The effect of geometrical relations on

spatial deductive reasoning

Ginette Boudreau, Ph.D.

Defence and Civil Institute of Environmental Medicine

1133 Sheppard Avenue West

P.O. Box 2000, Toronto, Ontario, Canada

M3M 3B9

In air, ground, or naval navigation, operators are required to represent the relative location of entities in three dimensions. They must also deduce, from their mental representation, the relative location of entities not explicitly related from the relationships that have been specified among the set of entities. Spatial reasoning is critical for the operators safety and the accomplishment of their mission.

This study addresses a particular type of spatial reasoning, namely spatial deductive reasoning as an aspect of formal logic. Spatial deductive reasoning comprises two components. The mental representation of spatial relations which consists in organising in memory the relative position of entities in 3D according to projective relations such as left, or below. Operators can construct such mental representations as mental models which depict relations, or as mental prepositions which describe such relations. The process of spatial deduction itself consists in inferring the relative location of entities not explicitly related from the mental representation of a given set of relative locations.

Theories of Spatial Deductive Reasoning

Cognitive scientists have proposed two main opponent theoretical views, Formal Rules and Mental Models, to account for the mechanisms by which humans represent and reason about relative locations.

Formal Rules theories and Mental Models theory

The Formal Rules theories argue that humans construct mental propositions to which formal rules of inference are applied to derive conclusions irrespective of an argument‚s content (Beth, 1971; Hagert, 1985; Hagert & Hansson, 1983, 1984; Piaget, 1972). The alternative Mental Models theory (Johnson-Laird, 1983; Johnson-Laird & Byrne, 1993) repudiates this view arguing that humans construct mental models from which conclusions are validated, as logically true or false, without the use of formal rules of inference or even content-specific rules (Cheng & Holyoak, 1985). Experiments have pitted the Formal Rules (Hagert, 1985) and Mental Models (Johnson-Laird, 1983) theories of spatial deductive reasoning, and the results have corroborated the Mental Models theory‚s predictions (Boudreau, Pigeau, & McCann; 1995, 1998; Byrne & Johnson-Laird, 1989).

Humans construct mental models of spatial relations. Given that assumption, then we can predict that mental models would reproduce the structure of the relations among entities whether these relations are perceived or conceived. In the area of spatial reasoning, this structural property implies that mental models would reproduce geometrical relations among entities, such as number of dimensions and the directions specifying coordinate axes. However, the geometrical properties of mental models and the mechanisms by which mental models reproduce such properties remain unknown. We will propose two opponent views based on predictions made by Spatial Reference Frames models.

Spatial Reference Frame Models

A first view, based on the Conceptual Reference Frame model (Logan, 1995), is to suppose that humans construct mental models relative to conceptual reference frames projected into space as it is perceived or conceived. Conceptual reference frames are co-ordinates axes derived from the egocentric physical axes: the vertical, horizontal, and line of sight axes. If humans construct mental models relative to conceptual reference frames then the difficulty of constructing mental models should vary with the co-ordinate axes. The vertical axis would be the easiest to construct, followed by the line of sight axis, and finally the horizontal axis.

The alternative view, based on the Equiavailability model (Franklin & Tversky, 1990; Levine, Jankovic, & Palij, 1982; Logan, 1995), is to argue that all three axes are equally available. Evidence for this model has been found for simple cognitive maps and viewed pictures (e.g., Levine, Jankovic, & Palij, 1982). If this view holds true, then geometrical relations should have no reliable effect on the difficulty of constructing mental models or inferring conclusions from them. However, given that mental models should reproduce geometrical relations, then humans should use conceptual reference frames to build such models.


To address the above issues, we conducted an experiment (Boudreau & Pigeau, 1997) in which we analysed the effects of geometrical relations on the mental representation and processes of spatial reasoning. We addressed these variables in light of the Mental Models theory in order to determine if mental models reproduce geometrical relations such as dimension and direction. We also investigated these variables in light of the Spatial Reference Frames models in order to specify whether humans create conceptual reference frames in order to build mental models. The construction of a conceptual reference frame would imply that geometrical relations would have a differential effect on the difficulty of spatial reasoning.


