7.5.6. Syllogisms

We can now establish the validity of the forms of argument that are syllogisms in the narrower of the traditional senses of the term. Syllogisms in this sense are two-premised arguments whose component sentences are analyzed as restricted generalizations or their denials with quantified and restricting predicates that are unanalyzed. A syllogism must contain exactly three such predicates, and each of these predicates must appear in exactly two of the component sentences. The generalizations that are asserted or denied in the premises and conclusion can be affirmative or negative but all must be direct.

The constraints on predicates leave four possible figures distinguished by whether each of restricting or quanitified predicates of the conclusion appears in the same or different role in the premise in which it also appears. For each choice of figure, there are 4 × 4 × 4 = 64 moods reflecting the choice of a form of sentence (an assertion or denial of an affirmative or negative generalization). There are thus 256 syllogisms; and, of these, 15 are valid. This number is small enough that they could all be named, and mnemonic names were introduced for them in the middle ages. These names were constructed to display the mood of the syllogism in their choice of vowels and, in some of their consonants, ways of establishing the validity of some syllogisms on the basis of others.

Below is a derivation for the best known of these patterns. The name of this syllogism, Barbara, is one of the few that does not sound like the artificial construction it is.

The letters chosen for predicates in the analysis are designed to highlight the figure. Notice that the restricting and quantified predicates of the conclusion (R and Q) play the same roles when they appear in the premises. The third predicate (M) is traditionally known as the middle term. An example is All humans are mortal, All philosophers are human ⊨ All philosophers are mortal.

Middle terms do not always stand between the other two in the range of their application (as does human between philosopher and mortal); but, in all valid syllogisms, the middle term nevertheless provides the basis for the relation that the conclusion asserts between the other two predicates. It can thus be understood to connect these predicates and stand between them in this sense.

This derivation also provides an example of the form that will be taken by arguments involving restricted universals when they are reformulated using unrestricted quantifiers. Were we to have special rules for restricted universals, one kind of exploitation rule would have the effect of the sort of combination of UI and MPP seen in stages 3 and 4 and again in stages 5 and 6 above. The planning role for a restricted universal goal would have the effect of the sort of combination of UG and CP in stages 1 and 2; in short, it would introduce a general argument with a supposition that predicated the restricting predicate of the generalization to the independent term and a goal that predicated the quantified predicate to the same term.