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Theory of adaptation: application of symbolic logic
Ecological Modelling 107 (1998) 35 – 50
Theory of adaptation: application of symbolic logic Edward M. Hulburt * Woods Hole Oceanographic Institution, Woods Hole, MA 02543, USA Received 22 April 1997; accepted 18 November 1997
Abstract The theory of adaptation takes up, first, completely symbolized expressions in which the basic element is the dyadic term Axy, x is adapted to y. When this is combined with a monadic term such as Ey, y is a moist environment, in (y)(Axy³ Ey), validity can be achieved by reversal and denial (transposition): (y)(Axy³ Ey) (y)( Ey³ Axy). This provides a full and complete description when applied as follows: the North American forest x is adapted only to moist environments if and only if it is not adapted to any non-moist environments. Then dyadic – dyadic, monadic–dyadic–dyadic, and monadic–dyadic–monadic – dyadic combinations are presented and exemplified. The theory of adaptation takes up, second, incompletely symbolized expressions such as:(x)(Fx v Hx), (x)[(Fx³ Gx) · (Hx³Ix)], (x)(Gx v Ix)%, which provides the structure for this description of temperate insects: all undergo diapause or nondiapause; all, if undergoing diapause, are winter adapted, and if undergoing nondiapause, are summer adapted; therefore, all are winter adapted or summer adapted. The phrases winter adapted, Gx, and summer adapted, Ix, are incomplete symbolizations since they are brief versions of (×y)(Wy · Axy), x is adapted to winter, and of (×y)(Sy · Axy), x is adapted to summer. The brief versions, however, make elaborations and extensions of the basic formulation possible. The formulas of complete symbolization provide for affirmation or denial of adaptation or no decision. Thus, in addition to the example of the North American forest other examples are species of the lizard Anolis, each of which is well adapted in its niche but not well adapted outside its niche, the associations of the North American forest which are tall, densely packed and adapted in moist regions but short, scattered, and unadapted in non-moist regions, and species of Gilia which are adapted to their habitats unless these habitats are not adapted to them. Then in small-to-moderate forest gaps regrowth of prevalent, adapted primary species contrasts with the regrowth of unadapted, non-prevalent pioneer species. But it cannot be decided whether the lizard Anolis allogus in a smaller niche on Cuba is adapted to this or is not adapted to this when comparing it to Anolis oculatus in its larger niche on Dominica. The formulas of incomplete symbolization provide only affirmation of adaptation. Examples in addition to the diapause—nondiapause case are for diatoms which are dark or light adapted and ciliates which are myxotrophically or heterotrophically adapted. Then, evolutionary pathways to warm bloodedness of fish include being adapted to fast swimming in tuna fish, or northern distribution in lamnid sharks, or deep diving in swordfish.
* Tel.: +1 508 5483074. 0304-3800/98/$19.00 © 1998 Elsevier Science B.V. All rights reserved. PII S 0 3 0 4 - 3 8 0 0 ( 9 7 ) 0 0 1 9 9 - 3
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And finally four basic evolutionary pathways include anaerobic and aerobic photosynthesis and thus adaptation to the lighted planetary surface and include chemosynthesis in two ways and thus adaptation to the dark ocean floor. © 1998 Elsevier Science B.V. All rights reserved. Keywords: Adapted; Valid; Dyadic; Physicalistic; Interpretive
1. Introduction The concerns of this study are both small scale and large scale. The thrust will be predominantly ecological, for adaptation is ecological. The issues to be encompassed will be offshoots of two strands of thought. A single formulation will be at first dominant. The formulation is just that one thing is adapted to another, that x is adapted to y. Then a second formulation will stem from the simple precept that x is adapted in this way or that way or in some further way. From these simple formulations a richness of detail and outlook will emerge. Is there precedent for the outlook that is to come? Except in writings of the author there is little precedent. But it is the wish and interest of the author’s studies to make adaptation a deeply permeating part of biological material and ecological structure. This intent is insisted upon strenuously in the present study. There is nothing wrong in pointing out how some structure, the opposed digits of a bird, for example, makes the bird adapted in some respect or constitutes an adaptation that the bird has. But such adaptations are minor. Surely they do not encompass the adaptation of biological entities to the physical world and to each other. The broad adaptation of entities to the physical world and to each other is an old assumption, immersed in Western social and religious structure (Levins and Lewontin, 1985). It would seem difficult to avoid this old assumption, this old belief in the harmony of the world. Even though there are so many cases against it, so many vicissitudes that beset the world and that beset the belief in its harmony, in its adaptation. So what follows is split to a considerable degree between seeing adaptedness and seeing its lack. The method is simple, for if x is adapted to y then y is a certain condition of the environment, which is to say that if y is not this condition of the environment then x is not adapted to y. But this is a logical structure, a case of logical validity. Thus it is that the theory of adaptation to be presented is a logic structure theory. Some knowledge of elementary symbolic logic is advisable for perusal of what is to come—a knowledge obtainable of course from texts such as those of Kahane (1986), Copi (1979), and Quine (1972). But briefer instruction is all that is necessary, to be gotten from Hulburt (1992, 1995, 1996). Also, a glossary of terminology is given at the end; perhaps this is explanation enough.
2. Fully symbolized expressions: dyadic-monadic linkages The basic dyadic element is Axy, x is adapted to y. With this validity1 is: if Axy then Axy, Axy implies Axy (see glossary): Axy³ Axy.
1
Validity is superficially obvious in‘Axy³ Axy’. Validity is achieved by truth value analysis (see glossary).
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Axy is linked to the monadic term Ey, y is a moist environment, in a validity arrangement mimicking the above formula: (Axy³Ey)³(Axy ³Ey).
(1)
The right member in parentheses may be replaced by its transpositional equivalent Ey ³ Axy. And the left member may be similarly and separately replaced and the resulting expressions (both valid) may be joined by conjunction (by ‘and’, ‘ · ’): [(Axy ³ Ey)³( Ey ³ Axy)] · [( Ey ³ Axy)³ (Axy³ Ey)],
Conj.
which is: (Ey ³ Axy) (Axy ³ Ey)
Equiv.
(Axy³Ey) ( Ey ³ Axy)
Comm..
(2)
In partially quantified form Ex. (2) is: (y)(Axy ³Ey) (y)( Ey ³ Axy)
U.G..
(3)
A full and complete description thus eventuates. By way of example, with Ey still as y is a moist environment, Ex. (3) is: for every y, if the North American forest x is adapted to y then y is a moist environment—equivalent to: for every y, if y is not a moist environment then x is not adapted to y. Briefly, the North American forest is adapted only to moist environments if and only if it is not adapted to any non-moist environments. For the North American forest is composed of tall and densely packed species in the East and West where rainfall is \ 30 inches/year and exemplifies adapted forest. In the southwest US where rainfall is B 30 inches/year pinon pine, creosote bush, and other woody species are small and spaced apart (Oosting, 1948; Hulburt, 1995) and exemplify unadapted forest. Ex. (3) may have adapted denied with the environmental term undenied. This is shown next, in Ex. (5), in a description of the niche structure of species of the lizard Anolis on Hispaniola (Rand, 1962; Moermond, 1979; Williams, 1983). Going from the ground up the tree, there are the ground-trunk species A. cybotes, the trunk species A. distichus, the trunk-crown species A. chlorocyanus, and the crown species A. ricordii. These overlap each other considerably. This overlap is portrayed in Ex. (4), (5) and points (1) – (5) following. Abbreviations for these expressions are the following. Gy is y is a ground-trunk niche, Ty is y is a trunk niche, Ky is y is a trunk-crown niche, Cy is y is a crown niche, x is for A. cybotes, z is for A. distichus, u is for A. chlorocyanus, and 6 is for A. ricordii. We have first (note z in Ex. (5)): (y)(Axy ³ Gy) (y)( Gy ³ Axy)
(4)
(y)( Azy³ Gy) (y)( Gy ³Azy)
(5)
Ex. (4) says Cybotes is well adapted only to ground-trunk niches if and only if it is not well adapted to non-ground-trunk niches. Ex. (5) says Distichus is not well adapted only to ground-trunk niches if and only if it is well adapted to non-ground-trunk niches. Equating non-ground trunk niches to trunk niches, we can say next in 1 and 2: (1) (2)
(y)(Azy ³ Ty) (y)(Ty ³ Azy) (y)(Auy ³Ty) (y)(Ty ³ Auy)
(1) Distichus is well adapted only to trunk niches if and only if it is not well adapted to non-trunk niches; (2) Chlorocyanus is not well adapted only to trunk niches if and only if it is well adapted to non-trunk niches. Equating non-trunk niches to trunk-crown niches we say next:
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(3) (4) (5)
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(y)(Auy ³Ky) (y)(Ky ³ Auy) (y)(A6y ³ Ky) (y)(Ky ³A6y) (y)(A6y ³ Cy) (y)(Cy ³ A6y)
(3) Chlorocyanus is well adapted only to trunk-crown niches if and only if it is not well adapted to non-trunk-crown niches; (4) Ricordii is not well adapted only to trunk-crown niches if and only if it is well adapted to non-trunk-crown niches — which are the same as crown niches and (5) Ricordii is well adapted only to crown niches if and only if it is not well adapted to non- crown niches. The distinction between well adapted and not well adapted must be reiterated throughout the vertical niche structure. Well adapted cannot possibly be applied throughout this structure. This is a vital perception to be gleaned from the dividing up of the structural niche and the overlapping of the dividing up species and the attenuation of the lower species as they overlap the upper species. Repetition of the dyadic-monadic linkage of Ex. (1) is in order. Consideration of the North American forest suggests some linkages. No matter whether one considers the beech–maple association of eastern US, the spruce – fir association of eastern Canada, the Engelmann spruce–subalpine fir association of the Rocky mountains, the Sitka spruce dominated forest from Alaska to the luxuriant Olympic Peninsula where it is joined by western hemlock, arborvitae, and grand fir, or other associations outside the dry southwestern US (Oosting, 1948; Hulburt, 1995); all associations x are tall only in moist regions, (x)(y)(Txy ³My), all associations are densely packed only in moist regions, (x)(y)(Dxy ³My), all associations are competitive only in moist regions, (x)(y)(Cxy ³ My), all associations are high growth capable only in moist regions, (x)(y)(Gxy ³ My), all associations are adapted only in moist regions (x)(y)(Axy ³ My). But there are non-moist regions (×y) My in the southwest US. We have the following valid argument, with repeated use of modus tollens: (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16) (17) (18) (19) (20) (21)
(x)(y)(Txy ³ My) (x)(y)(Dxy ³My) (x)(y)(Cxy ³My) (x)(y)(Gxy ³ My) (x)(y)(Axy ³My) (×y)My My Txy³My Txy Dxy³My Dxy Cxy³My Cxy Gxy³My Gxy Axy³My Axy Txy · Dxy · Cxy · Gxy · Axy (×x)(Txy · Dxy · Cxy · Gxy · Axy) My · (×x)(Txy · Dxy · Cxy · Gxy · Axy) (×y)[My · (×x)(Txy · Dxy · Cxy · Gxy · Axy)]
p p p p p p 6, E.I. 1, U.I. 8, 7, M.T. 2, U.I. 10, 7, M.T. 3, U.I. 12, 7, M.T. 4, U.I. 14, 7, M.T. 5, U.I. 16, 7, M.T. 9, 11, 13, 15, 17, Conj. 18, E.G. 7, 19, Conj. 20, E.G.
