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If my plumbing plans do not meet the construction code, then I cannot build my house.

If I hire a licensed contractor, then my plumbing plans will not meet the construction code.

I hire a licensed contractor. Therefore I can build my house.

Prove by rule of inference is the argument valid or invalid :

below is my attempt

Premise 1: $\neg P \rightarrow \neg Q$
Premise 2: $R \rightarrow \neg P$
Premise 3: $R$
Conclusion: $Q$

and to further prove the validity i used this method :

Premise 4 : $P \rightarrow Q$ (Inverse of premise 1)
Premise 5 : $\neg P$ (Modus ponens of premise 2 + 3)
Premise 6 : $\ P$ (negation of premise 5)
$\ Q$ (Modus ponens of premise 4 and premise 6)

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  • $\begingroup$ Premise 4 doesn't follow from Premise 1. Inverse are not logically equivalent. Hint: Instead, of inverse, take the contrapositive , q implies p. Use rules of inference to show that not Q is true. Conclude that the original argument is invalid. Alternatively, assume Q is true, then not Q is false. Can you derive a contradiction? $\endgroup$ Commented Feb 17, 2023 at 0:12
  • $\begingroup$ Thank you do much , i just realised they were not logically equivalent !!! $\endgroup$
    – Learner
    Commented Feb 17, 2023 at 0:54

1 Answer 1

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Welcome! it's a good attempt, you almost have it.

Consider,

Premise 1: $\neg P \rightarrow \neg Q$
Premise 2: $R \rightarrow \neg P$
Premise 3: $R$

By premise (2) and (3) + Modus Ponens we have: $ \neg P$

So,

premise 4: $ \neg P$

by premise (1) and (4) + Modus Ponens

Conclusion: $\neg Q$

What can you say, about the truth of Q, given the above argument?

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    $\begingroup$ thank you so much for the guidance! I have figured out by myself the final solution to the problem $\endgroup$
    – Learner
    Commented Feb 18, 2023 at 2:20

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