Epistemological Significance of Models in Science
5
Relations between the quantities of the set X at resolution level u will
be ri (X)u, the set of all relations being
(2)
The system S is the set of the relations between the external quantities:
(3)
This very simplified definition of a system may be brought into accord with
other definitions [8], all on the basis of set theory. Because the definitions
of systems (or models) are based on the set theory, it should be mentioned
that a system is distinguishable from the pure set by its structure. Moreover,
an important feature of a system, namely its "wholeness", may also be
described in terms of set theor\.
A definition similar to set theon" may in principle be specified on the
basis of the logic of predicates and classes [12]. This latter definition
becomes necessary if one is dealing with more than a first predicate logical
step, viz. with properties of properties, properties of relations, relations of
relations, etc. We rna \" distinguish between a predicate logical one-step
system S(I) and a predicate logical k-step system S(k); the first of which
may be formalized as follows:
\(1) = .1 U 1 Al U 1 A2 U , ... , UlAn
(4)
A: basic set of elements; k An: class of n-occupancy predicates of the
k-th step.
Models in Formal Respect
With the formalization of a model, the three main functions of models
with respect to sYstems, namely mapping, reducing and subjectivization,
have, of course, to be included. The first of them, mapping, may be described
as follows [31: Given are 2 sets; of these B with predicates QI' ... , Qk is
a model of a set "oj with predicates PI' ... , Pk if there exists a mapping
rp I A -+ B of /1 into R and a mapping rp I Pj -+ Qj of Pj into Qj such that
cp Pj (Xl' . . . , Xn) ~. Q i (q Xl' . . . , rp xn) for) = 1, ... , n and all Xl' . . . , Xn
E A x, ... , X A. In this definition cp becomes a homomorphism with
respect to the structure S which will be in both sets (A and B) defined as a
collection of predicates of one or several variables ranging over the set.
A homomorphism with respect to the structure :s is called a mapping cp if
for every predicate P on .0cJ x, ... , X A a predictae cp P defined on
B x, ... , >" B is associated.
Reduction may be considered in a model definition in which the uniunique mapping refers to predicates which comprise only subclasses of the
5
Relations between the quantities of the set X at resolution level u will
be ri (X)u, the set of all relations being
(2)
The system S is the set of the relations between the external quantities:
(3)
This very simplified definition of a system may be brought into accord with
other definitions [8], all on the basis of set theory. Because the definitions
of systems (or models) are based on the set theory, it should be mentioned
that a system is distinguishable from the pure set by its structure. Moreover,
an important feature of a system, namely its "wholeness", may also be
described in terms of set theor\.
A definition similar to set theon" may in principle be specified on the
basis of the logic of predicates and classes [12]. This latter definition
becomes necessary if one is dealing with more than a first predicate logical
step, viz. with properties of properties, properties of relations, relations of
relations, etc. We rna \" distinguish between a predicate logical one-step
system S(I) and a predicate logical k-step system S(k); the first of which
may be formalized as follows:
\(1) = .1 U 1 Al U 1 A2 U , ... , UlAn
(4)
A: basic set of elements; k An: class of n-occupancy predicates of the
k-th step.
Models in Formal Respect
With the formalization of a model, the three main functions of models
with respect to sYstems, namely mapping, reducing and subjectivization,
have, of course, to be included. The first of them, mapping, may be described
as follows [31: Given are 2 sets; of these B with predicates QI' ... , Qk is
a model of a set "oj with predicates PI' ... , Pk if there exists a mapping
rp I A -+ B of /1 into R and a mapping rp I Pj -+ Qj of Pj into Qj such that
cp Pj (Xl' . . . , Xn) ~. Q i (q Xl' . . . , rp xn) for) = 1, ... , n and all Xl' . . . , Xn
E A x, ... , X A. In this definition cp becomes a homomorphism with
respect to the structure S which will be in both sets (A and B) defined as a
collection of predicates of one or several variables ranging over the set.
A homomorphism with respect to the structure :s is called a mapping cp if
for every predicate P on .0cJ x, ... , X A a predictae cp P defined on
B x, ... , >" B is associated.
Reduction may be considered in a model definition in which the uniunique mapping refers to predicates which comprise only subclasses of the