Teaching syllogistic reasoning is often perceived as teaching pupils the purely formal rules of deductive inference. According to this common conception, such reasoning is a highly abstract skill, one that is carried out by the processing of syntactically encoded representations of the premises. This paper argues that syllogistic reasoning may, indeed, keep clear of the concrete contents of the premises, but is realized by a skill that is less abstract than rule-following. It argues that reasoning is continuous with our other skills and is realized by our capacities to deal with spatial situations. This explains why the use of Venn diagrams, a much-used technique for evaluating syllogistic inferences, is effective: the spatial layout expressed by the diagrams directly activates the actual mechanism reasoners use. Teachers are therefore right when they teach the use of such diagrams. This paper also argues that using tools that tap into our capacities to deal with a three-dimensional world will be even more effective and corroborates this argument with an experiment in which three-dimensional Venn diagrams were used to train high school pupils.