Popper - Two Autonomous Axiom Systems for the Calculus of Probabilities, STUDIA, Filozofia nauki, Filozofia Nauki
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The British Society for the Philosophy of Science
Two Autonomous Axiom Systems for the Calculus of Probabilities
Author(s): Karl R. Popper
Reviewed work(s):
Source: The British Journal for the Philosophy of Science, Vol. 6, No. 21 (May, 1955), pp. 51-
57
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 NOTES AND COMMENTS
Two AutonomousAxiom
Systems
for
the Calculus
of
Probabilities
I
InformalExplanations
I
sHALL
introduce,
in this
paper,
two ideas. The first is that of absolute
probability:
p(x)
= r
(A)
which
may
be read 'the
(absolute)probability
of x
equals
the real number
r .
The second is that of relative
probability
:
p(x, y)=r (R)
which
may
be
read'
the
probability
of x
given y equals
the realnumber r'.
Both
p(x)
and
p(x, y)
are numerical functions of non-numerical
variablesor
arguments.
The
argumentsmay
be
interpreted
in various
ways.
Thus 'x' and
'y' may
be
interpreted
as
(variable)
names: for
example,
names of statements,or of classes,or of
predicates,
or of events.
My
intention is to construct
a
system
that is
formal
in the sense that its
interpretation
is
left open.
This does not mean that in
constructing
the
system
I have no
interpretations
in mind-on the
contrary,
I have a con-
siderablenumber of
diferentinterpretations
in mind.
Moreover,
the
system
is autonomous
in the sensethat it is
designed
in such
a
way
that it allows the calculationof
probabilities,
but does not assumea
calculusfor
operations
with the
arguments.
What is meant
by
this
may
be
explained
as follows.
Most writers on the
subject
assume,
before
proceeding
to
give
their
axioms for
probability,
that certainlaws are valid for their
arguments,
for
example,
the commutative and associativelaws
:
=
Xy
yX
(I.I)
(xy)z
=
(1.2)
X(yz)
Assuming substitutivity
of identical
terms,
they
derive from these
p(xy)
=
p(yx),
(I.la)
p((xy)Z)
=
p(x(yZ)), (I.2a)
and similar
equations
for relative
probabilities.
It is
usually
assumed that
all the rules of Boolean
algebra
hold for the
arguments,
or that the rules
of the calculus of
propositions
hold for them.
Since I am interestedin
constructingsystems
in whichnot moreis assumed
thana bare
minimum
necessary
I shall not make
for
the calculus
of probabilities,
any
such
assumption
as
I.I
and 1.2 but shall instead use axioms of the
5S
KARL
R. POPPER
form
I.I
and
1.2a.
These are
very
much
weaker;
for from
I.I
and
I.2
we can
obtain,
as
indicated,
I.Ia
and
1.2"
by
substitution,but not the other
way
round. A
system
in which
only
formulae such as
I.Ia
and
1.2
are
used and
no
formulae
for the
arguments
alone, I
call an 'autonomous
system
for the calculusof
probabilities
'.
Some authors believe that
only
the idea of relative
probability
makes
sense,
and that absolute
probability
is
meaningless.
But after
introducing
autonomously
'xy
', interpretable
as the
conjunction
or meet of x and
y,
and
'x ', interpretable
as the
negation
or
complement
of
x,
it is
always
possible
to define absolute
probability
in terms of relative
probability by
a definition like
p(x)
=
p(x,xx)
(I.3)
This definition
gives meaning
to absolute
probability,
in terms of relative
probability.
Similarly,
we can define relative
probability
in terms of absolute
prob-
ability, by
a
customary
definition like
Let
p(y)
o;
then
p(x, y)
=
p(xy)/p(y) (I.4)
This defines relative
probability provided p(y) ==o;
for
p(y)= o,
relative
probability
is undefined. This is
very
awkward,
especially
if we
intend
to allow for a
logicalinterpretation
of the calculus. For in such a
oftheformalsystem
we wish to have a rule like the follow-
ing
rule
of interpretation
logicalinterpretation
:
If x follows from
y,
then
p(x, y)
=
I.
