Testing Categorical Syllogisms for Validity
There are several methods for testing for validity. A popular method is that of Venn diagrams. Another makes use of rules that depend on the notion of distribution.
The method we will use is that of refutation by logical analogy. We do so because it makes use of the central concept of deductive logic–validity. An argument is said to be valid if it is impossible to have true premises and a false conclusion. In other words, if there is any argument that has true premises and a false conclusion, then one knows that the form is invalid.
Identifying examples of argument patterns that show invalidity is to make use of a counter example procedure. By identifying a single instance in which a particular syllogistic pattern produces a false conclusion from true premises, the method of refutation by logical analogy shows that a syllogism is unreliable or invalid. It cannot be counted on 100 percent. It will sometimes yield a false conclusion.
It is not the case that every argument with an invalid form will have only the combination of true premises and a false conclusion. There can be several possibilites:
true major premise, false minor premise and true conclusion false major premise, false minor premise and false conclusion true major premise, true minor premise and true conclusion
These are just some of the possibilities. There is only one impossible situation: valid argument pattern, true premises and a false conclusion. If one finds a particular argument form that has true premises and a false conclusion–even if sometimes this same form has some other combination of true and false premises and conclusion–then one knows that this form is invalid. It just takes one occurrence of true premises and a false conclusion to show that an argument pattern is invalid. Here is how it works. Say we have the following pattern:
No plates are dogs. No rooms are plates. Therefore no rooms are dogs.
Each of the statements is true. Perhaps the syllogism is valid. But, then, upon experimentation we produce the following analogous syllogism. It has exactly the same form, EEE-1, but we use different terms:
No kangaroos are cows. No Jerseys are kangaroos. Thus no Jerseys are cows.
Here we have true premises, but a false conclusion! It is not the case that “No Jerseys are cows.”
So our analogous syllogism has produced a counter example. The EEE-1 form has produced a false conclusion from true premises. It is not just this syllogism that is the problem; the form itself is unreliable.
The promise of a deductive argument is that one can trust that if one has true premises then one will get a true conclusion. But, as this instance shows, the form EEE-1 has failed! It only takes one counter example to show a form to be invalid. Remember we said we expected 100 percent reliability. So if fails a single time, then it is invalid–even though it may work many, if not most of the, times.
Limitations of the Method
Given enough imagination and patience one could work through all 256 possible categorical syllogisms, discovering those that are invalid. The remaining ones would be valid. But this is the method’s limitation. It often requires exceeding imagination and persistence to discover that a syllogism is invalid. One might think of a counter example right away, but often one has to do considerable experimenting to discover one.
Accordingly, other, more mechanical methods have been devised, such as the traditional rules (the middle term must be distributed at least once, if a term is distributed in the conclusion it must be distributed in the premise, etc.) and Venn diagrams.
But to learn these rules requires one to invest more effort than what I think useful in this course. I would prefer us to build on the notion of validity presented thus far and proceed to the propositional calculus, which is one part of modern, symbolic logic.
But before doing so we will spend some time practicing the method of refutation by logical analogy.