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The Logical Study of Science
Author(s): Johan Van Benthem
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Source: Synthese, Vol. 51, No. 3 (Jun., 1982), pp. 431-472
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VAN
JOHAN
BENTHEM
STUDY
THE LOGICAL
OF SCIENCE*
taken for
often
of science,
between
The
relation
logic and philosophy
of
usefulness
of
the
criticisms
fashionable
current
in
fact
is
Although
problematic.
granted,
should be taken seriously
which
there are indeed difficulties
logic are usually mistaken,
in the two
mentalities"
"scientific
different
other
to do, amongst
things, with
having
of
logic is, or should be, a vital part of the theory
1). Nevertheless,
(section
disciplines
a
of
notion
to
the
devoted
is
this
of
bulk
the
this
key
make
To
paper
science.
clear,
of this
formal
various
in a logical perspective.
explications
First,
"scientific
theory"
3).
(section
logical theory is discussed
notion are reviewed
2), then their further
(section
or
in mathematics
like those of Klein
of grand
In the absence
programs
inspiring
one
can
here.
do
best
is
the
this
in metamathematics,
Hubert
ground-work
preparatory
of the
and merits
ends on a philosophical
note,
The paper
discussing
applicability
ABSTRACT.
formal
of
to the study
approach
science
(section
4).
CONTENTS
1 Introduction
of Science
1.1 Logic
and Philosophy
of Science
1.2 Logicians
and Philosophers
1.3 L?gica Magna
2 Formal
Notions
of 'Theory'
from Hubert
History
David Hilbert
2.1 A Short
2.1.1
2.1.2
Frank
2.1.3
Marian
Ramsey
Przetecki
2.1.4 Joseph
2.2 A Systematic
2.2.1 Syntax
Sneed
Logical
2.2.2
Structures
2.2.3
Semantics
2.2.4
3 Formal
to Sneed
Perspective
Pragmatics
Questions
3.1 Properties
3.2 Relations
concerning
of Theories
between
Theories
Theories
Aftermath
4 Philosophical
4.1 What
Is 'Application'?
of Formalism
4.2 In Praise
5 Technical
Appendix
Notes
References
Synthese 51 (1982) 431^172. 0039-7857/82/0513-0431 $04.20
Copyright
?
1982 by D. Reidel
Publishing
Co., Dordrecht,
Holland,
and Boston,
U.S.A.
432
JOHAN
1.
1.1 Logic
VAN
BENTHEM
INTRODUCTION
and Philosophy
of Science
an intimate
exists
connection
between
logic and the
seem
to
of
would
be
science
obvious,
given the fact that
philosophy
and Mill
the two subjects were still one until quite recently. Bolzano
even
in
in
this
their
the
twentieth
work; but,
century, main
exemplify
or
currents
like Hempel's
view
of
science,
'hypothetico-deductive'
an
'falsificationism'
if
often
presuppose
obvious,
Popper's
implicit,
link with
is true to an even greater extent of Carnap's
logic. This
That
there
of science,
in con
still echoing
'logical reconstruction'
that
of
1971.
like
Sneed
what
is
the
research
So,
temporary
problem?
contacts
These
rather superficial - going no
are, on the whole,
tradition is a
deeper than elementary
logic. The Carnap-Suppes-Sneed
as
there
advanced
of
favourable
but,
well,
applications
exception;
one
encounters
Padoa's
occasionally,
logic remain isolated examples:
Theorem
Method
(1953) or Craig's Theorem
(1953).
(1901), Beth's
or non
like
of
modern
Cohen's
technique
Highlights
logic,
forcing
program
of
at all.1 Moreover,
standard model
theory, have found no applications
the technical work which
is being done often seems to lack contact
with actual science.
in an otherwise
Are
these
problems
just maturation
promising
or
is
there
wrong
calling for a
marriage,
something
fundamentally
is given by more and more people,
rapid divorce? The latter diagnosis
In the philosophy
of
Kuhn
and, especially,
Feyerabend.
following
the
of
view'
is
or, at
science,
they say,
'logical point
inappropriate,
extent
the
of
The
scientific
best, inadequate
(to
reality
being useless).
or too 'com
one should be concerned
with
is either too 'dynamic',
plex' to be captured by formal tools.2
This type of criticism will not be discussed
here. Either
it amounts
to such authors
for
different
stating general
personal
preferences
of science - indeed quite honour
say history or sociology
approaches,
are adduced,
able subjects - or, when
these will
specific complaints
so
to
not
be
found
illustrate
much
the inadequacy
of logic
invariably
as that of the author's
to be treated
The problem
logical maturity.3
here is rather that logicians and philosophers
of good will have not
to
yet been able to get a successful
enterprise
going comparable
foundational
in mathematics.
research
THE
LOGICAL
STUDY
OF
SCIENCE
433
In the final analysis,
there may be deep
(and, no doubt, dark)
a non-trivial
reasons for this failure - prohibiting
logic of science. And
a struggle,
to
in fact, some logicians
defeat
without
accept
prefer
the cover of manoeuvres
like the following.
under
is the
'Logic
or
of
mathematical
deductive
the
of
science,
philosophy
philosophy
is for the philosophers.'
natural
science
Or again:
of
'Philosophy
the
of
Science
is, by definition,
component
(applied) logic'.
pragmatic
Both ways,
the logician can remain at home. As I see it, however,
it is
never
a
at
far too early for such conclusions.
had
the
Logic
good try
- and this
to
is
of
science
devoted
the
paper
theory
ground for
clearing
once said, the important
such attempts. Like some bearded German
the problematic
situation, but to change it.
thing is not to re-interpret
1.2 Logicians
and Philosophers
of Science
It would be an interesting historical
the adventures
project to describe
as compared
of logicians
in mathematics
with
those of formal
in their fields of study. Subterranean
of science
grum
philosophers
has been quite hospitable
to logic - even
mathematics
blings apart,
like set theory, model
absorbing whole
logical sub-disciplines
theory
or recursion
has
No
taken place,
similar
theory.
development
or biology.
Far
from
physics
experiencing
common
about
should
this
worry
making
Schadenfreude,
logicians
cause with those philosophers
or
of science
in
foundational
engaged
not
al
has
this
research.4
methodological
Why
long ago
happened
causes may be advanced
but there are also
ready? Again, historical
some serious methodological
These
should be mentioned
obstacles.
first, so as to look each other straight in the face.
in 'mentality' between
To begin with, there is a difference
logicians
however,
in,
e.g.,
of science.
philosophers
Briefly,
logicians want
seem
content
these
often
with
philosophers
more
than
usual
This
observation
reveals
the
acade
definitions.
just
between
related disciplines.
To see this, it
mic animosity
closely
an (admirable) book like Reichenbach
suffices to compare
1956 with
and many
theorems
formal
where
1973. Reichenbach
discusses
various formal issues concerning
Suppes
ever formulating
solu
Time without
problems
admitting of deductive
tions in the form of elegant
In the Suppes
in
theorems.
volume,
one
to
finds
contrast,
(and probably
conceptual
analyses
leading
formal
results - say representation
theorems
guided by) beautiful
434
JOHAN
VAN
BENTHEM
Robb's
causal
of Space-Time
with
the Special
coupling
analysis
of
Theory
Relativity.
This difference
in mentality
in goals:
may well reflect a difference
as
an
formal
sense
in
versus
aim
in
itself
say,
'explications'
Carnap's
as a means
formal definitions
for obtaining
desired
theorems.
Put
another
did not formulate
their formal con
way, Frege and Hubert
of
in
to
theories
order
called
'the logical
ception
papers
publish
structure
of arithmetic',
but in order to carry out their programs
as laws of pure logic; proofs
statements
(derivation of all arithmetical
of consistency).
not
does
the
of science know such
But,
philosophy
as
of
that
the
In a
say
well,
guiding programs
'Unity of Science'?
a
but
is
there
subtle
difference.
The
above
sense, yes,
mentioned
and indeed they
logical programs made claims which were falsifiable5;
were falsified - witness,
G?del's
It is
Theorems.
e.g.,
Incompleteness
so
this
characteristic
which
made
them
fruitful.
the
mainly
(Compare
case
of
the
see
In
to
it
is
hard
contrast,
analogous
circle'.)
'squaring
how programs
like the Unity of Science,
or even its implementations
like 'fitting each scientific
could be
theory into the Sneed formalism'6
refuted at all. Who
is going to unfold the really inspiring banner?
To a certain extent, the preceding
to a polite
amounted
paragraphs
invitation to philosophers
of science:
invest more heavily
in technical
there looms
the equally
logical theory. But, on the other hand,
obstacle
of the self-imposed
isolation of logic. Certainly,
deplorable
are
first-rate
out
many
nowadays
logicians
opening up to problems
side the familiar circle of mathematics,
of
those
the
semantics
notably
of natural language. But the boundaries
of logic should be set 'wider
or so I will now try to argue.
still and wider',
1.3 L?gica
Magna
both as 'mathematical
Contemporary
logic is a flourishing
discipline,
the
handbook
Barwise
logic' (witness
1977) and as 'philosophical
schemes
like the one to be presented
logic'. Therefore,
organisational
below might well be thought superfluous - being rather the symptom of
a subject
in its infancy.
of logic
(The most
grandiose
conceptions
were drawn up in times of logical stagnation.) The only excuse for the
is its modest
to make
realize
aim, namely
following
people
(or
more
or
what
could
be.
remember)
logic is,
THE
LOGICAL
STUDY
OF
435
SCIENCE
and however
it
Logic I take to be the study of reasoning, wherever
occurs. Thus,
an ideal logician is interested both in that
in principle,
and its products,
both in its normative
and its descriptive
activity
in
both
inductive
and
deductive
aspects,
argument.7 That, in all this,
she is looking for stable patterns
if one wishes)
to study is
('forms',
but innocent:
the assumption
of regularity underlies
inevitable,
any
science. These patterns assume various shapes: the 'logical form' of a
sentence
of inference,
the 'logical structure'
of a book or theory,
or
in
rules'
discourse
debate.
'logical
Given any specific field of reasoning,
the ideal logician chooses
her
Which
level of complexity
will be attacked:
weapons.
sentences,
is
theories?8
which perspective
texts, books,
inferences,
Furthermore,
most suitable; syntactic,
semantic or pragmatic?
which
tools
Finally,
are to be used in the given perspective:
which formal language, which
type of theory of inference and of which
strength? Thus she decides,
e.g., to study certain ethical texts using a tensed deontic
predicate
- or a
semantics
"world course"
logic with a Kripkean
theory of
a prag
quantum mechanics
using a propositional
language receiving
matic
in
terms
of
Even
verification
so,
games.
many
interpretation
aspects of the chosen field may remain untouched
by such analyses,
of course.
dis
then, there are various
Fortunately,
neighbouring
to
related
interests
be
with
consulted.
ciplines
On this view there is no occasion
for border clashes, but rather for
mutual
trade with not just philosophy,
mathematics
and linguistics,
but also, e.g., with psychology
and law. These
are not idle recom
but important
tasks. An enlightened
like Beth,
mendations,
logician
for instance,
realized the danger of intellectual
sterility in a standard
of knowledge
the genesis
in advance
from its
gambit like separating
to beyond
the logical horizon,
(thus removing psychology
justification
- witness
Beth and Piaget
1966. Another
by definition)
type of project
which
should become
is the
among
culturally
respectable
logicians
of mathematical
and juridical modes
of
systematic
comparison
itself
(cf. Toulmin
1958). But, not even mathematical
reasoning
logic
covers
its chosen field in its entirety. A book like Lakatos
1976 makes
- on the
it clear how eminently
view
of logic,
present
logical subjects
that is - have fallen out of fashion with the orthodox mathematico
logical
community.
it may
Finally,
be noted
that
such
cross-connections
would
also
436
JOHAN
provide
supposed
in pure
the dishes
which,
to flavour.
form
Only
VAN
BENTHEM
after all, a spice like 'logical awareness'
is
the most
insensitive
tolerate
palates
spices
....
2.
