494 Paul Thompson
tion of informal theories, of which scientific theories are instances. Suppes wrote a
number of papers during the 1960’s in which he argued that scientific theories were
better represented as a set-theoretical predicate (see, for example [Suppes, 1962]).
In 1967, he wrote a brief non-technical account of his view [Suppes, 1967]. In this
paper, he quite clearly sets out his reasons for rejecting what he calls the “stan-
dard sketch of scientific theories” and for adopting a semantic account. His central
conclusion is that scientific theories are not appropriately or usefully formalised as
axiomatisations in first-order predicate logic but rather in set theory.
The main thrust of his argument was that correspondence rules (he calls them
co-ordinating definitions) “do not in the sense of modern logic provide an adequate
semantics for the formal calculus” [Suppes 1967, p.57]. One should instead talk
about models of the theory. These models are non-linguistic entities that are
Axiom 1: The setsPandA−are finite and non-empty.
Axiom 2: For anya∈Pandl, 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 anya, b∈Pandl, m∈Lsuch thatf(a, l)andf(b, m),g(a, l) is indepen-
dent ofg(b, m).
WherePandA−are 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 atI
in 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 formalization of population genetic theory in its simple Mendelian system form.
Characterizing Mendelian genetics using a state space approach is more complicated. A theory
on this view consists of the specification of a collection of mathematical entities (numbers, vectors,
functions) used to represent states of the system within a state space (a topological structure),
the behaviour of which is specified by three functions. These functions are commonly called laws
but are not the same as laws in the received view. In the received view, laws are statements
describing the behaviour of entities in the world. In the semantic view, laws are descriptions of
the behaviour of mathematical systems of mathematical entities. The three laws are: laws of
coexistence (which specify the physically possible set of states of the system); laws of succession
(which specify the possible histories of the system); and laws of interaction (which specify the
behaviour of the system under conditions of inputs from interaction with other systems). A
theory also requires the specification of a set of measurable magnitudes (variables) (represented
by a function defined on the state space). Statements which formulate propositions to the effect
that a particular magnitude has a particular value at a particular time are elementary statements.
A satisfaction function determines the sets of states which satisfy the assignment of a value to a
physical magnitude.
For population genetic theory, the state space will be a Cartesiann-space where ‘n’ is a function
of the number of possible pairs of alleles in the population. A law of coexistence to the effect that
only alleles at the same locus can form pairs will select the class of physically possible pairs of
alleles. States of the system (genotype frequencies of populations) are n-tuples of real numbers
from zero to one and are represented in the state space as points. These are the measurable
magnitudes. An example of a satisfaction function for the elementary statement “genotypeAa
occurs with a frequency of 0.5” would specify the set of states in the state space that satisfy the
statement. In this case, the set of states would be a Cartesian (n−I)-space, which is a subset of
the state space. For population genetic theory, a central law of succession is the Hardy-Weinberg
law.