Question

Use rules of implication, rules of replacement, conditional proofs and/or indirect proofs, to prove the arguments (or tautologies) valid. Be sure to justify every line you provide with both line numbers and the applicable rule.

1. A ∨ B
2. C ⊃ D
3. A ⊃ C
4. ~ D / B
a. B
b. A
c. ~B
d. C

Answer 1

The argument is proven valid by using Modus Tollens on premises 3 and 4 to infer ~A, which, when combined with premise 1 (A ∨ B) using Disjunctive Syllogism, leads to the conclusion B.

Step-by-step explanation:

To prove the argument valid using rules of implication, rules of replacement, conditional proofs, and indirect proofs, let's follow the given premises to reach the conclusion B:

1. A ∨ B (Given)
2. C ⊃ D (Given)
3. A ⊃ C (Given)
4. ~ D (Given)
5. ~ A (From 4 and 3, using Modus Tollens)
6. B (From 5 and 1, using Disjunctive Syllogism)

The given argument utilizes the structure of logical deduction to prove the validity of the conclusion. The presented syllogisms and inferences ensure that if the premises are true, the conclusion necessarily follows.