[FOM] First-order arithmetical truth

Hartley Slater slaterbh at cyllene.uwa.edu.au
Sat Oct 21 19:50:55 EDT 2006

Arnon Avron says:

>However, I prefer
>the "ancestral logic" version, because the notion of "ancestor"
>is part of everybody's logic, 100% understood also by
>non-mathematicians. Needless to say, the Natural numbers
>can be categorically characterized in this language.

That sounds like he means no more than that the natural numbers can 
be characterised 'up to isomorphism'.  So his 'ancestral logic' is no 
better than second order logic in specifying the one model required, 
i.e. the one that contains the natural numbers rather than some other 
objects with the same structure. (I made this point about second 
order logic before, see 
http://www.cs.nyu.edu/pipermail/fom/2006-October/010974.html )

The problem with identifying the natural numbers ought to defeat 
Avron, since it is the same basic problem as arises in connection 
with the Liar: getting hold of the standard interpretation of some 
symbols.  I said in connection with the Liar, amongst other things 
(see http://www.cs.nyu.edu/pipermail/fom/2006-October/010874.html ):

>The point is that it is *not* the sentence which has the property of
>truth; at the very least it is the sentence when interpreted a given

Avron replied to that (see 
http://www.cs.nyu.edu/pipermail/fom/2006-October/010875.html ):

>Personally, it seems to me that almost everything Slater
>is saying below (whether these are sentences or something
>else which supposed to have the property of truth)
>is neither true nor false. The reason may be that being
>a machine (while Slater is not), I cannot attach any meaning
>to this sequence of sentences...

But if he cannot attach a meaning to a sentence then he cannot attach 
a referent to a numeral, or a range of values to the variables in a 
formal Arithmetic.
Barry Hartley Slater
Honorary Senior Research Fellow
Philosophy, M207 School of Humanities
University of Western Australia
35 Stirling Highway
Crawley WA 6009, Australia
Ph: (08) 6488 1246 (W), 9386 4812 (H)
Fax: (08) 6488 1057
Url: http://www.philosophy.uwa.edu.au/staff/slater

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