Question

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."

Short answer

The rule of interference used to show the given argument which implies the given conclusion can be shown.

Step-by-step solution

Find the form of argument

Let’s take p be the proposition “Randy works hard” and q be the “he is a dull boy” and r be the “Randy will not get the job”. The argument is in the form of abstract. If p then q; If q then r; If p then r. The argument is in the form of Hypothetical syllogism.

Construct the truth table

To construct the truth table for proving of validity,

p	q	r	First premise \(p \to q\)	Second premise \(q \to r\)	Conclusion \(p \to r\)
T	T	T	T	T	T
T	T	F	T	F	F
T	F	T	F	T	T
T	F	F	F	T	F
F	T	T	T	T	T
F	T	F	T	F	T
F	F	T	T	T	T
F	F	F	T	T	T

There are no cases where there are all true premises and also false conclusion. They have affirmed the argument validity and the argument will give conclusion.