Difference between logic, fuzzy logic, probabilistic logic, quantum logic?
Russell and Norvig catalogue a few formal languages in the following table[*]:

\begin{array} {lll}
\hline 
\text{Language} &amp; 
\text{Ontological Commitment} &amp; 
\text{Epistemological Commitment} 
\\
&amp; 
\text{(What exists in the world)} &amp; 
\text{(What an agent believes about facts)} 
\\
\hline 
\text{Propositional logic} &amp; 
\text{facts} &amp; 
\text{true/false/unknown} 
\\ 
\text{First-order logic} &amp; 
\text{facts, objects, relations} &amp; 
\text{true/false/unknown} 
\\ 
\text{Temporal logic} &amp; 
\text{facts, objects, relations, times} &amp; 
\text{true/false/unknown} 
\\ 
\text{Probability theory} &amp; 
\text{facts} &amp; 
\text{degree of belief $\in [0, 1]$}
\\ 
\text{Fuzzy logic} &amp; 
\text{facts with degree of truth $\in [0, 1]$} &amp; 
\text{known interval value} 
\\ 
\hline  
\end{array}

(Note this is within the context of artificial intelligence.)

So the most important difference between fuzzy logic and probabilistic logic seems to be that in probabilistic logic you do not deal with degrees of truth, only degrees of belief.

I couldn't tell you very much about quantum logic other than that it appears to have originated in this paper by Birkhoff and von Neumann.

[*] Russell, S. and Norvig, P. (2009). Artificial Intelligence: A Modern Approach (third edition). Pearson. Chapter 8, First-Order Logic.