Formalisations of Evolutionary Biology 491important.^12 As an example of a syntactic formalisation of evolutionary theory
it fails. Indeed, in addition to axiomatising only one part of a robust theory of
evolution, her axiomatisation draws heavily on set theory, not first-order predicate
logic. In the next section, it will become clear that set-theoretical axiomatisations
fall within the semantic conception of theories and more broadly into the Galilean
conception of theories.^13 In addition, in Part II, it will become clear that Williams’
axiomatisation as a set-theoretical axiomatisation is an important contribution to
the formalisation of a robust evolutionary theory.
By the late 1970’s, logical empiricism had declined in influence. The early
criticisms of Quine and later of Kuhn had taken a significant toll on the credibil-
ity of the programme. But the difficulties continued to emerge. Correspondence
rules, for example, were totally inadequate to the task of providing a semantics
for scientific theories. The most insightful, clear and devastating critique of corre-
spondence rules was provided by Kenneth Schaffner [1969]. He highlighted the fact
that relating a theory to phenomena requires the employment of laws from other
independent theories, a fact that correspondence rules could not accommodate.
Although employing correspondence rules was the logical empiricist’s dominant
method of providing a semantics for well-formed formulae and, more broadly,
syntactic systems, there is a large body of literature exploring a variety of ways
in which the semantics can be provided. In the context of scientific theories, all
are aimed at connecting the terms in the symbolic structure to phenomena (to the
empirical world).
An alternative proposed by Alfred Tarski [1944; 1953; 1956] employs models.
For Tarski, a “model” designates a structure through which the abstract symbolic
formulae (wff’s) of the theory can be given meaning, “A possible realization in
which all valid sentences of a theoryTare satisfied^14 is called amodelofT.” [1953,
p. 11]. This is a robust and important method of providing a meaning structure
for a theory formalised in first-order predicate logic and is a stepping-stone to the
semantic account of theories. The model is a “true” interpretation of axiomatic-
deductive systems. It is the fact that the model provides atrue interpretation
of the set of symbolic formulae that qualifies it as a meaning-structure for that
(^12) Rosenberg and Williams published two joint papers [1985b; 1986] which continue the per-
spective Rosenberg took in his 1985 book [1985a]. One reasonably might assume from these joint
ventures that Williams endorses Rosenberg’s interpretation of her work. However, Williams’ con-
tribution can, and I claim should, be severed from this interpretation. Hers is not a first-order
predicate logic axiomatisation interpreted via correspondence rules. She has provided an archety-
pal set-theoretic axiomatisation and it is better interpreted as providing a central component of
a semantic or Galilean conception of the formalisation of evolutionary theory.
(^13) The subtitle of Williams’ paper, “A Mathematical Model,” correctly suggests that she is
constructing a mathematical model, not a first-order predicate logic axiomatisation. As a math-
ematical model it falls within a Galilean conception of theories.
(^14) “Satisfied” means “rendered true.” Hence, a set of expressions in a formal syntax is satisfied
if all the expressions are true and, hence, consistent. Hence, for Tarski a model for the syntactic
formalisation of a theoryTis a realisation (an interpretation) ofTin which all the expressions
ofTare true. IfTis an axiomatic-deductive structure, an interpretation of the axioms ofTthat
renders them all true constitutes a model ofT. All other expressions are a deductive consequence
of the axioms and, hence, will be true in that interpretation.