As a domain expert in the field of mathematical logic, I am delighted to provide an in-depth explanation of what constitutes a model of logic. Logic, as a branch of mathematics and philosophy, deals with the principles of correct reasoning. It is the study of the principles of valid inference and argumentation. A model of logic, therefore, is a critical concept that helps us understand the structure and validity of logical systems.
Step 1: English Explanation
A model in the context of logic is a representation of a logical system that satisfies certain conditions or axioms. It is a way to interpret the abstract constructs of a formal language and to provide a framework for understanding the truth or falsity of statements within that system. Models are essential in various areas of logic, including model theory, proof theory, and computational logic.
### Formal Languages and Theories
To understand models, it is crucial to first comprehend formal languages and theories. A formal language is a set of strings made up of symbols from a specific vocabulary, which are structured according to a set of rules. These languages are used to express mathematical and logical concepts unambiguously.
A theory in this context is a collection of sentences in a formal language that are considered to be true within that system. Theories are often defined by a set of axioms, which are the basic building blocks or starting points for the theory. Axioms are statements that are accepted without proof and serve as the foundation for deriving other truths within the theory.
### Interpretations and Structures
A model of a theory is an interpretation or a structure that gives meaning to the symbols of the formal language in a way that makes the sentences of the theory true. This structure typically includes a domain of discourse, which is the set of entities that the model is referring to, and a set of assignments that define the relationships between the symbols and the elements of this domain.
### Satisfaction and Truth
When we say that a model satisfies a theory, we mean that every sentence in the theory is true when interpreted under the model. This satisfaction is what validates the model as a correct representation of the theory. Conversely, if a model makes a sentence of the theory false, it is considered not to satisfy the theory and is therefore not a valid model for that theory.
### Example: Arithmetic Models
To illustrate, consider the theory of arithmetic. This theory includes axioms such as the Peano axioms, which define the properties of natural numbers. A model for this theory could be the set of all natural numbers with the standard operations of addition and multiplication. This model satisfies the theory because all the sentences derived from the Peano axioms are true when interpreted over this set.
### Model Theory
Model theory, a branch of mathematical logic, is particularly concerned with the study of models and their relationship to theories. It explores questions like:
- What are the necessary and sufficient conditions for a structure to be a model of a given theory?
- How do different models of the same theory relate to each other?
- What can be inferred about a theory based on the properties of its models?
Model theory has deep connections with other areas of mathematics and has been applied to algebra, geometry, and analysis, leading to profound results like Gödel's incompleteness theorems.
### Universal Algebra and Logic
Universal algebra extends the ideas of abstract algebra to study algebraic structures in a more general setting. When combined with logic, it forms a powerful framework for understanding and analyzing the algebraic properties of models. This intersection of universal algebra and logic is known as model theory, which provides a comprehensive toolkit for studying the algebraic structures that underlie logical systems.
### Conclusion
In summary, a model of logic is a fundamental concept that allows us to ground abstract logical theories in concrete interpretations. It is through models that we can explore the truth and validity of logical statements and understand the implications of different axiomatic systems. The study of models is not just a theoretical exercise but has practical applications in fields such as computer science, where it is used to design and analyze formal systems and algorithms.

**read more >>

The objects of study are models of theories in a formal language. A set of sentences in a formal language is called a theory; a model of a theory is a structure (e.g. an interpretation) that satisfies the sentences of that theory. ... universal algebra + logic = model theory.read more >>