486 Paul Thompson
Consequently, a theory can be formalised in first-order predicate logic with
identity, in set theory, in matrix algebra and, indeed, any branch of mathematics
satisfying (1) and (2).
A theory (for example Euclidean geometry or arithmetic) also requires:
- a complete specification of a set of primitive formulae of the theory (those
which cannot be inferred from any others and from which all other formulae
of the theory will be inferred.^4 )
Ascientifictheory also has to include:- an interpretation (a semantics) for the formal calculus (the syntax).
How the semantics is provided differs in different conceptions of the formalisa-
tion of scientific theories.
1 THE PATH TO A GALILEAN CONCEPTION OF SCIENTIFIC
THEORIESIn the early part of the 20th-century, formalisation in science was virtually synony-
mous with the representation of a theory in first-order predicate logic with identity.
This was a result of the dominance through to the 1950’s of logical positivism and
its descendent logical empiricism in philosophy of science. As noted below, this
view has various names of which “the syntactic conception” is the most descriptive.
The logical empiricist conception provided the semantics through correspondence
rules. Through the 1950’s and into the1960’s, serious logical and conceptual dif-
ficulties with the logical empiricist conception were forcefully articulated. In the
forefront of this wave of critical material was Willard van Orman Quine’s, “Two
Dogmas of Empiricism” [Quine, 1951]. One of the most influential criticisms was
Thomas Kuhn’s,The Structure of Scientific Revolutions[Kuhn, 1962]. A common
thread in much of the criticism was the failure of logical empiricism to capture the
holist nature of scientific theories. Much of the blame for this rested with the
employment of correspondence rules to provide a semantics for the theory.
In the latter part of the 1960’s, an alternative conception was developed in
which theories were represented in set theory or in a state space. This conception
was widely called “the semantic conception.” It builds on earlier work on model-
theoretic semantics of formal systems (e.g., Alfred Tarski and Evert Beth). This
conception treats scientific theories holistically, avoiding, thereby, one of the cen-
tral defects of the logical empiricist conception of theories. It is important to be
clear that a syntactic conception of theories is not wedded to providing semantics
by means of correspondence rules (about which I shall have more to say later). In-
deed, following Taski, one could provide a model-theoretic semantics and, thereby,
capture the holistic character of theories. Those who espouse the semantic concep-
tion accept that this provides a more robust syntactic conception of theories, but
(^4) Kurt G ̈odel established that formal systems cannot be both complete and consistent.