RUSSELL'S SET VERSUS THE UNIVERSAL SET
IN PARACONSISTENT SET THEORY*
Ayda I. ARRUDA
and
Diderik BATENS
1. Introduction
Russell's set lead to the well-known troubles in Cantor's set theory.
It was ruled out from classical set theory. Later, the paraconsistent
logicians thought it possible and interesting to have it in paraconsistent set theory. The objective of this paper is to show that Russell's
set causes other troubles in paraconsistent set theory than the ones
discussed in 1 I1 and 121.
In [2] one of us proved that in some of da Costa's paraconsistent set
theories with Russell's set, R = 5ʻc - t(x e x), U U R is the universal set.
In th is paper we prove th a t th is result i s va lid i n a ll 'strong'
paraconsistent set theories (particularly in all da Costa's paraconsistent set theories). Answering a question in Section 6 of 121 we also
prove that in all strong paraconsistent set theories with Russell's set
UR is the universal set. Finally, we also prove that in some 'weak'
paraconsistent set theories the existence of Russell's set implies the
existence of the universal set.
A paraconsistent set theory is a set theory which is inconsistent but
non-trivial; in other words, some formula is a theorem together with
its negation, but nevertheless not a ll formulas are theorems. Th e
underlying logic o f a paraconsistent set themy should clearly be a
paraconsistent logic, i.e., a logic in which there is a symbol o f
negation, say --,, such that it is not possible in general to obtain any
formula B from some formula A and its negation
One o f the prime motivations fo r constructing paraconsistent set
theories resides in the attempt to articulate nontrivial set theories in
which Russell's class is a set. I t is well-known that set theories
without universal set are richer than the ones with universal set in that
the former allow for a larger number of distinct sets. Moreover, the
122
A
R
R
U
D
A
and D. BATENS
former are more interesting for independent reasons. In view of this
situation it is important to find out whether it is possible to devise
paraconsistent set theories with Russell's set but without universal
set, a problem already considered in 121. The present paper contributes to the solution o f this problem.
In vie w o f the multitude o f logics already developed, we shall
distinguish between strong and weak paraconsistent set theories. To
set a boundary we stipulate that minimal strong paraconsistent set
theories are those that have as underlying logic the positive intuitionistic first-order logic with equality to which excluded middle, formulated as A v —A, is added. Notice that the propositional fragment
of this logic is even weaker than the basic logic PI of [41. Some weak
paraconsistent set theories will be characterized as they are needed in
the sequel of this paper.
In a ll paraconsistent set theories considered in th is paper we
suppose that Russell's class is a set.
The results presented in this paper might seem t o lead t o the
conclusion that the paraconsistent programme is bound to fail in the
context of set theory. In the last section we shall argue why we do not
subscribe to this conclusion.
2. The universal set in strong paraconsistent set theory
In this section we first prove that U U R is the universal set, i.e., that
(x) x e U U R in minimal strong paraconsistent set theories. Th is
result may be obtained as a corollary to the second theorem in this
section, (x) x e U R, but we prove it as a theorem because it was this
result that gave rise t o a ll other results presented in th is paper.
Moreover. Theorem 2.2 may be proved as Theorem 3.1, but as the
proof of the latter is very long we give here a shorter proof of the
former.
For the following proofs we need some properties o f equality and
some results o f set theory. The properties o f equality are the usual
ones. The results of set theory we need concern the unitary set, ix },
and the union, Ux and x U y. A ll o f them are easily proved in our
minimal strong paraconsistent set theories.
RUSSELL'S SET VERSUS THE UNI VERSAL SET...
L EMMA 2.1. F- (x, y, z) : x, y ( z )
D
1
2
3
X = y.
L EMMA 2.2. 1 . H R E R
.
11. - - - 1 ( R ER)
THEOREM 2.1. ( x ) x c U UR
PROOF.
(1) ( R U{x}) E {R U{x}} h y p .
(2) { R i x ) ) R U {
(3)
R E R U (x)
f r o m
x
(4)
U {
}
f R r Eo(R m
x( ) ) 1R
(5)
f r o m
)R U(x)
fa {r nR oU (x))
(6)
d e R f r o m
m
(7)
L —e, ( { RmU (x)}me U { x } ) ) h y p .
(a 1 2R U(x)) e R f r o m
(8)
(9)
)2 { R. U(x))1 E R f r o m
(10)
{ Rn U ix )) g UR f r o m
a
.
