492 Paul Thompson
set of formulae. The force of the requirement that the interpretation be “true”
is to demand that the model “satisfy” “all” the formula in the set. A formula is
satisfied if the interpretation given by the model yields a specification that is well
founded, that is, understood and widely believed by those in a position to judge.
In the case of a scientific theory, the resulting specification must be empirically
acceptable. That is, it is consistent with the phenomena within its scope.
Consider a simple example of this concept of satisfaction. The formula:
(x)(y)(((x=J)&(y=B))→((Txy→Lyx)&(Tyx→Lxy)))is satisfied by the statement:
If John speaks to Bill then Bill listens carefully to John and if Bill speaks to
John then John listens carefully to Bill.
This is a simplistic example since it provides anordinary-languageinterpretation
of asingle formula. Model interpretations of scientific theories are most often
formulated in a mathematical language. In addition, since a theoryTwill almost
always be an interconnected set of formulae, the model ofTmust simultaneously
satisfyallthe formulae of the theory.
Of crucial importance to the use of models for interpreting formal systems is
the fact that more than one interpretation can be given for a formula or set of
formulae. Game theory, for example, has been successfully employed in ecology,
economics, and international relations. The abstract structure of game theory can
be expressed as an abstract formal system. In each of the discipline-specific appli-
cations, different models of the abstract system are being provided. In each case,
the meanings of the terms as well as the empirical truth of the interpreted formulas
can be different. And, in each case, the model satisfies (provides a semantics for)
the abstract formal system if the resulting expressions are empirically acceptable.
This method of providing an interpretation for an abstract formal system turns
out to be extremely powerful in the sciences and in mathematics. One of the sec-
ondary effects of its development was to call into question the utility of abstract
formal structures in the context of scientific theories. Several features of scientific
theories suggested that the syntactic formalisation was unachievable and, more
importantly, unnecessary. A few of the features are: that anything approaching a
full formal account of any actual scientific theory has proved elusive, correspon-
dence rules, as indicated, as a method of interpretation have significant problems
and almost all actual scientific theories are model-theoretic structures. As a result,
some philosophers of science began to question the added value of a formalisation
in first-order predicate logic given that the appropriate models that interpret the
formal system could be specified directly. It was considered unimportant whether
one could specify the formal system that a given model satisfied, why would one
want to? As long as the model can be adequately specified without reference to
the formal system and as long as the model can be used to achieve everything
for which one might appeal to scientific theory, there is no added value. The se-
mantic account of theories emerged from this view that scientific theories could be
construed directly as models.