FORMAL
NOTIONS
OF
'THEORY'
seem to be that of a
in the study of science would
concept
in
with
the
remarks
made
In
accordance
section
1.2, we will
'theory'.9
in
mind
in
this
section
both
its
formal
study
keeping
approach
of
results. First, a short historical
and pleasant
definitions
sequence
most
than
which
be
richer
is given
definitions
may
(2.1),
already
are aware of. Nevertheless,
it should be emphasized
that
logicians
a
but
tale
for
historical
this is not a representative
didactical
account,
a
a rather special
how
is
review
there
Then,
purpose.
showing
an arsenal modern
for the sys
versatile
and flexible
logic provides
The
key
tematic development
of such definitions
(2.2). Further
logical theoriz
of the
to
is
the
in
order
results,
subsequent
subject
produce
ing,
enter
intellectual
interest
of
the
the
section
(3). Hopefully,
following
as
we
will
clear
become
advocated
here
proceed.
prise
2.1 A Short History
from Hilbert
to Sneed
'the logical view' of a theory
is that of a formal
To many
people,
a
a
are
set of axioms and
formal
whose
components
language,
system,
an apparatus
It is one of
theorems
from
these.
of deduction
deriving
that they
of the early modern
achievements
the amazing
logicians
so
to
unrealistic
with
this
much
notion.
do
extremely
managed
to
for the
the
formalists
fashionable
blame
it
has
become
Nowadays,
with
actual
this
when
of
(that
concept,
practice
compared
'poverty'
never have denied
this is conveniently
But,
forgotten).
they would
one
them for their happy
should
rather
their
aims,
congratulate
given
so often
Like
notion.
in the
but fruitful
of this austere
choice
to
it
be
of
science,
simple-minded.
paid
development
aims may call for richer concepts.
different
in
Nevertheless,
E.g.,
cases
one
a
needs
the
observation
that
additional
many
developing
a
in
consists
of
and
theory
judicious
interplay
proof
definition. Thus,
a logical concern
becomes
of equal
with
definability
importance
The complications
to be considered
here are of a different
derivability.
- as will
however
whose
from
the
kind,
appear
sequence,
following
THE
LOGICAL
STUDY
to account
for
is how
theme
in
theories
natural
science.
empirical
mathematics
proper:
main
2.1.1 David
OF
437
SCIENCE
of
additional
complexity
our
inside
Still,
story begins
the
Hilbert
As
is well-known,
Hubert's
of consistency
Program
proofs presup
view
of
the
above-mentioned
mathematical
theories, which had
posed
in the course
of millennia
of geometrical
studies. But,
developed
a
was
as consis
it
view
based
of
mathematics
moreover,
upon
global
a
core
more
of
'finitistic'
surrounded
hull
theories
like
abstract
by
ting
or
core
set
concrete
The
consists
of
theory.
analysis
simple
manipu
in some fragment of arithmetic.
lations with numbers;
say, encoded
to become
extended
human comprehen
beyond
threatening
are invented,
theories
infinite
sion, 'higher'
adding
(possibly
objects)
to
and
indeed
the
of
others,
very process
amongst
speed up proofs
arithmetical
(Cf. Smorynski
1977.)
discovery.
one could
a typical mathematical
as a
formalize
Thus,
theory
a
affair:
'concrete'
translated
two-stage
part Tx (with language Lx)
into some 'abstract superstructure'
T2 (with language L2), or maybe
in some mixed
contained
theory TX2 (with language Lx + L2). These
These
two set-ups are obviously
discussed
henceforth.
related:
for convenience,
the latter will
be
the consistency
of Tx was beyond doubt: but that of
For Hilbert,
- whence
was
the
not10
the
attempt to prove it by means within
Tu
was
matter
of
Another
of
the
out
side
Kreisel:
Tx.
range
pointed
by
Hilbert
assumed
that such abstract
extensions
did not create new
concrete
to discover
it easier
for
(they only make
insights
proofs
this means
that Tu is a conservative
extension
of Tx:
them). Formally,
if TXt2\-cp
then
Tx\-<p,for
all
Lx-sentences
<p.
this was refuted by G?del.
and
(Take for Tx: Peano Arithmetic,
for Tu, say, Zermelo-Fraenkel
Set Theory - <pbeing the relevant Liar
the above notion of 'theory' remains
inter
Nevertheless,
Sentence.)
in
view
of
later
esting, especially
developments.
Before
views is to be
turning to these, one more aspect of Hilbert's
noted. It would be rash to assume
that the underlying
logic of Tx will
one
one
be predicate
For
want
to
have a more
logic.
thing,
might
- but this is not the issue here. It is
at
this
constructive
level11
logic
All
438
JOHAN
VAN
BENTHEM
that
the complexity
of Li-statements
is important.
Clearly,
like 7 + 5= 13 are 'concrete' - but what about
statements,
seems to have allowed
Hilbert
universal
statements,
quantification?
but
combinations
like Vxyx + y <x
like Vx
not, e.g., quantifier
y;
- unless
reformulated
these
could
be
That
3y<p(x, y)
constructively.
are rather 'abstract' is brought out by the Skolem
such combinations
theoretical
functions:
equivalence
introducing
rather
individual
Vx3ycp(x,
y)^3/Vx<p(x,/x).
say Vx3yx < y as
Only when such an / may be given constructively
Vxx < Sx
is this permitted. Thus, one uses only a fragment of predicate
Ti.1213
logic in formulating
2.2.2
Frank
Hilbert's
Ramsey
view
of
theories
was
One could
inspired by mathematics.
con
'finistic'
introduces
say, perhaps,
part
empirical
concrete
siderations
of
but
(actual manipulation
symbols),
only in a
sense.
a
A
in
wrote
weak
little
short
very
later,
1929, Ramsey
paper
called
for a long time. (Cf. Ramsey
'Theories', which
lay dormant
also and notably
the role of theories,
1978). In it, he describes
as
follows.
there
is
the
so-called
theories,
First,
'primary
empirical
the prima
facie descriptions
of our various
language'
containing
These descriptions
may be observed
(laws)
regularities
experiences.
or reports of individual observations
Then, in order to
(consequences).
a so-called
this complex
field, one introduces
'secondary
systematize
a
of
A
will trans
with
level
abstraction.
language',
higher
dictionary
that
the
into 'secondary'
ones, and the latter will be
concepts
some
means
set
of
of axioms,
from which
the
simple
organized
by
are
laws
and
derivable.
consequences
original
for
there are similarities with the Hilbert
view. Notice,
Evidently,
to
statements.
'laws'
the
above
universal
how
instance,
correspond
than in
this restriction
is even more reasonable
For empirical
theories,
at
context. E.g., one may observe
the original mathematical
particles
or state universal
from
like
'in
certain positions,
these,
extrapolations
at most
is occupied
this trajectory
each position
thrice'. But, a
late
'primary'
at each
is somewhere
localization
like 'each particle
principle
to the theoretical
amounts
that a position
claim
function
uses of various conditions
lies in the possible
Another
analogy
instant'
exists.
on the
THE
LOGICAL
OF
STUDY
439
SCIENCE
In the Hilbert
relevant Skolem
functions.
like these
case, one would
to be 'constructive'.
In the empirical case, one would
like them to be
or
continuous
of constructive
(and hence
by means
approximable
measurable
functions).14
In order to bring out the postulated
character
of the secondary
a
as an
that
stressed
could
be
language, Ramsey
theory
regarded
second-order
viz.
&
sentence,
existentially
quantified
3Y(A(Y)
Y
A
is
the
where
the
and
D(Y,X))secondary
vocabulary,
'axioms',
D the 'dictionary' coupling Y to the primary vocabulary X. Thus, not
too much ontological
to theoretical
entities:
reality should be ascribed
a methodological
turn which
is quite typical for Ramsey's
philosophy
in general.
simple
Notice
that the whole
is based
point
upon
the following
logical observation:
then 3Y(A(Y) & D(Y,X))hS(X);
ifA(Y),D(Y,X)\-S(X),
where S is any primary
statement.
There
is a connection
here with
Hubert's
if
have
you
(cf. Smorynski
1977):
Program
'primary' reasons
for believing
in 3Y(A(Y)
& D(Y,X))
then you have
(consistency)
reasons
for
in
the
truth
of
any primary consequence
primary
believing
S(X) of the whole
theory (conservation).
secondary
This syntactic point of view is quite true to the spirit of Ramsey's
in the pre-semantical
work, which appeared
stage of modern
logic. Its
full impact only became
as we
clear in a semantical
setting, however,
shall see later on.
In authoritative
text books on the philosophy
of science,
one
like Nagel
finds
the
so-called
'statement
of
scientific
view'
1961,
an observational
theories. There,
as
such a theory is taken to possess
well as a theoretical
The
first
is
actual
vocabulary.
interpreted
by
Intermezzo.
inspection,
interpretation
respondence
measurement,
derived
and
from
so
on
the
-
the
second
receives
a
'partial'
first
so-called
'cor
through
in
for
(taken
Ramsey's
granted
to implement
precisely.
E.g.,
This division
principles'.
turns
out
to
be difficult
approach)
like explicit
of
original
strong proposals,
definability
('reduction')
to
un
theoretical
out
to
observational
turned
be
terms,
vocabulary
tenable. Nagel
is not unaware of relevant
logical work. He discusses
mean
to
which
be
taken
may
Craig's Theorem,
that, if one can find a
recursive
for some mixed
axiomatization
'observational/theoretical'
for its obser
theory, then one can also find such an axiomatization
440
JOHAN
VAN
BENTHEM
form of reduction.15 Although
there are
part alone: a weaker
in this picture,
'semantic'
elements
its main emphasis
is still
syntactical.
to find that Nagel does refer to Ramsey's
It is interesting
views, be
it in passing,
in his discussion
of instrumentalist
views of theories
versus
sentence was desig
realist ones. On his account,
the Ramsey
statement
ned to turn an otherwise
undetermined
theoretical
form
vational
certain
as a 'convenient
short-hand'
for some
instrumentalistically
a
set
statement.
into
of
determinate
observation
complex
reports)
too
turned things the other way around: an otherwise
(Our account
was
a
statement
to
theoretical
about
entities
weakened
strong
specific
statement.
in
existential
is
Nevertheless,
merely
Nagel
right
classify
on the instrumentalist
side - and it is ironical that the
ing Ramsey
(meant
revived
'semantical'
Ramsey
has
become
popular
with
latter
day
realists.)
In order
to pick up the main thread of our story, we go back to
a 'semantic conception
when
E.
W. Beth proposed
of theories'
1947,
still
in
of
Beth
alive
the
work
authors
like
F.
(cf.
1947),
Suppe or B. C.
van Fraassen.
as describing
to
Beth
view
theories
proposes
Briefly,
or
certain structures
what
you like), on
'histories',
systems,
(physical
the pattern of Tarski semantics. Thus, e.g., laws of the theory may be
on the trajectories
viewed as restrictions
open to the system in phase
- and that before the first
were
These
ideas
space.
prophetic
flowering
of logical model
theory.
A more
influential author propagating
similar views
is P. Suppes,
was
to
whose
theories
'set-theoretic
formulated
approach
predicate'
and applied
in the fifties already.
1957, Suppes
(Cf. Suppes
I960.)16
would
be defined as a class of
E.g., Newtonian
particle mechanics
structures
certain
set-theoretic
systems)
(time-dependent
satisfying
in set-theoretic
Newtonian
conditions
formulated
These
terminology.
- set
axioms of mechanics
may be taken as the (syntactic)
theory then
as
of
but
this is not
the
apparatus
deduction;
serving
underlying
- let alone a full first-order
made
predicate-logical
explicit
usually
We will have more to say about such a ianguage-free'
axiomatization.
(as it is sometimes
called) below. For the moment,
just note
approach
in
which
the historical
of
Klein's
precedent
'Erlanger Program'
was
the
geometry
approached
purely
through
study of
'structurally'
on
of
certain
transformations
(invariants of)
groups
spaces.
geometric
Suppes
did
use
model-theoretic
tools
occasionally
e.g.,
Padoa's
THE
LOGICAL
STUDY
OF
SCIENCE
44?
the
of Mach
in his discussion
method
(cf. Suppes
1957). Nevertheless,
not
fit
in
the
does
with
set-theoretic
smoothly
predicate
approach
on
structural
the
model
biased
of
theory
being
perspective
logical
our next author is more
in line with the
side, so to speak. Therefore,
present
exposition.
2.1.3 Marian
Przelecki
in formal semantics
tradition
of scientific
is a strong Polish
an elegant
1969
little
the
book
and
theories,
provides
Przet?cki
this.
is
derived
from
earlier
work by
of
(Its inspiration
partly
example
a
The
author
model
Cf.
1978.)
presents
Ajdukiewicz.
