## Thinking Mathematically (6th Edition)

Use letters to represent both simple statements in the argument. p: You only spoke when spoken to, and I only speak when spoken to. q: Nobody would ever say anything. Express the premises and the conclusion symbolically. $p\to q$ If you only spoke when spoken to, and I only speak when spoken to, then nobody would ever say anything. $\frac{\tilde{\ }q}{\therefore \tilde{\ }p}$$\frac{\text{Everyone would say something}}{\text{You wouldn }\!\!'\!\!\text{ t speak when spoken to, and I always speak}}$ Thus, the symbolic form of the provided argument is \begin{align} & p\to q \\ & \frac{\tilde{\ }q}{\therefore \tilde{\ }p} \\ \end{align} Write a symbolic statement of the form $\left[ \left( \text{premise 1} \right)\wedge \left( \text{premise 2} \right) \right]\to \text{conclusion}$. The symbolic statement is$\left[ \left( p\to q \right)\wedge \tilde{\ }q \right]\to \,\tilde{\ }p$. Construct a truth table for the conditional statement $\left[ \left( p\to q \right)\wedge \tilde{\ }q \right]\to \,\tilde{\ }p$ as shown below: It can be seen from the truth table that the symbolic form $\left[ \left( p\to q \right)\wedge \tilde{\ }q \right]\to \,\tilde{\ }p$ is always true. Thus, the valid argument is people sometimes speak without being spoken to. Hence, the valid argument is,people sometimes speak without being spoken to.