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Signatures of Formal Theories of Logic

A formal signature of a theory of logic typically consists of two parts:

  1. A set of symbols and operators used in the theory, along with their associated arities (i.e., the number of arguments they take). These symbols and operators might include logical connectives like "and" and "or", quantifiers like "forall" and "exists", and other symbols like variables and constants.

  2. A set of axioms and rules of inference that govern how the symbols and operators can be combined and manipulated to form valid logical arguments. These axioms and rules of inference are usually specified in a precise, formal language like first-order logic or propositional logic.

By specifying a formal signature, a theory of logic can provide a rigorous, mathematical framework for reasoning about logical concepts and arguments. This can be useful in a wide range of applications, from computer programming and artificial intelligence to philosophy and mathematics.

Logical Signatures and Interpretation

A formal signature of a theory of logic does not necessarily imply an interpretation of that theory.

A formal signature simply 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 does not specify any particular interpretation or meaning for these symbols and operators.

An interpretation of a formal theory of logic involves assigning a specific meaning to the symbols and operators used in the theory. This can be done in many different ways, depending on the context and application of the theory. For example, the same formal theory of logic could be interpreted in a classical, intuitionistic, or modal logic framework, each with different meanings assigned to the logical symbols and operators.

So while a formal signature is necessary for specifying a theory of logic, it does not by itself provide an interpretation of that theory. An interpretation must be separately specified in order to give the theory any specific meaning.