(11)
d R U ( x ) e UR f r o m
(12)
R 3 U (x) g U UR f r o m
(
(13)
x
E. U U R
f r o m
)
Lemma 2.2 part I.
(4) and Lemma 2.1.
(2), (5) and Lemma 2.2 part
(7) and the definition of R.
(1)-(6) and (7)-(8)
(9).
(10).
(11).
(12). •
Now we shall prove that UR is the universal set. To do this we need
two lemmas.
L EMMA 2.3. I . H x ER { x ) ER
II.
y
ER D{x , y) ER
PROOF.
I. I f {x } E {x} then {
x obtain f x ) ER. I f
e
{x)) then ix ) ER •
} If—Ix, y ) xe (x,
. y ) then f x , = x v {x, y ) = y. Thus, b y the
T hypothesis
h u s that
,
x, y ER, we obtain (x, y) ER. I f --,({x, y) E (x,
b y)), ythen ix, y) ER
t
h
e
LhEMMA
y p2.4.oH(xt ) i ( x ,
R
PROOF.
h e s i s
(1)
t k i x h, R)) E {
(2)
(a { { x ,t R
)xx, R ) }
)he = y (p
.
xR ,
R,
)
fw r
oe
m
(
1
)
124
(3)
(4)
(5)
(6)
(7)
(8)
(9)
A
R
R
U
D
A
x = R
f
r
o
x , RER
f
r
o
{ x . RI E R
f r o
{{x ,
R
f r o
— 4 { { x , R}I e {ix , RI ))
{ { x , R}I e R
f r o
( {x, RI ) e R
f r o
and D. BATENS
m
m
m
m
h yp .
m
m
(2) and Lemma 2.1.
(3) and Lemma 2.2, part
(4) and Lemma 2.3, part
(6) and Lemma 2.3, part
(7) and the definition of R.
(1)-(6) and (7)-(8). •
THEOREM 2.2. ( x ) . x E U R .
PROOF. First suppose that --, ((x, RI E i x , R
RI
) E R by the definition of R and consequently x E UR. Suppose next
) . (x,
that
I nRI E t {x,h II).
i Hence
s
ceither
a {x,s RIe= R or (x. = x. I f {x,
RI
Rx then, ( x, E R by Lemma 2.2, part I. If. on the other hand,
I
{x,
x
then ( x . RI I = I x } and consequently (x ) ER by Lemma 2.4. In both cases, xE UR . •
Originally we found a more complicated proof o f Theorem 2.2.
proceeding along the lines of the proof of Theorem 4.1 in [2], but with
the empty set defined as ( y ) x ey. The present proof has the
advantage of showing very clearly why Theorem 2.2 holds, viz, that
any set x is a member of some sets, viz. {x, RI and {x}, o f which at
least one is itself a member o f R f o r 'obvious' reasons. A s fas as
propositional logic is concerned, all we need for the present proof is
Modus Ponens (A, A DB / B), Excluded Middle (A V a n d Proof
by Cases (From a., a V B , a U (A I H-C, and a U (13},-C to derive
C).
3. The universal set in the set theories based on P
The weakest paraconsistent logic already developed is the Arruda
and da Costa system P (see 131). In this section we prove that even in
the weak paraconsistent set theories based on P, the existence o f
Russell's set implies that UR is the universal set.
The postulates of P are the following:
A l. A - ) A R I .
A , A -* I 3 / B
A I A & B -> A A - > I 3 , 1 3 - 4
C / A
- C
RUSSELL'S SET VERSUS THE UNI VERSAL SET...
A3.
A4.
A5.
A6.
A7.
A & B -> B
AA V B
A V13
A V -I A
- , - - , A -> A
1
2
5
R3. A , B / A & B
R4. A - * B. A - ) C / A - ) B & C
R5. A & B 4 D , A & C - 1 3 / A & ( B vC)-> D
D . A H =df - > B1SE( B
-> A )
To obtain the corresponding predicate calculus P . it was proposed
in 131 that the following postulates be added, where the restrictions are
the usual ones (as in Kleene151):
QI. (x ) A(x)-> A(y) Q 4 . A (x)--* C I (Ex) A (x )-› C
Q2. A (y )-) (Ex) A(x) Q 5 . (x ) C A ( x ) : : C V(x) A(x)
Q3. C - * A(x) / C ( x ) A(x) Q 6 . C & (Ex) A(x) : : (Ex) C & A ( x )
To obtain the corresponding predicate calculus with equality. P .
we furthermore add the following postulates:
11. x = x
12. x - y - > y = x
13. x y & A ( x ) - A (y )
L EMMA 3.1. The following statements hold true in P
T1.
