Epimenides Formalised
Gödel formalised Epimenides’ paradox (also known as the Liar Paradox) as follows:
[R(q); q]
(Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I, p. 175).
Writing the paradox like this requires dropping the hexameter of Κρῆτες ἀει ψεῦσται, κακὰ θηρία, γαστέρες ἀργαί and casting aside its entire Cretan context. In short: the expression [R(q); q] is an abstraction. As such, it applies to other paradoxes as well (such as Russell’s Barber Paradox, the Richard Antinomy, and Turing’s Halting Problem). The expression [R(q); q], in other words, denotes the “general case” encompassing all paradoxes of this type, including the Liar Paradox. But does Epimenides’ Paradox really fall under this “general case”, together with the other paradoxes I just mentioned, and which are not from the 6th century BC, but products of the 20th century? To rephrase this question: are all abstractions formalisations, and moreover, are all formalisations abstractions?
No.
For already (and even particularly) in its classical formulation, Κρῆτες ἀει ψεῦσται, κακὰ θηρία, γαστέρες ἀργαί, as I will explain in upcoming posts, is a formalisation—yet, very clearly, it is not an abstraction. Philosophically speaking, formalisations and abstractions are not (or at least not necessarily) the same. Philosophical formalisations are never general but remain particular: they retain their historical origins, and retain them in such a way that these origins are able to concern, to move the reader or listener.
The Liar Paradox is such a philosophical formalisation: able to concern, move—to dislodge, bring about the peculiar kind of displacement which the ancient Greeks circumscribed by using the adjective ἄτοπος (atopos). Κρῆτες ἀει ψεῦσται, κακὰ θηρία, γαστέρες ἀργαί, in its classical formulation, is ἄτοπος. The expression [R(q); q], on the other hand, is not ἄτοπος. For something to be ἄτοπος, a τόπος is needed, and [R(q); q] is without τόπος, if only because it disregards the Cretan context of the Liar. So, without τόπος, it is but distopic. Welcome to the twentieth century.