of presenting a reasoning task (e.g., Markovits and Potvin, 2001). The argument
and the evidence for this view are quite instructive about how to go about matching choice of logic (on the
automation side) with choice of reasoning task (on the psychological side). CL is a model of
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
Q!X:%!,)..%,1+).!>%19%%.!PY!&.*!;(5,:)')4+,&'!+.1%=;=%1&1+).!(:)-'*!.)1!>%!(%%.!&(!%W:&-(1+3%!@1:+(!9+''!>%,)0%!
)>3+)-(!+.!1:%!-(%!)/!PY!*%(,=+>%*!+.!$%,1+).!6A7![1!+(!V-(1!&.!&(;%,1!1:&1!9%!%0;:&(+(%!+.!1:%!,).1%W1!)/!1:%!,-==%.1!
&=4-0%.17!!

!
J!
demonstration, or dispute resolution given a fixed interpretation for the set of premises. A proponent puts
forward for consideration a conclusion as following from agreed premises, while her opponent claims that
the conclusion does not follow. The proponent must then construct the proof such that the opponent is left
no rational way for disagreement. This is very different from a habitual conversation or from story-telling,
where participants must first reach a mutual understanding of the stuff that they are talking about. Yet the
standard laboratory syllogistic task presents two premises and asks the subject to draw a conclusion, or to
assess the validity of a presented conclusion. There is no information about who proposes these premises
and conclusion: presumably it is the experimenter, who seems a helpful sort of guy, but very often she is
also the one who marks the exam at the end of term. Agreement seems a good first strategy, and hence LP
reasoning in terms of intended model a good logic. However, when Achourioti et al. (2014) tells the
subjects that the origin of the premises is a nefarious character called Harry-the-Snake, that their task is to
decide whether or not to accept Harry’s bet that the conclusion is valid, and that they must produce a
counterexample if they decide to bet against him, then we see radically different data, better fitting classical
logic. Instead of a hugely asymmetrical tendency to get problems with valid conclusions correct, and ones
without wrong, in the draw-a-conclusion task, subjects behave symmetrically with Harry. If anything, they
bet against him more than they should. Their counterexamples are also highly informative. They are by no
means tactically expert at constructing them, but they do rely on some paradoxical properties of CL in their
construction. For example, subjects often propose correct counterexamples which rely on interpreting
‘empty antecedent’ conditionals as true; essentially they reason in accordance with the paradoxes of
material implication. When Harry bets that it follows from All B are A and No B are C that No C are A,
they often bet against Harry with the countermodel {A, ~B, C}. This is only a model of the premises if they
accept that empty-antecedent conditionals are true, i.e., that All B are A and No B are C are true because
there are no Bs in this model. The counterexample is valid because indeed No C are A under the
interpretation {A, ~B, C}. A common error is the production of adequate models of the premises, which
however do not falsify the conclusion. Perhaps the task of solving 32 problems under time-pressure, and
the accumulated tiredness lead to such errors in subjects’ semantic reasoning; this is a question to be
addressed in follow-up studies. The crucial point we wish to make is that when put in a dispute, they do
exhibit some conceptual grasp of CL reasoning, even if they are not tactically expert at calculating all the
angles in this fragment under experimental conditions involving time pressure and fatigue.
What morals can be drawn from this? Any automated reasoning researcher knows that CL is a case in
which the problem is one of constructing the theorem prover which works over the logic. In psychological
terms this comes down to understanding that there is conceptual knowledge of CL and tactical knowledge
of theorem proving and theory must account for both. All the psychological theories of human reasoning,
many of which fervently deny that logic plays any role, are best construed as theorem provers for very
small fragments of CL (or sometimes now probability). In this respect, LP is a rather odd case. Because of
its computational origins, it comes with at least a very abstract theorem prover built in. Nevertheless,
distinguishing theorem prover from logic is as crucial as distinguishing tactical or procedural knowledge
from conceptual or declarative knowledge. Just as the two kinds of knowledge manifest themselves in
human minds, people may also use logics where theorem proving is the main issue. In support of this
claim we mentioned the study of Achourioti et al. (2014) which has presented evidence that CL is a logic

!
Z!
humans use in certain circumstances, and have some conceptual understanding of. Such evidence suggests
a program of collaboration where the goal is to study theorem proving within CL as a model of human
reasoning in some appropriate context.
It is also clear that multiple formalisms are required to model the data. Telling a story is a different kind
of discourse than providing a proof, and discourse interpretation proceeds from different grounds. It is true
that at a meta-level, a story may be making and argument, and at a meta-level a long mathematical proof
may tell a story (perhaps a thriller?), but if we do not get the basic object level difference, then we cannot
understand what the subjects are doing. What they are in fact doing is determined by the epistemic goals
that call for particular kinds of reasoning and thus determine the adopted logic.
When the psychology of reasoning had to acknowledge the mismatch between subjects’ reasoning and
CL, the new paradigm claimed that all reasoning was probability (Over, 2009, Oaksford and Chater, 2010).
What can be said about nonmonotonic reasoning and probability? There is, of course, a syllogism:
Probability is nonmonotonic.
Human reasoning is nonmonotonic.
So, human reasoning is probability.
Although it appears attractive, students arguing with Harry-the-Snake, can tell you this is a bad syllogism.
