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/s TC<j> /s CCp,q,r> ^ T C r »
peK
T.
a
set
the
1
vocabulary In
V
fact.,
formula
and
it (3)
is
closed
follows
from
under
the
relation
assumption
CI),
of
concatenation.
Conclusion
1.3a
and
then
the
that
C4>
K c If . 1
On
the
definition
other C3>
of
hand, W
and
if
p,q
formula
e
W
^
yield
cXp,Q,r>, T, of
THE AXIOMATIC THEORY TLTk OF LABEL TOKENS
'^
c o n c a ' t e n a t . l o n , i.e..
From f o r m u l a s and C5>, u n d e r C o n c l u s i o n I.3c, we o b t a i n W ^ W, which. In a c c o r d a n c e
wit-h , y i e l d s
the
thesis
be
principle
of
the
theorem
being proved. Theorem the
least
the
I.l set
for
This
is
with
the
is
thus
the
set
of
induction
v o c a b u l a r y and i s and
hence
the
closed
for under
principle
of
W.
n-conq>onentlal words; l e n g t h o f word
simple so
of
of
given
by
of
follows
compound initial
their
mereology>, concept
language
which
any
those
determined
the
a
words,
because
use
Lei^niewski's define
the
concatenation,
Compound w o r d s of
the
of
S e c . 1.6.
set
termed
which c o n t a i n s
relation
induction
may
are
from
word
words.
are
finitely
Definition
is
simple
componential
Cin t h e words. word
the
of
W.
exclusively
"structure"
parts
from I.l
recorded
The
component
which
generated
of
words
sense Before
we
refer
of we to
t h e usually adopted convention: C o n v e n t i o n or
without
subscripts,
1.3. L e t t e r s are
variables
Ars o f naturail n u m b e r s w i t h o u t We s h a l l
now
introduce
the
i,
J,
which
k,
I,
m, n,...,
range
over
the
concept
with
the
set
of
the
zero.
definition
of
18
CHAPTER I
set
V of
will
be
they
all
called
are
n—compoTiential
xaord
n-componential
xiiorcl
usually termed
"n-letter
tokens.
Elements of
tokens.
In
that,
formal
words". The s e t
set
languages
V is
defined
by i n d u c t i o n a s f o l l o w s < s e e G.Bryll, S.Miklos [19773): DEFINITION
1.2
a.
*K -
b.
r e
A
"""V «•
ial
We s h a i l
now
definition
\Dord token
c.
token is
word
is
a label
token
formulate adopted
n-componentlal
V
word
an n-componential
the
V
one-componential
n-H-component of
V,
and
word t o k e n s .
theorems
and
Their
tohich
a simple
several
aibove
a
describe inductive
simple
word;
is
concatenation
a
word
token.
which
result
the
an
from
properties
proofs
are
of
given
in
t h e Annex. N o t e f i r s t t h a t t h e f o l l o w i n g t h e o r e m holds: THEOREM
Every
1.7
n-componential
word
token
is
a word
token.
Theorem 1.5 has its analogon in THEOREM
1.8
pe A word token
is
token also
which
l^^q%p^<je is
equiform
an n-componential
Two other theorems follow:
with word
K. an
token.
n-componential
word
THE AXIOMATIC THEORY TLTlc OF LABEL TOKENS THEOREM
19
1.9
/N CCp,Q,T> If
a
la&el
m—componential
token one
and
an m*n-componential THEOREM
is
label
is
is
I.IO
then
that
can
It
essential as
replace
shown,
a
word
of
a simple
token
shall
that-
Definition
definition
whether
an
concatenation
of
now
I.2b
from
example,
only
and
an
because,
as
this
n+f-componential an
word
n-componential
formulate
the
theorem
which
theorem. token
word
and
is a
word. refers
to
the
Section.
1.11a
p e V =• V p e n word
In t h e
is
and
token
above,
follows
c o m m e n t made i n t h e i n i t i a l p a r t o f t h i s THEOREM
if
word
s i m p l e word o r a s i m p l e word and a n n - c o m p o n e n t i a l
Every
label
token.
be
We
words.
\ccp,.
a cOTtcateruztion
can easily not
**A
an n*1-componential
word
Theorem
obtained
one,
two
vord.
token
n-componential
is
of
1.10
A label that
V.
concatenation
n-componential
'-"''^
if
a
^ r e
token
proof of
of
is
a fini
this
Theorem
tely
componential
theorem
I.l ,
that
V.
we is
avail the
word.
ourselves, principle
by
of
way
of
induction
f o r W. Note fact
that
follows
the from
present
theorem
Definition
I.2a.
is
true Let
us
for
any
now
peK, assume
which "by
20
CHAPTER I
Induction" t h a t
it
is
true
for
word
tokens
p
ami
p
1
that
it
is
true
for
the
word r
which i s
their n
follows
from
the
assumption
that
p
1
e
hence by Theorem 1.9 we have that V j. e I.l
1
and
show
2
concatenation. n
2
V and
p
It
e
2
V,
and
K. By applying; Theorem
we find that the theorem being proved is true for any word
token. Theorems 1.11a and 1.7 yield THEOKEM
1.11
W-
U
V.
k.~1
The
set
component
of
ial
utord
Theorems in
a
I.ll
set
of
a
set
such
affirmative
all
xDord
tokens
is
the
set
of
all
finitely
tokens. arul
I.lla
finitely
have
shown
componential
determined
that
words.
every Is
unambiguously?
word
the
The
token
is
membership
in
answer
in
the
i s p r o v i d e d by
THEOREM I . l i b
p € »•' ^ Vi p e n "^ Every one
set
word of
token
finitely
is
a nnrd
xnhich
componential
V. is
a meitJber
of
precisely
words.
The p r o o f i s g i v e n i n t h e Annex. By
this
theorem
"componentiaiity"
an
other
n-componential
than
n,
and
hence
word
cannot
catfinot
be,
have
e.g.,
an
replaced
by
m - c o m p o n e n t i a l word i f m ^ n. The
rather
clumsy
word
"componentiaiity"
will
be
THE AXIOMATIC THEORY TLTk OF LABEL TOKENS •the
word
commonly uKtrd that
"lengt.h". used
token
p
concept
in
the
will
be
takes
DEFINITION
In
on
21
accordance
literature recorded the
of as
the
symbolism
subject
\p\.
In
the
TLTk
the
most
length
of
definition
a. of
form:
1.3
word
n-componential The
the
following
p € V •» I P I ..-, \r\
€ W ^ CCp, /s CCq,p,t) •» \r\ of
word
tokens
p and
length.
