A new approach to semantics, based on ordered Banach spaces, is proposed. The Banach spaces semantics arises as a generalization of the four particular cases: the Giles' approach to belief structures, its generalization to the non-Boolean case, and fuzzy extensions of Boolean as well as of non-Boolean semantics.

The paper describes in detail the procedure of identification of the inner language and an inner logico of a physical theory. The procedure is a generalization of the original ideas of J. von Neuman and G. Birkhoff about quantum logic.