Question: Use the quantifier negation rule together with the eighteen rules of inference to derive the conclusions of the following symbolized arguments. Do not use either

Use the quantifier negation rule together with the eighteen rules of inference to derive the conclusions of the following symbolized arguments. Do not use either conditional proof or indirect proof.

★(1) 1. (x)Ax ⊃ (∃x)Bx 2. (x)∼Bx / (∃x)∼Ax

(2) 1. (∃x)∼Ax ⋁ (∃x)∼Bx 2. (x)Bx / ∼(x)Ax

(3) 1. ∼(∃x)Ax / (x)(Ax ⊃ Bx)

★(4) 1. (∃x)Ax ⋁ (∃x)(Bx Cx)

2. ∼(∃x)Bx / (∃x)Ax

(5) 1. (x)(Ax Bx) ⋁ (x)(Cx Dx)

2. ∼(x)Dx / (x)Bx

(6) 1. (∃x)∼Ax ⊃ (x)(Bx ⊃ Cx)

2. ∼(x)(Ax ⋁ Cx) / ∼(x)Bx

★(7) 1. (x)(Ax ⊃ Bx)

2. ∼(x)Cx ⋁ (x)Ax 3. ∼(x)Bx / (∃x)∼Cx

(8) 1. (x)Ax ⊃ (∃x)∼Bx 2. ∼(x)Bx ⊃ (∃x)∼Cx / (x)Cx ⊃ (∃x)∼Ax

(9) 1. (∃x)(Ax ⋁ Bx) ⊃ (x)Cx 2. (∃x)∼Cx / ∼(∃x)Ax

★(10) 1. ∼(∃x)(Ax ∼Bx)

2. ∼(∃x)(Bx ∼Cx) / (x)(Ax ⊃ Cx)

(11) 1. ∼(∃x)(Ax ∼Bx)

2. ∼(∃x)(Ax ∼Cx) / (x)[Ax ⊃ (Bx Cx)]

(12) 1. (x)[(Ax Bx) ⊃ Cx]

2. ∼(x)(Ax ⊃ Cx) / ∼(x)Bx

★(13) 1. (x)(Ax ∼Bx) ⊃ (∃x)Cx 2. ∼(∃x)(Cx ⋁ Bx) / ∼(x)Ax

(14) 1. (∃x)∼Ax ⊃ (x)∼Bx 2. (∃x)∼Ax ⊃ (∃x)Bx 3. (x)(Ax ⊃ Cx) / (x)Cx (15) 1. ∼(∃x)(Ax ⋁ Bx)
2. (∃x)Cx ⊃ (∃x)Ax 3. (∃x)Dx ⊃ (∃x)Bx / ∼(∃x)(Cx ⋁ Dx)
★(16) 1. (∃x)(Ax Bx) ⊃ (x)(Cx Dx)
2. (x)[(Ax ⋁ Ex) (Bx ⋁ Fx)]
3. ∼(x)Dx / (x)(Ex ⋁ Fx)
(17) 1. (∃x)(Ax Bx) ⋁ (∃x)(Cx ⋁ Dx)
2. (∃x)(Ax ⋁ Cx) ⊃ (x)Ex 3. ∼En / (∃x)Dx (18) 1. (∃x)Ax ⊃ [(∃x)Bx ⋁ (x)Cx]
2. (∃x)(Ax ∼Cx)
3. ∼(x)Cx ⊃ [(x)Fx ⊃ (x)∼Bx] / (∃x)∼Fx ★(19) 1. (∃x)(Ax Bx) ⊃ (x)(Bx ⊃ Cx)
2. Bn ∼Cn / ∼(x)Ax (20) 1. (∃x)Ax ⊃ ∼(∃x)(Bx Ax)
2. ∼(x)Bx ⊃ ∼(∃x)(Ex ∼Bx)
3. An / ∼(x)Ex

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