The Fregean Axiom and Polish Mathematical Logic in the 1920s
Author(s): Roman Suszko
Reviewed work(s):
Source: Studia Logica: An International Journal for Symbolic Logic, Vol. 36, No. 4 (1977), pp.
377-380
Published by: Springer
Stable URL: http://www.jstor.org/stable/20014872 .
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eoman
xhe
Axiom
Fregean
Suszko
and
Polish Mathematical
in the 1920s.
Summary of the talk given
(Poland), July 5-9, 1976.
Logic
on the History
to the 22nd Conference
so
Man,
the
of Logic,
as
far
only
Cracow
I know,
animal
is
capable
of lying to himself.
EOBEET AEDEEY
that
assumption
the same, i.e., have
Axiom.
Another
the Fregean
The
semantical
describe
sentences
is called
law of materially
salva
equivalents
true
all
all false)
(and,
similarly,
referent
(BEDEUTUNG)
a common
formulation
of
it is the
relacement
several
ontologioal
It also has
veritate.
e.g.,
versions,
(AF) VpVq((po q) =>{&(p)o 0(g))).
The main
(1)
of this
thesis
The
construction
was
the effective
deceit
is twofold:
of so called many-valued
abolition
of the Fregean
by Jan
logics
Lukasiewicz
Axiom]
the chief perpetrator
of a magnificent
out in mathematical
logic to the present
is
Lukasiewicz
(2)
talk
lasting
conceptual
day.
to tell you started very
1910.
story I am going
early,
just before
on the other hand, my
on
of
it
But
is
based
certain
results
presentation
and Tarski
with
Lindenbaum
dated much
originated
and,
later,
just
1930.
before
The
and
Lindenbaum
an absolutely
of the whole
connections
Tarski
free or, anarchic
of
class K(J2?)
j?? and
between
observed
that
algebraic
all algebraic
any structure
so called
of S? to s?
the
JS? is
language
the
fountain
hence,
similar
to j??. The
formalized
structure
and,
structures
s? in K(if)
conditions
are given by maps
here as
labelled
and
satisfying
morphism
are admissible
valuations
reference
algebraic
of 3? over s?. They
assign?
ments.
The domain
of all expressions
of them consists
of definite
syntactic
terms
formulas
and diverse
of for
kinds
category:
(sentences),
(names)
mators.
limited.
The
In
size
of
particular,
(addmissible
Algebraic
when
supplied
referents).
structures
with
of
codomaius
the
formulas
in K(??)
or,
"distinguished"
algebraic
may
valuations
have
many
in a suitably
chosen
sets of admissible
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is not
algebraic
subclass
referents
a priori
values
of K(??)
of for
378
Boman
into models
turn
mu?as
to
valuations
quence
any
find
operations)
logic considered
sets
here
where
ax,
logical
...,
a1,
can
the
an,b
...,
on
an
an
inference
of
conse?
(finitary
in case of
Consequently,
relations
|? , compare
[1], one can
for all formulas
defined
and,
the
with
following
adequacy
property:
n =0,1,2,...
then :
(? b if and only if for all t in V,
= 1 whenever
t(b)
t(a1) =...==
two-valued.
(logically)
in case of Lukasiewicz's
an
of algebraic
truth
and
satisfaction,
relation
functions
with
valuations,
are any formulas
use
making
able,
of inference
theory
set of all formulas.
valued
is
every
short,
logic
be easily
exemplified
logic, sez.
The
adequate
was
notions
algebraic
of ??.
formulas
the
as
zero-one
of
V
called
In
define
to
relating
also the
created
consequence,
Tarski
Tarski
and,
Suszko
t(an)
=
1.
This
statement
general
three-valued
sentential
the
(V1 ? V ? V2) between
one
sets
set V1 and the largest
Some
smallest
72.
adequate
adequate
are better,
some other are worse.
