A theory of logic must be interpretable from the sentences constructed by that theory. This means that every sentence constructed using the symbols and rules of inference of the theory must have a clear and unambiguous interpretation within the theory. In other words, there must be a way to determine the truth or falsity of every sentence constructed within the theory based on the rules of inference and the axioms of the theory.

The reason for this is that the fundamental purpose of a theory of logic is to provide a formal system for reasoning and deduction. In order to perform this function, the theory must be capable of expressing statements and drawing conclusions based on those statements. This requires that every statement that can be constructed using the symbols and rules of inference of the theory must have a well-defined truth value within the theory.

Theorems are expressions of logical truths that are provable within a theory of logic. Theorems are expressed as sentences within the theory because sentences are the basic building blocks of the theory. Every statement that can be made within the theory must be expressible as a sentence. Theorems are special sentences that have been proven within the theory to be true based on the axioms and rules of inference of the theory.

The ability to express theorems as sentences within a theory of logic is important because it allows us to reason about the logical implications of our assumptions and to draw conclusions based on those assumptions. If a theory of logic were unable to express theorems as sentences, it would be incapable of providing a formal system for reasoning and deduction. This would severely limit the usefulness of the theory for applications such as mathematics, computer science, and philosophy.

In summary, a theory of logic must be interpretable from the sentences constructed by that theory because the purpose of the theory is to provide a formal system for reasoning and deduction. Theorems are expressed as sentences within the theory because sentences are the basic building blocks of the theory, and the ability to express theorems as sentences is important for reasoning about the logical implications of our assumptions and drawing conclusions based on those assumptions.