[Inquiry] Simple Meanings In Limnal Expressions

[Jon Awbrey]

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SMILE.  Note 1

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Peirce List,

I need to tease out a thread that's gotten too tangled
in talking about the futures of logical graphs to take
up there but that seems to demand an independent focus.

It's a turbulent bunch of topics, and I had my trouble
thinking up a truly apt title to tie them together and
take them up under -- I'll try a tentative one for now.

In thinking about this unruly gang of issues I found myself
drawn back to a time when I fretted about the bonds between
math and logic, and to a place where Peirce tries to tackle,
in his own inimitable way, of course, quite a few facets of
the question that appears to lurk within or underlie it all.
And so I was tempted to take up my theme under the limnings
of "Simple Mathematics In Logical Expressions", but then it
struck me that prior notions about logic or math might make
that script just a bit too constrained to give a free reign.

Anyway, taking things in no particular order, the first trial text
that comes to mind is lodged in the framing of "Qualitative Logic".

Here is one place where Peirce draws distinctions
among a "language", an "algebra", and a "calculus".
Of course, nothing about this means that he is not
free to draw them differently in different places:

| Here then we have a written language for relations of dependence.
| We have only to bear in mind the meaning of the symbol ^+ (not by
| translating it into 'if' and 'then', but by associating it directly
| with the conception of the relation which it signifies), in order to
| reason as well in this language as in the vernacular, -- and, indeed,
| much better.
|
| So far, we have a 'language' but still no 'algebra'.
| For an algebra is a language with a code of formal
| rules for the transformation of expressions, by
| which we are enabled to draw conclusions without
| the trouble of attending to the meaning of the
| language we use.
|
| Our algebraic rules must enable us to prove the two propositions:
|       _____
|    If 'a' + 'b', then if 'a' then 'b';  and
|                                  _____
|    If from 'a' follows 'b', then 'a' + 'b'.
|
| Any rules which will prove these propositions will evidently enable us to
| prove every conclusion.  But that is not enough;  for we require that the
| rules should enable us to dispense with all reasoning in our proofs except
| the mere substitution of particular expressions in general formulae.
|
| We do not as yet demand rules which shall enable us to dispense with
| difficult reasoning in discovering the truth and in inventing modes
| of proof, -- that would be demanding more than an algebra, namely
| a 'calculus', but we do require that in the proofs themselves
| nothing but simple substitutions shall be called for.
|
| The first of the above propositions, however, namely,
| that if 'a' and 'a' ^+ 'b' then 'b', being nothing but
| the 'modus ponens', might stand as a fundamental rule.
|
| The second proposition may be divided into two,
| each of which follows from it while it follows
| from them dilemmatically.  They are:
|                     _____
|    If not 'a', then 'a' + 'b';  and
|                 _____
|    If 'b', then 'a' + 'b'.
|
| C.S. Peirce, "Qualitative Logic", NEM 4, 107-108
|
| C.S. Peirce, "Qualitative Logic", MS 736, pp. 101-115 in:
| Carolyn Eisele (ed.), 'The New Elements of Mathematics by
| Charles S. Peirce, Volume 4, Mathematical Philosophy',
| Mouton, The Hague, 1976.
|
| Cf. C.S. Peirce, "Qualitative Logic", MS 582 (Fall-Winter 1886), pp. 323-371 in:
|'Writings of Charles S. Peirce:  A Chronological Edition, Volume 5, 1884-1886',
| Peirce Edition Project, Indiana University Press, Bloomington, IN, 1993.

NB.  In this transcription, I am using the compound symbol ^+
     as an in-line version of Peirce's streamer-cross symbol:
     ___
       +

By way of context, here are some previous citations and discussions:

QUAL.  Qualitative Logic
QUAL.  http://stderr.org/pipermail/inquiry/2005-May/thread.html#2697
QUAL.  http://stderr.org/pipermail/inquiry/2005-June/thread.html#2776

QUAL.  Qualitative Logic -- Commentary
QUAL.  http://stderr.org/pipermail/inquiry/2005-May/thread.html#2704
QUAL.  http://stderr.org/pipermail/inquiry/2005-June/thread.html#2788
QUAL.  http://stderr.org/pipermail/inquiry/2005-July/thread.html#2848

QUAL.  Qualitative Logic -- Discussion
QUAL.  http://stderr.org/pipermail/inquiry/2005-May/thread.html#2706
QUAL.  http://stderr.org/pipermail/inquiry/2005-July/thread.html#2867

Jon Awbrey

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inquiry e-lab: http://stderr.org/pipermail/inquiry/
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