Essay:Quine, Tarski, and the Liar's Antinomy

Introduction
I am largely indebted to W.V.O Quine's essay Truth(1994, ibid §.Paradox, pp. 423-24) wherein he discusses the so called "liar antinomy", and how both Tarski's explication of Truth and Tarski's Indefinability theorem, can be used to elucidate, and to avoid, the vicious and counterintuitive problems associated with the liar's antinomy. I was originally planning on putting this on the Douglas Hofstadter talk-page, but in the end decided against it, as it seemed more appropriate in the format of an essay.

I will include as an epilegomena some of my own perceptions and inferences regarding the connection(s) between the liar's antinomy, Quine's New Foundations, and Combinatory Logic. I would like to show how combinatory logic, in conjunction with Quine's comprehension schema, can deal with the Liar's paradox without invoking a Russelian hierarchy viz. a hierarchy of types of truth.

The liar's antinomy, or the problem of the truth predicate
The liar paradox arises from three devices: quotation, appending, and the truth predicate. Consider the following sentence: If we carry out the instruction presented in (2) by appending this nine-word sentence to its own quotation, we have: Hence (2) says that (1) is not true—but (1) and (2) are the same sentence! (1) is saying that (1) is not true ...
 * 1) 'Is not true when appended to its own quotation.'
 * 1) 'Is not true when appended to its own quotation'is not true when appended to its own quotation.

The antinomy arises when the truth predicate is included in the object-language alongside such innoxious devices as quotation and appending. For instance, the following object-language sentence does not include the truth predicate, and hence, evinces no such antinomy: Indeed, this problem pertaining to the truth predicate, can be tackled using the notion of disquotation. The truth predicate can be legally said to disquote a sentence S—if and only if—the schema: comes out as true when S is named in the first blank and written in the second blank. Notice that in (4) the truth predicate cannot legally occur in the first blank.
 * 1) 'Snow is white.'
 * 1) '____'is true if and only if____

The truth predicate is incomplete: it cannot refer to itself or to equivalent parapahrases (e.g. 'is the case that') on pain of contradiction. On the other hand, a sentence that can be legally substituted into (4) is (3), thus: Indeed (5) is extravagant, for it is equivalent to just asserting 'Snow is white'. However, these perplexities do not imply that the truth predicate is useless, for even in elementary logic we may deal with infinite conjunctions (especially in maths) where, obviously, we cannot assert each and every conjunct. As it happens, we can circumvene this antinomy by utilising an infinite hierarchy of truth predicates, whereby as we venture up the hierarchy we encounter stronger, and stronger, truth predicates—each one capturing more truths than the last, but alas never all.
 * 1) 'Snow is white'—is true if and only if—Snow is white.

The method is this: sentences that contain no truth-predicate, such as 'Snow is white', belong to the bottom level of the hierarchy; call the predicates of this level truth0. Truth0 sentences, are disquoted in the fashion of (5) tout court: At the next level of the hierarchy we encounter true1 truth predicates, which disquote all the sentences in true0. For example: Notice how in (7), on the left blank, we are naming true0—everything that occurred at the first level of the hierarchy which is signified by the double quotation marks; whereas on the right blank we are simply writing, unencumbered by additional quotation marks, the name of the truth predicate from the first level.
 * 1) 'Snow is white'—is true if and only if—Snow is white.
 * 1) "Snow is white'—is true if and only if—Snow is white'—is true if and only if—'Snow is white'—is true if and only if—Snow is white.

And so on-up this process of disquotation continues, with each level allowing us to disquote even more truths than the last.

This process of disquotation is largely due to Tarski's explication of truth and his indefinability theorem, but it also bares some of the hallmarks of Russell's early theory of types by which he, in his own way, tried to use to abrogate the liar's antinomy.

Epilegomena
An epilegomena is a formal essay at occurring at the end of a work, or more precisely an essay occurring upon and in addition to a prior essay. It aims to extend, albeit tentatively, the work of the previous essay, towards a future goal e.g. another essay, a book, a monograph. Above all, epilegomena indicates that the work is meant to be systematic—it should be the opposite of myopic.

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Quine's comprehension schema—from his axiomatic Class theory New Foundations(NF, henceforth)—employs the notion of a stratified formula, as opposed to Zermelo's restrictions, or Russell's theory of types. Both of the latter theories---Zermelo's and Russell's---employ restrictions on predicativity that prevent us from being able to explore a Universal Class in Zermelo's case; and in Russell's case, his prodigality has left us with a bloated, unusable theory which Randall Holmes wryly noted is akin to a "Hall of Mirrors" effect—Russell's theory of types, both simple and ramified, begets a Universal set at every level of the type hierarchy.

Quine's motive in NF is to try and avoid the problems engendered by the Russell Class without restricting our mathematics in a way that does violence to the insights and aspirations of naive class theory. However, it should be noted that ... NF is not meant to supersede ZFC; the hope is, rather (in and amongst other things) that the combination of NF with ZFC could provide a richer, synoptic image of mathematics—allowing scientists to fuse into one stereoscopic image, both the image of maths with a universal class and that of mathematics with a cumulative-hierarchy.

Crucially though, the importance of NF and ZFC lies in their utility for science, for applied mathematics.

NF axiom schema of comprehension
Formulae are said to be stratified if and only if there is a function that maps segments of syntax to an initial sequence of the natural numbers, such that we have the following comprehension schema:
 * 1) {xεy: f(y)=f(x)+1}.

Moreover, as was discovered by Halperin (1944), the reference to types implicit in the notion of stratification can be eliminated by demonstrating that there is an injective function that can substitute a finite conjunction of terms as instances of 'x'; this procedure allows for the finite axiomatisation of NF.

Combinatory logic
What follows is my use of Quine's comprehension schema with Hailperin's modification, to demonstrate how the problems of the truth predicate can be avoided without having to embrace a hierarchy of types: where y = f(x+1).
 * 1) ((S) x z (K K) y)
 * 2) (S x z (K K y)
 * 3) (S x z (K y(K y)))
 * 4) (x z ( = y)
 * 5)  x z = y