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 isn’t truth-functional, so their form cannot be evaluated fully using only the tools of propositional logic. Because of this, propositional logic sometimes yields the wrong verdict when assessing the validity of an argument. For example:
All philosophers think. Socrates is a philosopher. Therefore, Socrates thinks.
Obviously, this argument is valid: there is no way for the conclusion to be false if both premises are true. However, neither the premises nor the conclusion contain any truth-functional connectives. So, if we symbolize the argument in propositional logic, we get:
| A |
| S |
| ∴ T |
which is an invalid argument form.
Thus, propositional logic sometimes implies that an argument is invalid when in fact it’s valid. (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.)
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.
Fortunately, 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 propositions involving possibility and necessity—i.e., claims about what could be true or what must be true. Then, in the following chapter, we’ll learn about another system called predicate logic. Predicate logic enables us to represent the logical structure of subject-predicate relations within simple propositions—propositions that contain no truth-functional connectives (like the ones in the argument about Socrates, above).
Both modal logic and predicate logic employ the same five truth-functional connectives used in propositional logic, so the effort you’ve invested learning about these connectives won’t go to waste! However, these more advanced systems add additional symbols, using more complex syntax to represent the additional logical structure. Modal logic, for example, adds just two new symbols: one to represent possibility, and one to represent necessity. But those modest additions add tremendous power to represent complex logical relations, as we’ll see in the next chapter.