Section 2.3: Problem 1 Solution

Working problems is a crucial part of learning mathematics. No one can learn... merely by poring over the definitions, theorems, and examples that are worked out in the text. One must work part of it out for oneself. To provide that opportunity is the purpose of the exercises.
James R. Munkres
Show that for a proper initial segment $\alpha'$ of a wff $\alpha$ , we have $K(\alpha')<1$ .
Here, for any wff $\delta$ , by $\delta'$ we denote an initial proper segment of $\delta$ .
For an atomic formula $\alpha'$ consists of an $n$ -place predicate symbol ($1-n$ ) followed by at most $n-1$ complete terms () and, possibly, an initial proper segment of the subsequent term (). Overall, . Then, by induction.
For $\alpha=(\neg\beta)$ , $\alpha'$ is one of the following, $($ ($K=-1$ ), $(\neg$ ($K=-1$ ), $(\neg\beta'$ ($K\le-1$ ), $(\neg\beta$ ($K=0$ ).
For $\alpha=(\beta\rightarrow\gamma)$ , $\alpha'$ is one of the following, $($ ($K=-1$ ), $(\beta'$ ($K\le-1$ ), $(\beta$ ($K=0$ ), $(\beta\rightarrow$ ($K=-1$ ), $(\beta\rightarrow\gamma'$ ($K\le-1$ ), $(\beta\rightarrow\gamma$ ($K=0$ ).
For $\alpha=\forall v_{i}\beta$ , $\alpha'$ is one of the following, $\forall$ ($K=-1$ ), $\forall v_{i}$ ($K=0$ ), $\forall v_{i}\beta'$ ($K\le0$ ).