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How does metalogic study the syntax and semantics of formal languages?

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Metalogic is the study and analysis of the syntax and semantics of formal languages and systems.1

Syntax

Syntax focuses on the relations among expressions, independent of their meaning. It involves:1
* Formation Rules: A mechanical specification of symbols and their combinations that define well-formed formulas.2
* Formal Systems: The introduction of axioms and rules of inference to derive theorems.2
Syntax is closely related to proof theory and is often viewed as a branch of number theory.2

Semantics

Semantics examines the relations between expressions and their meanings. It involves:1
* Interpretation: Assigning meanings to atomic sentences relative to a domain of objects, which determines the truth-value of sentences.2
* Truth and Denotation: Concepts such as truth, meaning, and denotation are central to semantics.2
Semantics is closely related to model theory and is often viewed as a branch of set theory.2

Kurt Gödel demonstrated that semantics cannot be reduced to syntax, as not all true sentences are necessarily provable as theorems in significant formal systems.2