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Step (21) says therefore, there is a non-moist environment in which there is an association which is not tall and not densely packed and not competitive and not high growth capable and not adapted. The proof of (21) from (1)–(6) has adapted in (5) and not adapted in (21). These are interpretations put on the physicalisticly descriptive tall, densely packed, competitive, and high growth capable, and their negatives. If all the physicalisticly descriptive predicates are denied in the southwest USA, it would be best to go along with denying the interpretive word adapted too. Modus tollens used five times as in: Txy³ My My Txy is close to transposition, since transposition leads to modus tollens as follows: (1) (Txy³My) (My ³Txy) (2) [(Txy ³ My)³(My ³Txy)] · [(My ³Txy)³ (Txy³ My)] (3) (Txy³My)³ (My ³Txy) (4) [(Txy ³My) · My] ³Txy
1, Comm., Equiv. 2, Simp. 3, Exp.
Step (4) is modus tollens written horizontally.
3. Fully symbolized expressions: dyadic – dyadic linkages The basic dyadic element, Axy, has reversal of variables in Ayx. Thus if x is adapted to y then y is adapted to x (Hulburt, 1996): Axy³ Ayx.
(6)
For example, if symbiont x is adapted to symbiont y then symbiont y is adapted to symbiont x. This is a case of symmetry. Another case of symmetry is that of plant species x and it’s habitat y. Again validity is gotten through transposition: (Axy³ Ayx) ( Ayx ³ Axy)
(7)
saying if plant species x is adapted to habitat y then y is adapted to x, if and only if, if habitat y is not adapted to species x then x is not adapted to y. Surely if habitat y is not adapted to species x, how could x flourish there and be adapted to y? The annual herbs Gilia splendens and G. caruifolia are allopatric, the first occurring in the south Coast range and the San Gabriel, San Bernardino and San Jacinto ranges in southern California and the second ranging southward from the Cuyamaca, Laguna and Palommor mountains to the San Pedro Martin range (Grant and Grant, 1954). So these two species are examples of Ex. (7), when their names are put for x and their habitats are put for y. Gilia australis occurs too in very southern California but in hotter, drier habitats of sandy marshes of the foothills and plains below the pine belt. Where Hy is y is a hotter, drier habitat and x is put for G. australis we have: (y)(Hy ³Axy)³ (y)(Hy ³Ayx)
(8)
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which says if australis, x, is adapted to hotter, drier habitats, then hotter, drier habitats are adapted to it2. This and the transpositional equivalent ((5) below) make a valid whole. The equivalent is gotten as follows: (1) (2) (3) (4) (5)
(y)(Hy ³ Axy) ³ (y)(Hy ³ Ayx) (y)(Hy ³ Ayx) ³(y)(Hy ³Axy) (×y)(Hy ³ Ayx) ³ (×y)(Hy ³ Axy) (×y)(Hy · Ayx) ³ (×y)(Hy · Axy) (×y)(Hy · Ayx) ³ (×y)(Hy · Axy)
1, 2, 3, 4,
Trans. Q.N. Imp. D.N.
Step (5) says that if there is a hotter, drier habitat and it is not adapted to australis, x, then there is a hotter, drier habitat and australis is not adapted to it. For suppose that a hotter, drier habitat is not adapted to australis, then how ever could australis be adapted to it?
4. Fully symbolized expressions: monadic – dyadic –dyadic connections The basic dyadic element, Axy, is preceded by another dyadic term, Pxy, as Pxy³Axy. This linkage is joined to an existentially quantified monadic term, (×y)[Gy · (Pxy ³ Axy)]
(9)
Let us take a particular case, that of regrowth in forest gaps from fallen trees at Barro Colorado, Panama (Brokaw, 1982, 1985). In the smaller and moderate gaps saplings of primary species, those that persist in gaps of all sizes, are very prevalent; in the very large gaps saplings of pioneer species, those that persist only in large gaps, are modestly prevalent. In general primary species are prevalent whereas pioneer species are not in small-to-moderate gaps. Where (x) takes in all species, primary and pioneer both, we have: (x)(×y)[(Gy · (Pxy³ Axy)] (x)(×y)[Gy · ( Axy³ Pxy)]
(10)
saying all species, if prevalent in, are well adapted to small-to-moderate gaps; if and only if all species, if not well adapted to, are not prevalent in small-to-moderate gaps. This is brief for the detailed: for every species x there is a y such that y is a small-to-moderate gap and if x is prevalent in y then x is adapted to y; if and only if …. A crucial feature of Ex. (10) is overlap, since under the same conditions of small-to-moderate gaps prevalent, well adapted species and not well adapted, non-prevalent species overlap. A striking feature of Ex. (10) is that the dyadic term Pxy provides a criterion for x’s being adapted to y. There are many cases that exemplify Ex. (10) or various versions of Ex. (10). Some of these cases are given in Hulburt (1992).
2 The (y)(Hy³Axy) part of Ex. (8) says: for every y, if y is a hotter, drier habitat then x is adapted to y, which is: all hotter, drier habitats x is adapted to, which is: x is adapted to hotter, drier habitats.
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5. Fully symbolized expressions: monadic – dyadic –monadic–dyadic connections The basic dyadic element Axy is last in a series of terms in this manner: (×y)( Ry · Ixy) ³(×y)( Ry · Axy).
(11)
.Proceeding with transposition yields two results —two disastrous results—as follows: (1) (2) (3) (4) (5)
(×y)(Ry · Ixy) ³ (×y)(Ry · Axy) (×y)(Ry · Axy) ³(×y)(Ry · Ixy) (y)(Ry · Axy) ³ (y)(Ry · Ixy) (y)(Ry v Axy) ³(y)(Ry v Ixy) (y)(Ry ³Axy) ³ (y)(Ry ³ Ixy)
Eq. (11) 1, Trans. 2, Q.N. 3, De.M. 4, D.N., Imp.
Thus, with Ry for y is an unrestricted niche and Ixy for species x is interfered with in y (1) says if species x is not interfered with in an unrestricted niche then it is adapted to an unrestricted niche. And the equivalent (5) says if species x is not adapted to any unrestricted niches then it is interfered with in the unrestricted niche. The lizard Anolis oculatus on Dominica confirms Ex. (11), for it occurs throughout the 10°C range of the island uninterfered with by any other species. But on Cuba A. allogus confirms (5) because it does not occur through the 10°C niche due to interference from other species that overlap it (Ruibal and Philibosian, 1970; Ruibal, 1961) and so it is not adapted to the unrestricted 10°C niche. But the following also pertains: (6) (7) (8) (9) (10)
(×y)(Ixy · Ry) ³(×y)(Axy · Ry) (×y)(Axy · Ry) ³(×y)(Ixy · Ry) (y)(Axy · Ry) ³(y)(Ixy · Ry) (y)(Axy v Ry) ³(y)(Ixy v Ry) (y)(Axy ³ Ry) ³(y)(Ixy ³Ry)
1, 6, 7, 8, 9,
Comm. Trans. Q.N. De.M. Imp., D.N.
Step (10) says if species x is adapted only to restricted niches then it is uninterfered with only in restricted niches. That is, A. allogus by confirming this can be considered to be adapted to the restricted B 10°C niche imposed on it by the other competing species. The point here is that you can not decide about the adaptation of A. allogus on Cuba by comparing it with A. oculatus on Dominica. If oculatus and allogus occurred together on Dominica, oculatus covering the unrestricted 10°C range but allogus covering less than this 10°C range, then you would probably want to say that oculatus was adapted, or better adapted, to this niche range and that allogus was not. But the two species do no overlap on the same island, and it can not be decided whether allogus is adapted or not.
6. Adaptational decision The use of transposition and modus tollens are decision procedures, providing for adapted and its denial, not adapted — or well adapted and not well adapted. But this decision procedure depends on the applicability of two or three part structure only, as in Ex. (1) and Ex. (9). When four part structure of the type in Ex. (11) is applied no decision is possible.
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7. Incompletely symbolized expressions The monadic terms, Fx, Gx, Hx, and Ix will be used in a varied manner. Some will be used simply, as previously done. Others will be used to express relationships, so that in a sense the relationships are incompletely symbolized. The terms can be used in the following development. To the valid: Fx³ Fx there corresponds the valid: (Fx³ Gx) ³ (Fx ³Gx) which can be further changed by these equivalents: [(Fx ³ Gx) · Fx] ³Gx,
Exp.
[Fx · (Fx ³Gx)]³ Gx.
Comm.
Presented vertically as a deduction the last is (a): (1) (2) (3)
Fx Fx ³ Gx Gx
2,1, M.P.
A repetition of this modus ponens deduction is a second modus ponens, (b): (1) (2) (3)
Hx Hx ³Ix Ix
2,1, M.P.
