SUO: irreducible triadicity
I know I said I wouldnt say any more on this topic here, but several people have asked me for the review I mentioned, so here is a brief summary of the pith and essence, shorn of references to Burch's formalism and terminology.
On irreducible triadicity.
There are three distinct domains to consider. The first is a purely mathematical notion of a connected graph; the second is formal relational languages; and the third, somewhat murkier, is concerned with the metaphysical nature of relations. Peirce proved a simple result in graph theory to which he and others attribute metaphysical significance; but it does so only if one chooses a particular connection between the first and third of our domains via the second. Other connections are possible, under which the result, while of course still true, seems to have much less significance.
The result in graph theory is, roughly, that any connected graph can be built using nodes of degree 1,2 and 3, but that one cannot do it with nodes only of degree 1 and 2. (Degree of a node is the number of arcs connected to it.) More precisely, it is that every connected graph can be 'expanded' to a graph built using such nodes, where one expands a graph by replacing some of its nodes by connected subgraphs whose external arcs are isomorphic to the arcs to that node in the first graph. In effect, the expanded graph has a subgraph which can be collapsed (by identifying all its nodes and any connections between them) into a single node of the original graph. In modern terms one might say that the expanded graph is an implementation of the first graph in which some nodes are implemented by datastructures which are themselves connected graphs: imagine using LISP to encode a graph, for example.
It is easy to see how to construct such a datastructure for any node of degree n >3; one simply takes n-2 degree-3 nodes and links them together in a chain: (view the following in a fixed-width font)
| | | | |
---1------2------3------4---....---n-2---
which has n outgoing arcs to which the nodes linked to the original node can be attached. Also it is pretty obvious that you couldn't do it with nodes of degree 2, since the process of linking them together would itself use up all but the two spare 'ends', so you can only get another degree-2 graph.
That is the sum total of the notion of 'irreducible triadicity'. It is an interesting result in graph theory, but why is it considered to have any deep importance in semantics and semiotics? To see why Peirce thought it did, one has to appreciate how Peirce would have interpreted a semantic network.
A semantic network is a way of interpreting labelled graphs as sets of assertions: the nodes of the graph correspond to things, and the arcs of the graph correspond to relations. Only binary relations can be directly accomodated in this way, so to encode the fact that a relation R of greater arity (eg a trinary relation) holds, one introduces a new node to be an 'R-fact', and links it to the arguments of the trinary relation by arcs (which can be labelled with relations like 'first', 'second', etc., if no intuitively sensible relation names suggest themselves.) This corresponds to the logic translation I gave in an earlier message, and it has been widely used in KR work for many years. (To give a full translation of FOL into graphs requires other devices, including some way to represent quantifier scope. John Sowa's CG's are an excellent example; I will ignore this from now on as it is orthogonal to the 'triadicity' issue.)
With this encoding of relational language into graphs, the irreducible triadicity result is not important, since the degree of a node plays no special role in the interpretation (it corresponds to the number of times a name is used in a logical expression). Peirce however had a rather different way of interpreting such a graph. In Peirce's graphical notation, the nodes of the graph, not the arcs, are thought of as indicating relations, so that the degree of the node is the number of arguments the relation has. With this interpretation the degree of a node is clearly of much greater importance, and the graph-theoretic result takes on a new significance. It is therefore interesting to investigate this other 'Peircian' way of interpreting of a graph.
One might object immediately that if the nodes are relations, what part of the graph indicates the things the relations hold between, ie the things related by the relations? This question gets to the heart of the Peirce/Whitehead notion of the world being in some sense made of process, rather than things; 'things' are thought of here as a kind of convenient illusion (one which arises, in fact, from noticing relations.) The relations are seen as the the metaphysical ground on which the notion of individual is itself built. I do not want here to get involved in this metaphysical discussion, but will just remark that it is completely divorced from what might be called the intellectual mainstream of the last century, in which mathematics and formal semantics have been based on set theory, which in turn is rooted in the idea of collecting together things, a notion which depends on the idea of one individual thing being distinct from another. I mention all this only to partly motivate what would otherwise seem to be a very odd answer to the question about what in the the graph denotes the individuals, which is Peirce's answer: nothing. There are no individuals being related; there are only relations. The basic 'connection' between relations expressed by the arcs in the graph is now rather mysterious, but it can be thought of as a kind of existential connection: it says that the two relations share a kind of factual bonding at this point; they are mutually instantiated. In modern terms one would write this as an explicit existential claim using a quantifier:
(exists x)(R1(...x...) & R2(.....x...))
where the dots indicate that the other arguments are filled in with other, different, variables.
