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115FS_PHIL 1011.001, Midterm #2, Study Guide.
Tuesday, 3 november, 2015

• Hurley – Chapters 6 & 7
• Total points = 200. Plus 20 bonus points.
• There are Five (5) sections and one (1) bonus section.
• Make sure that you have the following concepts committed to memory:

o The truth-tables for the following truth-functional connectives:
• Conjunction :
o (p • q)
• Disjunction:
o (p ? q)
• Negation:
o ~p
• Conditional:
o (p ? q)
• Bi-conditional:
o (p ? q)

o The following six (9) inference rules (m & n stand for line numbers):

1. Modus Ponens (M.P.)
m (p ? q)
n p
? q m,n M.P.
2. Modus Tollens (M.T.)
m (p ? q)
n ~q
? ~p m,n M.T.
3. Disjunctive Syllogisim (D.S.)
m (p ? q)
n ~p
? q m,n D.S.
m p
? (p ? q) m Add.
5. Simplification (Simp.)
m (p • q)
? p m Simp.
6. Conjunction (Conj.)
m p
n q
? (p • q) m,n Conj.

7. Hypothetical Syllogism (H.S.)
m (p ? q)
n (q ? r)
? (p ? r) m, n H.S.

8. Constructive Dilemma (C.D.)
m (p ? q) • (r ? s)
n (p ? r)
? (q ? s)) m, n C.D.

9. Absorption (Abs.)
m (p ? q)
? p ? (p • q) m Abs..

1. (20 points) Determine the truth-value of the given sentences using {Conj, Disj, Neg, Cond, and Bicond}, and the given truth values of the simple sentences.

For example:
• Flipper is a dolphin = TRUE
• Fred is a snake = FALSE
~[(Flipper is a dolphin ? ~Fred is a snake) • Fred is a snake] ? Fred is a snake
~[(T ? ~F) • F] ? F
~[(T ? T) • F] ? F
~[(T) • F] ? F
~[F] ? F
T ? F
F
2. (20 points) Translations of English sentences into Boolean truth-functional sentences.
i. Translate the sentence(s) into a truth-functional symbolic sentence.
ii. Produce a truth-table for the symbolic sentence.

For example: If Steve runs and Ayca does not read poetry, then Muk Yan will play badminton.

R = Steve runs.
B = Muk Yan plays badminton.

(R • ~P) ? B

R P B (R • ~P) ? B
T T T T F F T T
T T F T F F T F
T F T T T T T T
T F F T T T F F
F T T F F F T T
F T F F F F T F
F F T F F T T T
F F F F F T T F
3. (70 points) Using a truth-table for the given symbolic argument:
• Assess the validity of the argument. Mark the rows that demonstrate validity, or demonstrate invalidity.
• For each argument: conjoin the premises and imply the conclusion (i.e. create a conditional in which the antecedent is the conjunction of all the premises and the consequent in the conclusion). If your assessment is “valid” then this should result in a tautology.

For example: P1 (p ? q) ? q
P2 p
? p

Ref Col: p Ref Col: q P1 (p ? q) ? q P2 p ? p [((p ? q) ? q) • p] ? p
T T (T) T (T) T T * [(T) T (T)] T (T)
T F (T) F (F) T T [(F) F (T)] T (T)
F T (T) T (T) F F [(T) F (F)] T (F)
F F (F) T (F) F F [(T) F (F)] T (F)
Argument Assessment: VALID.

4. (40 points) Via a truth-table, assess whether the given sentences are logically equivalent.

Sentences:
i. (W ? R)
ii. (R ? W) • ( W ? R)
Ref Col: W

Ref Col: R Sentence 1
(W ? R)
Sentence 2
(R ? W) • ( W ? R)
T T (T) T (T) T (T) T (T)
T F (T) F (F) T (T) F (F)
F T (F) F (T) T (F) F (T)
F F (F) T (F) T (T) T (T)

? Sentence 1 and Sentence 2 are logically equivalent.
5. (50 Points)
Using the nine rules of inference and the given completed valid proof cite each non-premise line with the appropriate rule of inference and the line(s) that justify the cited rule of inference.

