Chapter 7: Introduction to Quantificational Logic (QL)

In Propositional Logic, the smallest unit of language  is a declarative sentence which does not contain the operators "and," "or," "not," "if...then...," and "....if and only if..."  We called those sentences "atomic."   It's easy to see that some of the valid arguments which we can formulate in English are clearly invalid according to Propositional Logic, because they consist solely of atomic sentences.    Consider the following argument, for example:

There are no connectives in the propositions which make up this argument. So our only choice is to translate each proposition as an atomic proposition. The argument is:

But clearly this argument is invalid. There's a row of the truth-table where the premises are true and the conclusion false. This should conflict with your intution about the validity of the argument. If all basketball players can dribble and Tyler is a member of the class of basketball players, he should also count among the dribblers. So what went wrong?

The problem is that Propositional Logic only gets at the propositional structure, the relation among propositions expressed in terms of the propositional operators. There are no "if"s "and"s "or"s or "but"s in this argument, hence no analyzable structure. But there is some structure, and we recognize that structure when we intuitively (and correctly) think that it is a valid argument. By the end of this chapter, you'll be able to translate this argument into a new logical language, Quantificational Logic, which is an extension of Propositional Logic. Even better, you'll be able to show that this argument is valid.