\begin{frame}[t]{At Most Two} \examplemodel \pause\bigskip \sentence{At most two $P$-values:} \pause \begin{talign} \model{\amodel} \malt[1]{\gray{\satisfiesblack}}{\satisfies} \formula{\forallst{x}{\forallst{y}{\forallst{z}{( \unpred{\mediumblue{P}}{x} \logandinf \unpred{\mediumblue{P}}{y} \logandinf \unpred{\mediumblue{P}}{z} \logimpinf \equalto{x}{y} \logorinf \equalto{x}{z} \logorinf \equalto{y}{z} )}}}} \end{talign} \updatepause \sentence{Not at least three $P$-values:} \begin{talign} \model{\amodel} \malt[1]{\gray{\satisfiesblack}}{\satisfies} \formula{\lognot{\existsst{x}{\existsst{y}{\existsst{z}{( \unpred{\mediumblue{P}}{x} \logandinf \unpred{\mediumblue{P}}{y} \logandinf \unpred{\mediumblue{P}}{z} \logandinf \notequalto{x}{y} \logandinf \notequalto{x}{z} \logandinf \notequalto{y}{z} )}}}}} \end{talign} \updatepause Both are equivalent because: \begin{scenter} \sentence{at most two} ($\le 2$) $\iff$ \sentence{not: at least three} (not $\ge 3$) \end{scenter} \end{frame}