Substitutivity

Substitutivity is a principle in logic, mathematics, and related disciplines stating that expressions referring to the same object can be substituted for one another within larger expressions without changing the truth value or reference of the overall statement. It underpins many forms of logical reasoning and algebraic manipulation.

In logic and mathematics, substitutivity is most prominently associated with the substitution property of equality. Leibniz’s law (the principle of the identity of indiscernibles) is often stated as: if a=b, then any context or function applied to a yields the same result as applying it to b. Formally, for any expression F in which a and b occur, F(a) = F(b) if a=b. This extends to predicates, relations, and function applications, provided that variables are handled properly to avoid capture. Substitutivity enables equational reasoning, term rewriting, and the transfer of properties across interchangeable terms.

In programming and computer science, substitutivity underpins referential transparency: an expression can be replaced by its computed value without changing program behavior. This allows compiler optimizations, reasoning about code, and modular design.

Substitutivity has limits. In intensional contexts—such as beliefs, desires, knowledge, and necessity—substituting co-referential terms can alter truth conditions. For example, in beliefs, substituting “Morning Star” with “Evening Star” may change the truth value of a belief ascription, even though both terms refer to Venus. Such failures are central to discussions of intensional semantics.

See also: Leibniz’s law, equality, referential transparency.