In AI, the field tends to be described homogenously as ‘reasoning in uncertainty’. This is useful
shorthand, but since Donald Rumsfeld, we know that there is more than one kind of uncertainty: the known
unknowns and the unknown unknowns, at least. We can express the difference between probability and LP
roughly as uncertainty about truth (or the known unknowns), and uncertainty about interpretation (or the
unknown unknowns). Probability requires an algebra of propositions and the probability distributions
between their combinations; LP however is beneficial for selecting the propositions of concern. We wish to
make it crystal-clear that we are not arguing that probability cannot be a good model of any form of human
reasoning; what we are doing is to plead for more accurate characterisation of its place among the various
logics that are needed to model that many-splendoured activity, and to raise attention to its computational
complexity which does not fare well with realistic modelling of everyday reasoning phenomena.
We get to the interpretation of the ‘whole problem’ by reasoning in LP or some such intensional
formalism7!This is what we actually live through, more likely than a sequence of choices on the basis of
hugely complex probabilistic computations. And it is the mental process of interpreting newly arriving data
that we are here interested in characterising. The next Section describes modelling causal reasoning by
means of LP and heuristic meta-data about conditional frequencies. Subjects’ judgments about frequencies
make the basis for fast and frugal heuristics to combine the properties of ‘defeaters’ of inferences to make
predictions about subjects’ confidence in inference.
4. Logic for modelling implicit reasoning.
In a series of seminal studies with the head-touch task (Gergely et al., 2002, Király et al., 2013), pre-
linguistic infants have been shown to engage in selective imitative learning. We first introduce the
experiment. After showing behavioral signs of being cold and wrapping a scarf around her shoulders, an
adult demonstrates to 14-month-olds an unfamiliar head touch as a new means to activate a light-box. Half
the infants see that the demonstrator’s hands are occupied holding the scarf while executing the head action

!
Q!
(Hands-Occupied condition – HO), the other half observe her acting with hands visibly free after having
knotted the scarf (Hands-Free condition – HF). After a one-week delay subjects are given the chance to act
upon the light-box themselves. They all attempt to light-up the lamp; however reenactment of the observed
novel means action with the head is selective: 69% of the infants in the HF, and only 21% in the HO. More,
Király et al. (2013) have shown that selectivity is contingent on a communicative action demonstration.
This involves that throughout the demonstration session the experimenter behaves prosocially towards the
infant, using both verbal and non-verbal communicative-referential cues. When the action was presented in
a communicative context, the previous results were replicated. However, when the novel action is
performed aloof, without infant-directed gaze or speech, the reenactment rate is always below chance level,
and there is no significant difference between the HO and HF conditions. Gergely and his colleagues
propose that infants’ selectivity is underlain by a normative understanding of human actions with respect to
goals. That is, infants learn some means actions but not others depending on the interpretation in terms of
goals (teleological) afforded by the observed agential context.
The model set forth in Varga (2013) adopts this inferential perspective from the standpoint of multi-
level teleology, i.e., a broad representation of goals that covers a whole range from physical goals (e.g.,
turning on a light-box) to higher-order intentions and meta-goals (e.g., the adult’s teaching intention,
infants’ intentions to understand and to learn what is new and relevant)10. The inferential engine is
constraint logic programming (CLP). The model gives voice to infants’ interpretation of observations and
to planning their own actions in the test phase. This voice is spelled out in the language of the event
calculus (van Lambalgen and Hamm, 2004) – 14-month-olds’ observations and relevant bits of causal
knowledge are represented as event calculus program clauses, e.g., Initially(communication) – agent
exhibits infant-directed communicative behaviour, Terminates(contact, light-activity, tk11) – contact is the
culminating point of the light-box directed activity. Their teleological processing is called for and guided
by the epistemic goals to understand and to learn, represented as integrity constraints (Kowalski, 2011, van
Lambalgen and Hamm, 2004). CLP allows to express higher-order goals as integrity constraints. These are
peculiar conditional clauses which impose local (contextual) norms on the computations involved in goal
clause resolution; they are universally quantified (but see footnote 13). For instance, IF
?Initially(communication) succeeds THEN ?HoldsAt(teachf , t) succeeds12!expresses the assignment of a
pedagogical intention to the observed agent conditional on her infant directed communicative behavior.
When the antecedent is made true by the environment, i.e., in the communicative conditions, the young
reasoner must act such that the goal expressed in the consequent becomes true. “teachf” is a parameterised
fluent, i.e., a variable that must be specialized to a constant in the course of resolution. Infants’ propensity
for teleological understanding has been represented as an unconditional integrity constraint, namely
?Happens(x,t), Initiates(x,f(x),t), gx = f(x) succeeds. It demands assigning a concrete goal to an observed
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
"O!E-'1+2'%3%'!1%'%)')45!+(!>&(%*!).!M)9&'(R+C(!@#O""A!*+(1+.,1+).!>%19%%.!&,:+%3%0%.1!;:5(+,&'!4)&'(?!&.*!0&+.1%.&.,%!
4)&'(?!9:)(%!-.*%=;+..+.4(!:&3%!>%%.!%W;=%((%*!+.!,)0;-1&1+).&'!')4+,7!!
""!X:+(!+(!&!1%0;)=&'!,).(1&.17!
"#!\)1%!1:&1!1:%!(%0&.1+,(!)/!1:%!,).*+1+).&'!+.!+.1%4=+15!,).(1=&+.1(!+(!&!1:)=.5?!-.(%11'%*!+((-%!@M)9&'(R+?!#O""A7!]&=4&!
