Theory

A theory says something. If we allow that "says" is continuous with "does", then a theory does something. From the perspective of a model, it is "what does something". Importantly, a theory is the fundamental or most general example of an operator (what does something) or hypothesis (what says something). In this way, the concept of a theory defines the concept of all operators, hypotheses, or any other class of "actions" or "transformations".

Often we will need to refer to what a theory says, i.e. its result, which we sometimes call a presentation (for operators) or sometimes a sentence (for hypotheses). What is the relation of a theory to any sentence? A theory "says" that sentence...

In the provisional formalism, a theory is written as $$R_\infty$$. In the domain formalism, a theory is a map from an undefined domain to an undefined codomain, written $$\mathbb{T}:\;\uparrow \; \to \; \uparrow$$.

Consistency
See the article on consistency. We often characterize theories by a certain property called consistency. A theory is consistent if...

Generalized theories
In a logical model (or just logic), a theory is a set of sentences, where a sentence is a well-formed formula without free variables. Naturally, we can also speak of generalized theories whose objects are well-formed formulas with free variables.

Relational theories
See the article on abstractions for a discussion of equivalence, modulus, and relative type.

Intuitive theories
See Spelke and Tenenbaum on intuitive theories.

We act inside the world on the basis of intuitive theories---of space, of time, of objects, and so on. These theories embody our conscious and unconscious assumptions about the world. But suppose that the real world is ultimately divorced from the world that we perceive. Then we can think of these intuitive theories, which presume to say something about the real world, as "what define the undefinable."