COMPLETENESS PROOFS FOR THE
SYSTEMS RM.3 AND BN4
Ross T. BRADY
*I. Introduction
In III. on pp. 420-426 (written by Dunn) and on pp. 469-4704writ ten
by Meyer). two axiomatizations of RM3 are given and both arc shown
to be complete with respect to the following 3-valued matrix set MI :
(CI
st f
•h h
s
t
t
h
b
t
b
t b
f
b
b f
1 '1 'f
a l
There arc two designated values. t and b. The method used in both
of these completeness proofs relies on Dunn's Extension Theorem
p, 42E4; Every consistent proper extension of RM has a finite
characteristic matrix set, Thus the given completeness proofs for
RM3 are specialized proofs which rely on RM3 being a consistent
proper extension of RM.
In this paper, I present a completeness proof of an axiomatizat ion of
RM1 with respect to the matrix set M
on
I Darift'S Extension Theorem. nor on any connection RM3 has with
RM.
, a The
n dmethod
t hofi this
s proof is fairly general and it is hoped that the
proof
to enable axiontalization% of other matrix sets
p r can
o beo modified
f
C
dobe obtained.
o
e
s
n I alsoopresentt a completeness proof of the same axiomatizat ion of
RM3
with
d
e respect
p
eto a 2nset-updmodel structure, thereby showing that
M
formulae.
s
Again, the method of proof is fairly general and it is hoped
that
.
other model structures can be axiontatized by appropriate modifications
to the proof.
a
n
d
t
h
i
s
2
s
e
t
-
ID
R
O
S
S
T. BRADY
Further. I consider the following 4-valued matrix set M. :
&
•i
•13
n
I
f
b
n
t
I b
• i
t
• b b b
n
n
I
f
n
b n
l f
I ri
f f
t b
f
I
f
f
4 , 1
s b
n
f
n
I
t f
b
A i n
t
There are two designated values, i and b. The matrix set M. is an
adaptation of the Smiley matrixes. given in f 11 on pp. 161-162, which
have only the one designated value. t. and have the occurrence of •b•
in the - m a t r i x of M, replaced by 1'. and the occurrences of 'n• in
the - • m a t r i x of M
4hence
r eMi.
p lisacharacteristic
c e d
for the system E
h
s s y but M
ments,
degree.
T
, io s fh t ae u t o l o g i c a l
4
S
e There
m
anmd is
t eaiaclose
il relationship
l between M
Can
be
understood
as
an
extensional
logic of sentences which take one
em 4 oya nr d
i . e .
or
both
of
the
values
truth
and
filikillY
m
e w ha e t r e a s
1
cx a ofnsentences
b which
e
take one. both or neither of the
rextensional
s MM
u i4i logic
u
d
r falsity.
s t oThis
o can
d he seen front the respective single.
svalues
t 3an truth
b ee and
a
given in §2 and §4_
tset-up
l e semantics,
.s
a
I
will
give
n
an
axiomatization
for a system. which I will call BN4.(t)
af
o r
that this axiomativaion is complete with respect to the
iand
r l will
a show
d
above
s
y4-valued matrix set. As lot MI D
.s I tw i l lfor BN4
a lis complete
s o
axiomatization
with respect to a 2 set-up model
s
h
o
w
structure.
Both
of
these
compietenns
proofs require refinements OD
e
tm methods
h
a of proof
t
the
of the corresponding completeness theorems
tw
Oven
for
h RM3, eand these refinements are then available for use in
proving
completeness theorems for other matrix sets and model
i
structures.
t
h l will also sketch similar completeness proofs for the Lukasiewicz
3-valued
f
o logic and state simple relationships between the axiomatizalions
of
the three systems_
r
m
u
l
a
e
o
f
a
n
y
COMPLETENESS PROOFS FOR THE SYSTEMS RM3 AND ati4
*2. Corn/401mm of RA13 with respect to At
AXI M Pla t iZ a t iO n i f R t 1 . t . (
2
)
Primitives: & ,
Definitions:
A
A - B -ditA
BJ 1 &(1 3 -6 At.
d
r-(--A
&
- 1 3
Axioms
I.A -6 A
2. AdociA--613)-0B
3. A & B . AA & B 4.
.5.H(A-6 1 3 )&(A-1 6 0 -6 .
