The consistency of spatial descriptions is relevant to tasks ranging from navigation to architecture. In contrast to studies of deduction in which a conclusion is drawn from premises, there have been only a few investigations into how human reasoners decide whether or not a description is consistent. We report results corroborating the theory that reasoners make such judgments usually relying on a single initial mental model of the description. As a result, the task is difficult if it calls for an alternative model of the assertions that must be revised. Especially the model construction process and the way of how information is integrated into a model can explain errors in evaluating problems as consistent. Implications for other theories of reasoning are discussed.
Inconsistency in a set of beliefs or assertions is dangerous, and can have disastrous consequences. Its importance therefore raises the question of how individuals assess consistency – that is, the assertions or beliefs can all hold at the same time. We have investigated this problem using descriptions of spatial layouts, which have everyday analogs in architecture, route finding, and design. Consider, for example, the following problem about a fruit and veg stall: (1) The box of apples is left of the box of pears.
The box of kiwis is right of the box of the pears.
The box of apples is right of the box of kiwis.
The answer to this question is “No”, as there is no arrangement of the three boxes in a line integrating all information in the description. We therefore refer to a set of assertions as consistent if there it has at least one model sattisfying all the assertions in the set. A general test for inconsistency based on formal logic works as follows: Choose any assertion from the set of assertions, and prove that its negation follows from the remaining assertions in the set. Only if no such proof exists is the set consistent.
Hence, consistency depends on the failure of an exhaustive search for all possible proofs. – a process that is computationally intractable. An alternative process could be based on the new paradigm of probabilistic logic. Adams (1998) formulated a notion of p-consistency, according to which a set of assertions is consistent if each assertion in the set can have a high probability. No psychologists, as far as we know, have endorsed this procedure. One difficulty is to specify how people determine the relative constraints on the probabilities of assertions in a set. So, we need an alternative account. In the next section, we therefore describe the mental model theory – the “model” theory for short – and we derive its predictions for assessments of consistency. We then report an experiment that tested these predictions. Finally, we discuss the implications of these results.
(2) The apple is to the left of the pear.
The pear is to the left of the kiwi. The pear is to the left of the orange. The orange is to the left of the mango.
The kiwi is to the left of the orange.
You may have formed a visual image of the arrangment, or
your representation may have been more abstract. It needs
only to represent the spatial relations among the objects
. . . where the ellipsis denotes implicit models – in this case the other models that can be found in Table 1. The fifth assertion is consistent with these possibilities because it holds in the explicit mental models, and the model that is present to the participants is the same as the explicit mental models (all other models in Table 1), and yields the response: “yes (all assertions are consistent).” In contrast, consider the following problem: (3) The apple is to the left of the pear.
The kiwi is to the right of the pear.
The orange is to the right of the pear.
The orange is to the left of the mango.
The mango is to the left of the kiwi.
Again, can these five assertions all be true at the same time? This set of five assertions is not consistent with the initial model (apple pear kiwi orange mango). The fifth assertion ‘The mango is to the left of the kiwi’ forces the reasoner to revise the recently constructed initial model to the model apple pear orange mango kiwi as the fifth assertion conflicts the previous built model. The fifth assertion (in problem 3) does not hold in the preferred or initially constructed model (built after the assertions 1-4, cp. Table 1), but only in the model constructed according to the leftmost insertion principle. In this sense we would have a mismatch between the initial model (the model: apple pear kiwi orange mango) and the fifth assertion (“The mango is to the left of the kiwi”). Of course, all five statements are consistent.
Thus, the theory of mental models predicts that reasoners may have some difficulty in inferring that problems such as 3 are consistent, as the participants will have a conflict with their initial model they constructed after the first four assertions. The fifth assertion does not correspond with the initial model of the four assertions and so individuals should respond, “no” – in contrast to Problem 2.
Human reasoners tend to evaluate a given set of assertions
as inconsistent if it does not match the initially built mental
model, which is constructed according to the right-most
insertion principle (see Table 1) – that is an alternative
name for the first-free-fit principle
If participants construct initial models then the way
information is integrated should have as well an influence.
This effect has been investigated for deductive reasoning in
the premise order effect.