For example:

6.
i. (Bonus Section – 20 points)

Use only the nine rules of inference. In addition, you may use either CP or IP to give a proof of the valid arguments below.

For example:
Introduction to Logic.
15FS-PHIL-1011.001
Midterm #2.
Tuesday, 3 November, 2015
Total points = 200 (20 bonus points).

Name: ________________________________________________________________

Signature: ______________________________________________________________

1. (20 Points/10 each) Determine the truth-value of the following sentences. The truth-values of the simple sentences are given. Substitute the truth-values of the simple sentences into the Boolean sentence.

• Michael Jordan played outfield for the White Sox. = TRUE
• Michael Jordan played guard for the Bulls. = TRUE
• Lebron James plays third base for the Indians. = FALSE
• Joan Benoit was an Olympic marathoner. = TRUE

a. ~[ (Michael Jordan played guard for the Bulls ? ~Michael Jordan played guard for the Bulls) • Michael Jordan played outfield for the White Sox] ? Lebron James plays third base for the Indians.

b. ~[~(Michael Jordan played outfield for the White Sox ? Michael Jordan played guard for the Bulls) = (Lebron James plays third base for the Indians • ~Joan Benoit was an Olympic marathoner)]

2. (20 points/10 each)
Translations of English sentences into Boolean truth-functional sentences:

i. Translate the sentence(s) into a truth-functional symbolic sentence.
ii. Produce a truth-table for the symbolic sentence.
iii. Please use the propositional variables assigned to the simple sentences (below).

R = Steve runs.
B = Muk Yan plays badminton.

a. If Steve does not runs and Ayca reads poetry, then Muk Yan plays badminton and Steve runs.

b. Muk Yan plays badminton if and only if Ayca reads poetry, unless Steve does not run.

3. (70 points) Using a truth-table for the following symbolic argument:
i. Assess the validity of the argument.
ii. For each argument: conjoin the premises and imply the conclusion (i.e. create a conditional in which the antecedent is the conjunction of all the premises and the consequent is the conclusion). If your assessment is “valid” then this should result in a tautology, else not a tautology. (i.e. (P1 • P2 • .. • Pn) ? ?).

a. (35 points) 1. (p ? q) ? r
2. ~[(~p • ~q) ? r]
? ~r

b. (35 points) 1. ~[~(p • q) • ~(~p ? ~q)]
2. ~(~p ? ~q)
? q
4. (40 points) Using a truth-table for the given symbolic sentences determine if the two sentences are logically equivalent. Remember to place the biconditional symbol between the two sentences, and derive the truth conditions for the biconditional.

a. (20 points)
i. [(p • q) ? ~(p • q)]
ii. ~[ ~(p • q) • (p • q)]

b. (20 points)
i. p ? q
ii. ~(p • q) ? ~(p ? q)
5. (50 points total)

• For the following completed valid proof:
o Determine the inference rule used for each line of the proof.
o Determine the line number(s) from the premises or the derived statements that justify the inference rule cited.
a) (30 points)

1. A ? (B ? D)
2. ~C ? (D ? E)
3. A ? C
4. B
5. ~C
6. ? E ? D
7. ~A
8. B ? D
9. D ? E
10. B ? E
11. (B ? E) • (B ? D)
12. B ? B
13. E ? D

b) (20 points)

1. C ? D
2. ~D
3. ~C ? ~E
4. E ? F
? F
5. ~C
6. ~E
7. F

6. Bonus Section (20 points)

? For the following are valid arguments:
Derive a proof for each using only the basic nine.

a) (10 points)

1. A ? B
2. ~A
3. B ? (C ? Z)
? C ? (C • Z)
b) (10 points)

1. (B ? D) • (A ? (E ? F))
2. B ? A
3. ~D
4. ~F
? ~E ? R
use this webpage: http://www.cengagebrain.com/content/hurley34172_0840034172_02.01_chapter01.pdf

to work on the questions or any other source ..

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