6. A t
17. A --6 --A-. --A
8.
k (AB -6 -B B
9.
- -A -6 A
V C
to.
) - &- B - 0 . A -6 B
II.
- 6*-A -6 A V (A B )
( A
Rules
&
I.HA, A-4 1 3 .1 3
2.i A , B
,3.) A -6B. C -6 D B --10C• A ---*D
4V• A &
(
Theorem.
B
A
& A —6AVB
2_ A—pH VA
C
)3. A v -A
4. A - - A
5. --A -6 --4A B )
6. --B
&
B)
7- & B) - - A V --B.
8. - B & (A -6 13)-6 --A
9- A V - B V (A B ) . (Use Az. 10.)
10. 13-* •-la V (A -4 B). (Use Az. I I . )
VB =
) ;
i l
12
R
O
S
S
T. BRADY
11.
- ( A
-sB).
12.- A l v --(A-. 13)-s
, ABec---(A
13.
. ( U Bs) -eo - B , (Ilse AX.
A * ,
Derlivelf
I
L Ridire
)
I. A — B ,
2. A A - . •
3„
,
( 7A- di. •B-,
A C.
- A,C
4.
A
-. C. (Ilse Ax. 6)
1 . B & C •
Single
ln order to faci Wale the completeness proof, the 1-valued matrix sel
S e
M3 i replaced by the following single set•up SeMantICS with values
t truth (T) and falsity (1:). one or both of which can be assigned to a
u p
formula under interpretation.
S r
A votriation 1,••• assigns a non-empty subset o f T , F ) to each
m
sentential variable.(
a n
3
f iV(p) • (T h ( F ) . V(p) ( T . F), etc.
)
c s
S The
y mvaluations
b o l i zV are extended to interpretatiem o f all formulae.
, f
e
inductively,
as follows:
o r
t h i s
6) l i p ) • V(p).
R
a
s
(ii)
T e 1(--A)c4 E R A ) .
7
f o l l F *U.-A)
o
e*T d- 1(A).
s
w s (iii): T F l(A ê 111) ,)11' F 1(A) and T
f
It A Ar H) )1F e RA) or F1(11).
3
(itt) T I C A H ) t4 (Tt HA) or T I I B))
and (F41(8) or Fe It A)).
F E l(A B ) c l T e 1(A) and F e 1(B).
A formula A is valid in this semantics 1ff T El(Al• for all valuations
V.
'Graham Priest. in I n has also used thi% semantics for RMII. I
1proceed to show that M
formulae.
I a n Let
d Vt h i s
i csasbefollows:
V.
e m a For
n all
t sentential
i
variables p.
t ch se
M
r h
a
v
e
v ta l u ha t
e
i o
s n
a
m
c eo r r
e v s p ao
l
n i d i dn
g
t
o
CO MP I LI E NE s s itHoOPS FOR THE SYSTEMS HMI AMU BN1
1 3
N1 ,
4
V
(p)
s
st
t• ( p
Let I
(g )
m& and •—•, as defined by M
s=
p
or bb.
I efor all valuations V ,
)T f
, ts. i A
n c e
il .
t 1.entina
hfh o er 1m. For
u l all
a formulae A . ( T c 1 ( A ) t - I
i 4
m
(F
v m
a l u a t i o
eA
r 1
b
.n 4
eisA
s
( c
1P
V
,xvr e f. aThe
, l proof
i is
d by induction on formulae. The cases for
p 1variables p. and formulae —A and A & B are clear. The ease
4
sentential
(a t i o
r rn
c b
l
a
r i d
) 1 is as follows:
T
for
A
c e1Ai - • B
t 1 h .a u
a(
V
)s n1 t T
i ICA
v i B):Nro(T#
cf f. l ItA o r T e RH))
(n c
p
c su and
d) D
p
4 i ( (A
) = I
)F1ael• t l
1 o and
la
m(I
f )n
, n m
o
ʻ( A
) I)
Vd
4 o (rn
B
m
)
( F (LI A) = f r b and I
d
4 f
IF
(f AIm
, E
C
AV o
cll=m
eF
(( 1A3. B) ) t 3T
= c 1(A) and FclIBP
) )o
) M r(V
frA t(
oor br and 1413)= f or b
(, cA)1,4A)=t
= t
1
I)p c
I) )h
1
)
A
f o m
.) 1
m4.