Theorem I.4a and C o n c l u s i o n 1.7 yield
Q and
-
|t|. q and
p have
the
THE AXIOMATIC THEORY TLTk OF LABEL TOKENS CONCLUSION
23
1.10
r e W\V «» ^ V
cc ^ | r | -
| p | + |.
p , ^fcPr
i4 compo-und
word, token
fino }))or 1. THEOREM
1.12
reW .^ | r | - 7. > 1 • •
^A^
p,^W
^ ' ^ ' " k ^ \--,P
C
simple Just ,
words.
s o f a r do n o t allow
that
down t h e t h e o r e m
I.lla>
componential
a length n
CTheorem
an
n-componential
n > 1, i s word
a
label
tokens.
formulated
which
In
above
denotes
the
o f n+2 a r g u m e n t s . +.,»P^
is
read:
p
is
a
24
CHAPTER I
concat.enat.ion tokens
of
n-K
P.>Pj>—>P ^^-
t o be t h e
successive
Oeneraiized
element.s,
The
labels
coinponents
relation
namely
of
of
successive
P^»P_»-..>P ^^
of
that
are
label
then
said
concatenation.
concatenation
c
l>
is
defined
by i n d u c t i o n . DEFINITION
1.4
a.
n " 1 •• Cc
(-P ,p ./>> «• CCp ,p ,p>>,
b.
n > 1 =• Cc"
«*
In
accordance
concatenation "ordinary"
of
with two
this
label
concatenation
n+1-element
l>
which
n-element
Pl'P2'
'^n ^"^ ^ label
only
those
if
p
means
it
of is
Cn>l!)
is
the
two
concatenation
2und
an
tokens of
P^>P^>->P >P ^A i f 1 2 n n+1 is
definition
a
a
two-element
same
as
labels;
p
a
p
is is
label
an
tokens
concatenation
concatenation
an
of
label
of
labels
p^_^^.
And here is the announced THEOREM
p ^ W^
I.13a
\p\
A \jtoTd. token concatenation In
m n+1 ^
of
length
of
certain
accordance
with
possible
to
token
a
is
formulate word
V
which
n*1
is
simple
C
a
label
• itord
Conclusion the is
following a
Cp ,p ,...,p
which
is
^ ,p).
an n+1
-element
tokens.
I.5b,
Theorem
conclusion:
concatenation
of
a
1.13a a
makes
compound
finite
it
word
number
of
THE AXIOMATIC THEORY TLTk OF LABEL TOKENS
25
component.iaI
simple
word
t.okens,
component.ial
simple
words
equals
and t.he
t-he
number
len(;Ch
of
of
the
t>hose
word
in
quest-ion. The p r o o f o f T h e o r e m 1.13a i s c i v e n i n t.he Annex. The f o l l o w i n g t-heorem a l s o h o l d s : THEOKEM
A
1.13b
compound,
coTicatenation The
xaord. of
a finito
simple
Conclusion
tohsn
is n-vmber
implication
I.5b,
a.
Theorem
of
in
by
lemma
induction,
I.13a,
Pi
and
a
certain
tokens. 1.13b
follows
Conclusion
to refer
I.3a.
To
from prove
to
is
a
^ ^ ^ """"^^^Pi'Pz'- 'f>n^'P^ generalization
based
on
that
D e f i n i t i o n I.4a,b, i s l e f t t o t h e Conclusions theorem
word
is
I.l
l ^ p,r,
a label
This
is
theorem
e V ^ p ^ r ^ q S : s ,
which i s a w e a k e n e d f o r m o f T h e o r e m 1.3: THEOREM
1.14
//
one
and
generalized
the
sajne
word
concatenation
tokens, in
the
l < v < n + l fi'^fi
then same
the
of
components
places
are
^ ^ -» l, e x c e p t
language
those
the in
of
any
C; S >
the
i.e.,
The
sixtuple
of
of
expression
the
successively,
the
operators.
of
be
expression
conditions
Idea
simplified
language in t h e form of t h e < Lb;
of
will
t o his ideas.
gave
CJ?>
the
and
£
by
certain
categorial
chapter,
cat&gorial
by
with
this
bound
simple
languages
simple
language
variables
a
of in
fixed
determined
r e s e c u ^ c h e r s who r e f e r r e d In
but
Include
is
TSCL
constructed
here
tokens,
categorial
theory
arbitrary
called
syntax
the
of
all
tokens,
the
and
expression
label
the
tokens.
tokens,
relation set
of
All
of all the
last
one,
are
described
to
Chap.I.
The
set
S
well-formed
concept
which 32
is
of
all
defined
in
the
be
the
theory
THE AXIOMATIC SYSTEM TSCL OF SIMPLE CATEGORIAL LANGUAGES TSCL to
and
the
tokens The
adso
<see
categoTial which
theories
may include
of
of
the
because
categorial
S
set
of
used
the
in
expression of
syntactic
connexion
of
expressions
They
are
categories to
carry
of
categories The set
I
o/ -^
o elements
of
functoral,
of
all
I
categorial
are
The
are
are
and
is
auxiliary they
are
in
the
the
syntactic
Secs.II.8
the
S
all
well-formed
Csee
of
of
useful
language
C t CS>
indices
indices.
used
to
The
indices
is
and
syntactic
and enable into
us
syntactic
generated
basic
indicate
remaining
univocally
termed
symbolized language
given
Sees.II.2
a
indicating
cisissif i c a t i o n
basic
syntactically
function
a
both
because
language
in
into
an
of
them.
the
indices
basic
determine
from
the
which
are
categories the
of
derivative,
categories.
Indices
words
logical
of
only
that
given all
expressions
all
expressions.
given
a
correctness,
above
perform
and
by
foundation
concept
III.4>
a
Csee
the
not
the
expression
bound
indices
they
and
of
of
C s e e S e c s . I I . 6 a n d 111.53.
set
I
used
the
out
of
pertaining
introduction
are
essentiaJ
Secs.II.5
checking
III.6>.
the
indices
definition
(see
languages
categorial
is
TSC«»-L,
a n d variables
Although
role
theory
i.e.,
require
languages. their
the
operators
categorial
function,
in
<ji-langvages,
definitions
II.3>,
Ghap,III>
33
by and
that
categorial
the
L. T h e its have
assigned
definite
language
/-unction set
E
of
subset
S
will
indices,
to
that
of
by
word
means
indication
all
is,
selected from
of
of
expression
be
tokens
token from
the
a
of
certain
indices of the
domain
a
and a
given
set of
of the
34
CHAPTER II
function A
t.
cert-ain
put-
synt.act.ic
t-oget.hei>,
concatenation, that and
have is
useful
vocabulary ainalogon (p
4>
the
obvioijs
that
well-formed
not
any
every
greatly
TSCL
language
constructed
in
show
and £, the
that
< Lh;
V, V; •^, C; I
satisfy
is
has
any
its
two
word
which
have
have
this
and
an
of
whose
,
postulated
in
that
THE AXIOMATIC SYSTEM TSCL OF SIMPLE CATEGORIAL LANGUAGES
35
t-heory. It, any
is
wort/h
noting
ci>-languaee,
form
of
an
to
that
which
ordered
the
the
system
syntactic
theory much
characterization
TSCw-L
more
pertains,
complex
than
of
has
the
CJP >
(see
Sec.III.l). The the
analyses
theory
III.5>
in
cairrled
of
syntactic
two
theorems
of
out
TSCL
which
theory
and
categories
theorems
t/ie
In
of
TSC«»~L c l e a r l y
and
might
syntac
tic
result
by
(see
termed
categories,
refer
to
Secs.n.7, /'undamiental
aibbreviated
as
rttsc. Their
formulation
is based
Cay
on t h e
expression
rCpyq^s,
whose
corresponding
Important The from
definitions
expression
the
by t h e Now
CcO
is
s
by
expression
p.
