Analyzing Arguments with Truth Tables

QUESTION 1: 4-1 Analyzing Arguments with Truth Tables

1. (M V G) ⸫ H

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.

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

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)

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)

3. Fa (CP)
4. Fa → Sa 1, UO
5. Ra → Sa 2, UO
6. Sa →Ra 5, CN
7. Fa →Ra 4, 6 CH
8. (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.