Logique A n a ly s e 161-162-163 (1998), 145-153
INVERSE NEGATION A ND CLASSICAL IMPLICATIVE LOGIC
V.M. POPOV
*
Abstract
The central result (theorem 5) of this paper arose within the context of
a study o f the problem o f transfer from implicative-negative logical
systems. This paper has been inspired by one of the ideas of the paper
[I] by A. Arruda. This idea is that a propositional variable is interpreted as some implicative formula, but propositional variable negation is interpreted as inversion o f the formula which interprets this
propositional variable. As we see it, the given interpretation of propositional variables and their negations opens the opportunity to enrich
implication theories by such negations, the nature o f which is essentially defined by these theories. We are building a calculus C ' / : and
showing (see theorem 5) that the set of all formulas being deduced in
C/ 7 ' is the least implicative-negative logic with inverse negation, the
latter being adequate to the implication o f the classical implicative
logic. In this paper a sequential version G Cn
presented,
the decidability and paraconsistency of Cl i s established.
i
the
connection
of Cl7 with the calculus VI o f A. Arruda [ I1 and with
the
calculus
PIL
,: o f
t h ofe D. Batens [2] is considered.
c a l c u l u s
C
I
i
s
Let LD and LD —, be standardly defined propositional languages. The alphabet o f the language L D (correspondingly the alphabet o f the language
LD ) contains only propositional variables pi, p2, , binary logical connection D, implication (correspondingly logical connection D, implication
and unary logical connection —
of
formula
is standard.
1,a n
e g aint these
i o n languages
)
We
call
implicative
logic
(correspondingly
implicative-negative logic)
a
n
d
any
non-empty,
closed
as
regards
the
rules
of
modus
ponens and substitup a r e n t h e s
tion,
set
of
formulas
in
LD
(correspondingly
in
LD
)
.
The set KD —, o f all
e s .
classical
tautologies
in
L
T
h
e
LD
are
correspondingly
implicative-negative and implicative logics. Using
D
,
a
n
d
d
e
f
i
n
i
tt hi eo
n
s
e
t
K * This work
D is sponsored by INTAS-REER. project No. 95-365.
o
f
a
l
l
c
l
a
s
s
i
c
a
l
t
a
u
t
o
146
V
.
M
.
POPOV
standard terminology, we call KD --, the classical logic and KD the classical
implicative logic. We call paraconsistent implicative-negative logic any
implicative-negative logic L such that for some formula A in L D-, the closure L u t A ,
LD
W e call any formula of LD-, a simple formula if this formula does
not
subformulas o f —
1 A contain
)
1
following
A
mapping
a
n dT of the
—set of all simple formulas of the language L
b y
into
the
set
of
all
formulas
of the language LD:
1
u
(
A
m o d
,D
u s
T(Pn)) (P 2 n
B o
p
n
- i 1 nD
ke n
s B)) =d (T(A). D T(B))
T((A
P 2 D
1 1e
W
i
)
d
e
f
i
swhere p
,
T
n
e
n
n is
toMain( Definition.
h
Let fo r implicative logic LD and implicative-negative
a n(
e
tlogic
LD-,
the
Jo/lowing
condition be satisfied: for any simple formula A
a r Pb
ei t r
AD
n — E) L if f T
qa r ( yA
) the logic LD-, is called implicative-negative logic with inverse negauThen
v a)= E
and its negation is called adequate to implication o f implicative logic
ation
rL u . i L( l D
ta b
P
ol e
It is 2known that the sets K u and K
ta
following
correspondently call C P and C I D ' . The calu
,n c calculi
a n whichb we e
hculus
n xD Ci Po is m
a standard
a
a t i calculus
z e dof Hilbert type over Lu; the calculus CP
ehas
d
modus ponens and substitution, and only
t only
Ph two
2r deduction
o
u rules:
g
sthree
axioms:
(P
i
D
(p2
D
p
i)),
((p i D (p2 D p3)) D ((p i D p2) D (p D
A
h n
ea
p3))),
(((Pi
D
p2)
D
i
n
)
D
p
1).
