Introduction
Inductive reasoning is among people’s most important
cognitive skills, allowing them to draw on prior knowledge
to make predictions under uncertainty. Induction is both
simple and complex. Induction is simple in the sense that
we can easily and automatically associate causes with their
effects, generalize properties to a category from a single
instance, or from one instance to another, and select actions
in complex situations by recognizing commonalities with
past experiences. Conversely, inductive reasoning can be
complex: how we generalize a property from a given
exemplar depends on the nature of the property in question,
the circumstances under which the property is observed, and
the nature of the relationships between categories.
Theories of category-based induction may be
distinguished in a similar manner. Simple, or associative,
models
(Kruschke, 1992; Rescorla & Wagner, 1972; Rogers
& McClelland, 2004; Sloman, 1993; Sloutsky & Fisher,
2008)
rely on similarity, contiguity, or co-occurrence
between instances, and are often modelled using
connectionist neural networks. Conversely, structured
relational models take advantage of more sophisticated
knowledge about the world, including directional causal
relationships, and domain specific rules
(Griffiths &
Tenenbaum, 2009; Heit, 1998; Kemp & Tenenbaum, 2009;
Murphy & Medin, 1985; Osherson, Smith, Wilkie, Lopez,
& Shafir, 1990)
.
Evaluating the strengths and weaknesses of structured and
relational models,
Bright and Feeney (2015)
proposed that
induction relies on two forms of knowledge: simple,
associative representations that are retrieved easily and
automatically, and more complex, structured relational
knowledge, including causal relationships, and domain
specific intuitive theories, that require cognitive effort. A
prediction of this hybrid model, investigated in the present
article, is that when associative and structured knowledge
come into conflict it often becomes necessary for the
reasoner to inhibit an incorrect inference, generated
automatically from associative knowledge, in order to
reason on the basis of more complicated relational
information. To test this prediction, we employed a
wellestablished mouse-tracking paradigm
(Freeman, Dale &
Farmer, 2011)
, allowing us to monitor participants'
momentby-moment movements toward responses that are cued by
associative and structured knowledge in a forced-choice
inductive reasoning task.
Knowledge types in category-based induction
Associative knowledge features most prominently in
connectionist, or neural network-based, models of
categorybased induction
(Rogers & McClelland, 2004; Sloman,
1993; Sloutsky & Fisher, 2008)
. In Sloman's (1993)
influential feature-based model, the known features of a
category can be represented as an activation vector applied
to the input nodes of a feed-forward neural network. Each
node is activated when its corresponding feature is
possessed by the target category. The network can be
presented with the premises of an inductive argument by
training it to activate its output node when presented with
the features of categories which do have a novel property.
The outcomes of the trained network (i.e., its inferences)
can be elicited by probing the input nodes with the features
of a novel category. The activation of the output node
corresponds to the degree to which the network believes the
novel category will have the property in question.
Simple associative architectures are capable of capturing
many aspects of human performance
(Rolison, Evans,
Dennis, & Walsh, 2012)
. For instance, like human subjects,
an associative network can be trained to rate similar
categories (which share many features) as more likely to
share a novel property, and properties that are present in a
diverse range of categories as more likely to be found in a
novel category. Adding additional layers to the neural
networks allows them to account for further characteristics
of human inference, such as sensitivity to different property
types, such as “is a” and “has a” properties
(Rogers &
McClelland, 2004)
, as well as basic context-dependent
inferences
(Sloutsky & Fisher, 2008)
.
Murphy and Medin (1985)
argue, however, that
similarity-based approaches fail to capture the full flexibility
of people's intuitive theories about the relationships between
categories in specific domains. In particular, participants
have been shown to be sensitive to property effects when
reasoning inductively, such that the strength of an argument
is dependent on the kind of property projected.
(Heit &
Rubinstein, 1994; Shafto, Coley, & Baldwin, 2007; Shafto,
Kemp, Baraff, Coley, & Tenenbaum, 2005)
. For instance,
transmittable properties such as infectious diseases are
thought to be shared by animals that are related
ecologically, such as predators and prey in a food chain,
whereas biological properties such as genes are shared only
by animals that are close together in their evolutionary
taxonomic tree. Such intuitions, in one domain, are captured
by the similarity-coverage model
(Osherson et al., 1990)
,
which uses a taxonomic tree to capture intuitions about how
properties are shared by related species in the natural world.
More recently, structured Bayesian models have been
introduced
(Griffiths & Tenenbaum, 2009; Kemp &
Tenenbaum, 2009; Shafto, Kemp, Bonawitz, Coley, &
Tenenbaum, 2008; Tenenbaum, Griffiths, & Kemp, 2006)
,
which are capable of describing flexible human performance
in a range of domains. These models require that, for each
domain, a specific structure is generated to capture
relationships between categories, such as food chains or
taxonomic trees, along with a generic probabilistic process
by which properties can be transmitted. In the biological
domain, such structures include unidirectional causal links
connecting prey to predators, or distance from a common
ancestor in a biological taxonomy.
