[FOM] Self-reference in natual, languages (re >>this sentence, cannot be proven true<<)"

[Haim Gaifman]

Quine's construction of self-referential sentences in English
simulates the Goedel-Carnap construction for formal languages.

The simulation works.
Contrary to what Slater argues (if I understand him correctly),
there is no ambiguity that undermines it.
Apparently the issue could do with some clarification.

For this purpose, consider first a
hybrid construct in a language obtained by adding to English
variables, 'x', 'y',..., ranging over
strings of symbols. If F and and G are expressions (strings of
symbols), and F contains a single variable,
define the result of applying F to G to be the result
of substituting the variable in F by the quoted G.

Note that the result of applying x to x is the string obtained from the string
x by substituting the variable that occurs in x by the quoted x.

Now consider the sentence:

(1) The result of applying x to x is A

where 'A' stands for any adjective in the language, such as
'false', 'true', 'short', 'grammatical' etc.

Then (1) expresses a property of the string x. The sentence that asserts that
(1) (as string of symbols) has this property is:

(2) The result of applying 'The result of applying x to x is A' to 'The
result of applying x to x is A' is A.

Let R be the result of applying 'The result of applying x to x is A' to 'The
result of applying x to x is A'. Then (2) states that R is A.

Now it is easy to verify that R is equal to:

' The result of applying ' The result of applying x to x is A' to 'The
result of applying x to x is A' is A '.

And this string is (2). Consequently, if '....' stands for this string
we get:

.... <==> '....' is A
Namely we get a fixed point.

In order to convert this into an English example we have to get the
same effect without using variables. In the hybrid case we used the
substitution that converts F(x) to F('...'). Since in English the
subject precedes the predicate
the effect is achieved by appending the latter to the quoted noun-phrase.
The effect of variable substitution is achieved by using the descriptive scheme

'appended to its quote' [or, if one prefers it, 'when appended to its own 
quote'].

In this scheme 'it' (or 'its') does not
refer to anything. But this is OK, since we are considering operations on 
and properties of
arbitrary strings. All that is needed is that we get at the end an unambiguous
description. Indeed, the construction leads to:

(3) '....' appended to its quote

Here 'it' refers obviously to '....' The string that is described by (3) is:

' '....' .... '

Thus, 'appended to its quote is grammatical' can be used to describe
a property; '....' has this property iff ' '....' ....' is grammatical.

For comparison consider the expression 'together with her sister'. The
pronoun does not refer. But it does refer unambiguously in:


(4) Mary together with her sister

Thus 'together with her sister took a trip' can be used to describe the
property that is true of Mary iff Mary together with her sister took a trip.

One should note that the possibility of finding a natural language
simulation of the Goedel-Carnap construction may depend on linguistic
features of the language in question. For example, Quine's trick does
not carry over to Hebrew; it seems that a simulation there would
require terminological innovation.
The question may have some linguistic interest, but certainly
not a philosophical one.

The construction that is based on indexicals is not ambiguous either.
It is true that the references of the names depend, eventually, on
some demonstratives. The self-reference is achieved because certain
facts hold in the actual world (e.g., a certain sentence is written in
a certain place). So what? The paradox is a paradox in our actual world,
invoking possible worlds is of no help and irrelevant.

Finally, it is true that, on mereological grounds, a sentence cannot
refer to itself by containing as a proper part its own quote. But it can
refer to itself by containing as a proper part a descriptive name of
itself, e.g., 'the sentence stored in location X is not true', refers to
itself, provided that this sentence is the only sentence stored in
location X. It seems that in Slater's argument he infers 'a' = 'b' from a = 
b. But this is
in general wrong. We can have:
The sentence stored in location X = The sentence uttered by John at
time t,
but surely,
The sentence stored in location X =/= The sentence uttered by John at
time t.

Haim Gaifman

Hartley Slater wrote:
Following on from the previous discussion, Laureano Luna, in a
private email, has asked me to show how ambiguity arises in the
following, Quine-type case: " 'appended to its own quotation
expresses no true proposition' appended to its own quotation
expresses no true proposition".  The point is of wider interest.

In fact I have discussed the original Quinean case already (see 'A
Poor Concept Script', Australasian Journal of Logic (2004)
(http://www.philosophy.unimelb.edu.au/ajl/2004/2004_4.pdf).  The
ambiguous term in it is the pronoun 'its', which gains its reference
from some preceeding expression, but which is left dangling without
such in the doubly-quoted place.  More generally, one can give a
number, or individuating description, locating some sentence or
phrase '....its....', but if the pronoun's antecedent is not within
that sentence or phrase then the sentence or phrase will not have a
fixed meaning, nor therefore, if it is a sentence, a truth-value.
The case is like the case of heterologicality, where the predicate
'is not self-applicable' contains another pronoun, 'self', which is
without its antecedent in the predicate.  Hence that predicate does
not express a fixed property.  I pointed that out even as far back as
1973, in the journal MIND, and the point is repeated in the above
publication (see also, for instance,'Choice and Logic', Journal of
Philosophical Logic. 43 (2005), 207-216).

Those two cases, along with the elementary 'Liar' case 'this sentence
expreses a proposition that is false', where ambiguity is showable by
direct argument clearly give good inductive grounds for the belief
that there is ambiguity in all cases, on top of the (separately
conclusive) indirect argument, via Reductio, that I presented before.
But the most valuable, additional benefit of looking at them is that
they show what kind of thing has been missed by people who have been
blind to the ambiguity in them: features of language, like
demonstratives and pronouns, which were not studied within logic
until recently.  Indeed, at one time, Quine's own 'eternal sentences'
were supposed to be able to eliminate all such contextual items from
sentences which could enter into logical forms.

The problem is that the contextuality of things like ''the sentence
at the top of p. n of book X does not express a true proposition' (in
this case its variability of truth value with respect to different
possible worlds) cannot be removed.  If it could be then we would
have something not just of the form "a = 'a does not express a true
proposition' ", but of the form
'p' = " 'p' does not express a true proposition",
where the descriptive phrase (or number) 'a' is replaced with a
quotation name " 'p' ".  But mereology prevents this: " 'p' does not
express a true sentence" is longer than 'p', so they cannot be the
same sentence.