Categorical Propositions Affirm Or Deny Relationships Between Classes

Glenn MacyPHI 26042011-1

Logic practices:

Categorical propositions affirm or deny relationships between classes.

Classes—the collection of all objects which have some specified object in common.

For the following propositions identify the subject and predicate and name the form:

1. Some historians are extremely gifted writers whose works read like first-rate novels. I
2. No athletes who have ever accepted pay for participating in sports are amateurs. E
3. No dogs which are without pedigrees are candidates for blue ribbons in official dog shows which are sponsored by the American Kennel Society. E
4. All satellites which are presently in orbits less than ten thousand miles high are very delicate devices that cost many thousands of dollars to manufacture.A
5. Some members of families that are rich and famous are not persons of wealth or distinction. O

A proposition distributes a term if it refers to all members of the class designated by the term.

Name the quality and quantity of each of the following propositions and state whether the subject and predicate terms are distributed or undistributed.

1. Some presidential candidates will be sadly disappointed people. I, affirmative, particular UU
2. Some recent rulings of the Supreme Courtwere politically motivated decisions which flouted the entire history of American legal practice.. I, affirmative, particular UU
3. No harmful pesticides or chemical defoliants were genuine contributions to the long-range agricultural goals of the nation.E, negative, universal DD
4. Some advocates of major political, social, and economic reforms are not responsible people who have a stake in maintaining the status quo. O, negative, particular UD
5. All shiny new high-performance automobiles are major sources of pollution.A, affirmative, universal DU

Square of Opposition

Contradictories—have opposite truth values

Contraries—the truth of one entails the other is false.

Subcontraries—can both be true but can’t both be false.

Opposition is a difference in quality or quantity, not disagreement.

Further immediate inferences:

Conversion—interchanging the subject and predicate

E: No S is P ≡ No P is S

I: Some S is P ≡ Some P is S

A: Not equivalent BUT Conversion by limitation isAll S is P ≡ Some P is S

O: Not equivalent

Obversion—change the quality and replace the predicate with its complement. The complementary class or complement is the collection of all things that do not belong to the original class. (If the original class is voter then its complement is nonvoter.)

Contrapositive—replace the subject with the complement of the predicate and replace the predicate with the complement of the subject.

A: All S is P ≡ All non-P is non-S

O: Some S is not P ≡ Some non-P is not non-S

E and I are not equivalent.

Taking the contrapositive is the same as taking an obversion, a conversion, and an obversion.

State the converse and indicate whether the two propositions are equivalent.

1. No people who are considerate of others are reckless drivers who pay no attention to traffic regulations. No reckless….regulations are people…others. equivalent
2. All graduates of West Point are commissioned officers in the United States Army. All commissioned….Army are graduates….Point. not equivalent.
3. Some European cars are overpriced and underpowered automobiles. Some overpriced…automobiles are European cars. equivalent
4. Some professional wrestlers are elderly persons who are incapable of doing an honest day’s work.

State the contrapositive and indicate whether the two propositions are equivalent.

1. All journalists are pessimists.All nonpessimists are nonjournalists. Equivalent
2. All scholars are nondegenerates. All degenerates are nonscholars. Equivalent
3. Some clergymen are not abstainers. Some nonabstainers are nonclergymen. Equivalent

If “All socialists are pacifists” is true, what may be inferred about the truth or falsehood of of the following propositions: WE WILL DO MORE of these!

1. No socialists are nonpacifists. (Solved for you). Give reason and truth value for each step.

1. All Sis P Given True (ALWAYS START WITH THE GIVEN).

2. No S is non-P Obverse True

1. Some nonpacifists are not nonsocialists.
2. No nonpacifists are socialists.
3. No nonpacifists are nonsocialists.
4. All nonsocialists are nonpacifists.
5. All pacifists are socialists.
6. Some socialists are not pacifists.
7. Some nonpacifists are socialists.
8. No pacifists are nonsocialists.
9. All nonpacifists are nonsocialists.

If “Some socialists are pacifists” is true, what may be inferred about the truth or falsehood of the following propositions:

1. No socialists are pacifists.
2. Some socialist are not pacifists.
3. Some nonpacifists are non socialists.
4. All nonpacifists are socialists.
5. No nonpacifists are nonsocialists.