linguistic:voice_change, structural:reformatting
General theories (as expressed in formal language)

When defining theories for foundational purposes, additional care must be taken, as normal set-theoretic language may not be appropriate.

The construction of a theory begins by specifying a definite non-empty conceptual class E {\displaystyle {\mathcal {E}}}, the elements of which are called statements. These initial statements are often called the primitive elements or elementary statements of the theory—to distinguish them from other statements that may be derived from them.

A theory T {\displaystyle {\mathcal {T}}} is a conceptual class consisting of certain of these elementary statements. The elementary statements that belong to T {\displaystyle {\mathcal {T}}} are called the elementary theorems of T {\displaystyle {\mathcal {T}}} and are said to be true. In this way, a theory can be seen as a way of designating a subset of E {\displaystyle {\mathcal {E}}} that only contain statements that are true.

This general way of designating a theory stipulates that the truth of any of its elementary statements is not known without reference to T {\displaystyle {\mathcal {T}}}. Thus the same elementary statement may be true with respect to one theory but false with respect to another. This is reminiscent of the case in ordinary language where statements such as "He is an honest person" cannot be judged true or false without interpreting who "he" is, and, for that matter, what an "honest person" is under this theory.