Use rules of inference to show that the hypotheses “All lions are fierce”and “Some lions do not drink coffee” imply the conclusion “Some fierce creatures do not drinkcoffee.”
Question
Use rules of inference to show that the hypotheses “All lions are fierce”and “Some lions do not drink coffee” imply the conclusion “Some fierce creatures do not drinkcoffee.”
Solution
Sure, here is the step by step solution:
- All lions are fierce. (Premise 1)
- Some lions do not drink coffee. (Premise 2)
We can translate these premises into logical form:
- ∀x(Lx → Fx) (For all x, if x is a lion, then x is fierce)
- ∃x(Lx ∧ ¬Cx) (There exists an x such that x is a lion and x does not drink coffee)
We want to prove: ∃x(Fx ∧ ¬Cx) (There exists an x such that x is fierce and x does not drink coffee)
Proof:
From premise 2, we know there exists an x such that x is a lion and x does not drink coffee. Let's call this x, a.
So, La ∧ ¬Ca (a is a lion and a does not drink coffee)
From premise 1, we know if a is a lion, then a is fierce. So, La → Fa
Since we know La, we can infer Fa (a is fierce) by Modus Ponens.
So now we have Fa and ¬Ca (a is fierce and a does not drink coffee)
Therefore, ∃x(Fx ∧ ¬Cx) (There exists an x such that x is fierce and x does not drink coffee)
So, the conclusion "Some fierce creatures do not drink coffee" is valid.
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