[FOM] More than a Typo in Friedman's Concept Calculus Paper

[Frode Bjørdal]

Friedman's typographical adjustment does not rectify the Concept Calculus.

1: The definition Friedman now suggests, y>exPhi iff Forallz(y>z iff
Existsx(Phi and (x=z or z>x))), is not materially adequate as the R.H. may
hold on account of Phi holding for x much smaller than y whereas we would
not want the L.H. to be true in such cases.

2: Moreover, Phi may hold for some x incomparable with y, and so my
objection to the proof of Lemma 8.2 retains its full force.

3: It seems clear that the amendment I suggested is a formalization which
better captures the informal account invoked by Friedman in justifying what
he took to be just a typographical error.

4: However, a more elegant recommendation is that one in defining y>exPhi
includes the additional conjunct y>Phi. It may well be that the Concept
Calculus is rectifiable with the latter amendment of what was taken as a
typographical error, but this should be checked.

Frode Bjørdal



2013/8/15 Harvey Friedman <hmflogic at gmail.com>

> I just saw a typo in the definition of y >ex phi. Change x > z to z > x.
>
> This is an obvious typo because of the informal definition of "exactly
> better" in the section Better Than, Much Better Than, which reads:
>
> "We say that x is exactly better than a given range of things if and only
> if x is better than every element of that range of things, and everything
> that something in that range of things is better than, and nothing else."
>
> Harvey Friedman
>
> On Mon, Aug 12, 2013 at 2:18 PM, Frode Bjørdal <frode.bjordal at ifikk.uio.no
> > wrote:
>
>> As I have attempted to come to more clarity on Friedman's Concept
>> Calculus it turns out that Lemma 8.2. of
>> http://www.math.osu.edu/~friedman.8/pdf/ConcCalcInf103109.pdf is false.
>>
>> In defining y>exPhi on page 13 Friedman has Forallz(y>z iff Existsx(Phi
>> and (x=z or x>z))).
>>
>> However, in proving Lemma 8.2 on page 28 it is presupposed that y>exPhi
>> iff Forallz(y>z iff Existsx(Phi and (x=z or y>x>z))).
>>
>> But the proof of Lemma 8.2 cannot be rectified even if one changes the
>> definition og y>exPhi to the one presupposed in the text. For the step
>> indicated by «Suppose Phi. Then x < y.» is a non sequitur, as nothing
>> prevents that Phi holds for an x which is incomparable with y.
>>
>> It may be that a rectification may be had by presupposing the more
>> elaborate definition y>exPhi iff Forallz(y>z iff Existsx(Phi and (x=z or
>> y>x>z)) and not Existsx(Phi and x incomparable with y)).
>>
>> In my previous post Remarks on the Concept Calculus I make a suggestion
>> for domain for MBT which is not quite apt; nevertheless, it may be that a
>> rectified Concept Calculus may be modelled by elaborating appropriately
>> upon my suggestion.
>>
>>
>> Professor Dr. Frode Bjørdal
>> Universitetet i Oslo Universidade Federal do Rio Grande do Norte
>> quicumque vult hinc potest accedere ad paginam virtualem meam<http://www.hf.uio.no/ifikk/personer/vit/fbjordal/index.html>
>>
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