F-A&B=B&A
T2. I - - (A &B) & C &
(B & C)
T3.
x
=y y x
DR1.
B
DR2. A - > B 1 - - A & C - > B & C
A-*13 H A & C & D - > B & C
DR4.
B
A & C - ) B&C
DR5. A - 4 C , B--*C H A VB C
DR6. A B H A & C
-DR7.
A=BHC&A=C&B
-DR8.
B &A
CB H A V C B V C
DR9. A B H C V A C v B
DRIO. A=1 3 . A - ) C F-B ->C
DRII. A = 1 3 , C . A I A B , (x)A H (x)B
DRI2.
-C->B
DRI3.
A
B
PROOF.
,
Left to the reader (cf. also 131). •
( E x
) A
( E
x )
B
126
A
.
I
.
A R R U D A and D . B A T E N S
L EMMA 3.2. The following rule is derivable in P '
EQ+ 1 f H A -= B and there is an occurrence o f A in C outside the
scope o f a symbol fo r negation - , , and D is the result o f
replacing that occurrence o f A in C by B. then C H D.
PROOF. By induction on the depth o f A in C and by application of
DR6-DRI3. •
Let us consider now the paraconsistent set theory S P ' based on
P . Fo r our needs the following specific postulates will do:
SI. (E x)(x) x ey ( x Ex) ( E x i s t e n c e o f Russell's
set, R.)
S2. (z)(Ey)(x) x E y x = z ( E x i s t e n c e of the unitary
set, i z l . )
S3. (u , v)(Ey)(x) : x Ey . x = u V x = v (Existence of the pair
vl. )
S4. (u )(Ey)(x) : x ey ( E z ) . x e z & z e u (Existence o f the union
of u, u u.)
S2 is obviously P'-derivable fro m S3. We list it separately fo r
future reference.
L EMMA 3.3. 1 . H R E R
11. x E ix}
III. x E{x, R}
IV. i
PROOF.
I. We obtain —1(R
E
R
AR
l. ) R (R
Hence,
f r RE-RV
o m G R )
R R by DR.S. But R E R v —1(R ER)
x
derives
S
from
I
A6.
, Consequently. we obtain R ER by RI.
,
a FromnS2 andd
IIIRand IV. FroR
m S3 and 11. •
)
E
LEMMA
3.4.
F- x c y & y e z
x
R
PROOF.
From S4 and Q2 by R2. •
R
R
f In order to
r simplify the following proofs we shall denote formulas of
the
o forms xmEX and (x E x) v ( x ex) b y the expressions o f the
RUSSELL'S SET VERSUS THE UNI VERSAL SET...
1
2
7
corresponding forms K x and Tx respectively. We also shall denote
x EZ & y Ez by x, y ez.
LEMMA 3.5. H x ER & T(x ) - > ( x ) ER.
PROOF.
(I) K ( x ) - ) x ( x )
S
2
,
EQ-1-.
(2) K (x ) & x ER
( x )
& x ER (I ), DR2.
(3) x = (x ) & x ER ( x ) ER 1 3 .
(4) K {x } &
{
(5)
S
I
.
x --7 K {x}--) Oc ER
(6)
&
X
ER
{
(5), DRI.
}
xE x ER
(7)
R & T l x ) -• * l x ) ER
(4),(6), R5, EQ+. •
}(
2E ) R ,
( EMMA
L
3
)3.6., H X. y ER & x . y) ( x , y) ER
PROOF.
R
(I)
2 K (x , (
x. K
, (x , y) & x, y ER . - ) ( ( x , y) x v(x, y) = y) & x, y ER
(2)
(I), DR2.
y )
(3)
= (x , y) = x & x, y ER ( x , y
A2, 13, R2, EQ-+
}x E
(4)
l x , Ry) = y & x, y ER .-0 (
A3, 13, R2, EQ-i
xV , ((xiy, y)
(5)
) x EV ix,Ry) = y) & X. y ER
(
(6)
(2), (5), R2.
x K ,(x , y) & x, y ER (
x
SI.
x= , y y ) ) ( x , y )9 R
(7)
,
(7), DRI.
E
(8)
--TK(x,
R
y)
&
x,
y
E
R.-0
(
x
,
y
l
ER
y
y
(9)
(6), (8), R5. EQS x, y ER & Tlx , y) ( x , y) ER
)
3
c
.L EMMA 3.7. H (x) (
R
(
PROOF.