@#O"DA!&*);1%*!&!,'&((+,&'!(%0&.1+,(7!

!
I!
instrumental behaviour, i.e. finding a value for the Skolem function13 f(x). The requirement succeeds
makes an existential claim with respect to a physical goal, i.e. there is such a state as g, which is a function
f(x) of an action x.
Reasoning to an interpretation of the observed action context amounts to finding its means – ends
structure; it is meant to ensure contextual understanding. Given the program clause
Initially(communication) in the communicative condition, infants assign the adult the pedagogical intention
expressed in the consequent of the constraint; further computations must unify parameter f with a concrete
observed fluent, which is deemed to count as new and relevant information. Infants goal assignment to the
agent’s object-directed activity is done by resolving the unconditional constraint mentioned above. A
successful unification is sought by specializing the function f(x) to a constant fluent from the narrative of
events, given an evaluation of the causal relations available in the contextual causal model. The model
shows how backward derivations from the constraint output the solution that the state light-on is the goal of
contacting the light-box with the head, which is the culminating point of the observed activity. This
represents infants’ teleological conjecture, expected to render the action context understandable.
Interpretation is then subserved by a plan simulation algorithm – infants verify the goal conjecture by
considering what they themselves would have done in order to achieve the goal light-on. This view of
inferential plan simulation, and not merely motor simulation as traditionally construed (e.g., Meltzoff,
2007), is one of the main innovations brought about by this use of CLP for modelling. In the HO condition
the mismatch between infants’ closed-world plan calling for default hand contact, and observation of head
contact is resolved by reasoning that the adult must use her hands for another goal, i.e., to hold the scarf in
order not to be cold. The situation is fully understandable, hence infants specialize parameter f in
?HoldsAt(teachf ,t) to the object’s newly inferred function, light-on.
The HO simulationist explanation does not work in the HF condition – the adult’s free hands are not
required to fulfill any different goal, so why it is that she does not use them to activate the object? Infants
then integrate the adult’s previously assigned pedagogical intentions in the explanatory attempt. Assigning
a pedagogical intention to the reliable adult’s otherwise incomprehensible head action renders it worth
learning. Although touching a light box with the head in order to light it up may not be the most efficient
action for the physical goal, the model proposes that it is considered efficient (and thereby reenacted) with
respect to the adult’s intention to share knowledge and the infant’s corresponding intention to learn.
In the test phase, upon re-encountering the light-box, infants plan their actions. The integrity constraint
that guides their computations is ?HoldsAt(learnf ,t), Happens(f’,t) succeeds;!it corresponds to the adult’s
pedagogical intention, and it expresses a ‘learning by doing’ kind of requirement. The outcome of
interpretation, i.e., the means - ends structure of observations and the corresponding specialization of
parameter f, modulate the constraint resolution. It sets up the physical goals that infants act upon in the test
phase – either learn the new object’s function in HO (upon specialization of f to light-on), or also learn how
to activate it in HF (upon specialization of f to contacthead). These goals are reduced to basic actions
through the CLP resolution rule of backwards reasoning, which prescribes their observed behaviour. In the
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
"D!X:+(!+(!.%%*%*!1)!:&.*'%!1:%!,)0>+.&1+).!)/!-.+3%=(&'!&.*!%W+(1%.1+&'!^-&.1+/+,&1+).!_!1:%!%W+(1%.1+&''5!^-&.1+/+%*!
3&=+&>'%!9+1:+.!1:%!(,);%!)/!&!-.+3%=(&'!^-&.1+/+%=!+(!=%;'&,%*!9+1:!1:%!3&'-%!)/!&!/-.,1+).!)/!1:%!-.+3%=(&''5!^-&.1+/+%*!
3&=+&>'%7!

!
"O!
HF condition thus, infants act upon two goals, learning the function and learning the means. The former
goal is reduced to default hand actions (as required by closed-world reasoning), whereas the latter – to the
novel head action. This explains infants’ performance of both hand and head actions. Reenactment of the
head action can be described as ‘behavioural abduction’, a continuation in behavioural terms of the
unsatisfactory explanatory reasoning.
The CLP model of observational imitative learning corroborates developmentalists’
argument that infants’ acquisition of practical knowledge from observation of adult agents is an instance of
instrumental rationality. It does so by providing a concrete example of pre-linguistic reasoning to an
interpretation, and of planning from the inferred means – ends structure of the situation. A logic is thus
shown to be helpful in formalizing a quasi-automatic kind of reasoning, very different from the traditional
understandings whereby playing chess, or proving mathematical theorems are the paradigmatic cases of
reasoning.
We end this Section with some comments about the bearings of this modelling result on the prominent
framework of dual-process theories of cognition (for a more extended discussion see Varga and
Hamburger, 2014). The ‘received view’ of the theories (Evans, 2012) draws a qualitative distinction
between processes of Type 1 – intuitive, fast, automatic, associative, unconscious, effortless,
contextualized, biased, and of Type 2 – reflective, slow, deliberate, rule-based, cogitative, effortful,
decontextualized, normatively correct. Each of these features are pairwise dichotomic – e.g., biased vs.
normatively correct, fast vs. slow. Infants’ reasoning in the head-touch task, because it is implicit, does not
hinge on linguistic abilities, and does not require conscious awareness, would be classified as Type 1.