Co) i s
common
the
has
r,
the s
p, the
r
the
are
its
expressions
schema
of
the
replacement
of
The second
as
II.
has
the
TSC«-L
its
r
its
is
are
fttsc.
One
second
same
syntactic
in
the
form
of
of
the
same
syntactic
theorem
under
y.
implication a y^ y
^
ft
of
q
those
assumption:
the
the
obtained
constituent
of
a ^ ft ^ ftlsc
of
both
implication I.
and
expression
expression
thesis
TSCL
the
assumption
expressions
has
in
theories.
read:
following
are
theorem
Cj'J Thus
in t h o s e
expression
theorems
That
definitions
category.
expression: category.
consideration
has
the
36
CHAPTER II
as
it,s
schema.
Under
the
assumpt.ion
Now Cft^ h a s Cft'i
The
t - h e t w o
its
concretlzatlon
r,s
are
expressions
of
sentences.
fttsc
In t h e
of
the
aa-e
converse
theorems.
expression:
syntactic
category
implications
CIO
a y^ ft' •* r
or dll}
are
a ^
schemata
theory
of
categories requires
of the definitions
syntactic as
of
both
belongs
the
ft'y
C;'>, often
The
expressions
introduction
categories to
of
^
of
categories.
sets
the
Cf
into
Cot) and
syntactic
of
the
of
syntactic
replaceable
In
sentences
of
syntactic
theory
concept
category
in
treatment
the
the
adopted
of
expression
sentences, that
that
is, the
concept of sentence. Including the concept of
the
theory
of
of sentence
syntactic
in the
categories
possibility of defining it. It seems
that
primitive
would
concepts
preclude
It is the
the
defining
of
the concept of sentence that should be one of the fundamental tasks
of
natural definition
the and
theory symbolic
of
sentence
of
syntactic
categories
languages.
It
seems
should
satisfy
the
as also
applied that
condition
to the
which
enables us to formulate the algorithm of checking the syntactic connectedness
of
.^Jdukiewicz. But condition, when
a
expressions, definition
CI'> and
anadogous
of
CIII) are
sentence treated
to which
that
given
satisfies
ais a schema
of
by this the
THE AXIOMATIC SYSTEM TSCL OF SIMPLE CATEGORIAL LANGUAGES
definition
of
belonging
syntactic
categor-y,
circle.
constructing
In
languages is
we
treated
certain chapter.
try
as
constructed
expression
in
involve
essential
IV.6
we
the
with
the
purpose
DTSCL
and
ideas
in
more
vicious
endeavour and the
next
They
underlie
describing
its
theories
DTSCw-L.
which
same
categorial
TSCL
outlined two
of
a
Our
theory
discuss
the
of
difficulties.
of
the
syntactically
TSCw-L,
languages:
accordance
for
goal
to
danger
of
those
theory
and
the
theory
overcome
the
expressions
of are
TSCL
languages
of
types.
S e c . II.2.
theory.
the
two
the
categorial
TSC«»-L
Now
to
Secs.IV.S
syntactically
and
would
modification, In
of
37
C o n n e c t i o n s b e t w e e n TLTk a n d TSCL
TSCL Hence
substitution,
like
TLTK
all are
is
logical valid
based rules,
on in
classical
logic
and
set
particular
the
rule
of
in TSCL b e c a u s e
TSCL i s
superstructured
o v e r TLTk. Hence a g a i n REMARK II.1.
The c o n s t a n t s y m b o l s o f t h e v o c a b u l a r y o f TLTk Lb,
y,
W, %, C
a r e c o n s t a n t s y m b o l s o f t h e v o c a b u l a r y of
TSCL.
The v a r i a b l e s y m b o l s o f t h e v o c a b u l a r y o f TSCL i n c l u d e C-u i
p, i^ T, s,
t, XI, x>, w i t h o r w i t h o u t
subscripts,
and Cu )
X, w i t h o r w i t h o u t and
Cv > r a n g e ,
subscript;
respectively
Csee
Convention
I.la,b>,
over
38
CHAPTER II
Lb a n d t,he f a m i l y 2 The
variables
of
all it.s subset^s.
list-ed
under
a n d
do
1
list,
of
t-he
variable
symbols
of
not
exhaust,
t-he
Z
t-he
vocabulary
of
t-he
language
o f TSCL < s e e Sec.II.S, C o n v e n t i o n I1.3b>. We o b s e r v e t h e p r i n c i p l e Csee C o n v e n t i o n I.2> C o n v e n t i o n whose
numbers
are
II.1.
marked
Those t h e s e s with
an
Caccepted
asterisk
are
m e t a t h e o r y o f TSCL, a n d n o t e c o r r e s p o n d i n g l y THEOREM II.l . a thesis
of
Both tokens. being the
In
shall
both
now
under
theories
it
account.
From
into
ff*, b u t
see
thosis
sentences)
theses
in
the
that
Cacc&pted
sentence)
of
TLTk
is
TSCL.
theories
taken
set
Every
that
also t h e
CSec.II.3,
set
consideration is
Lb that
I of
Theorem
which set
all
II.l),
/
pertain
is we
the
has
out
set
not
indices;
no
word
universal
single
categorial
to
common
only
as
we
elements
w i t h W. Now recorded
/
is with
TSCL, w h i c h
a the
defined use
denotes
of
the
concept a set
new of
in
TSCL.
primitive all
Its
term
basic
definition
is
I , specific
of
indices
CI
includes o
a t leaist one t e c h n i c a l symbol). The
set
/
will
be
defined
with
reference
to
the
set
/
in o
t h e f o l l o w i n g m a n n e r Ccf. D e f i n i t i o n
I.la):
THE AXIOMATIC SYSTEM TSCL OF SIMPLE CATEGORIAL LANGUAGES
DEFINITION
II.la
XCXy'^ ** I The set and
only
of if
relation
39
Z X ^
label it
of
^
Cp.QS
tokens
incltides
X ^ c ^ r e X).
satisfies the
set
the I
expression
and
is
\<Xy
closed
under
if the
concatenation.
Hence we have . The set
of
all
categorial
tokens
tthich
satisfies
By
these
definitions
which
includes
indices
the
/
is
expression
I
is
and
is
/
are
the
\CX>
the
set
of
label
.
least
closed
least
set
of
under
the
label
tokens
relation
of
o concatenation. The three
properties of
the
of
following
described
five
above
axioms
of
all TSCL
by
the
Ccf.
first Axioms
I.9-I.13>: AXIOM
II.l
o The set AXIOM
of
indices
is
non-empty
iTidex.
of
label
tokens.
II.2
p e l A label
set
tnhich
is
etfuiform
y \ q S i p ^ q e I . with
a basic
index
is
also
a
basic
40
CHAFFER II AXIOM
II.3
cCp,q,r> A basic
ind»x
is
not
a
" • r e / .
concatenation
of
any
two
label
a concatenation
of
tokens. AXIOM
II.4
p, ^ p,qi € // those
a categorial labels
are
We s h a l l I
of
index also
now
all
is
a concatenation
categorial
introduce
/.
a
n—componential
of
two
labels,
then
indices. definition
by
categorial
induction
indices
of
the
Ccf.
set
Definition
I.2a,b>. DEFINITION a.