In general,
set of logical
each adequate
a natural
which
turns
out
to be
valuations
bears
(dual) pre-topology
sets
V
form
interval
cases.
a genuine
in certain
important
topology
are morphisms
the logical valuations
On the other hand,
some
to the zero-one
in
cases,
only.
exceptional
model)
Thus,
of quite
ialsity
play
which
(of formulas
valuations
the
valuations
and algebraic
logical
The
relate
nature.
former
to
different
conceptual
the refence
represent
assignments.
and, the latter
a double
semantical
role,
it into the
of our thesis
in
general.
is
It
the
are
the
functions
truth
and
The
formulas
Fregean
Axiom
inseparable
unity.
amalgamates
runs as follows.
In case of the truth?
the proof
Now,
valuations
coincide
valuations
and
s-functional
the
algebraic
logical
logic,
are
and
the Fregean
by 1 and 0.
Axiom)
represented
(in accord with
Obvio?
the
connective.
is
the
material
identity
equivalence
then,
Clearly,
a
in
mad
idea
values
is
of
any multiplication
and,
fact,
logical
usly,
did
Lukasiewicz
logic
i^3 making
not
actualize
essential
on the three element
use
it.
Indeed,
of algebraic
set {1,1/2,0}
with
he
defined
valuations
his
three-valued
to a suitable
1 as the
algebra
sole "distinguished"
of JSf3
the
but
defined
Lukasiewicz
system
logical
a
as an
one
in
natural
reformulate
way
may
??3
However,
-tautologies.
of Luka?
the following
features
inference
relation
[2]). Then,
(compare
can
revealed
be
siewicz's
[3]:
<??3
logic
element.
(a)
(b)
(c)
Actually,
as stated previously.
two-valued
J2?3 is logically
a
the sense of [4],
in
is
classical
if3
logic
a
is
j?f8
of SCI,
i.e., the sentential
strenghtening
particular
with identity (compare [5], [6]).
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calculus
The Fregean
axiom
and Polish
mathematical
379
logic in the 1920s
values
the
values
are, by no means,
1, 1/2,0
algebraic
logical
for
rather
referents
of formulas,
admissible
^f3. They
represent
typical
also say that Lukasiewicz
constructed
the logic 3?z under
by ??3. One may
to the Fregean
that
there
the assumption
exist
(contradicting
Axiom)
The
three admissible
exactly
two do not.
the other
of formulas.
referents
of them
One
obtains
and
has been
the Fregean
Axiom
abolished
constructively
Thus,
by Jan
as
not
he
new
he
did
create
could
Lukasiewicz.
any
However,
not,
logical
To be sure, POSSIBILITY
the truth
and falsity.
besides
value
is our
of all our failures.
It is, however,
neither
only hope and the headspring
nor what
a logical value
formulas
refer to.
may
of
Because
creative
freedom
confuse
the
was
were
truth
his
and
the
and
possibility
how
could he
But,
sentences
How
describe?
personality,
intellectual
idols.
dearest
with
falsity
the
that
it possible
unusual
Lukasiewicz's
what
the
of many
humbug
values
logical
over
persisted
the last fifty years'?
seem
may
in technical
the
To many
logicians
affair
consists
problem
in a shift
siewicz's
to be
philosophical
and mathematical.
scientific philosophy
simple:
This
terminology.
Luka?
is true
school in logic was both
but it is not the whole truth, I think. The Polish
It was
quite
a pioneer
so
in
called
modern
(logical empiricism of some kind) and also had a good
of classical
On the
of problems
philosophy.
Polish
of
mathematicians
set-theoretical
thinking
sense
on minds
of their
fellows
pressure
to the intense
movement
intellectual
in
logic.
in Polish
other
the growing
hand,
a considerable
exerted
factors
Many
contributed
logic at these times. Conse?
in a quite unconventional
the
tendentiously
changed
terminology
quently,
an
was
the semantical
because
of
abuse
words
it
way.