However the model shows in detail how it proceeds according to logical rules – indeed a weaker logic than
classical logic, but nevertheless a logical system. This result throws doubt onto the qualitative dichotomic
distinction between the two Types of reasoning processes. It thus strengthens the critical arguments with
respect to dual-process theories from the perspective of ‘logical intutitions’ (De Neys, 2002). Although it
does not provide conclusive arguments against the theories, it does bring to the fore the need to better
specify the grounds of the distinction. The black-or-white distinction along the dimension ‘underpinned by
rules’ is shown not to hold the water – if anything, the rules that support different processes might be said
to be representable in distinct formalisms. This latter proposal is a further specification in logical terms of
Kruglanski and Gigerenzer’s (2011), that both types of processes are rule-based. More research is needed
in modelling other instances of fast and automatic reasoning processes, evidence of which is on the rise
(e.g., Day and Gentner, 2007); logical systems that instantiate closed-world reasoning are noteworthy
candidates.
5. A joint enterprise of Logic Programming and fast and frugal heuristics for reasoning
and decision-making.
We now show how a combined use of LP and its meta-analysis extension for counting can provide an
account of causal reasoning. Martignon et al.’s (in prep) replication of Cummins’s (1995) seminal results
was taken as an empirical proof that subjects’ judgments expressed in heuristic terms (i.e., the tallying
heuristic) predicts their confidence in conditional inferences. The authors propose that the use of fast and
frugal heuristics thus provides a method of nonmonotonic reasoning to interpretations.

!
""!
Let us begin with a brief account of heuristics as analyzed in Gigerenzer et al. (1999). An
acknowledgment of the systematic use of heuristics goes back to Archimedes. The meaning of the word
varies in different contexts. Some consider heuristics very broadly to be rules of thumb, without a specified
set of rules that provide ‘recipes’ for their use. In the context of the ABC group, heuristics have inherited
the meaning given by Einstein in his famous paper of 1905 (Einstein, 1905). That is, they are fast and
frugal algorithms that “make us smart” (cf. Gigerenzer et al., 1999), because of their simplicity and not in
spite of it. In the field of judgement and decision-making they are specified as very simple linear models
for combining cues or features in tasks like comparison, estimation or categorization. Empirical evidence
of their use has been provided by several researchers, eminently Bröder (2003, 2009) and Rieskamp and
Otto (2006). Typical examples of these heuristics are “Take The Best” – a linear model with non-
compensatory weights, “Tallying” – a linear model with all weights equal to 1, or WADD – the weighted
additive heuristic (Payne, Bettman and Johnson, 1993) whose weights are the validities (or
‘diagnosticities’) of the cues.
Martignon et al. (in prep) have recently set forth an impressive analogy between the use of heuristics for
combining cues in decision-making, and people’s use of defeaters in reasoning. Cummins had shown
empirically (Cummins, 1995) in the conditional reasoning paradigm, that a tallying of the defeaters
generated by subjects for a given causal conditional is an excellent inverse predictor of their endorsement
of the causal link between the antecedent and the consequent in the Modus Ponens inference. Consider for
instance the causal conditional “If the brake was depressed then the car slowed down”; defeaters are cases
when although the brake is depressed, the car does not slow down, e.g., the brake is broken. Cummins
(1995) provided evidence that the more defeaters are generated, the less likely people are to endorse the
classically valid conclusion of Modus Ponens. Martignon and colleagues recognized that it is precisely the
tallying heuristic on a profile of defeaters that is used for combining them in further inferences. This same
heuristic is used in the field of judgement and decision-making. In the typical comparison task analysed by
Gigerenzer et al. (1999), subjects must decide which of two German cities has a larger population, based on
cues like “city A has a license plate with one letter, while city B has a license plate with two letters”, or
“city A has a soccer team in the Bundesliga and city B does not”, etc. In cases in which cues are abundant,
subjects tend to tally them in order to make the comparison (Martignon and Hoffrage, 2002). On the other
hand, when cues are scarce, they tend to rank them and use the heuristics Take The Best, i.e., use the first
cue that discriminates the cities and chooses the city with the highest value.
So far the ranking of cues for judgements such as comparisons, say, by means of “Take The Best”, has
been modeled in a Bayesian framework. Such ranking for using “Take The Best” assumes that for each
cue, like “city A has a soccer team in the national league and city B does not”, its validity (diagnosticity)
is given by the probability that a city with such a soccer team is larger than one without such a soccer
team; that is, a cue is valid when probability is larger than 0.5. This probabilistic computation has always
been seen as cumbersome in the theory of fast and frugal heuristics (Dougherty et al., 2008), leading to
serious doubts that probabilities can provide realistic process models. LP can offer a much simpler way
for ranking cues, be it in a comparison or a categorization task. It should be easy to see that a broken
brake, for instance, can be represented as an abnormality condition in the LP representation of the
conditional as p & ~ab
!
q. The simpler way for ranking cues thus amounts to counting abnormalities for

!
"#!
the conditional “If city A has a license plate with one letter and city B has a license plate with two or more
letters then city A is larger than city B”. Here a simple tallying of defeaters will provide a good
approximation of the conditional validity without actually computing it. In a similar vein, Martignon et al.
