Secondary theories can be created or interpreted from a subset of the set of all possible sentences of a theory of logic. In fact, many theories of logic are built by considering only a subset of the possible sentences that can be formed from a given set of symbols.

For example, propositional logic is a theory of logic that is based on a subset of the possible sentences that can be formed from a set of logical connectives (e.g., "and", "or", "not"). Propositional logic considers only sentences that are composed of propositional variables (i.e., variables that can be either true or false), logical connectives, and parentheses. By focusing on this subset of sentences, propositional logic is able to capture the basic logical relationships between propositions, without considering the more complex structure of predicates and quantifiers that are used in predicate logic.

Similarly, modal logic is a theory of logic that is based on a subset of sentences that involve modal operators (e.g., "necessarily", "possibly"). Modal logic considers only sentences that involve these modal operators, along with logical connectives and propositional variables. By focusing on this subset of sentences, modal logic is able to capture the formal structure of modal concepts such as necessity and possibility.

In general, the choice of which subset of sentences to consider depends on the intended domain of the theory and the specific problems that it is meant to address. By selecting an appropriate subset of sentences, it is possible to create a secondary theory that captures the relevant features of the domain, without being burdened by the complexities of the full set of possible sentences.