We may construct a proof to demonstrate its validity, relying simply on the rules from chapter 8. Moreover, as before, equivalence rules may still be applied to parts of a statement. In this section we will consider implicational rules as they are applied to quantified statements. Thus we will learn rules regarding the elimination of quantifiers, as well as rules regarding the creation of quantified statements. One of the first arguments we considered in this chapter was about Socrates.
All humans are mortal.
Socrates is human. Therefore, Socrates is mortal. While we may be tempted to apply Modus Ponens to lines 1 and 2, you'll note that it does not fit the proper form. Modus Ponens allows us to infer the consequent of a conditional when we know that the conditional is true and the antecedent is true. Line 1, however, is not a conditional.