(in prep) have also showed that other simple heuristics, like “Best Cue” (Holte, 1993) or WADD (Payne,
Bettman and Johnson, 1993) contribute effectively to predict subjects’ confidence. The crucial message is
that LP as a modelling tool can solve one aspect of the heuristics for judgement and decision-making that
have been criticized by other authors, namely relying on a Bayesian computation of cue validities
(Dougherty et al., 2008). Relatedly, it helps to make progress with respect to the problem of heuristic
selection, compared with previously proposed modelling frameworks (Marewski and Schooler, 2001). So
this might be a rather general method of nonmonotonic reasoning, which in fact may give a computational
model of how the interpretations necessary for further probabilistic reasoning are arrived at.
It is a fascinating result that precisely the same heuristics that function so well for cue combination in
judgment and decision-making are excellent for defeater combination in conditional reasoning. LP can
easily model both an interpretation of causal conditionals taking into account defeaters (remember the
defeaters that hinder the car stopping despite pressing the brake), and of the conditional expression of
typical cues for decision-making (If city A has a soccer in the Bundesliga and city B does not, city A is
larger than city B). The use of LP in modelling thus allows a unified framework that integrates the theory
of (causal) reasoning, with that of judgment and decision-making. This result aligns with recent similar
‘unificationist’ approaches in the new paradigm of psychology of reasoning, e.g., Evans et al., 1993,
Bonnefon, 2009.
6. Conclusions: wrapping-up and further-on.
Despite the fact that gaps such as the one that gives the theme of the workshop are not easy to see in
the raw data of the psychology of reasoning lab, to begin with however, their possibility must be
acknowledged in order to allow for bridging. We started by presenting interpretation as an intrinsic, sine
qua non stage of reasoning; this acknowledgement constrains realistic modelling endeavours to take it into
account. We reviewed evidence that an approach to modelling which does take intensionality seriously by
use of an expressive yet simple (at most linear on the name of nodes) formalism contributes to the
theoretical integration of reasoning with judgement and decision-making. We also presented a
computational model of pre-linguistic reasoning based on data from developmental psychology, and
mentioned some consequences of this result for the ongoing debate with respect to dual-process theories
of cognition.
With respect to future prospects for modelling applications of Logic Programming, we highlight the
need for hypotheses of different domains where interpretation via minimal model construction may be
adequate, and model that in terms of formalisms with minimal model semantics. The methodological
implication of the multiple-logics proposal is a research program where modellers, given the properties of
a particular formalism, hypothesise what kind of reasoning task it might model, and collaborate with
experimenters to test those predictions; or observe properties of a reasoning task, hypothesise an
appropriate formalisation, and test its empirical generalisations. With respect to LP, for instance, we
propose that minimal model construction accurately models people’s interpretation of conditionals

!
"D!
uttered in a conversation setting (Varga and Gazzo Castañeda, in prep.); investigations concerning other
cases of cooperative reasoning, e.g., joint planning, joint intentionality, are current work in progress.
Throughout the paper we used LP to instantiate the multiple-logic proposal. Some other examples of
applying non-deductive logics to human reasoning are Diderik Batens’s program of adaptive logics
(Batens, 2007), or Fariba Sadri’s review of work on intention recognition (Sadri, 2010). It is noteworthy
that both are essentially multiple-logic approaches. Consequently, last and most importantly, we wish to
encourage pursuit of a multiple-system approach in research concerned with human reasoning. Our
concrete suggestion concerns research on combining a logic that might appropriately model interpretation
under computational constraints, i.e., in realistic cases of reasoning, with other formalisms, e.g.,
probability (Demolombe and Fernandez, 2006). One envisaged result is an alleviation of the problem of
the priors (e.g., Pearl, 2014) by means of an intensional perspective offered by logics of interpretation.
Such endeavour would bridge the gap between logical and AI systems for engineered reasoning, on the
one hand, and empirical human reasoning research.
References
1. T. Achourioti, A. Fugard, and K. Stenning (2014). The empirical study of norms is just what we are
missing. Frontiers in Cognitive Science 5, 1159, doi: 10.3389/fpsyg.2014.01159.
2. G. Baggio, M. van Lambalgen, and P. Hagoort (2014). Logic as Marr's computational level: Four case
studies. Topics in Cognitive Science.
3. D. Batens (2007). A universal logic approach to adaptive logics. Logica Universalis, 1:221–242.
4. J. F. Bonnefon (2009). A Theory of Utility Conditionals: Paralogical Reasoning From Decision-Theoretic
Leakage. Psychological Review, 116(4): 888-907.
5. J. F. Bonnefon, and D. J. Hilton (2002). The suppression of Modus Ponens as a case of pragmatic
preconditional reasoning. Thinking & Reasoning, 8(1): 21-40.
6. A. Bröder (2000). Assessing the empirical validity of the Take The Best heuristics as a model of human
probabilistic inference. Journal of Experimental Psychology: Learning, Memory and Cognition. 26(5): 1332-
1346.
7. A. Bröder (2003). Decision making with the adaptive toolbox. Influence of environmental structure,
intelligence and working memory load. Journal of Experimental Psychology: Learning, Memory and Cognition.
29(4): 601-625.
8. R. J. Byrne (1989). Suppressing valid inferences with conditionals. Cognition, 31: 61–83.
9. D. D. Cummins (1995). Naïve theories and causal deduction. Memory and Cognition, 23(5): 646-658.
10. S. B. Day, and D. Gentner (2007). Nonintentional analogical inference in text comprehension. Memory and
Cognition, 35(1): 39-49.
