Introductory Logic Test #2 March 20, 2006 Name: ________________________ R. Hammack Score: _________

1.
(20 points) Translate the following sentences into symbolic form. Use capital letters to represent simple statements.
For each letter you use, please indicate what statement it stands for (e.g. N = "my nose itches").

 Y ⊃ (L ∨ E) Y = "Your test grade will be dropped." L = "It is your lowest grade." E = "You have an excused absence."

 (b) If it does not rain soon, then the risk of forest fires will be great and cuation will be necessary.

 ~R ⊃ (F • C) R = "It rains soon." F = "The risk of forest fires will be great." C = "Cuation will be necessary."

 (c) Oregon does not have a sales tax, but Virginia does.

 ~O • V O = "Oregon has a sales tax." V = "Virginia has a sales tax."

 (d) If affirmative action programs are dropped, then if new programs are not created, then minority applicants will suffer.

 A ⊃ (~N ⊃ M) A = "Affirmative action programs are dropped." N = "New programs are created." M = "Minority applicants will suffer."

 (e) Yosemite and Kings Canyon restrict vehicle traffic unless Bryce and Zion do not.

 ~(~B • ~Z) ⊃ (Y • K) OR    (B ∨ Z) ⊃ (Y • K) OR    (Y • K) ∨ (~B • ~Z) Y = "Yosemite restricts vehicle traffic" K = "Kings Canyonrestricts vehicle traffic" B = "Bryce restricts vehicle traffic" Z = "Zion restricts vehicle traffic"

2.
(16 points) Write out the truth tables for the following propositions. For each proposition, say if it is tautologous, self-contradictory, or contingent.

(a)
 ~ [ ( M ⊃ P ) ⊃ M ] ≡ ( P ⊃ M ) F T T T T T F T T T F T F F T T F F T T T F T T F F F T F F T F T F F F T F T F
It is CONTINGENT.

(b)
 ( Q ⊃ ~ P ) ∨ ( P • Q ) T F F T T T T T T T T F T F F T F T F T T T F F F T T F T F F F
It is TAUTOLOGOUS.

3. (24 points) Determine if the following pairs of statements are logically equivalent, contradictory, consistent, or inconsistent.

(a)
 A ⊃ B ~ B ⊃ ~ A T T T F T T F T T F F T F F F T F T T F T T T F F T F T F T T F
They are LOGICALLY EQUIVALENT and CONSISTENT.

(b)
 ~ ( A ∨ B ) ~ A ∨ ~ B F T T T F T F F T F T T F F T T T F F F T T T F T F T T F F F T F T T F
They are CONSISTENT.

(c)
 ( A ∨ B ) ⊃ A ~ B ∨ A T T T T T F T T T T T F T T T F T T F T T F F F T F F F F F T F T F T F
They are LOGICALLY EQUIVALENT and CONSISTENT.

4. (20 points) Use indirect truth tables to decide if the following sets of statements are consistent or inconsistent
 (a) P ⊃ ( R ≡ A ) / A ⊃ ( W • ~ R ) / R ≡ ( W ∨ K ) / P • U / U ⊃ K T T T T T T T ? F F T T T ? T T T T T T T T
When the table is filled out, there is a value for every letter except W. But regardless of W's value, there is a contradiction (highlighted). Thus the statements are INCONSISTENT.

 (b) M ∨ B / ~ B / M • A / B ⊃ M / A ∨ B T T F T F T T T F T T T T F
There are no contradictions. Thus the statements are CONSISTENT.

5. (20 points) Use any technique from Chapter 6 to decide if the following arguments are valid or invalid.
(a)
 M ⊃ ( C ∨ D ) ~X ∨ M ( D ∨ C ) ⊃ X M ≡ X

 M ⊃ ( C ∨ D ) / ~ X ∨ M / ( D ∨ C ) ⊃ X // M ≡ X T T F F F T F T T F F F T F T F F F T F T T F T T F F T

There are two ways for the conclusion to be false, and a line is filled in for each way. Notice that there is a contradiction on both lines, so the argument is VALID

(b)
 M ⊃ ( L ⊃ K ) P ⊃ M ~S S ∨ L K ⊃ P

 M ⊃ ( L ⊃ K ) / P ⊃ M / ~ S / S ∨ L // K ⊃ P ? T T T T F T ? T F F T T T F F

Notice that when the table is filled out, the value of M cannot be determined. However, if you set M=T (or F), all the premises are true and the conclusion is false. Therefore the argument is INVALID.