Towards The Logical Structure of Hamilton and Darwin's Arguments.
- From: "John Edser" <edser@xxxxxxxxxx>
- Date: Thu, 1 Dec 2005 00:21:19 -0500 (EST)
http://plato.stanford.edu/entries/square/
I argue that a consistent failure exists wherein Hamilton et al utterly fail
to preserve or correctly convert fitness at one level of selection into
another. Therefore I suggest that a revision of basic logic may be in order
to help settle the question. Traditional logic still forms the basis of all
meaning. Below is a representation of "The Traditional Square of Opposition"
put into more familiar evolutionary theory terms where the subject (S) is
"genes" and the predicate (P) "selected at the organism level". The four
corners of the square are traditionally marked: A,E,I and O. It should be
noted that all valid statements require a subject (S) and a predicate (P)
which must form a NON reversible logical linkages within any non
tautological propositions, e.g. "the cat sat on the mat" does not have the
same meaning as "the mat sat on the cat" because their subject to predicate
relationship is non reversible. OTOH all felines are cats only because all
cats are felines, remains an entirely reversible tautological proposition.
Within the sciences a tautological statement is only valid if it is just a
definition i.e. it is OK to define all felines as cats if and only if
felines can be _empirically_ separated from non felines via a theory of
nature.
2
(A)All genes selected at the organism level ------------- No genes
selected at the organism level(E)
(All S are P) | |
(No S is P)
|
|
1| |3
< -- (A mirror can be placed across sides 1 and 3)
|
|
|
|
(I)Some genes are selected at the organism level------------- Some genes
are not selected at the organism level (O)
(Some S are P) 4
(Some S are not P)
^
|
(A mirror can be placed
across sides 4 and 2)
Arrows above indicates two places where a mirror can be placed to form a
reflection within the traditional square of opposition. The mirror cannot be
placed on any of the diagonals because the corners represent contradictions
which cannot reflect anything about each other because they have nothing in
common to reflect.
Note that the A,E,I,O, propositions could be validly represented as Venn
diagrams (sets): http://www.cs.uni.edu/~campbell/stat/venn.html
The subject S is "genes" and the predicate P is "selected at the organism
level".
The two single level of selection propositions:
(a) The A proposition: All S are P is the union of set S and set P
describing just the one, same independent level of selection.
(b) The E proposition: No S is P is the non union (entire
separation) of set S and set P describing two independent levels that have
no relationship to each other.
The two multi levels of selection propositions:
(c) The I proposition: Some S are P is the intersection of set S
with set P where the intersection is not empty describing two independent
levels that have a social interaction.
(d) The O proposition: Some S is not P is the intersection of set S
with set P where the intersection remains empty describing two independent
levels which could have but do not have any social interaction.
CONTRADICTIONS:
The only two CONTRADICTIONS that exist represent absolute opposing
propositions which are represented DIAGONALLY, i.e. A contradicts O and I
contradicts E.
1) A contradicts O: All genes are selected at the organism level (all S are
P) contests some genes are not selected at the organism level (some S are
not P).
2) I contradicts E: No gene is selected at the organism level (no S are P)
contests some genes are selected at the organism level (some S are P).
The contradiction in both cases is that if S and P are entirely merged then
they cannot possibly be intersecting because these events are self
exclusive, i.e. sets that relate are either intersecting or merged.
Therefore two propositions can only be proven to be contradictory if both
cannot be true and both cannot be false, i.e. at least one proposition must
be true. Either the sets relating have become merged or they have become
intersected where one must remain true.
NON Contradictions
The other four propositions: two sub contrary and two subaltern represent
relatively opposed propositions which can validly form reflections of each
other.
1) All genes are selected at the organism level and no genes are selected at
the organism level are contraries (A reflects E).
2) Some genes are selected at the organism level and some genes are not
selected at the organism level are subcontraries (I reflects O).
3) Some genes selected at the organism level is a subaltern of all genes
selected at the organism level (I reflects A)
4) Some genes are not selected at the organism level is a subaltern of no
genes selected at the organism level (O reflects I)
Note that:
Two propositions are contraries if they cannot both be true but can both be
false. Compare this to a contradiction where at least one proposition must
remain true.
Two propositions are sub contraries if they cannot both be false but can
both be true. A proposition is a subaltern of another if it must be true if
its superaltern is true and the superaltern must be false if the subaltern
is false.
The Darwinian Argument is empirically just mono centric: All S are P which
is the A proposition. As far as genes and organisms are concerned all genes
are dependently selected at just the one organism level of selection within
Darwinism. The only valid way that gene selection can be modeled is via the
assumption of gene fitness as a dependent nested set within organism fitness
and not as an independent gene fitness intersecting with an independent
organism fitness set. Note that any one proposition only has 3 others to
refer to: two reflections and one contradiction. In Popperian Terms with
reference to just the mono-centric A Darwinian proposition:
i) A non verifies E and E non verifies A
ii) A refutes O and O refutes E
iv) A verifies I and I verifies A
The Hamiltonian Argument is heuristically multi centric: Some S are not P
because some genes are supposed to be selected at the gene level forcing
altruism at the organism level which is the contradictory O proposition to
the Darwinian A. Therefore Hamilton *OR* Darwin can be true but not both
because they stand in contradiction to each bother. Note that within the
Hamiltonian argument O is verified by E but E non verifies A which is the
Darwinian proposition for gene selection. Likewise, I non verifies O which
is the Hamiltonian proposition but verifies A which is the Darwinian
proposition.
For those who find this a bit hard going (if only epistemology was as easy
as population genetics) then here is a simple visualization. Imagine the
square of opposition to be a boxing match where the two boxers come out
fighting from just the A and O corners. Pushing the other onto any of the
ropes is not definitive. The winner must KO the opponent (winning by points
is not allowed because it is just an argument from authority) by knocking
them into the opponents corner. I will leave the argument here for reader's
to contemplate.
Regards,
John Edser
Independent Researcher
edser@xxxxxxxxxx
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