11. R. Demolombe, and A. M. O. Fernandez (2006). Intention recognition in the situation calculus and
probability theory frameworks. In Proceedings of Computational Logic in Multi-agent Systems (CLIMA).

!
"6!
12. W. De Neys (2002). Bias and conflict – a case for logical intuitions. Perspectives on Psychological Science,
7(1): 28-38.
13. M. R. Dougherty, A.M. Franco-Watkins, and R. Thomas (2008). Psychological plausibility of the theory of
probabilistic mental models and the fast and frugal heuristics. Psychological Review, 115(1): 199-211.
14. A. Einstein (1905). Über einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen
Gesichtspunkt. Annalen der Physik, 4, 132-147.
15. J.St.B.T. Evans (2012). Dual-process theories of deductive reasoning: Facts and fallacies. In K. Holyoak
and R. Morrison (eds.), The Oxford Handbook of Thinking and Reasoning (pp. 115–133). Oxford University
Press, New York.
16. J.St.B.T. Evans, J.L. Barston, and P. Pollard (1983). On the conflict between logic and belief in syllogistic
reasoning. Memory and Cognition, 11: 295-306.
17. J.St.B.T. Evans, D.E. Over, and K.I. Manktelow (1993). Reasoning, decision making and rationality.
Cognition 49: 165-187.
18. G. Gergely, H. Bekkering, and I. Király (2002). Rational imitation in preverbal infants. Nature, 415: 755
19. G. Gigerenzer, P.M. Todd, and the ABC research group (1999). Simple heuristics that make us smart.
Oxford University Press, New York.
20. S. Hölldobler, and C.D.P. Kencana Ramli (2009). Logic Programs under Three-Valued Lukasiewicz’s
Semantics. In P.M. Hill and D.S. Warren (eds.), Logic Programming, LNCS series, vol. 5649, pp. 464-478,
Springer-Verlag Berlin Heidelberg.
21. R.C. Holte (1993). Very simple rules perform well on mostly used data sets. Machine Learning, 11(1): 63-
91.
22. M. Henle (1962). On the relation between logic and thinking. Psychological Review, 69: 366-378.
23. R. Hertwig, B. Benz, and S. Krauss (2008). The conjunction fallacy and the many meanings of and.
Cognition, 108: 740-753.
24. I. Király, G. Csibra, and G. Gergely (2013). Beyond rational imitation: Learning arbitrary means actions
from communicative demonstrations. Journal of Experimental Child Psychology, 116: 471-486.
25. F. Knight (1921). Risk, Uncertainty and Profit. Miffin: Boston, New York.
26. R. Kowalski (1988). The Early Years of Logic Programming. Commun. ACM 31(1): 38-43.
27. R. Kowalski (2011). Computational Logic and Human Thinking: How to be Artificially Intelligent.
Cambridge University Press, New York Cambridge University Press, New York Cambridge University Press,
New York.
28. A. Kruglanski, and G. Gigerenzer (2011). Intuitive and deliberate judgments are based on common
principles. Psychological Review, 118: 97-109.
29. J. Marevski, and L.J. Schooler (2001). Cognitive niches: an ecological model of strategy selection.
Psychological Review,118(3): 393-437.
30. H. Markovits, and F. Potvin (2001). Suppression of valid inferences and knowledge structures: The curious
effect of producing alternative antecedents on reasoning with causal conditionals. Memory and Cognition, 29(5):
736-744.
31. L. Martignon, and U. Hoffrage (2002). Fast, frugal and fit: Simple heuristics for pair comparison. Theory
and Decision, 52(1): 29-71.
32. L. Martignon, K. Stenning, and M. van Lambalgen (in prep.). Qualitative models of reasoning and
judgment: theoretical analysis and experimental exploration.
33. J. McCarthy, and P. Hayes (1969). Some philosophical problems from the standpoint of Artificial
Intelligence. In Machine Intelligence, pp. 463-502, Edinburgh University Press.

!
"K!
34. A. N. Meltzoff (2007). The ‘like me’ framework for recognizing and becoming an intentional agent. Acta
Psychologica, 124(1): 106–128.
35. M. Oaksford, and C. Chater (2010). Editors’ introduction. In Oaksford, M. (ed.), Cognition and
conditionals: Probability and logic in human thought, pp. 3-36, Oxford University Press, Oxford.
36. D.E. Over (2009). New paradigm psychology of reasoning. Thinking & Reasoning, 15(4): 431-438.
37. J.W. Payne, J.R. Bettman, and E.J. Johnson (1993). The adaptive decision maker. Cambridge University
Presss, Cambridge.
38. J. Pearl (2014). Probabilistic reasoning in intelligent systems: Networks of plausible inference. Morgan
Kaufman Publishers, San Francisco, CA.
39. J. Pijnacker, B. Geurts, M.van Lambalgen, J. Buitelaar, and P. Hagoort (2010). Exceptions and anomalies:
An ERP study on context sensitivity in autism. Neuropsychologia, 48(10):2940-2951.
40. G. Politzer (2003). Premise interpretation in conditional reasoning. In D. Hardmann and L. Macchi (eds.),
Thinking: Psychological Perspectives on Reasoning, Judgment, and Decision Making. pp. 79-93.
41. J. Rieskamp, and P.E. Otto (2006). SSL: A theory of how people learn to select strategies. Journal of
Experimental Psychology: General, 135(2): 207-236.