*/
II.2 -
/
, o
b.
r e
"**/ • •
V
V
pe I ^ A one-componential n-t-1-component concatenation
ial of
cCp,q,r>. o
categorial categorial
index
index
an n-componential
is
a
is
a
label
categorial
basic token index
index; xahich and
a
an is
a
basic
i ndex. The II.2a,b
syntactic above
respectively,
with. which
analogies
of
Definitions are,
Definitions I.la,
I.l,
successively,
and
II.la,
II.1,
I.2a,b
of
definitions
of
atfid TLTk, the
THE AXIOMATIC SYSTEM TSCL OF SIMPLE CATEGORIAL LANGUAGES expression
\CX>,
The
same
the
correspondine
those of
a
set,
W
to
the
analogies
appiies
Axioms
observations dv.al
analogon
DEFINITION TLTk
the
which
II.l . next
t-he
set
V,
between
are
self-evident.
Axioms
11.1-11.5
I.9-I.13
of
TLTk.
We
shall
more
precisely
by
making
use
of
of
sxpression
an
/ /
to
and
41
(p
is
of
any
variables
formulate
the
concept
TLTk.
expression
may
and
of
include
the
the
language
symbol
Lb
of
and
the
only
if
symbols
K,
then
the
obtained place
from
of
¥. is
a
tp f>y
symbols
dual tf*-^
analogon
of
replacement,
Cl> respectively
I . I,
'^
"v.
for
any
i
e
i. " c ,
ma.,
(.Co.) "ft".,
We e n l a r g e
Convention
with
zero.
Convention concerning
•
d,
the
range
t"c.,
of
the
i»J»fc>i »"»>'>»•••> (see
(.<s>
//u
tCp.) d.
(.C<j> " 6 ,
I.3>
II . 3 a
the
to
the
is
set
not
in
variables
tCs.)«d..
variables
with
or
NLKO>
of
without all
naturail
contradiction
of
the
type
subscripts,
with ^'*'^^
numbers
Remarlc Csee
II.1
Sec.II.2>
because CONCLUSION
II.5C
C S V .
The alI
set
word
of
all
expression
tokens
is
included
in
tfie
set
of
tokens.
T h i s f o l l o w s f r o m C o n c l u s i o n s Il.Sb and II.la. The
correctness
of
Convention
11.3c
follows
from
Conventions
II.3a,b and C o n c l u s i o n II.5b. We now -formed
formulate
expression
£.
In
doing
of
the
of
order
set n
so
the tokens
we
avail
S consisting CTI>0>.
definition of
the
a given simple
ourselves of
of
all
of
the
well-formed
set
S of
all
categorial
welllanguage
inductive
definition
expression
tokens
THE AXIOMATIC SYSTEM TSCL OF SIMPLE CATEGORIAL LANGUAGES DEFINITION
53
II .4
s b.
p e
$
^
p e
S ^
V Po^Pj,
,P„€*5
00
c.
S -
By
Oefinlt,ion
well-formed
U "S. n-0
II.4a
a
expression
well-formed
simple
of
expression
order
token
of
is
£
are
has
with
already an
p satisfies the
c accordance
functor
of
of
index
the
that
or
II.4b
set
is
a
every
of
S, a b o u t which we a
Is
order assume
compound
expression
that
all
its
order
h,
S. F u r t h e r ,
the
such
tokens the
Definition
is
Pr^yPtyP'.'-fP
to
^
of
words
that
is
expression
condition
m in
If
the set
expression
which b e l o n g
By
of
o r d e r fc+1 e i t h e r
defined,
expansion
well-formed
expressions
been
0.
p of
k, and t h u s i s an e l e m e n t o f that
expression
Ca,a^,...,a^,ajj>,
with
p, t h a t a
functor;
which is
the
and t h e in
m
the
categorial
a-forming
indices the
of
index
functor, all
a.
index is
of a
be
main
concatenation
successive
should
the
arguments followed
of
by
a
P definite
technical
AJdukiewicz's
index.
symbolism
f u n c t o r of p, t h a t i s f r a c t i o n a/at.,a.,...,a . \ Z n By D e f i n i t i o n
II.4c,
Note the
the S is
in
categorial
a-forming the
this
sum
connection index
of
Index, h a s t h e of
all
sets
of
that the
form
in main
of
the
well-formed
54
CHAPTER II
expression
tokens
of
a
finit-e
order
Cgreat-er
about-
well-formed
t^han o r
equal
to
tokens
we
zero). Instead
of
speaking
s h a l l h e n c e f o r t h r e f e r Jvist t o w e l l - f o r m e d The
set
following
S
of
we
well-formed
AJdukiewlcz
syntactically Now
all
in
S
can
adopt
be
a
expressions.
expressions
that
connected
expression
respect
-
of
J? we
call
the
shall se t
of
expressions.
defined
otherwise.
successive
To
substantiate
definition,
which
is
this
claim
auxiliary
In
character. DEFINITION
II.S
s
p n>l
p^,p^,
. . . ,p^eX
A, C The set
X satisfies
a superset
of
expansion condi
uhose
tion
We
the
m
is
P
also
give
tfte set
all also the
E words in
Ca,a.,...,a
condi
and
'^
tion
'^ ^O '^l ,a > •* p e
SCXy
if
e.very compo-and are
that
in
*^n
X and
Xi.
and
only
if
expression which
it
is
with
an
satisfies
the
set.
theorem
which
could
replace
the
adopted
definition of S. THEOREM
II.2
S - n < X I 6<X> >. S is The
the
least
proof
ourselves of
is
set given
that in
satisfies the
the
Annex. In
condi that
tion
6CX>.
proof
we
avail
THE AXIOMATIC SYSTEM TSCL OF SIMPLE CATEGORIAL LANGUAGES LEMMA
II.1
S S
m 0, would also
in
be
be
empty,
accordance
Definition
empty,
which
with
too.
would
II.4b,
each
H e n c e , by
II.4a
the
Definition
contradict
Definitions
of
Conclusion
and
II.3a
we
at
CONCLUSION
II.9C
° S - £• m s The s e t that
by
have
of
vords
of
an index,
is
£,
r) y !H 0.
DCL)
that
is
the
set
simple
iiiord
tokens
non-empty.
C o n c l u s i o n s II.9a,b a n d II.6a,b y i e l d t h e CONCLUSION
of
II.9d.
The
sets
DCiy,
successive f)Ci.J>\K,
E,
and
E
are c
non-empty. We shall now adopt definitions of two sets: the set B of basic
expressions
of £
saxd the s*f
F of
all
functors
all
of that
language. DEFINITION
II.6
'^ The
set
of
xoell-formed
that It II/4
set
have
'
o
expressions of
is
£ vthich
have
the
set
a basic
of
all
those
index.
II.7
of an
follows and
basic
expressions
DEFINITION
The
all
II.3
'^ is the
functors index
which
from
the
that
is
not
' set
a basic
definitions
the
index
o well-formed
of
of of
expressions
index. B and a
F and
functor
from is
Axioms
always
a
THE AXIOMATIC SYSTEM TSCL OF SIMPLE CATEGORIAL LANGUAGES
concatenation whereas
of
the
They
of
are
immediately
least
index
concatenation sets.
Cat
of
indices. also
from
two
-
cf.
a
basic
This
shows
non-empty.
