Chapter 1 - Section 1.8 - Proof Methods and Strategy - Supplementary Exercises - Page 112: 8

The provided statements are inconsistent as after performing some rules of inference like Modes Ponens the statement "Sergei gets a higher Salary" becomes both true and false. So the statements are inconsistent

We can translate the given statements using the specified interpretations. The statements are inconsistent if we obtain that one variable in both true and false (which of course cannot be true). Let us assume: $p$= “Sergei takes the job offer” $q$= “Sergei gets a signing bonus” $r$= “Sergei gets a higher salary” STEP REASON $p \rightarrow q$ premise $p \rightarrow r$ premise $q \rightarrow \neg r$ premise $p$ premise $q$ Modus ponens from $(1)$ and $(4)$ $r $ modus ponens from $(2)$ and $(4)$ $\neg r$ modus ponens from $(3)$ and $(5)$ We obtained that $r$ has to be true and $\neg r$ has to be true. However $\neg r$ is true when $r $ is false. Thus we obtained that $r$ is both true and false, which is impossible and thus the statements are inconsistent.