42. F. Sadri (2010). Logic-based approaches to intention recognition. In Handbook of Research on Ambient
Intelligence: Trends and Perspectives, pp. 346-375, Citeseer.
43. K. Stenning, and M. van Lambalgen (2004). A little logic goes a long way: Basing experiment on semantic
theory in the cognitive science of conditional reasoning. Cognitive Science, 28(4): 481-530.
44. K. Stenning, and M. van Lambalgen (2008). Human reasoning and cognitive science. MIT University
Press, Cambridge, MA.
45. K. Stenning, and A. Varga (under review). Many logics for the many things that people do in reasoning. In
International Handbook of Thinking and Reasoning, L. Ball and V. Thompson (eds.), Psychology Press, London,
UK.
46. G. Störring (1908). Experimentelle Untersuchungen über einfache Schlussprozesse. Archiv für die Gesamte
Psychologie, 11: 1-127.
47. A. Tversky, and D. Kahneman (1983). Extensional versus intuitive reasoning: The conjunction fallacy in
probability judgment. Psychological Review, 90(4): 293-315.
48. M. van Lambalgen, and F. Hamm (2004). The Proper Treatment of Events. Blackwell Publishing.
49. A. Varga (2013). A Formal Model of Infants’ Acquisition of Practical Knowledge from Observation. PhD
thesis, Central European University, Budapest.
50. A. Varga, and K. Hamburger (2014). Beyond type 1 vs. type 2 processing: the tri-dimensional way.
Frontiers in Psychology 5, 993. doi: 10.3389/fpsyg.2014.00993.
51. A. Varga, and E. Gazzo Castañeda (in prep). Yes, you may Affirm the Consequent. In a conversation
setting.
52. P.C. Wason (1968). Reasoning about a rule. The Quarterly Journal of Experimental Psychology, 20(3):
273-281.

KLEENE 3-VALUED SEMANTICS FOR LOGIC PROGRAMS IN
NON-MONOTONIC REASONING
KEITH STENNING AND MICHIEL VAN LAMBALGEN
In Chapter 7 of our Human reasoning and cognitive science we use def-
inite logic programs to represent non-monotonic reasoning with condition-
als. The main technical tool is the interpretation of conditionals via the
immediate consequence operator: the semantics is procedural, not declar-
ative. This is because in a cooperative setting the truth of a conditional is
not an issue, only what can be inferred from the conditional. This has con-
sequences for what we mean by ‘model of a program’. One may interpret
the ‘→’ in program clauses truth-functionally, and say that M |=3ϕ→q
(where Mis a 3-valued model) if the truth value of ϕ→qequals 1. We
do not think that truth-functionality is appropriate, since it would license
nested occurrences of ‘→’, whereas nesting is not allowed by the syntax of
logic programs, and hardly ever occur in natural language. Furthermore in
our setting conditionals are never false, but apparent counterexamples are
absorbed as ‘abnormalities’. It follows that we cannot give the expression
‘model of a program’ its literal meaning. We therefore use ‘model of a
program P’ in a different sense, outlined below.
We start with the simpler case of positive programs. Recall that a positive
logic program has clauses of the form p1∧. . . ∧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 monotone operator:
Definition 1. The operator TPassociated to a positive logic program P
transforms a valuation 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) = 1 if there exists a set of proposition letters C, true on
M, such that VC→v∈P
(2) TP(M)(v)=0otherwise.
Definition 2. An ordering ⊆on (two-valued) models is given by: M⊆N
if all proposition letters true in Mare true in N.
Lemma 1. If Pis a positive logic program, TPis monotone in the sense
that M⊆Nimplies TP(M)⊆TP(N).
Now consider the completion comp(P).
Definition 3. Let Mbe a valuation. We say that Mis a model of Pif
M |=comp(P).
1

2 KEITH STENNING AND MICHIEL VAN LAMBALGEN
Again we see that program clauses are not interpreted as truth functional
implications, but rather as closure conditions on a model. This idea is best
expressed using the operator TP.
Lemma 2. Suppose M |=comp(P). Then TP(M)⊆ M.
PROOF. Application of TPresults in changing the truth value of atoms
for which there is no immediate ground in the program Pfrom 1 to 0. 
Definition 4. A model Msuch that TP(M)⊆ M is called a pre-fixpoint
of TP. It is fixpoint if TP(M) = M.
We next investigate the relation between completion, pre-fixpoints and
fixpoints.
Lemma 3. (Knaster-Tarski) A monotone operator defined on a directed
complete partial order with bottom element (dcpo) has a least fixed point.
In the simple situation considered (no negation), a model of the comple-
tion is a fixpoint of TPand conversely, but this will no longer be true once
negation is taken into account. Models of the completion comp(P) figure
mostly when studying semantic consequences of the program P, therefore
the following theorem provides all one needs:
Theorem 1. Let Pbe a positive program, then there exists a fixpoint TP(M) =
Msuch that for every positive formula1F:
comp(P)|=F⇐⇒ M |=F.
PROOF.⇐Choose a model K |=comp(P). The set of models {B |
B ≤ K} is a dcpo, hence TPhas a least fixed point M⊆Khere. Indeed,
if 0denotes the bottom element of the dcpo, then 0⊆ K implies TP(0)⊆
TP(K)⊆ K, whence it follows that the least fixpoint of TPis a submodel
of any K |=comp(P). By hypothesis M |=F. Since Fis positive and
M⊆K,K |=F, whence comp(P)|=F.