These two structures will be needed. And a third structure will be needed too. Thus these valid pairs will be needed, (c): (Fx · Hx)³(Fx v Hx) (Gx · Ix)³(Gx v Ix) These can be expressed vertically this way: (1) (2) (3)
Fx · Hx Fx Fx v Hx
(1) (2) (3)
Gx · Ix Gx Gx v Ix
1, Simp. 2, Add.
Implications, presented horizontally, and their vertical counterparts (inferences) portray a tight holding together of the universe — as in if earring then ear, if predator then prey, if diapause then winter adaptation. We wish to hold together tightly the valid (a), (b), and (c) in the following valid structure (Kahane, 1986): [(Fx v Hx) · (Fx ³Gx) · (Hx ³Ix)] ³ (Gx v Ix).
(12)
There is then this deductive development to derive the structure of Ex. (12): (1) (2)
Fx Hx
p p
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(3) (4) (5) (6) (7) (8) (9) (10) (11)
(Fx v Hx) · (Fx ³Gx) · (Hx ³Ix) Fx v Hx Fx ³Gx Gx Hx³ Ix Ix Gx · Ix Gx v Ix [(Fx v Hx) · (Fx³ Gx) · (Hx³ Ix)]³ (Gx v Ix)
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Ap 3, Simp.; 1, Add. 3, Simp. 5, 1, M.P. 3, Simp. 7, 2, M.P. 6, 8, Conj. 9, Implication (c) 3–10, C.P.
Step (11) can be expressed vertically in several ways; one is: (1) (2) (3)
Fx v Hx (Fx³ Gx) · (Hx³ Ix) Gx v Ix
(13)
In quantified form Ex. (13) is (Hulburt, 1995): (1) (2) (3)
(x)(Fx v Hx) (x)[(Fx ³Gx) · (Hx ³Ix)] (x)(Gx v Ix)
U.G. U.G. U.G.
(14)
Repetition of the whole formulation of Ex. (13) or Ex. (14) with the conclusion step (3) the first premiss of another formulation is possible — plan A. Additions to Gx and Ix in step (2) of Ex. (14) can be done as Gx v … and Ix v … — plan B. Repetition of modus ponens of steps (5) (1) and (6), occurs in steps (7) (2) and (8) of the development. Any number of repetitions are possible, so that (4) could be Fx v Hx v … and (10) could be Gx v Ix v … — plan C. Ex. (14) can be illustrated by the following description of temperate insects (Howard, 1937; Saunders, 1976; Beck, 1980). 1. All undergo diapause or nondiapause. 2. All, if undergoing diapause, are winter adapted, and if undergoing nondiapause, are summer adapted. 3. Therefore, all are winter adapted or are summer adapted. The clause x is winter adapted is symbolized by Gx. But this is an incomplete symbolization, since really what is said is: x is adapted to winter (×y)(y is winter and x is adapted to y), (×y)(Wy · Axy). So the crucial, dyadic, relational term Axy is present still, though hidden. Likewise for summer adapted. From here on the clauses concerned with adaptation will be shown by Ai x, where the subscript i changes from one situation to another. Thus it will be clear that there is a distinction between physicalistic description in clauses without subscripts and interpretive description in adaptational clauses with subscripts. Ex. (13) or Ex. (14) can be expanded very much by plans A, B, and C. Application of plan A to diapause of the coastal marine plankton copepod Labidocera aesti6a is as follows. Marcus (1979, 1980, 1982) found that parent copepods exposed to long-day experimental regimes produced eggs that hatched
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right away at summer temperatures and that parents under short-day experimental regimes hatched only after being kept at 5°C for 40 days or more then warmed to summer temperatures. So the short-day diapause eggs of fall overwintered in the mud bottom and hatched during the warmth of June and the quickly hatching, subitaneous, eggs of summer produced the continuously present plankton copepods from June to fall in coastal water. In terms of one egg x this supports the iterated formula of Ex. (13) of plan A: (1) (2) (3) (4) (5)
Dx v Sx (Dx³Awx) · (Sx³ Asx) Awx v Asx (Awx ³ Atx) · (Asx³ A1x) Atx v Alx
1. 2. 3. 4.
Egg x is diapause or egg x is subitaneous. If egg x is diapause then it is winter adapted, and if egg x is subitaneous then it is summer adapted. Therefore, egg x is winter adapted or egg x is summer adapted. If egg x is winter adapted then it is from parent copepods adapted to short-day photoperiod, and if egg x is summer adapted then it is from parent copepods adapted to long-day photoperiod. 5. Therefore, egg x is from parents adapted to short-day photoperiod or egg x is from parents adapted to long-day photoperiod. Ex. (14) can be expanded by plan B, by additions at step (2). Experimentally, cells from four diatom species grow slowly in the dark when organic compounds such as sodium glutamate, glucose, lactate, and others are provided. In the light they grow considerably faster, experimentally (Lewin and Hellebust, 1975, 1976; Hellebust and Lewin, 1977). Cells of ciliate protozoa grow better, experimentally, in the light if they contain chloroplasts of ingested algae than they grow in the dark with the same algal food (Stoecker et al., 1988 – 1989); so they are myxotrophic or heterotrophic. With Adx as x is dark adapted, Alx as x is light adapted, Amx as x is myxotrophically adapted, and Ahx as x is heterotrophically adapted the common bond of a varied nutritional adaptation emerges from the following: (1) (2) (3)
(x)(Dx v Cx) (x){[Dx ³ (Adx v Alx)] · [Cx³ (Amx v Ahx)]} (x)(Adx v Alx v Amx v Ahx)
Step (1) is: all cells are from one of the diatom growers or are ciliate growers. Step (3) is: therefore, all cells are dark or light or myxotrophically or heterotrophically adapted. Ex. (14) can be expanded by plan C, by having more than two modus ponens structures in conjunction. Two cases will be presented of this expansion — one case with a small coverage and a second with a large coverage of the world’s biota. The first is concerned with multiple evolutionary pathways leading to the adaptation of fish being warm under varied ecological conditions. In tuna fish dark muscle with profusely branching blood vessels in the body wall forms a heat exchanger keeping the central body 10°C higher than the ends of the fish (at water temperature) (Carey, 1973). In Carey’s words, ‘a major adaptive advantage of an elevated body temperature is greatly enhanced muscle power and this must surely be connected to the adaptive advantage of increased swimming speed’. Thus the tuna fish can swim faster and supposedly catch fast prey better in cold water than it could if it were at water temperature. Lamnid sharks with increasing amounts of red, axial, heat exchanger muscle have higher mid-body temperatures and range farther north in the ocean (Carey et al., 1985). The brain of the swordfish is heated 10–14°C above water temperatures by brown thermogenic tissue next to the brain (Carey, 1982) and this is an advantage in deep diving, up to 600 m, where the water can be 19°C less than at the surface (Carey and
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Robinson, 1981). So there are three evolutionary pathways to heat adaptation. Where (x) is: for all evolutionary pathways x, or briefly pathways, we have: (1) (2) (3)
(x)(Dx v Rx v Tx) (x)[(Dx ³ Adx) · (Rx³Arx) · (Tx³ Atx)] (x)(Adx v Arx v Atx)
1. Pathways are by body wall dark muscle, Dx, or by axial red muscle, Rx, or by brown thermogenic tissue next to the brain, Tx. 2. Pathways, if by body wall dark muscle, led to the adaptive advantage of fast swimming, and if by axial red muscle, led to the adaptive advantage of allowing northern distribution, and if by brain thermogenic tissue, led to the adaptive advantage of deep diving. 3. Therefore, pathways led to the adaptive advantage of fast swimming or northern distribution or deep diving. Multiple pathways of evolution are exemplified by the equations for CO2 fixation (Jannasch and Mottl, 1985): hv
2CO2 + H2S+ 2H2O 2[CH2O] +H2SO4 (nonoxygenic photoautolithotrophy, purple and green bacteria) hv
(A1)
CO2 + H2O [CH2O] +O2 (oxygenic photoautolithotrophy, green plants)
(A2)
CO2 + H2S +O2 +H2O [CH2O] +H2SO4 (aerobic chemoautolithotrophy, bacteria)
(A3)
2CO2 + 6H2 [CH2O] +CH4 +3H2O (anaerobic chemoautolithotrophy, bacteria).
(A4)
The first occurs terrestrially; the second occurs terrestrially and in seaweeds and the phytoplankton of the sea surface; the third and fourth occur in the hydrothermal vents of the sea floor. More simply, multiple pathways of CO2 fixation are by bacterial photosynthesis, by plant photosynthesis, and two by bacterial chemosynthesis. More simply still, there are two pathways on the planetary surface and two pathways on the sea floor. With (x) for all pathways x or just pathways and Fx for pathway x is by nonoxygenic photoautolithotrophy, etc. and Apx for x is adapted to the lighted planetary surface, etc. we have: (1) (2) (3)
(x)(Fx v Gx v Hx v Ix) (x)[(Fx ³ Apx) · (Gx ³Apx) · (Hx³ Asx)³ (Ix³ Asx) (x)(Apx v Asx)
1. Pathways are by Eq. (A1) or by Eq. (A2) or by Eq. (A3) or by Eq. (A4). 2. Pathways if by Eq. (A1) are adapted to the lighted planetary surface and if by Eq. (A2) are adapted to the lighted planetary surface and if by Eq. (A3) are adapted to the dark sea floor and if by Eq. (A4) are adapted to the dark sea floor. 3. Therefore, pathways are adapted to the lighted planetary surface or are adapted to the dark sea floor. It is clear that this is large scale adaptation. It is crucial, because adaptation is to be looked upon as deeply permeating biological material and ecological structure.
