Cathy and Pierre,
I'd like to add a some comments to your discussion,
which may help to clarify some aspects of Thirdness,
metalevels, signs, and their relevance to ontology.
CL> As you can see, the key is the ability to perform 'hypostatic
> abstraction' on any relation to make it itself into a relatum.
> The way Pierluigi put things, in terms of a continually extensible
> 'triangulation', was actually a very good explication of this
> (thanks, Pierluigi). But in order for such triangulation to work,
> the identity relation must be transitive, and transitivity is
> an essentially triadic relation. (Thus the irreducibility of
PG> Again, I'm not sure what the significance and value of these
> reflexions are, say speaking as an ontologist. It is also odd
> to think of transitivy as a relation in a rigorous way. It seems
> to me to be a property of certain relations. It involves three
> terms, at any rate three signs (is that what you mean?),
> for each of the relata.
Yes, transitivity is a property of relations -- i.e., it is
a metalevel property. It may be considered as a one-word
abbreviation that a certain axiom applies to a relation:
For any relation R, transitive(R) <=>
(Ax)(Ay)(Az)((xRy & yRz) => xRz).
From one point of view, this statement could be considered
a second-order proposition, since it has a quantifier over
relations. But when you perform "hypostatic abstraction",
you convert relations into "things", which may then be used
as the arguments of other relations. In that sense, the
axiom is a metalevel statement about how the relations may
be applied to other things at the object level.
Peirce's notion of Thirdness is in effect a metametalevel
property about a entire class of metalevel relations (such as
transitivity, sign-relation, intentionality, etc.) which obey
axioms that define how they relate entities at the object level.
PG> So I don't quite clearly see how issues of numerical
> identity about signs and so on are so crucial for ontology.
Peirce himself had many doubts about why a numerical property
should be so crucial for ontology. He first discovered the
notion when he was about 30 years old, and he admitted that
he often had dobuts about it over the next 30 years or so.
But he kept coming back to it because it kept reappearing
in many different guises over the years. He finally accepted
it in his later years, when it became the cornerstone of his
As another example, the attached diagram (feedback.gif) shows
a triadic relation that is very important for engineers:
the feedback loop. The classic discovery, which occurred
in the mid 19th century, was the regulator or governor for
steam engines. Before it was invented, steam engines were
difficult to use because some human being constantly had
to adjust the input fuel and the steam pressure in order to
keep the speed relatively constant.
A similar example is cruise contol for a car. As feedback.gif
illustrates, may be considered a dyadic relation between
the fuel and the speed. Without cruise control, the driver
must constantly adjust the fuel to maintain constant speed.
But cruise control is a triadic relation that constantly
monitors the speed and adjusts the amount of fuel to maintain
constant speed. (This diagram is simplified because many
different factors affect speed, but they can be represented
by multiple dyadic relations. Feedback involves an
The connection between purpose or intention and triads,
which is illustrated by the feedback loop for cruise control,
also appears in other human inventions, such as thermostats,
alarm clocks, and the refill mechanism in a toilet tank.
In nature, triadic relationships most commonly appear in
living organisms, which involve feedback loops to control
every iportant aspect of life.
There are, however, some inanimate things in nature that
involve feedback, and interestingly, they are often
personified as living beings. One example is a hurricane,
which has a complex feedback loop involving wind, water,
and speed. That feedback loop enables the hurricane to
maintain a fairly constant speed and configuration over
a period of many days. The resulting behavior is so
"life-like" that people give them names, such as the
current hurricane named Isabelle, which is supposed
to hit the eastern US tomorrow.
Summary: Thirdness is a metametalevel notion, which
characterizes the metalevel axioms that define a large
number of important ontological relations. Thirdness
is essential to all the relations involving signs and
representations of any kind, to most relations that
are characteristic of living organisms, to any human
artifacts that involve feedback loops, and to some
inanimate processes, such as hurricanes. The connection
between Thirdness and life is so common that people
tend to personify inanimate things that involve
naturally occurring feedback loops.
Bottom line: Thirdness is important for ontology
because an irreducible triadic relation is usually
a central or characteristic feature of whatever
entity is associated with it.