In the presentations of fuzzy logic that I've seen, $A \vdash B$ is understood as meaning that in any interpretation where $A$ is 1, $B$ is also 1. If we are reasoning meta-theoretically about fuzzy logic, we could make much more complicated statements about how the truth value of $A$ relates to the truth value of $B$. For instance, we could say that in any interpretation, the truth value of $B$ is at least the truth value of $A$. We could say that the truth value of $B$ is always within $\epsilon$ of the truth value of $A$. We could say that the truth value of $B$ can get arbitrarily close to the truth value of $A$.
These specific examples aren't so much my point; rather they illustrate the kind of thing I'm interested in. I would like to hear about approaches to deduction for fuzzy logic which allow the expression of more complicated relationships between propositions than preservation of being valued 1. Could you please provide some sources on this topic?
