Exam 11: Predicate Logic

In the following proof, which justification is correct for line 3? \begin{array}{llcc} \text { (1) (x) \mathrm{M} x } && \text {Premise } \\ \text { (2) \( \mathrm{Hi} \) } & \text { \(/ \mathrm{Hi} \cdot \mathrm{Mi}\) }& \text {Premise/Conclusion }\\ \text {(3) \( \mathrm{Mi} \) } &\\\end{array}

In the following proof, which justification is correct for line 3? \begin{array}{llcc} \text { (1) \mathrm{Ca} \supset(x)(\mathrm{D} x \supset \mathrm{E} x) } && \text { Premise } \\ \text { (2) \( \mathrm{Da} \cdot \mathrm{Ca} \) } & \text {\(/ \mathrm{Ea}\) } & \text {\quad\quad \quad\quad \quad\quad Premise/Conclusion } \\ \text {(3) \( \mathrm{Ca} \) } &\\\end{array}

In the following proof, which justification is correct for line 11? \begin{array}{llcc} \text { (1) (x)(\mathrm{C} x \supset \mathrm{D} x) } &&\text { Premise } \\ \text { (2) \( (\exists x) \mathrm{C} x \)} &&\text { Premise } \\ \text { (3) \( (x)(\sim \mathrm{E} x \supset \sim \mathrm{D} x) \) } &\text { \(E(\exists x) \mathrm{E} x\) } &\text { Premise/Conclusion } \\ \text { (4) \( \mathrm{Ca} \) } &&\text { \( 2 \mathrm{EI} \)} \\ \text {(5) \( \mathrm{Ca} \Rightarrow \mathrm{Da} \) } &&\text { 1 UI } \\ \text { (6) \( \mathrm{Da} \) } &&\text { 4, \( 5 \mathrm{MP} \)} \\\text {(7) \( \sim \mathrm{Ea} \supset \sim \mathrm{Da} \) } &&\text { \( 3 \mathrm{UI} \) } \\ \text { (8) \( -\mathrm{Da} \) } &\text {\( 6 \mathrm{DN} \) } & \\ \text { (9) \( \sim \mathrm{Ea} \) } &&\text { \( 7,8 \mathrm{MT} \) } \\ \text { (10) \( \mathrm{Ea} \)} &&\text { \( 9 \mathrm{DN} \) } \\ \text { (11) \( (\exists x) E x \)} &\\ \text { } &\\\end{array}

The following inference is an application of which rule? ( $\exists$ x)(y)Bxy.Therefore, (y)Bay.

Identify the type of statement represented by the following. ( $\exists$ x)Fx

In the following proof, which justification is correct for line 4? \begin{array}{llcc} \text { (1) (x)[(\mathrm{C} x \cdot \mathrm{D} x) \supset \sim \mathrm{E} x] } && \text {Premise } \\ \text { (2) \( \mathrm{Ea} \) } & \text {\(I \sim \mathrm{Ca} v \sim \mathrm{Da}\) } & \text { Premise/Conclusion } \\ \text { (3) \( (\mathrm{Ca} \cdot \mathrm{Da}) \supset \sim \mathrm{Ea} \)} && \text {\(1 \mathrm{UI}\) } \\ \text { (4) \( \sim \mathrm{Ea} \) } &\\\end{array}

Symbolize the following statement form. Some A are not B.