The dual of a logical linear programme

A Linear Programme (LP) involves a conjunction of linear constraints and has a well defined dual. It is shown that if we allow the full set of Boolean connectives {and, or, sim} applied to a set of linear constraints we get a model which we define as a Logical Linear Programme (LLP). This also has a well defined dual preserving most of the properties of LP duality. Generalisations of the connectives are also considered together with the relationship with Integer Programming formulation