8. Discussion The theory of adaptation presented is a static analysis approach. The expressions developed are for describing situations just as they are. However, some of the expressions are ideal for describing in a
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dynamic way evolutionary change. Thus, expression Ex. (10) describes the partial evolutionary change to warm blooded from cold blooded terrestrial temperate vertebrates in: all, if active year-round are adapted to year-round temperature if and only if all, if not adapted to year-round temperature, are not active year-round—the evolutionary change is to adapted from unadapted. Thus, too, expression Ex. (14) describes the termination of evolutionary change in that temperate insects are diapause, winter adapted or are nondiapause, summer adapted. The guiding feature in the theory of adaptation presented is the selection of the basic element Axy. Then elaboration follows a narrow path at first, in which validity dictates Axy in (Axy³ Ey) ( Ey³ Axy). The affirmation of adaptation and its denial go hand in hand. They portray perfectly various parts of the biological world, whether the parts are small as the vertical niches of Anolis species or are large as the North American forest. The elaboration then buries the dyadic relational Axy in a simpler structure, the monadic Aix, and diversifies this in the affirmation of adaptation. The purpose then is just to put together two or more modus ponens structures by conjunction and then to change parts of these to disjunctions. Thus diapause and nondiapause are closely related but exclude each other and are ‘or’ related, are diapause or nondiapause. These intimate winter adaptation or summer adaptation. But these lead, too, to multiple pathways of evolution which are or related. Thus adaptation is seen ecologically primarily, with evolution as a process of value in holding together configurations of nature that have both similarity and the bond of being adapted. Thus the theory presented has two divergent formulations. But these come from the same starting formula. The sequence of steps from the starting formula may be viewed as follows. Where p is for Axy and q for Ey the first sequence is: (1) (2) (3) (4) (5) (6)
p³p (p ³ q) ³ (p ³q) [(p ³ q) ³ (q ³ p)] · [(q ³ p)³(p³q)] (q ³p) (p ³ q) (p ³ q) (q ³p) [(p ³ q) · q]³ p
Where p is for Dx, q for Awx, r for Sx, and s for Asx (as in the Labidocera example) the second sequence is: (1) (2) (3) (4) (5) (6)
p ³p (p ³q) ³ (p ³q) [(p ³ q) · p]³q [(p ³q) · p] ³ q · [(r³ s) · r] ³s [p · (p ³ q)] ³ q · [r · (r³s)]³s [(p v r) · (p ³q) · (r³s)]³(q v s)
Steps (1) and (2) are the same in each sequence; at step (3) there is diverging structure. All steps have only valid structure. Each step follows validly from the last3. Each sequence is valid as a whole. Validity is not mere overlay on the description of nature. Validity permeates nature and must be sought in nature. We see validly, what we see is valid. But this separation is inconsequential. Nature is valid.
3
Except step (6) in the first sequence, which comes from step (3) by simplification and exportation.
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The model for incompletely symbolized expression blocks the denial of adaptation, in the sense that validity is achieved without denial. In another study (Hulburt, 1996) validity is achieved without denial by use of the formula [(p ³q) · (q³ p)] (q p). Elaboration of this formula occurs in p. Thus if x and y help each other (symbionts), or x dominates y (plant and habitat), or x suppresses y (phytoplankton and nutrient), or x preys on y (predator and prey), or x enslaves y (ciliate and ingested algal chloroplast)—all this in place of p — then x and y are adapted to each other—this last for q; thus p ³q. Though denial can be entertained in q — if plant x is adapted to habitat y then y is adapted to x (x and y are adapted to each other), but suppose habitat y is not adapted to x then how ever could plant x be adapted to y — such lurking denial is overwhelmed by the large-scale affirmation of adaptation available from the elaborated formulation. The theory separates physicalistic description from interpretive adaptational description haphazardly in the fully symbolized expressions. But both physicalistic and interpretive description are clearly distinguished in the model for incompletely symbolized expressions. Here in the modus ponens structure the factual statement is put first, diapause, and then the interpretive ascent: if diapause then winter adaptation is put second. First diatom grower, then the interpretive ascent in: if diatom grower then dark adapted or light adapted. First body wall dark muscle, then the interpretive ascent in: if body wall dark muscle then adaptive advantage of fast swimming. First pathway of evolution by nonoxygenic photoautolithotrophy, then the interpretive ascent in: if pathway of evolution by nonoxygenic photoautolithotrophy then adaptation to the lighted planetary surface. Nature is valid, but this validity spans from the physicalistic to the interpretive, to adaptation.
9. Glossary Connectives: · And v Or ³ Goes between a clause beginning with if and a clause beginning with then; thus the if clause implies the then clause
Equivalent to, if and only if Not Therefore p Premiss statement Ap Assumed premiss — when if is put before a statement to be followed by a then statement later, indicated by a broken arrow and C.P. x, y Variables—they stand for subject and object in a clause or phrase (x) For every x, all x (the universal quantifier) (×x) There is an x such that (the existential quantifier) Validity is truth by truth value analysis, wherein a true ‘if’ statement implying a true ‘then’ statement makes a true whole, a false implying a true makes a true whole and a false implying a false makes a true whole. So Axy ³Axy is true³true and false³ false, both true as a whole. Only true implying false is false as a whole. So in Ex. (1) Axy ³ Ey could be false, but false³ false would happen for the whole of Ex. (1), so Ex. (1) would be true as a whole for this case—and for the other cases of true and false for Axy and Ey, so Ex. (1) is only true, is valid.
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With the added rule that when equivalent expressions are alike in truth value, are true true or false false, both cases are true as wholes, and noting that denial ( ) of true is false and denial of false is true and disregarding the quantifier (y), Ex. (2), Ex. (3), Ex. (4), Ex. (5), and points (1)–(5) are found valid. Ex. (10) is seen to be valid because the equivalents in parentheses are in conjunction with Gy which if true makes both sides true or false as the equivalents are true or false and which if false makes both sides false—so that both sides are alike in truth value and are thus equivalent. Beginning with the North American forest a different method of showing validity is used, a method that depends on the following procedures.
10. Rules of inference Rules of substitution are such that p, q, r may have substitutions of Axy, Ey, (Txy³ My), (y)(Hy ³ Axy), etc. in place of one of p, q, r: 1
Modus ponens (M.P.) p ³ q p q
2
Modus tollens (M.T.) p³q q p
3
Conjunction (Conj.) p q p ·q
4
Simplification (Simp.) p ·q p
5
Addition (Add.) p pvq
Rules of replacement: any of the following equivalent expressions can replace each other wherever they occur: 6 7 8 9 10 11 12 13
Transposition (Trans.) Commutation (Comm.) De Morgan’s theorem (De. M.) Implication (Imp.) Exportation (Exp.) Double negation (D.N.) Equivalence (Equiv.) Quantifier negation (Q.N.)
(p ³ q) (q ³ p) (p · q) (q · p) (p · q) (p v q) (p³ q) (p · q), (p v q) (p ³ q) [p³ (q³ r)] [(p · q)³ r] p p [(p ³q) · (q³ p)] (q p) (×y) (y) (y) (×y)
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10.1. Uni6ersal generalization (U.G) Fx (x)Fx
10.2. Uni6ersal instantiation (U.I.) (x)Fx Fx
10.3. Existential generalization (E.G.) Fx (×x)Fx
10.4. Existential instantiation (E.I.) (×x)Fx Fx Meaning of some terms: Conjunction Disjunction Modus ponens Modus tollens Monadic Dyadic Diapause Nondiapause Physicalistic Interpretive
The joining of two clauses, or statements, by ‘and’ The joining of two clauses, or statements, by ‘or’ An argument of this form: if this then that, this; therefore that An argument of this form: if this then that, not that; therefore not this A clause, or statement, with only predicate and subject, such as Ey A clause, or statement, with predicate, subject, and object, such as Axy An overwintering, non-developing stage of an insect, either egg or larva, brought about by short day-length A spring-summer mature stage of an insect brought about by long day-length (plus warmth) Description that is close to the facts Description that adds a connotative, personalistic element to the facts
References Beck, S.D., 1980. Insect Photoperiodism. Academic Press, New York, p. 378. Brokaw, N.V.L., 1982. Treefalls: frequency, timing, consequences. In: E.G. Leigh, Jr., A.S. Rand, D.M. Winsor (Eds.), The Ecology of a Tropical Forest: Seasonal Rhythms and Long-term Changes. Smithsonian Institution Press, Washington, D.C., pp. 101 – 8. Brokaw, N.V.L., 1985. Gap-phase regeneration in a tropical forest. Ecology 66, 682 – 687. Carey, F.G., 1973. Fishes with warm bodies. Sci. Am. 228, 36 – 44.
