Section 2.4: Problem 3 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

(a) Let $\mathfrak{A}$ be a structure and let $s:V\rightarrow|\mathfrak{A}|$ . Define a truth assignment $v$ on the set of prime formulas by $\begin{eqnarray*} v(\alpha)=T & \mbox{iff} & \vDash_{\mathfrak{A}}\alpha[s]. \end{eqnarray*}$

Show that for any formula (prime or not), $\begin{eqnarray*} \overline{v}(\alpha)=T & \mbox{iff} & \vDash_{\mathfrak{A}}\alpha[s]. \end{eqnarray*}$

Remark: This result reflects the fact that $\neg$ and $\rightarrow$ were treated in Chapter 2 the same way as in Chapter 1.

(b) Conclude that if $\Gamma$ tautologically implies $\phi$ , then $\Gamma$ logically implies $\phi$ .

(a) We show the statement by induction. For any prime formula $\alpha$ considered as a sentence symbol, $\overline{v}(\alpha)=v(\alpha)=T$ iff $\vDash_{\mathfrak{A}}\alpha[s]$ . Now, $\overline{v}(\neg\alpha)=T$ iff $\overline{v}(\alpha)=F$ iff (by induction) $\not\vDash_{\mathfrak{A}}\alpha[s]$ iff $\vDash_{\mathfrak{A}}\neg\alpha[s]$ , and $\overline{v}(\alpha\rightarrow\beta)=T$ iff $\overline{v}(\alpha)=F$ or $\overline{v}(\beta)=T$ iff (by induction) $\not\vDash_{\mathfrak{A}}\alpha[s]$ or $\vDash_{\mathfrak{A}}\beta[s]$ iff $\vDash_{\mathfrak{A}}\alpha\rightarrow\beta[s]$ . We conclude, that for every formula $\alpha$ , $\overline{v}(\alpha)=T$ iff $\vDash_{\mathfrak{A}}\alpha[s]$ .

(b) If $\Gamma$ tautologically implies $\phi$ then for every structure $\mathfrak{A}$ and $s:V\rightarrow|\mathfrak{A}|$ , if for every $\alpha\in\Gamma$ , $\overline{v}(\alpha)=T$ , then $\overline{v}(\phi)=T$ . Therefore, for every structure $\mathfrak{A}$ and $s:V\rightarrow|\mathfrak{A}|$ , if for every $\alpha\in\Gamma$ , $\vDash_{\mathfrak{A}}\alpha[s]$ , then $\vDash_{\mathfrak{A}}\phi[s]$ . Hence, $\Gamma\vDash\phi$ .

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Section 2.4: Problem 3 Solution

Section 2.4: Problem 3 Solution