Definition
Theorem
I.
expression that
The
II.6
59
never
B and
Axiom
is
F are
non-emptiness
and
indices,
of
11.11
a
disjoint B
follows
because
we
have CONCLUSION
11,10 CSN^'sJ n
There
is
basic
a compound
S ^
well-formed
0.
expression
of
X which
is
a
expression.
The
non-emptlness
II.9a,
Definitions
of
F
follows
II.4c,a,b,
immediately
Convention
from
Conclusion
II.3b,
Theorem
1.13'^b,
Definitions
II.6
and
II.7,
that
the
sum
and D e f i n i t i o n II.7, It
also
Conclusion
follows I1.6a,
immediately
Axiom
II.7,
from
and
Conclusion
1.3
o f B and F e q u a l s t h e whole s e t S. Hence we h a v e THEOREM
II.4
S - B u f / ^ B i « 0 / N F ^ 0 ^ B n F « i 0 . The
set
of
two
non-empty
and
that Since,
sets
k
of by
0 S \ S,
demonstrated
all and
all
well-formed disjoint
func
on t h e
setst
that
of
of
all
£
is
the
basic
sum
of
expressions
tors.
Conclusion where
expressions
fe
II.9a, >
0,
strength
II.1. T h i s i s why t h e s e t
S\ S is of
Is
non-empty,
non-empty, Definitions
too,
each which
II.4c,b,a
SN S, which i s equal t o t h e
and
set
of can
the be
Lemma
60
CHAPTER II
^ c
Ca,a.,...,a
,a^» >
1
71
O
c a n n o t be e m p t y . Hence t h e r e a
finite
Theorem
Is a p
number
e
of
1.13 b, t h e r e
S such t h a t
indices. Is
a
Hence,
is
by
p e S such
s o t h a t , by D e f i n i t i o n II.7, p
a
that
a concatenation
Definition
II.4c
P^^eS aind a
e
of and
I\I
,
find
a
X;
it
Is a f u n c t o r . This yields 0
CONCLUSION
11.11
^S n F J' 0. There It
in £ a simple
follows
basic
from
expression
follows that
is
Conclusion aunong
uhich
11.10
that
the
compound
Conclusion
II.lt
and
always
a functor
fi'om
we s h a l l
expression
find
is
we
a /unc
shall
always
expressions
Definitions among
the
tor.
of
II.4a
and
elements
of
11.3a the
v o c a b u l a r y o f Jf. The
following
theorems
have
close
connections
with
Theorem
II.3: THEOREM II.5a
peB^Cf^p^qcB. An expression
and
expression is
of also
£
a basic
wttich
is
equi/orm
expression.
with
its
basic
THE AXIOMATIC SYSTEM TSCL OF SIMPLE CATEGORIAL LANGUAGES THEOREM
Il.Sb
An expression also
61
a f tine
of
£
vttich
is
etfuiform
with
its
functor
is
Theorem
II.3,
tor.
Theorem
II.5a
follows
Conclusion
II.7
and
Deflnlt-loti
n.7.
Theorem
from
Axiom
Definition
II.6,
Theorem
II.5b
II.2;
II.3,
Conclusion
II.7,
follows
from
Theorem
1.4 a,
and Axiom 1.7.
S e c . II.6.
Synlact.ic c a t e g o r i e s
The
concept.
E.Husserl's 11900/013.
syntactic
category
Bedoufungskategorien Husserl
the point of sentence,
of
and
used
interchangeable
CSemantic
distinguished
view o f
his
semantic
s y n t a x , namely t h e
them
originated
with
role
reference
Creplaceable>
to
from
categories)
categories
from
t h e y c a n play i n classes
expressions
of
in
a
mutually sentential
contexts. A similar semantic
intuitive
category
syntactic In
that
of
the
was associated
by S.Le:^niewskl,
categories
C193S1.
sense
course
"syntactic
119291, of
the
A.Tarski
time
the
category"
the
author
of
11933),
and
more
was
with
the
of
theory
of
K.AJdukiewicz
adequate
adopted
concept
term,
by
namely
I.M.Bochetiskl
11949], y.Bar-Hlllel 11930,1964], and c o n t e m p o r a r y l o g i c i a n s . The l a t t e r agrees
with
do n o t its
always u s e
origin.
In
that
term
in t h e
the
present
manner
book,
too,
which when
62
CHAPTER II
formulating usual
t-he
definition
intuitive
sense,
of
which
that is
term,
due
we
to
deviate
certain
from
its
difficulties
in
symbolized
by
d e f i n i n g t h a t t e r m , o u t l i n e d i n Sec.II.1. A syntou:tic Ct
.
We
category
observe
an
Conventions
definition of t h a t DEFINITION
with,
index
^
II.3a-c
will
and
be
adopt
the
following
concept:
II.8
Ct^ - < p I a % ? >. A syntac
tic
category
of
£ whose
expressions The
intuitions
category
defined
with, index
is
connected in this
an
index etfuiform
with
way
f
the
are
is
the
with
f.
concept
explained
by
set
of
of
syntactic
the
those
two
fttac
given in Sec.II.7. Now Ct
will be used to denote the family
categories
of
expressions
of
Jf,
in
of
all
syntac
tic
accordance
with
the
following DEFINITION
II.9
Ct m i ct The is
the
indices By
family
of
family of
of
syntactic
all
expressions
Conclusion
demonstrate
Ct that
agreement.
categories
syntactic of
Il.Pd
that
which d e t e r m i n e s categorial
all
I ? e t<Ey >. of
categories
expressions whose
of
X
indices
are
We
shall
£. the
is
set its
partition The
E
is
logical is
the
definition
of
non-empty. partition.
relation the
The « of
relation
relation syntactic
»
hais
the
THE AXIOMATIC SYSTEM TSCL OF SIMPLE CATEGORIAL LANGUAGES
following
form:
DEFINITION
11.10
P ^ 9 «• The
63
expression
p >• q i s
y P>9 e Ct .
r e a d : p and
q belong
to
the
saune
c
syntactic In to
category.
accordance
the
same
with
Definition
syntactic
category
11.10
if
two
and only
expressions if
they
belong
belong
to
a
s y n t a c i c c a t e g o r y w i t h a c e r t a i n index. Definitions I.la-c
11.10
aind
II.8,
Conventions
II.3a-c,
and
Axioms
yield
CONCLUSION
11.12
p - % t .
i4 const i/went equiform
with
o/
that
order
zero
of
a
given
expression
expression.
< e C* •» V "^ . „, X fP " pCt„,t^,...,t > ^ p n>l tj^,t.,...,t eD0=».CfeC«» ^
A constituent finite
is
11.16
t e C P t ^ C P
^ Q ^ a p ^ t e C , «J x v t i i % ; t * v e C . P
of
a
68
CHAPTER II Pi>om
11.5b
Oefinlt,ion
and
II.2a
11.12,
we
arrive,
Convent.lon by
II.3a
applying
and
Conclusions
mat.heinat.ical
induction,
at. CONCLUSION II.17a
P
P
in accordance wit.h which auny const.it.uent.s of
an expression
of
£
are words of £ which have a cate^^orlal Index. If
p
belong
a
compound
t.o the
belong II.9>
is
t.o
it.s
are
applying
set. E
well-formed
equal
induction
det.ermined
expressions
and
by
Theorem II.3 we easily arrive
t,hen
p does
S, and t.he const.it.uent.s
unambiguously
well-formed
expression,
by
of
expansion
referring
to
p which
Csee
Definilions
not.