The definition o f deduction in C/ D is
t
h
e
tstandard.
The
calculus
CP
i
s
a
standard
calculus of Hilbert type over the
n 1
)
L
u
;
the
calculus
Cl
h
a
s
only
two deduction rules: modus
olanguage
d ,
and
substitution,
and
only
four
axioms
—all axioms of C/D and the
fponens
B
axiom
(((-1/21)
D
(-1
p
2
))
D
(p2
D
p
i)).
The
definition o f deduction in
aa
C/D -7 is standard.
lr
le We define the calculus C /
Lu.-., which satisfies the following conditions:
fs z
s
a
oi 7• thea calculus
t a n d aCl r h da s only two deductive rules: modus ponens and
rm s substitution;
H i l b e r
m
p t• the calculus Cl7 has four axioms: all axioms o f CP and the axiom
ul
((pi D P2) D (((
D
P2) D P2));
y
p
le t• the definition
of deduction in C' I
af en i
l
c
u
so c7 i a
s
s t a n d a r d
l
u
s
ir [
3
]
.
v
nm o
e
r
u
l
INVERSE NEGATION AND CLASSICAL IMPLICATIVE LOGIC 1 4 7
We define the sequential calculus GCn, a s a standard sequential calculus
over LD sa t isf yin g the following conditions:
• the calculus GC'17 has as the main sequences all the sequences of the
A - ) A type (here A is a formula in LD ) and only these sequences;
• the calculus GC1 h a s as the initial rules only the fo llo win g nine
rules (here and further A and B are formulas in L 3 , and F, A,
are finite (perhaps empty) sequences of formulas in LD ) :
1"--3 A, A,B4O A , A , 1 " 0
0 , A ,
A,T (1 )
r -> 0
r
>
A,A B , - › 0 A ,
r
A
,
A , T " - “ J ' r - - - 3 0 , A ' ( A D B ), F , E -> A , 0 - ›
B
A, r--- o•
r
A
'
A , ( A ) B )'
o , (-1A)
• the definition of deduction in G CV ,
- i s s t a n d a r d
Note
f 1.oThe rsystem obtaining from GC17 by the addition of t
the
l rule
sequential
o f classical i m p l i c a t i
- s ineitia q
u ise a n
t i a version
l
logic.
directly
follows
from
the
fundamental
results of G. Gentzen [3].
(- ) ce . ItaA
l ac s u
l
i
.
vtit
Note
2. Using Gentzen's method [3] one can prove that the cut rule A. I-)0
r
is
admissible
in GC71
e ig tive
Lemma
1. I f Cr: I- A, then G g ' ; F- › A.
',,
This lemma can be proved by induction on the length of deduction in the
calculus u s i n g the admissibility o f the cu t rule in the calculus
GCl
.
Lemma 2. I f GCC7F F -) t h e n CL,'
Here and further i s the mapping of the set of all sequences over LD
into the set of all formulas of this language, satisfying the following conditions:
• (1)(A1., , A „ ) ( A l D (••• (An D ( ( p 1 D PO)) -));
• q)(A 1, A „ B I ) = (A I D (... (A„D BI) ...));
• (P(A 1, , A
n ...)
- D B
n
› iB
m ) D)
I B
D„ , )
( )
• . .
• . )
• )
Ow
148
V
.
M
.
P
O
P
O
V
Examples:
0
qb(
( —> B i, B2) =((B1 DB 2 ) D B2),
cP(Al
— ---> /31),) ( A l
Lemma
D
) 2 can be proved by means of induction on the height of deduction
in GC1
B
1 n )sid. e rin g that if GC17 F r - , .1 then 3 is not empty. The
= co
latter- is easily proved by means of induction on the height of deduction in
GOP
— (
Corollary (from Lemma 2). I f GC1 — > A, then CC7 I- A.
