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.
On
The best way to understand this is by using the existential and universal quantifiers, then understanding the rules of negating those. The existential qualifier $\exists$ is used to say at least one thing exists that fulfills a given statement; the universal quantifier $\forall$ is used to say that all things fulfill a given statement. Translating your statements into logic:
Some people are honest and some people are not honest.
Let $H(x)$ denote that a person $x$ is honest. Then we have $(\exists{x}:H(x)) \wedge (\exists{x}:\neg H(x)) $.
No one loves everybody.
Let $L(x, y)$ denote that $x$ loves $y$. Then we have $\neg \exists{x} \forall{y}:L(x, y)$. (It is not the case that there exists an $x$ who loves all $y$)
Now for negations, we use the rules that $\neg \forall{x}P(x) \equiv \exists{x} \neg P(x)$ and $\neg \exists{x}P(x) \equiv \forall{x} \neg P(x)$. Negating the first statement, we have
\begin{align} \neg(\exists{x}:H(x)) \wedge (\exists{x}:\neg H(x)) &\equiv (\neg\exists{x}:H(x)) \vee (\neg\exists{x}:\neg H(x)) &\text{(De Morgan's law)} \\ &\equiv(\forall{x}:\neg H(x)) \vee (\forall{x}:H(x)) & \end{align} In plain English,
either everyone is honest or everyone is not honest
which makes sense as the negation of the original statement.
Negating the second statement is straightforward; $\neg \neg \exists{x} \forall{y}:L(x, y) \equiv \exists{x} \forall{y}:L(x, y)$, meaning
there exists some person $x$ who loves all people $y$.
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)).