Fitch Logic Proof

Question

I am stumped on this proof. I have attached a link with my proof so far.

I'm not sure how to derive a contradiction from WeakPref(a,b) on line 12.

Answer 1

Yes, that's the way to go!   You are almost spot on the ball.   What you need to do at line 12 is assume $\def\S{\mathsf{StrongPref}}\def\W{\mathsf{WeakPref}}\W(a,b)$, assume $\S(b,a)$, derive a contradiction, therefore introducing a negation so you may thereby use it in the disjunction elimination. $$\def\fitch#1#2{~~\begin{array}{|l} #1\\\hline #2\end{array}}\def\S{\mathsf{StrongPref}}\def\W{\mathsf{WeakPref}}\def\too{\leftrightarrow} \fitch{~5.~\S(a,b)}{~6.~\S (a,b)\too\neg \W(b,a)\\~7.~\neg\W(b,a)\\~8.~\W(a,b)\vee \W(b,a)\\\fitch{~9.~\W(b,a)}{10.~\bot\\11.~\neg \S(b,a)}\\{\color{red}{\fitch{12.~\W(a,b)}{\fitch{13.~\S(b,a)}{~~\vdots\\~~\vdots\\16.~\bot}\\17.~\lnot\S(b,a)}}}\\18.~\neg \S(b,a)}$$