Ordinary semantic compositionality (meaning of whole determined from meanings of parts plus composition) can serve to explain how a hearer manages to assign an appropriate meaning to a new sentence. But it does not serve to explain how the speaker manages to find an appropriate sentence for expressing a new thought. For this we would need a principle of inverse compositionality, by which the expression of a complex content is determined by the expressions of its parts and the mode of composition. But this presupposes that contents have constituent structure, and this cannot be taken for granted. However, it can be proved that if a certain principle of substitutivity is valid for a particular languages, then the meanings expressed by its sentences can justifiably be treated as structured. In its simplest form, this principle says that if in a complex expression a constituent is replaced by another constituent with a different meaning, the new complex has a meaning different from the original. This principle is again inversely related to the normal compositional principle fo substitutivity. The combination of ordinary and inverse compositionality is here called 'strong compositionality'. The proof is carried out in the algebraic framework developed by Wilfrid Hodges and Dag Westerstahl.
