Example: Show that the premises "A student in this class has not read the book," and "Everyone in this class passed the first exam" imply the conclusion "Someone who passed the first exam has not read the book."

Solution: Let

C(x) be "x is in this class,"

B(x) be "x has read the book,"

P(x) be "x passed the first exam."

the premises are $\displaystyle \exists x(C(x) \wedge \neg B(x)) $ and $\displaystyle \forall x (C(x) \rightarrow P(x))$. The conclusion is $\displaystyle \exists x(P(x) \wedge \neg B(x))$. The steps can be used to establish the conclusion from the premises.

** Step________________________****Reason**

1. $\displaystyle \exists x(C(x) \wedge \neg B(x))$____________Premise

2. $\displaystyle C(a) \wedge \neg B(a)$________________Existential instantiation from (1)

3. $\displaystyle C(a)$________________________Simplification from (2)

4. $\displaystyle \forall x(C(x) \rightarrow P(x))$____________Premise

5. $\displaystyle C(a) \rightarrow P(a)$________________Universal Instantiation from (4)

6. P(a) ________________________Modus ponens from (3) and (5)

7. $\displaystyle \neg B(a)$_______________________Simplification from (2)

8. $\displaystyle P(a) \wedge \neg B(a)$________________Conjunction from (6) and (7)

9. $\displaystyle \exists x(P(x) \wedge \neg B(x))$____________ Existential generalization from (8)