4
o Theorem
a ( (0AL- For all formulae A. A is valid in the single set-up
,
8
rsemantics
l
A -3
1•for 1 1 1 4 1 A is valid in M .
)
bl
) ,1
3
=
) Proof.
f
8
f By
t) ) Lemma I . T H A ) v ' a l
,hence.
l
m
oAfo
=is1valid in thi$ semantics for RM3 4.)A is valid in M3.
o
( A ) r .o
o r
rt 1o
Theorem
r
r (Soundness). For all formulae A.
b
. ro 2.
b
Fil
) b
r t
l o
,b o
this to the reader.
u
m Proof,
/I leave
(
, l
b
Aa
y
t c
M
A,
s
i u
)
s s
.
v i
a n
l g
i t
d h
i e
14 R O 5 5 T .
a k A D Y
Completeetell Prd,of-PrelominorteN
The following preliminaries are needed for the forthcoming completeness theorem, the proof of which is modelled on the RoutleyMeyer completeness proof for relevant and para-consistent
which can be found in [51. Chapter 4 and also inP1111.
An KV3-theory a is a set o f formulae closed under provable
R.M3--implication and adjunction. i.e. IT„, A --4
and
AC:a and ecia A Si: BE a.
,
1 3AnaRM3-theory
n d
A ae is aprime i f . fo r a ll formulae A a nd B .
A
V
B
e
a
4
1
t
E
E
I
O r B 2a$ 3 .
B
e
,
A set b of formulae is diVitnetively RA13-derivabl(' from a set a of
formulae, written a i u4 3 b, i f, for some A
l
some
11,, 1 1 , . E b(n 1 1
.0 , 4
A—set of formulae a Es 11.413•maxiowl ilT a +
,A 1 .0AE a t , m ;
l Leetimo
, , , a). Z. I &
f the set
a a isnRM13-maximal
d
then a is a prime RM3&
theory.
B
V
B Pr1,4.. (i ) Le, t l i
contradicting
- Rh11-masimality.
. m o A the
,
tii) L e t A ea a n d B ea a nd A & )14 Et T h e n . since
0 : 1
A &13-• A &B. a ) a (iii)
n d
i t m L e t A v B ea. A r e a a n d B l a. T h e n s i n c e
A
1 -r , , a .
c
a
o c o n t r a d i
a
n
Lemma
I f a)
mj c t3,i (Extension).
n g
d that a 625, bç a' and a' is KM3-masimal.
such
,A
t
h
e
H t h e n
•) - R
M 3 t a hmEnumerate
e a . rx ei the
IProof.
m formulae of RI1/13; A ,
T
,iIDefine
Aa .sets
sl h
A
of
formulae
i t y . a, and b„ for i =0. 1, 24 r e c ur s i v e l y as
e
n
follows:
ia
I
,s
.ae=a.
. b „t
A
a
'
o
f
.
e
fA
o l(i)
Ir f a m
u
o b
lv
a (ii)
I
e
f
a,
ti
(A1..
.4 U
Let a =1A a
R
)1 a
.l
4 1w l
a
n
d
b
,
a
1 t) h e n
b
•
a
)
- .- . .
=
t
m
-y
m l
b
1
A
p
im
1b 4
1
.
.j
)
eat
aH
.
COMPLETENESS PROOFS FOR THE SYSTEMS RM1 AND B M
1 5
a' Ub' s e t of all formulae. We need to show the following:
-It.)
b
„
For all
The proof is by induction on i,
at, )pl 1 5 . , , by assumption.
Let a
l
* p(a) Suppose a
a lw
b
il l 4b A , . . )
and
itb ) — B
Then
I
, np &
i1where
m Bs = B, & & B,, a n d
C • C. v C, S i n c e
&
.v h
tb
a
b
1
,A s
iV
some
. disjunction C ' of elements o f b
l T
H
d
itC
putting
Bh" — & 13' a nd C ' ' —C 't• C', t i
B' C
.i,es er, B
u
' n "& v A L
and
o
+
Theorems
1
I Ca nd
, o fl RM3
in
b
.A
„
a
v
,
R
.