Giedymin
account
in the following
theories
theoretic
of empirical
steps. To
an
whose
there
is
observational
language L0
predicates
begin with,
There
in a certain concrete
of osten
domain U0 by means
interpreted
means
cases
and
of
(More precisely,
by
extrapola
paradigmatic
tion from these.) This domain
l/0 is gradually
enlarged by us to an
and galax
(Thus, electrons
empirical domain U of 'physical objects'.
will be all those structures
ies join dishes and babies.) L0-structures
as a
with domain U which
contain
l/0 (with its fixed interpretation)
a
substructure.
is
theoretical
Lt
Next,
added, calling for
vocabulary
of the L0-structures.
The relevant model-theoretic
suitable enrichment
of
is that of expansion,
concept
consisting
yielding L0+ Lt-structures
some Lo-structure
to which
have
been
added
(without
interpretations
Yet not all such expansions
changing the domain U) for the Lrterms.
are necessary
will do:
in terms of the Li
'meaning
postulates'
is a set T of
the
of
the
part
vocabulary.
Finally,
syntactic
theory
some
class
of
first-order
L0+ Lrstruc
L0+ Lrsentences
defining
are
si?n.
tures.
rise to various
e.g.
logical
questions,
in terms of
of definability
of Lt-vocabulary
a wealth
and
of notions
the book contains
Indeed,
L0-vocabulary.
results which cannot be reviewed
here. (Recall that this is not meant
to be a comprehensive
the most
historical
survey.) Probably
striking
an 'analytical'
a
T
into
one is Przet?cki's
to
theory
attempt
separate
This
framework
different
concerning
gives
kinds
without
part A, containing meaning
import, and a
postulates
empirical
claims
the
real
work
of
S
'synthetical'
part
making
empirical
doing
of
Poland
certain
The
L0-models).
(by excluding
People's
Republic
=
has invested many years of research
into this elusive
equation T
442
JOHAN
VAN
BENTHEM
A + S, without
answers.
in this
finding
satisfactory
Nevertheless,
some
have been
raised. E.g., A
connection,
questions
interesting
should clearly
be 'semantically
in the sense
that it
non-creative';
should not exclude L0-structures:
each L0-structure
must be expand
able to an L0 +Li-structure
is a model
which
for A. A true model
wants
to
find
the
side
of this coin. Is it
theorist, Przet?cki
syntactic
that A
is 'syntactically
in the sense
that the only
non-creative',
derivable
from it are the universally
valid ones?17 This
Lo-sentences
is the precursor
of a more general one (treated by
type of question
in
related
to
which we now turn.
Przet?cki
publications)
The L0 + Li-theory
T has a class of models MOD(T).
Accordingly
be said
it may
structures which
to describe
the
can be expanded
?L0
class MOD(T)
to L0+Lt-structures
of
those L0
in which
T
holds. Notice
the way
in which Ramsey's
idea is implemented:
this
a 'second-order
class of Lo-structures
receives
existential'
description
in this way. Now,
let us return to the original
of
version
syntactic
form. We
have an L0-theory
ideas, in the following
Ramsey's
T0
which
is contained
in the L0+ L?-theory
T; i.e.,
(1)
if T0\-(p then T\-<p, for each L0-sentence
<p.(Extension)
what
about
the connection
between
Then,
MOD(T0)
(those Lo
are models
structures which
of T0) and the class MOD(T)
?L0 des
at least, it should hold that T does not exclude
cribed by T? Clearly,
models
of T0:
(2)
MOD(To)
?5contained
inMOD(T)
\ L0. (Ramsey Extension)
But, the above syntactic
T has Lo-consequences
Hilbert
(3)
requirement
does not guarantee
requirement
outside
of T0. Now,
there was
of conservative
extension,
reading
then T0hp,
if Thp
Extension)
for
each L^sentence
indeed, (2) implies (3), as is easily
imply (1).) Do the two conditions match
answer
is negative.
And,
seen.
<p. (Conservative
(Notice
precisely?
this: maybe
also the old
that (2) does
Unfortunately,
not
the
Let T0 be the complete
first-order
Counter-example:
theory stating
that Time has a beginning
from there via a 1-1
0, and proceeds
successor
never making
S ('to-morrow')
operation
loops. (Infinitely
THE
LOGICAL
STUDY
OF
SCIENCE
443
to secure this.) The latter phenomenon
is
axioms are needed
a
T
in
the
transitive
theory
having
(finitely axiomatizable)
explained
such that Vxy(Sx = y -> Bxy), as well as a pro
relation B ('before')
& lEY),
such that EO, Vx(Ex-^ESx),
perty E ('early')
Vx3y(Bxy
to
from
'late', never to
(Time goes
Vx(~IEx-?Vy(Bxy
'early'
->~lEy)).
a
form
model
for T0
numbers
the
natural
(N, 0, S)
return.) Now,
to a model
cannot be expanded
for T.18 (More
which
'empirical'
many
in full here.)
take too much space to present
would
counter-examples
a
to
this
establish
model-theoretic
Thus,
attempt
duality has
typical
failed. Still, there is a result which one can prove, viz. the equivalence
version
of Conservative
with
the following
weaker
of
Extension
Extension:
Ramsey
(4)
each L0-structure
in MOD(T0)
to some L0+ Lt-structure
extension
an L0-elementary
has
in MOD(T).
the condition
that the original
situations
(Lo-struc
empirical
be
to L0+ Li
with
'enrichable'
theoretical
entities
suitable
tures)
structures which are models
for T, is now related to their being thus
'extendable' - addition of individuals becoming
allowed as well. Given
the fact that this is a not unknown
in
procedure
physics
(postulating
new particles,
or whole planets) model-theoretic
curiosity has led to a
Thus,
not
unreasonable
amendment
on
Ramsey's
views.
will have given an impression
of Przet?cki's
considerations
These
we
model-theoretic
turn
to
the most
finally,
maybe
spirit. Now,
influential author on formal philosophy
of science
in recent years:
2.1.4
Joseph
Sneed
In his book Sneed 1971, this author gave an analysis of classical particle
mechanics
which
transcends
the above
in
using a formal machinery
various
Since
this
work
has
become
well-known
respects.
interesting
cf. Stegm?ller
(largely through allied work;
1979), it will suffice to
are
mention
to
those
ideas
which
relevant
the present discussion.
only
In the above picture of a theory, with ingredients T0, MOD(T0),
T,
it is implicitly assumed
that a division
into non-theoretical
MOD(T),
if you wish) vocabulary
vocabu
('observational',
(L0) and theoretical
has
been
In
effected
this is
lary (Lt)
satisfactorily.
practice, however,
a difficult problem,
as we have seen - and Sneed gives an ingenious
us here).
solution
of which
need not concern
(the exact nature
444
JOHAN
VAN
BENTHEM
on
of his work
Another
fruitful practical perspective
is the emphasis
to apply a theory to a given empirical
what it means
situation. On the
to show
this becomes:
how
view,
Ramsey
by Sneed,
adopted
theoretical
be
introduced
may
concepts
(forces,
('super-natural')
and so on) turning the situation
into a model
for the whole
wishes,
theory T.
will any
Will
such an addition always be possible? More precisely,
model of T tL0 = {(pE L0|Th<p} (clearly, the most'fitting'choice
for the
a
answer
to
is
be
model
for
T?
The
part T0)
expandable
empirical
a
we
this
T
extension
of
and
is
conservative
T0,
negative:
evidently,
are asking if it is also a Ramsey
remains
extension.
Thus, the problem
like
in the preceding
In Sneed's
sub-section.
then, the
terminology,
not
need
be
eliminable'
from
'Ramsey
vocabulary
always
to
If one wishes
it
will
have
be
this,
postulated.19
theoretical
the theory.
we
Next,
most'.
turn
from
Will
questions
about
'at
least'
to questions
do?
theoretical
'at
about
that Przet?cki
(Recall
just any
expansion
at this point.) This seems implausible.
'meaning postulates'
are to be formulated.
Sneed considers
the
suitable restrictions
Thus,
a
there
be
for
model
T0,
expan
that, given any
unique
requirement
sion of it to a model
for T. Model Theory gives one the cash value of
this proposal
for a large class of languages.
By Beth's Definability
amounts
to
this
of
Theorem,
explicit definability
'implicit definability'
introduced
reduc
Lt in terms of L0 (on the basis of T). Thus, old-fashioned
tionism would
re-emerge!
as Sneed
This
line of thought
should not be pursued,
realized,
one wants
because
it is too 'local'. The kind of restriction
is a more
one,
'global'
cross-connections
concerning
between
expansions
of
What we have in reality is a class of
empirical
are
in such a
which
situations,
simultaneously
empirical
expanded
are obeyed.
of
that certain
'constraints'
Two typical examples
way
are the following
such constraints
different
situations.
in different
particles
the same mass values
(Cl)
('Mass'
Thus,
Milky
the earth
Way.
is a theoretical
gets
the
empirical
situations
should
receive
through different expansions.
function
in Sneed's
analysis.)
same mass
in the Solar
System
as
in the
THE
LOGICAL
the physical
(C2)
mass
separate
STUDY
OF
join of two particles
445
SCIENCE
receives
the sum of their
values
(whether present
situation considered,
- at some
or not).
earlier
or later time -
in the
the picture
of an empirical
say, (T0,
theory now becomes,
- where C is the set of constraints.
T, MOD(T),
C)
MOD(T0),
Applying
such a theory to a class E of empirical
situations means
finding a
E
of
C
simultaneous
such
that
the
expansion
satisfying
resulting class
is contained
at
in MOD(T).
E
will
have to be a
least,
(Obviously,
'suitable' candidate,
in the sense that it is contained
in MOD(T0).)
Thus,
In later publications
is
(cf. Balzer and Sneed
1977/8), this picture
a
of
its
set-theoretic
'lan
stripped
purely
linguistic content,
leaving
tradition. E.g., MOD(T)
in the Suppes
is
guage free' formulation
an
a
X
class
of
C
becomes
while
Lt-structures
replaced by
arbitrary
con
class of subclasses
of the L0+ Lt-universe
certain
satisfying
ditions.
inclusion
is one of these.) This
(Closure under set-theoretic
move will be discussed
in section 2.2.
of 'constraint'
the
notion
which
complicates
a model-theoretic
of
view.
it
is
Thus,
point
a
about
In
little
alternatives.
fact,
possible
speculate
be several promising
escape
routes, open to further
The first is to have one single domain
for the language L0
exploration.
the 'empirical
in
1969), considering
(as already proposed
Przet?cki
as its subdomains.
situations'
would
then apply to
Ramsey
expansion
it
Clearly,
above
picture
to
worthwhile
there seem to
the
whole
is the
from
'universal'
L0-structure:
immediately
guaranteeing
con
straints
like (Cl) above.20 Possible
be that this
objections
might
too global, and also that it is not obvious how all constraints
becomes
further complications.
As for the 'globality',
may be captured without
notice
that one need not think of a literally
domain:
it
'universal'
a class of L0-structures
would
suffice to consider
directed by model
their direct union for convenience
theoretic
inclusion - introducing
a
This
is conceptually
observation
second route which
opens up
only.
as well.
attractive
a
Up to now, we have followed
theory in assigning
ordinary model
class MOD(To)
in mathematical
classes
of
'isolated' L0-structures
practice, one is normally
of
such models,
with
relations
to a theory T0. But, already
confronted
with 'structured'
and operations
connecting
446
JOHAN
VAN
BENTHEM
be natural,
it would
e.g., to replace MOD(T0)
by the
mor
of
their
natural
T0-models
category
CAT(T0)
plus
connecting
of
then
choice
will
be
The
dictated
phisms.
particular
morphisms
by
of the theory considered.
think of
the necessities
E.g., one might
On this view, constraints
(as substructures).
isomorphic
embeddings
come
to expand a class of L0-structures
in as follows.
One wants
structure:
class should
i.e., the expanded
together with its categorial
- the
be an Lt-subcategory
original morphisms
remaining morphisms.21
them.
Thus,
Put
the forgetful
should yield CAT(T0) when
fashionably,
functor
to
Some
reflection
shows
that this takes care of the
CAT(T).
applied
as
As
it
the
will even be
constraint
well.22
stands,
(Cl)
requirement
too strong in many cases. E.g.,
it seems reasonable
that isomorphic
trans
situations
up to a co-ordinate
empirical
(being equivalent
but not each single
say) receive
isomorphic
expansions;
an
remain
need
Lo-isorphism
L0-\-Lt-isomorphism!