T o simp lif y t h e p ro o f w e sh a ll( denote x , R Eix,
R)
x,& R
R ER
) &) T(x, R) & TI (x. R b y H.
3
(I)
K
i
l
x
,
E
R . RI ) ( l x , RI ) = (
)
x , K (R(x,) R
(2)
,
2
.
}S ((x . R
(3)
(
(4)
S
24
.
}) x& E l(x, R }1 -) x = (x. R)
(5)
K
((x
,
RI
)
&
H
x
(
x
,
(
2
)
,
(3), (4), R2.
)H = .
)
(6)
f- x-K ((x
>, , R)) & H ( i x , RI ) = {
,
x(7)
R E ix,
R
ER(ix, RI ) 1 3 , EQ+.
( , l i xR, }R)) = &l x , R) & R
&
S
25
.
l(8)
, R11-3
RR =
1 (x,
x. Rx E lix,
K
f
l
x
,
R
I
)
&
H
R
=
{
((9)
1
)
,
R
,
x) , R) R}
D
3
,
E
(x=
6Q
) + ,
E
.
Q
(c(
7
)
,
((x
8
)
,(,
R
2
xR
.,)
&
R
)x
128
A
.
I
.
A RRUDA and D. BATENS
(10) K ((x , RI I & H
x
= {x , & R = i x ,
(11) x = {x, & R { x , RI . -> x = R
(12) K f {x , RI I & H
x
= R
(13) K {{x ,
&
H
x
= R & R ER
(14) x R & R ER
x
ER
(15) x = R & R E R . - - * . x , R E R
(16) K ( i x , RI I & H x , R ER
(17) K {{x ,
&
H • x, R E R & T i x , RI
(18) x, R ER & T{x, - - 4
R ) ER
(19) K {f x ,
&
H
1 Z
ER
(20) K ({x , RI } & H i x , RI ER & T({x,
(21) l x , RI ER & T{{x, R
}
(22)
K {{x , RI I & H . -> ( { x , Ry) ER
(23)
E R
}
{-1 iK (ixx , ,RI ) & H { { x ,
(24) TI{{x , } RI I & H { ( x ,
E R
R
(25)
E T f {xR, R}I & H
(26) f {x , RI } ER
(5), (9), R4.
13, EQ+.
(10), (I I ), R2.
(12), DR4.
13, EQ+.
(14), DR4, EQ-H.
(13), (15), R2.
(16), DR4, EQ+.
Lemma 3.6.
(17), (18), R2.
(19), DR4.
Lemma 3.5.
(20), (21), R2.
Si, DR1.
(22), (23), R5, EQ+.
A6, Lemma 3.3, R3.
(24), (25), RI. •
THEOREM 3.1. ( x ) . x e U R .
PROOF. We shall denote f l x , R I ) ER & x ( x ) & R ER & x e {x.
RI by G.
(I) K {x ,
( x ,
= x V ix. RI = R S 3 .
(2) K {x, RI & G ( { x ,
= x v (x , R} = R) & G
(I), DR2.
(3) i x , RI =
x
{ { x , RI I
S
2
,
A3, EQ+.
(4) l x , RI = x & G
x
E {{x, R}I St i{x, Rr
(3), DR3.
i E R
(5) x c ({x , RI I & i{x , R}I ER
x
U R L e m m a 3.4.
(6) {x , = x & G
x
e UR
( 4
) ,
(5), R2.
(7) i x , = R & G ( x ,
= R & R ER A 3 , EQ+.
(8) (x , = R & R ER i x , RI ER 1 3 , EQ-1-.
(9) (x , = R &G l x , RI ER ( 7 ) , (8). R2.
(10) l x ,
R
&G ( x , E R & x e (x, ( 9 ) , DR4.
(11) (x , RI ER & x E ix, RI
x
e UR L e m m a 3.4, EQ+.
(12) {x , RI = R &G • x
(
1
0
)
,
(11),
(13) ((x , = x v (x , = R )& G X E UR (6), (12), R5, EQ+.
(14) K (x , RI & G . X E UR
(
2
)
.
(13), R2.
RUSSELL'S SET VERSUS THE UNI VERSAL SET...
1
2
9
(15) —11(x, R) l x , RI ER
S
i
.
(16)
R I
& G { x , R} ER & x Eix, RI (15), DR3, EQ+.
(17) {x , R} ER & x e {x, R} . X E UR L e m m a 3.4.
(18)
G
- 3 X E UR ( 1 6 ) , (17), R2.
