## Abstract

The analogy between spatial inclusion and set membership offers the opportunity to go beyond the study of how we establish analogical mappings to the investigation of the impact of those mappings on mental process. This analogy is the basis of longstanding didactic techniques for teaching elementary logic (Eulers Circles and Venn Diagrams) and therefore bears on psychological theories of human deductive reasoning (e.g. Erickson 1974).

An adequate cognitive theory of analogy must explain how mental processing is aided by analogical mappings. In the case of spatial inclusion as an analogy for set membership this comes down to requiring a theory of how graphical representations aid processing. Here we sketch a general theory in terms of the specicity of graphics which we have developed further elsewhere (see Stenning Oberlander 1991a, 1991b). This general approach to graphical representations rests on the observation that graphical representation systems limit abstraction and thereby purchase computational tractability.

In applying this general theory to the case of graphical methods of syllogism solution we sketch a method of using Eulers Circles and show how interpretative conventions for graphical objects allow the necessary abstractions. In particular we show that even choosing the least expressive system of diagrams limited to circles, still gives the requisite expressive power.

The Euler's Circle method sketched is then used as a guide to connectionist implementation of corresponding internal representations Graphical representations share some general computational properties with connectionist systems. In the domain of the syllogism the central issue is the implementation of the binding of attributes into type descriptions. Various proposals for connectionist variable binding in the literature are assessed as partial implementations. Some proposals are made for implementation of novel graphical features of the Eulers Circle algorithm. This case study of an analogy suggests the hypothesis that one important role that analogy may play is providing a representation which is more limited in expressive power and therefore more computationally tractable than the literal representations analogies supplant.

An adequate cognitive theory of analogy must explain how mental processing is aided by analogical mappings. In the case of spatial inclusion as an analogy for set membership this comes down to requiring a theory of how graphical representations aid processing. Here we sketch a general theory in terms of the specicity of graphics which we have developed further elsewhere (see Stenning Oberlander 1991a, 1991b). This general approach to graphical representations rests on the observation that graphical representation systems limit abstraction and thereby purchase computational tractability.

In applying this general theory to the case of graphical methods of syllogism solution we sketch a method of using Eulers Circles and show how interpretative conventions for graphical objects allow the necessary abstractions. In particular we show that even choosing the least expressive system of diagrams limited to circles, still gives the requisite expressive power.

The Euler's Circle method sketched is then used as a guide to connectionist implementation of corresponding internal representations Graphical representations share some general computational properties with connectionist systems. In the domain of the syllogism the central issue is the implementation of the binding of attributes into type descriptions. Various proposals for connectionist variable binding in the literature are assessed as partial implementations. Some proposals are made for implementation of novel graphical features of the Eulers Circle algorithm. This case study of an analogy suggests the hypothesis that one important role that analogy may play is providing a representation which is more limited in expressive power and therefore more computationally tractable than the literal representations analogies supplant.

Original language | English |
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Title of host publication | Analogical Connections |

Editors | John A Barnden, Keith J Holyoak |

Place of Publication | Hillsdale, New Jersey |

Publisher | Lawrence Erlbaum Associates |

Pages | 446-486 |

Number of pages | 31 |

Volume | 2 |

ISBN (Print) | 978-156-750-039-4 |

Publication status | Published - May 1994 |

### Publication series

Name | Advances in Connectionist and Neural Computation |
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Publisher | Lawrence Erlbaum Associates |

Volume | 2 |