1
From( Lemma 1 and this Corollary we get the following:
0
Theorem
1 1. C l
7 1DTheorem
A P 2. Calculus Cl i s decidable.
fo llo ws fro m Theorem I and the fact that the calculus
i This
f I f theorem
G
G
C/
,
C )
zen's
-h ) method of reduced sequences following [31.
:— ,
Theorem
3. Calculus C C is paraconsistent in the sense that the set
i) s
Tr(C1
7
,)=
[A I C I - A [ is a paracons'istent implicative-negative logic.
d
A e
Proof
I
t
follows
from the definitions that Tr(Ct f ,
c. i
tive
logic.
That
is
why to prove our theorem it is sufficient to show that p2
n
d
a not belong to the closure as regards modus ponens of the set T r ( C )
does
7
n
i m p l i c a t i v e bu) (p
l i i.s (— a
1)1)1.
To show this, use the matrix MI which is a submatrix o f
n
e
g
a
e
.
Arruda's
matrix MI [11:
T
h MI =({0 ,1 ,2 }, (1,2), D, -1 )
e
where
the operations are defined by the table
d
e
0
I
2
D B)
(
c
o
I
I
1
and
—10
i
1
0
I
1
I 01
d
2
0
1
1
A 12
a
)
b
It's not difficult to check that for any evaluation y of LD i n MI the folilowing conditions are done:
l
i
t
y
o
f
G
C
f ill' •
INVERSE NEGATION A ND CLASSICAL I MPLI CATI VE LOGIC
1 4 9
I.
C
IA I v(A) = I ),
2. t h e set IA v (A ) E 1 , 2 H is closed as regards modus ponens.
Let w be an evaluation of LD-, in MI such that w(pi) = 2 and w(pi) = 0, for
i 0 1. Then, by the condition 1 we have
Tr(cV
n
Therefore, by condition 2 it follows that the closure as regards modus
7) o f the set T r ( C I Z ) u { p i , (
ponens
{1,21),
—
u but by the definition of w we have p
,I Hence,
p i ) )p2 does
i not
s belong to the closure as regards modus pon ens of the
t
p
l
set
T
r(C
I
7
)u
{p
i,
w
i-) n e c l ( uA d e (I d
( Theorem
A, n
) ( 3 is proved.
E
i—
need
1 Now
, we
1 the auxiliary
) calculus VI which is defined as a stanIT
—
,A 2will
dard
Hilbert
type
calculus
over
L D, , satisfying the following conditions:
).
w
p
1
() . 1. V I Ahas only two deduction rules modus ponens and substitution,
) 2. )a x1io ms of VI are the following formulas:
E c (a) a ll the axioms of C l ,
t (b) a l l the formulas (a D ( (
LD w h i c h is not a propositional variable,
A-(c) t h e formulas ((pi D ( —
1 a )
D
w1p ( piP2),
2 ) D)
p 2 ) ) )
3. (t hD
e
definition
of
in VI is standard.
w h e( r deduction
e
Aa1 3.pLet1cp)be)any
Lemma
, mapping o f the set of al/formulas in L D-, into iti
s
self )satisfying
the
following
three conditions:
(a (
1
(
po ) r
m
u
I. E
f of r any
propositional
variable
pi the formula cp(pi) is not a proposiD
l
a
I tional variable,
p
i
n )
2. (,(:)((
— i
1
)
3.
I Acp
) ) ((A D B)) = ((p(A) D cp(B)).
( , Dt
Therefore
( 2
A ) if Cfl h A then Vi cp (A ).
The
) , lemma can be proved by means of induction on the length of deduction 1
in C I D ' .
We
1 define the sequential calculus GVI as a standard sequential calculus
over LD -,, satisfying the following conditions:
•
I. t h e main sequences of G1
7
1 a r e
t h e
s a m e
a
s
o
f
G
C
C
7
,
150
V
.
