[FOM] consistency and completeness in natural language

[Hartley Slater]

Torkel Franzen (FOM Digest Vol 3 Issue 20) asks:

>Could you elaborate on what you take to be the illuminating argument 
>or conclusion put forward in [Neil Tennant's July 2002] paper? You
>mention only that Tennant has argued that we know that reflection
>principles hold for the systems in question, but this can hardly be
>what you regard as the specific contribution of the paper.

Of course it is not just a matter of knowing the reflection 
principles hold, but more fully of obtaining 'Goedel Phenomena' by 
'Doing without the truth-predicate by using reflective extensions [to 
system S]' (section 8.4).  Tennant's major point, as I see it, is 
'one does not need T at all' (p571).  Certainly there are others:

>The central argument of the paper turns on whether we make essential
>use of a "thick" notion of truth in arriving at the truth of the Godel
>sentence for a system S. Tennant argues against what he calls the
>"substantialist dogma" according to which "the way in which the
>semantical argument establishes the truth of the G–del sentence
>requires that the notion of truth be substantial", by giving a
>"deflationary" way of carrying out the argument, using a reflection
>principle.

Franzen goes on to argue against the idea that a 'thick' or 'fat' 
notion of truth was ever involved:

>My impression is that the argument of the paper is based on a
>misunderstanding.  Tennant presents a "Semantical argument for the
>truth of the Godel sentence" in a formulation that he attributes to
>Dummett, and this is the argument he wishes to replace. But this
>"semantical argument" is an odd one, and I don't agree that it is the
>argument put forward by Dummett.

But my interest starts with questioning whether Tennant's 
replacements involve (even) a 'deflationary' or 'thin' notion of 
truth.  Continuing the metaphor, the replacements he provides have 
zero thickness!  Trivially, and vacuously, one can replace the used 
sentence, P, on the right-hand side of a reflection principle, with 
TP, where 'T' is a truth operator (i.e. the null or identity operator 
in the modal system T), but my interest starts from the recognition 
that the reflection principles Tennant is concerned with have 
abandoned entirely any truth-predicate.  That puts the emphasis on 
the simple contrast between the mention and the use of the same 
formula as in for instance (p572):
	(n)(Prov-in-S 'Psi(_n_)'  -> Psi n).
It is the further significance of that pure shift from mention to use 
which I was concerned with in my previous posting (FOM Digest Vol 3 
Issue 18).

There is also the shift from the numerals within the system (like 
'_n_') to the possible replacements for variables outside the system 
(like 'n'), since the latter are not necessarily confined to 
numerals.  I will be reading a paper at this year's LOGICA conference 
on this matter, if anyone on the list is also going to Kravsko.  It 
is called 'Hilbert and Goedel versus Turing and Penrose'.  My point 
there (summarised) is that, in formulations of Arithmetic using the 
epsilon calculus rather than simply the predicate calculus, epsilon 
terms are possible replacements for the variables outside the system. 
But then, from '(n)Psi n' one can obtain, in particular, 'Psi en~Psi 
n' (where 'e' is epsilon), which is, of course, '(n)Psi n'.  So that 
is how we humans can beat Turing machines, and get universal 
conclusions they cannot:  From "(n)(Prov-in-S 'Psi(_n_)')" there does 
not follow "Prov-in-S '(n)Psi n'", simply because, although in 
epsilon Arithmetics epsilon terms refer to numbers, they are not 
numerals, and only numerals are used in the goedel numbering of 
formulae in the system.  If the formal system could directly speak 
about numbers, things would be different, but it is confined to 
speaking about numerals.

That point is also linked to the defeat of formalism inherent in 
Goedel's Theorems.  Numerals are all attributive terms, in fact one 
can determine their referents from the terms themselves.  If one 
thinks of Arithmetic as essentially involving such terms, then it 
becomes plausible that we can learn all facts about numbers from 
facts about the signs which refer to them; i.e. we can be persuaded 
of a form of formalism.  But epsilon terms are not necessarily 
attributive, as has appeared, for instance, in previous postings of 
mine on FOM.  Expanding our language to include them enlarges what 
becomes provable in several ways, not just the one indicated above.
-- 
Barry Hartley Slater
Honorary Senior Research Fellow
Philosophy, School of Humanities
University of Western Australia
35 Stirling Highway
Crawley WA 6009, Australia
Ph: (08) 9380 1246 (W), 9386 4812 (H)
Fax: (08) 9380 1057
Url: http://www.arts.uwa.edu.au/PhilosWWW/Staff/slater.html