Topological relations such as inside, outside, or intersection are ubiquitous to our spatial thinking. Here, we examined how people reason deductively with topological relations between points, lines, and circles in geometric diagrams. We hypothesized in particular that a counterexample search generally underlies this type of reasoning. We first verified that educated adults without specific math training were able to produce correct diagrammatic representations contained in the premisses of an inference. Our first experiment then revealed that subjects who correctly judged an inference as invalid almost always produced a counterexample to support their answer. Noticeably, even if the counterexample always bore a certain level of similarity to the initial diagram, we observed that an object was more likely to be varied between the two drawings if it was present in the conclusion of the inference. Experiments 2 and 3 then directly probed counterexample search. While participants were asked to evaluate a conclusion on the basis of a given diagram and some premisses, we modulated the difficulty of reaching a counterexample from the diagram. Our results indicate that both decreasing the counterexample density and increasing the counterexample distance impaired reasoning performance. Taken together, our results suggest that a search procedure for counterexamples, which proceeds object‐wise, could underlie diagram‐based geometric reasoning. Transposing points, lines, and circles to our spatial environment, the present study may ultimately provide insights on how humans reason about topological relations between positions, paths, and regions.

中文翻译：

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基于图的几何推理中的反例搜索

内部、外部或交叉等拓扑关系在我们的空间思维中无处不在。在这里，我们研究了人们如何通过几何图中点、线和圆之间的拓扑关系进行演绎推理。我们特别假设反例搜索通常是这种推理的基础。我们首先验证了没有经过特定数学训练的受过教育的成年人能够生成包含在推理前提中的正确图解表示。然后我们的第一个实验表明，正确判断推理无效的受试者几乎总是会产生反例来支持他们的答案。值得注意的是，即使反例总是与初始图有一定程度的相似性，我们观察到，如果一个物体出现在推理的结论中，那么它更有可能在两张图之间发生变化。然后，实验 2 和 3 直接探讨了反例搜索。虽然要求参与者根据给定的图表和一些前提评估结论，但我们调整了从图表得出反例的难度。我们的结果表明，降低反例密度和增加反例距离都会损害推理性能。综上所述，我们的结果表明，以对象方式进行的反例搜索程序可以作为基于图的几何推理的基础。将点、线和圆转换到我们的空间环境中，