M
.
POPOV
2. G V 1 has as initial rules only the following rules:
(a) a ll the initial rules of G C C ,
(b) th e rule a , r-)o wh e re a is a formula in L a -, which is not a
propositit;fiaVallable,
3. t h e definition o f deduction in Glit is standard for sequential calculi
Note 3. B y means of Gentzen's method [3] it is possible to prove that the
cut rule rA,- E--,e
'
Lemma
4. I f VI I- A, then Gill 1- -) A.
6
This
lemma
can be proved by induction on the length of deduction in the
• A
calculus
lit,
using
the admissibility of the cut rule in the calculus G171.
i s
a d m
Lemma 5. I f G1/1 - ) A. then Vi F0 ( T -* A ).
i s s i
Lemma 5 can be proved by means of induction on the height of deducbtionl in
e GV1.
i
nCorollary (from Lemma 5). I f GI/11- -•) A, then 1/1 F A.
G
V following theorem is obtained from Lemma 3 and Corollary fro m
The
1Lemma 5.
.
Theorem 4. VI A i f f
GV1
F >
Lemma
A
. 6. I f A is a simple formula, then VI I- A if ic Cr I- A.
Proof According to Theorem I and Theorem 2, it is sufficient to prove that
if A is a simple formula, then
GI/1 H -> A if f G q ,
Let us
; H name- a )sequence F A a simple sequence, if F and A consist o f
simple
only.
A formulas
.
It is easily proved by induction on the height of deduction in GV1 that if
S is a simple sequence and GV1 F S, then G CC'F S. On the other hand, it
is evident that any sequence deduced in GCI i s deduced in GI/1. Therefore if S is a simple sequence, then GV1 S iff GCL:F S. In particular, if
A is a simple formula, then GV1 F -› A if f G q
- Lemma 6 is proved.
L Let us define two mappings * and § which are adaptations to L a -, o f the
from I ll. The conditions for * are:
,mappings
F
- of
) the same
A name
.
INVERSE NEGATION A ND CLASSICAL I MPLI CATI VE LOGIC
1 5 1
(Pn)
*
((-111))*
= (P2)1 D P217-1), , , ii ff A
p a propositional variable.
=
A=
is not
n
( P
The ,conditions for § are:
2
n (P2n) ( (
t
—
D
( n )
'( P
P
-D
21 7. Fo r any formula A in L D -„ if VI I
Lemma
P n
cannbe proved
- This
An
t h e
C 1 by
D induction on the length of deduction in the
)1, lemma
, CI
calculus
D'
.
)
- 1(1 P
I
A ,2) ) *n
.
Lemma
(§ 8. Fo r any formula A in L D ,
- 1 )
(=
§VI I
A
=(D
-lemma can be proved by induction on the number of occurrences o f
ThisC
((
B
o1( connections in A.
logical
)(
nA
)A 9. Fo r any formula A in L D , if C 1
Lemma
D
)
Proof:
9 - 1*()* )1 - ( A ) * ,
t h e n
=
§
V -§ I 1
I
(assumption)
) C1. D
A I. (P
2.(,D VI
I- ((A)*)§
(by 1, the definition of § and Lemma 3)
-1
A F )F (((A)*)§ D A)
3.n
A
(by Lemma 8)
( VI
)
, 1- A
4.)) VI
(by
2,
3,
modus
ponens).
( (
a
*A A
n 9 is proved.
Lemma
D
D) *
d
(B
From
V lemmas 7 and 9 we obtain
B
)
))I
Lemma
l a Fo r any formula A in LD-,,
*§F
)=(VI I- A zffCl I - (A)*.
,(A
D
Letnma
I l . Fo r any simple formula A in L D(
(
i A
(
, ) VI F A if f (A)* E K .
A
D
)
(
*
B
)
)
§
§
)
)
.
.
152
V
.
M
.
POPOV
Proof
I.
