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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 ?

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    I presume the "Intersection" represents a set. Negating a set doesn't make sense. Instead you could translate "E is a subset of F" as "For all x, x in E implies x in F", and then try negating that. – 2017-01-09
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    The formula nust be read "the intersection of $s$ and $t$ is **not** a subset of $s$", i.e. $\lnot ((s \cap t) \subseteq s)$. – 2017-01-09
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    The *negation* sign applies to a *formula*, like: $(s \cap t) ⊆ s)$ and not to *terms* (i.e. "names") like: $s,t$ and $s \cap t$. – 2017-01-09

2 Answers 2

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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

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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 $t$.