# Show $\neg (W \to W), (W \leftrightarrow W) \wedge W, E \vee (W \to \neg (E \wedge W))$ are jointly inconsistent.

Working on P.D. Magnus. forallX: an Introduction to Formal Logic (pp. 182, exercise B. 9):

$$\def\fitch#1#2{\quad\begin{array}{|l}#1\\\hline#2\end{array}} \fitch{\neg (W \to W)\\(W \leftrightarrow W) \wedge W\\E \vee (W \to \neg (E \wedge W))}{ \fitch{W}{ W }\\ W \to W\\ \bot }$$

As I understand, I need to derive a contradiction assuming those sentences as premises. Is this the right approach?

• Correct........ – Mauro ALLEGRANZA Mar 23 at 17:09

You can also argue that the first premise is by itself a contradiction (because it cannot be satisfied by any interpretation of $$W$$), so therefore all three premises do form a jointly inconsistent set (since there is no interpretation of $$\{E,W\}$$ that will simultaneously satisfy all of them).