On a Derivation of the Necessity of Identity - Princeton University

Here (1), universal necessary self-identity, is simply postulated. (2) is an instance
the axiom scheme of the indiscernibility of identicals from classical identity logic,
also known as substitutivity of identicals and as Leibniz’ Law: 3
(4) ∀x∀y(x = y → (Φ(x/z) → Φ(y/z)))
(3) follows by classical predicate logic. 4 The derivation (1)-(3) is one of those
things that, though it may appear obvious once pointed out, requires some
ingenuity for its initial discovery.
Kripke makes it clear in remark (A) that the ingenuity in question is not to
be attributed to himself. He mentions one earlier source, [Wiggins 1965], but
leaves us wondering: Who first found the derivation (1)-(3)? But let us first ask:
Who first found the conclusion (3)? More specifically, let us ask after the sources
of (3) and (1)-(3) in formal systems of quantified modal logic.
To begin at the beginning, the study of formal systems of quantified modal
logic was launched by three papers that the late Ruth Marcus published, under the
name Ruth Barcan, in the Journal of Symbolic Logic (JSL) in 1946-47. These
derive from her Yale dissertation, written under the supervision of Frederic Fitch.
Two systems are developed, based on the modal sentential logics S2 and S4. The
first Barcan paper [Marcus 1946a] is best remembered for the controversial Barcan
schemes, converse and direct, one given as a theorem, the other taken as an axiom.
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