Does the category framework permit new logics? It appears to me that a topos permits a broader concept of subsets than the yes/no decission of a characteristic function in a set theory setting. Probably because the subobject classifier doesn't have to be {0,1}. 
But I wonder, aren't all the multivalued logics also part of/can be modeled in set theory? Is there some new logic coming in with topoi which weren't there before? Did it just help discovering new ideas? Fuzzy stuff etc. are all existent in "conventional set theory mathematics" already, right? 

No, you can't model all multivalued logics in set theory.  Set theory models classical propositional logic, but it does not model a logic where say the principle of contradiction fails and its negation fail also.  All formal theorems of any multivalued logic exist within classical logic in the sense that if A comes as a formula in multivalued logic, it will also happen in classical logic and thus can get modeled by set theory (the converse does seem to hold for some multivalued logics, but hardly all that many of them).  But, the domain of truth values differs for a formula in a multivalued logic than in classical logic.
Fuzzy stuff does NOT exist withing conventional set theory mathematics.  The axiom of extensionality does not hold for fuzzy sets. 
