The consequence of this fact is that when we wish to enumerate the sphere of a term — a process termed division — or when we wish to run over the content of a term — a process called definition — since we cannot take the elements of our enumeration singly but must take them in groups, there is danger that we shall take some element twice over, or that we shall omit some. Hence the extension and comprehension which we know will be somewhat indeterminate. But we must distinguish two kinds of these quantities. If we were to subtilize we might make other distinctions but I shall be content with two. They are the extension and comprehension relatively to our actual knowledge, and what these would be were our knowledge perfect.
Peirce, CE 1, 462
Peirce, C.S., “The Logic of Science; or, Induction and Hypothesis”, [Lowell Lectures of 1866], pp. 357–504 in Writings of Charles S. Peirce : A Chronological Edition, Volume 1, 1857–1866, Peirce Edition Project, Indiana University Press, Bloomington, IN, 1982.
Additional References
- Excerpts on Definition
- Excerpts on Determination
- Relation Theory • OEIS Wiki • PlanetMath
Incidental References
- Cathy O’Neil • The Art Of Definition
- Peter Cameron • (1) • (2) • (3) • (4)
cc: Inquiry List • Peirce List (1) (2) (3)
Inquiry Into Inquiry 
“ … every part however small is divisible … Hence the extension and comprehension which we know will be somewhat indeterminate.”
“Suffice it to say that a sign endeavors to represent, in part at least, an Object, which is therefore in a sense the cause, or determinant, of the sign even if the sign represents its object falsely.”
(Peirce)
Typically signs are objects of higher compression (or lower resolution or complexity) than the objects they represent, but in some cases the reverse may be true.
I think of knowledge as consisting of networks of associations. If each association has a probability and each network has a geometry, knowledge isn’t built much differently from physical stuff.
Presumably the brain uses a wide variety of relatively specialized algorithms and heuristics (evolved and learned) depending on the kinds of signs, objects or data types and structures involved in a task.
How useful Is fractal geometry for representing recursive networks of objects-associations?
Mirror reflections are not yet critical reflections. You have to keep an eye out to keep on comparing and contrasting images.
Dyadic correspondences between objects and signs may serve as stepping stones to the necessary triadic co-respondence among objects, signs, and interpretant signs, but it takes a leap to cross the gap to a comprehensive grasp of the object — and that leap is not of necessity reducible to the steps that led up to it.
A mirror of silver and glass makes no critical analysis of what it reflects, but we think that few (if any) reflections are perfect. When light reflected from objects in our field of vision enters the eye a series of additional reflections are created by our optical and visual systems. The eye and brain apply adaptive-corrective algorithms all along the way. Among these are associations with previously recorded and computed objects, signs, rules, etc.that provide context and some critical analysis — a comprehensive (in some degree) grasp. Many leaps and grasps have occurred before we are consciously aware of an image at all.
What we know or comprehend about something is largely (wholly?) based on contrasts and comparisons (associations) with other stuff we already know. The original vestige of prior knowledge presumably comes somehow from the DNA and possibly other materials of the fertilized embryonic germ cell and this is inherited by the first neuroblasts that go on to form the brain.
Contrasts and comparisons are fundamental operations of the biochemical machines that run up and down the DNA chains making DNA repairs, copying it, building proteins based on it, etc. At the biochemical level reflections (associations) typically come in the form of positively or negatively matching shapes and electromagnetic charges.
I think we agree that the basic geometry of an association is triadic. I tend to think of it most often in terms of two nodes and a connecting line, the basic unit of a network; rather than as a triangle. A triangle, it seems, has more than three parts.
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This caught my attention in particular, as division (genetic vs. coordinate, in analysis vs. in reality) and definite vs. determinate were all terms/ideas that played a major role in Auxier and my “The Quantum of Explanation”.
That Collection Of Source Materials (COSM) from Peirce and others goes back to the time when I turned to working on Inquiry Driven Systems from a decidedly cybernetic and systems-theoretic standpoint.
There are several places where Peirce glosses this complex of kindred terms. His usages of choice typically reflect the entire history of relevant concepts going back to Medieval and Classical times.
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