Exercise 8: (6 POINTS) 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
Exercise 8: (6 POINTS) 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:
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Hypothesis 1: All lions are fierce. This can be represented as: ∀x(L(x) → F(x)), where L(x) represents 'x is a lion' and F(x) represents 'x is fierce'.
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Hypothesis 2: Some lions do not drink coffee. This can be represented as: ∃x(L(x) ∧ ¬C(x)), where C(x) represents 'x drinks coffee'.
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We need to prove: Some fierce creatures do not drink coffee. This can be represented as: ∃x(F(x) ∧ ¬C(x)).
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From Hypothesis 2, we know there exists a lion that does not drink coffee. Let's call this lion 'a'. So, we have: L(a) ∧ ¬C(a).
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From Hypothesis 1, we know that if 'a' is a lion, then 'a' is fierce. So, we have: L(a) → F(a).
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By Modus Ponens (if P then Q, P is true, therefore Q is true), from steps 4 and 5, we can infer that 'a' is fierce and 'a' does not drink coffee. So, we have: F(a) ∧ ¬C(a).
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Therefore, there exists a fierce creature that does not drink coffee, which is what we needed to prove. So, the hypotheses imply the conclusion.
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