8.2 Translating Categorical Propositions

Types of Categorical Propositions

Categorical propositions are statements about classes of objects. They make claims about whether members of one class belong (or don't belong) to another class. There are exactly four types, traditionally labeled A, E, I, and O. Translating them into predicate logic lets you work with their structure precisely, but there's a critical pattern to notice: universals use conditionals ($\to$), while particulars use conjunctions ($\wedge$). Getting that distinction right is the single most important thing in this section.

Universal Propositions

A (Universal Affirmative): "All S are P"

Symbolized as: $(\forall x)(Sx \to Px)$

This says: for every object $x$, if $x$ is an S, then $x$ is a P. The conditional is doing important work here. You're not claiming everything in the universe is an S or a P. You're only saying that among things that are S, every one of them is also P.

• Example: "All cats are mammals" becomes $(\forall x)(Cx \to Mx)$
• Read it as: for any $x$, if $x$ is a cat, then $x$ is a mammal

E (Universal Negative): "No S are P"

Symbolized as: $(\forall x)(Sx \to \neg Px)$

This says: for every object $x$, if $x$ is an S, then $x$ is not a P. The structure is the same as the A proposition, but with a negation on the predicate.

• Example: "No dogs are reptiles" becomes $(\forall x)(Dx \to \neg Rx)$
• Read it as: for any $x$, if $x$ is a dog, then $x$ is not a reptile

Particular Propositions

I (Particular Affirmative): "Some S are P"

Symbolized as: $(\exists x)(Sx \wedge Px)$

This says: there exists at least one object $x$ such that $x$ is both an S and a P. Notice the switch from $\to$ to $\wedge$. You need conjunction here because you're asserting that some specific thing actually is both S and P, not just setting up a hypothetical.

• Example: "Some birds can fly" becomes $(\exists x)(Bx \wedge Fx)$
• Read it as: there is at least one $x$ that is a bird and can fly

O (Particular Negative): "Some S are not P"

Symbolized as: $(\exists x)(Sx \wedge \neg Px)$

This says: there exists at least one object $x$ such that $x$ is an S but is not a P. Same structure as the I proposition, but with a negation on the predicate.

• Example: "Some animals are not mammals" becomes $(\exists x)(Ax \wedge \neg Mx)$
• Read it as: there is at least one $x$ that is an animal and is not a mammal

Why Universals Use $\to$ and Particulars Use $\wedge$

This is the most common mistake students make, so it's worth understanding why the connectives differ.

If you tried to translate "All cats are mammals" as $(\forall x)(Cx \wedge Mx)$, you'd be saying everything in the entire universe is both a cat and a mammal. That's wildly wrong. The conditional restricts your claim: you're only talking about things that happen to be cats.

If you tried to translate "Some birds can fly" as $(\exists x)(Bx \to Fx)$, you'd get a statement that's true whenever there exists anything that isn't a bird (since a conditional with a false antecedent is true). That would make the sentence trivially true and meaningless. The conjunction forces the thing to actually be a bird and also fly.

Quantifiers and Logical Connectives

Quantifiers

The universal quantifier $\forall$ expresses that something holds for all members of the domain. It's read as "for all" or "for every." When you write $(\forall x)(Px)$, you're claiming every object $x$ in the domain has property P.

The existential quantifier $\exists$ expresses that something holds for at least one member of the domain. It's read as "there exists" or "for some." When you write $(\exists x)(Px)$, you're claiming at least one object $x$ in the domain has property P.

Both quantifiers are followed by a variable (usually $x$) that gets "bound" by the quantifier. That variable then appears inside the parentheses that follow, referring back to the same object.

Logical Connectives

Conditional ($\to$): Expresses an if-then relationship. $P \to Q$ asserts that whenever $P$ is true, $Q$ must also be true. It says nothing about cases where $P$ is false. "If it is raining, then the ground is wet" only makes a claim about rainy situations.

Conjunction ($\wedge$): Asserts that both parts are true simultaneously. $P \wedge Q$ is true only when $P$ and $Q$ are both true.

Negation ($\neg$): Flips the truth value. $\neg P$ is true when $P$ is false, and false when $P$ is true.

Biconditional ($\leftrightarrow$): Expresses logical equivalence. $P \leftrightarrow Q$ asserts that $P$ and $Q$ always share the same truth value. "A figure is a square if and only if it is a rectangle with four equal sides" means being a square and being an equal-sided rectangle are the same condition. You can think of it as a conditional that works in both directions.