Bright and Feeney (2015)
argue that neither associative
nor structured models are sufficient to account for all of the
phenomena observed in category-based induction. They
propose a hybrid theory in which both associative and
structured knowledge can be used in reasoning. They
provide evidence that the two kinds of knowledge can be
dissociated. Namely, measures of the strength of the
association between two categories predicts participant
ratings of the strength of inductive arguments made under
cognitive load, and under time pressure. Conversely, a
measure of structured knowledge predicts ratings of
argument strength otherwise
Conflict in category-based induction
If both associative and structured knowledge play a role in
induction, a natural question is how associative and
structured knowledge interact when they support conflicting
beliefs. For instance, upon learning that a biological
property is true of salmon, does one decide that this is also
true of grizzly bears (strongly associated, but no structured
means of transmission for biological properties) or of
goldfish (weakly associated, but related taxonomically)?
Clearly, the decision depends on what kind of knowledge is
recruited, with purely associative knowledge in this case
leading to a non-normative inference. One possibility is that
one or other representation is activated, depending on
available time and cognitive capacity, in what
Evans (2007)
labels a “preemptive conflict resolution” model.
Alternatively, both representations may compete, either with
associative knowledge being recruited by default, which
must be inhibited in order for structured representations to
come online (“default interventionist models”), or with both
representations activated in parallel (“parallel-competitive
models”), leading to a conflict. Bright and Feeney (in prep.)
offer evidence that associative and structured knowledge do
conflict during category-based induction. In a triad task
(Gelman & Markman, 1986)
, in which participants were
asked which of two target species was most likely to share a
biological property given that it was found in a third base
species, participants were more likely to fail to select a
structurally (i.e. taxonomically) related target when the
alternative response was strongly associated to the base.
Crucially, participants were less able to inhibit the
association-driven response under cognitive load, or when
lacking in semantic inhibitory control or working memory
capacity.
Although the above results provide some support for a
proposal that associative and structured knowledge can
compete during inductive reasoning, these conclusions are
drawn from analysis of participants' responses – the end
product of the reasoning process – and thus constitute only
an indirect measure of the underlying processes. The
mousetracking paradigm
(Spivey, Grosjean, and Knoblich, 2005;
Freeman et al., 2011)
, on the other hand, provides a
powerful tool for measuring these processes as they unfold
during cognition. Monitoring the location of the mouse
cursor whilst participants are choosing between choice
options located on opposite sides of the computer monitor,
this method allows us to track the time-course of reasoning
that leads to an inference. Mouse-tracking has been used to
reveal parallel competition effects on a range of simple
cognitive and perceptual tasks
(i.e. Freeman, Ambady, Rule,
& Johnson, 2008; O’Hora, Dale, Piiroinen, & Connolly,
2013; Spivey et al., 2005)
, in which participants are shown
to be attracted simultaneously to competing response
options. In more complicated tasks, participants also have
been found to exhibit more discrete “changes of mind”
tendencies by switching between choice options mid-trial
(Dale & Duran, 2011; Freeman, 2013)
. In the present study,
we use this technique to test for conflict between associative
and structured knowledge in the triad task. Participants were
asked to choose between projecting a biological property
from a base species to a correct target species belonging to
the same taxonomic group, or to an unrelated foil species.
Critically, the strength of the association between the base
and the foil species is varied within subjects. If, as
suggested by
Bright and Feeney (2015)
, responses cued by
associative knowledge must be inhibited in order to reason
on the basis of structured relations, we should expect to
observe an initial attraction toward the foil that is
proportional to the strength of the associative connection
between the foil and the target. On the other hand, if
participants recruit one or other form of knowledge, we
should not expect to find an initial attraction to the foil,
regardless of the strength of association between the target
and foil.
Method
Stimuli
Participants were presented with a version of the inductive
triad task
(Gelman & Markman, 1986)
. On each trial,
participants were informed that a particular gene is
possessed by a given base species and were asked to decide
which of two candidate target species was most likely to
possess the same gene (see Figure 1). The correct response
was the species belonging to the same taxonomic group as
the base (mammals, birds, insects, reptiles, or plants). The
foil response belonged to a different taxonomic group than
the base and was weakly, moderately, or strongly associated
with the base. The strength of association was determined
by prior testing. Across 27 experimental trials, nine base
species were each presented three times, paired with the
same correct response species but a different foil species on
each occasion. An additional 27 filler trials were included,
in which the property to be generalized was susceptibility to
a given disease.
