## Relations & Their Relatives : 8

At this point we can distinguish two forms of decomposability or reducibility — along with their corresponding negations, indecomposability or irreducibility – that commonly arise.

• Reducibility under relational composition $P \circ Q.$

All triadic relations are irreducible in this sense.  This is because relational compositions of monadic and dyadic relations can produce only more monadic and dyadic relations.

• Reducibility under projections.  For this we need a few definitions:

Every triadic relation, say $L$ contained as a subset of the cartesian product $X \times Y \times Z,$ determines three dyadic relations, namely, the projections of $L$ on the three “planes” $X \times Y,$ $X \times Z,$ and $Y \times Z.$

In particular:

Every sign relation, say $M$ contained as a subset of the cartesian product $O \times S \times I,$ those being the sets of objects, signs, and interpretant signs respectively under discussion, determines three dyadic relations, which we may notate as follows:

• $\mathrm{proj}_{OS}{M},$ the projection of $M$ on the $O \times S$ plane;
• $\mathrm{proj}_{OI}{M},$ the projection of $M$ on the $O \times I$ plane;
• $\mathrm{proj}_{SI}{M},$ the projection of $M$ on the $S \times I$ plane.

The following Figure illustrates the situation:

Here is the critical point.  The triadic relation always determines the three dyadic projections but the three dyadic projections may or may not determine the triadic relation.  Thus we have two cases:

• If the dyadic projections determine the triadic relation, that is, there is only one triadic relation that has those three projections, then the triadic relation is said to be projectively reducible to those three dyadic relations.
• If the dyadic projections do not determine the triadic relation, that is, there is more than one triadic relation that has those same three projections, then the triadic relation is said to be projectively irreducible.

See the following article for concrete examples of both possibilities, a pair of generic triadic relations that are projectively irreducible and a pair of triadic sign relations that are projectively reducible to their dyadic projections.

## Relations & Their Relatives : 7

We use this or that species of diagrams to represent a fraction of the properties, hardly ever all the properties, of the objects in an object domain.  The diagrams that Peirce developed to represent propositions about relations are quite handy so long as one grasps the conventions of representation, manipulation, and interpretation.  They are not all that different in kind from Feynman interaction diagrams or Penrose twistor diagrams.  Iconicity is nice when you can get it but one has to keep in mind that the map is not the territory, as the saying goes.

What do I see in a picture like this?

         s
/
o---<L
\
i


The $L"$ brings to mind a triadic relation $L,$ which collateral knowledge tells me is a set of triples.  What sort of triples?  The picture sets a place for them by means of the place-names $o", s", i",$ in no particular order.  Without loss of generality I can take them up in the ordered triple $(o, s, i).$  All of this is just mnemonic machination meant to say that a typical element is $(o, s, i)$ in $L.$  It’s up to me to remember that $L$ is a subset of $O \times S \times I,$ with $o \in O,$ $s \in S,$ and $i \in I.$  The diagram is just a mnemonic catalyst.  You have to know the codebook to decode it.

Pictures can victimize people, as Wittgenstein stated and as often exemplified.  One way that people fall victim to pictures like the one depicted above is when they confuse a relation with a single one of its tuples.  That would represent a misunderstanding of what the picture is intended to represent.

## Animated Logical Graphs : 6

At root we are dealing with a genre of very abstract formal systems.  They have grammars that determine their well-formed expressions and rules that determine the permissible transformations among expressions, but they lack all logical meaning until we supply an interpretation.

The formal system Peirce developed for propositional logic and Spencer Brown resurrected for his Laws of Form admits a formal duality that allows it to be fleshed out with logical meanings in two distinct ways.  These interpretations are employed in Peirce’s entitative graphs and existential graphs, respectively.  It is clear from everything they write that both authors are well aware of both interpretations, but Peirce would come to found his later developments on the existential sense while Spencer Brown favored the entitative sense in his expositions.

See the following readings for further discussion:

## Peirce’s 1880 “Algebra Of Logic” Chapter 3 • Selection 8

### Chapter 3. The Logic of Relatives (cont.)

#### §4. Classification of Relatives (cont.)

227.   These different classes have the following relations.  Every negative of a concurrent and every alio-relative is both an opponent and the negative of a self-relative.  Every concurrent and every negative of an alio-relative is both a self-relative and the negative of an opponent.

There is only one relative which is both a concurrent and the negative of an alio-relative;  this is ‘identical with ──’.

There is only one relative which is at once an alio-relative and the negative of a concurrent;  this is the negative of the last, namely, ‘other than ──’.

The following pairs of classes are mutually exclusive, and divide all relatives between them:

Alio-relatives and self-relatives,
Concurrents and opponents,
Negatives of alio-relatives and negatives of self-relatives,
Negatives of concurrents and negatives of opponents.

No relative can be at once either an alio-relative or the negative of a concurrent, and at the same time either a concurrent or the negative of an alio-relative.

228.   We may append to the symbol of any relative a semicolon to convert it into an alio-relative of a higher order.  Thus $(l;\!)$ will denote a ‘lover of ── that is not ──’.

### References

• Peirce, C.S. (1880), “On the Algebra of Logic”, American Journal of Mathematics 3, 15–57.  Collected Papers (CP 3.154–251), Chronological Edition (CE 4, 163–209).
• Peirce, C.S., Collected Papers of Charles Sanders Peirce, vols. 1–6, Charles Hartshorne and Paul Weiss (eds.), vols. 7–8, Arthur W. Burks (ed.), Harvard University Press, Cambridge, MA, 1931–1935, 1958.  Volume 3 : Exact Logic, 1933.
• Peirce, C.S., Writings of Charles S. Peirce : A Chronological Edition, Peirce Edition Project (eds.), Indiana University Press, Bloomington and Indianapolis, IN, 1981–.  Volume 4 (1879–1884), 1986.

## Scientific Attitude : 1

There is an outlook on the world that I call the Scientific Attitude (SA).  There are times when the “A” is better served by apperception, approach, or attunement, but attitude will do for a start.

The scientific attitude accepts appearances, as appearances, but it does not stop there — it inquires after the possible realities that would both save and solve the appearances.

Reality is what persists and the scientific attitude accepts its limitation to what persists.  Thisness and thatness may come and go, but scientific knowledge rests on results that are reproducible.  It is knowledge of particulars in general terms.

## Relations & Their Relatives : 6

In the best mathematical terms, a triadic relation is a cartesian product of three sets together with a specified subset of that cartesian product.

Alternatively, one may think of a triadic relation as a set of 3-tuples contained in a specified cartesian product.

It is important to recognize that sets have formal properties that their elements do not.  The greatest number of category mistakes that bedevil errant discussions of relations and especially triadic sign relations arise from a failure to grasp this fact.

For example, the irreducibility (or indecomposability) of triadic relations is a property of sets-of-triples, not of individual triples.

See the articles under the following heading for concrete examples and further discussion: