154 Logic as a Tool C\(A∪B) C (A∩B)\C A∩B∩C A B Figure 3.2 Three-set Venn diagram × S P S × P S • P • S P Figure 3.3 Venn diagrams depicting the four types of categorical syllogistic propositions From left to right: “All S are P”, “All S are not P”, “Some S are P”, and “Some S are not P” of the eight regions represents a particular set-theoretic combination of the three sets, for example, the region in the middle represents A ∩ B ∩ C while the region above this represents C \ (A ∪ B ), etc Venn diagrams can be used to represent the terms in a categorical syllogism by sets and indicate the relations between them expressed by the premises and the conclusion In particular, each of the four types of categorical syllogistic propositions can be easily illustrated by Venn diagrams as in Figure 3.3 The circles correspond to the three terms; the barcoded and shaded regions indicate those referred to by the premises; a cross × in a region means that that region is empty; a bullet • in a region means that that region is non-empty; and no cross or bullet implies that the region could be either empty or non-empty Each of the three propositions of a categorical syllogism can therefore be represented on a three-set Venn diagram, corresponding to the three terms For instance, the syllogisms in Example 121(1) and 121(3) are represented on Figure 3.4 as follows In the Venn diagram on the left corresponding to Example 121(1), the circle M represents the set of mortals (the major term), the circle L represents the set of all logicians (the minor term), and the circle H represents the set of all humans (the middle term) The diagram depicts the two premises of Example 121(1) by indicating the regions H \ M that corresponds to “humans who are not mortals”, and L \ H that corresponds to “logicians who are not humans”, as empty Likewise, in the Venn diagram on the right corresponding to Example 121(3), C represents the set of crooks (the major term), P o represents the set of all politicians (the minor term), and P h represents the set of all philosophers (the middle term) The diagram depicts the two premises of Example 121(3) by indicating the region P h ∩ C corresponding to all “philosophers who are crooks” as empty, while indicating the region P h ∩ P o corresponding to all “philosophers who are politicians” as non-empty (the bullet)