We start by presenting the satisfiability problem for Łukasiewicz infinitely-valued logic  (Ł∞-SAT).  We discuss how this can be made using Mixed Integer Programming, which, in fact, highlights the NP-completeness of the problem.  We then use this gives rise to a polyhedral semantics for a parameterised family of tractable logics whis approximate Ł∞ such that each logic in the family is associated to a polynomial-time linear program.y. In this way, we provide a tractable decision problem for each intermediate logic in the path to obtaining  Ł∞-SAT. We highlight some properties of the logic system derived from polyhedral semantics and the details of an algorithm for the approximation process.