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Carey, F.G., 1982. A brain heater in the swordfish. Science 216, 1327 – 1329. Carey, F.G., Robinson, B.H., 1981. Daily patterns in the activities of swordfish, Xiphias gladius, observed by acoustic telemetry. Fish. Bull. Fish. Wild. Serv. 79, 277–292. Carey, F.G., Casey, J.G., Pratt, H.L., Urquhart, D., McCosker, J.E., 1985. Temperature, heat production, and heat exchange in Laminid sharks. Mem. South Californian Acad. Sci. 9, 92 – 108. Copi, I.M., 1979. Symbolic Logic. MacMillan, New York, p. 398. Grant, A., Grant, V., 1954. Genetic and taxonomic studies in Gilia. VII. The woodland Gilias. El Aliso 3, 59 – 91. Hellebust, J.A., Lewin, J., 1977. Heterotrophic nutrition. In: Werner, D. (Ed.), The Biology of Diatoms. University of California Press, Berkeley, pp. 169–97, p. 498. Howard, L.O., 1937. The Insect Book. Doubleday, Doran, Garden City, p. 429. Hulburt, E.M., 1992. Equivalence and the adaptationist program. Ecol. Model. 64, 305 – 329. Hulburt, E.M., 1995. Adaptation, Its Logic, and the Sentimentalization of Nature. Woods Hole Oceanographic Institution, Woods Hole, MA, p. 118. Hulburt, E.M., 1996. The symmetry of adaptation in predominantly asymmetrical contexts. Ecol. Model. 85, 173 – 185. Jannasch, H.W., Mottl, M.J., 1985. Geomicrobiology of deep-sea hydrothermal vents. Science 229, 717 – 725. Kahane, H., 1986. Logic and Philosophy. Wadsworth, Belmont, California, p. 502. Lewin, J., Hellebust, J.A., 1975. Heterotrophic nutrition of the marine diatom Nitzchia pa6illardi Hustedt. Can. J. Microbiol. 21, 1335 – 1342. Lewin, J., Hellebust, J.A., 1976. Heterotrophic nutrition of the marine pennate diatom Nitzschia angularis var. afinis. Mar. Biol. 36, 313 – 320. Levins, R., Lewontin, R., 1985. The Dialectical Biologist. Harvard University Press, Cambridge, MA, p. 303. Marcus, N.H., 1979. On the population biology and nature of diapause of Labidocera aesti6e (Copepoda: Calanoida). Biol. Bull. 157, 297 – 305. Marcus, N.H., 1980. Photoperiodic control of diapause in the marine calanoid copepod Labidocera aesti6a. Biol. Bull. 159, 311 – 318. Marcus, N.H., 1982. Photoperiodic and temperature regulation of diapause in Labidocera aesti6a (Copepods: Calanoida). Biol. Bull. 162, 45 – 52. Moermond, T.C., 1979. Habitat constraints on the behavior, morphology, and community of Anolis lizards. Ecology 60, 152 – 164. Oosting, H.J., 1948. The Study of Plant Communities. W.H. Freeman, San Francisco, p. 389. Quine, W.V., 1972. Methods of Logic. Holt, Rinehart and Winston, New York, p. 280. Rand, A.S., 1962. Notes on Hispaniolan herpetology. 5. the natural history of three sympatric species of Anolis. Breviora. Mus. Comp. Zool. 154, 15. Ruibal, R., 1961. Thermal relations of five species of tropical lizards. Evolution 15, 91 – 111. Ruibal, R., Philibosian, R., 1970. Eurythermy and niche expansion in lizards. Copeia 4, 645 – 653. Saunders, D.S., 1976. Insect clocks. Pergamon, Oxford, New York, p. 279. Stoecker, D.K., Silver, M.W., Michaels, A.E., Davis, L.H., 1988 – 1989. Enslavement of algal chloroplasts by four Strombidium spp. (Ciliophora Oligotrichida). Mar. Micro. Food Webs 3, 79 – 100. Williams, E.E., 1983. Ecomorphs, faunas, island size and diverse end points in island radiations of Anolis. In: Huey, R.B., Pianka, E., Schoener, T. (Eds.), Lizard Ecology, Studies of a Model Organism. Harvard University Press, Cambridge, MA, pp. 326 – 70.
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Theory of adaptation: application of symbolic logic Edward M. Hulburt * Woods Hole Oceanographic Institution, Woods Hole, MA 02543, USA Received 22 April 1997; accepted 18 November 1997
Abstract The theory of adaptation takes up, first, completely symbolized expressions in which the basic element is the dyadic term Axy, x is adapted to y. When this is combined with a monadic term such as Ey, y is a moist environment, in (y)(Axy³ Ey), validity can be achieved by reversal and denial (transposition): (y)(Axy³ Ey) (y)( Ey³ Axy). This provides a full and complete description when applied as follows: the North American forest x is adapted only to moist environments if and only if it is not adapted to any non-moist environments. Then dyadic – dyadic, monadic–dyadic–dyadic, and monadic–dyadic–monadic – dyadic combinations are presented and exemplified. The theory of adaptation takes up, second, incompletely symbolized expressions such as:(x)(Fx v Hx), (x)[(Fx³ Gx) · (Hx³Ix)], (x)(Gx v Ix)%, which provides the structure for this description of temperate insects: all undergo diapause or nondiapause; all, if undergoing diapause, are winter adapted, and if undergoing nondiapause, are summer adapted; therefore, all are winter adapted or summer adapted. The phrases winter adapted, Gx, and summer adapted, Ix, are incomplete symbolizations since they are brief versions of (×y)(Wy · Axy), x is adapted to winter, and of (×y)(Sy · Axy), x is adapted to summer. The brief versions, however, make elaborations and extensions of the basic formulation possible. The formulas of complete symbolization provide for affirmation or denial of adaptation or no decision. Thus, in addition to the example of the North American forest other examples are species of the lizard Anolis, each of which is well adapted in its niche but not well adapted outside its niche, the associations of the North American forest which are tall, densely packed and adapted in moist regions but short, scattered, and unadapted in non-moist regions, and species of Gilia which are adapted to their habitats unless these habitats are not adapted to them. Then in small-to-moderate forest gaps regrowth of prevalent, adapted primary species contrasts with the regrowth of unadapted, non-prevalent pioneer species. But it cannot be decided whether the lizard Anolis allogus in a smaller niche on Cuba is adapted to this or is not adapted to this when comparing it to Anolis oculatus in its larger niche on Dominica. The formulas of incomplete symbolization provide only affirmation of adaptation. Examples in addition to the diapause—nondiapause case are for diatoms which are dark or light adapted and ciliates which are myxotrophically or heterotrophically adapted. Then, evolutionary pathways to warm bloodedness of fish include being adapted to fast swimming in tuna fish, or northern distribution in lamnid sharks, or deep diving in swordfish.
* Tel.: +1 508 5483074. 0304-3800/98/$19.00 © 1998 Elsevier Science B.V. All rights reserved. PII S 0 3 0 4 - 3 8 0 0 ( 9 7 ) 0 0 1 9 9 - 3
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And finally four basic evolutionary pathways include anaerobic and aerobic photosynthesis and thus adaptation to the lighted planetary surface and include chemosynthesis in two ways and thus adaptation to the dark ocean floor. © 1998 Elsevier Science B.V. All rights reserved. Keywords: Adapted; Valid; Dyadic; Physicalistic; Interpretive
1. Introduction The concerns of this study are both small scale and large scale. The thrust will be predominantly ecological, for adaptation is ecological. The issues to be encompassed will be offshoots of two strands of thought. A single formulation will be at first dominant. The formulation is just that one thing is adapted to another, that x is adapted to y. Then a second formulation will stem from the simple precept that x is adapted in this way or that way or in some further way. From these simple formulations a richness of detail and outlook will emerge. Is there precedent for the outlook that is to come? Except in writings of the author there is little precedent. But it is the wish and interest of the author’s studies to make adaptation a deeply permeating part of biological material and ecological structure. This intent is insisted upon strenuously in the present study. There is nothing wrong in pointing out how some structure, the opposed digits of a bird, for example, makes the bird adapted in some respect or constitutes an adaptation that the bird has. But such adaptations are minor. Surely they do not encompass the adaptation of biological entities to the physical world and to each other. The broad adaptation of entities to the physical world and to each other is an old assumption, immersed in Western social and religious structure (Levins and Lewontin, 1985). It would seem difficult to avoid this old assumption, this old belief in the harmony of the world. Even though there are so many cases against it, so many vicissitudes that beset the world and that beset the belief in its harmony, in its adaptation. So what follows is split to a considerable degree between seeing adaptedness and seeing its lack. The method is simple, for if x is adapted to y then y is a certain condition of the environment, which is to say that if y is not this condition of the environment then x is not adapted to y. But this is a logical structure, a case of logical validity. Thus it is that the theory of adaptation to be presented is a logic structure theory. Some knowledge of elementary symbolic logic is advisable for perusal of what is to come—a knowledge obtainable of course from texts such as those of Kahane (1986), Copi (1979), and Quine (1972). But briefer instruction is all that is necessary, to be gotten from Hulburt (1992, 1995, 1996). Also, a glossary of terminology is given at the end; perhaps this is explanation enough.
2. Fully symbolized expressions: dyadic-monadic linkages The basic dyadic element is Axy, x is adapted to y. With this validity1 is: if Axy then Axy, Axy implies Axy (see glossary): Axy³ Axy.
1
Validity is superficially obvious in‘Axy³ Axy’. Validity is achieved by truth value analysis (see glossary).
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Axy is linked to the monadic term Ey, y is a moist environment, in a validity arrangement mimicking the above formula: (Axy³Ey)³(Axy ³Ey).
(1)
The right member in parentheses may be replaced by its transpositional equivalent Ey ³ Axy. And the left member may be similarly and separately replaced and the resulting expressions (both valid) may be joined by conjunction (by ‘and’, ‘ · ’): [(Axy ³ Ey)³( Ey ³ Axy)] · [( Ey ³ Axy)³ (Axy³ Ey)],
Conj.
which is: (Ey ³ Axy) (Axy ³ Ey)
Equiv.
(Axy³Ey) ( Ey ³ Axy)
Comm..
(2)
In partially quantified form Ex. (2) is: (y)(Axy ³Ey) (y)( Ey ³ Axy)
U.G..
