# A predicate within a predicate?

I'm new here. I wish to ask a question regarding predicate logic:

I was given three predicates:

parent(p,q): p is the parent of q.

female(p): p is a female.

p = q: p and q are the same person.

Now, I was tasked with translating this sentence: Alice has a daughter.

My answer was: There exists a q such that parent(Alice,female(q)).

The answer given is: There exists a q such that female(q) AND parent(Alice,q).

Is it correct to have a predicate (in this case, female) within another predicate (in this case, parent)?

Much appreciated.

• No; a prdicate letter is used to write a "statement": $Red(x)$ used with the term $book$ gives $Red(book)$. Terms are variables, constants ore "complex" terms built-up with function letters. Thus, the syntax of a bynari predicate like $Parent(x,y)$ needs two "names" (i.e. terms) to be "completed". Sep 20, 2016 at 9:44
• Compare with arithmetic : $<$ is a (binary) predicate (or relation), while $+$ is a (bynary) function. The "result" of $2 < 4$ is true, while the "result" of $2+4$ is $6$, a number. Thus, the expression $2+4$ is a "name" for the number $6$, while $2 < 4$ is an arithmetical sentence. Sep 20, 2016 at 9:48
• Perhaps your confusion stems from the fact that you read $\mathrm{female}(q)$ word by word, as the English phrase "the female $q$". But, a predicate is not an adjective; you should read $\mathrm{female}(q)$ as the sentence "$q$ is female". Then it is clear that "Alice is the parent of '$q$ is female'" doesn't make a lot of sense. Sep 20, 2016 at 10:32

Intuitively, a predicate in a predicate doesn't make sense; predicates only take terms as arguments. Using the normal convention of abbreviating predicates and terms by letters ($P(p,q)$ for parent, $F(q)$ for female, $a$ for Alice), your example is $$\exists q\ P(a,F(q))$$ which is interpreted as "Alice is the parent of true" – absurd, since children aren't truth values. In other words, your attempt does not produce a well-formed formula.

The given answer translates as $$\exists q\ F(q)\land P(a,q)$$ which is well-formed.

No, it is not in this particular case. You conclude that by substitution.

female(q) can have two values : true or false.

• For female q, female(q)=true you have : parent(Alice,true).
• For non-female q, female(q)=false you have : parent(Alice,false).

Since false and true are not humans (they are truth values) the proposition parent(Alice,female(q)) is false for all values of q.

So there does not exist a q such that parent(Alice,female(q)).

Your predicate within a predicate is a well-formed formula. True and false can be used as terms of a predicate. But the formula does not express the required concept.

• What does this add to the existing two-year-old answer? Apr 26, 2018 at 17:35
• @Noah An explanation in the same terms used by the OP. Furthermore, I've edited it to clarify that OP's formula is well-formed. Apr 26, 2018 at 18:08