498 Paul Thompson
least squares method^23 sufficiently corresponds to the relevant elements of the
theory. One of the most straightforward ways of assessing the correspondence is
to represent the data in an ‘observation space’ analogous to the ‘mathematical
space’ in which the theory is represented. In these spaces, the states of both
the empirical system and the theories are points in the respective spaces. In the
easiest case, namely that of a deterministic linear system, the comparison of the
observation space and the phase space of the theory is uncomplicated and based
on an identity relation — the observation space and the phase space will have
the same dimensionality and the points representing states of the system will be
identically located within the space.
There are two prominent versions of the semantic conception of theories: a
set-theoretical version and a state space version.
An example of each can be given for Mendelian genetics. The set-theoretical
version of the formulation of Mendelian genetics is:
T: A systemβ=〈P, A, f, g〉is a Mendelian breeding system if and only
if the following axioms are satisfied:
Axiom 1: The setsPandAare finite and non empty.
Axiom 2: For anya∈P andl, m∈A, f(a, I)&f(a, m)iffI=m.
Axiom 3: For anya, b,∈PandI∈A, g(a, I)&g(b, I)iffa=b.
Axiom 4: For anya, b∈Pandl∈Lsuch thatf(a, l)andf(b, l),g(a, l)
is independent ofg(b, l).
Axiom 5: For any a, b ∈ P and l, m ∈ L such that f(a, l)and
f(b, m),g(a, l) is independent ofg(b, m).WherePandAare sets andfandgare functions. Pis the set of all alleles
in the populations,Ais the set of all loci in the population. Ifa∈PandI∈A,
thenf(a, l) is an assignment, in a diploid phase of a cell, ofatoI(i.e.,fis a
function that assignsaas an alternative allele at locusl). Ifa∈P,andI∈A,
theng(a, I) is the gamete formed, by meiosis, withabeing atIin the gamete (the
haploid phase of the cell). Although more sophistication could be introduced into
this example (to take account, for example, of meiotic drive, selection, linkage,
crossing over, etc.), the example as it stands illustrates adequately the nature of
a set-theoretical approach to the formalisation of population genetic theory in its
simple Mendelian system form.
Characterizing Mendelian genetics using a state space approach is more com-
plicated. A theory on this view consists of the specification of a collection of
1990].
(^23) The Gauss-Markov conditions provide a high level of confidence that the estimate of the
parameters is reliable. Hence, it allows a determination of when least squares is a good method.
The Gauss-Markov conditions are:
E(εi)=0,for alli
var(εi)=E(εi−E(εi))^2 =E(ε^2 i)=σ^2
E(εiεj) = 0 for alli=j
whereεisandεjs are errors,Eis expectation andσis variance.