Question

Use either indirect proof or conditional proof (or both) and the eighteen rules of inference to derive the conclusions of the symbolized argument below.

NOTE: Include the numbers of the first and last indented premises when listing the premises that you draw upon to support the premises of your proof that you identify with CP or IP.

1. H ⊃ (L ⊃ K)

L ⊃ (K ⊃ ∼L) / ∼H ∨ ∼L

Answer 1

Using both Conditional Proof and Indirect Proof, we have derived ∼H ∨ ∼L from premises 1 and 2.

Step-by-step explanation:

To prove ∼H ∨ ∼L from the premises using Conditional Proof (CP) and Indirect Proof (IP), let's break it down step by step:

• H ⊃ (L ⊃ K) (Premise)
• L ⊃ (K ⊃ ∼L) (Premise)

We want to derive ∼H ∨ ∼L using CP or IP.

Using Conditional Proof (CP):

Assume H (Assumption for CP)

• L ⊃ K (From 1, 3; Modus Ponens)
• K ⊃ ∼L (From 2, Modus Ponens)
• K (From 4, 3; Modus Ponens)
• ∼L (From 6, 5; Modus Ponens)
• ∼H ∨ ∼L (From 3-7; Disjunction Introduction)

Now, let's use Indirect Proof (IP) to derive ∼H ∨ ∼L:

Assume ∼(∼H ∨ ∼L) (Assumption for IP)

• ∼H ∧ L (Assume ∼(∼H ∨ ∼L) and use De Morgan's Law)
• L (From 10; Simplification)
• K (From 2, 11; Modus Ponens)
• H (Assume ∼(∼H ∨ ∼L) and use Reductio ad Absurdum)
• ∼H (From 9-13; Indirect Proof)
• ∼H ∨ ∼L (From 14; Addition)

Therefore, using both Conditional Proof and Indirect Proof, we have derived ∼H ∨ ∼L from premises 1 and 2.