Based on two assumptions, decide whether the logical conclusion is correct:
Assumptions:
- George had eggs or salami for breakfast.
- George had salami for breakfast.
Conclusion: George did not have eggs for breakfast.
Oliver solved the task in the following steps:
(1) George wrote down each assumption and conclusion using two elementary statements, which he labelled $E$ and $S$:
$E$: George had eggs for breakfast.
$S$: George had salami for breakfast.
$E\lor S$: George had eggs or salami for breakfast.
$\neg E$: George did not have eggs for breakfast.
(2) He compiled a truth table, in which he distinguished the assumptions (green) from the conclusion (blue): $$\begin{array}{|c|c|c|c|}\hline E &\color{green}S &\color{green}E\lor S &\color{blue}\neg E\cr \hline 1 &\color{green}1 &\color{green}1 &\color{blue}0\cr\hline 1 &\color{green}0 &\color{green}1 &\color{blue}0\cr\hline 0 &\color{green}1 &\color{green}1 &\color{blue}1\cr\hline \qquad0\qquad &\color{green}\qquad0\qquad &\color{green}\qquad0\qquad &\color{blue}\qquad1\qquad\cr\hline \end{array}$$ (3) He marked (red) the rows in the table where both assumptions are true: $$\begin{array}{|c|c|c|c|}\hline E &\color{green}S &\color{green}E\lor S &\color{blue}\neg E\cr \hline \color{red}1 &\color{red}1 &\color{red}1 &\color{red}0\cr\hline 1 &\color{green}0 &\color{green}1 &\color{blue}0\cr\hline \color{red}0 &\color{red}1 &\color{red}1 &\color{red}1\cr\hline \qquad0\qquad &\color{green}\qquad0\qquad &\color{green}\qquad0\qquad &\color{blue}\qquad1\qquad\cr\hline \end{array}$$ (4) He evaluated the truth table, concluding that if some of the conclusions is true for all the assumptions satisfied, then overall conclusion is logically correct.
Is Oliver's solution correct? If not, identify where Oliver made a mistake in the procedure.
Oliver's solution is correct.
The mistake is in step (1). Oliver incorrectly rewrote one of the given assumptions/conclusions using symbols for statements $E$ and $S$.
The mistake is in step (2). Oliver incorrectly determined the truth value for one assumption/conclusion in the truth table.
The mistake is in step (4). Oliver made an incorrect judgment from the correct truth table.
Oliver evaluated the truth table incorrectly. The overall conclusion is true only if, in every row of the table where all assumptions are satisfied, the conclusion is also satisfied. However, this does not hold true in the first row of the table, making his judgment logically incorrect.