% 2A
" . B y Derived Rul e 4 , w e
V
A ", . , &
.C
have
,
u
B
"
s
—
i
n
*
C
g
"
.
Hence
b
,
, d contradicting our assumption.
c . t ,h " . u
i,a
T
s a ,
n
.,iel ,oD e r i v
d
a , Li ) p l ae (b)
fin
=e Suppose
b...
,
b.... a. 1.1 b
g muSince
3 bl
l.t e
b
o
aR=Is,.
tradiction
fre
r
a
1
e
s
l ,
Hence,
,s
l .a1 w a
T h
, h i e c nh
.r—
tio
By
2 (4s), a rnb' -4 . since.a i f a' 11b• + then, for some A and for
0. =
e
n
%
.c
some
,B, a
, nand A -6 h
1
s
m
da* I,oA Ca.
In
addition,
a' )1
m
ia
)e
I
n
.
)
,v
p
1
k
w
that
.
a,
,"
itm)
h
,b
a
n
d
- (d A
,
.
b
e
c
A
required.
i.h
i- ,. 3e In)sc %e
c
o
6
1
,a
a. i, n c ne
t4ms
n
Lemma
Let T he the set of theorems of RM3 and let A
.j1
w
hx (Priming).
»
m
A i pc 4.
i)s .dh
he
of RM3. Then there is an RM3-thivry T'inch that
a
c
o
n tt, or
b
i non-theorem
(. ia
v
u
1
T
T
.
A
t
T'
and
a
'
s
a
d
i
c
t
H
cT' is prime.
, me
sn
A
e
n
1
t
h
rc
s
e
lIb 3e .
Proof. Immediate from Lemmas 2 and 3.
q
1
b
.. eau + n '
ty
'I" which is prime and contains all RM3 theorems
'n
h
1
r . RM34heory
•ii1
,. Any
induces
a
valuation
V . as in the semantics for RM3 given above, on
e
r
e
H
d
e
), d
3
o
the
sentential
variables
as follows:
e
.n
n
,.. 4 c
e
.L i . ,
B
a
'„e s
'o
.t a
if.t n
s
.h i
e
R
l.e s ,
M
/f u 3
e
c
m
a
o
3
m
t hA
i
L6
R
O
S
S
T. BRADY
Tolit(p)igtpET'_
F 0 / ( p ) * - p c T",
Voterequire for such a YalUaltiOn that. for all sentential variables p.
T E Vip) or F W O , Here, this is so, since T
T
u T ' and hence p v -p a n d . by primeness of T . p C
-p
T
, ' Co r
T
3.V{Interpretation).
41Theorem
A
- . A The valu.ition V induced by T i s
extended
. •
t o a n interpretation I o f all formulae, such that
( T h e o r e m
(T e 1(A) el ET') and c 1(A) I
3
)
,
) Proof,
- A When
e T ' ( )T .e 1(A)114A * T') and ( f • II(A) cl --A e T') are
shown, the interpretation I will have the required property. T 6 1(A) or
e I4A1, since, as above, A v -A e T and, A T ' Or --A ET'.
The proof is by induction on formulae.
(I) T c l(p)
T I(-A)
c
c
)T E V ( M
p ET'.
cVip)
-p c
FeI(A)
C
HA)
4T
* A E: T' (induction hypothesis)
e*
-40 - - A T ' . by Axiom 9 and Theorem
4
4 of RM3. and the fact thatATP is an 1043- theory.
(iii) T ERA /CIO t -e) T • 11(A) and T 1 ( 8 )
T A T ' a n d B T ' (i nduc ti on
'
hypothesis)
(e
i- an KM3-theory.
