2 
CATENAS OF ATTRIBUTES AND RELATIONS 
2.1 
BEYOND FORMAL CONNECTEDNESS 
2.1.1 
A MATTER OF BEING HEAVY, EQUALLY HEAVY OR LIGHTER 
Formal logic is the logic of validity in a chosen domain
of discourse  'the logic of the chosen domain'. The choice of
domain itself, and the translation from ordinary language or
informal argument to formal symbolism is, strictly speaking, not a
subject of formal logic. These are questions of interpretation,
of fields of study like ontology and linguistic pragmatics. Yet,
from a more general point of view, the 'logic of domain
choice', which precedes the logic of the chosen domain, is not
less important. This reflection on the choice of formal domain
itself should give us more insight into the presuppositions
made explicitly or implicitly, and into the relevance or
irrelevance of the distinctions drawn thereby. The question of
what is relevant or not we have to leave until later; suffice it
to say at this place that someone's reasoning may be as valid as
it ever can be within the formal logic of a chosen domain, but
that this does not finally settle an issue if the scope of the
domain itself is selected on the basis of an irrelevant factor.
Objects or other things would then be taken in or left out which
ought to be excluded or included for a proper argument.
(Consider, for example, a utilitarian who proposes that a
hedonic balance of good or pleasure over evil or pain should be
maximized and who exclusively chooses human beings as the objects
whose happiness is to be taken account of.)
Some 'discoveries' of those working with formal logical
systems depend entirely on the presuppositions made and the
foreknowledge at hand before selecting the domain. Without such
a priori knowledge, also if merely conventional or linguistic,
it would not even have been possible to choose a domain of the kind
in question. As an example, let us have a look at a formal
analysis of the relations heavier than and as heavy as.
The domain is said to consist of material objects only. The
conversation is therefore about concrete primary things.
Heavier than and as heavy as denote twoplace relations
and, moreover, are comparative notions. Such concepts enable us
to order individuals on the basis of the relation concerned. Let
Hxy be the relation heavier than and Exy the relation as
heavy as. Hab means, then, a is heavier than b and
Eab, a is as heavy as b. Now, to 'conclude' that b is as
heavy as a if Eab is true, or that b is less heavy or lighter than
a if Hab is true, already requires a certain foreknowledge.
Logically speaking, it is first necessary to define E as an
equivalence relation, that is, a relation which is symmetrical,
reflexive and transitive: 'symmetrical' in that Exy implies
Eyx and vice versa (if a is as heavy as b, then b is as heavy as
a); 'reflexive' in that for every object x Exx is true
(every x is as heavy as itself); and 'transitive' in that
Exz is true if Exy and Eyz are true (if a is as
heavy as b, and b as heavy as c, then a is as heavy as c).
The logical properties of the relation H are that it is
transitive as well, that it is irreflexive with respect to E and
connected with respect to E: 'irreflexive' with respect to E
in that Hxy is not true if Exy is true (if a is as heavy
as b, then a is not heavier than b); and 'connected' with respect
to E in that always Hxy or Hyx is true if Exy is not
true (a is heavier than b, or b is heavier than a, if a is not as
heavy as b ). If we take the relation R, heavier than or as
heavy as, then there is a strong form of connectedness in that
for every x and for every y of the domain concerned, Rxy or
Ryx is true. (In a weak sense a relation R is connected over
the class C of things if, and only if, whenever x and y belong
to C, and are distinct, either Rxy or Ryx is true.)
With these formal requirements for E and H it is possible to
arrange all individuals to which they apply in a certain order
 a quasilinear order in which there is exactly one place for
every individual, but sometimes with several individuals (which
are equally heavy) at one place. But what is the class of
individuals to which these formal requirements apply? And what
is the foundation of these formal requirements themselves? Have
they been miraculously supplied to spiritually nourish the
absolutely blank minds of logicians and scientists before they
started to think of qualitative, comparative and quantitative
notions at all? Of course not: before anyone had ever heard of
notions like strong or weak connectedness in their logical
sense, expressions like being heavier than, being lighter
than, being as heavy as and being as light as were
already intimately connected in common parlance. All the comparative
notions of the same name are linguistically even unthinkable
without the corresponding 'qualitative' (or 'classificatory')
notions of heavy and light (even when heaviness and
lightness are derelativized relations rather than genuinely
nonrelative attributes or qualities). At least from the perspective
of language, speaking of "heavy" and "light" precedes all
speaking of "heavier than", "lighter than" and "as heavy as".
