Many strategies could be employed to distinguish between valid and invalid categorical syllogisms:
- random choice (not a very scientific basis for decision-making at any time, but particularly when the chance of winning is only 24/256);
- memorization, an old, laborious standby;
- knowing where the answer can be found (Table 9);
- recognition that the correct solutions all obey a few rules (only five rules are needed for successful separation of the 24 valid syllogisms from the 232 invalid ones);
- sketching Venn diagrams;
- substitution, in which we recognize that the problem structure is identical to one whose answer is known.
All except for the ‘random choice’ option are acceptable solutions to the problem, but memorization and substitution have the strong advantage of much greater speed. In the remainder of this section, I list the valid syllogisms for easy reference, and then I describe substitution -- the easiest closed-book technique for evaluating syllogisms.
Substitution is an easy way to evaluate categorical syllogisms. As with the evaluation of any formal logic, the validity of the form is independent of the actual terms used. If we insert familiar terms into the syllogism, choosing ones that yield true premises, then an untrue conclusion must indicate an invalid syllogism. For evaluation of categorical syllogisms, I select substitutions from the following classification tree:
animals
/ \
mammals reptiles
/ \ / \
dogs cats snakes turtles
The danger of substitution is that a true conclusion does not prove that the logic is valid, as we saw above for the syllogism “Some mammals are dogs; some mammals are cats; therefore no cats are dogs.” Substitution can prove that an argument is invalid but, unfortunately, cannot prove that it is valid. If the premises are true, a substitution that yields a true conclusion may or may not be of valid form. In contrast, a substitution with true premises and false conclusion must be of invalid form. Thus one needs to consider several substitutions, to see whether any case can prove invalidity. For example, the following argument is not disproved by the first substitution but is disproved by the second one:
Some physicists are theoreticians. Some astronomers are theoreticians. Therefore some physicists are astronomers.
Some dogs are animals. Some mammals are animals. Therefore some dogs are mammals.