⇒Since Mis the least fixpoint of TP,M |=comp(P), whence M |=
F.
Definition 5. A model K |=comp(P)is called minimal if there is no N
which is a proper submodel of K(i.e. makes fewer atoms true).
Lemma 4. The least fixpoint of TPis the unique minimal model of comp(P).
PROOF. Let Mbe the least fixpoint of TP(which is obviously minimal).
Let K |=comp(P)be another minimal model. Then since the bottom el-
ement 0⊆ K and hence TP(0)⊆TP(K)⊆ K, it follows that M⊆K,
which by minimality implies M⊆K.We will sometimes abuse
language by talking about the ‘minimal model of the program P’, meaning
the minimal model of the completion of P. Again, the difference is that
to specify a model for P, one would need a declarative semantics for the
1A formula containing only ∨,∧.

KLEENE 3-VALUED SEMANTICS FOR LOGIC PROGRAMS IN NON-MONOTONIC REASONING3
arrow of logic programming, whereas no such thing is required in defining
a model for the completion of P.
The logic programs that we need must allow negation in the body of
a clause, since we model the natural language 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 approach to the least fix-
point. Our preferred solution is to replace the classical two-valued logic by
Kleene’s strong three-valued logic, for which see figure ?? in Chapter 2.
We also define an equivalence ↔by assigning 1 to ϕ↔ψif ϕ,ψhave the
same truth value (in {u, 0,1}) , and 0 otherwise.
We show how to construct models for definite programs, as fixed points
of a three-valued consequence operator T3
P. We will drop the superscript
when there is no danger of confusing it with its two-valued relative defined
above.
Definition 6. 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.
Lemma 5. Let Fa formula not containing ↔, with connectives interpreted
using strong Kleene 3-valued logic; in particular →is defined using ¬and
∨. Let M≤N, then truthM(F)≤truthN(F).
Definition 7. Let Pbe a program.
a. The operator TPapplied to formulas constructed using only ¬,∧
and ∨is determined by the strong Kleene 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) = 0 iff there is a clause ϕ→qin Pand for all such
clauses, M |=¬ϕ
(c) TP(M)(q) = uotherwise
The preceding definition ensures that unrestricted negation as failure ap-
plies only to proposition letters qwhich occur in a formula ⊥ → q; other
proposition letters about which there is no information at all may remain un-
decided.2This will be useful later, when we will sometimes want to restrict
the operation of negation as failure to ab. Once a literal has been assigned
value 0 or 1 by T3
P, 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 6. If Pis a definite logic program, TPis monotone in the sense
that M≤Nimplies TP(M)≤TP(N).
2This parallels the similar proviso in the definition of the completion.

4 KEITH STENNING AND MICHIEL VAN LAMBALGEN
Lemma 7. Let Pbe a definite program.
(1) The operator T3
Phas a least fixpoint, obtained by starting from the
model M0in which all proposition letters have the value u. By
abuse of language, the least fixpoint of T3
Pwill be called the minimal
model of P.
(2) There exists a fixpoint T3
P(M) = Msuch that for every formula F
not containing ↔:
comp(P)|=F⇐⇒ M |=F;
for Mwe may take the least fixpoint of T3
P.
PROOF OF (2). The argument is similar to that in the proof of theorem 1.
⇐Choose a model Kwith K |=comp(P). We have T3
P(K)≤ K:
(i) suppose ris assigned 1 by T3
P(K), then there exists a program clause
θ→rin Psuch that Kassigns 1 to θ. Since K |=comp(P), in particular
K |=r↔Def (r), and since θ→Def (r), it follows that ris true on K.
(ii) suppose ris assigned 0 by T3
P(K), then there exists a program clause
θ→rin Pand for all such clauses, Kassigns 0 to their bodies. It follows
that Def (r)is assigned 0 by K, hence the same holds for r.
(iii) if rhas value uin T3
P(K), this means neither (i) nor (ii) applies and
there exists no program clause θ→rin Pwith θeither 0 or 1. It follows
that θmust have value u, hence ras well.
Note that we may have T3
P(K)<K, for instance in case P={q→r}and
K |=comp(P),Kmakes r, q false, then T3
P(K)makes qundecided.
The set of models {B | B ≤ K} is a dcpo, hence T3
Phas a least fixpoint
M⊆Khere. Indeed, if 0denotes the bottom element of the dcpo, then 0≤
Kimplies T3
P(0)≤T3
P(K)≤ K, whence it follows that the least fixpoint of
T3
Pis a submodel of any Ksuch that K |=comp(P). By hypothesis M |=
F. Since Fis monotone and M≤K,K |=F, whence comp(P)|=F.
⇒Since Mis the least fixpoint of T3
P,M |=comp(P), whence M |=
F.One step in the proof deserves special mention
Lemma 8. For any model Kwith K |=comp(P)one has T3
P(K)≤ K. In
other words, a model of the completion is a pre-ficpoint of the consequence
operator.
Lemma 4(3) in Chapter 7 of Human reasoning and cognitive science in-
advertently stated that every model for the completion is a fixpoint. This
doesn’t affect the cognitive applications however, which are couched in
terms of least fixpoints; and as we have seen entailment is determined by
the least fixpoint.