|
|
I thank you both Abo and Srinivasan for your responses.
But none of you tell why D is ill-formed; you just go to the
contradiction in the result. Note that Priest would not be convinced by
this kind of argumentation.
Srinivasan says he doesn't accept a definition containing 'this
definition'. But 'the last letter in the written form of this
definition' looks like an admissible definition of the letter 'n' or at
least an expression unambiguously identifying it.
It seems obvious that we should reject D on the grounds of circularity.
Perhaps what is not admissible is that D contains not just 'this
definition' but 'defined by this definition' , and this renders evident
that D means to be built upon itself, so to say.
Regards
R. Srinivasan wrote:
> On Oct 22, 11:45 pm, "R. Srinivasan" <sradh...@xxxxxxxxxx> wrote:
> > On Oct 22, 9:12 pm, "LauLuna" <laureanol...@xxxxxxxx> wrote:
> >
> >
> >
> >
> >
> > > Graham Priest in 'Is Arithmetic Consistent?' (Mind v. 103, 411, p.
> > > 337-349) proposes what he calls 'Petersen's argument' to prove there is
> > > some natural n such that n=n+1.
> >
> > > The following is a simplified version that I think retains all the
> > > essential.
> >
> > > Call 'D' the following definition:
> >
> > > 'the natural number which this definition defines (or 0 if it fails to
> > > define) plus 1'
> >
> > > Now it seems evident that D defines some natural, for if it didn't,
> > > then according to D itself, it would define 1 (0+1).
> >
> > > Call 'n' the number defined by D. According to D, n equals the number
> > > defined by D plus 1. So n = n+1.
> >
> > > How and why should this argument be rejected? I'm interested in your
> > > way to put it.Graham Priest seems to have plenty of time for stuff like
> > > this, but not
> > even one minute to spare to look at my work on NAFL (I had sent him an
> > email or two a couple of years ago, but got no reply). Surprising,
> > since NAFL is close to a paraconsistent logic.
> >
> > Anyway, I agree with abo, D is just not a "definition" of anything. And
> > I don't accept using "this definition" within a purported definition.
> > Also note that D already assumes that the natural numbers are
> > pre-defined. So it is just a play with words, like "guess what natural
> > number this sentence refers to".
> >
> > Anyway, playing along, I would use the following arguments.
> >
> > (1) D does not define a unique natural number. Assuming that D defines
> > some natural number x results in the conclusion that D also defines
> > x+1. Conversely, assuming that D defines more than one natural number
> > also results in the same conclusion. So if at all D defines any natural
> > number, it defines more than one.
> >
> > (2) D defines at least one natural number. Assuming that D defines
> > nothing or that D defines something that is not a natural number
> > results in the conclusion that D defines the number 1.
> >
> > (3) If D defines the number n, D also defines the number n+1. By
> > induction, D defines all natural numbers > n.
> >
> > (4) The assumption that D defines a number n>1 also leads to the
> > conclusion that D defines n-1, n-2, ....1. How? Suppose D defines 5.
> > Looking at D, it is clear that D could define 5 only after first
> > defining 4 and adding 1. Similarly D must already have defined 3, 2, 1.
> >
> > (5) From the above, conclude that D defines the natural numbers 1, 2,
> > 3, .....
> >
> > (6) D could define the number 1 only if also defined some entity X that
> > is not a natural number. Substituting X into D leads to the conclusion
> > that D defines the natural number 1 (since X is not a natural, we
> > substitute 0 for X as per the prescription and add 1 to it).
> >
> > (7) Conclusion: D defines at least one unspecified entity X that is not
> > a natural number as well as the natural numbers 1, 2, 3, ....
> >
>
> The entity X could be the class of all positive natural numbers
> {1,2,3,....}. At least from the point of view of NAFL, defining each of
> 1, 2, 3, ... implies that {1,2,3,....} has also been defined.
>
> Regards, RS
|
|