Basic Structure Requirement
But the realist program just enunciated must be extended. There
must be a separation between an entity and its productive or defensive
properties. There must be a distinction between an entity as a particular
and the essential, productive or defensive properties that it has.
There must be, too, an environmental particular or property that
the first particular’s essential property adaptedness is related
to. See appendix II. This will be illustrated next.
Set Theory: Relations
The application of relations in set theory presented next is taken
from Lipschutz (1998, pp. 64-93). There are many mathematical texts
on set theory, of little applied use; but the text just mentioned
is replete with examples of principles. These principles will be
married to the four principles of adaptation in search of a greater
perspicuity in delineating the four principles.
The Mussel and the Oyster and the First Principle
of Adaptation
To back-up the first principle of adaptation of two different entities
under the same condition with one adapted and the other not adapted,
there is this presentation. This presentation will include the basic
structure requirement of separation of a particular from its essential
properties (including adaptedness) and from an environmental condition.
This presentation will numericalize the elements of a set.
The property of being open (in order to pump water) of the mussel,
and the property of pumping water of the oyster has been determined
experimentally for different temperatures and is presented graphically
in Hulburt (2002). From this graph were read off values of percentage
of time being open, the property of being open, of the mussel at
experimental temperatures of 1º, 5º, 10º, 15º
and 20ºC, Table 1.
This constitutes a set, A, of five elements, each one represented
as ai. There is a second set of five water temperatures,
B, in coastal southern New England (U.S.A.) where mussels live.
Each one of these five elements is represented as bi.
The product set A x B is:
1) A x B = {(a1, b1),
(a2, b2), (a3, b3),
(a4, b4), (a5, b5)}
This shows how each pair, (a, b), is a matching
of the mussels experimental time open at a given temperature and
an ocean temperature. Where a A
means a belongs to the set A and likewise for bB;
where : is such that, 
is and, A x B is :
2) A x B = {(a, b) : aA
bB}
2) says that the product set A x B equals the
set of pairs (a, b) such that aA
and b
B. There is a further formulation where R is the
relation of adaptation between A and B. This formulation
is:
3) (a, b)R
Each (a, b) belongs to R. Thence we
say “a is related by adaptation to b”,
that is, a has adaptedness to b : a R b.
The mussel’s experimental time open at a single temperature,
ai, is an instance of the property of being open, a property
the mussel has. Thus the mussel’s five instances of the property
of being open and the five ocean temperatures corresponding to the
experimental temperatures compose this next set of five adaptednesses,
where the first reads “the mussel’s experimental time
open at 1ºC has adaptedness to the water temperature of 1ºC”:
4) {(a1R1b1), (a2R2b2),
(a3R3b3), (a4R4b4),
(a5R5b5)}
But the oyster is different. Table 1 shows the set of four experimental
pumping rates at four temperatures for the oyster. The four sets
of pumping rates and the four sets of ocean temperatures have two
instances where there is no or very little pumping and so there
is no or very little adaptedness to the ocean temperatures of 1º
or 7º. There are two instances of moderate or marked pumping
rate; thus the oyster has only two adaptednesses. We have the set:
5) {(a1not R1b1), (a2not R2b2),
(a3R3b3), (a4R4b4)}
4) and 5) are a set theory presentation to portray perspicuously
the first principle’s dictum that two quite different entities
under the same condition have in one case, 4), all adaptednesses
and in the other case, 5), not all adaptednesses. The R
of set theory when applied here is the relation of adaptedness between
mussel or oyster and environment – but R is an essential
property of mussel or oyster, which is separated from mussel or
oyster in accord with the basic structure requirement.
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