And this rule
(which
does not
belong
to the autonomous calculusbut to
one of its
interpretations)
shouldhold
quite independently
of
any assumption
as to
p(y)
not
being zero,
for at least two reasons. First because,if
y
is a
universallaw and x a
singular
instanceof
it,
then
p(x, y)
should be
I,
even
ifp(y)
equals
zero.
Secondly
because
p(yJ)
=
o
; but since
every
x
follows
from
yy,
we wish
p(x,
yy)
to be
equal
to
I.
For these reasons, 1.4
is not
quite adequate
as a definition of relative
probability
in terms of absolute
probability;
and the same
can
be
said of
p(x, y)p(y)
=
p(xy)
(I.5)
which is
easily
seen to be
equivalent
to
(I.4.)
On the other hand, it
would
be
quite
inadequate
to
postulate
1
that, if
p(y)
=
o,
p(x, y)
should
always
For take the case of a universallaw
y
with
p(y)
=
o,
and of an x
be
I.
1
This is a
possibility
discussedand
rejectedby
Carnap,
in
Logical
Foundations
of
1949,
p.
295 sq.
Catnap's
reasonsfor
rejecting
it are
quite
different
frommine.
Probability,
52
TWO
AUTONOMOUS
AXIOM
SYSTEMS
which contradicts
y.
In such a case we wish
p(x,
y)
to be zero rather
than
one,
even
though p(y)
=
o.
I shall
give
below a definition of
p
(x,
y)
in terms of
p(x)
which
gets
over these difficulties. We can
easily
interpret
this definition
by purely uni-
versaland
purely
existentialsentences. But if we wish to extend the method
in orderto
interpret
the calculus
by
sentenceswith mixed
prefixes
of universal
and existential
operators,
then it soon becomes
unwieldy,
to
say
the least.
For such
purposes
it is
preferable
to build
up
the calculusas a calculusof
relatwve
probabilities,
and to
introduce absolute
probabilitiesby
a definition
such as
(1.3),
or some
equivalent
formula.
In what
follows,
I shall
give
two
systems,
one that
takesabsolute
prob-
ability
as
fundamental,
and one that takes relative
probability
as funda-
mental. Both
systems
are
demonstrably
consistent
and
independent.
2
A Formal
Systemof
AbsoluteProbabilities
Postulate
I.
If x is an element of
the
system
S, then
p (x)
is a realnumber.
Postulate2. If
x, y
and z are elements
of the
system S,
then
xy
and
3c
are
elements of the
system S,
and the
following
axiomshold:
A.I
p(xy)
>
p(yx)
(Commutation)
p((xy)z)
>
p(x(yz))
A.2
(Association)
p(xx)
>
p(x)
A.3
(Tautology)
p(x) >
p(xy)
B.I
(Monotony)
p(x)
=
p(xy) +
p(xy,)
B.2
(Complement)
B.3 (x)(Ey)(p(y)
o &
p(xy)
=
p(x)p(y))
(Existence
&
Multiplication)
This last axiom
1
demands for
exery
element x the existence of an
element
y
which has a
probability
=
o and which is
independent
of x.
At the
same time, it is
clearly
a much weakened
consequence
of
(1.3)
and
(I.4),
since it
demandsfor
every
x
a
y
such that
p(x)
=
p(xy)/p(y),
that is to
say,
in view of
(I.4),
such
that
p(x)
=
p(x, y).
If
(I.3)
and
(I.4)
are admitted
1
It
replacesA4 andB3 of my old
system
in
Mind,47 (N.S.), 1938,
p. 275 sq.