The participants’ task was to evaluate whether or not spatial descriptions were consistent. Half of the problems were determinate and half of them had a local inderminacy that the fifth assertion resolved. One third of the indeterminate problems asked for the preferred model (no mental model revisions necessary), one third asked for the alternative models with a revision distance 1 from the preferred model and one third asked for the alternative models with a revision distance 2 from the preferred model. The problems consisted of five assertions stating spatial relations of five objects. Each of the first four successive assertions introduced a new object, which was randomly inserted from a list of either fruits (apple, peach, orange, etc.) or “breakfast items” (toast, bagel, biscuit, etc.). The forty problems differed in five independent variables with respect to consistency, determinacy of the description, premise order, distance, and type of relation.
Half of the problems were consistent (e.g., Problem 2) and half of the problems inconsistent (e.g., Problem 1). We also manipulated the determinacy of the description, so that half of the problems’ descriptions are determinate (see, Table 2), i.e., they allow for only one model after four presented assertions, and half of them were indeterminate, i.e., they allowed for three possible arrangements of the objects. This indeterminacy appears, however, only in the first four assertions, after the fifth assertion, each problem description is determinate. The indeterminate problems differed variable in the revision distance of the initial model to integrate the last assertion. It can require zero vs. one vs. two spatial operations from the initially built mental (see Table 1). The third variable is the sequence, i.e., are the first two assertions continuous or semi continuous (see Table 2). We manipulated this by exchanging the first two assertions. We counterbalanced the problems regarding the type of relation, i.e., half of the problems used horizontal relations like left and right and half of the problems vertical relations (above and under).
Our 40 problems differed in the three variables outlined above: Determinate vs. indeterminate problems, consistent vs. inconsistent, and the sequence of the premises. In order to examine possible effects of revising initial models, we only considered the consistent indeterminate tasks and differentiated the type of conflict to the initial model. First, only for consistent tasks we can interpret a negative response as an indicator of difficulty. Second, for these problems a conflict to the initial model does not mean a conflict to its consistency. In the condition ‘0’ (the rightmost insertion principle in Table 1), the representation of an initial model was possible till the last introduced assertion. In 1-step-revised-model condition (the mixleft/right-insertion principle in Table 1), a revision of the model only required the revision of two objects of the initial model. In the 2-revised-model condition (the leftmostinsertion principle in Table 1), more operations were needed to revise the initial model and detect the problems’ consistency. Assuming the representation of an initial model, we expected increasing difficulty related to the number of operations necessary.
Each participant received all 40 problems and acted as their own controls. Each problem and each assertion were presented self-paced. After the participants received the fifth assertion, which either was consistent or inconsistent with the four preceding assertions (consistent vs. inconsistent), they had to consider the consistency of each problem. Participants could verify (“y”) or reject (“n”) all the assertions by answering the question “Could all of these assertions be true at the same time?” The problems were presented in a randomized order. Reading and response times were recorded as well as the correctness of answers and analyzed as dependent variables.
Six participants were excluded from the analysis, as their
accuracy did not differ significantly from chance. The
remaining 21 participants solved an average of 85% of all
problems correctly. Table 3 presents the percentages of
correct response in each of the 4 main conditions.
Reasoners correctly identified as consistent determinate
descriptions (90%) more often than descriptions that were
locally indeterminate (72%, Wilcoxon test, z = 3.08, p < .01
[1-tailed]) and the same pattern holds for the reaction times
(29.0 vs. 23.4, Wilcoxon test, z = 2.88, p < .01 [1-tailed],
see Figure 1). The participants identified inconsistent
problems (89%) in general significantly more often than
consistent ones (82%, z = 2.12, p < .05 [2-tailed], r = -.46).
