SUO: An SUO 'Bookshelf' of Results
Everyone,
I have not been following the details of the triadicity discussion, but I was
delighted to see Pat Hayes' detailed explication of it. I'm guessing that it is
to a significant extent, a condensed and synthesized view of this thread so
far. When this issue dies down, and there is reasonable agreement on things,
including what to disagree on, then it would be possible to record this as an
'official' outcome of the SUO group.
I propose that a kind of SUO bookshelf be set up on our web pages. It would
contain detailed technical summaries similar to what Pat has produced. This
would be an
EFFECTIVE VEHICLE FOR DISSEMINATION OF THE
SIGNIFICANT PROGRESS WE HAVE MADE ON MANY FRONTS,
which to date is virtually inaccessible, buried in the discussion archives.
One can think of this as real-time knowledge mining of our archives.
This would affect many folk:
1. The world at large can see our progress to date.
2. Newcomers can much more easily get up to speed.
3. Regulars can skip the details and be assured of a good summary, in time.
4. All participants can be more efficient in the time they spend reading SUO,
being able to keep up to speed on things on their own time, without fear
of missing key information.
Personally, I have on many occasions been on the verge of getting off this list
because it is overwhelming. Yet, there are MANY things that are of great
interest and relevance to me, which I lack the time to follow and contribute in
detail to. A kind of SUO bookshelf, I hope would make everyone make more
efficient use of their SUO-time.
I propose the following process/format for how this might happen.
1. Discussion threads take place in the usual fashion.
2. When things settle down a bit, and points of agreement are reached, these
should be summarized and fed back to the group. This would also include points
where people agree to disagree.
3. When the main participants in the discussion reach agreement on the
summary, then it could be submitted to the overall group for placement
on the 'bookshelf'. The assumption is that things DO go on the bookshelf.
I don't think we want a formal voting process for this. The idea of going to
the group at large, is to see if there are any further suggestions or
improvements to the document.
4. We announce the existence of each new entry to the bookshelf to a SUO
interest group which is only intersted in the outcomes and timely announements,
rather than the day to day discussions. We may already have such list, I do
not know whether the creation of SUO sub-lists some months ago, actually
worked.
I would hope or expect that in many cases, these summaries could be the basis
for quality technical papers that are publishable.
Can we submit a journal arcitle with SUO as author? :-)
Mike Uschold
------------
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Subject: SUO: irreducible triadicity
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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
---------------------------------------------------------------------
IHMC (850)434 8903 home
40 South Alcaniz St. (850)202 4416 office
Pensacola, FL 32501 (850)202 4440 fax
phayes@ai.uwf.edu
http://www.coginst.uwf.edu/~phayes
--============_-1227507994==_ma============
Content-Type: text/enriched; charset="us-ascii"
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)
<fixed> | | | | |
---1------2------3------4---....---n-2---
</fixed>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:
<fixed> P
|
|
R----I3 ----Q
</fixed>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
---------------------------------------------------------------------
IHMC (850)434 8903 home
40 South Alcaniz St. (850)202 4416 office
Pensacola, FL 32501 (850)202 4440 fax
phayes@ai.uwf.edu http://www.coginst.uwf.edu/~phayes
--============_-1227507994==_ma============--