[FOM] On >>this sentence cannot be proven true<<

[Hartley Slater]

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.


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