CONVERSION

One feature of Aristotle’s logic that is represented very clearly by Carroll diagrams is the idea of conversion.

The converse of a categorical statement is what you get when you switch the subject and predicate terms.

For instance, the converse of

(1) Some SOILS are not LOAMS.

is

(2) Some LOAMS are not SOILS.

We say that a statement is validly convertible iff it is logically equivalent to its converse. In the example given (1) is not validly convertible into (2). Indeed no Î-statement is validly convertible, nor is any À-statement. This is obvious from their Carroll diagrams. If we switch the terms, this makes no difference if and only if the diagram is symmetrical about the diagonal from the top left to the bottom right. Thus the diagrams of both E- and !-statements are symmetrical, and therefore E-statements are validly convertible, and so are !-statements. For instance,

(3) No IMAMS are BA’ATHISTS.

is logically equivalent to

(4) No BA’ATHISTS are IMAMS.

SUMMARY

• The class to which all the individuals in question belong is called the universe of discourse, (UD). This class includes all those individuals to which the predicates may or may not apply.

• Each category or predicate term, such as “foxes” or “animals of the dog family,” corresponds to a class of individuals. A Carroll diagram is a rectangular array in which statements involving categories may be represented. Any individuals of a given category C, or of which a given predicate C is true, will lie in the areas opposite the C. Any individuals of which the predicate is not true, i.e., those of category not-C or, will lie in the remaining area.

• In such a diagram, a zero 0 in a region indicates that there are no individuals there, an X indicates that there is at least one individual there, and an x straddling two regions means that there is at least one individual in one or the other region, or both.

• The converse of a categorical statement is what you get when you switch the subject and predicate terms. A categorical statement is validly convertible iff it is logically equivalent to its converse, !-statements are validly convertible, and so are E-statements. À-statements and Î-statements are not.

• À-statements and O-statements are contradictories of one another. So are !-state­ments and E-statements.

EXERCISES 15.2

3. (a)-(k): Construct a Carroll diagram for each statement in exercise 2 above.

4. Identify (i) which pairs of the following statements are converses of each other, and

(ii) which pairs are validly convertible into one other:

(a) All COMPUTERS are INTELLIGENT.

(b) Some COMPUTERS are INTELLIGENT.

(c) Some COMPUTERS are not INTELLIGENT.

(d) There are INTELLIGENT things that are not COMPUTERS.

(e) No MICROORGANISMS are SENTIENT beings.

(f) No INTELLIGENT things are COMPUTERS.

(g) No COMPUTERS are INTELLIGENT.

(h) Some SENTIENT beings are MICROORGANISMS.

5. Identify which pairs of statements in 4 are contradictories of each other.

15.3