Symbolizing Complex Categorical Propositions
When symbolizing a sentence in predicate logic, it is often helpful to begin by paraphrasing the sentence as a categorical proposition, even if the sentence contains additional logical structure. For example, consider the sentence Any student who studies either logic or philosophy learns to think carefully and reason cogently. This sentence contains the truth-functional connectives ‘or’ and ‘and,’ but the sentence as a whole can be paraphrased as follows:
Everything in the category of students who study either logic or philosophy is in the category of things that learn to think carefully and reason cogently.
This is a categorical proposition, and it relates the following two categories:
- students who study either logic or philosophy
- things that learn to think carefully and reason cogently
The first category can be represented as a logical combination of several different categories: the category of students, the category of things that study logic, and the category of things that study philosophy. Let’s represent these simpler categories using the following predicate letters:
Sx = x is a student
Lx = x studies logic
Px = x studies philosophy
Putting those together with truth-functional connectives, we can represent the category students who study either logic or philosophy as follows:
- (Sx • (Lx ∨ Px)) = x is a student and either x studies logic or x studies philosophy
The formulas ‘(Sx • (Lx ∨ Px))’ and ‘(Tx • Rx)’ are not WFFs, because they contain unbound variables. However, they can be used as parts of a WFF when placed into the scope of a matching quantifier, as shown below.
Similarly, the category things that learn to think carefully and reason cogently can be represented using ‘Tx’ and ‘Rx,’ as follows:
- (Tx • Rx) = x learns to think carefully and x learns to reason cogently
Now that we see how to represent the logical structure of the two categories used in the categorical proposition above, we can symbolize the whole proposition as follows:
(∀x)((Sx • (Lx ∨ Px)) ⊃ (Tx • Rx))
In the well-formed formula above, the antecedent of the conditional (highlighted in yellow) represents the first complex category used in the categorical proposition; the consequent of the conditional (highlighted in green) represents the second complex category.