Axiom
II.4c,a,b.
Definition
By
11.12
and
at
CONCLUSION II .17b
p « S ^ c " £ C P P in
accordance
with
which
any
=S,
constituents
of
well-formed
expressions are well-formed expressions. Every some
constituent
syntactic
of
a
position.
compound Further,
expression to
every
occupies
in
constituent
it
of
a
given
compound
sequence ot makes
it
expression
we
oaxx assign
a
finite
natural numbers greater than or equal t o
possible
to
determine
its
syntactic
ordered
zero
position
which
In
that
of
th.6
expression. Let consti
the tuents
symbol of
the
C expression
Cn>0>
denote
the
p Cof order other
set
than 0) u^^lc/l
THE AXIOMATIC SYSTEM TSCL OF SIMPLE CATEGORIAL LANGUAGES
occupy
in
It
the
adopt, t h e f o l l o w i n g
syntactic
definition
69
position of this
y-Jt,J^,->J^>-
We
term:
D E F I N I T I O N 11.13
** p e p word of the expansion of p,
the
a.
f < s C
b.
Ac > 0
^ Ct e
Juj.^""*- word of
C ^ P
'^
E c
^
t
is
eaxiiform
C P
4* t
the expansion
of
II.13a
and
with
is
the
J.-th 1
»quifoT«t
with
of
set
a constituent
the
" >.
From
Definitions
and
b
II.12b
and
c
we
easily
arrive by induction at the following conclusions: CONCLUSION
11.18
n>0=*C
* ^
" S C " P
P The
constituent
determined
by
of
the
the constituent
p
n-term
of n-th
which
occupies
setfuence
order
of
of
in
it
natural
the
posi
tion
numbers Cn>Oy is
p.
C O N C L U S I O N 11.19 N.
rt
^^
^n
*
P Every the
consti tuent
syntactic
natural
We further
.
order
determined
Cn>0) of by
1
^
n ^
P
an
p occupies
n-term
in
seifuence
it of
numbers.
now proceed analyses,
replacement
of
a
to formulate of
the definition, important for
the four-argument
consti
tuent
of
an
relation
expression
of token.
Its
70
CHAPTER II
formulat.lon reltxtion 0)
of
The
requires
adopt.ion,
Cy"> of
ttxs Teplacsmeixt
a givon
expression
expression
obt.alned
t.he
from
of
lnduct.ion,
a constituent
Is
read:
expression
s
by
t.he
expression
t.he
n-th
order
the
expression
rCp/tfys
is
read
r
replacemen-t
c o n s t i t u e n t Q o f n - t h o r d e r by t h e e x p r e s s i o n The
of
of
tohen.
rCp/q> s
the
by
Is
of
its
s,
but
p.
analogically
as
rK.p/'tfi
w i t h t h e o m i s s i o n o f t h e p h r a s e "of n—th o r d e r " . Here
are
the
definitions
of
the
relations
under
consideration: DEFINITION
a.
rCp/<j> s 4* s,r
The
expression
replacement expression such
11.14
that
of
its
p if
and
tf is
is
obtained,
from,
constituent only
eQuiform
if
the
*.
replacement obtained
s
s
equifornu
obtained
r
the
of
expression
s «•
constitvent
from
first
is
expansion
of
are
its
obtained
the
expansions
k > 0 ^ CrCpyqy
The
lt.s
the
the
q arid having
determines
numbers other
c.
with
in
71
s
by
consti the
Cgreater readings
Just.ificat.lon
obtained
in
tvent
from
the
q
p
by
replacement than
or
of
t^he
the
equal labels
conclusions
of
expression if
and
its
only
by if
constituent
to 0^ by
the r
is
q of
a
p.
rCp/(}> s which
s
and
follow
rs directly
has from
D e f i n i t i o n s 11.14 and 11.12 amd C o n c l u s i o n II.6a: CONCLUSION
11.20
a.
rCp^q)
s -^ q e C
b.
rCpyq^s
^ q G C
In a c c o r d a n c e replaced
in
a
with given
^ p e C , ^ p G C .
these
conclusions
expression
is
the its
expression
which
constituent
is Cits
72
CHAPTER II
c:onst.it.uent. a
of
definit,e
const.it.uent.
expression
Cits
that
same
order.
obtained
replacing
Moreover,
expressions
and
const.lt.uent.
is
replaced and the
ordei>>,
of by
the
are
replacing
a
definite
are
order)
of
Further,
constituents
constituents
same
expression
replacement.
constituents
they
occupy
t,he
which
syntactic
of
is the the the
in
compound
position
because
Definitions II.14b and c, 11.13a and b yield by induction CONCLUSION
11.21
r 0 " » , .
.
. ^
rCp/g>s
^ r « s ^ p « q c
//
an
category
expression as
replacement
and
that C2)
r,s e S
and
that
there
are
s^,s
expressions
,...,s
e
S
Cn >1>
and
e S such that
r«
C4)
and t h a t f o r a c e r t a i n n a t u r a l number j '
such
that
•'I
we
0 < jf
< n
have
if ^
s
A
and C7>
p % r
and
also
Since s formulas II.9
-
and r a r e C2>-C4!>,
they
than
zero
well-formed
Conclusion
belong
II.4a> and a r e a t
an
A
to
the
unambiguously,
Hence and
II.6a, set
Definition
SN S
<see
II.4c>.
the in
By
Axiom
expansions
view
of
of
11.3b,
and
Definitions
t h e same time e x p r e s s i o n s
CDefinition
injection.
and compound e x p r e s s i o n s
of
II.9 s
and
Definition
II.4b
r
and
the
atnd
greater
function are
by
Axiom
II.3a
an o r d e r
the
-
p
is
determined formulas
THE AXIOMATIC SYSTEM TSCL OF SIMPLE CATEGORIAL LANGUAGES
C3>
and
C4>
s
respectively.
and
r
Under
sat.isfv
t.he
Convention
75
condlt,ions
II.3c
we
tn s
atnd
record
tn , r
these
conditions, respectively. In the form COS
c
cd,d^,d^,...,d^
C
on
the
s t r e n g t h of Conclusion II.7, t h a t
*"'
0,
follows
that
it
view
on
76 p
CHAPTER II -
this
lemma
s and r ^
holds
Tor
k>0
and
n«l.
n—h. A s s u m e
s.
c
We s h a l l d e m o n s t r a t e hence t h a t
it
follows
that
from
under t h e s e
the
a s s u m p t i o n s p ^ Q, amd c
assumption
that
f o r n-M 0> t h a t i t a l s o h o l d s f o r n-ft+1. By Definition II.4c it follows from r
s
that
s
and
k rCv-yty
there
1 w Cp/«j> t .
the assumption stating V Cp/ (
are
expressions From
that
r
and Lemma II .4 h a s
the
m s c
Lemma II.4
the
v
and
inductive
we i n f e r
assumption t
such
that
assumption
that
been proved for
holds
v
n^l,
and
• t 1 c* 1
Since
we h a v e
that
P - rCp/9)s If
an
expression
r
of
const
replacement syntac
tic
its
category
syntactic
as
is
^pmif'^rms.
obtained i txtent
if,
from
an
expression
ef by p tiihich
then
r
and.
s
s
belongs belong
to to
by the
the same
the
same
Definition
II.14d
category.
