## Categorical Propositions

A categorical proposition is a proposition that relates two categories of things in any of the following 4 ways:

1. Everything in category A is in category B.
2. Nothing in category A is in category B.
3. At least one thing in category A is in category B.
4. At least one thing in category A is not in category B.

Using ‘Ax’ to mean x is in category A and ‘Bx’ to mean x is in category B, we can symbolize these four types of categorical propositions as shown in the following table:

Categorical proposition Symbolization How to read the formula aloud
Everything in category A is in category B. (∀x)(Ax ⊃ Bx) “For all x, if x is A then x is B.”
Nothing in category A is in category B. ~(∃x)(Ax • Bx)
or
(∀x)(Ax ⊃ ~Bx)
“There is no x such that x is both A and B.”

“For all x, if x is A then x is not B.”
At least one thing in category A is in category B. (∃x)(Ax • Bx) “There exists an x such that x is both A and B.”
At least one thing in category A is not in category B. (∃x)(Ax • ~Bx)
or
~(∀x)(Ax ⊃ Bx)
“There exists an x such that x is A but not B.”

“It is false that for all x, if x is A then x is B.”
As indicated in the table above, there are two ways to symbolize the 2nd type of categorical proposition: (∀x)(Ax ⊃ ~Bx) is logically equivalent to ~(∃x)(Ax • Bx). Similarly, there are two ways to symbolize the 4th type of categorical proposition: (∃x)(Ax • ~Bx) is logically equivalent to ~(∀x)(Ax ⊃ Bx). These and other equivalences rules of predicate logic will be introduced later in the chapter. For now, just keep in mind that there are two equivalent ways to symbolize the 2nd and 4th types of categorical propositions.

English sentences often can be symbolized using the forms above, even if the notion of a category isn’t explicitly mentioned. For example, consider the sentence All Athenians are brilliant. We can paraphrase this sentence as a categorical proposition, using one category for the subject and another category for the predicate. The sentence means that all members of the category Athenians also belong to the category of brilliant things. So, it can be symbolized as the first type of categorical proposition listed above:

(∀x)(Ax ⊃ Bx)

A wide variety of English sentences can be paraphrased and symbolized in a similar way. For another example, consider the following sentence:

All philosophy students learn to think carefully.

The subject refers to the category of philosophy students, and the predicate says that things in that first category also belong to another category, namely, the category of things that learn to think carefully. Thus, we can paraphrase the sentence as a categorical proposition:

Everything in the category of philosophy students is in the category of things that learn to think carefully.

Like the previous example, this is an instance of the first type of categorical proposition, so it can be symbolized as follows:

(∀x)(Px ⊃ Lx)

In this context, ‘Px’ means x is a philosophy student. (In other words, x is in the category of philosophy students.) Similarly, ‘Lx’ means x learns to think carefully. (In other words, x is in the category of things that learn to think carefully.) So, the symbolization above can be read like this: “For all x, if x is a philosophy student, then x learns to think carefully.” That means the same thing as the original sentence. Thus, the symbolization above provides an accurate representation of the logical structure, or form, of the proposition All philosophy students learn to think carefully.

Many English sentences are ambiguous and could be symbolized in different ways depending on how the sentence is understood. For instance, consider the sentence “Philosophy students learn to think carefully.” Is this sentence referring to all philosophy students, or only to some of them? Context clues might help us to determine which interpretation is intended; but if not, then the sentence could be interpreted either way. If an ambiguous sentence appears within an argument, apply the principle of charity: interpret it in the way that makes the argument most reasonable and compelling.

In general, if you can find a way to paraphrase an English sentence as one of the four types of categorical propositions listed above, then you can symbolize it as a categorical proposition. The following table provides several examples of each type of categorical proposition, using the categories P and L from the previous example:

English sentence Paraphrase Symbolization
All philosophy students learn to think carefully.
Philosophy students all learn to think carefully.
Every philosophy student learns to think carefully.
Each philosophy student learns to think carefully.
Any philosophy student learns to think carefully.
Everything in category P is in category L. (∀x)(Px ⊃ Lx)
Philosophy students don’t learn to think carefully.
No philosophy students learn to think carefully.
No philosophy student learns to think carefully.
Nothing in category P is in category L. ~(∃x)(Px • Lx)
or
(∀x)(Px ⊃ ~Lx)
At least one philosophy student learns to think carefully.
SomeThe word “some” is symbolized the same way as “at least one.” See below. philosophy students learn to think carefully.
At least one thing in category P is in category L. (∃x)(Px • Lx)
Some philosophy students don’t learn to think carefully.
Not all philosophy students learn to think carefully.
Philosophy students don’t all learn to think carefully.
At least one thing in category P is not in category L. (∃x)(Px • ~Lx)
or
~(∀x)(Px ⊃ Lx)

A couple of important points are worth highlighting in the above examples:

• First, notice that the word “some” is symbolized the same way as “at least one.” In English, the word “some” usually means more than one. There are ways to represent the distinction between at least one and more than one in predicate logic, using more sophisticated symbolizations. For now, though, we’ll keep things simple and ignore the distinction.
• Second, when symbolizing categorical propositions, it is important to remember that the universal quantifier ‘∀’ always goes with the conditional (symbolized with a horseshoe, ‘⊃’), whereas the existential quantifier ‘∃’ always goes with the conjunction (symbolized with a dot, ‘•’). There is a good reason for this. Although the rules of syntax for predicate logic do permit the universal quantifier to attach to other kinds of propositions besides conditionals, the result will not be a categorical proposition.

To illustrate the second point, suppose we erroneously tried to symbolize the categorical proposition all philosophers think with the dot instead of the horseshoe, like this: (∀x)(Px • Tx). The resulting proposition says that everything that exists (including rocks, trees, etc.) is both a philosopher and a thinker! Obviously, that is not an accurate representation of what the original proposition meant.

Similarly, if we mistakenly attach the existential quantifier to a conditional instead of a conjunction, the result will be a WFF that does not accurately represent any categorical proposition. For example, suppose we want to symbolize the controversial sentence some animals have souls. The correct symbolization is (∃x)(Ax • Sx), but if we incorrectly use a horseshoe instead of the dot with the existential quantifier, the resulting WFF is (∃x)(Ax ⊃ Sx). The latter formula expresses a claim that isn’t controversial at all: it says there is at least one thing such that if it’s an animal, then it has a soul. But that claim is made true by things that aren’t even animals! It is true of a pencil, for example. Since the antecedent of the conditional is false (the pencil isn’t an animal), the conditional is automatically true. Thus, attaching the existential quantifier to a conditional instead of a conjunction doesn’t accurately capture the meaning of the proposition we wanted to represent.