In the indeterminate condition the first four assertions
allowed for three different models (cp. Table 1), while with
the fifth assertion the set of assertions was again a
determinate description and, hence, allowed for one model
only. The more transformation steps the fifth assertion
requires from the initial model the lower was the correctness
rates and the higher are the response times: If there is no
revision step necessary (i.e., if the fifth assertion is
consistent with the initial model the correctness is 93%); if
the fifth assertion requires one revision-step of the initial
model (cp. Table 1) the accuracy and response times are
lower than in the initial model condition (80%, Wilcoxon
test, z = 1.98, p < .01; response times: 24.3s vs. 30.7s,
Wilcoxon test, z = 1.68, p < .05). The same pattern holds if
two model revisions are necessary (correctness 49%,
Wilcoxon test, z = 3.36, p < .001; response times: 24.3s vs.
31.3s, Wilcoxon test, z = 1.90, p < .05). The manipulation
of the first two assertions’ order (the continuous order vs.
semi-continuous order cp. Table 2) did not effect mean
correctness to a significant extent (85% vs. 84%, Wilcoxon
test, z = .34, p = .735). However, participants needed
significantly more time to generate answers, in the
semicontinuous case (when the first two assertions were
exchanged; 25.1s vs. 28.4s, Wilcoxon test, z = 2.17, p < .05
[1-tailed], r = -.46). This delay can be traced back to longer
reading times for the third assertion in the semi-continuous
order in contrast to the continuous order. This supports
again a continuous integration of information into a model
during the reasoning process. In accordance with previous
results
In contrast to an evaluation in formal logic, the order of assertions has a major effect on human evaluations of consistency. This finding occurs even when a description is determinate and consistent, e.g.:
The pear is to the left of the kiwi.
The pear is to the left of the orange.
The orange is to the left of the mango.
The kiwi is to the left of the orange.
Can all five of these assertions be true at the same time? Reasoners can construct a model of the first two assertions: apple pear kiwi But, how are they to interpret the third assertion? The orange could be to the right of the kiwi or it could be between the pear and the kiwi. The final assertion in the description resolves the indeterminacy, but nevertheless its local occurrence impedes the evaluation of the description as consistent. Indeed, if a subsequent assertion contradicts an initial model, then the chances increase that reasoners will err and evaluate the description as inconsistent. Similar difficulties occur when the referents in a description are ordered discontinuously, e.g.:
The second assertion cannot be integrated into the model of the first assertion until reasoners interpret the third assertion. This discontinuity contributes to the difficulty of evaluating consistency. But, theories that do not postulate the construction of mental models have difficulty in explaining the phenomenon.
A byproduct of our investigation was the finding that human
reasoners find it slightly easier to work in a vertical
direction, e.g., A is above B, than in a horizontal direction,
e.g., A is to the left of B
The two main alternatives to the model theory – mental
logic and probability logic – have not addressed reasoning
about the consistency of spatial descriptions. The only
general method for testing consistency in mental logic is to
negate one assertion, and to try to prove that it follows from
the remaining assertions
The difference in difficulty between problems in a vertical
dimension and those in a horizontal dimension makes sense
in the model theory: a confusion between left and right
should be echoed in the construction of models. But, it is
inexplicable for theories based on formal rules: there is no
reason why the transitivity of above should be easier to
grasp than the transitivity of left. The concept of lexical
marking according to which marked items are harder to
work with than unmarked items
Probabilistic approaches can explain deductions from
conditional and quantified premises
In contrast, to its alternatives, the model theory provides a simple explanation of the phenomena. If, and only if, individuals can build a model of a set of assertions then they judge them to be consistent. An initial model may clash with a subsequent assertion. Reasoners may then search for an alternative model to accommodate the assertion. Even if they find one, the task is harder than when the initial model accommodates all the subsequent assertions. Likewise, the task will be harder when there is a discontinuity in the referents. Reasoners have to bear in mind two separate spatial relations, which they can integrate only in the light of a subsequent assertion. Once again, this factor adds to the difficulty of evaluating consistency.
In conclusion, reasoning about the consistency of descriptions is important in everyday life. The model theory provides an account of how naïve reasoners carry out this task, and our investigation has corroborated its main predictions.
The research was partially supported by the DFG with a Heisenberg grant (RA 1934/3-1) and a DFG research grant (RA 1934/2-1) in the SPP 1516 “New frameworks of Rationality” to the first author. The authors are grateful to Stephanie Schwenke for proof-reading and Eva-Maria Steinlein for comments and discussions.