(E.g., not each
a metrical
of
remain
need
purely
topological
automorphism
Space
in
with
when
is
metricized
the
accordance
Space
isomorphism,
This
second
well
have
considerable
model
may
possibility
topology.)
or
it accommodates
all constraints
interest in itself, whether
theoretic
formation,
not.
section 5.)
(An example will be found in the appendix,
our historical
A
This concludes
very rich and promising
survey.
formal notion of 'theory' has emerged - whose
logical study is still in
the fate of the usual properties
of and
its infancy. For example,
terms
defined
in
of
relations between
such theories
(normally
formal
and dis
is still to be explored.
Some relevant
systems)
suggestions
it should be remarked that
cussions will be found in section 3. Finally,
of contemporary
this survey
is by no means
logical
representative
a
of theories,
model-theoretic
views
standpoint.
being
inspired by
our scope
outside
tracks have remained
different
Thus,
interesting
are
new
Wessel
That
there
frontiers
for
e.g.,
(cf.,
1977).
logicians will
was
our
have become
and
that
clear, however;
abundantly
goal.
2.2 A Systematic
Logical
Perspective
in that it
A phrase like 'the logical structure' of a theory is misleading
one
one
to
the
suggests
study of
single (or
preferred)
logical approach
but
there is one single logical research mentality,
science. Certainly,
an aspect of that is precisely
the plurality
of available
logical ap
on one's
Thus, depending
specific aims, there are many
proaches.
THE
LOGICAL
STUDY
choices available
1.3). For
(cf. section
in particular,
the following
perspectives
2.2.1
OF
SCIENCE
447
the study of scientific
theories,
should be kept in mind.
Syntax
in 2.1 already how the syntactic
notion of a formal
remarked
a
to
out
be
for many enquiries.
It was
system turned
happy choice
a
arose
from
also recalled
that this notion
of
long development
- in which the axiomatic
of deductive
Euclidean
organisation
geometry
its value as a means
for efficient organisation
and a
knowledge
proved
stimulus for rapid development
of deductive
Even so, one
knowledge.
has a variety of syntactic
in the actual computer
approaches.
E.g.,
in the Eindhoven
formalization
of mathematics
AUTOMATH
project
a syntactic
text logic
(cf. Van Benthem
1979), one needed
Jutting
lambda-calculus
in a natural
of a version
of the typed
consisting
a
from providing
of truths,
deduction
Apart
catalogue
presentation.
It was
a model
such a system also provides
for actual mathematical
prose.
are a natural and interesting
(Indeed, modern
computer
languages
at the text level of the syntactic approach
extension
in logic.) Similar
no doubt
for the natural
sciences would
projects
inject invaluable
a priori debate
into the now
often
practical
experience
purely
concerning
2.2.2
the possibilities
of their formalization.
Structures
was made
no distinction
in geometry
be
the advent of
and spatial intuition. After
led to the concept
of
this separation
non-Euclidean
geometry,
structures
certain
syntactic
('spaces') modelling
purely
geometrical
sentences.
is exemplarily
sets of geometrical
The resulting
interplay
1959. But, purely
'structural'
views
of
illustrated
in, e.g., Tarski
arose very early too: witness
Klein's
theories
'Erlanger Program'
to
in section 2.1, or Poincar?'s
mentioned
approach
group-theoretic
Before
tween
the
19th century,
derivation
syntactic
and mechanics.
geometry
In the natural
sciences,
lip service was paid for a long time to
in
structural views
lie even more at
axiomatic
ideals; but,
practice,
hand there. E.g., Newtonian
mechanics
be
identified with
might easily
a certain class of real 'mechanical
Which
brings one to the
systems'.
are
use in the logic
most
for
'structures'
suitable
which
question
just
of natural
science.
448
JOHAN
VAN
BENTHEM
a start, let us quickly
of
dispose
is
abstract
'Mathematics
about
objection.
This
is true;
about non-formal
Reality'.
or
deal with are models
natural scientists
For
a
'philosophical'
pseudo
natural science
structures,
it is too true. What
only
of reality,
representations
models?
which
these go into the logical study of science.
So,
seems
of
'structure'
rich
model-theoretic
notion
the
True,
ordinary
to
kind
of
in
accommodate
any
system
(cf. Suppes
enough
principle
- but it would also be natural to
1974)
try and work
1960, Montague
of their
in the sense of general system theory, because
with systems
Padulo
and
Arbib
1974.)
(Cf.
time-dependent
presentation.
explicit
and
E.g.,
representation
system-theoretical
theory
contains
many
'logical'
lessons.23
it may be of interest to observe
that a classical
In this connection,
its main
formulates
of
in
the
mechanics
paper
quantum
logical study
a
struc
in
with
result on 'hidden variables'
(a topic
syntactical
ring)
as
a
and
than
result
about
classical
rather
tural Hilbert
space terms,
Kochen
and
axiomatic
theories.
(Cf.
Specker
quantum-mechanical
1967.)
form can be
not all this mean
that the ladder of linguistic
Does
thrown away, now that one has arrived at the structural
'reality'
seem to be the 'set-theoretic
theories? This would
scientific
behind
indeed for many problems
this is a
earlier on. And
view' discussed
- witness
and
work
Sneed
to
much
sensible
do
by
perfectly
thing
of
which
the
obvious
systems
problem
E.g.,
presentations
Suppes.
the same 'real' system
describe
co-ordinates,
(in different
say) is a
about
classes.
suitable
structural
Or, when
isomorphism
problem
finite
machines
relations
of
between
talks
about
(cf.
analogy
Ashby
are
machines
the
that
1976),
mutually
homomorphic
Ashby
insight
owes
theories.
to underlying
Other
syntactic
nothing
full
the
per
may
structural-linguistic
require
questions
interesting
as
will
be
illustrated
below.
however:
spective,
be
observes
the friction one sometimes
If this be so, then why
and the 'model-theoretic'
tween
the 'set-theoretic'
(Cf.
approach?
in such cases, there are priority disputes
1974.) Inevitably,
Przet?cki
is 'better'V* E.g., on the
the atmosphere:
whose perspective
poisoning
one
loses
that
it
is
claimed
set-theoretic
side,
except maybe
nothing
models
for mathematical
troubles
theories, which
(with non-standard
one did not want anyway).
however. Why
This seems short-sighted,
isomorphic
not
leave
the
door
open,
e.g.,
to
'non-standard
mechanics',
when
THE
LOGICAL
STUDY
OF
SCIENCE
449
are just discovering
the delights
of non-standard
analysis?
ladder forever means
away the linguistic
Throwing
cutting oneself
off from the original motivation
of many
structural
A
concepts.25
similar warning was given by Dieudonn?
to mathematicians
wanting
to apply logical gadgets
like ultraproducts
without
being bored with
mathematicians
stories about their logico-linguistic
of course,
origin. This is possible,
but one deprives
oneself
from understanding
as
their full importance,
well as the heuristic
fuel for further discoveries
in the same vein.
2.2.3
Semantics
Since logical model
field (cf. Chang and
theory is such a well-known
Keisler
to just a few
1973), it will suffice here to draw attention
relevant
To begin with,
here
is a short list of questions
points.
are meaningful
theories which
per
concerning
only in a semantical
spective.
as a structural notion - that is,
'determinism'
(1) Does mechanical
is possible
into the future
given ? past history, only one continuation
as a linguistic
(in the relevant set of structures)
imply 'determinism'
at time t + 1 are explicitly
state variables
notion:
i.e., the relevant
definable
in terms of those at time t - on the basis of the mechanical
axioms?
1976: 'Deterministic
Theories'.)
(Cf. Montague
"a point relation
is said to be objective
if it is
(2) In geometry,
invariant with respect to every automorphism.
In this sense the basic
are objective,
relations
and so is any relation
in
defined
logically
terms of these.
relation may be so
[...] Whether
every objective
defined raises a question
of logical completeness
[...]' (Weyl 1963, p.
to be sure. (Cf. the appendix
73): a model-theoretic
(section 5)
question,
for some relevant results.) Finally,
the direction may also be reversed,
as in
- viz. that
like Einstein's
(3) Do
principles'
'relativity
linguistic
transformation
be invariant under the Lorentz
'physical equations'
admit of a structural characterization?
One could ask scores of similar questions,
say about the relation
between
'linear' functions
and 'linear' polynomial
forms, all of them
to look for a
the same peculiarity
of model
illustrating
theory:
systematic
Another
and structural
duality between
syntactic
constant
attention
point which deserves
points of view.
is the following.
450
JOHAN
VAN
BENTHEM
one
semantics
should not make
of plain Tarski
great success
link
structures
in
between
and
is
any
language
forget that,
principle,
interest. Thus, e.g., non-classical
semantics
of model-theoretic
of the
or the 'vague' semantics
of Fine
1975 are equally
forcing variety,
a
In
the
choice
of
certain
class of
then,
respectable.
particular,
structures does not commit one yet to a fixed underlying
apparatus of
The
or otherwise.26 This observation,
deduction:
classical,
intuitionist,
by
one with an excellent
to make more
the way, provides
opportunity
exact sense of the famous
in Quine
Thesis'
1951.27
'Revisability
2.2.4
Pragmatics
them, syntax and structures
(and hence: Model
Theory)
seem to exhaust all possible
upon theories.
logical perspectives
our actual hand
themes concerning
But, moreover,
many pragmatic
can
notions
of
above
be
studied by logical means.
the
Thus,
ling
as
activities
rather
than products
theories
scientific
of such activities
are not irrevocably
outside
the scope of logic. Just a few examples
Between
would
have to provide
the backing for this claim here.
In the purely syntactic
there is already
the matter
of
perspective,
of proofs.
the heuristic
Lakatos
in
mentioned
section
1976,
1.3,
(Cf.
or Hintikka
and Remes
the logical study of the
1974.) Moreover,
or attack of sentences
in dialogues
actual defense
has been initiated in
works
like Lorenzen
and Lorenz
1978.
In the semantic
stu
too, there is room for pragmatic
perspective
will
that successful
dies. E.g., model
has
theory presupposes
interpretation
seman
taken place already. How? Here
is where
'game-theoretical
tics' in the style of Hintikka
useful
may become
(cf. Saarinen
1979).
There is also a connection
here with actual measurement
(cf. note 15),
as well
as with
to physical
Giles'
theories
(cf. Giles
1979).
approach
an
1979
volume
Hooker
the
Indeed,
signals
interesting
'logico-prag
turn in the study of physical
it will be
matic'
theories. Nevertheless,
are only the first landmarks
clear that these references
in a hopefully
fruitful new area of logic.
an appropriate
the pragmatic perspective
occasion
Finally,
provides
for stating a concluding
remark about the notions
here.
considered
Exclusive
concentration
reveal a lot about the
upon
'edifice
as
theories
of science',
intellectual
but it may
products may
also generate
THE
LOGICAL
STUDY
OF
451
SCIENCE
in jeopardy when
of science
pseudo-problems.
E.g., is the rationality
and
two of its beams do not fit exactly - say, Newtonian
mechanics
be
Will
'continuous
Einsteinian
mechanics?
Of
growth'
endangered?
course
is guaranteed
not: the continuity
of science
the
(re-)
by
worker's
know-how
of common
intellectual
construction
rules, stra
tegies,
whole
and the
theories:
3.
like. Take away specific
axioms,
languages,
this cat will still retain its rational grin.
FORMAL
QUESTIONS
CONCERNING
or even
THEORIES
as the
illustrates
how logic may be viewed
section
preceding
a
non-trivial
scientific
More
of
whole
theories.28
study
specifically
and tools was
A theory
reviewed.
range of logical considerations
should always be studied with specific questions
in mind and these
This brings us back to
call for a suitable choice of logical perspective.
The
one
made
in section
1.2 about logical research
the observations
on
the
the
lure
of
theorems
horizon. Here a related
being guided by
will
treated.
What
becomes
of
the
be
question
logician's pet theorems
when her subject
is used in the present
context?
