[FOM] Retraction: no objection to Friedman's Concept Calculus Paper

Frode Bjørdal frode.bjordal at ifikk.uio.no
Sun Aug 18 15:33:42 EDT 2013

Friedman in private communication points out that his original definition
of y>exPhi works. He is absolutely right! It is indeed straightforward to
show that y>exPhi under Friedman's original definition entails y>Phi, and
this takes the sting out of my earlier posted objection.

My interest in the Concept Calculus was indicated in my post Remarks on the
Concept Calculus,
http://www.cs.nyu.edu/pipermail/fom/2013-August/017473.html, and stems from
my own use of an important earlier 1973 work by Friedman referred to there
in order to show that my own foundational system librationism,
*soberly* interprets ZF if ZF has a standard model; I lectured on this at
the UNILOG2013 in April in Rio de Janeiro and material is available for
those interested.

Friedman usefully points out that there are some matters to learn from the
exchange. Firstly, inquire offlist before publishing criticism as the
moderator may not always exercise sufficient caution, and, secondly, do not
automatically assume that there is merit to criticism. One may consider
that it had been useful to me, and perhaps to others, had y>exPhi => y>Phi
been included as a lemma or an easy exercise.

I am still interested in whether the Concept Calculus is most naturally
interpreted by something like Friedman's S of 1973.

Frode Bjørdal

2013/8/16 Frode Bjørdal <frode.bjordal at ifikk.uio.no>

> 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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