What is the denial of a statement in logic math? I'm trying to get the hang of denials in logic in math. I would like to use these two examples: "Some people are honest and some people are not honest. (All people)" "No one loves everybody. (All people)" What would be the denial in both English and logic and why? Thanks.

The crucial point in both formal and natural language is that the negation of a universal statement is an existential statement, and vice versa; specifically, "not everybody" is equivalent to "there exists someone who does not" and "nobody" is equivalent to "everybody does not" although English grammar permits two interpretations of the last expression; I use it to mean "for everybody holds that they do not...".

_While doing the exercises, also keep in mind that the negation of a conjunction is a disjunction and vice versa; e.g., not (A and B) is equivalent to ((not A) or (not B))._