Formalisations of Evolutionary Biology 489According to this conception of theories, at the core of a theory are a few axioms
(fundamental laws of that domain of science). The axioms can be expressed in
ordinary language but their formal and definitive expression is in the language of
symbolic logic. The axioms are laws of the highest generality within the theory.
Ideally, they constitute a consistent set, no one of which can be derived from any
subset of the others. From these axioms in their symbolic-logic formulation all the
rest of the formula (laws) of the domain of science, in principle, can be deduced.^10
These deductions employ the inferential machinery of first-order predicate logic.
Invariably, in the case of scientific theories, numerous subsidiary assumptions must
be added in any deduction in order to derive a law from the axioms; it is held
that perfect knowledge, although unattainable, would make these assumptions
unnecessary.
Just as the terms used in ordinary language need to be given meaning (to be
defined) in order for a grammatically correct sentence to be understood, so too
do the terms in a well-formed formula in symbolic logic need to be given meaning
(be defined). In the logical empiricist formulation, this deductively related set
of statements is given empirical meaning by definitions — called correspondence
rules — whichultimatelylink theoretical terms (e.g., population, fertile, disease,
motility, polymorphonuclear, chemotaxis, gene, and so forth) to observations (e.g.,
a theoretical term like “fertile” is partially defined by reference to the outcomes
of numerous sexual events of a specified kind under specified conditions). Some
theoretical terms are defined by reference to one or more other theoretical terms.
Ultimately, any chain of such definitions must end in theoretical terms that are
defined by reference to observations. In this way, the theory as a whole is given
empirical meaning. Because of this complex interconnection of theoretical terms,
the meaning of any one term is seldom independent of the meaning of many if
not all of the other terms of the theory. Hence, theories have a global meaning
structure: changes to the meaning of one term will have consequences for the
meaning of many and usually all the other terms of the theory.
The influence of this conception on the discussions of theory structure in biology
was profound. The earliest impact was on theoretically engaged biologists. For
example, very early in its development, J.H. Woodger [1937; 1939] argued for an
application of the axiomatic method in biology and provided his own axiomatic
account of selected biological theories. C.H. Waddington [1968–72] also promoted
(^10) Two important features of an entirely satisfactory formal system (and, hence, a scientific
theory) are completeness and consistency. A formal system is complete if, within its domain,
all the possible formulaeortheir negations are derivable (provable). A theory is consistent if
there is no formulaandits negation that can both be derived. Unfortunately, Kurt G ̈odel, in
a now famous proof, established that no formal system can be both complete (every formula
or its negation being provable) and consistent (no formula and its negation being provable) (a
translation of the original 1931 paper can be found in [Van Heijenoort, 1966], and an excellent
exposition of the proof is in [Nagel and Newman, 1958]). G ̈odel’s proof was directed at the
problem of the consistency of arithmetic but it applies to all formal systems. This feature of
formal systems is not one that for most purposes outside mathematics matters profoundly. If
either completeness or consistency has to be sacrificed in order to employ a formal system, it is
usually advantageous to relax the requirement of completeness.