P r o o f .
The
second
fttsc
follows
from
r
S/%p" hold, we h a v e A
d
* c
0i and
conclude
that
*
k'
hold
d
%
c,
we
may
so
refer
that,
by
C o n c l u s i o n 11.12 and C o n v e n t i o n II.3c we h a v e t h a t T ^ c It
remains
to
prove
that
if
Lemma
to
II .5 h o l d s
Theorem
Axiom
I.lb.
s.
for
a
natural
number n"fc 0>, i t a l s o h o l d s f o r a n a t u r a l number n"*+l. Assume
accordingly
that
t h e same applies
shall
demonstrate
that to
the
the
inductive
formulas
assumption
r
k+1
s
c
we
can
infer,
by
Definition
II.14c,
that
there
are
expressions
78
CHAFTCRII
V, and 1
t^ s u c h 1
t.hat.
It
rCu^/t^) s 1 1
1
and
-U-Cp/q) t^. 1 1
1
Since
v^Cp/Q> t , 1 1
h o l d s we i m m e d i a t e l y h a v e t h a t , v^ • t^ b e c a u s e we h a v e a s s u m e d 1 c 1 t h a t P " <J and Lemma II.5 h a s b e e n p r o v e d f o r nmi. Since c
r ( u / t ) s,
r
m
1 1
s
follows
immediately
from
the
inductive
c
assumption. The t w o f t t s c THEOREM
t a k e n J o i n t l y make i t p o s s i b l e t o
prove
II.7
rs ^ Cp — <j • • r » s ) . c
Two expressions if
and only
-formed which
if
of
on replacing
expression belongs
belonging confirms
of
to
Theorem
II.7
of
X heloixg
the
one
to
the
same
of
them
by
£ u>e ohtain same
cannot,
syntac
of
tic
correctness
to
the
be
other
as
used
same
the
tic
category in a
xnell-
expression
category
the
of
syntac
a well-formed
course,
two expressions
the
c
the
as
a
X
former. definition
syntactic
definition
of
category. of
of It
syntactic
c a t e g o r y a d o p t e d i n S e c t i o n 11.6.
S e c . II.8.
The a l g o r i t h m o f c h e c k i n g t h e
syntactic
correctness of expressions
The l a n g u a g e theory
TSCL,
should
be
expression of
X.
The
procedure
X,
which
ahould
an
be
is
to
algorithm defined
in
object
decidable,
algorithm
belongs
the
which
enabling
the
set
to
be
us
S of
investigation is
to
say
to
check
the
well-formed
described
AJduklewicz
of
below
C1935]
that
whether
is and
by
a
the
there given
expressions
a
generalized
intended
to
THE AXIOMATIC SYSTEM TSCL OF SIMPLE CATEGORIAL LANGUAGES
investigate or
the
syntactic
certain
ideas
pertains
syntactic
connectedness,
correctness
of
to
in
be
found
about
precision,
but
it
we
there
shall
AJdukiewicz
not
seems
i.e.
expressions.
t o compoiind e x p r e s s i o n s o f
comments
79
It
also
refers
[19603. T h a t
be
observe
no
point
the
in
to
algorithm
Jf. When f o r m u l a t i n g
always
to
well-formedness
further
rigour
making
of
those
comments more p r e c i s e . We
begin
algorithm, which
with
several
which,
are
it
usually
general
seems, made
are
on
remarks
somewhat
such
an
on
the
concept
different
occasion.
from
We
do
of
those
not,
of
operations
or
c o u r s e , claim them t o be original. An algorithm operations directives.
is
(IP)
fini
and
the
of
should
Cthe
determined
should
have
number
indicate
earlier one
the
them the
directive of
should a
given
should
define
successive
the
essentiad
directives
should
are
appropriate
several
performance
of
operations about
by
Ceach
effective
Cone o f
should
inform
psychophysical
machine,
an
teness
the
of
per/ormaibility of
teness
others
results
directive
as
defini
a
directives
possibility
operation),
operation,
by
Those such
the
finite>,
sequence
performed
properties secure
a
the
first
operation
known,
and
characteristic
be
once
the
last
feature
of
t h e final operation of a given algorithm). When
formulating
algorithm of a given about
language
the
lad>el o f
checking
the
Jf we
symbolism
Jf i s
the
of
directives
which
well-formedness
avail £.
ourselves First
of
recorded with a finite
of
from all
we
determine
the
the
expressions
of
several
assumptions
assume
that
number o f
every
s y m b o l s . Next
we
80
CHAPTER II
assume they
t.hat.
are
them i s that the
about,
any
two
equiform,
and
about
the
concatenation
about
any
vocabulary
and w h e t h e r for
any
parts to
it
it
and
the
additional and
second is
labels,
remaining
decide
it
that
of
£.
Further
sense
its
of
X
we
Le^niewski's
which
of of
parts
that
of
Jf we c a n
amy
word
has
an
indicate
that which
correspond
corresponds
we
the
index,
aussume
correspond
Finally
from
indicate
of .it
of
assume
a
we
can
it
functor.
is
mereology>
e x p a n s i o n , which p a r t and
whether
whether
of
whether
t w o . We aJso
an e x p r e s s i o n
expression
decide
to
to
the
assume
that
categorial
index
it.
in
with
by the
the
way
label,
treated
belonging t o
the
a
is
made not
not
functors which
structurally
following
to
checking:
calculus, we
force
compound
the
subject
because
binding
a word
above
sentential
x+y • z ,
the
main
are
the
label
are
however, as
of
about
labels
them
example,
notation
information
in
assumptions
of
arithmetical
+. T h e s e
indicate
can
word
expression
laJ>els,. c h o s e n label
three
the
can
we
a
compound
In a c c o r d a n c e
the
of
£
is
Cin t h e
sissigned t o
any
whether
arguments
every
of
it
functor,
successive
we
£,
is
words of
main
for
of
given
of
the
Its
label
labels
of
their
has
ambiguous
an
index name
The s e q u e n c e o f c a t e g o r i a l
I s c a l l e d t h e intermBctiats
conventions. indices
sxpansion
of
tfte
exprossion
the
functors we
cannot
arguments.
arithmetic.
