So far every predicate has said something about one thing: $Fa$. Many claims link two or more things: "Ava helps Ben", "7 is greater than 3", "the library is between the park and the school". Relational predicates handle them. They extend the translations of Translating Harder Quantified Sentences.
Key idea
Why it works
Worked examples
Worked example: Order
- The one who likes goes first, the one who is liked second.
Answer: $Lba$
Worked example: A quantifier in one place
- "Everyone likes Ava": for every $x$, $x$ likes $a$.
- "Ava likes everyone": for every $x$, $a$ likes $x$.
Answer: $\forall x Lxa$ and $\forall x Lax$
Worked example: With a negation
- It is not the case that someone likes Ben.
Answer: $\neg \exists x Lxb$, or $\forall x\neg Lxb$
Common mistake
| English | Symbols |
|---|---|
| Ava helps herself. | $Haa$ |
| Someone helps Ben. | $\exists x Hxb$ |
| Ben helps someone. | $\exists x Hbx$ |
| Ava helps Ben, but Ben does not help Ava. | $Hab \land \neg Hba$ |
Relations can have more than two places. "The library is between the park and the school" uses a three-place predicate: $Blps$, with the order fixed by the key "$Bxyz$: $x$ is between $y$ and $z$". The same care applies: decide the order of the places once, write it down, and stick to it.
Relations with quantifiers in two places, like "everyone likes someone", need two quantifiers, and their order turns out to matter a great deal. That is the next lesson.