Substitution (Open Logic Project, 2026)

This Open Logic Project module is the complete official two-page build titled Substitution, revision 9620cc7 dated 2026-07-12 under CC BY 4.0. Both pages were read from native text and visually inspected at 200 dpi. The archived file is byte-identical to the official build; its SHA-256 is ff7400025e178833adea2613282a2f18ccfc28cc654e37e9af953af29309e010.

What the module establishes #

The module distinguishes substitution into terms from substitution into formulas. Its formula operation replaces only free occurrences and descends through a quantifier only when the replacement term is free for the target variable. That side condition is the load-bearing rule: without it, a variable that was free in the replacement term can become bound by a quantifier in the receiving formula.

The source's examples correctly separate three cases:

• harmless replacement where no binder captures a variable from the term;
• vacuous replacement where the target has no relevant free occurrence;
• an inadmissible replacement that would change meaning through capture.

The notation $\varphi(t)$ is only conventional shorthand for an admissible substitution instance. It is not ordinary function application in the object language.

Scope boundary #

This is a guarded partial substitution definition. The module does not define a total capture-avoiding implementation that first chooses fresh variable names, does not cover simultaneous substitutions, and does not prove the semantic substitution lemma. Alpha-renaming and the link between syntactic substitution and assignment updates therefore require separate justification.

No mathematical defect was found in the narrow mechanism. The PDF does contain visible build anchors, empty template headings, blank metadata, and extraction that flattens indexed formulas; the rendered equations, not the raw text dump, control the audit.