Logical transductions are not sufficient for notational equivalence

Logical transductions are an invaluable tool for the comparison of linguistic theory. However, they do not fully capture all linguistically relevant information that may be present in two theories of representation, and thus a transduction alone is not sufficient for claiming equivalence in a linguistically relevant way. This work shows that two theories of segmental representation can be logically equivalent while still predicting different sets of natural classes. Since phonological theory by and large operates over natural classes, the notion of natural class preservation is defined as a crucial domain of comparison for linguistic theories, on top of the formal results of the transduction itself. This opens the door for additional domains of phonologically relevant information to be evaluated in the same way, ultimately building towards a notion of a strong generative capacity for phonological theories.