(19) Tf x , & G
x
e UR
( 1 4 ) ,
(18), R5. EQ-F.
(20) T(x , R) & G
A
6
,
lemmas 3.3 and 3.7, R3
(21) X E UR
(
1
9
)
,
(20), RI. •
The system P
A7,- Q3,
i s Q4, Q5 and Q6. Anderson and Belnap's systems E and R and
Routley's
r a t hsyste
e ms D M O a n d D K a re extensions o f P , a n d
consequently
all these systems lead to the existence o f the universal
r
setwin view
e ao f S I-S4. We do, of course, consider only set theories in
which
k .all theorems of the underlying logic hold.
Y
e
t
,
4. Some
remarks on a vety weak paraconsistent set theory
w
e
In this section we show that the universal set cannot be avoided
d in ve
i ry weak paraconsistent set theories with Russell's set,
even
d one
n avoids that either —,(x ER) or x Ex determine a set. To
unless
'
t
prove this we consider an extremely weak logic and. consequently. an
e
extremely
weak set theory, in which we take as postulates exactly
v we need to prove Theorem 4.1 below. The absence of excluded
what
e
middle
should n o t be taken too serious: i f both A --3 A V B and
n
B -3 A V B were added, excluded middle would be derivable.
n L v be a first order logic characterized by:
Let
e
RI. A — ) B , B — >
e
C/A
R2.
A —> B, --,A B / B
d
—
R3.>AC(x) / (x)A(x).
A
In
4 order to define Russell's set we need equivalence, which is not
explicitly
definable in the present logic. However, to show our point
,
weAdo not even need Russell's set properly. We consider a very poor
set5theory SL V, based on L V. in which there is a set (still called R)
that
, contains at least all members of Russell's set, and in which, for
any two sets x and y, there is a set (still denoted b y ' x Uy•) that
contains at least all members o f x and o f y. The postulates are the
following (where x, y and z are sets):
130
A
R
R
U
D
A
and D. BATENS
SI. —1(x ex)—> x ER
S2. z e x , Z EX Uy
S3. z e y , z Ex Uy
Let us now suppose that either ----t(x ER) or x Ex 'determine' a set
in SL V, X and Y respectively, such that:
F.
R
)
,
x X
or
x E X —› X e Y .
THEOREM 4.1. 1 . H(x ) . x E R U X in SL Vp/us F.
11. H ( x ) . x 04' UR in SL V phis E"
PROOF.
S2.
I. (I ) x ER—) x ER UX
F.
(2) —,(x ER)—> x EX
S3.
(3) xE X —) x ER UX
(4) --1(x E R ) , x ER U X
(2), (3), RI.
(5) x ER UX
(I), (4), R2.
(6) (x) x ER UX
( 5 ) ,
R3.
H. The proof is similar to that of part l . 0
In a sense the present results are not surprising. Although the
negation in SL v is very weak, its occurrence in F guarantees that X
contains all sets which are non-members of R, i.e. that the former is 'a
complement' of the latter; and its occurrence in SI guarantees that R
is 'a complement' o f Y. It is obvious that the universal set cannot be
avoided in the presence of 'union' and 'complement'.
5. Concluding remarks
Any strong paraconsistent set theory wit h Russell's se t is an
extension of the minimal strong paraconsistent set theory considered
in Section 2. Thus, UR is the universal set in any strong paraconsistent set theory. In Section 3 we have seen furthermore that even in
RUSSELL'S SET VERSUS THE UNI VERSAL SET...
1
3
1
weak paraconsistent set theories which contain Russell's set and are
based on a paraconsistent extension of P, UR is the universal set.
It should be pointed out that P is very weak, and that paraconsistent
set theories based on a further weakening of P will be extremely poor
as far as theorems are concerned. Th e ir mathematical usefulness is
dubious. Yet, we have shown that even in paraconsistent set theories
which are weaker than SP= (presented in Section 3) troubles arise.
Although we were unable to show that the presence o f Russell's set
leads by itself to the existence of the universal set, we have demonstrated that Russell's set together with either the set 'X --,(x E R) or the
set k (x e x) leads to the existence of the universal set in a very weak
fragment of SP= T h e same obviously holds in all richer paraconsistent set theories.