To take just one
is rather dear to her: does Lindstr?m's
curious weed which
Theorem
a
in this more
have any significance
In
area?
sense, yes,
general
insofar as this theorem
in general)
is
(and abstract model
theory
of
concerned
semantics.
with
the general
of logical
properties
languages
plus
It has been argued that scientific
theories may be based
upon quite different
logics; and abstract model
theory may help
reasons
meta
for
choices,
by telling one which
provide
pleasant
theorems will be gained or lost in each case. (Cf. Pearce
1980A for an
of abstract model
application
theory to the Sneed framework.)
some
More
favourite
systematically,
topics and results of logical
now (be it very sketchily) with respect
research will be reviewed
to
to the study of scientific
their possible
relevance
theories
This will serve as a first measure
of the required angle
in general.
of logical
re-orientation.
To be sure, there exist some important
logical topics already in the
or analyticity.
of empirical theories, like causal explanation
metascience
The point is that we want more.
To begin with, classical research into the foundations
of mathematics
area.
has produced mainly
results in the following
452
JOHAN
VAN
BENTHEM
3.1 Properties
of Theories
In the syntactic
of notions with a logical
tradition, there is a multitude
us
text book
Let
follow
the
traditional
list: special
just
pedigree.
then
the
familiar
triad
of axioms,
derivability,
definability,
forms
and, finally, decidability
independence,
consistency,
(or
completeness,
of the well-known
standard
of the theorem
set). Many
complexity
results of logic are in this area: Craig's Theorem,
G?del's
Theorems,
Church's
Theorem.
Some
of the original
these
problems
inspiring
seem to become
less urgent
in the case of empirical
would
theories.
E.g.,
consistency
will
be
a minor
concern
-
the much-praised
'physical
one might want
intuition' being there to keep us straight.29 Similarly,
so
to 'neutralize'
to
G?del's
rather for
results,
speak,
searching
our
the
of
non-mathematical
theories.
for
parts
'partial completeness'
are
ones in
in
the
structural
tradition
often
mathematical
Questions
even
their own right, having become
in
science
respectable
empirical
a
before
the adolescence
of modern
One
of
logic.
example
pertinent
result
which
be called
in spirit is the theorem
in
might
'logical'
1964 to the effect
the that the only mappings
between
are those
reference
frames preserving
causal connectibility
physical
of the Lorentz
group.
Some
relevant model-theoretic
in
topics were mentioned
already
section 2.2.3. A more systematic
list follows
the semantic duals of the
of axioms
above
of
(more generally:
topics.30 Forms
complexity
are
so
with
connected
the
to
theorems
dear
definition),
preservation
con
This
the
of
relation
between
structural
closure
topic
logicians.
seems of wide
ditions and explicit
In this con
interest.
definability
Zeeman
like Keisler's
characterization
of elementary
out
first-order
of
structural
data in
definitions
classes,
up
conjuring
terms of closure under isomorphisms
and ultraproducts,
should have
value. Next,
theorems
great inspirational
completeness
linking syn
tactic derivability
with
structural
retain an obvious
consequence
is the finding in Giles
interest. (A nice example
1979 that the logic of a
nection,
key
results
Lukasiewicz's
certain
system Loo.)
'physical game theory' is precisely
and Beth's Theorem
Padoa's Method
tying up explicit
definability
with structural determination
have already been applied repeatedly
in
the theory of empirical
science
1960, Montague
1974,
(cf. Suppes
Rantala
retreats into
1977). Of the above-mentioned
triad, consistency
- the
for theories being presup
the back-ground
of models
possession
THE
LOGICAL
STUDY
OF
SCIENCE
453
has fallen out of
Moreover,
independence
posed model-theoretically.
themselves
favour with the logicians
already, being of purely acade
with the important
is connected
mic interest. Completeness,
however,
im
of categoricity,
which may have wider
structural phenomenon
portance.
clear from this quick look is not that all these logical
becomes
as they stand - only that this type of result
remain interesting
and proven in the
interest if it can be formulated
remains of potential
as
more complex
in section
theories
of
developed
empirical
setting
Take
Tennenbaum's
chosen.
2.1. Here
is one example,
arbitrarily
are the
theorem
beautiful
stating that the standard natural numbers
What
results
with recursive operations
of Peano Arithmetic
only countable model
Here
is a more
of addition and multiplication.
'empirical' one in the
same spirit. The only countable
models
for linearly ordered Time
the
which are homogeneous
(in some natural sense) are the rationals,
van
that
and
their
order).
(Cf.
(in
product
integers
lexicographic
Benthem
about a result in this spirit for mechanics?
1980.) What
or the succession
of scientific
Interest
in the edifice of science,
soon
one
will
lead
if
into the
theories
wishes)
('scientific
progress',
area.
less
settled
following
3.2 Relations
between
Theories
in
have been discussed
theories
relations
between
Some
syntactic
extension.
conservative
extension
and
2.1 already,
section
notably
Important relevant results are, e.g., the theorem of Craig and Vaught
1958 stating that each recursively
axiomatized
theory ( in a language
or
additional
with identity) is finitely axiomatizable
using
predicates;
trans
well
be
results. Such questions
G?del's
non-conservation
may
a con
is classical mechanics
sciences.
ferred to the empirical
E.g.,
would
this
be an
of classical
servative
extension
not,
(If
analysis?
to applied theory.)
from application
interesting example of 'feed-back'
These notions of extension may be regarded as special cases of the
relations
of interpretability
and embeddability
whose
(respectively)
are as follows. Tx (in Lx) is interpretable
formal definitions
in T2 (in L2) if
U with one free variable (the "universe")
there exists some L2-formula
as well as some effective
translation t from the non-logical
constants
in
to
for
such
that
Lx
fitting (possibly complex) L2-expressions,
T2\-(r(a))u
each axiom a of Tx. (Here r(a) is the result of replacing each non-logical
454
JOHAN
VAN
BENTHEM
"U" indicates
while the superscript
in a by its r-counterpart;
to
in
of
all
relativisation
U.) For technical
r(a)
quantifiers
subsequent
that (r((p))u
follows
It
one
then
that
also
reasons,
T2|-3xU(x).
requires
case this
In
in T2 for each theorem
of
derivable
becomes
Ti.
<p
are
can
be
also
i.e., only Tptheorems
reversed,
provable
implication
in T2. These arose already in
(through t, U) in T2, Tx has been embedded
constant
versus non-Euclidean
the early history of Euclidean
Notice,
geometries.
an inter
establishes
'circle interior' model
for example,
that Klein's
of
into a conservative
extension
from hyperbolic
geometry
pretation
a
an
constant
for
individual
Euclidean
(obtained by adding
geometry
a
a
an
kind
of
here
is
As
of
relevant
circle).
logical result,
example
theorem for inner models'
1980a): "If Tx
(cf. van Benthem
'compactness
are
that
each
theories
such
(in Li), T2 (in L2)
finite part of Tx is
D
translated
in
in
Lx
L2
T2 (constants
being
identically),
interpretable
extension
in some conservative
then the whole theory Tx is interpretable
these
of T2".31 In general,
there is a scarcity of results concerning
like
notions. Lately,
however,
logicians have been studying questions
of Peano Arithmetic,
the above for special cases. E.g., for extensions
one has the stronger 'Orey Compactness
Theorem'
1979).
(cf. Lindstr?m
is replete with
science
'reduction
structurally,
empirical
Speaking
versus
sta
for
thermodynamics
calling
logical analysis:
phenomena'
versus relativistic
or quantum
classical mechanics
tistical mechanics,
etc. Very
often no uniform
mechanics,
satisfactory
logical notions
have been forged yet to get a good grip on these. (Cf. Pearce
1981,
however.)
illustrate
Time. Sup
this, here is a small example
concerning
- on the
one
has
worked
with
dense
time
pose
pattern of the
always
at
it
that
is discrete
rationals
Now
becomes
clear
time
say.
<Q, <),
some deeper level, on the pattern on the integers (Z, <). The resulting
structure
of a
could be the lexicographic
product Q x Z consisting
To
unbounded
linear sequence
of copies of the integers. Notice
is that of discrete
that the theory of this structure
time! Should one
new
Q
is
to
the
Q x Z - and, if
that
the
reducible
'richer'
say
original
to choose
from. E.g.,
sense? There are various relations
so, in which
Q is clearly
embeddable
into
QxZ.
the
isomorphic ally
Conversely,
^
a
is
obvious
contraction
QxZ
onto
from
mapping
-homomorphism
dense
in its backward
the 'p-morphism'
clause of
direction,
Q, satisfying,
that
the
is not a
modal
contraction
1971). (Notice
logic (cf. Segerberg
< -homomorphism.)
seem
to be
would
The correct view of the situation
THE
LOGICAL
STUDY
OF
SCIENCE
455
the following,
however: Q is reducible
to a certain 'level' of Q x Z, in
a
to
the sense of being
quotient
isomorphic
of the latter structure
some suitable
relation.
under
that the above
(Notice
equivalence
was
&
not
in
the
model-theoretic
congruence
sense.) This
equivalence
seems of potential wide application.32
notion of reduction
two
Third comes the model-theoretic
upon the previous
perspective
sets of notions.
relations
between
'conservative
Duality
syntactic
were
extension'
and structural
extension'
'Ramsey
already treated in
an interesting duality result will be
section 2.1. As for interpretability,
found in Pearce
to
1980B, which also contains many useful references
literature. (Even so, powerful
criteria are still lacking to
the technical
or embeddability
in given cases.) One partial
disprove
interpretability
is
result
Theorem:
the following
of the Compactness
consequence
easy
in T2 up to
'A finitely axiomatized
is
theory Tx
interpretable
a
for
contains
and
if
model
if
each
substructure
T2
only
disjunction33
is a model
for TV'
with L2-definable
domain and predicates
which
to
is
'local'
structural
Thus,
equivalent
'global' definability
definability
A simple syntactic
trick enables one to get rid of
up to disjunction.
the latter phrase:
a 'single' one -
any 'disjunctive'
may be replaced by
interpretation
result for inter
and so we have a second duality
its
of a V3-form
'structural'
is
clause
pretability.
Unfortunately,
which does not make for easy counter-examples
(unlike in the case of
or
Beth's
Cf.
G?del's
Theorem
Theorem).
Definability
Completeness
(section 5) for more related results.
this unsatisfactory
there
is an
state,
interesting
a
to
be
observed
'reversal
of direction'.
here,
phenomenon
namely
is a common
event
in semantics:
the experienced
This
reader will
account
like Sneed's
of how to
have pondered
already about cases
content and applicability
threaten to become
'apply' a theory, where
In the present case, the relevant observation
is
inversely proportional.
to structural
from Tx to T2 amounts
that syntactic
'reducibility'
from models
of T2 to models
of Ti.34 Thus, e.g., struc
'reducibility'
the appendix
Even
in
of orthomodular
relations
like the nonembeddability
Hilbert
and
do
Kochen
space lattices into Boolean
1967)
(cf.
Specker
algebras
not correspond
to
of
the
quantum
smoothly
non-interpretability
into that of classical
mechanics!
mechanical
axiomatic
theory
in
1960
thesis that
the
of
the
structural
formulation
Similarly,
Suppes
to
in
be
the
terms:
'for any
reduced
following
biology may
physics'
an
to
construct
it
model of a biological
[is] possible
theory
isomorphic
tural
456
JOHAN
VAN
BENTHEM
within physical
theory' is very hard to fathom in syntactic terms.
at such an elementary
these uncertainties
level, we will
the far more
involved
from formulating
(and numerous)
of 'theory'
of 'reducibility'
arising from the richer concept
ones
out
in
2.
the
fruitful
section
Nevertheless,
picking
developed
model
Given
refrain
notions
and Sneed
the many
Sneedian
among
(cf. Balzer
proposals
one
tasks.
will
first
be
of
the
1977/8)
logicians'
to the above
let us return once more
To conclude,
temporal
Q and QxZ
illustration.
The structural
reduction
relation between
was that of division by the equivalence
relation of 'being only finitely
a corresponding
What
about
discrete
many
steps apart'.
syntactic
seems
the two syntactical
At
first
this
formulation?
sight,
hopeless,
a
versus
discreteness.
inconsistent:
theories
So,
density
being
be a spectacular,
That would
but evidently
revolution?
scientific
from
A better description
is the following.
From
rather shallow conclusion.
dense
linear Time
the theory of unbounded
(Tx, language Lx) one
a
on
to
two-sorted
passes
theory T2 retaining Tx at one level and
linear Time at the other discrete
unbounded
the theory of
having
the two sorts.
between
'bridge principles'
together with some obvious
concern
matter
the
Further
model-theoretic
just
questions
might
in terms of the 'lower
definable
when
the 'upper sort' is explicitly
too aristocratic
for that.)
it is clearly
sort'. (In the present
example,
4.