However,B3
of the
presentsystemmay
be
splitup (as
in thenotein
Mind)
intotwo
:
(x)(Ey)(p(y)
>
p(x)
&
p(xy)
=
p(x)p(y))
B.3-
(Ex)(Ey)p(x)
t
p(y)
E.I
Axiom
B.3- (Multiplication
and
Independence)
to
B.3
and
CG2
of the
relative
system.
I
may perhapssay
herethat I still
uphold
the views of
my
note
in
Mind,except
thatI now
regard
the
relative
system
as
preferable
corresponds
to the absolute
system:
at the time I believedthata
postulate
I
(asI callit here;
in my
note
in
Mind
it
is
called,most
misleadingly,
'Axiom of
Uniqueness')
couldnot, without
contradiction,
be formulated
for the relative
system
as
strongly
and
unconditionally
as for
the absolute
system.
53
KARL
R.
POPPER
to be
intuitive, then,
clearly,
all axioms of the
system
are intuitive. This is
interesting
in view of the fact that the
system
allows the deduction of the
laws of the
upper
and lower bounds which have been often assertedto be
conventional
(or
non-intuitive
1).
In
conjunction
with
(I.4),
our
system
allows the derivation of all the
formulae of the
customary systems
of
probability.
However, I aim at a
definition which is
stronger
than
(1.4).
In order to define
p(x, y)
on
the
basis
of these axioms,
I make use
of the
concepts
of
a
finite
system;
of
a
sequence
of finite
systems
S,;
and of
the limit
system
of this
sequence.
The chain of four definitions
is a
little
complicated,
but its
function
is
mainly
heuristic
:
it is to lead
up
to a
system
of relative
probabilities
that
represents
a
very powerful generalisation
of the
customarysystems.
In
preparation
for
my
definitions,
I first
introduce,
informally,
the idea
of
an
atomicelement,
or
atom,
of
S, corresponding
to an atomicsentence
(and
interpretableby it).
It
may
be based in its turn
upon
the idea of structural
equivalence
of two elements x and
y, symbolised
here
by
x
= y,
and to be
distinguished
from
anything
like Boolean
equivalence
: 'ab
=
ba' will
be,
in
general,
false
(except
if a
=
b) ;
and both 'a
=
aa' and ' a
=
a'
will
always
be
false. Thus x
=
y
holds
only
if x and
y
are
composed by
the
same
operations
of the same elements in the same order. We
may
then
say
: if x
is in
S,
then x is a
compound
elementof S if and
only
if there are
in S elements
y
and
z
such that either x
=
yz
or x
= j;
otherwise x is
an atomof S. Now we can define
:
DefinitionI.
A
system
S is
calledfinite
if and
only
if
(i)
there exists
a finite set A of real
numbers such that for
every
element x of
S, p(x)
is in
A; (ii)
if
x,
.
.
., x,
are differentatoms, or
complements
of
different atoms,
of
S,
then
p((.
.
.
(xxx,)
. .
)x,)
+
o.
(Conse-
quently,
if x is an atom, then
I
=
p(x)
+
o.)
Definition
2.
Let S
be
finite,
and let
x
and
y
be in S.
Then we
define:
Ifp(y)
=
o,
then
p(x, y)
=
I
;
Ifp(y)
4
o,
then
p(x, y)
=
p(xy)/p(y).
Definition
3.
Let there be for
every positive integer
n
a finite
system S,
such that
(a)
So
is the
empty system, (b) every
element u of
S,_1
is an element of
S,
(so
that,for
example,
if for
every
i,
ui
is in
Si,
then
w,
is in
S,
where
w,
is defined as
w,_l1u,;
w,
is, of course, also
in
S,)
;
(c) ifp(u)
=
r is true of u as an element of
S,_1,
then the same
is true of the same
u
as an element of
S, (or
in other words,
p,_1
(u)
-
p,(u),
so that the indices afterthe letters
'p' may
be
omitted).
Then
we call the S, an
(infinite) increasingsequence
of
systems.
1
Cf. Carnap,op. cit. p.
286
54
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