Structural Adaptation
Structural adaptation is the theme of ten previous studies by the author (Hulburt, 1982, 1984, 1985a, 1985b, 1988-1989, 1991, 1996, 1998, 2001, 2002). The last outlines approximately the content of structural adaptation. In structural adaptation there are four principles of adaptation. These principles are empirically based. The principles and their empirical content are delineated by logically valid expressions, which are valid truth functionally and axiomatically. The principles rely on following the precepts of philosophical realism, on use of universals. Universals are properties, traits, characteristics.

The four principles of adaptation in this essay will be given added perspicuity, it is hoped, by recasting them within the structure of set theory. Specifically, relations and functions of set theory will be employed. A new world opens up when scientific data and principles are portrayed in the context of set theory’s relations and functions.

The Four Principles of Adaptation
The first principle of adaptation is that if two quite different entities occur under the same condition, then one is adapted and the other is not adapted to this condition. Thus the warm-blooded vertebrate is adapted to year-round temperature in temperate regions because it is behaviorally active year-round (except hibernators), whereas the cold-blooded vertebrate is not adapted to year-round temperature because it is not behaviorally active year-round (plus, hibernators). Put in logically valid form this is : there’s a year-round temperature range y and if x is behaviorally active throughout y, then x is in an animal adapted to y, that has adaptedness to y – equivalent to : there’s a year-round temperature range y and if x is in an animal not adapted to y, that does not have adaptedness to y, then x is not behaviorally active throughout y. This is represented by : (y)[Ty (BxyAxy)] (y)[Ty (~Axy~Bxy)]. (See Appendix I).

The second principle of adaptation is that if one entity occurs under two quite different conditions, then it is adapted to one condition but is not adapted to the other condition. Thus the North American forest is adapted to moist condition in the east and west, but is not adapted to the non-moist, semi-desert conditions of the southwest. Put in logically valid form this is : no matter what y is chosen, if the forest x is adapted to y then y is a moist condition – equivalent to : for any y if y is not a moist condition then the forest x is not adapted to y. Briefly this is : the forest is adapted (has adaptedness) only to moist conditions if and only if it is not adapted (does not have adaptedness) to non-moist conditions. These are symbolized as follows: (y)(AxyMy) (y) (~My~Axy).

The third principle of adaptation is that if one entity is adapted to a second, then the second is adapted to the first. – if one entity has adaptedness to a second, then the second has adaptedness to the first. Thus, if the California redwood x is adapted to its locale y, then its locale is adapted to it; furthermore, if the locale y is adapted to the redwood x, then the redwood is adapted to the locale; and so, redwood and locale are adapted to each other. All of which is : [(AxyAyx) (AyxAxy)] (Axy Ayx).

The fourth principle of adaptation is that if two or more quite different entities occur under two or more quite different conditions, then each is adapted, has adaptedness, to its particular condition. For example and in a logically valid structure, there is diapause or there is non-diapause in insects; if there is diapause (an insect in an overwintering larval stage), then there is winter adaptedness, and if there is non-diapause (the same insect is in a winged stage), then there is summer adaptedness; therefore, there is winter adaptedness or there is summer adaptedness. All of this may be gathered together as:
p v r; (pq) (rs); .. q v s.

In the first two principles the underlying structure is contraposition: (pq) (~q~p). There is reversal and denial of the parts p and q. This means that there is both affirmation and denial of adaptation. In the third principle the underlying structure is equivalence : [(pq) (qp)] (p q). Here there is only affirmation of adaptation. The fourth principle presented in bare structure above is called constructive dilemma, and in it only affirmation of adaptation is the result. These formulations are explained fully in Kahane (1986, pp. 50-68), wherein a truth-functional explanation is presented. An axiomatic explanation is given in Hulburt (2002).

Nominalism and Realism
For the nominalist, to be is to be a particular. For the realist, to be is to be a particular and to be a particular is to have a property, a universal. This distinction has already been made, when “is adapted” is paralleled by “has adaptedness”. Take the example of the third principle: 1) if the California redwood species, a particular, is adapted to its locale, another particular, then its locale is adapted to it – versus : 2) if the California redwood species has adaptedness to its locale, then its locale has adaptedness to it. In the first no structure of the redwood species is required. But in the second the possession of the property of adaptedness by the redwood species strongly suggests that there may be many properties, all those properties that collected together in one bundle compose each redwood tree and that bridge from tree to tree, so that all the trees of the species by having nearly the same properties are the same taxonomically. The properties that all redwoods have may be productive; all have tallness, all have dense-packedness, all have competitiveness, all have high growth capacity, and all have adaptedness to moist conditions. Here these properties define five sets, which are identical in having the same elements, all redwood trees. Properties are productive in most of the cases that will be presented. But in one case, the rocky shore snails, properties will be defensive. (For full explanation of realism see Loux (1970), Armstrong (1978, 1989), Wolterstorff (1970), and Moreland (2001)).

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