Re: R.J. Lipton • Why Is Discrete Math Hard To Teach?
The Liberal Arts trivium of Grammar, Logic, Rhetoric received a latter day echo in the Unified Science trivium of Syntax, Semantics, Pragmatics, which was in turn the way Charles Morris transmogrified C.S. Peirce’s theory of triadic sign relations.
Re: R.J. Lipton • Why Is Discrete Math Hard To Teach?
Rhetoric deals with forms of argument that consider the interpreter. As considerate reason, it is involved in the style of training the Greeks dubbed education, “leading out”, and it leads in our time to both semiotics and computer programming.
Re: Peirce List Discussion • HR
I would not want the dyadic case to detain us too long, as often happens when we frame a simple example for the purpose of illustration and then fail to rise beyond it.
I raised the example of biblical brothers simply as a way of illustrating the distinction between a relation proper, like that symbolized by the formula “x is y’s brother” and any of its elementary relations, like the ordered pair (Cain, Abel).
There are, however, a few more points that could be illustrated within the scope of this simple example.
Recall that we had a universe of discourse X consisting of biblical figures and a 2-place relation B forming a subset of the cartesian product X × X such that (x, y) is in B if and only if x is a brother of y.
The “biblical brother relation” B would contain a large number of elementary dyadic relations or ordered pairs (x, y), for example:
(Abel, Cain), (Isaac, Ishmael), (Esau, Jacob), (Benjamin, Joseph), …
(Cain, Abel), (Ishmael, Isaac), (Jacob, Esau), (Joseph, Benjamin), …
Because B is a symmetric relation, each unordered pair {x, y} makes its appearance as two ordered pairs, (x, y) and (y, x).
The extension of the elder brother relation E would have the pairs:
(Cain, Abel), (Ishmael, Isaac), (Esau, Jacob), (Joseph, Benjamin), …
Peirce regarded a set of tuples as an “aggregate” or “logical sum” and would have written the above subset of B in the following way:
B = Abel:Cain +, Isaac:Ishmael +, Esau:Jacob +, Benjamin:Joseph +, …
+, Cain:Abel +, Ishmael:Isaac +, Jacob:Esau +, Joseph:Benjamin +, …
So what does all this — the distinction between relations in general and elementary relations plus the analysis of relations in general as sets or sums of elementary relations — imply for the case of triadic relations in general and sign relations in particular?
It means that non-trivial examples of triadic relations are aggregates, logical sums, or sets of many elementary triadic relations or triples.
As a result, the classification of single triples and their components gets us only so far in the classification of triadic relations proper, and except in very special cases not very far at all.
Re: Peirce List Discussion • HR
The immediate task is to get clear about the critical relationship between relations as sets and elementary relations as elements of those sets. What’s at stake is understanding the extensional aspect of relations. Beyond its theoretical importance, the extensional aspect of relations is the interface where relations make contact with empirical phenomena and ground logical theories in observational data.
The relationship between tokens and types, under one pair of terms or another, has been pervasive in science and knowledge-oriented philosophy from the time of Plato and Aristotle at least, arising from the observation that knowledge is of forms and generalities, not haecceities or individuals in themselves.
There is a communication problem that arises here, because the words “token” and “type” tend to be used differently outside Peirce studies, referring to objects that aren’t always signs. So I have found it less confusing to use more neutral terms, like “instance of a type” or “element of a set”.
In that sense, we can say that the ordered pair (Cain, Abel) is an instance of the type B, where B is a particular subset of all ordered pairs of biblical figures.
Re: Peirce List Discussion • GF • JBD • HR
I think a few people are making this harder than it needs to be.
Let’s put aside potential subtleties about elementary vs. individual vs. infinitesimal relatives and simply use “elementary relative” to cover all cases at a first approximation. One of the advantages of this approach is the analogy it highlights between elementary relations in the logic of relatives and elementary transformations in linear algebra, affording a bridge to practical applications of relation theory.
The time has come for a concrete example. Suppose we have a universe of discourse X consisting of biblical figures.
Linguistic phrases like “brother of __” or “x is y’s brother” and many others may be used to indicate a dyadic relation B forming a subset of X × X such that (x, y) is in B if and only if x is a brother of y.
