The four categorical forms from The Four Forms: A, E, I, O are the first sentences most predicate-logic translations are built from. Each becomes a quantifier with a connective inside, using what you learned in The Quantifiers ∀ and ∃.
The four forms in predicate logic.
Form
English
Symbols
A
All $S$ are $P$.
$\forall x(Sx \to Px)$
E
No $S$ are $P$.
$\forall x(Sx \to \neg Px)$
I
Some $S$ are $P$.
$\exists x(Sx \land Px)$
O
Some $S$ are not $P$.
$\exists x(Sx \land \neg Px)$
🔑 Key idea
Universal statements (A and E) use $\forall$ with $\to$: "for anything, if it is an $S$, then …". Particular statements (I and O) use $\exists$ with $\land$: "there is something that is an $S$ and …".
💡 Why it works
"All dogs are mammals" is not about everything being a dog; it says that anything which is a dog is a mammal. That "if" is the arrow. "Some dogs are pets" says there is a thing that is both a dog and a pet; that "both" is the $\land$. Swapping them gives nonsense: $\forall x(Dx \land Mx)$ says everything in the domain is a dog and a mammal, and $\exists x(Dx \to Px)$ is true as soon as one thing in the domain is not a dog, since a conditional with a false antecedent is true.
Rex is a pet dog, Wolf a wild dog, Tom a pet cat and Rock a rock. Compare the right forms with the wrong ones at the bottom.
Worked examples
✏️ Worked example: An A statement
Let $Wx$: $x$ is a whale; $Mx$: $x$ is a mammal. Translate "Every whale is a mammal."
Universal, affirmative: $\forall$ with $\to$.
Answer: $\forall x(Wx \to Mx)$
✏️ Worked example: An O statement in disguise
Translate "Not every whale is a fish" ($Fx$: $x$ is a fish).
"Not every" denies an A statement: $\neg \forall x(Wx \to Fx)$.
That is equivalent to an O statement: some whale is not a fish.
Pairing $\forall$ with $\land$ or $\exists$ with $\to$. $\forall x(Sx \land Px)$ says everything is both $S$ and $P$, which is far too strong. $\exists x(Sx \to Px)$ is far too weak: it is true if anything at all is not an $S$. Memorize the pairs: all goes with if, some goes with and.
English offers many ways to say each form. "Every", "each" and "any" usually mark an A statement; "no" and "not a single" an E statement; "some", "there is" and "at least one" an I statement; "not every" and "not all" an O statement. Decide which of the four forms the sentence expresses before you write any symbols, and the pairing of quantifier and connective follows.
Predicate logic reads A and E statements without existential import: $\forall x(Sx \to Px)$ is true when there are no $S$ at all. That is the modern reading taught in Existential Import.
Practice
Pass with 3 right answers, the last 2 in a row. Each right answer earns 2 XP.
Patrick J. Hurley and Lori Watson, A Concise Introduction to Logic, 13th ed., section 8.1, translating categorical statements
forall x: Calgary, an open introduction to formal logic (P. D. Magnus, Tim Button and others; Open Logic Project), chapter on the four Aristotelian forms