PHILOSOPHICAL
AFTERMATH
not only in its background
This paper has been extremely
general,
but
in
its technical
considerations
also
(section
1),
development
as
as
to follow).
the
well
2, 3;
(sections
Nevertheless,
appendix
of more
is not advocated
here - to the
production
logical generalities
author has taken a vow that this paper has
(The present
contrary.
the observation
Recall
been his last sin against his own precepts.)
made
in section
one wants
1.2:
should be guided by specific
logical research
to obtain. Now,
it has often been noted that both
from its
the internal dynamic of a theoretical
subject and the demands
source
set
of inspiration
such goals. The
latter
may
applications
at the present
deserves
the main
also
emphasis
juncture: witness
in Suppes
editorial
exhortation
1973 to carry the banners
Suppes'
to the logical study of Space, Time and Kinema
from mathematics
route
is not the same as a Cause, certainly;
A
of
march
tics.
but, it is
results
THE
STUDY
LOGICAL
457
SCIENCE
of a general Program. The remainder
to some feelings
about the scope of
all we have to propose by way
of this section will be devoted
such an undertaking.
4.1 What
OF
Is
'Application'?
of the a priori
1.1 it was argued, as against proponents
out
to
far logic will
how
find
the
that
of
only way
logic,
inadequacy
to avoid
us
in
order
to
and
is
of
science
in
the
take
go
try. But,
study
some
meant
is
what
about
by
unnecessary
disillusions,
prior thought
its frequent
of logic seems appropriate.
Quite generally,
'application'
at the academic
use in scientific
talk - and its usefulness
money
- should not hide the fact that
is a very diffuse term, in
tap
'application'
For the latter purpose,
indeed, a separate
urgent need of clarification.
In section
paper (or book)
will be made.
would
be
required:
here
only
a few
relevant
points
scientific
theory means
logic' in the study of a certain
'Applying
a
statement.
un-informative
but
For, these
neat,
using logical tools:
mere
to
notions
and
theorems
from
methods
'tools' may be anything
even
esoteric
or notations.
'tool'
would
be
that
the
Sometimes,
only
else
everything
(though real) quality called
'logical sophistication'
not
in
is
This
situation
different
work.
hard
prin
being supplied by
are
some engineers
ciple from that of any formal discipline. Maybe
- but
as
a
wand
kind
of
to
to
able
magic
problems
'apply' mathematics
on the
of new mathematics
most
require the creation
applications
spot.
to imply that
should not be misunderstood
The preceding
paragraph
if
matters.
For
some vague
is
of
all
that
view'
instance,
'logical point
one merely
then what
in the study of science,
used logical notions
sauce
would
be the difference
say, the information-theoretic
with,
so
infant
has
which
many
'receiver',
'channel')
poisoned
('sender',
labels for analyses?
in the cradle by substituting
(Cf. the
note
a
to
in
renascent
referred
Aristoteleanism
of
6.) It was
danger
of
1.2
1.1
that
and
the
of
sections
contention
application
precisely
not
notions
here:
should
of
science
in
the
stop
logic
philosophy
or
theorems
should be tied up with
existing,
(already
'regulative'
created especially
for the purpose). And - an esoteric
logical sophisti
these requires a very down-to-earth
cation able to produce
training in
sciences
technical
Thus,
logic....
the question
arises what
it means
to apply
a logical
theorem.
458
JOHAN
VAN
BENTHEM
traffic lights'.
Very often, these will serve as no more than 'methodical
us
are
tells
too weak
to
recursion
that
certain
grammars
E.g.,
theory
are
whereas
others
worth
the
generate
languages,
given
trying. (Cf.
famous
in Chomsky
Beth's
1957.) Similarly,
ascending
hierarchy
us that a certain
Theorem
warned
formulation
in
of constraints
to Reductionism.
section
3.1 paved
the way
Notice
the familiar
here: failure is the only definitive
form of
Popperian
point exemplified
Positive
results may
be less informative,
because
the
success.35
reason
for their success
remains
note
unclear.
27
about
the
(Cf.
success of 'classical' calculi of deduction.)
It may be of interest to observe how critics of modern
logic fail to
this
the
influential
1976
Perelman
acknowledge
point. E.g.,
polemic
the fundamental
establishes
of formal logic in the study
inadequacy
of juridical reasoning by means of examples
like the following.
In the
were
French
tradition
after
the
Code
legal
Napol?on
judges
required
on the presuppositions
to give their verdicts
that the Law contained
neither
'conflicts' nor 'gaps', while a decision
could be reached
in a
simple methodical
objective
was
behind
this strictness
motive
way. (By the way, the honourable
to minimize
thus
juridical arbitrariness;
out
this
ideal
turned
untenable
the
citizens.
Nevertheless,
protecting
in practice:
there remained an irreducible component
of interpretation
as for the methodological
on the part of judges.) Now,
of
background
this ideal, Perelman
remarks (correctly)
to requiring
that it amounts
to be a consistent,
the Law
and decidable
complete
theory: a rare
species
even
in mathematics,
and
so...
one
more
example
of
the
of logic. In the light of the preceding
inadequacy
paragraph, however,
an application
of logic has been given, demonstrating
the inherent
room for
of certain
limitations
of
thus
Law;
conceptions
creating
more
or
con
methodical
sophisticated
conceptions,
supplementary
of content
(rather than mere form).
reason why
results may
be of doubtful
ap
positive
their
nature.
is
plicability
(necessarily)
general
E.g., having a general
from practice - which
results
recursion
theory requires some distance
in paradigms
of 'constructivity'
like primitive
recursive
functions
siderations
Another
easily becoming
Many
physically
non-computable.
logical existence
as
theorems are proven by means of quite un-realistic
'constructions',
was noted already
in connection
note
with Craig's Theorem
(cf.
15).
in such proofs) may be largely
Thus,
(as embodied
logical methods
inspirational.
THE
LOGICAL
OF
STUDY
459
SCIENCE
an optimistic
search
In the present author's
however,
experience,
in
will usually
be a successful
instances
for manageable
strategy
cases. Reality may be harsh, but she is not mean.
concrete
4.2
In Praise
of Formalism
will be converted
devoid of 'logical sensitivity'
by apologies
Nobody
never make
one
them to un
should
for formal methods.
Thus,
so
to
do
to
one
in order
has
audiences.
(Unless
sympathetic
make a living, of course.) As one need not make them to sympathetic
audiences,
a
rather
silent
conclusion
seems
to
follow
....
a few virtues of 'the formal
it may concern,
Still, to whomever
are
will
which
be
extolled
here,
praised too seldom in the
approach'
or defenders).
I do
cultural
climate
(whether by opponents
prevailing
to be a road to instant
not take 'formal' or 'exact' philosophy
a means
for
It is, if anything,
and rock-bottom
insights.
rationality
afloat on Neurath's
ocean), gauging the
re-fitting our raft (precariously
our
the
underneath.
of
intellectual
Accordingly,
ignorance
depth
of
work
choice for this type ?f
(instead
all-embracing
philosophies)
rather than
has always seemed to me a matter of intellectual
honesty
poverty.
is a virtue, of course, but not a particularly
exciting one.
Honesty
Let me, therefore,
be more explicit about the merits of being formal
and precise - or, better, about making
things a little more formal and
us
one
to
become
For
clearer about cherished
it
forces
thing,
precise.
intuitions - teaching us the invaluable art of being wrong.
(Saying that
means
and
the world
is an 'Organic Whole'
learning, nothing:
risking,
is a heroic and instructive
saying that it is a Finite State Machine
as we unravel our concepts,
is
their real wealth
mistake.) Moreover,
are
Hence
and
hitherto
unveiled,
opened up.
possibilities
unexplored
- as has been
to creative
is a stimulus
formal precision
phantasy
of
'creative
stressed
in Piaget
1973. The
fashionable
opposition
freedom'
and 'logical armour' does not do justice to logic (nor, one
fears, to creativity).
I take to be of
a task of logic emerges which
is also where
Here
a
to the philosophy
It should provide
of science.
vital importance
are
out
under
tested
ideal
cir
where
ideas
'conceptual
laboratory'
cumstances.
too, there should be a 'conceptual
Very
importantly
sanctuary'
(or 'mental
asylum'?),
where
old discarded
scientific
ideas
460
JOHAN
VAN
BENTHEM
are kept alive36 - like strains in a pollen bank for grain, which may be
needed again at some future time. Hopefully,
in this way,
logic could
a
in
the
the worlds
deal
between
gap
great
closing
self-imposed
help
sense and science from which our culture is suffering.
of common
5.
TECHNICAL
APPENDIX
text of this paper
is rather sketchy,
main
detailed
technical
In order to establish at least some
arguments
having been suppressed.
here are some samples
of more
technical work
logical credentials,
around our general theme.
The
I. Invariance
From
a Model-Theoretic
A typical model-theoretic
a certain
starts studying
Perspective
line of research
might be the following. One
the
discrete
two-dimensional
domain,
say
a certain
lattice ZxZ;
endowed
with
structure,
say the ternary
comes
a
relation
'further
than'.
First
structural
(from)
question.
Which
of the lattice are automorphisms
with respect to this
bijections
structure?
the group of
(The answer would be, in this case, precisely
rotations
and
These
translations,
reflections.)
automorphisms
give rise
on
to invariants:
are
relations
the
domain
onto
which
n-ary
mapped
themselves
all
in
the
remarks
group. (Cf. previous
by
automorphisms
about Klein's
These
invariants
form a very
Program.)
interesting
or
closed
under
like
intersection
class, being
operations
complement,
Could,
then, all invariants be characterized
projection.
linguistically
some
at least, the original relation is
suitable
using
language? Clearly,
an invariant,
are predicate-logically
together with all relations which
definable from it. (Cf. the earlier quotation
from Weyl
the
1949.) Does
converse
is a difficult question,
hold too? This
for
'internal'
calling
to some mother
results (relativized
rather than
definability
structure),
'external' ones like Keisler's
of elementary
characterization
classes.
Here
come
a few pertinent
results.
(Cf. also Rantala
1977a.)
1. In finite structures,
PROPOSITION
invariance
for automorphisms
first-order
implies
definability.
some invariant.
Proof: Let 3) be some finite structure, and AQDn
a
A direct combinatorial
first-order
definition.
argument will produce
Here
we
just
observe,
however,
that
this
also
follows
from
pro
THE
position
3 to be proven
(L-I-A)-saturated.
STUDY
LOGICAL
if we
below,
Thus,
let
x
be
any
OF
461
SCIENCE
can only
show
of,
sequence
say,
that
(3, A)
is
s variables,
some set of (L + A)-formulas
and 2 = 2(x)
finitely many
involving
in 3. Saturation
satisfiable
D
is finitely
in
which
requires
parameters
it were not. Then,
for
satisfiable.
that 2 be simultaneously
Suppose
each s-tuple d in Ds, there exists some formula cr? in 2 such that o\j is
it would
such s-tuples,
false at d. But, there being only finitely many
in 3 : contradic
follow that some finite subset of 2 is non-satisfiable
on 2. Q.E.D.
ting the original assumption
1 may
Counter
fail, however.
proposition
infinite structures,
natural
of
the
structure
numbers
In
the
consisting
IN0Z
example:
IN is
a
in
the
usual
of
the
followed
ordering,
copy
by
integers,
in
definable
first-order
without
invariant for automorphisms,
being
terms of the ordering relation.
results arise only when single structures are
Often, model-theoretic
let T be some first-order theory in a
theories.
their
Thus,
replaced by
some
A
is
where
Indeed, a
n-ary relation symbol.
language L-l-A,
In
duality
between
invariance
and definability
2. A2 is an L-automorphic
PROPOSITION
definable
for T if and only if A is explicitly
if
to
left, clearly,
Proof: From right
ThVx(Ax~?i((x))
is now
forthcoming:
invariant in each model
in T up to disjunction.
v ... v Vx(Ax^?m(x))
where 8\,...,
then, in each model for T, A will be
8m are L-formulas
some
it will be invariant in the above
first-order defined by
6?.Hence,
sense.
Theorem may be applied
Svenonius'
for the converse,
Secondly,
Let
3 be an L-structure
theorem
and
Keisler
1973;
5.3.3).
