13.1 Metalogic: Syntax vs. Semantics

Metalogic examines logical systems from a higher perspective. Syntax deals with the formal rules and structure, while semantics focuses on meaning and interpretation. These concepts are crucial for understanding the foundations of logical systems.

In this part, we'll explore the distinction between object language and metalanguage, well-formed formulas, and how interpretations and models work. These ideas help us analyze and evaluate logical arguments more effectively.

Syntax and Metalanguage

Defining Metalogic and Syntax

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• Metalogic studies the properties of logical systems from a higher level of abstraction
• Syntax refers to the formal rules and structure of a logical system that govern how formulas can be constructed
• Specifies the alphabet of symbols used in the language and the rules for combining these symbols to create well-formed formulas
• Syntax is purely formal and does not consider the meaning or interpretation of the symbols

Object Language and Metalanguage

• Object language is the formal language being studied or discussed, consisting of a set of symbols and rules for combining them
• Metalanguage is the language used to talk about and analyze the properties of the object language
• Metalanguage operates at a higher level of abstraction than the object language and includes terms for describing syntax, semantics, and other properties of the object language
• Distinguishing between object language and metalanguage helps avoid confusion and allows for precise analysis of logical systems

Well-Formed Formulas

• A well-formed formula (wff) is a string of symbols from the alphabet of a formal language that conforms to the syntactic rules of the language
• The rules for constructing well-formed formulas are part of the syntax of the language and ensure that formulas are structured in a meaningful way
• Examples of well-formed formulas in propositional logic: $(p \land q), ((p \rightarrow q) \lor (q \rightarrow p))$
• Formulas that violate the syntactic rules, such as $(p \land)$ or $(\rightarrow p q)$, are not well-formed and are considered meaningless within the formal system

Semantics and Interpretation

Defining Semantics

• Semantics refers to the meaning and interpretation assigned to the symbols and formulas of a formal language
• Provides a way to determine the truth values of formulas based on the meanings assigned to the symbols
• Semantics connects the formal syntax of a language with the underlying mathematical structures or real-world concepts being represented

Interpretation and Models

• An interpretation is a function that assigns meanings to the symbols of a formal language
• In propositional logic, an interpretation assigns truth values (true or false) to the propositional variables
• A model is an interpretation that makes a formula or set of formulas true
• Example: Consider the formula $(p \land q)$. An interpretation that assigns true to both $p$ and $q$ is a model of the formula, while an interpretation that assigns false to either $p$ or $q$ is not a model

Truth Values and Semantic Concepts

• Truth values are the possible values that can be assigned to formulas in a logical system, typically true and false
• The truth value of a complex formula is determined by the truth values of its constituent parts and the semantic rules for the logical connectives
• Semantic concepts, such as validity, satisfiability, and logical consequence, are defined in terms of truth values and interpretations
• A formula is valid if it is true under all interpretations, satisfiable if there exists at least one model, and a logical consequence of a set of formulas if it is true in every model of the set