The set-theoretic multiverse

I came across this somewhat intriguing idea of a set theoretic multiverse. I want to try to find out the links between a set theoretical multiverse and logics. As i look at it, I see logic as a theory based on the rules of set theory to propagate labels such as truth/false. What I am curious about is whether if there are more than 1 foundational set theory - does that lead to many logics .

http://lumiere.ens.fr/~dbonnay/files/talks/hamkins.pdf