Answer
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:
Work Step by Step
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.
