  In general, providing an axiomatization for an arbitrary logic is a task that
may require some ingenuity. In the case of logics defined by a finite logical
matrix (three-valued logics being a particularly simple example), the
generation of suitable finite axiomatizations can be completely automatized,
essentially by expressing the matrix tables via inference rules. In this
chapter we illustrate how two formalisms, the 3-labelled calculi of Baaz,
Ferm\"uller and Zach and the multiple-conclusion (or Set-Set) Hilbert-style
calculi of Shoesmith and Smiley, may be uniformly employed to axiomatize logics
defined by a three-valued logical matrix. The generating procedure common to
both formalisms can be described as follows: first (i) convert the matrix
semantics into rule form (we refer to this step as the generating subprocedure)
and then (ii) simplify the set of rules thus obtained, essentially relying on
the defining properties of any Tarskian consequence relation (we refer to this
step as the streamlining subprocedure). We illustrate through some examples
that, if a minimal expressiveness assumption is met (namely, if the matrix
defining the logic is monadic), then it is straightforward to define effective
translations guaranteeing the equivalence between the 3-labelled and the
Set-Set approach.