The quantifier is not really needed, and one can just use 'anonymous' names of free variables:
R1(...x...) & R2(.....x...)
This is what an arc in the graph translates into in Peirce's view of graphs as assertions. Notice that this is an exact dual of the semantic network view of the graph: here, the arcs are the 'things' and the nodes are the relations.
If one thinks of relations as atoms, then this bonding is rather like chemical valency, and the connected graphs which result are analogous to molecules; and logic becomes a kind of relational chemistry. This metaphor is superficially attractive, and in particular it has the merit, if one feels that these 'things' are best kept out of sight, of disposing of the connections as real things in themselves.
This view of what a graph means, however, has some limitations. Notice that in the translation into logic sketched above, it is possible to use any variable (or individual name) at most twice, corresponding to the two ends of the arc in the graph. This produces a curiously attenuated logic, in which for example it is not possible to say that something has three properties:
P(x) & Q(x) & R(x)
has no Peircian graph corresponding to it (it would need an arc with three ends). To overcome this, Peirce introduces a special class of relations, called identity relations. Then the graph which would say that this P-Q-R-ish thing existed would have the following translation:
P(x) & Q(y) & R(z) & I3(x, y, z) (notice each variable is only in 2 places)
where I3 is the special relation of three-way identity, which Peirce called 'teridentity'. The graph looks like this:
P
|
|
R----I3 ----Q
There can be such special relations I-n for all finite n, but in fact we only need I3, since the others can be 'implemented' using I3 in the way outlined at the beginning, by chaining enough (n-2) I3's together. Using a modern translation, for example, I5 can be defined as
I5(x y z u v) <==> I3(x y A) & I3(A z B) & I3(B u v)
where the A and B links are 'private' to this little subgraph. Peirce attributed great significance to the identity relations, as well he might, since once they are put in place they clearly play the role of individuals. To assert a relation of identity is to say that something exists, and linking the identity to other relations says that they hold of thing that exists.
The above reduction-to-I3 trick doesnt work if you try to use just I2, ie good old equals. You might think that it would be easy, since it is easy to write it in modern logic:
I3(x y z) <==> (x=y) & (y=z)
but if you now try to use this rewrite, the 'y' has been used up since it occurs twice in the definition already - no name can be used more than twice - so you can't say that anything else equals y; so this is really just I2(x,z). The two-ended nature of graph arcs has got you cornered. Triadicity really is irreducible in Peircian graph language.
Notice that the only triadic relation we really need is I3 itself, since we can string together a suitable implementation for any n-ary node using copies of I3, and even string an extra link to a node which holds the label of the original node as well, and use binary or unary relations for everything else; so any claim that some *particular* relation (other than I3) is itself irreducibly triadic must be based on some other criterion.
The interest of this irreducibility, however, is relative to how seriously one views the metaphysical consequences of the Peircian interpretation of graphs. I suggest that it isn't of much interest (other than historical), for several reasons. First, there's an obviously better interpretation available (semantic networks.) Second, the triadicity result applies to graphs, but it doesnt apply to a simple generalization which is just as mathematically respectable, if harder to draw, and which provides a more natural structure to interpret in the Peircian fashion. Hypergraphs are graphs where an arc (called a hyperarc) can link more than two nodes. These have a very direct Peircian-style interpretation which doesnt require the rather artificial 'identity relation' nodes; but triadicity is reducible in hypergraphs. Third, if one asks what the 'identity relation' really is saying, it is clear that it amounts to what would be expressed in modern terms as an existential assertion, ie it says that some THING exists; and if we can refer to that thing, it is obvious that any n-fold identity can be expressed as a conjunction of binary equality statements. The irreducibility of triadicity arises from a curiously obtuse combination of insisting that existence can be expressed by using an identity relation, and refusing to allow any way of referring to the thing that exists.
Pat Hayes
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