(Chang
for T.
with (L + A)-isomorphic
(3, A), (3), A') to models
expansions
=
+
A'
A
of
then
being
A,
((L A)-automorphisms
By the invariance
this implies
theorem
By the above-mentioned
L-automorphisms).
explicit
definability
up to disjunction.
Q.E.D.
like the
results to 'internal' problems
such changed
to
be
shown
have
one, it would
that, e.g., any
original geometrical
=
Th((Z x Z, A)) in
given invariant A in Z x Z gives rise to a theory T
In order
to apply
462
JOHAN
BENTHEM
VAN
the interpretation
of A is an invariant. We do
each of whose models
not solve this problem here.
invariance
inside which
there does exist a kind of structure
Finally,
- an observation
further
ado
which
without
implies explicit definability
will
below:
be used
invariant in an (L + A)
3. If A is an L-automorphic
PROPOSITION
in 3.
saturated structure 3, then it is first-order L-definable
an
relation.
A
be
Consider
this
Let
any n-ary
n-ary
Proof:
a of objects
of a in 3 cannot be
the L-type
in A. Now,
sequence
of the
satisfied outside of A. Otherwise,
(a consequence
L-homogeneity
not
of
3
an
would
saturation
L-automorphism
provide
property)
given
+
the
of
the
Put
onto
A
itself.
(L A)-set consisting
differently,
mapping
L-type of a together with the formula "~1Ax is not satisfiable in 3. Hence
- 3
some finite
subset of it will not be satisfiable
(L + A)
being
saturated.
Say
...,ts(x),
{ti(x),
is not
(1)
where
in 3
satisfiable
~1Ax}
;where
... &
ti &
ts.
since each a in A will
Moreover,
(L + A)-set
,ts are true of a. Equivalently,
is true in 3;
Vx(T?(x)-?Ax)
Ta=def
tx, ...
satisfy
some
such L-formula
r?, the
{-\Ta\aG A}U{Ax}
is not
finitely
many
t =
either.
ax,...,
sequences
-> t(x))
Vx(Ax
(2)
where
in 3
satisfiable
def t?,
In combination
v
. .. v
with
Vx(t(x)<-?Ax)
A has been L-defined.
II. The Categorial
Mode
there
Hence,
again by saturation,
at in A such that
is true in 3
exist
;
r?t.
(1), it then follows
that
is true in 3:
Q.E.D.
of Thinking
in connection
mentioned
The categorial
perspective
a
to
characterization
constraints
may be used
provide
with
Sneed's
of first-order
THE
LOGICAL
STUDY
OF
SCIENCE
463
inside a structure. To see this, let 3 be some L-structure,
definability
A some n-ary invariant in it. Now,
3 belongs
to the category CAT
all
of
with
(L)
L-structures,
(not neces
L-isomorphic
embeddings
its
for
and
the
construction
sarily surjective)
morphisms,
ultraproduct
as a characteristic
I do not know of any
operation.
(Unfortunately,
categorial
definition
behaviour.)
of ultraproducts
in terms of their
'morphological'
4. A is first-order
PROPOSITION
in 3
if and only if
L-definable
is simultaneously
a
to
of CAT
expandable
sub-category
such
that
all
and
of
(3, A)
containing
morphisms
operations
same
+
the
in
remain
CAT(L)
CAT(L
A).
CAT(L)
(L + A)
=
is first-order L-definable,
say A
<p?, then the obvious
of each L-structure
3' to (3', ??') will do. For, L-isomor
expansion
remain
(L + A)-isomorphisms
phisms
preser
by their fundamental
vation
and Los'
Theorem
the same
for
property,
guarantees
Proof:
If A
ultraproducts.
that CAT(L)
has been expanded
as indicated.
suppose
Conversely,
Let U be any countably
over some
ultrafilter
incomplete
a-good
=
=
a
index set I; where
the
|i| max(^V0,
|D|) and
|I|+. Consider
- which
11^ 3 in CAT(L + A)
expanded
ultrapower
equals Uv(3, A),
Notice
that this ultraproduct
is (L + A)-saturated;
by assumption.
by
AUu? is invariant in
1973, theorem 6.1.8. Moreover,
Chang and Keisler
the ultraproduct:
of this structure being automa
L-automorphisms
con
The promised
(L -I-A)-automorphisms,
tically
by assumption.
clusion then follows at once from proposition
3. Q.E.D.
this line of thought will yield more
Hopefully,
tion in the first draft of this paper was verified
III. Varieties
The
results. (This specula
in Pearce
1980B.)
of Reduction
of 'reduction' has been shown to admit of various
logical
in section 3.2. First, a syntactic
result of that section will
explications
be proven here.
Let Tx (T2) be a first-order
t
theory in language Lx (L2). Translations
from Lx to L2 will assign, possibly
to
complex,
Lr
L2-predicates
- those
in Lx n L2 being mapped
primitives
(Thus, one
identically.
the idea that a partial correspondence
is given
in ad
implements
notion
464
JOHAN
VAN
BENTHEM
of Tx in T2 means
that some translation
vanee.)
'Interpretation'
r(Tx)
be
in
to a unary L2-predicate
relativized
may
proven
T2, possibly
some sub-domain.
a
Since Tx may be infinitely axiomatized,
denoting
theorem'
would
be
was
here.
in fact, ?s
'compactness
But,
helpful
stated in note 31, no such result holds in general.
(Cf. Lindstr?m
1979,
however.) What we do have is the following:
PROPOSITION
5. If each finite subset of Tx is interpretable
in T2,
then the whole
in some conservative
exten
theory Tx is interpretable
sion of T2.
in either
Proof: Take a new unary predicate constant U not occurring
consider T2 U T V (where all quantifiers
in Tx
Li or L2. Now,
occurring
are relativized
to U). Tx is trivially interpretable
in this new theory. Thus,
to show that the latter is a conservative
it only matters
extension
of T2.
To see this, let <pbe any L2-sentence
such that T2U T^Vy. It follows
that,
for
ax,...,
many
finitely
ak E
T2 U
Tx,
{aV,
, ?u
}h<p.
of these formulas,
taking a to be the conjunction
nor
since
neither
U
the
of a
Then,
(Lx
L2)-vocabulary
T2\-au^xp.
occur
in T2, it follows
for the universal
second-order
closure
u ->
that T2hV(a u-?(p).
V(a
<p) taken with respect to U and Lx-L2on Tx and T2, there exists some translation
Next, by the assumption
t of Lx into L2, as well as some unary L2-predicate
? such that
From
the previous
that
follows
it then
T2\-(r(a))?.
paragraph,
Equivalently,
->
T2\-(j(a)?
<p, and
hence
T2h<p.
Q.E.D.
In section 3.2 the following model-theoretic
theories Tx:
stated, for finitely axiomatized
interpretation
result was
6. Tx is interpretable
in T2 up to disjunction
if and
an L2-definable
for T2 contains
sub-domain
which
is a model
for Tx.
together with L2-definable
predicates
to
From
left
if Tx is not
Proof:
right, this is obvious.
Conversely,
as indicated,
then T2 united with the set of all negations
interpretable
of L2-definable
translations
of Tx (relativized
to all possible
L2
definable
is a finitely
set of formulas.
satisfiable
domains)
By com
In other words, T2
satisfiable.
then, it will be simultaneously
pactness,
a model without
submodel
for Tx. Q.E.D.
possesses
any definable
PROPOSITION
if each
only
Next,
the
model
'syntactic
trick' mentioned
in section
3.2 works
as exem
THE
LOGICAL
by the following
plified
special
a 3x1/0
T2Htx(Tx)u>
the first disjunct
Call
U(x)
Notice
=
OF
STUDY
case.
465
SCIENCE
Suppose
that
v (T2(TX)V> a 3x17z).
a, and set
dei(Ux(x) a a) v (U2(x)
a la).
for arbitrary
Then,
T2h3xU(x).
= def
a a) v (t2(P)(x)
(ti(P)(x)
T(P)(X)
that
P,
Li-predicates
set
a la).
that T2\-r(Tx)u.
A simple argument about models
of T2 establishes
'cross-structural'
conditions
Further
refinements,
using
categorial
as in II above, yield a duality result for interpretability
'tout court'.
Instead of embarking upon this course, however, we prefer to review
the whole
topic
from a different
angle,
namely
that of combination
of
theories.
One has
theories
Tx (in Lx) and T2 (in L2),
got two first-order
that both
Lx n L2. Let us assume
possibly
sharing some vocabulary
are intended to describe
form
the same kind of objects. The weakest
seem to be:
of combination
would
(1)
the deductive
union
of Tx and T2 is consistent
in Lx + L2.
Theorem
Joint Consistency
tells us that this occurs
just
n
\
when Tx Lx
L2 and T2\ LXC\L2 contain no mutually
contradictory
that Tx U T2 need not be a conservative
theorems. Notice
extension
of
either Tx or T2: both theories may have learnt in the process.
Some
reflection upon Robinson's
proof shows that, e.g., Tx U T2 will be a
of T2 if and only if T? ?Lx n L2 is contained
in
conservative
extension
Robinson's
T2\ LXD L2.
from left to right is obvious. Conversely,
for any
(For, the direction
if
then T2U{"l<p} has got a model
3. By the
L2-sentence
<p, Ttf?,
of 3 united with Tx will be
then, the (Lx n L2)-theory
assumption,
- and hence it has a model 3'.
satisfiable
finitely
Starting from 3 and
one
builds
unions may be
3',
chains, whose
elementary
alternating
folded
into an (Lx + L2)-structure
Tx U T2 while
together
verifying
and
Keisler
<p.Cf. Chang
1973.)
falsifying
Next,
addition,
(2)
connections
stronger
such as:
Tx U T2 is contained
between
in some
Tx and T2 may
definitional
arise
extension
upon
of T2.
466
JOHAN
VAN
BENTHEM
?
of (Lx L2)-vocabulary,
I.e., given certain L2-definitions
Tx becomes
from T2. This
is the earlier definition
derivable
of 'interpretation'
of possible
to sub-domains,
relativization
leaving out the complication
that is. In this case, automatically
extension
Tx U T2 is a conservative
of T2 (cf. the proof of proposition
5 above),
of
though not necessarily
that
Tx. The latter need not even happen on the stronger connection
occurs:
'definitional
reducibility'
Tx U T2 coincides
(3)
Contrary
Extension
theories
with
some
definitional
extension
of T2.
to our experience
with the mis-match
between Conservative
and Ramsey
the latter connection
between
Extension,
admits of an elegant structural characterization:
7. Tx is definitionally
PROPOSITION
to T2 if and only if
reducible
each model
for T2 admits of exactly one expansion
to a model
for Tx.
to T2 - say,
that Tx is definitionally
reducible
Proof: First, suppose
8 as above. Now,
Tx U T2 is axiomatizable
by T2 plus L2-definitions
consider any model 3 for T2. By interpreting
the (Lx L2)-vocabulary
an
+
to
3+ which
satisfies
through 8, 3 is expanded
(Lx L2)-structure
for Tx:
(T2 plus 8 and hence) Tx U T2. A fortiori, then, 2>+ is a model
as for 'at most'.
and the 'at least' has been taken care of. Next,
If 3+,
3+f are any two expansions
of 3 to models
for Tx, then (Tx U T2, and
the respective
of
hence) 8 will hold in both. Therefore,
interpretations
the
-
(Li
L2)-vocabulary
must
coincide:
i.e.,
3+=
3*'.
that the above structural condition
holds.
Its
suppose
Conversely,
'at most'
side means
that the implicit definability
clause of Beth's
Theorem
is satisfied - and hence explicit L2-definitions
8 of the (Lx
are
U
in
derivable
It
follows
that
Tx
T2.
Tx U T2 may be
L2)-vocabulary
as
of
its
the
U
?
union
(re-)axiomatized
L2-part (Tx T2) L2 with 8. For,
both of these are derivable
from Tx U T2. Moreover,
vice
clearly,
if
U
U
where
versa,
Tx
L2)
'?(Li)'
T2f-<p(L,, L2), then T,
T2\-?(8(Lx),
refers to suitable L2-replacements
the L2-formula
through 8-, whence
to (Tx U T2) \ L2. In conjunction
with 8, then, the
?(8(Lx), L2) belongs
latter theory reproduces
the original formula
<p(Lx,L2).
to
in
order
of Tx to T2, it now
extension
Thus,
prove definitional
to
suffices
show that (Tx U T2) ?L2 coincides
with T2. Put differently,
it remains to be shown that Tx U T2 is a conservative
extension
of T2.