A total of forty subjects (30 males, 10 females) from DCIEM and the Canadian Armed Forces participated in the experiment. Each subject solved a set of 64 spatial deductive reasoning problems. Each problem consisted of a set offour premises and a question type. Each premise described the relative position of two objects (e.g., "star left of cross"). Together, the premises described the relative position of five objects (circle, square, cross, triangle, star). The subject‚s task was to deduce the relative location of two objects not explicitly related from the relationships specified in the premise set.

Problems varied in their geometrical content, that is, by the dimensions (two-dimensions, three-dimensions) and directions (right/left, above/below, front/behind) specified in the premise sets. These variables were ascribed as within-subject conditions.

Table 1

Between-subject conditions illustrated by a simple example of premise

Symbolic structure

Symbolic content





star left of cross

* left of +


star cross

* +

Problems were displayed using sentences or diagrams to represent the relations among entities, and nouns or images to represent the entities. The sentences described the relative location of each pair of entities using a projective preposition such as "left of" (e.g., star left of cross; * left of + ). The diagrams displayed each pair of entities according to their relative location without using projective prepositions (e.g., star cross; * + ). These variables were specified as between-subject conditions (see Table 1): diagrams-images, diagrams-nouns, sentences-images, sentences-nouns.

We measured the effects of the above conditions on three dependant variables: the premise‚s inspection times, the responses to the questions, and the response times. Each dependant variable was subject paced. The premise inspection times aimed at elucidating the nature of the mental representation of spatial relations. The responses and the response times aimed at clarifying the process of deduction.

Subjects participated individually using a 386 PC for problem generation and a two-button mouse for responses. Each premise and each question was presented individually according to predefined spatial and temporal parameters. Subjects controlled the inspection time of each premise and each question once completely displayed on the computer screen.

Analyses and Results

We carried out three separate analyses of variance for repeated measures, one on the premises‚ inspection times, a second on the percentages of correct responses, and a third one on the response times obtained for the correct responses. The level of significance was set at the probability of 0.01.

Effects of the number of dimensions

The effects of number of dimensions on the premises‚ inspection times indicate that the 3D models were as easy to construct as the 2D ones (p > 0.01) thus suggesting that the co-ordinate axes were equally available. However, spatial deductions were systematically more difficult to make from the 3D models than from the 2D ones. Moreover, the difficulty of the deductions increased reliably with the number of dimensions that subjects had to locate between two objects. The results indicate that subjects created conceptual reference frames to construct mental models although all three co-ordinate axes were equally accessible.

Effects of direction

The effects of direction on the premise‚s inspection times (P1 & P2) indicate that the horizontal axis was no more difficult to construct than the vertical axis (p > 0.01). However, once subjects had constructed the first axis ( e.g., * + O ) of their mental model, the difficulty of the directions varied for each adjuvant axis. Thus, subjects found it easier to represent an object below a reference object (located on the horizontal axis) rather than above. For the 3D models, subjects found it easier to represent an object in front of a reference object rather than behind. And, for the 2D and 3D, subjects found it easier to represent an object to the left of a reference object (located on the vertical axis) rather than to the right. The results confirmed the predictions made by the Conceptual Reference Frames model.

Effects of the problems‚ symbolic display

The inspection times of premises displayed as diagrams were consistently shorter than of those displayed as sentences. Questions also took significantly less time to answer when presented as diagrams than as sentences. The inspection times of premises displayed as sentences were reliably longuer when images rather than nouns were used as the lexical tokens. Spatial deductions were also reliably more difficult to make when sentences symbolised the entities using images rather than nouns. The use of nouns or images as lexical tokens in diagrams had no reliable effect (p > 0.01) on the mental representation or process of spatial deduction. These results suggest that subjects symbolised the entities of diagrams using nouns as easily as images. The above effects of dimension and direction generalised to both diagrams and sentences.


The results suggest that mental models reproduced the euclidian (number of dimensions) and projective (directions) relations among entities in a way similar to which visual images preserve metric relations (Kosslyn, Ball, & Reiser, 1978). The differential effect of the geometrical relations indicate that humans construct mental models relative to conceptual reference frames. The Conceptual Reference Frame model (Logan, 1995) thus generalised to spatial deductive reasoning while providing an account of the structural properties of mental models. Spatial deductive reasoning was reliably easier with diagrams than with sentences. These results further support the Mental Models theory while contradicting the Formal Rules theory‚s opposite predictions.


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