Formalisation

Formalisation is the process of translating informal, natural-language material into a formal language with precise syntax and semantics. The goal is to make assumptions explicit, eliminate ambiguity, and enable rigorous reasoning, proof, and automated verification. A formalisation typically begins by choosing a formal language (such as first-order logic or a specification language), identifying the basic objects of interest, and specifying axioms, definitions, and inference rules that govern how statements may be derived.

In mathematics and logic, formalisation underpins theory-building and proofs. A theorem is derived from axioms by valid inference steps; proof theory and model theory study the relationship between formal statements and their interpretations, and investigate properties such as consistency, completeness, and decidability.

In computer science and engineering, formal methods apply formalisation to specify, design, and verify systems. Formal specification languages (for example Z or VDM) and temporal logics are used to describe requirements and behavior, which can then be checked by proof systems or model checkers to verify properties such as safety, liveness, or correctness.

Applications extend to linguistics, law, and knowledge representation, where formalisation codifies rules and relations. Limitations include the difficulty of capturing all relevant aspects in a formal system, the risk of misinterpretation or loss of nuance, and the resource costs of formalisation. Some domains resist complete formalisation, while others benefit from partial formalisation that captures essential properties.

Historically, formalisation emerged from logic in the 19th and early 20th centuries with Frege, Peano, and later Hilbert; its methods were advanced by Godel, Tarski, and modern computer science through Hoare logic, Z notation, and automated theorem proving.