(3)
A full and complete description thus eventuates. By way of example, with Ey still as y is a moist environment, Ex. (3) is: for every y, if the North American forest x is adapted to y then y is a moist environment—equivalent to: for every y, if y is not a moist environment then x is not adapted to y. Briefly, the North American forest is adapted only to moist environments if and only if it is not adapted to any non-moist environments. For the North American forest is composed of tall and densely packed species in the East and West where rainfall is \ 30 inches/year and exemplifies adapted forest. In the southwest US where rainfall is B 30 inches/year pinon pine, creosote bush, and other woody species are small and spaced apart (Oosting, 1948; Hulburt, 1995) and exemplify unadapted forest. Ex. (3) may have adapted denied with the environmental term undenied. This is shown next, in Ex. (5), in a description of the niche structure of species of the lizard Anolis on Hispaniola (Rand, 1962; Moermond, 1979; Williams, 1983). Going from the ground up the tree, there are the ground-trunk species A. cybotes, the trunk species A. distichus, the trunk-crown species A. chlorocyanus, and the crown species A. ricordii. These overlap each other considerably. This overlap is portrayed in Ex. (4), (5) and points (1) – (5) following. Abbreviations for these expressions are the following. Gy is y is a ground-trunk niche, Ty is y is a trunk niche, Ky is y is a trunk-crown niche, Cy is y is a crown niche, x is for A. cybotes, z is for A. distichus, u is for A. chlorocyanus, and 6 is for A. ricordii. We have first (note z in Ex. (5)): (y)(Axy ³ Gy) (y)( Gy ³ Axy)
(4)
(y)( Azy³ Gy) (y)( Gy ³Azy)
(5)
Ex. (4) says Cybotes is well adapted only to ground-trunk niches if and only if it is not well adapted to non-ground-trunk niches. Ex. (5) says Distichus is not well adapted only to ground-trunk niches if and only if it is well adapted to non-ground-trunk niches. Equating non-ground trunk niches to trunk niches, we can say next in 1 and 2: (1) (2)
(y)(Azy ³ Ty) (y)(Ty ³ Azy) (y)(Auy ³Ty) (y)(Ty ³ Auy)
(1) Distichus is well adapted only to trunk niches if and only if it is not well adapted to non-trunk niches; (2) Chlorocyanus is not well adapted only to trunk niches if and only if it is well adapted to non-trunk niches. Equating non-trunk niches to trunk-crown niches we say next:
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(3) (4) (5)
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(y)(Auy ³Ky) (y)(Ky ³ Auy) (y)(A6y ³ Ky) (y)(Ky ³A6y) (y)(A6y ³ Cy) (y)(Cy ³ A6y)
(3) Chlorocyanus is well adapted only to trunk-crown niches if and only if it is not well adapted to non-trunk-crown niches; (4) Ricordii is not well adapted only to trunk-crown niches if and only if it is well adapted to non-trunk-crown niches — which are the same as crown niches and (5) Ricordii is well adapted only to crown niches if and only if it is not well adapted to non- crown niches. The distinction between well adapted and not well adapted must be reiterated throughout the vertical niche structure. Well adapted cannot possibly be applied throughout this structure. This is a vital perception to be gleaned from the dividing up of the structural niche and the overlapping of the dividing up species and the attenuation of the lower species as they overlap the upper species. Repetition of the dyadic-monadic linkage of Ex. (1) is in order. Consideration of the North American forest suggests some linkages. No matter whether one considers the beech–maple association of eastern US, the spruce – fir association of eastern Canada, the Engelmann spruce–subalpine fir association of the Rocky mountains, the Sitka spruce dominated forest from Alaska to the luxuriant Olympic Peninsula where it is joined by western hemlock, arborvitae, and grand fir, or other associations outside the dry southwestern US (Oosting, 1948; Hulburt, 1995); all associations x are tall only in moist regions, (x)(y)(Txy ³My), all associations are densely packed only in moist regions, (x)(y)(Dxy ³My), all associations are competitive only in moist regions, (x)(y)(Cxy ³ My), all associations are high growth capable only in moist regions, (x)(y)(Gxy ³ My), all associations are adapted only in moist regions (x)(y)(Axy ³ My). But there are non-moist regions (×y) My in the southwest US. We have the following valid argument, with repeated use of modus tollens: (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16) (17) (18) (19) (20) (21)
(x)(y)(Txy ³ My) (x)(y)(Dxy ³My) (x)(y)(Cxy ³My) (x)(y)(Gxy ³ My) (x)(y)(Axy ³My) (×y)My My Txy³My Txy Dxy³My Dxy Cxy³My Cxy Gxy³My Gxy Axy³My Axy Txy · Dxy · Cxy · Gxy · Axy (×x)(Txy · Dxy · Cxy · Gxy · Axy) My · (×x)(Txy · Dxy · Cxy · Gxy · Axy) (×y)[My · (×x)(Txy · Dxy · Cxy · Gxy · Axy)]
p p p p p p 6, E.I. 1, U.I. 8, 7, M.T. 2, U.I. 10, 7, M.T. 3, U.I. 12, 7, M.T. 4, U.I. 14, 7, M.T. 5, U.I. 16, 7, M.T. 9, 11, 13, 15, 17, Conj. 18, E.G. 7, 19, Conj. 20, E.G.
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Step (21) says therefore, there is a non-moist environment in which there is an association which is not tall and not densely packed and not competitive and not high growth capable and not adapted. The proof of (21) from (1)–(6) has adapted in (5) and not adapted in (21). These are interpretations put on the physicalisticly descriptive tall, densely packed, competitive, and high growth capable, and their negatives. If all the physicalisticly descriptive predicates are denied in the southwest USA, it would be best to go along with denying the interpretive word adapted too. Modus tollens used five times as in: Txy³ My My Txy is close to transposition, since transposition leads to modus tollens as follows: (1) (Txy³My) (My ³Txy) (2) [(Txy ³ My)³(My ³Txy)] · [(My ³Txy)³ (Txy³ My)] (3) (Txy³My)³ (My ³Txy) (4) [(Txy ³My) · My] ³Txy
1, Comm., Equiv. 2, Simp. 3, Exp.
Step (4) is modus tollens written horizontally.
3. Fully symbolized expressions: dyadic – dyadic linkages The basic dyadic element, Axy, has reversal of variables in Ayx. Thus if x is adapted to y then y is adapted to x (Hulburt, 1996): Axy³ Ayx.
(6)
For example, if symbiont x is adapted to symbiont y then symbiont y is adapted to symbiont x. This is a case of symmetry. Another case of symmetry is that of plant species x and it’s habitat y. Again validity is gotten through transposition: (Axy³ Ayx) ( Ayx ³ Axy)
(7)
saying if plant species x is adapted to habitat y then y is adapted to x, if and only if, if habitat y is not adapted to species x then x is not adapted to y. Surely if habitat y is not adapted to species x, how could x flourish there and be adapted to y? The annual herbs Gilia splendens and G. caruifolia are allopatric, the first occurring in the south Coast range and the San Gabriel, San Bernardino and San Jacinto ranges in southern California and the second ranging southward from the Cuyamaca, Laguna and Palommor mountains to the San Pedro Martin range (Grant and Grant, 1954). So these two species are examples of Ex. (7), when their names are put for x and their habitats are put for y. Gilia australis occurs too in very southern California but in hotter, drier habitats of sandy marshes of the foothills and plains below the pine belt. Where Hy is y is a hotter, drier habitat and x is put for G. australis we have: (y)(Hy ³Axy)³ (y)(Hy ³Ayx)
(8)
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which says if australis, x, is adapted to hotter, drier habitats, then hotter, drier habitats are adapted to it2. This and the transpositional equivalent ((5) below) make a valid whole. The equivalent is gotten as follows: (1) (2) (3) (4) (5)
(y)(Hy ³ Axy) ³ (y)(Hy ³ Ayx) (y)(Hy ³ Ayx) ³(y)(Hy ³Axy) (×y)(Hy ³ Ayx) ³ (×y)(Hy ³ Axy) (×y)(Hy · Ayx) ³ (×y)(Hy · Axy) (×y)(Hy · Ayx) ³ (×y)(Hy · Axy)
1, 2, 3, 4,
Trans. Q.N. Imp. D.N.
Step (5) says that if there is a hotter, drier habitat and it is not adapted to australis, x, then there is a hotter, drier habitat and australis is not adapted to it. For suppose that a hotter, drier habitat is not adapted to australis, then how ever could australis be adapted to it?
4. Fully symbolized expressions: monadic – dyadic –dyadic connections The basic dyadic element, Axy, is preceded by another dyadic term, Pxy, as Pxy³Axy. This linkage is joined to an existentially quantified monadic term, (×y)[Gy · (Pxy ³ Axy)]
(9)
Let us take a particular case, that of regrowth in forest gaps from fallen trees at Barro Colorado, Panama (Brokaw, 1982, 1985). In the smaller and moderate gaps saplings of primary species, those that persist in gaps of all sizes, are very prevalent; in the very large gaps saplings of pioneer species, those that persist only in large gaps, are modestly prevalent. In general primary species are prevalent whereas pioneer species are not in small-to-moderate gaps. Where (x) takes in all species, primary and pioneer both, we have: (x)(×y)[(Gy · (Pxy³ Axy)] (x)(×y)[Gy · ( Axy³ Pxy)]
(10)
saying all species, if prevalent in, are well adapted to small-to-moderate gaps; if and only if all species, if not well adapted to, are not prevalent in small-to-moderate gaps. This is brief for the detailed: for every species x there is a y such that y is a small-to-moderate gap and if x is prevalent in y then x is adapted to y; if and only if …. A crucial feature of Ex. (10) is overlap, since under the same conditions of small-to-moderate gaps prevalent, well adapted species and not well adapted, non-prevalent species overlap. A striking feature of Ex. (10) is that the dyadic term Pxy provides a criterion for x’s being adapted to y. There are many cases that exemplify Ex. (10) or various versions of Ex. (10). Some of these cases are given in Hulburt (1992).
2 The (y)(Hy³Axy) part of Ex. (8) says: for every y, if y is a hotter, drier habitat then x is adapted to y, which is: all hotter, drier habitats x is adapted to, which is: x is adapted to hotter, drier habitats.
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5. Fully symbolized expressions: monadic – dyadic –monadic–dyadic connections The basic dyadic element Axy is last in a series of terms in this manner: (×y)( Ry · Ixy) ³(×y)( Ry · Axy).
(11)
.Proceeding with transposition yields two results —two disastrous results—as follows: (1) (2) (3) (4) (5)
(×y)(Ry · Ixy) ³ (×y)(Ry · Axy) (×y)(Ry · Axy) ³(×y)(Ry · Ixy) (y)(Ry · Axy) ³ (y)(Ry · Ixy) (y)(Ry v Axy) ³(y)(Ry v Ixy) (y)(Ry ³Axy) ³ (y)(Ry ³ Ixy)
Eq. (11) 1, Trans. 2, Q.N. 3, De.M. 4, D.N., Imp.
Thus, with Ry for y is an unrestricted niche and Ixy for species x is interfered with in y (1) says if species x is not interfered with in an unrestricted niche then it is adapted to an unrestricted niche. And the equivalent (5) says if species x is not adapted to any unrestricted niches then it is interfered with in the unrestricted niche. The lizard Anolis oculatus on Dominica confirms Ex. (11), for it occurs throughout the 10°C range of the island uninterfered with by any other species. But on Cuba A. allogus confirms (5) because it does not occur through the 10°C niche due to interference from other species that overlap it (Ruibal and Philibosian, 1970; Ruibal, 1961) and so it is not adapted to the unrestricted 10°C niche. But the following also pertains: (6) (7) (8) (9) (10)
(×y)(Ixy · Ry) ³(×y)(Axy · Ry) (×y)(Axy · Ry) ³(×y)(Ixy · Ry) (y)(Axy · Ry) ³(y)(Ixy · Ry) (y)(Axy v Ry) ³(y)(Ixy v Ry) (y)(Axy ³ Ry) ³(y)(Ixy ³Ry)
1, 6, 7, 8, 9,
Comm. Trans. Q.N. De.M. Imp., D.N.
