Consider the following list of premises:
a ^ :b ! c; :a ! :d; Show that g is a valid conclusion by constructing a step-by-step argument
To construct a step-by-step argument to show that g is a valid conclusion, we need to use the given premises and apply logical rules to derive the conclusion. Here's how you can proceed:
1. Apply the rule of simplification (∧-elimination) to the first premise "a ∧ :b ⇒ c" by splitting it into two separate statements:
- Statement 1: "a"
- Statement 2: ":b ⇒ c" (equivalent to ":c ⇒ b" by contrapositive)
2. Apply the rule of modus tollens to the second premise ":a ⇒ :d":
- By contrapositive, this can be rewritten as "d ⇒ a".
3. Apply the rule of hypothetical syllogism to combine Statement 1 from step 1 and the derived conclusion from step 2:
- "d ⇒ a"
- "a"
- Therefore, "d"
4. Apply the rule of disjunctive syllogism (also known as the rule of elimination) to Statement 2 from step 1 and "d" from step 3:
- ":c ⇒ b"
- "d"
- Therefore, b
5. Apply the rule of disjunctive syllogism again to Statement 2 from step 1 and "d" from step 3:
- ":c ⇒ b"
- "d"
- Therefore, :c
6. Combine the derived conclusions from steps 4 and 5:
- "b"
- ":c"
7. Apply the rule of conjunction (∧-introduction) to combine the results from step 6:
- "b ∧ :c"
Since we obtained "b ∧ :c" (which can be expressed as "g") as a valid conclusion from the given premises, we have successfully constructed a step-by-step argument to demonstrate that g is valid.