ONT Re: Question Of Logic
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Joseph Ransdell wrote (JR):
Seth Sharpless wrote (SS):
Jon Awbrey wrote (JA):
SS, quoting CSP:
| Logical principles of inference are merely rules for the illative transformation
| of the symbols of the particular system employed. If the system is essentially
| changed, they will be quite different. (CP 2.599).
JA: Yes, but Peirce would be rather acutely aware that the significant features
of transformations on any space, whether algebraic, geometric, or syntactic,
would be the invariant properties, the kinds of "objective qualities" that
are not affected by the vicissitudes of the arbitrary metamorphoses in the
conventionally arbitrated bases of representation.
JR: Jon is surely correct on this, and the surrounding context for those
sentences actually seems as if designed to make that clear. Here is
a part of that context, the point at issue there being that it is one
thing to reason and quite another to represent an argument logically
for purposes of critical assessment.
JR, quoting CSP:
| [A reasoner] does not, in strict accuracy, reason in any form of syllogism.
| For his reasoning moves in first intentions, while the forms of logic are
| constructions of second intention. They are diagrammatic representations
| of the intellectual relation between the facts from which he reasons and
| the fact which he infers, this diagram necessarily making use of a particular
| system of symbols -- a perfectly regular and very limited kind of language.
| It may be a part of a logician's duty to show how ordinary ways of speaking
| and of thinking are to be translated into that symbolism of formal logic;
| but it is no part of syllogistic itself. Logical principles of inference
| are merely rules for the illative transformation of the symbols of the
| particular system employed. If the system is essentially changed,
| they will be quite different. (CP 2.599, 1902).
JR: The reason for the reference to syllogistic is that Peirce was
illustrating the point with use of two different figures of the
syllogism. (Also, Peirce sometimes uses the word "syllogistic"
and its cognates in its original sense, which just meant any
deductive form: the Aristotelian form so-called is actually
just a special case of syllogism, in that sense.) The same
point could be made by contrasting the transformation rules
of his Existential Graphs system and, say, those appropriate
to his earlier Entitative Graph system, which differed from
the later system chiefly in treating conjoint inscription on
the sheet of assertion as conjunction rather than inclusive
alternation. Or, more to the point, perhaps, the difference
between the Existential Graph system and any of the variations
on the algebraic type of notation as regards transformation rules
since one of the motives of developing the graph system was to be
able to develop a notation for logical representation that required
no inference principles that were required merely to compensate for
the peculiarities of the notation itself, such as, say, the principle
that enables you to switch the positions of p and q in conjunctive
expressions, which is simply gratuitous in the graphical system
he developed.
JR: I trust that someone will correct me if I am wrong, but my understanding
is that logic as usually construed since Peirce cannot even distinguish
between an inference principle (rule) which is an artifact required
only because of a peculiarity of the notation and a principle
required by logical considerations regardless of notation.
Seth, Joe, many thanks for this! Seth's citing of the initial passage,
Joe's highlighting of the context, and his further remarks about the
difference between 1st and 2nd intentions help me to elucidate what
I have been struggling to express for several days with regard to
a related question on the OCA List, for instance, where I say:
JA: We seem to be about the business of marking explicitly what's given implicitly.
But is it a mark against our ephemeris that celestial bodies do not consult it
for their itineraries, nor have the eyes in their orbs to read our fine prints,
that the planets are enlightened by the sun on their courses and their destiny
by some adumbration other than differential equations in plain black and white?
JA: The point here is that a different order of being comes into play
when being begins relating itself to itself via the automediation
of signs. Being a material and a natural thinker, I see this all
taking place within the order of matter and nature, as an utterly
internal development, differentiation, and "ontologogenesis" of a
cosmos, if you catch my drift, but nothing about saying this does
anything to diminish the import of the formal aspect of its being.
I also agree with Joe's assessment of all the other points, for instance,
I believe that I understand Peirce's attitude in thinking how ridiculous
it is to call a rule like "A&B = B&A" a "law of logic" (LOL), being what
is merely a rule that relates the (in)convenience of a particular syntax
to its "logical equivalence class" (LEC), at the gate of which only does
logic proper commence. Peirce's watching of the line between syntax and
semantics amounts to an acutely conscious recognition that we notary and
signatory species craft our languages to our own specifications but that
our capricious fashions of productivity only serve to populate the forms
that constitute an abstract, ideal, and logically invariant archetexture.
Mathematical logicians since at least the time of Gödel have been rather
acutely aware that we currently lack logical calculi and languages for
the "theory of recursive functions" (TORF) that enjoy the brands of
invariance properties that would normally be de rigueur in maturer
fields of mathematics. And so I would agree that the majority of
systems in widespread use today have the indiscretionary quality
that Joe notices, even if it is the business of a study that is
not too coincidentally called "Category Theory" whose regular
duty is to find the proper forms of natural ideals, and whose
application to logic has only just begun to make any inroads
against the defect. Examples of work in this direction are:
| Lambek, J. & Scott, P.J.,
|'Introduction to Higher Order Categorical Logic',
| Cambridge Studies in Advanced Mathematics 7,
| Cambridge University Press, Cambridge, UK, 1986.
| Barr, Michael & Wells, Charles,
|'Category Theory for Computing Science',
| Prentice Hall International Series in Computer Science,
| Prentice Hall, Hemel Hempstead Hertfordshire, UK, 1990.
Jon Awbrey
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