Step (10) says if species x is adapted only to restricted niches then it is uninterfered with only in restricted niches. That is, A. allogus by confirming this can be considered to be adapted to the restricted B 10°C niche imposed on it by the other competing species. The point here is that you can not decide about the adaptation of A. allogus on Cuba by comparing it with A. oculatus on Dominica. If oculatus and allogus occurred together on Dominica, oculatus covering the unrestricted 10°C range but allogus covering less than this 10°C range, then you would probably want to say that oculatus was adapted, or better adapted, to this niche range and that allogus was not. But the two species do no overlap on the same island, and it can not be decided whether allogus is adapted or not.
6. Adaptational decision The use of transposition and modus tollens are decision procedures, providing for adapted and its denial, not adapted — or well adapted and not well adapted. But this decision procedure depends on the applicability of two or three part structure only, as in Ex. (1) and Ex. (9). When four part structure of the type in Ex. (11) is applied no decision is possible.
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7. Incompletely symbolized expressions The monadic terms, Fx, Gx, Hx, and Ix will be used in a varied manner. Some will be used simply, as previously done. Others will be used to express relationships, so that in a sense the relationships are incompletely symbolized. The terms can be used in the following development. To the valid: Fx³ Fx there corresponds the valid: (Fx³ Gx) ³ (Fx ³Gx) which can be further changed by these equivalents: [(Fx ³ Gx) · Fx] ³Gx,
Exp.
[Fx · (Fx ³Gx)]³ Gx.
Comm.
Presented vertically as a deduction the last is (a): (1) (2) (3)
Fx Fx ³ Gx Gx
2,1, M.P.
A repetition of this modus ponens deduction is a second modus ponens, (b): (1) (2) (3)
Hx Hx ³Ix Ix
2,1, M.P.
These two structures will be needed. And a third structure will be needed too. Thus these valid pairs will be needed, (c): (Fx · Hx)³(Fx v Hx) (Gx · Ix)³(Gx v Ix) These can be expressed vertically this way: (1) (2) (3)
Fx · Hx Fx Fx v Hx
(1) (2) (3)
Gx · Ix Gx Gx v Ix
1, Simp. 2, Add.
Implications, presented horizontally, and their vertical counterparts (inferences) portray a tight holding together of the universe — as in if earring then ear, if predator then prey, if diapause then winter adaptation. We wish to hold together tightly the valid (a), (b), and (c) in the following valid structure (Kahane, 1986): [(Fx v Hx) · (Fx ³Gx) · (Hx ³Ix)] ³ (Gx v Ix).
(12)
There is then this deductive development to derive the structure of Ex. (12): (1) (2)
Fx Hx
p p
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(3) (4) (5) (6) (7) (8) (9) (10) (11)
(Fx v Hx) · (Fx ³Gx) · (Hx ³Ix) Fx v Hx Fx ³Gx Gx Hx³ Ix Ix Gx · Ix Gx v Ix [(Fx v Hx) · (Fx³ Gx) · (Hx³ Ix)]³ (Gx v Ix)
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Ap 3, Simp.; 1, Add. 3, Simp. 5, 1, M.P. 3, Simp. 7, 2, M.P. 6, 8, Conj. 9, Implication (c) 3–10, C.P.
Step (11) can be expressed vertically in several ways; one is: (1) (2) (3)
Fx v Hx (Fx³ Gx) · (Hx³ Ix) Gx v Ix
(13)
In quantified form Ex. (13) is (Hulburt, 1995): (1) (2) (3)
(x)(Fx v Hx) (x)[(Fx ³Gx) · (Hx ³Ix)] (x)(Gx v Ix)
U.G. U.G. U.G.
(14)
Repetition of the whole formulation of Ex. (13) or Ex. (14) with the conclusion step (3) the first premiss of another formulation is possible — plan A. Additions to Gx and Ix in step (2) of Ex. (14) can be done as Gx v … and Ix v … — plan B. Repetition of modus ponens of steps (5) (1) and (6), occurs in steps (7) (2) and (8) of the development. Any number of repetitions are possible, so that (4) could be Fx v Hx v … and (10) could be Gx v Ix v … — plan C. Ex. (14) can be illustrated by the following description of temperate insects (Howard, 1937; Saunders, 1976; Beck, 1980). 1. All undergo diapause or nondiapause. 2. All, if undergoing diapause, are winter adapted, and if undergoing nondiapause, are summer adapted. 3. Therefore, all are winter adapted or are summer adapted. The clause x is winter adapted is symbolized by Gx. But this is an incomplete symbolization, since really what is said is: x is adapted to winter (×y)(y is winter and x is adapted to y), (×y)(Wy · Axy). So the crucial, dyadic, relational term Axy is present still, though hidden. Likewise for summer adapted. From here on the clauses concerned with adaptation will be shown by Ai x, where the subscript i changes from one situation to another. Thus it will be clear that there is a distinction between physicalistic description in clauses without subscripts and interpretive description in adaptational clauses with subscripts. Ex. (13) or Ex. (14) can be expanded very much by plans A, B, and C. Application of plan A to diapause of the coastal marine plankton copepod Labidocera aesti6a is as follows. Marcus (1979, 1980, 1982) found that parent copepods exposed to long-day experimental regimes produced eggs that hatched
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right away at summer temperatures and that parents under short-day experimental regimes hatched only after being kept at 5°C for 40 days or more then warmed to summer temperatures. So the short-day diapause eggs of fall overwintered in the mud bottom and hatched during the warmth of June and the quickly hatching, subitaneous, eggs of summer produced the continuously present plankton copepods from June to fall in coastal water. In terms of one egg x this supports the iterated formula of Ex. (13) of plan A: (1) (2) (3) (4) (5)
Dx v Sx (Dx³Awx) · (Sx³ Asx) Awx v Asx (Awx ³ Atx) · (Asx³ A1x) Atx v Alx
1. 2. 3. 4.
Egg x is diapause or egg x is subitaneous. If egg x is diapause then it is winter adapted, and if egg x is subitaneous then it is summer adapted. Therefore, egg x is winter adapted or egg x is summer adapted. If egg x is winter adapted then it is from parent copepods adapted to short-day photoperiod, and if egg x is summer adapted then it is from parent copepods adapted to long-day photoperiod. 5. Therefore, egg x is from parents adapted to short-day photoperiod or egg x is from parents adapted to long-day photoperiod. Ex. (14) can be expanded by plan B, by additions at step (2). Experimentally, cells from four diatom species grow slowly in the dark when organic compounds such as sodium glutamate, glucose, lactate, and others are provided. In the light they grow considerably faster, experimentally (Lewin and Hellebust, 1975, 1976; Hellebust and Lewin, 1977). Cells of ciliate protozoa grow better, experimentally, in the light if they contain chloroplasts of ingested algae than they grow in the dark with the same algal food (Stoecker et al., 1988 – 1989); so they are myxotrophic or heterotrophic. With Adx as x is dark adapted, Alx as x is light adapted, Amx as x is myxotrophically adapted, and Ahx as x is heterotrophically adapted the common bond of a varied nutritional adaptation emerges from the following: (1) (2) (3)
(x)(Dx v Cx) (x){[Dx ³ (Adx v Alx)] · [Cx³ (Amx v Ahx)]} (x)(Adx v Alx v Amx v Ahx)
Step (1) is: all cells are from one of the diatom growers or are ciliate growers. Step (3) is: therefore, all cells are dark or light or myxotrophically or heterotrophically adapted. Ex. (14) can be expanded by plan C, by having more than two modus ponens structures in conjunction. Two cases will be presented of this expansion — one case with a small coverage and a second with a large coverage of the world’s biota. The first is concerned with multiple evolutionary pathways leading to the adaptation of fish being warm under varied ecological conditions. In tuna fish dark muscle with profusely branching blood vessels in the body wall forms a heat exchanger keeping the central body 10°C higher than the ends of the fish (at water temperature) (Carey, 1973). In Carey’s words, ‘a major adaptive advantage of an elevated body temperature is greatly enhanced muscle power and this must surely be connected to the adaptive advantage of increased swimming speed’. Thus the tuna fish can swim faster and supposedly catch fast prey better in cold water than it could if it were at water temperature. Lamnid sharks with increasing amounts of red, axial, heat exchanger muscle have higher mid-body temperatures and range farther north in the ocean (Carey et al., 1985). The brain of the swordfish is heated 10–14°C above water temperatures by brown thermogenic tissue next to the brain (Carey, 1982) and this is an advantage in deep diving, up to 600 m, where the water can be 19°C less than at the surface (Carey and
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Robinson, 1981). So there are three evolutionary pathways to heat adaptation. Where (x) is: for all evolutionary pathways x, or briefly pathways, we have: (1) (2) (3)
(x)(Dx v Rx v Tx) (x)[(Dx ³ Adx) · (Rx³Arx) · (Tx³ Atx)] (x)(Adx v Arx v Atx)
1. Pathways are by body wall dark muscle, Dx, or by axial red muscle, Rx, or by brown thermogenic tissue next to the brain, Tx. 2. Pathways, if by body wall dark muscle, led to the adaptive advantage of fast swimming, and if by axial red muscle, led to the adaptive advantage of allowing northern distribution, and if by brain thermogenic tissue, led to the adaptive advantage of deep diving. 3. Therefore, pathways led to the adaptive advantage of fast swimming or northern distribution or deep diving. Multiple pathways of evolution are exemplified by the equations for CO2 fixation (Jannasch and Mottl, 1985): hv
2CO2 + H2S+ 2H2O 2[CH2O] +H2SO4 (nonoxygenic photoautolithotrophy, purple and green bacteria) hv
(A1)
CO2 + H2O [CH2O] +O2 (oxygenic photoautolithotrophy, green plants)
(A2)
CO2 + H2S +O2 +H2O [CH2O] +H2SO4 (aerobic chemoautolithotrophy, bacteria)
(A3)
2CO2 + 6H2 [CH2O] +CH4 +3H2O (anaerobic chemoautolithotrophy, bacteria).