But this follows
from the 'at least' side of the above
structural
condition.
such that T2?<p, then T2 U {l<p}
For, if <p is any L2-formula
THE
LOGICAL
STUDY
OF
SCIENCE
a model:
can be expanded
have
which
U
Ti U T2 U {l<p} whence
Tx
T2V(p. Q.E.D.
to
will
were
These
notions
pleasing
the theories
-
but,
rather
467
some
un-realistic.
to be added will
refer
for
model
in actual
For,
to different
(cf. Tarski
kinds
practice,
usually
of objects.
Consider,
e.g., Elementary
1959)
Geometry
with
'betweenness'
and (quaternary)
(ternary)
'equidis
primitives
=
tance' for Tx, and the algebraic
theory of the reals (IR
(R, 0,1, +, ))
for T2. As was noted already in section 3.2, a two-sorted
combination
the main
is more appropriate
sortal
connections
becoming
possible
issue. E.g., in this case, equidistance
induces an equivalence
relation
a
between
of
for
notion
of
points
allowing
'length'
pairs
bridge
certain bridge principles
for 'addition'
satisfying
(etc.). The relation
between
such more complex
notions
of reduction
and the
syntactic
theorems will not be investigated
usual representation
here.
Rijksuniversiteit
Groningen
NOTES
*
1
I would
like to thank David
Pearce
Rantala
and Veikko
comments.
for their helpful
to this rule is V. Rantala's
and Non-Standard
paper
'Correspondence
exception
As
for
366-378.
A Case
in Acta
Fennica
30 (1979),
Study',
Philosophica
a variety
of publications
there
is, of course,
by Robinson
'ordinary'
applications,
An
Models:
himself
and co-workers.
2
to justify
their
have used similar excuses
defeatist
(or elitist)
Regrettably,
logicians
exclusive
concentration
upon pure mathematics.
3
in
text book version
of logic. E.g.,
What
is usually
is some static elementary
criticized
the otherwise
work
London,
very
(Batsford,
of Reasoning
interesting
Psychology
to show
that
and P.N.
Johnson-Laird
1972), P.C. Wason
example
give the following
two sentences
'if prices
'actual
transcends
formal
The
any
modelling.'
reasoning
increase
the firm goes bankrupt'
and 'prices
increase,
only if the firm goes bankrupt'
are stated
to be 'logically
the same
'i->B\
in
equivalent',
having
But,
logical form
feel a difference,
due to additional
and so ....
people
practice,
temporal or causal content,
seem unaware
The authors
of the fact that observations
like these are precisely
at the basis
of modern
B'
and
'I
?
logical
semantics.
E.g.,
in tense
past B\
respectively.
And that without
red carpets.
expecting
such philosophers
to regard
of science
set theory,
many:
game theory,
topology,
5
This
is Popper's
main
insight
applied
logic,
There
logic
the two logical
seems
as
just
forms would
to be a growing
one auxiliary
system
theory,
to the philosophy
be
tendency
discipline
etc.
of
science
'I -? future
itself.
among
among
JOHAN VAN
468
BENTHEM
6
one sometimes
In this connection,
is in the air,
fears
that a New
Aristoteleanism
with understanding
them.
confusing
labeling phenomena
7
It almost
that her interest will
also cover
the most
diverse
goes without
saying
of reasoning:
of information,
extraction
refu
rationalization,
purposes
justification,
or
link with Hempelian
In particular,
the latter two form an obvious
tation, explanation.
Popperian
8
Notice
of science.
philosophy
on this view,
there is a continuous
to that of scientific
theories.
language
how,
(natural)
9
Yet more
have
entities
complex
(Lakatos).
programs'
even
But,
been
then,
proposed,
the level of
spectrum
from
the
logical
like 'theory nets' (Sneed) or
a convenient
theories
remains
study
of
'research
starting
point.
10
Speaking
this view may have been
that only
doctrine
historically,
inspired by Kant's
to (possible)
which
is related
is free from contradiction,
whereas
knowledge
experience
runs the constant
risk of antinomies.
'pure reason'
11
one argue
to the opposite
Or should
effect:
non-constructive
for
logic is harmless
one has
once
non -constructive
to be more
domains:
careful
constructively
given
domains
are considered
(the
intuitionist
position)?
As
so often,
metaphors
either
point
way.
12
There
If T\
is axiomatized
be a problem
here.
of universal
might
by means
to use universal
statements
and one is only allowed
in proofs,
could not it
statements,
be that certain universal
theorems
become
their predicate-logical
because
unprovable:
intermediate
the
steps of higher
requires
quantifier
complexity?
Fortunately,
proof
answer
as may be seen, for instance,
is negative:
semantic
tableaus.
by using
13
case is, e.g., Goldbach's
to the effect
An interesting
that each
border-line
Conjecture
even natural number
do not know any
than two is the sum of two primes. We
greater
n equals
such that each even
functions
number
only primes
explicit
/, g yielding
there are (trivially)
recursive
functions
But,
/(n) + g(n).
/, g such that, if n can be
as a sum
written
make
of
two
Goldbach's
Conjecture
recursive
Skolem
primitive
14
A relevant
such
Or
a sum. Does
should
one
this
require
In the real numbers,
it is
is the following.
(A.S. Troelstra)
+ x = 0 has solutions
that the equation
for each value of
provable
y3-3y
x. But, there is no continuous
function
such that
/: IR-?IR
example
constructively
the parameter
Vx(/x)3-3(/x)
(A 'cusp catastrophe'
15
here
By the way,
can
at all, then f(n) + g(n) will be
primes
an acceptable
'concrete'
statement?
functions?
+ x=0.
occurs.)
is an instructive
to the claim
that philosophers
counter-example
a knowledge
of just logical results. For, as it happens,
Craig's method
a 'tricky' set of axioms
for the observational
without
any
produces
sub-theory,
use. Thus,
a
it is good to know
that a 'reduction'
in principle,
exists
afthough
get by with
of proof
practical
lot of work
remains
16
This contribution
to be done to produce
and interesting
convincing
examples.
in the philosophy
exhausts
of
by no means
Suppes'
important work
To mention
his papers on measurement
science.
just one other example,
theory have a
terms.
clear interest
for the semantics
of 'observational'
17
is negative,
Cf. Przet?cki
1969.
The answer
maybe
surprisingly.
18
Notice
that, in this case, T\-<p if and only if T0h<p, for each L0-sentence
<p (T0 being
THE
Hence,
complete).
be
and (3) would
LOGICAL
STUDY
also
the example
a syntactic
to
at least, it would
be a decidable
matter,
This
eliminable.
is Ramsey
conjecture
1978.
in van Benthem
however,
20
As for C2, it would
have to be decomposed
U y)
Vxy m(x
=
469
SCIENCE
of (1)
that the conjunction
the conjecture
the structural
condition
that MOD(To)
refutes
counterpart
\ L0.
equals MOD(T)
19
was
it
that,
hoped
Originally,
T, if its theoretical
vocabulary
OF
given
was
a theory
refuted,
into a law
4- m(y)
m(x)
(with 'U! denoting
physical
join) plus the identity constraint
21
from a quite
of this type of requirement
For an example
Kreowski
and P. Padawitz:
H.-J.
'Stepwise
Specification
Cl.
different
and
area,
cf. H. Ehrig,
of
Implementation
Lecture
Notes
5 (1978),
in Proceedings
in
Data
ICALP
Springer
Types',
are algebraic
the morphisms
Science
62, 205-226. Here
homomorphisms.
Computer
22
a common
and Di will have
in both D\
and Di. Now,
Let p occur
D\
empirical
at the theoretical
D containing
this relation must be preserved
p. Since
superstructure
Abstract
level,
D2.
23
p will
receive
would
One
also
of
semantics
the same
theoretical
function
like to see
the Kripke
structures
of modern
'empirical
structures'
theories.
scientific
(The
values
in the expansions
tense
of
logic
recent
of D, D\
applied
work
and
in the
by
R.
are of
this kind.)
discussions
parallel
W?jcicki
24
There
are
the status
of natural
laws:
linguistic
concerning
or structural
Cf. Suppe
1976.
relations'?
'sequencing
expressions
25
on Sneed's
is intelligible
set-theoretic
constraints
the 'subset condition'
(and
E.g.,
a universal
even
formulation.
has
('had'?)
correct)
explicit
only
if the constraint
occur
in at least one
like 'any two particles
'Universal-existential'
constraints
together
situation'
empirical
Another
finitely
case
many
undefinability
the ultimate
for us
the condition.
satisfy
with
in equilibrium
in Sneed
1971 of a balance
is the example
on
will
take
the predicate-logical
it. Some
structuralists
objects
to be
of this simple
of finiteness
situation)
type of empirical
(and hence
verdict
is force
do not
in point
one
do
such an analysis would
But, what
case. E.g., a
'finite' in this particular
on the
a recursion
would
involve
example
formalisation.
upon first-order
to think about the cash value
of the above
formalisation
simple first-order
in a natural
number
of objects,
corresponding,
on the balance.
weights
of
fashion,
to progressive
addition
of
that the incorporation
into
in Pearce
of linguistic
1980B it is argued
aspects
Finally,
it 'still more
and comprehensive'.
would make
the Sneedian
frame-work
expressive
26
one to classical
Cf. Vaught
It need not even commit
semantics.
logic on the Tarski
true in all countable
structures
with
the class
of formulas
1960, where
classically
to be non-recursively
is shown
axiomatizable.
decidable
predicates
27
the preferred
role of classical
strength
logic may be due largely to its excessive
E.g.,
a way from premises
to conclusions
that
ignore the possibility
blasting
making
people
for
the same theory might
have been organized
allowing
logical means,
using weaker
subtler
metic,
as a
distinctions
which
enables
(mathematical)
to be made.
one
One
to derive
theorem.
A
traditional
the Law
recent
example
of Excluded
exciting
example
Arith
Heyting
statements
for atomic
is intuitionist
Middle
is the proof
in M.
Dunn:
JOHAN VAN
470
Mathematics'
'Quantum
Arithmetic
axiomatized
laws as mathematical
28
like to see at least
One would
theme.
29
Although
of
(Department
with
quantum
theorems.
one
BENTHEM
Philosophy,
logic
produces
text book
... would
you board a rocket
in the theory of differential
inconsistency
30
of Chang
The organisational
principle
to Mars,
Bloomington,
all additional
of model
theory
immediately
1980) that Peano
classical
logical
organized
after
around
the publication
this
of an
equations?
and Keisler
1973, viz. that of various methods
seems
less suitable
from the present
construction,
(cf. note 28).
perspective
a feeling
awareness
for the ontological
of such
Still,
principles'
gives
'proliferation
universe.
of the structural
abundance
31
is necessary:
without
The
latter addition
be given. E.g.,
it, counter-examples
may
=
take for T2 the first-order
theory of (N, O, S) and for T, :T2 plus {c# s'0, i 0, 1,2,...}.
is not as trivial as it seems at first sight.)
calculation
(The required
32
with
To get better acquainted
these various
notice
that Z is isomorphically
notions,
in Q, as well as being
to one of its quotients
all 'pins'
embeddable
(contract
isomorphic
of model
(n, n +
1] for
reducibility
of Q- and
converses
fail. As
for
n). Both
integer
to be introduced
Z is not isomorphic
below,
the converse
the model-theoretic
to any
definable
notion
of
substructure
as well.
fails
as translations
ti, ..., Tk of the
,Uk as well
U\,...
a 3xUi)
into
such
that
L,
L2
non-logical
T2(-(ti(Ti)Ui
v ... v (Tk(Tx)Uk a 3xUk).
34
One might
dualities
embed
try to restore
symmetry
by formulating
concerning
rather than interpretability.
'for each Li-sentence
if
<p,Tih<p if and only
dability
E.g.,
on MOD(Ti),
to: [some symmetric
is equivalent
condition
t, U]\
MOD(T2),
T2Kt(<p))1/,
35
once remarked
to his political
As the famous Dutch
Calvinist
prime minister
Colijn
'In negation
lies our strength'.
opponents:
36
A prime example
is the Leibnizian
view of Space
and Time;
cf. Suppes
'relational'
1973.
exist
there
33I.e.,
L2-formulas
constants
in
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