We a d o p t t w o t e r m i n o l o g i c a l
lack
the
expressions;
and
^-i,
because
The it
expression
THE AXIOMATIC SYSTEM TSCL OF SIMPLE CATEGORIAL LANGUAGES
c6y
pcp^.p^
81
p^j)
which has i t s expauislon in t.he sequence of e x p r e s s i o n s
If and only i f f o r any i CO t h a t
it
satisfies
tKe
tnp i f and only' i f t h e index f^ ^0 i s t h e concatenation index a of t h e e x p r e s s i o n p and of t h e indices
the
?-,?_,...,? 1 ^ n, s o t h a t t h e following holds: It
is
assumed
indices t h a t
it
about
every
i s possible t o
finite
sequence
of
decide whether i t
categoriad
does, or
does
not, s a t i s f y t h e condition Itt . Note t h a t i f of
the
t h e sequence
expression
condition
*«,r-. i f
C^> i s
C£> t h e n
the
and
if
only
an intermediate
latter that
sequence
condition
expansion
satisfies
is
the
satisfied
by
CC? Csee Condition I.ll>. Before
we
algorithm
for
shall the
describe t r e e of
compound
formulate checking an
the the
auxiliary
derivation
expression
directives
well-formedness al{;orithm,
Such
a
termed t r e e I of a given expression.
of
namely
Calso termed t h e token.
which
determine
the
expressions,
we
that
of
dendrite)
of
diagram
will
drawing a
given
henceforth
be
82
CHAPTER II
By r e w r i t i n g we
the
want, t-o c h e c k
writing to
we o b t a i n
immediately
write,
the
the
expansion
the
second
compound
below,
expression
the
on one
words equiform of
the
line
of
and t h e
with t h e
expression tree
first
I.
If
whose
well-rormedness
line
same
of
level,
successive
under
tree from
such
By left
components
consideration
every
I.
word
of
we
obtain
a
simple
is
e x p r e s s i o n o r i f s o m e o f t h e m i s a compound w o r d which i s n o t compound
expressions
decomposed tree the if
I of
into the
its
Cand main
not
every
compound w o r d s
of
words
which
expansion, way
we
until
are
in t h e
obtain we
perform
of
the
tree
I
that
such
an
follows
from
one
a
simple
expansion
of
the
a
line
are
whose
expression
case
I.
We
all
on
of
we
words
line
proceed
of
its
In
this
this
way
two. in
elements
write
are
simple
t h e m i s a compound w o r d s which i s n o t By w r i t i n g
the
is
fact is
tree
If,
given expression
successive
of
the
and
the
three
then
expression,
with
element
there
is
in t h e
be
completed.
equiform
compound
the
been
such
line
we
has
word
way a s
cannot
arguments),
each
compound e x p r e s s i o n s . line
such
its
that
under
e x p r e s s i o n s or some of
a
and
word
then
same
obtain
a
in q u e s t i o n
the
compound e x p r e s s i o n s ,
is
functor
expression
contrairy,
hence
last
down t h e
operation
expression obtained that
in
an element
in
in
a
last
of
element
the
finite line
which i s
of
The
a
such
construction
question.
each
a
of
conclusion
number
of
except
the
steps last
r e c o r d e d by m e a n s
a g r e a t e r number o f s i m p l e w o r d s t h a n e v e r y e l e m e n t o f t h e
of
next
line. If
in
tree
I
of
a
given
expression
some
of
its
final
nodes
THE AXIOMATIC SYSTEM TSCL OF SIMPLE CATEGORIAL LANGUAGES
is
not
a simple
expression), of
that
then
Ajduklewicz final
simple
the
is
nodes
is
not
tl93S]
it of
thoroughly
of
operations
.
expansion
out
of
the
of
checking
been performed,
consideration
is
of
the
a
accordance they
Once
we
the
C0
zero.
85
of
addiMon,
We
proceed
t,he t-o
relat.ion
check
t-he
expression
X + (y
< 0>.
We c o n s t r u c t t r e e I o f t h e e x p r e s s i o n CTT1> Csee Flg.II.l). .
Fig.Il.l. As
can
be
well-arranged.
T r e e I o f t h e e x p r e s s i o n
seen, We
x+Cy
construct
is
tree
thoroughly II
of
the
c a t e g o r l a l I n d i c e s o f t h e e x p r e s s i o n Cnl> Csee Pig.II.2>.
•> n
s/'nn
Fig.II.2.
of
T r e e II o f t h e e x p r e s s i o n (rTl>
Trees
I and II o f
three
l i n e s e a c h . The l i n e s o f
expansions: t h a t expression equiform
being
of
the
the
and
letters
under
tree
expressions
checked
w i t h y Its
of
as
concatenation
of
arguments
E x a m p l e
a
certain
indicate,
the
of
typographical
operations
Crtl)
indices
belong
linear"
functor
functor
successive
which
of
arguments
does
category
For
slant
of
respectively,
the
/
symbol
essentiad
tree
of
the
the
given
expression
index
expression
a
with
successive
The
t-o
Cconcatenations
well-formedness
by
are,
expressions
that
the
the
-
expressions.
before
after
of
of
equiform
occurs
occur
II
"quasi-fractional
are
expression
which
indices
the
tree
belong
functors
in
which
of
which
the
the Hence
is
the
index
of
that
indices
of
the
the
expression
expression.
the
syntactic
correctness
of
calculus:
p •• C-i . tree
expression shown in
I
Fig.II.4.
is
shown IS
in
Fig.II.3.
thoroughly
It
can
well-arranged.
be
seen Its
that
tree
the II
is
87
THE AXIOMATIC SYSTEM TSCL OF SIMPLE CATEGORIAL LANGUAGES p-»C-iC-i>vp>
Pie.II.3.
T r e e I o f t h e e x p r e s s i o n
sySS
•. S
OS
s/s
Flg.II.4. Trees now
under
that
accordance first
index
of
index
of
expression that
and
II
T r e e II o f t h e e x p r e s s i o n Cn2> of
the
consideration
expression
carrying
the
I
out with
the
of
expression
The
three
of
satisfies
functor
second
of
lines
intermediate
those the
of
the
desired is
that
Indices o f
expression each.
the
intermediate
a
Tree
II
adgorithm -
we
condition
so
forms
successive expso^sion
-
In that
that
the
of
the
Cl.e.,
the
arguments also
of
When
find
concatenation
functor
Cn2)
expansions.
expansions
implication which
calculus
four
operations
order
Crr2>> and t h e
functor.
consist
basic
the
expansion the
sentential
determines the
s/s
of
satisfies
CHAPTER [I
88
the
desired
the
contrary,
condition
condition the
tn
-
In t h i s
third
.
Hence
expansion we
under c o n s i d e r a t i o n i s n o t E
x
a
m
p
l
e
case
can
the
does
c o n d i t i o n TO , -
not
conclude
satisfy that
the
the
On
desired
expression
well-formed. II.3.
We
shall
nov
examine
the
w e l l - f o r m e d n e s s o f a n o t h e r a r i t h m e t i c a l e x p r e s s i o n , namely
X • Cy + < - « » -
In r e c o r d i n g pertaining the
that
to
the
brackets.
expression
determines eight
v> + <x • C-e)>.
expression
we
observe
constants
•,
+,
-,
tree
I
We
aind
Cx
construct
note
that
each
the
the
tree
usual
conventions
variables
and has
tree
x,y,s,
II
five
of
this
lines
and
expansions.
X- < y - K - « ) ) - C x y ) + < x - C-«>)
Cxy)+Cx
Flg.II.5.
T r e e I o f t h e e x p r e s s i o n C7T3)
and
p
in
which
eqxjlforni the
the with
category
operators, the of
a n d '^ X = X
namely
symbol
^,
sentences
the
bind,
and
a
as
X,
universad
respectively, variable
of
quantifiers,
a
the
variaJile
of
category
of
names. The
elimination
ambiguity" o f of into
a
of
operators
syntactically diversiform
conventions
P^>P^> of
the
operator p ,
p
are
called,
expression
•» •»