(A4)
The first occurs terrestrially; the second occurs terrestrially and in seaweeds and the phytoplankton of the sea surface; the third and fourth occur in the hydrothermal vents of the sea floor. More simply, multiple pathways of CO2 fixation are by bacterial photosynthesis, by plant photosynthesis, and two by bacterial chemosynthesis. More simply still, there are two pathways on the planetary surface and two pathways on the sea floor. With (x) for all pathways x or just pathways and Fx for pathway x is by nonoxygenic photoautolithotrophy, etc. and Apx for x is adapted to the lighted planetary surface, etc. we have: (1) (2) (3)
(x)(Fx v Gx v Hx v Ix) (x)[(Fx ³ Apx) · (Gx ³Apx) · (Hx³ Asx)³ (Ix³ Asx) (x)(Apx v Asx)
1. Pathways are by Eq. (A1) or by Eq. (A2) or by Eq. (A3) or by Eq. (A4). 2. Pathways if by Eq. (A1) are adapted to the lighted planetary surface and if by Eq. (A2) are adapted to the lighted planetary surface and if by Eq. (A3) are adapted to the dark sea floor and if by Eq. (A4) are adapted to the dark sea floor. 3. Therefore, pathways are adapted to the lighted planetary surface or are adapted to the dark sea floor. It is clear that this is large scale adaptation. It is crucial, because adaptation is to be looked upon as deeply permeating biological material and ecological structure.
8. Discussion The theory of adaptation presented is a static analysis approach. The expressions developed are for describing situations just as they are. However, some of the expressions are ideal for describing in a
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dynamic way evolutionary change. Thus, expression Ex. (10) describes the partial evolutionary change to warm blooded from cold blooded terrestrial temperate vertebrates in: all, if active year-round are adapted to year-round temperature if and only if all, if not adapted to year-round temperature, are not active year-round—the evolutionary change is to adapted from unadapted. Thus, too, expression Ex. (14) describes the termination of evolutionary change in that temperate insects are diapause, winter adapted or are nondiapause, summer adapted. The guiding feature in the theory of adaptation presented is the selection of the basic element Axy. Then elaboration follows a narrow path at first, in which validity dictates Axy in (Axy³ Ey) ( Ey³ Axy). The affirmation of adaptation and its denial go hand in hand. They portray perfectly various parts of the biological world, whether the parts are small as the vertical niches of Anolis species or are large as the North American forest. The elaboration then buries the dyadic relational Axy in a simpler structure, the monadic Aix, and diversifies this in the affirmation of adaptation. The purpose then is just to put together two or more modus ponens structures by conjunction and then to change parts of these to disjunctions. Thus diapause and nondiapause are closely related but exclude each other and are ‘or’ related, are diapause or nondiapause. These intimate winter adaptation or summer adaptation. But these lead, too, to multiple pathways of evolution which are or related. Thus adaptation is seen ecologically primarily, with evolution as a process of value in holding together configurations of nature that have both similarity and the bond of being adapted. Thus the theory presented has two divergent formulations. But these come from the same starting formula. The sequence of steps from the starting formula may be viewed as follows. Where p is for Axy and q for Ey the first sequence is: (1) (2) (3) (4) (5) (6)
p³p (p ³ q) ³ (p ³q) [(p ³ q) ³ (q ³ p)] · [(q ³ p)³(p³q)] (q ³p) (p ³ q) (p ³ q) (q ³p) [(p ³ q) · q]³ p
Where p is for Dx, q for Awx, r for Sx, and s for Asx (as in the Labidocera example) the second sequence is: (1) (2) (3) (4) (5) (6)
p ³p (p ³q) ³ (p ³q) [(p ³ q) · p]³q [(p ³q) · p] ³ q · [(r³ s) · r] ³s [p · (p ³ q)] ³ q · [r · (r³s)]³s [(p v r) · (p ³q) · (r³s)]³(q v s)
Steps (1) and (2) are the same in each sequence; at step (3) there is diverging structure. All steps have only valid structure. Each step follows validly from the last3. Each sequence is valid as a whole. Validity is not mere overlay on the description of nature. Validity permeates nature and must be sought in nature. We see validly, what we see is valid. But this separation is inconsequential. Nature is valid.
3
Except step (6) in the first sequence, which comes from step (3) by simplification and exportation.
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The model for incompletely symbolized expression blocks the denial of adaptation, in the sense that validity is achieved without denial. In another study (Hulburt, 1996) validity is achieved without denial by use of the formula [(p ³q) · (q³ p)] (q p). Elaboration of this formula occurs in p. Thus if x and y help each other (symbionts), or x dominates y (plant and habitat), or x suppresses y (phytoplankton and nutrient), or x preys on y (predator and prey), or x enslaves y (ciliate and ingested algal chloroplast)—all this in place of p — then x and y are adapted to each other—this last for q; thus p ³q. Though denial can be entertained in q — if plant x is adapted to habitat y then y is adapted to x (x and y are adapted to each other), but suppose habitat y is not adapted to x then how ever could plant x be adapted to y — such lurking denial is overwhelmed by the large-scale affirmation of adaptation available from the elaborated formulation. The theory separates physicalistic description from interpretive adaptational description haphazardly in the fully symbolized expressions. But both physicalistic and interpretive description are clearly distinguished in the model for incompletely symbolized expressions. Here in the modus ponens structure the factual statement is put first, diapause, and then the interpretive ascent: if diapause then winter adaptation is put second. First diatom grower, then the interpretive ascent in: if diatom grower then dark adapted or light adapted. First body wall dark muscle, then the interpretive ascent in: if body wall dark muscle then adaptive advantage of fast swimming. First pathway of evolution by nonoxygenic photoautolithotrophy, then the interpretive ascent in: if pathway of evolution by nonoxygenic photoautolithotrophy then adaptation to the lighted planetary surface. Nature is valid, but this validity spans from the physicalistic to the interpretive, to adaptation.
9. Glossary Connectives: · And v Or ³ Goes between a clause beginning with if and a clause beginning with then; thus the if clause implies the then clause
Equivalent to, if and only if Not Therefore p Premiss statement Ap Assumed premiss — when if is put before a statement to be followed by a then statement later, indicated by a broken arrow and C.P. x, y Variables—they stand for subject and object in a clause or phrase (x) For every x, all x (the universal quantifier) (×x) There is an x such that (the existential quantifier) Validity is truth by truth value analysis, wherein a true ‘if’ statement implying a true ‘then’ statement makes a true whole, a false implying a true makes a true whole and a false implying a false makes a true whole. So Axy ³Axy is true³true and false³ false, both true as a whole. Only true implying false is false as a whole. So in Ex. (1) Axy ³ Ey could be false, but false³ false would happen for the whole of Ex. (1), so Ex. (1) would be true as a whole for this case—and for the other cases of true and false for Axy and Ey, so Ex. (1) is only true, is valid.
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With the added rule that when equivalent expressions are alike in truth value, are true true or false false, both cases are true as wholes, and noting that denial ( ) of true is false and denial of false is true and disregarding the quantifier (y), Ex. (2), Ex. (3), Ex. (4), Ex. (5), and points (1)–(5) are found valid. Ex. (10) is seen to be valid because the equivalents in parentheses are in conjunction with Gy which if true makes both sides true or false as the equivalents are true or false and which if false makes both sides false—so that both sides are alike in truth value and are thus equivalent. Beginning with the North American forest a different method of showing validity is used, a method that depends on the following procedures.
10. Rules of inference Rules of substitution are such that p, q, r may have substitutions of Axy, Ey, (Txy³ My), (y)(Hy ³ Axy), etc. in place of one of p, q, r: 1
Modus ponens (M.P.) p ³ q p q
2
Modus tollens (M.T.) p³q q p
3
Conjunction (Conj.) p q p ·q
4
Simplification (Simp.) p ·q p
5
Addition (Add.) p pvq
Rules of replacement: any of the following equivalent expressions can replace each other wherever they occur: 6 7 8 9 10 11 12 13
Transposition (Trans.) Commutation (Comm.) De Morgan’s theorem (De. M.) Implication (Imp.) Exportation (Exp.) Double negation (D.N.) Equivalence (Equiv.) Quantifier negation (Q.N.)
(p ³ q) (q ³ p) (p · q) (q · p) (p · q) (p v q) (p³ q) (p · q), (p v q) (p ³ q) [p³ (q³ r)] [(p · q)³ r] p p [(p ³q) · (q³ p)] (q p) (×y) (y) (y) (×y)
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10.1. Uni6ersal generalization (U.G) Fx (x)Fx
10.2. Uni6ersal instantiation (U.I.) (x)Fx Fx
10.3. Existential generalization (E.G.) Fx (×x)Fx
10.4. Existential instantiation (E.I.) (×x)Fx Fx Meaning of some terms: Conjunction Disjunction Modus ponens Modus tollens Monadic Dyadic Diapause Nondiapause Physicalistic Interpretive
The joining of two clauses, or statements, by ‘and’ The joining of two clauses, or statements, by ‘or’ An argument of this form: if this then that, this; therefore that An argument of this form: if this then that, not that; therefore not this A clause, or statement, with only predicate and subject, such as Ey A clause, or statement, with predicate, subject, and object, such as Axy An overwintering, non-developing stage of an insect, either egg or larva, brought about by short day-length A spring-summer mature stage of an insect brought about by long day-length (plus warmth) Description that is close to the facts Description that adds a connotative, personalistic element to the facts
References Beck, S.D., 1980. Insect Photoperiodism. Academic Press, New York, p. 378. Brokaw, N.V.L., 1982. Treefalls: frequency, timing, consequences. In: E.G. Leigh, Jr., A.S. Rand, D.M. Winsor (Eds.), The Ecology of a Tropical Forest: Seasonal Rhythms and Long-term Changes. Smithsonian Institution Press, Washington, D.C., pp. 101 – 8. Brokaw, N.V.L., 1985. Gap-phase regeneration in a tropical forest. Ecology 66, 682 – 687. Carey, F.G., 1973. Fishes with warm bodies. Sci. Am. 228, 36 – 44.
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