Propositions may 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 the wrong verdict when assessing the validity of an argument. For example, consider the following argument:
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, if we symbolize the argument in propositional logic, we get:
| A |
| S |
| ∴ T |
which is an invalid argument form. So, propositional logic sometimes implies that an argument is invalid when in fact the argument is 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.) To accurately represent the structure of arguments like the one above, we’ll turn to a more powerful logical system called predicate logic.
Recall that a proposition is the meaning of a declarative sentence. A declarative sentence has a subject (the thing the sentence is talking about) and a predicate (the claim that is made about the subject). In the sentence Socrates lived in Athens, for example, ‘Socrates’ is the subject and ‘lived in Athens’ is the predicate.
In predicate logic (also called quantificational logic), a proposition’s predicate is represented by a capital letter called a predicate letter. If the subject of the sentence is a proper noun (i.e. a name like ‘Socrates’ or ‘Athens’), it is represented by a lowercase letter written to the right of the predicate letter. For example, the proposition “Socrates lived in Athens” can be represented ‘Ls’, where ‘s’ stands for Socrates and ‘L’ is the predicate ‘lived in Athens.’ Similarly, the proposition ‘Athens was the birthplace of a great philosopher’ could be represented ‘Ba,’ where ‘a’ stands for Athens and ‘B’ stands for the predicate ‘was the birthplace of a great philosopher.’
Lower-case letters ‘a’ through ‘w’ are called constants. They are used to represent subjects that refer to one specific thing by name. The letters ‘x’, ‘y’, and ‘z’ are reserved for a different purpose, which will be explained on the next page.
Only letters ‘a’ through ‘w’ can be used to represent names. For example, the letter ‘a’ could stand for Athens or Aristotle. But the letter ‘z’ should not be used to stand for Zeno or any other named individuals. If your name comes after ‘w,’ you’re out of luck. (Just substitute a different letter instead. Or file for a name change.)
A constant (lowercase letter ‘a’ through ‘w’) can only represent one specific thing by name. For example, ‘p’ could stand for Plato, but it could not stand for the subjects of either of the following sentences:
Whenever the subject of a sentence is a named individual, predicate logic symbolizations are similar to those of propositional logic, and we can use truth-functional connectives in the same way that we did before. For example, the sentence If Socrates was great, then Athens was the birthplace of a great philosopher could be symbolized:
Below are a few more examples of symbolizations involving predicate letters and constants.
Socrates is a philosopher is symbolized: Ps
Socrates thinks is symbolized: Ts
Socrates is a philosopher if and only if he thinks is symbolized: (Ps ≡ Ts)
If Socrates raises riddles and Plato ponders profound problems, then Aristotle studies sublime subjects is symbolized: ((Rs • Pp) ⊃ Sa)