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Philosophy 103: Introduction to Logic
Strategies for Successive Inferences

 Abstract: The technique of successive applications of logical relations drawn from the square of opposition and further immediate inferences is discussed and illustrated. I. Before attempting successive immediate inferences, be able to write out from memory all the logical relations from the Square of Opposition and the Further Immediate Inferences. A. The Square of Opposition includes (1) contradiction, (2) contrariety, (3) subcontrariety, and (4) subalternation. B. Further Immediate Inferences include (5) conversion, (6) obversion, and (7) contraposition. C. These seven logical relations compose a kind of "tool-kit" used to do inferences one after the other (hence the name "successively"). II. Successive immediate inferences are used to establish whether two different statements about some state of affairs are logically related in some way. A. The object of the exercise is to try to establish a truth value for a statement, if we know in advance the truth value of the other statement. B. Often, this technique is used to interpret the meaning of a statement which is initially difficult to understand. 1. For example, if a movie critic admits, "Some violent movies are events which do not cause aggressive behavior." 2. Does this admission indirectly imply that some violent movies cause aggressive behavior? We can prove the first statement does not entail the second statement as follows. Statement Reason T.V. 1. Some violent movies are events which do not cause aggressive behavior. given true 2. Some violent movies are not events which cause aggressive behavior. obversion true 3. Some violent movies are events which cause aggressive behavior. subcontrariety unknown 3. If the movie critic believes "Some violent movies are events which cause aggressive behavior," then the critic would have to try to establish its truth some other way. III. Rules of Thumb for solving Successive Immediate Inferences. A. A good initial strategy to solve such problems is to try the following approach. 1. Compare the subject and predicate classes of both statements and note the differences. Begin by using inferences to match exactly the subject and predicates of both statements. a. If there is a two-complementary class difference, contraposition is used in a step. b. If there is a one-complementary class difference, obversion is used in a step. c. If the classes are simply reversed in order, conversion is used in a step. 2. Once the exact classes are matched, use inferences from the Square of Opposition to match the statement form. 3. If the use of step (1) match classes and step (2) match statement fails to get a truth value, try reversing the order of steps. B. Some inferences, of course will result in an undetermined truth value because the two statements are not logically related. 1. Even so, you should try to maintain a truth value (i.e., true or false ), if possible, for each step. 2. The shortest route is generally the best. Nevertheless, if you get an undetermined truth value, go back through the problem and try reversing steps.

Example Problem:

Suppose we know "All stoics are philosophers" is true," what would be
the truth value of "Some nonphilosophers are not stoics"?

Compare the sequence of the application of the rules of thumb with the
statements below in the bulleted list.

PREVIOUS            NEXT

 Click "NEXT" above for an explanation  of the next step.

• (1) All stoics are philosophers.

¯  OBVERSION

• (2) No stoics are nonphilosophers.

¯ CONVERSION

• (3) No nonphilosophers are stoics.

¯  SUBALTERNATION

• (4) Some nonphilosophers are not stoics.

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