All the logics considered in this paper have the law o f excluded
middle as an axiom, and in the corresponding set theories Russell's
class is a set. The conclusion we may obtain from the above results is
that the la w o f excluded middle together with th e existence o f
Russell's set lead to the existence o f the universal set. Thus, if we
want to have Russell's set in a set theory without universal set, we
have either to eliminate the la w o f excluded middle fro m the underlying logic or to introduce Russell's set in an ad hoc way. Another
consequence o f the above results is that, in order t o avoid the
existence of the universal set in paraconsistent set theories with the
law o f excluded middle, we must prevent operating with union on
Russell's set. Perhaps, a similar problem will arise with other nonclassical sets.
Paraconsistent set theories without universal set ma y be ve ry
usefull for a deep analysis of the behavior of Russell's set and of other
non-classical sets. Nonetheless, many difficulties arise in the construction of such paraconsistent set theories, e.g., (i) to decide which
non-classical sets will be introduced apart from Russell's set, (ii) to
avoid paradoxes like those developed in III and 121, and (iii) to decide
which operations will be defined for the non-classical sets in order to
avoid the aforementioned problems. Needless to say that it does not
make much sense to introduce non-classical sets but to prevent at the
same time to operate with them. If one were to do so, the non-classical
sets would play the same role as the classes in classical set theories
and hence paraconsistent set theory would not present any new
132
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A RRUDA and D. BATENS
mathematical interest. Finally, it should be pointed out that it is still
an open question in which paraconsistent set theories Russell's set is
different from the universal set ; a very unpleasant situation indeed.
There can be no doubt that some expectations o f paraconsistent
logicians with respect to set theory did not came true. I t does not
follow fro m this that the development o f paraconsistent logics has
been useless. We even are convinced that one cannot conclude that
the paraconsistent research tradition has failed o r is bound to fail
completely with respect to set theory. We will conclude this paper by
offering two arguments in this connection.
It is a specific problem of paraconsistent logic to clarify the process
of deduction within a theory in which an unexpected inconsistency
has been derived and before a consistent alternative theory has been
developed. Even if there are no interesting paraconsistent set theories, it may turn out necessary to develop further a set theory which
originally wa s intended t o b e consistent b u t wa s proved t o b e
inconsistent later. A contribution to the solution o f this problem is
offered in [51 by the development of dynamic dialectical logics, viz.
logics which adapt themselves to the specific inconsistencies that
arise within some theory. This part of the paraconsistent programme
is not in any way affected by our present results.
For the sake of our second argument we first want to clarify a point
concerning the paraconsistent programme. None o f us has ever
claimed or even believed —and the same holds for the vast majority of
people working in the paraconsistent research traditon — that any set
of postulates, o r even a set o f postulates which relies on 'cle a r
mathematical intuitions', may be supplied with an underlying paraconsistent logic to the effect o f arriving at an interesting theory in
which specific unwanted results are avoided. Paraconsistent logic is
not a wonderful remedy. None of da Costa's set theories and no set
theory considered in this paper contains, e.g., the axiom of abstraction in its unrestricted form. I t is not excluded that there be some
restriction on S2-S4, perhaps even a restriction which is not ad hoc,
that results in an interesting paraconsistent set theory with Russell's
set but without universal set. Also, it has not been demonstrated that
RUSSELL'S SET VERSUS THE UNI VERSAL SET...
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an interesting set theory cannot be based on a logic in which excluded
middle fails.
Universidade Estadual de Catnpinas
Department o f Mathematics
A
Rijksuniversiteit Gent
Department o f Philosophy
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BATENS
REFERENCES
111 Arruda A l . *The Paradox of Russell in the Systems NF' . Proceedings of the Third
Brazilian Conference on Mathematical Logic. (Eds. A. I. Arruda, N. C. A. da
Costa and A . M. Sette), Sociedade Bras ileira de LOgica, S4o Paulo, 1980;
pp. 1-12.
121 Arruda A.!. 'Remarks on da Costa's Paraconsistent Set Theories'. To appear.
131Arruda A l . and da Costa N.C. A. 'On the Relevant Systems P and P* and some
Related Systems'. To appear.
141 Batens D. 'Paraconsistent Extensional Propositional Logics'. Logique et Analyse
1980; 90-91: pp. 195-234.
151 Batcns D. 'Dy namic Dialectical Logics'. Paraconsistent Logic (Eds. R. Routley
and G. Priest). To appear.
161 Kleene S.C. Introduction to Metamathematics, Van Nostrand, 1952.
* This paper is the consequence of the seminars developed at the University of Gent.
Belgium, when the first author was there as Vis iting Professor with a partial
grant (81/0522-4) of the Tundac ao de Amparo à Pesquisa do Estado de Sao
Paulo' (FAPESP), Brazil.