Design and Procedure
Forty four undergraduate students at Queen's University
Belfast participated for course credit. Stimuli were custom
programmed using the OpenSesame software package
(Mathôt, Schreij, & Theeuwes, 2011)
and were presented on
a computer monitor. The 27 experimental trials were
presented in three blocks of nine trials each, interspersed
with nine filler trials. Trials were randomly assigned to each
block with the constraints that each base species appeared
once in each block and each block contained three weakly,
three moderately, and three strongly associated foil trials.
Trial order within blocks was randomized with the
constraint that the same base could not appear twice within
three trials.
On each trial, participants were first primed with the kind
of property they were to reason about: “gene” for
experimental trials or “disease” for fillers. This prime
appeared in the center of the monitor for one second. For
each experimental trial, participants were then informed that
the given gene (i.e. “Gene r3P”) is found in the bodies of
one of the two target species, which appeared as labeled
images in the top left and right corners of the screen (Figure
1). The two species were randomly assigned to the left and
right positions on each trial and appeared for 1.6 seconds
each. The targets then remained visible and participants
were asked which species they believed was most likely to
possess the gene, given that it was possessed by another
species. Participants were then instructed to click a
“START” button located in the bottom center of the
monitor, after which a fixation cross appeared for 1.5
seconds, which was then replaced by a labeled image of the
base species (Figure 1). At this point, the mouse cursor was
reset to the center of the start button and participants were
given five seconds to respond by selecting one of the two
target species labels with their mouse cursor. Participants
were given five seconds to respond following presentation
of the base category. Additionally, in line with previous
mouse-tracking research, on trials in which participants did
not move the cursor away from the start button within 1.5
seconds of the onset of the base, they were shown a message
reminding them that they were under time pressure. This
was done to encourage participants to make their decision
while the mouse cursor was in motion.
Figure 1: Screenshot of the experiment screen following
onset of the base species (“Killer Whales”).
Analysis
Mouse trajectories were normalized to a standard
coordinate system, with all trials beginning at point [0, 0], and
ending at point [1, 1.5] in the top right corner. Trajectories
in which the chosen response was on the left were reflected
through the y-axis. For each trial, we calculated the time
from target onset to a response (response time), the time
from target onset to the beginning of the mouse movement
(initiation time), the deviation of mouse trajectory away
from a straight line to the response, measured in the
standard co-ordinate system (maximum deviation), and the
frequency of changes of mouse trajectory direction on the
xaxis (x-flips).
Our analysis was restricted to trials on which the
taxonomically-related species was chosen. Thus, our data
set was unbalanced. Therefore, we conducted random
effects linear regression modeling on our data. This analysis
accounted for the clustering in our data by allowing for
random intercepts at the subject and base species level
(Baayen, Davidson & Bates, 2008)
. Main effects were
assessed on the basis of the -2 log-likelihood model fit
improvement, tested using the chi-square statistic. Main
effects were followed-up with Tukey pairwise comparisons
between each group, with p values calculated using the
normal approximation.
Log transformations were used for analyses of response
time, initiation time, and maximum deviation due to
violations of normality. A Poisson regression model was
used for the count of x-flips. A logistic random effects
model was used for the analysis of choices in each
condition.
Results
Results did not differ appreciably between the stimuli
blocks, and so data were collapsed across blocks for
analysis. Participants selected the correct
taxonomicallyrelated response on 81% of trials when the foil was weakly
associated with the base, 61% of trials when moderately
associated, and 57% when strongly associated. This
suggests that participants were influenced by the associative
strength of the foil option, such that stronger associations
competed with structural knowledge. A logistic mixed
effects model indicated a main effect of foil strength (ΔAIC
= -86.4, -2LL χ2(2) = 90.4, p < .001). Pairwise comparisons
revealed significant differences between the weak and
moderate (t = 7.310, p < .001), and weak and strong foils (t
= 8.489, p < .001), but not between moderate and strong
foils (t = 1.428, p > .3).
Inspection of the mouse cursor data revealed two kinds of
mouse trajectory: movements directly toward the taxonomic
option, and initial movements toward the foil option that
changed direction toward the taxonomic option mid-trial.
Analysis of the distribution of the maximum deviation
statistic (Figure 2) revealed two normally-distributed
subpopulations of responses, one centered on a deviation
close to 0 (measured in the standard co-ordinate system) that
corresponded to movements directly toward the taxonomic
option, and a second centered around 1.4. The bimodality of
this distribution was confirmed by calculating its bimodality
coefficient
(Freeman & Dale, 2012)
, yielding a value of
.636, well above the threshold of .555 usually interpreted as
indicating bimodality. We therefore fitted a two-sample
finite mixture model to these maximum deviation values, in
order to classify trajectories as either “changes of mind”
(maximum deviation > .827), or “direct to taxonomic
option”. The two kinds of mouse trajectories are shown in
Figure 3.
Condition means and statistical tests for the measures
described above are shown in table 1.
