PART 2 MODULE 4 CATEGORICAL SYLLOGISMS AND DIAGRAMMING Consider the following argument: Some lawyers are judges. Some judges are politicians. Therefore, some lawyers are politicians. Although the premises and conclusion of this argument sound reasonable, and although the structure of the argument looks similar to transitive reasoning, this argument is invalid. In order to show that the argument is invalid, all we have to do is conceive of a situation in which the conclusion is false, while both premises are true. In order to do so, it helps if we imagine a world with a small population of lawyers, judges and politicians. Suppose there are only two lawyers, Alice and Bill, and that Bill is also a judge, but Alice isn't. Suppose that in addition to Bill there is only one other judge, Carla, and Carla is also a politician, but Bill isn't a politician. Finally, suppose there is one other politician, Don, who isn't a lawyer and isn't a politician. In this conceivable world, some lawyers are judges (Bill), and some judges are politicians (Carla), but no lawyers are politicians. Since it is possible to conceive of a situation in which the conclusion is false while both premises are true, this argument is invalid. The previous argument is an example of a CATEGORICAL SYLLOGISM, which is an argument involving two premises, both of which are categorical statements. Categorical statements are statements of the form "all are...," "none are..." or "some are..." A categorical statement of the form "all are..." is also called a positive universal statement. A categorical statement of the form "none are..." is also called a negative universal statement. A categorical statement of the form "some are..." or “some aren’t is also called an existential statement. In this discussion we are primarily concerned with categorical syllogisms in which at least one premise is an existential statement, because such arguments cannot be analyzed using the methods of Unit 2 Module 1.Existential statements A statement of the form "Some are...," such as "Some lawyers are judges," is conceptually quite different from a universal statement, in that it cannot be restated in terms of logical connectives in any way that is of practical use. Whereas a positive universal statement such as "All cats are mammals" can be informally restated as "If __ is a cat, then __ is a mammal," and whereas a negative universal statement such as "No cats are dogs" can be restated as "If __ is a cat, then __ isn't a dog," it is not possible to make such a transition with an existential statement such as “Some mammals are predators.” This means that the techniques of Unit 2 Module 1, which are based on truth tables and logical connectives, are of no use for arguments involving the existential statement. Diagramming categorical statements There is an extensive literature on the topic of categorical syllogisms, dating back to medieval scholarship and earlier. This includes an impressive body of special terminology, symbols, and characterizations of forms, which a student might encounter in a more intense study of the subject, such as in a history of philosophy course. This discussion will be limited to the presentation of a method of analyzing categorical syllogisms through the use of three-circle Venn diagrams. This method is called diagramming. Individual statements are diagrammed as follows. 1. Use shading to diagram universal statements, by shading out any region that is known to contain no elements. 2. Use an "X" to diagram an existential statement. If a region is known to contain at least one element, place an "X" in that region. If it is uncertain which of two regions must contain the element(s), then place the "X" on the boundary between those two regions. 3. If a region contains no marking, then it is uncertain whether or not that region contains any elements.The marked Venn diagram below illustrates these ideas. Diagramming categorical syllogisms To test the validity of a categorical syllogism, follow these steps. 1. In order to be valid, a categorical syllogism must have at least one premise that is a universal statement. If none of the premises is a universal statement, then the argument is invalid, and we are done. The following steps assume that at least one premise is a universal statement. 2. Begin by diagramming the universal premise(s). A universal statement will have the effect of shading (blotting out, so to speak) some region of the diagram, because a universal statement will always assert, directly or otherwise, that some region of the diagram has no elements. 3. Confining your attention to the part of the diagram that is unshaded, diagram an existential premise by placing an "X" in a region of the diagram that is known to contain at least one element. If it is uncertain which if two regions should contain the element(s), place the "X" on the boundary between those two regions.4. After diagramming the premises, if the diagram shows the conclusion of the argument to be true, then the argument is valid. If the diagram shows the conclusion to be uncertain or false, then the argument is invalid. 5. If all the statements in the argument are universal statements, then the argument can be analyzed in terms of transitive reasoning or false chains (see Unit 2 Module 1), and so diagramming is unnecessary. 6. If the both remises are universal statements but the conclusion is an existential statement, then the argument is invalid. No diagram is necessary. You cannot deduce “some” from “all” or “none.” EXAMPLE A Use diagramming to test the validity of this argument: No terriers are timid. Some bulldogs are terriers. Therefore, some bulldogs are not timid. SOLUTION We will mark this three-circle Venn diagram, which shows the sets "terriers," "bulldogs" and "timid (things):" First, diagram the negative universal premise "No terriers are timid." According to this premise, the overlap of those two sets contains no elements, so that part of the diagram is shaded, or "blotted out."Next, diagram the existential premise "Some bulldogs are terriers" by placing an "X" in the appropriate location in the unshaded portion of the diagram. Now that both premises have been diagrammed, check to see if the marked diagram shows the conclusion to be true.Because the marked diagram shows that the conclusion is true, the argument is valid. EXAMPLE B Use