2.
3.
4.
A is a simple formula
A* is a formula in LD
(A)* is T(A)
(A)* E K J -, if f (A)* E KD
5.
ClD " A
D
VI A if f
VI
a FA i f
VI
, - F A if f
VI
( AA ) if*f
6.
7.
8.
9.
if f (A)* E K
F
(A)* E KD
(A)* E KD
T(A) E KD
(assumption)
(by I and the definition of *)
(by I and the definitions of * and 1)
(by 2 and the definitions of the sets
K3 a n d K
(from the fact that
D C/ D" is an
axiomatization
of KD -1)
)
(Lemma 10)
(by 5 and 6)
(by 4 and 7)
(by 3 and 8).
Lemma I I is proved.
Corollary (from Lemma 6 and Lemma I I ). Fo r any simple fomuda A
A i f (A)* E KD.
Lemma 12. I f L is an implicative-negative logic with inverse negation adequate to implication of logic KD, then CC7F A implies A E L.
Proof Let L be an implicative-negative logic with inverse negation adequate to implication o f logic KD. Then for any simple formula A the following is true: A E L if f T(A) E KD. Let us show that the set of all theorems of the system i s included in L.
As L is closed as regards modus ponens and substitution, it is sufficient to
show that
(1) K 3 C L a n d ( 2 ) ((p i D p2) D ( ( ( p i ) D p2) D p2)) E L.
Let us prove (1). Let A E K D Th e n A is a simple formula without
occurrences of —
1is. a substitution instance o f a formula A. B y the rule o f substitution we
have
T h eT(A)
r eEf KD,
o r and then by the choice of L we obtain A E L.
e Let us
T prove (2). According to the choice of L and the simplicity of the
formula
( A ((p
) i D p2) D ((( ) D p2) D p2)) it's sufficient to show that
T(((pi
D
p
l
)
D 1)1))) E KD. The latter occurs because the
d
o p2)e
formula
((pi
D
p2)
D
(((-p
2 ) D p2)) is a classical tautology in L
s
and
hence belongs to KJ.
D
n Lemma
o 12 is proved.
,t
c
o
n
t
a
i
n
o
c
c
u
r
r
e
n
c
e
s
INVERSE NEGATION AND CLASSICAL IMPLICATIVE LOGIC 1 5 3
Theorem 5. The set o f all formulas to be deduced in C r is the least
implicative-negative logic with inverse negation, adequate to implication of
the classical implicative logic K .
This theorem follows from the Corollary from Lemmas 6 and t h a t has
been formulated above, and from Lemma 12.
In conclusion a few words about the connection of the calculus C I T with
the calculus VI o f A. Arruda [11 and with the calculus Pli, of D. Batens [2].
From Note 3, made earlier, and from the definition o f the system VI it
follows that i f Cl I - A then V I I- A Besides, fro m Note 3 and f ro m
Lemma 6 it follows that if A is a simple formula, then Cl I - A if f VI I-- A.
As regards the connection o f C / 7 with PIL, when the main wo rk at the
paper was over, the author found out that Cl c o in c id e s with the implicative-negative fragment of the calculus PIL. Thus, the suggested paper may
be considered as a study of the implicative-negative fragment of the calculus PIL of D. Batens.
Tver State University
Zhelyabova str., 33,170000 Tver, Russia
e-mail: chagrov@phys I .tversu.ac.ru
REFERENCES
[11 Arru d a A.I. On the imaginary logic o f N.A. Vasilev. Proceedings o f
Fourth Latin-American Symposium on Mathematical Logic, NorthHolland, 1979, pp. 1-41.
[2] Batens D. Inconsistency-Adaptive Logics. Centre for Logic and Philosophy of Science. Ghent University, Belgium. Preprint 19.
131 Gentzen G. Investigations in Logical Deductions. Mathematical Theory of Logical Deduction. Moscow, 1967 (in Russian), pp. 9-74.