The author offers a new look at such things as the fuzzy subalgebras and congruences of an algebra, the fuzzy ideals of a ring or a lattice, and similar entities, by exhibiting them as the models, in the chosen frame \(T\) of truth values, of naturally corresponding propositional theories. This provides a systematic approach to the study of the partially ordered sets formed by these various entities, and its usefulness is demonstrated by employing it to derive a number of results, old and new, concerning these partially ordered sets. For example, it is proved they are complete lattices, algebraic or continuous, depending on whether \(T\) is algebraic or continuous, respectively; they satisfy the same lattice identities for arbitrary \(T\) that hold in the case \(T=\{0,1\}\). It is also proved that the familiar classical situations where the congruences of an algebra correspond to certain other entities (i.e. normal subgroups, ideals of rings), extend to the fuzzy case by proving that the corresponding propositional theories are equivalent.