Spaces:
Running
on
Zero
Running
on
Zero
GenAi Evaluation 7
#72
by
Ravka
- opened
😃: Use rules of inference to show that if ∀x(P (x) ∨ Q(x)),
∀x(¬Q(x) ∨ S(x)), ∀x(R(x) → ¬S(x)), and ∃x¬P (x)
are true, then ∃x¬R(x) is true.
🤖: To demonstrate using the rules of inference in logic, we need to apply the appropriate inferences from the given premises:
- Modus Ponens: From ∀x(P(x) ∨ Q(x)) and P(a), conclude ∀x(P(x) ∧ a).
- Modus Tollens: From ¬Q(a) and ∀x(P(x) ∨ Q(x)), conclude ¬P(a).
- Hypothetical Syllogism: From ∀x(P(x) ∨ Q(x)), ∀x(¬Q(x) ∨ R(x)), and R(a), conclude ∀x(P(x) ∨ R(x)).
- Negation Introversion: From ∀x(P(x) ∨ Q(x)), conclude ¬∀x(¬P(x) ∨ ¬Q(x)).
- Existence Introduction: From ∃x¬P(x), conclude ∃xP(x) or ∃xQ(x).
- Double Negation Elimination: From ¬¬P(x) and ¬¬Q(x), conclude P(x) or Q(x).
- Conditional Introduction: From ∀x(R(x) → ¬S(x)), conclude ∀x(R(x) → S(x)).
- De Morgan's Law: From ¬P(x) and ¬Q(x), conclude P(x) or Q(x).
- Identity Introduction: From P(a), conclude P(a) or not P(a).
- Substitution: Replace x with y in P(y), get Py.
Now let us apply these rules to ...