It is often convenient to use Peirce’s notation x:y for the ordered pair (x, y). Among other things it’s easier to type on the phone.
In the universe X of biblical figures, Cain:Abel is an elementary relation in the brotherhood relation B.
But Cain:Abel also belongs to the relation E indicated by “elder brother of” and again to the relation S indicated by “slayer of”. So the elementary relation by itself does not completely determine the general relation or general relative term under which it may be considered.
This means that classifying relations is a task at a categorically higher level than classifying elementary relations.
In the special case of triadic sign relations, almost all the literature so far has tackled only the case of elementary sign relations.
Re: Peirce List Discussions • (1) • (2)
First off, we need to be clear about the difference between objects and signs:
Next, we need to be clear about the distinction between relatives (= general relatives) and elementary relatives:
Here is one place where Peirce exhibits his appreciation of the critical difference between relatives in general and elementary or individual relatives.
225. Individual relatives are of one or other of the two forms
and simple relatives are negatives of one or other of these two forms.
226. The forms of general relatives are of infinite variety, but the following may be particularly noticed.
Relatives may be divided into those all whose individual aggregants are of the form 
It needs to be appreciated that classifying relations is vastly more complex than classifying elementary or individual relations.
In particular, classifying sign relations is vastly more complex than classifying elementary or individual sign relations, which is just about all the entire literature on sign taxonomy has been able to touch from Peirce’s time to ours.
In this Survey of blog and wiki posts on Relation Theory, relations are viewed from the perspective of combinatorics, in other words, as a topic in discrete mathematics, with special attention to finite structures and concrete set-theoretic constructions, many of which arise quite naturally in applications. This approach to relation theory is distinct from, though closely related to, its study from the perspectives of abstract algebra on the one hand and formal logic on the other.
| Relation Construction | Relation Composition | Relation Reduction |
| Relative Term | Sign Relation | Triadic Relation |
| Logic of Relatives | Hypostatic Abstraction | Continuous Predicate |
Re: Peirce List
Recent travels and other travails (dental work) have scattered my thoughts to the four winds, so let me just document a few bits from my current state of mind in case I can get back to it someday.
Here is the figure I drew to illustrate a recurring theme from Peirce.
Normative science rests largely on phenomenology and on mathematics;
metaphysics on phenomenology and on normative science.
— Charles Sanders Peirce, Collected Papers, CP 1.186 (1903)
Syllabus : Classification of Sciences (CP 1.180–202, G-1903-2b)
Here are a few resources I find useful by way of establishing a foothold on these shores.
Re: Peirce List • Jeffrey Brian Downard • Gary Richmond • John Collier
According to Peirce, it is logic that draws on both mathematics and phenomenology.
At any rate, Peirce takes the distinctive position that normative science, which includes logic, “rests largely on” phenomenology and mathematics. Unless there is a case to be made for a practical difference between drawing on and resting on, as those phrases are intended in the present setting, I would have to say they mean the same thing.
I discussed the relationship among these sciences in a previous post and drew the following figure to illustrate it.
Normative science rests largely on phenomenology and on mathematics;
metaphysics on phenomenology and on normative science.
— Charles Sanders Peirce, Collected Papers, CP 1.186 (1903)
Syllabus : Classification of Sciences (CP 1.180–202, G-1903-2b)
The following post contains a longer excerpt from Peirce’s Classification of the Sciences.
Re: Peirce List • Jeffrey Brian Downard
Just from my experience, the best first approach to questions of firstness, secondness, thirdness, and so on is to regard k-ness as the property that all k-adic relations possess in common. There is more to say once this first point is appreciated but it is critical to begin from this understanding.
It is best to view k-adic relations as whole sets of k-tuples rather than fixating on single k-tuples at a time since all the most relevant properties of relations are “holistic” properties of whole sets or whole systems that are not reducible to properties of their individual elements.
A k-adic relation and its converses, numbering k! possibilities in all, each bears essentially the same information about the relation of its domains. This means that fixating on a particular ordering will tend to distract us with inessential features of a particular presentation rather than highlighting the essential properties of the relation in view.
Peirce demonstrates in several places that he appreciates the significance of these facts.
Just my k bits …
cc: Peirce List
Inquiry Into Inquiry 