Categorical Syllogisms and Venn Diagrams

Categorical Syllogisms
and
Venn Diagrams

Categorical syllogisms are two-premise arguments that conform to some very specific rules. The rules are so specific that only 256 of the infinite number of possible argument structures qualify as categorical syllogisms. Not all categorical syllogisms are good arguments, though; only 16 of the 256 forms are valid. In valid arguments, we recall, true premises guarantee a true conclusion.

A categorical syllogism contains two premises and a conclusion, all stated in standard-form categorical claims. Each standard-form claim, of course, deals with two categories. The two premises of a syllogism must make use of three different categories between them. In any syllogism, valid or invalid, one category will show up twice in the premises and the other two will each show up once in the premises and once in the conclusion. For example:

All vehicles that burn fossil fuel are polluters. Some vehicles that burn fossil fuel are vehicles that save lives. Therefore, some polluters are vehicles that save lives. As we see in the example above, the first two claims (the premises) each deal with two categories, but they share a concern for one of them (vehicles that burn fossil fuel). That shared category does not appear in the conclusion. In any reasonably complete argument, we expect every category that appears in the conclusion to show up in the premises, but not the other way around; there can be many categories that appear in the premises of an argument that do not appear in its conclusion.

The argument above seems quite reasonable, but is it a valid argument? This is not a matter of opinion, but of proof. One way of proving a syllogism valid is the Venn Diagram method. There are other ways to accomplish the same task, but the Venn Diagram method has the advantage of getting us to think visually as well as verbally. Experienced problem-solvers know that it's good to be able to conceptualize problems in a number of ways, since one setup or another may highlight a solution that otherwise might go unnoticed.

Formal Language

convErsIon (valid for E- and I-claims)

How it's done:

No X are Y --> No Y are X (valid)
All V are C --> All C are V (invalid)

contrApOsition (valid for A- and O-claims)

How it's done:

Some D are not S --> Some non-S are not non-D (valid)
Some F are B --> Some non-B are non-F (invalid)

obversion (valid for all four claim types)

How it's done:

All Z are Q --> No Z are non-Q (valid)
Some G are P --> Some G are not non-P (valid)

Natural Language

Conversion

Conversion is only valid for E- and I-claims. It's not an especially interesting operation when it's done right. Conversion is sometimes misapplied to universal affirmative (A-) claims. Here are a few of the possibilities of valid conversions that mean the same thing as their originals:

Original: No works of Russian literature are carried by this store.
Converse: None of the works carried by this store is Russian literature.
Original: There are bass in this lake.
Converse: Some of the fish in this lake are bass.

In unfamiliar situations, people sometimes make the mistake of assuming that if a statement is true, its converse is also true. This is, of course, a serious overestimation of the reliability of conversion in the cases of A- and O-claims.

Contraposition

Contraposition is valid only for A- and O-claims. The applications of contraposition are significantly more complex in practice than those of conversion. Here are a few contrapositions:

Original: All current customers will receive a loyalty discount.
Contrapositive: People who don't receive a loyalty discount are not current customers.
Original: If you go on the Germy Stringer Show, you'll immediately get involved in an ugly situation.
Contrapositive: If you haven't been involved an ugly situation, you haven't been on the Germy Stringer Show.

Original: There are some non-permanent workers here who aren't unreliable.
Contrapositive: Some reliable people aren't permanent workers here.

Obversion

Obversion always works if it's done right. The results obtained by the operation of obversion are the least intuitive of the three types of immediate inference.

Original: Only CDs with good videos sell really well.
Obverse: None of the CDs that sell really well don't have good videos.
Original: No one has entered the Forbidden Temple and returned.
Obverse: Everyone who has returned has not entered the Forbidden Temple.