Answer:
The given argument ∀x (S(x) ∧ (M(x) V D(x)) --> ¬ A(x)) ∧ S(Penelope) ∧ A(Penelope) -->¬ D(Penelope) is valid.
Explanation:
Solution:
Let us Consider following predicates:
M(x): x missed the class
D(x): x got a detention
S(x): x is a student in the class
A(x): x got an A
Now,
We Express the hypotheses and conclusion as:
The hypotheses: ∀x (S(x) ∧ (M(x) V D(x)) --> A(x)) and S(Penelope) and A(Penelope)
So,
The Conclusion: D(Penelope)
Thus,
The Argument:
∀x (S(x) ∧ (M(x) V D(x)) --> A(x)) ∧ S(Penelope) ∧ A(Penelope) -->D(Penelope)
Then,
The given argument is valid or correct and prove using the inference rule as follows:
Step Premises Reason (Rule used)
1. ∀x (S(x) ∧ (M(x) V D(x)) --> A(x)) Premise
2. S(Penelope) ∧ (M(Penelope)
V D(Penelope)) --> ¬ A(Penelope) Universal instantiation
3. S(Penelope) Premise
4. A(Penelope) Premise
5. ¬[S(Penelope) ∧ (M(Penelope)
V D(Penelope))] 2,4, Modus Tollens
6. ¬S(Penelope) V (¬M(Penelope)
∧ ¬D(Penelope)) De Morgan law
7.¬M(Penelope) ∧ ¬D(Penelope) 3,6,Disjunctive Syllogism
8¬D(Penelope) 7, Simplification
Therefore, the given argument ∀x (S(x) ∧ (M(x) V D(x)) --> ¬ A(x)) ∧ S(Penelope) ∧ A(Penelope) -->¬ D(Penelope) is valid.
