Complement versus Negation My earlier question became too long, so succinctly:
Suppose $P(C)=0.2.$ Its complement is $0.8;$ i.e., $P(C)^\complement=0.8.$
But what does $P(¬C)$ mean? I think I am mixing up the terms 'complement' and 'negation'?
 A: Informally, exactly the same thing is intended. We can think of events as sets (nowadays preferred) or as assertions (we got $4$ or more heads). The complement $A^c$ of an event $A$ is the set language version of "$A$ doesn't occur." The assertion version is to say that "not $A$" occurs, or, in symbols, that  $\lnot A$ occurs.
A similar thing can happen with conjunction. In probability, it is usual to think of events as sets, so we ordinarily write $A\cap B$ for "$A$ and $B$." But some people still write $A\land B$, where $\land$ is the "and" of logic. 
Thus, in your example, $\Pr(C^c)=\Pr(\lnot C)=0.8$. 
A: 
Suppose $P(C)=0.2.$ Its complement is $0.8,$ i.e., $P(C)^\complement=0.8.$ But what does $P(¬C)$ mean? I think I am mixing up the term complement and negation?

The negation of the sentence $$P(C)=0.2$$ is $$P(C)\ne0.2,$$ whereas the complement of the event (set) $$C$$ is $$C^\complement.$$

The negation of a tautology is a contradiction;
on the other hand, the complement of the set of tautologies contains contingent sentences as well as contradictions.
