ONT Re: Apposite Purposes Of Logical Languages Objectified (APOLLO)
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This an area where I am still trying to get things clear in my own head.
There are pieces of the puzzle that seem solid by themselves, taken one
at a time, but that I am still wrestling to fit into a coherent picture.
We are conducting an inquiry into the pragma of logic.
The praxis of logic being to address, bring about, or
capture this object, in other words, to fasten it for
a moment it in the forms of our human conception, one
anticipates that the sense of its ending will bear on
the mean-time conduct of logic as much as the mediate
struggle of reasoning bears on the pragma in question.
If we know the object of logic then it tells us about
the objectives that logical media ought to facilitate.
Logic being a normative science, an agent pursuing it
is duly invested with the power to critically reflect
on different ways of carrying out its procedures, and
duty bound to examine their comparative effectiveness
in achieving the overall object of logic. A critique
of anything depends on an ability to make comparisons,
and this involves an ability to set the thing studied
within a setting of generously generated alternatives.
In sum, a critically reflective logic must empower us
to consider alternative means for achieving its goals,
to place side by side a sample of different, but more
importantly, independent paths toward the same pragma.
Our search into the relation between logos and pragma
led us to examine the place of logic within a general
theory of signs:
We considered Aristotle's account of the sign relation:
The first impression that we have of any pragma is the
pathema that it strikes in our soul, and all the other
signs and symbols that we form of this object are only
so many chips off the imprinting of that initial block.
We glanced at the supreme silhouette of our entire subject
that the prescient kindergartener C.S. Peirce prescissorly
clipped out of spare materials found lying about the house
and pasted into one of the most constructive papers of all
those leaves that we see interleaved in his vast aftermath:
| We are now ready to assign logic its definition.
| For this purpose, it must be granted that logic is
|
| a symbol
| whose essential end is
| to test
| ( truth,
| <
| ( by reasons.
|
| CSP, CE 1, pages 328-329.
|
| Charles Sanders Peirce, "Harvard Lectures 'On the Logic of Science'", (1865),
|'Writings of Charles S. Peirce: A Chronological Edition, Volume 1, 1857-1866',
| Peirce Edition Project, Indiana University Press, Bloomington, IN, 1982.
And, more than twenty notes later, I profess that I am still not far enough
into my own trial of a symphony even to touch the grandiosity of that theme.
Though I take these two lights as "stars to steer by" in my inquiry,
it is not my place to sail among the stars, but far below them, and
so I must lean to the form of steerage that is my fated oar to bear.
There are many ways to explore a continent so vast as the one that we dub "Logic".
Some explorers will set foot on a beach at the mouth of a river, plant a standard
in the sand, grandly claim for crown and country the beach, the river, all of the
lands that are drained by its waters, and all of the contiguous territory even so
far as it may stretch to the adjacently bounding seas. Then they promptly rebark
on the barks that brought them, and sail on to the next beaconhead of convenience.
We might say that their expedition has expedited the quest of those who follow up,
but it is an altogether different brand of explorer who will pursue understanding
deep into the interior, and perhaps be able to converse with the natives who have
had their homelands thus "discovered".
So let me then return to my more pedestrian explorations,
all of whose issues have no doubt been resolved long ago,
and far away, back within the folds of a civilized world,
by advances in trigonometry, figurative or literal, that
are bound to have occurred since the days when I took or
even since the days when I myself taught this discipline.
If we stick to a fragment of logic like ZOL(p, q), we would probably
feel safe in saying that the purpose of this fragment is to describe
as clearly as possible, perhaps for our own sakes alone, but perhaps
for the sake of communicating or conveying to others, one particular
mathematical object, namely, the lattice of 16 functions, {B^2 -> B}.
But we would like our grasp, our conception of any such object to be
as abstract, categorical, generic, invariant, or universal as we can
possibly make it, which is to say that we desire our notions of such
objects to be formulated in such a fashion that they are not subject
to the fading fad of fashion, but in such a way that renders them as
independent of any particular language, medium, or representation as
we can tailor them.
That is where the category-theoretic idea of a "universal" comes in.
Jon Awbrey
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