Some Remarks on Indiscernibility

If α and α’ are distinct variables and ϕ and ϕ’ are open sentences of some language, where ϕ’ is the result of replacing one or more free occurrences of a in α with free occurrences of α’ in ϕ’, then a universal closure of ⌜(α=α’ → (ϕ → ϕ’))⌝, is an indiscernibility principle of that language. For instance, (1) is an indiscernibility principle.The existence of opaque constructions falsifies the familiar unrestricted principle of substitution which affirms that co-referential expressions are intersubstitutable in all contexts without change of truth-value. But indiscernibility principles are another matter. Not every counter-example to the unrestricted principle of substitution is a counter-example to some indiscernibility principle. Indeed, it is likely to be thought that there is no counter-example to any indiscernibility principle, and that the semantics of variables and objectual quantification ensures that all indiscernibility principles are true.