Real sentences add words to the four basic forms of Translating A, E, I and O: adjectives on the subject, two properties in the predicate, "only", "none but" and "all except" from Only, None But and All Except. The method stays the same: find the form, then translate the subject phrase and the predicate phrase as whole pieces.
| English | Symbols |
|---|---|
| Only $S$ are $P$. / None but $S$ are $P$. | $\forall x(Px \to Sx)$ |
| All young $S$ are $P$. | $\forall x((Yx \land Sx) \to Px)$ |
| Some young $S$ are $P$. | $\exists x((Yx \land Sx) \land Px)$ |
| Every $S$ is young and a $P$. | $\forall x(Sx \to (Yx \land Px))$ |
| $S$ and $Q$ are $P$ (cats and dogs are pets). | $\forall x((Sx \lor Qx) \to Px)$ |
| All except $S$ are $P$. | $\forall x(\neg Sx \to Px) \land \forall x(Sx \to \neg Px)$ |
Key idea
Why it works
Worked examples
Worked example: Only
- Anyone who votes is a member.
Answer: $\forall x(Vx \to Mx)$
Worked example: An adjective
- Subject: things that are young and birds.
- Something is a young bird and can fly.
Answer: $\exists x((Yx \land Bx) \land Fx)$
Worked example: A compound subject
- Anything that is a cat or a dog is a pet.
Answer: $\forall x((Cx \lor Dx) \to Px)$
Common mistake
Worked example: All except
- Two claims: everyone who is not a manager pays dues, and no manager pays dues.
- Join the two universal statements with $\land$.
Answer: $\forall x(\neg Mx \to Px) \land \forall x(Mx \to \neg Px)$, equivalently $\forall x(\neg Mx \leftrightarrow Px)$.
When a sentence feels tangled, rewrite it in "logician's English" first: "for any $x$, if $x$ is a young bird, then $x$ can fly." Once the English has that shape, the symbols follow almost word for word.