Significant main effects were observed for maximum
deviation (ΔAIC = -2.33, χ2(2) = 6.33, p = .042) and for
changes of mind (ΔAIC = -5.0, χ2(2) = 9.012, p = .011),
with a marginally significant main effect for x-flips (ΔAIC
= -1.471, χ2(2) = 5.471, p = .065). Pairwise comparisons
showed significant differences between weakly and strongly
associated foils, with greater signs of conflict when strongly
associated, for maximum deviation (t = 2.44, p = .038), and
for changes of mind (t = 2.69, p = .023), and a marginal
difference for x-flips (t = 2.23, p = .067). There was an
additional significant difference between moderately and
strongly associated foils for changes of mind only (t =
2.673, p = .020), with more changes of mind for strongly
associated foils.
Discussion
Bright and Feeney (2015)
showed that both associative and
structured knowledge can serve as the basis for inductive
reasoning. Bright and Feeney (in prep.) provide evidence
that both kinds of knowledge can conflict during reasoning.
Here, we found that participants generalized biological
properties from a base species to a target species from the
same taxonomic group, rather than to a foil species, on the
majority of trials when the target was weakly associated
with the foil. The stronger the association between the base
and foil species, however, the more likely participants were
to generalize the property to the foil instead. This was
despite the base and foil species belonging to different
taxonomic groups. Analysis of the mouse cursor trajectories
revealed that many participants were initially drawn to
strongly-associated foil responses, even when they
ultimately selected the option that corresponded to
structured knowledge.
Our findings provide evidence against theories of
inductive reasoning that describe either an associative or
structured knowledge account. Rather, our present findings
suggest that both forms of knowledge are engaged during
reasoning, and that both can influence a single decision. At
the outset, we raised the question of how precisely the two
forms of knowledge interact. Although our data do not
provide a definitive answer, the patterns in the mouse
trajectory data shown in Figure 3 suggest that the majority
of trajectories went directly to the taxonomically related
target without any evidence of conflict. To the extent that
participants detect conflict between the choice options, such
trajectories are consistent with pre-emptive conflict
resolution
(Evans, 2007)
. “Change of mind” responses, on
the other hand, indicate online resolution of conflict. Further
research will be required to determine (a) why conflict is
sometimes resolved pre-emptively and sometimes online,
and (b) whether, when conflict is resolved online, both types
of knowledge are activated in parallel or in sequence.
Bayesian models of inductive reasoning claim that people
represent structured relations between categories when
reasoning, appropriate to the domain in question. By placing
these structured representations in conflict with associative
knowledge, we have shown that in order to reason in a way
consistent with a Bayesian account, it is sometimes
necessary for people to inhibit the associative
representations which come to mind more easily. This is
consistent with
Bright and Feeney's (2015)
demonstration
that reasoning is consistent with the structured Bayesian
model when people have adequate time and mental
resources, but is driven by simpler associative knowledge
otherwise.
Our results may be challenged by theorists who favor
purely associative models of induction
(Rogers &
McClelland, 2004; Sloman, 1993; Sloutsky & Fisher, 2008)
.
Neural network models have been shown to capture some
context sensitivity effects by means of input nodes encoding
contextual features
(Sloutsky & Fisher, 2008)
. However, the
“change of mind” movements which characterized our
mouse trajectory data are difficult to explain within this
framework. Simple feedforward neural networks of the type
used in models of induction are static, in that they are
probed once, and produce a single output pattern, providing
no mechanism for reversals during a trial. More complex
recurrent networks, on the other hand, with input and output
changing over time, can capture the evolution of choices.
However, extensive mouse-tracking research has
demonstrated that conflict in such networks is continuous,
with participants partially drawn toward two competing
responses, selecting one response but curving toward the
alternative
(i.e. Freeman & Ambady, 2011; Freeman et al.,
2008; Spivey et al., 2005)
. Discrete reversals have been
demonstrated on tasks thought to involve the sequential
operation of two processes
(Barca & Pezzulo, 2015; Dale &
Duran, 2011; Freeman, 2013; Freeman & Dale, 2012; Hindy
& Spivey, 2008; Tomlinson, Bailey, & Bott, 2013)
,
suggesting that our results reflect the initial activation of
associative knowledge and the subsequent retrieval of
structured knowledge.
To conclude, we believe that neither associative nor
structured models alone are capable of describing the
processes underlying human inductive reasoning. Instead,
people draw upon two forms of knowledge representation,
one associative, and easily accessed, and one structured, and
requiring mental effort to utilize. Making use of structured
knowledge appears to require the inhibition of associative
information, and as a result, participants were more likely to
select a foil response if it was strongly associated with the
base. Uniquely, our mouse trajectory results reflect the
online inhibition of association-driven responses, necessary
to reason according to structured knowledge.