All presentable symbols of a theory are typically part of the signature of that theory. The signature of a formal theory consists of a collection of symbols that are used to represent the basic objects and operations of the theory. These symbols are typically defined within the syntax of the theory, and any interpretation of the theory must be based on these symbols and their associated meanings.
If a theory allows the presentation of data as part of the sentence presentation, then those symbols used to represent that data would also be part of the theory's signature. An automaton or any other interpreter of the theory would need to be aware of these symbols and their meanings in order to correctly interpret the sentences of the theory.

The standard interpretation of a theory is not entirely independent of the published syntax and semantics of that theory. The interpretation of a theory is based on a combination of the signature of the theory and the rules of inference that are used to derive new sentences from the existing ones.

However, the interpretation of a theory also depends on the specific model or context in which the theory is being interpreted and it is possible to form competing interpretations of a theory of logic. In fact, different interpretations of a theory can often coexist and be used for different purposes or in different contexts.

While a theory of logic typically consists of a set of axioms and rules of inference that determine how sentences can be derived from one another, the meaning of the symbols and sentences in the theory is not fixed; it depends on the interpretation that is given to them. Different interpretations can assign different meanings to the same symbols and sentences, leading to different conclusions about the theory.

For example, one interpretation of the theory of first-order logic might view its quantifiers as ranging over all individuals, while another interpretation might view them as ranging only over some subset of individuals. Both interpretations are valid, and they can lead to different conclusions about the theory.

Competing interpretations can arise for various reasons, such as differences in philosophical assumptions or differing views on the intended semantics of the theory. However, it is important to note that these interpretations must still be consistent with the published signature of the theory and must follow its axioms and rules of inference. Otherwise, they would not be interpretations of the same theory, but rather a different theory altogether.

More than one theory can use the same logical symbols, for instance. Logical symbols, such as logical connectives (e.g., "and", "or", "not") and quantifiers (e.g., "for all", "there exists"), are typically used to represent basic logical concepts that are common to many theories. As such, these symbols can be used in different theories with different meanings, depending on the context and the interpretation that is given to them.

For example, the symbol "+" is used in both arithmetic and set theory to represent addition, but the meaning of "+" in each theory is different. In arithmetic, "+" is interpreted as a binary operation on numbers, while in set theory, "+" is interpreted as a binary operation on sets.

Similarly, the symbol "=>" is used in both propositional logic and predicate logic to represent implication, but the meaning of "=>" in each logic is different. In propositional logic, "=>" is typically interpreted as a material implication, while in predicate logic, "=>" is typically interpreted as a logical consequence.

Despite these differences in meaning, the use of the same symbols across different theories can facilitate communication and comparison between those theories. By using a common set of symbols, it becomes easier to recognize and compare the logical structure and properties of different theories.
The published signature of a theory is typically an essential part of its definition, and any interpretation that ignores or deviates from the signature would not be an interpretation of the same theory, but rather a different theory altogether.