Irving, Gary, Malgosia, list,

Yes, the theorematic-vs.-corollarial distinction does not appear in the Peirce 
quote to depend on whether the premisses - _up until some lemma_ - already 
warrant presumption. 

BUT, but, but, the theorematic deduction does involve the introdution of that 
lemma, and the lemma needs to be proven (in terms of some postulate system), or 
at least include a definition (in remarkable cases supported by a "proper 
postulate") in order to stand as a premiss, and that is what Irving is 
referring to. 

The confusion seems to be that, though Peirce says that deduction doesn't care 
whether the premisses are true (though it obviously cares that they be at least 
formally consistent), evidently it does matter whether they warrant presumption 
- are they supported by postulates already in place?, do they need another 
postulate?, etc.  This concern with the postulate system seems something native 
to 'pure' math, 

The rest of Irving's response seems to show that his 
theoretical-vs.-computational distinction and Pratt's creator-vs.-consumer 
distinction are made in terms of whether mathematical discovery and originality 
are involved, and are also, so to speak, at another level of scale. In other 
words, one could have merely computational-purposed and unoriginal definitions 
uncontemplated in a thesis. Moreover the computational math is not always out 
to prove a thesis, but to find a solution.

Another problem with calling computational/consumptorial math merely 
"corollarial" - Peirce said "Corollarial deduction is where it is only 
necessary to imagine any case in which the premisses are true in order to 
perceive immediately that the conclusion holds in that case." That's certainly 
untrue of a whole lot of computational deductions! - long calculations whose 
answers are anything but immediately evident. This may just be matter of 
allowing that a corollarial deduction, even if not involving an immediately 
evident conclusion, should at least be analyzable into steps each of which 
makes the next step immediately evident. The price is an alteration of Peirce's 
definition, and a less steep price would be instead to call such a deduction 
"multi-corollarial." or some such.

Best, Ben

----- Original Message ----- 
From: "Gary Richmond" <>
Sent: Monday, March 12, 2012 12:22 AM
Subject: Re: [peirce-l] Mathematical terminology, was, review of Moore's Peirce 

I *strongly* agree with your analysis, Malgosia.



On 3/11/12, malgosia askanas <> wrote:
> Irving wrote, quoting Peirce MS L75:35-39:

>>"Deduction is only of value in tracing out the consequences of
>>hypotheses, which it regards as pure, or unfounded, hypotheses.
>>Deduction is divisible into sub-classes in various ways, of which the
>>most important is into corollarial and theorematic. Corollarial
>>deduction is where it is only necessary to imagine any case in which
>>the premisses are true in order to perceive immediately that the
>>conclusion holds in that case. Ordinary syllogisms and some deductions
>>in the logic of relatives belong to this class. Theorematic deduction
>>is deduction in which it is necessary to experiment in the imagination
>>upon the image of the premiss in order from the result of such
>>experiment to make corollarial deductions to the truth of the
>>conclusion. The subdivisions of theorematic deduction are of very high
>>theoretical importance. But I cannot go into them in this statement."

>>[...] Peirce's characterization of theorematic and corrolarial
>>deduction would seem, on the basis of this quote, to have to do with
>>whether the presumption that the premises of a deductive argument or
>>proof are true versus whether they require to be established to be
>>true [...]

> I would disagree with this reading of the Peirce passage.  It seems
> to me that the distinction he is making is, rather, between (1) the case
> where the conclusion can be seen to follow from the premisses
> by virtue of the "logical form" alone, as in "A function which is continuous
> on a closed interval is continuous on any subinterval of that interval"
> (whose truth is obvious without requiring us to imagine any continuous
> function or any interval), and (2) the case where the deduction of the
> conclusions from the premisses requires turning one's imagination
> upon, and experimenting with, the actual mathematical objects
> of which the theorem speaks, as in "A function which is continuous
> on a closed interval is bounded on that interval".

> -malgosia-- 

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