Recall that propositional logic enables us to determine whether an argument is valid or invalid, provided the argument’s form (logical structure) can be represented using truth-functional connectives. However, some arguments rely on additional logical structure that is not truth-functional, so their form cannot be evaluated fully using only the tools of propositional logic. Similarly, propositional logic enables us to examine logical properties of individual propositions (whether a proposition is a tautology, contradiction, or contingency) and logical relations between propositions (entailment, equivalence, and consistency); but this only works when those properties and relations depend on truth-functional connectives.
Propositions often contain logical structure that is not truth-functional, but propositional logic is unable to represent that additional structure. Because of this, propositional logic sometimes yields incorrect results: it may imply that an argument is invalid when in fact the argument is valid, for instance. (Fortunately, the converse is not true. If an argument is valid in propositional logic, then it is valid no matter what additional structure the argument may have.) These limitations of propositional logic can be overcome by employing more advanced logical systems. In the next chapter, we’ll examine one such system: modal logic, which is designed to deal with the logical structure of claims involving possibility and necessity. Then, in the following chapter, we’ll learn about another system called predicate logic, which can be used to represent the logical structure of subject-predicate relations within simple propositions (propositions that contain no truth-functional connectives).