Homework 17 – more immediate inferences 1

Logical Thought Fall 2004

Homework 17

State the converses of the following propositions, and indicate whether they are equivalent to the given propositions.

1. All graduates of West Point are commissioned officers in the U.S. Army.

Converse: All commissioned officers in the U.S. Army are graduates of West Point. Not equivalent.

2. Some European cars are overpriced and underpowered automobiles.

Converse: Some overpriced and underpowered automobiles are European cars. Equivalent.

State the obverses of the following propositions:

3. No organic compounds are metals.

All organic compounds are nonmetals.

4. Some clergy are not abstainers.

Some clergy are nonabstainers.

State the contrapositives of the following propositions and indicate which of them are equivalent to the given propositions.

5. Some soldiers are not officers.

Contrapositive: Some nonofficers are not nonsoldiers. Equivalent.

6. Some noncitizens are not nonresidents.

Contrapositive: Some residents are not citizens. Equivalent.

If “All socialists are pacifists’ is true, what may be inferred about the truth or falsehood of the following propositions? That is, which could be know to be true, which known to be false, and which would be undetermined?

Homework 17 – more immediate inferences 2

7. No socialists are non-pacifists.

No S are nonP is true, because it is the obverse of the original statement, and obversion preserves the truth value of the original statement.

8. All socialists are non-pacifists.

This has the form ‘All S are nonP’. It is false, as can be seen through the following steps:

(1) All S are P [true, by assumption] (2) No S are P [false, because it is the contrary of (1)]. (3) All S are nonP [false, because it is the obverse of (2)].

9. All non-socialists are non-pacifists.

This has the form ‘All non-S are non-P’, and it’s truth value is unknown, as can be seen through the following steps:

(1) All S are P [true, by assumption] (2) All nonP are nonS [true, because it is the contrapositive of (1)] (3) All nonS are nonP [unknown, because conversion does not work on form A]

10. Some socialists are non-pacifists.

This is false, because it is the obverse of the contradiction of the original sentence:

(1) All S are P [true, by assumption] (2) Some S are not P [false, because it contradicts (1)] (3) Some S are non-P [false, because it is the obverse of (2)]