# Use rules of inference to show that the hypotheses “Randy works hard,” “If Randy works hard, then he is a dull boy,” and “If Randy is a dull boy, then he will not get the job” imply the conclusion “Randy will not get the job.”

To solve exercises where you have to apply rules of inference, the first step is to know the rules of inference.

Once we know the rules, we can then start to apply them. So, let’s start solving the exercise.

“Randy works hard,” “If Randy works hard, then he is a dull boy,” and “If Randy is a dull boy, then he will not get the job” imply the conclusion “Randy will not get the job.”

Let p=” Randy works hard”, q=” Randy is a dull boy” and r=” Randy won’t get the job”.

Now, let’s write the hypothesis using the propositional variables p, q, and r.

(1) p          Premise

(2) p->q    Premise

(3) q->r    Premise

And the conclusion will be r.

At this point, we must start applying the rules of inference from the table above in such a way that after one (or several) rules, we can get the conclusion r. This is also known as constructing an argument that shows that the premises lead to the desired conclusion (r in this case).

If we apply Hypothetical Syllogism to (2) and (3), we can conclude that p->r. We write that as follows:

(4) p->r          Hypothetical Syllogism using (2) and (3),

(5) r                Modus ponens using (1) and (4)

And with the previous steps, we constructed a valid argument that shows that the premises p, p->q, and q->r lead to the conclusion r.

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