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Analyzing Arguments with Truth Tables

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QUESTION 1: 4-1 Analyzing Arguments with Truth Tables

  1. (M V G) ⸫ H
HMG MVGH
TTTTTTT
TTFTTFT
TFTFTTT
TFFFFFT
FTTTTTF
FTFTTFF
FFTFTTF
FFFFFFF

 

Concerning question 1 and drawing from the table, the argument is invalid. In row 4 is an entirely false premise, whereas its conclusion is true.

Below is the table abbreviated from question 1.

H M GM V GH
F T TT T TF
F
F
F

 

 

 

QUESTION 2: 5-1 Journal: Direct Proofs in Natural Deduction

Choosing 1 and 4;

  1. ~M, (~M • ~N) → (Q → P), P → R, ~N, therefore, Q → R
  2. (S v U) • ~U, S → [T • (F v G)], [T v (J • P)] → (~B • E), therefore, S • ~B

Choosing Argument 1

  1. ~M
  2. (~M ~N) → (Q → P)
  3. P → R
  4. ~N
  5. (~M ~N) 1 and 4 Conjunction
  6. (Q → P) 2 and 5 Modus Ponens
  7. (Q → R) 6 and 3 HS

\ Conclusion – Q → R is a derivative of premises

M: I am attending the summit

N: Prince is attending the summit

Q: Lilian will attend the summit

P: Frank will attend the summit

R: Catherine will attend the summit

Argument 1 filled

I am not attending the summit. If I am not attending the summit and Prince is not attending the summit, then if Lilian is attending the summit, Frank will attend the summit. If Frank attends the summit, then Catherine will attend the summit. Prince is not attending the summit. I am not attending the summit, and Prince is not attending the summit. If Lilian attends the summit, then Frank will attend the summit. If Lilian attends the summit, then Catherine will attend the summit.

Choosing Argument 4

  1. (S v U) • ~U
  2. S → [T • (F v G)]
  3. [T v (J • P)] → (~B • E)
  4. ~U 1 Simplification Rule
  5. (S v U) 1 Simplification Rule
  6. S
  7. T • (F v G) 2 and 6 Simplification Rule
  8. T 7 Simplification Rule
  9. T v (J • P) 8 Addition Rule
  10. (~B • E) 3 and 9 Modus ponens
  11. ~B 10 Simplification Rule
  12. S • ~B 6 and 11 Conjunction

\ conclusion S • ~B is derived from the premises

S: My cow’s tag is Beto.

U: My cow’s tag is Trevin.

T: My dove’s tag is Joy.

F: My ugly beast’s tag is Trevin.

G: My chick’s tag is Kwikwi.

J: My worm’s tag is Canker.

P: My tortoise’s tag is Prince.

B: My python’s tag is Hearty.

E: My bear’s tag is Clarice.

 

Argument 4 – Filled Out

My cow’s tag is Beto or Trevin, but not Trevin. If my cow’s tag is Beto, then my dove’s tag is Joy, and my python’s tag is Hearty, or my chick’s tag is Kwikwi. If my dove’s tag is Joy or my worm’s tag is Canker, and my tortoise’s tag is Prince. My cow’s tag is not Trevin. My cow’s tag is Beto, or my cow’s tag is Trevin. My cow’s tag is Beto. My dove’s tag is Joy, and my ugly beast’s tag is Trevin, or my chick’s tag is Kwikwi. My dove’s tag is Joy. My dove’s tag is Joy, or my worm’s tag is Canker, and my tortoise’s tag is Prince. My python’s tag is not Hearty, and my bear’s tag is Clarice. My python’s tag is not Hearty. Therefore, my cow’s tag is Beto, and my python’s tag is not Hearty.

 

 

Question 3: 6-1 Journal: Indirect Proofs in Natural Deduction

  1. (G • P) → K, E → Z, ~P → ~ Z, G → (E v L), therefore, (G • ~L) → K

 

Conditional proof

1(G•P)àK
2EàZ
3~Pà~Z
4GàE v L
(G•~L)àK
5G•~LThe assumption for Conditional Proof
6G5Simplification
7~L5Simplification
8E v L4, 6Modus Ponens
9E7, 8Disjunctive Syllogism
10Z2, 9Modus Ponens
11~~Z10Double Negation
12~~P3, 11Modus Tollens
13P14Double Negation
14G•P6, 13Conjunction
15K1, 14Modus Ponens
16(G•~L)àK5-15Conditional Proof

 

If RAA was used for the proof, now CP will be used to replace it; if CP had been used earlier, now RAA will be used. Then, construct an alternative proof.

Reductio Ad Absurdum (Contradiction)

1(G•P)àK
2EàZ
3~Pà~Z
4GàE v L
(G•~L)àK
5~[(G•~L)àK]
6~[~(G•~L) v K]
7~~(G•~L) •~K
8(G•~L) •~K
9~K
10G•~L
11G
12~L
13E v L
14E
15Z
16~~Z
17~~P
18P
19G•P
20K
21K•~K
22

 

 

 

Question 4: 7-1 Journal: Basic Predicate Logic

Problem 1.

(x)(Fx → Sx), (x)(Rx → Sx) ∴ (x)(Fx → Rx)

  1. (x)(Fx → Sx) (premise)
  2. (x)(Rx → Sx) (premise)

∴ (x)(Fx → Rx) (conclusion)

 

  1. Fa (CP)
  2. Fa → Sa 1, UO
  3. Ra → Sa 2, UO
  4. Sa →Ra 5, CN
  5. Fa →Ra 4, 6 CH
  6. (x)(Fx → Rx) 7, UI

 

Proof and symbolization

The Peloponnesian War is a familiar thing to every historian. Familiarization with the Peloponnesian War can result in dominance in a conversation. Hence, every historian can gain dominance over a conversation.

(x)(Hx → Fx), (x)(Fx → Dx) |- (x)(Hx → Dx)

  1. (x)(Hx → Fx) P
  2. (x)(Fx → Dx) P – want (x)(Hx → Dx)
  3. Ha* → Fa* 1 UO
  4. Fa* → Da* 2 UO
  5. Ha* → Da* 3,4 CH
  6. (x)(Hx → Dx)

 

 

 

References

Howard-Snyder, D., Howard-Snyder, F., & Wasserman, R. (2020). The power of logic.

 

 

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