The signature of a theory of logic specifies the syntax and semantics of the language used in the theory, including the set of symbols and operators used and the rules for constructing valid expressions and making inferences. It defines the structure and grammar of the theory, but does not specify what the theory means or how it should be used.

An interpretation of a theory of logic, on the other hand, involves assigning a specific meaning to the symbols and operators used in the theory. It involves specifying how the theory should be used and what it means in a particular context. An interpretation can be thought of as a mapping between the language of the theory and some external domain of discourse.

The relative importance of the signature and interpretation of a theory of logic depends on the intended use of the theory. For example, if the goal is to use the theory to prove theorems or make logical inferences, then the signature of the theory is of paramount importance. The signature defines the syntax and semantics of the language used in the theory, which is necessary for constructing valid expressions and making logical inferences.

However, if the goal is to apply the theory in a particular domain or context, then the interpretation of the theory becomes more important. The interpretation specifies how the theory should be used and what it means in the particular context, which is necessary for applying the theory effectively.

It is important to note that the signature and interpretation of a theory of logic are closely related and can impact each other. A change in the signature of a theory can lead to a change in its interpretation, and vice versa. For example, adding new symbols or operators to the signature of a theory can require a new interpretation to be assigned to those symbols or operators.

In summary, the signature of a theory of logic specifies the syntax and semantics of the language used in the theory, while the interpretation of the theory assigns a specific meaning to the symbols and operators used in the theory. Both the signature and interpretation are important for using the theory effectively, and the relative importance of each depends on the intended use of the theory.