Certainly,
duplicity
after
50
still face an
of formulas
years we
eventually
disappeared
and,
of many
and falsehoods.
truths
paradise
illogical
a phenomenon
was
school
in the twenties
Polish
which
logical
the
lines
of
the
of
for a deeper
ideas.
At
analysis
along
history
pre?
nature
formal
should be noted
four spicy items of virtually
(com?
The
calls
sent,
pare
[7], [8]).
of all, in the early twenties
He
with
did
identity.
equivalence
of
his
connective
logic
equivalence
First
Secondly,
independence
Lukasiewicz
openly
Axiom
to equate
with
respect
material
to
the
???3.
in his doctoral dissertation
of the Fregean
used
so even
AF.
(1923) Alfred Tarski states the
In fact,
Tarski
explicitely
com?
pared AF with the fifth postulate of Euclidean Geometry and proved
the independence of AF by means of the logic jS?3,that is, SGI (!).
it was
Thirdly,
viations
in favour
equivalence.
is a material
Lesniewski's
of definitional
to eliminate
all abbre?
idea, I conjecture,
built by means
axioms
of the material
Tarski's
It underlies
equivalence
theorem
paper. His fundamental
(TH. 11)
him
considered
by
and, generally
recognized
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380
Boman
a d?finition
as
to day
valencies
of conjunction.
Tarski's
underlying
of AF
equi?
as definitions
in view
of the
in Tarski's
system.
Lesniewski
confessedly
his protothetics
became
Stanislaw
Finally,
material
labelled
ponder on all those DEFINITIONS
(BEF.). One may
absence
several
Similarly,
are explicitely
system
Suszko
the Fregean
Axiom
accepted
into a trivial
transformed
consequently,
and,
1 and 0, compare
of two entities
Lesniew?
[9]. To support his views,
theory
a
to
have constructed
it possible
ski claimed
which makes
general method
O not
all functions
to eliminate
AF.
One
is eager to compare
satisfying
method
Lesniewski's
the
with
result
independence
by
own
his
pupil.
References
D.
[1]
E.
[2]
Brown
J.
102
E.
and
On
W?jcicki,
matrix
E.
Polish
of Logic,
4, No.
Vol.
L.
[4] S.
3,
S. L.
pp.
87-90.
D.
J.
and
102
Mathematicae,
[5]
Bloom
and
E.
(1973),
[7]
1975.
Jan
Lukasiewicz,
[8]
Alfred
A.
The
Studia
of
Section
Institute
Polish
Allatum
?
Logic
Grzegorczyk,
research,
The
Selected
und
Bulletin
the
of
of Philosophy
abstract
Grundlagen
Section
and Sociology,
logics, Dissertationes
into
the Sentential
169-239,
Works,
Verlag,
Springer
ed.
L.
Calculus
with
(1972), pp. 289-308.
in: Logic Colloquium,
Borkowski,
Identity,
Lecture
Berlin-Heidelberg-New
North
Holland
(Amster?
1970.
Oxford
Semantics?Metamathematics,
University
1965.
Press,
[9]
pp.
(Warsaw),
Tarski,
logic,
Classical
of Lukasiewicz's
operations
43-52.
Investigations
453,
and PWN
dam)
Mathematicae,
Logik
Institute
13
of Formal
Logic,
of the Fregean Axiom,
in Mathematics
York,
Brown,
pp.
Suszko,
Dame
Journal
Notre
Abolition
[6] E. Suszko,
Notes
consequence
three-valued
of Sciences,
Academy
1975,
Bloom
of
representation
on Lukasiewicz's
Bemarks
Suszko,
Dissertationes
Logics,
Zeitschrift
f?r Mathematische
19 (1973), pp. 239-247.
sentential calculi,
der Mathematik,
[3]
Abstract
Suszko,
7-41.
(1973), pp.
of
L?gica,
systems
of Lesniewski
77-95.
3 (1955),
pp.
in
relation
Logic
and
Philosophy
Academy
of
est 1 Junii
Studia L?gica XXXVI,
Sociology
Sciences
1976
4
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