The principle that if two expressions denote the same object (are identical in referent) then one may be substituted for the other salva veritate (without changing truth) in contexts where reference is transparent.

Principle

Interchangeability salva veritate: identity licenses replacement in all extensional/transparent contexts; formal systems capture this as identity-elimination or congruence rules.

Demonstration

In equational logic, from a = b and P(a) one infers P(b) when P is an extensional predicate; in first-order logic with rigid designators, substitution preserves truth in atomic and extensional contexts.

Misapplication

Blindly substituting in opaque or intensional contexts (e.g., 'Alice believes that ...') can produce false results because belief reports, modal operators, propositional attitudes, or definite descriptions may block substitutivity.

Consequence

Justifies identity-elimination rules, supports equational reasoning and congruence principles in algebra and formal proofs, and underwrites replacement rules in proof systems for extensional languages.

Reversal

Failure of substitutivity marks intensional or opaque contexts: identical referents may not be interchanged without altering truth value in such contexts, revealing limits of purely extensional reasoning.

Boundary

Valid in extensional languages and contexts with rigid reference; excludes intensional, hyperintensional, indexical, or context-sensitive settings unless additional constraints or distinctions (e.g., sense, mode of presentation) are introduced.

Semantic Tension

Tension between Leibniz's law and propositional attitude opacity: the law favors substitution based on referent identity, while intensional semantics emphasize senses, modes of presentation, or context that prevent substitution.

Synthesis

Substitutivity of identicals is the operational expression of identity in extensional reasoning: it enables replacement and congruence where reference is transparent, and its failures highlight the need for intensified semantic apparatus (senses, contexts, rigidity distinctions) in intensional domains.