The semantics of a formal theory of logic concerns the meaning of the logical symbols, and how they relate to the world. Specifically, the semantics of a formal theory of logic defines how the logical symbols are interpreted, and what the consequences of those interpretations are.

A formal theory of logic consists of a set of symbols and a set of rules for manipulating those symbols. The symbols in the theory are typically divided into two types: logical symbols and non-logical symbols. Logical symbols are those that are used to express logical relationships, such as "and", "or", "not", "if-then", and so on. Non-logical symbols are those that are used to represent objects in the world, such as numbers, sets, or propositions.

There are several different ways to define the semantics of a formal theory of logic, but one common approach is to use a model-theoretic semantics. In this approach, a model is a structure that satisfies certain conditions, which are determined by the logical axioms and rules of inference of the theory. The model provides an interpretation of the non-logical symbols, and a truth assignment for the logical symbols. The truth assignment is defined recursively in terms of the logical connectives and quantifiers of the theory.

Once a model has been constructed, it is used to evaluate the truth of sentences in the theory. A sentence is true in a model if and only if it is true under the truth assignment for the logical symbols in that model. The model-theoretic semantics provides a precise and formal definition of the meaning of sentences in the theory, and allows us to reason about the consequences of different assumptions and axioms.