Testing Syllogisms for Validity

Using Syllogistic Rules

In an earlier exercise, we asked whether there will be any valid syllogisms whose premises are both particular but whose conclusion is universal.  If you answered this question negatively, you were right.  Think of the Venn Diagram for this syllogism. If the premises are both particular, then there will be x's in some of the areas, but no shading. A universal conclusion asserts, however, that some areas are shaded.  So diagramming just particular premises will never give us a diagram of a universal conclusion.  In other words, any syllogism with two particular premises and a universal conclusion will be invalid.

The conclusion we just reached is a generalization about all syllogisms, and it tells us that a certain class of syllogisms cannot be valid.  The ancient logicians came up with a set of rules which give us necessary and sufficient conditions for valid syllogisms: A syllogism conforms to all the rules, if and only if the syllogism is valid.  What are the rules?   There are many possible versions of the rules, including versions which incorporate the insight of the first paragraph.  In order to understand them, however, we will have to introduce a technical notion, called distribution.  

A term, either a subject or predicate, is distributed if and only if the categorical proposition in which it occurs makes a claim about every member of the class referred to by that term.  

We'll look now at A, I, E and O sentence forms to see which terms are distributed in each.

A: "All dentists are off on Fridays."  The term "dentists" is distributed, because we're talking about all dentists. The predicate term "persons who are off on Friday", is not distributed, because we're not talking about all of those folks, just the dentists.

I: "Some dentists are off on Fridays."  We're not talking about all dentists, but just some. And we're not talking about everyone who is off on Friday.  So neither the subject nor the predicate is distributed.

E: "No dentists are off on Fridays."  We're talking all dentists here, saying, in effect that all of them are not off on Fridays, that none of entire class of dentists is off.  But we're also talking about everyone who is off on Friday, and we're saying that everyone of them is a non-dentist.

O: "Some dentists are not off on Fridays."  Here we're not referring to all dentists, but just to some.  We are, however, referring to those who are off on Friday.  All of those who are off on Friday are non-identical to some dentists, the ones who are not off on Fridays.

Thus A distributes the subject, E distributes both subject and predicate, I distributes no terms, and O distributes the predicate.  We summarize these conclusions:

One way of remembering this is to notice that universal categorical sentences distribute subjects, and negative categorical sentences distribute predicates.

Now we can state the rules for valid syllogisms:

1. If a syllogism is valid, then the middle term is distributed at least once.
2. If a syllogism is valid, then if a term is distributed in the conclusion, it must be distributed in a premise.
3. If a syllogism is valid, it does not have two negative premises.
4. If a syllogism is valid, then it has a negative premise, if and only if it has a negative conclusion.
5. If a syllogism is valid, then if its premises are universal, then its conclusion is universal.

Exercise: Test lots of syllogisms for validity using each of the three methods covered. Verify that valid syllogisms are indeed valid.  Finally, rewrite arguments as standard form syllogisms and test them for validity.