First Order Logic - Negation in embed functions

Question

Given the First Order Logic Sentence :

¬SubsetOf((Intersection(s,t),s) 

where Constants : s,t are Sets and Functions: SubsetOf(.,.) , Intersection(.,.) are Sets Functions

Should the negation be moved inwards , like that :

SubsetOf((¬Intersection(s,t),s) 

Or it should be left as it is ?

Answer 1

Since SubsetOf(x, y) is a predicate, negation inwards is not needed. Only if x, y are associated with $\lor$ and $\land$, you should use De Morgan's Law to move NOTs inwards repeatedly.

Wiki CNF

Answer 2

Truth-functional operators like the negation can only be applied to claims, e.g. I can say that the claim $P$ is not true by saying $\neg P$.

Such operators do not apply to objects (terms, to be exact), e.g. I can;t take an object $a$, and then claim to say something meaningful when I say 'not $a$'.

A typical mistake beginners of logic make is to translate 'a is not b' with $a = \neg b$, and again, this is a mistake since the negation is in front of an object, not a claim. The right translation should be $\neg a = b$.

You are making a similar mistake when you say $\neg Intersection(s,t)$, since $Intersection(s,t)$ is not a claim, but a term: it is the object that results from intersecting $s$ and $t$.