Meaning of correlation In probability books it is sometimes mentioned that correlation is a measure of linearity of a relationship between random variables. 
This claim is supported by the observation that $\left| \rho(X,Y)\right|=1 \iff X=b+aY$.
But let consider a less extreme case: $X,Y,Z$ are random variables such that $\rho(X,Y)=0.5$ and $\rho(Y,Z)=0.8$. Does it mean that the relationship between $X$ and $Y$ is "less" linear than between $Y$ and $Z$? How at all the degree of linearity in a relationship can be defined (especially given the fact, that a relationship between random variables can be non-functional, so one can not always use the second derivative)?    
Edit:
I am talking about Pearson correlation here.

It depends on what measure of correlation you use.
The Pearson correlation (the most commonly used one) measures the linearity of a relationship between two random variables.
The Spearman rank correlation however, also measures nonlinear (monotonic) relationships. This is defined as the Pearson correlation coefficient between the ranked variables.
