Phi 8 Logic Pt 12 Categorical Syllogisms

The twelfth lecture concludes the substance of deductive logic by extending a consideration of categoricals from simple propositions to syllogisms. A categorical syllogism is in standard form when all of four conditions are met: 1) all three propositions are standard-form categoricals; 2) the two occurrences of each term are identical (no paraphrasing); 3) each term is used in the same sense (no equivocation) (The Four Terms Fallacy); and 4) the Major Premise is first, the minor premise second, the conclusion last. Recall that whereas the conditional syllogism has only 2 terms, the categorical has three. The Major Term is the predicate term (P) of the conclusion (and also occurs once in the Major Premise); the minor term is the subject term of the conclusion (S) (and also occurs once in the minor premise). The third term is the middle term (M), and it is that term occurring once in each of the premises but never in the conclusion. This means the placement of the middle term may vary: it might be either subject or predicate term; and that, in the Major Premise or the minor premise. To sort this, figure and mood are important. Figure refers to the respective placement of the middle term in any given categorical proposition. There are four possible permutations; we number these 1-4, and the best mnemonic is the so-called “shirt-collar model.” Mood is determined by which sort of categorical proposition is in the premises and the conclusion. For example, the famous Barbara syllogism has an A proposition for all three (thus its mood would be AAA); and its figure is 1. So another, more analytic, name for Barbara would be AAA-1. Combining figure and mood in this way allows us a discrete designator for each of the categorical syllogisms (and you may recall from ch 1, there are 256 of them!). Of the 256, however, no more than 24 are valid. (So it is much easier to screw up validity with categoricals than with either conditionals or disjunctives!) How do we decide which are valid and which invalid? By using Venn diagrams. Recall that the Venn for simple categorical propositions is merely an hieroglyphic, but the Venn for the categorical syllogisms is a machine. And what work does this machine do? It allows us to see, at a glance, whether any given one of the 256 syllogisms is valid or invalid. The procedure is: draw the complex Venn for any given syllogism; then draw the simple Venn for the conclusion statement alone. If the former says AT LEAST as much as the latter, then the syllogism is valid; if not, it is invalid. The test is just that straightforward. From ch 1 you already know substitution instances (the universal test for validity); the Venn diagrams provide a second test, but specific to categoricals only. There is a third test (also for categoricals only): rules. If any categorical syllogism violates even one of the following five rules, it is invalid. (And no further testing is needed to show this.) The rules are: 1) the middle term must be distributed at least once; 2) if a term is distributed in the conclusion, it must be distributed in a premise; 3) two negative premises are not allowed; 4) a negative premise requires a negative conclusion, and a negative conclusion requires a negative premise; and 5) if both premises are universal, the conclusion cannot be particular.