Both the absolute and the relative conceptions apply to the
class of all things that are heavy or light or something in
between (namely medium heavy or light). Two distinct things of
this class are either equally heavy (or light) or not, and if
so, then both ways  otherwise the word equal would simply
not be appropriate. If they are not equally heavy, the one is
heavier than the other, and the one which is not heavier is
lighter than the other as long as the two things have some weight.
Formal systems do not establish these trivial truths; they
start from them.
Now, we are going to choose a domain restricted to material
things. What is 'material', however? There are two possibilities
to be taken into consideration here: (1) material is defined
in terms of weight; or (2) it is defined in other physical
terms. In the first instance a material thing is a thing which
has a weight. This, in turn, may be defined in an absolute
(qualitative) or in a relative (comparative ) way. On the
absolute account things which have a weight are light, heavy or
something in between, and on the relative account they are
things which are lighter than, as heavy as or heavier than other
things. It is immediately obvious, when the relative approach is
selected, that the connectedness of being heavier than or as
heavy as (or lighter than) was presupposed or foreknown in the
definition of materialness. The knowledge that the relation
obtained by putting together heavier than and as heavy as
in one disjunctive predicate (expression) would be connected over
the class of material things and not over any class comprising
immaterial things was a prerequisite to the very division
between material and immaterial things itself.
Weight is not a relative notion. Just as length must be
read as longness rather than longerness, so weight
must be read as heaviness rather than heavierness. But
also on the absolute interpretation one must have foreknowledge of the
connectedness as somewhere in between cannot literally be
read as neither light nor heavy, since immaterial things, too,
are neither light nor heavy. Neither light nor heavy is to
refer to the borderline, or rather the fuzzy zone, between the
'light' and the 'heavy', that is, to the 'medium light' or the
'medium heavy'. Yet, strictly speaking, this may imply the
connectedness of the qualities or attributes but not the
connectedness of a relation in the logical sense. Somehow it
must also be given a priori that everything between light and
heavy is heavier than light and lighter than heavy. Granted that
things are as heavy as themselves or as other things belonging
to the same set of the three qualitative sets distinguished, we
have the same conditions as on the relative interpretation
(assuming, again, that we allow ourselves to construct the
disjunctive relation which is connected by now).
Instead of in terms of weight, material may be defined as
having a speed, as being at rest or moving (in a positive or
negative direction) or in other spatiotemporal or physical
terms. The crucial assumption is now that everything that is
material in this sense (that is at rest or moves, for instance)
is an object which is heavier than, as heavy as or lighter than
other material things. It is implicitly taken for granted, then,
that there could be no material object which is heavier than
another one without there being any other material object being
lighter than at least one other material object. Hence, also in
this case the connectedness of the relation heavier than or as
heavy as (or lighter than) over the class of all, and only,
material objects is presupposed or part of our foreknowledge.
Any 'discovery' of this formal connectedness in such a domain of
material objects begs the question. (This is not to play down
the issue of whether and why the class of all things that are
heavier, as heavy as or lighter than one another is identical to
the class of things that are, for example, at rest or in motion.)
There is another, perhaps more serious, objection to overrating
a formal, logical property such as connectedness and that is
that it may be a necessary condition but that it is not a
sufficient condition to express the typical 'connection' between
notions like heavier than and as heavy as. Imagine a
domain or a world in which all objects that are arranged in a
quasilinear order from 'light' to 'heavy' are arranged in a
parallel order from 'fast' to 'slow': every object that is
lighter happens to be faster (and necessarily faster if also
dealing with modal conditions) and every object that is heavier
happens to be slower (and necessarily slower). Our actual world
is not like this, but one may think of other attributes or
relations thus correlated. (The example is similar to that of
all cordate beings that are renate, and vice versa, with the
impossibility of identifying predicates on the basis of their
extension alone, whether they are proper or improper.) On our
assumption not only heavier than or as heavy as is connected
over the domain in question but also heavier than or as slow
as and faster than or as light as. Altho people are
free to fabricate any fancy, disjunctive, or other, predicate
expression they wish in formal logic, this certainly has no
bearing on reality. If the (logically) informal language we
communicate in is to mean anything, the concepts heavier than
and lighter than are related to the concept as heavy as
in a way they are not related to any other concept (whether it be as
slow as or something else). The formal, logical requirements
for H and E do not suffice: there must also be a special
relationship between these relations which connects them in a
way in which they are not or cannot be connected to any other relation.
But the moment we recognize such a relationship (like that between
heavier than and as heavy as) we do not need to emphasize
the other requirements for our purposes anymore.