RM3, and the fact that T' is
n
t
01(A e
d
•el - A e o r - 1 3 T ' (i nduc ti on
u
A
hypothesis)
t
c
&
cs
•
tB - ( A & 8) ET', this step being justified
as
ic follows:
I t
o
A )
T
n
o
'
h
r
,
y
F
b
p
y
E
o
A
1
tx
1
h
i
(
e
o
B
s
. I (-A)
COMPLETEN RAS PROOFS FOR THE SYSTEMS KM3 AND BN4
1 7
By Theorems 5 and l oi n n and the fact that T' is an R SI 3-theory,
if --A e o r -11 E
T By
t Theorem
h e n
o f RM3 and the fact that T is a prime 1043-theory. if
- t A- & B
( ) AT ' then - A v -B T ' and hence - A T ' or -13 T '
&
( IV ) I
c4 ( f f H A ) or T E 10311
B
-)
and
(F f 1(131 or F FI(A))E
T
'
„
1
(Air T' o r B a n d ( - B r j T ' o r
( A
--A4w T') (induction hypothesis)
1
3
A T
)
follows:
1
First Direction: 13) Axioni 2, A. - • B cTi -5, T ' or B c
t h
i EsT '
By Theorem 8 of 10t0. A - . B
T '
or - A ET ' .
s
Second Direction: We need to prove: (Ai T' or B c r ) and ( - B ( T'
t
or - A 1 - 1 1
e
- This is equivalent to:
p
)
b
A - 4 1( M
E IT' ' and --134• T') or ( M T o r -. A E T') or ( B
e
.- 1 3 t TEl o r 413 T ' and --A c r 1i -5 A-. 1 3 E r _
T' a n d
This is in turn eunivalent to:n
( M T ' and - B V I " - 5 A g
-. 8 ET
j
.(A t T
)13
(1
, E T' and - B 4 l ' ' - 5 A u-. B E T), and
1
3
1
s
(aBnnE d
t
d
.
T
By Theorem
9 of BNB. A E T'i or -B ET' or A -4 B c a n d hence
-a
(
1
1
T' and
e l
f
A
n
By
- Axiom
11. - A E T' . A ri T' or A E 'Ir% and hence A4 T'
i
and, - Adt T' e
f
5 ABy
- Bh i 10c of RM:1, 13 E T'
wTheorem
d -5-43 O r A 1 3 4
B
E
B L- h
T
T' and
-' , I 3 r T ' 5 A -.13 E T'
a
T
w
, By
-wi T
hAxiom
h'A•i i csah
10,nB d
E
i T' sand
h -eAsEn T c e
'
c( 13(h e) I , )
) E11(A -it B) 4-* T r, ICA) and F
i - 5. r A
B
E
1 T ' und - B ET' (induction hypothesis)
. . E 1 ( 1 3 A
s T -.
a
B) ET'. this step being justified as
4 w 2h i
c
h follows:
-n
) i .
s
,
d
(
4
)
,
A
B
E
l
'
)
Ig
R
O
S
1
4 First
1 3 Y Dire c tion; B
S
T. BR ,
y The ore m 1 1 o f R 1
ET'.
1
13.
A
E
r
Second
Direction:
We need
to prove : A I 3 ) f I ' A F 1" and
—B
a T'• n
d
This is equivalent to: (—(A 1 3 ) c T' A c
and
(
6
)
l ' ) (—IA . B ) c T' —> c
By Theorem 1 2 o f RM3 , —A E T' and ---(A .1 3 ) c T
Hence,
1 - . 0- A- BO
E ET'
T .'6 .
, — A ET', a n d h e n c e — A
or
(— AT IT T' ,' )E T• which
—(A.13)
A
is (5).
o ByrTT h'e w. A
E
Hence.
or —B T ' . Since II r or —BET ' ,
E - h --tAt
T
T e B ') nc '1".BCTr
.
.1 4e o n
- 1 i3 n
S
c
4. (Completeness). For all formulae A. A is valid in the
eT "Theorem
o f
- -R
- set-up
A
single
N
semantics for 1043 1 -i
, I 3
tV
-.1Proof.
,—
,
ALet A ,be a non-theorem of KM). By Lemma 4, there is a
prime
T'. containing all the theorems tif RM3 and such
3. B
A
1
1 RM3-theory
A .
that
A
i
T'.
By
Theorem
3, T ' induces an interpretation I on all
T
E c
formulae
B
such
that
T
E
1(3) B F T . Hence T f IIA) and A is
' T
T
invalid
i n the semantics, since the induced interpretation is a n
1. '
extension
of a valuation V of the semantics,
a a
nn
For all formulae A. A c4 A is valid in M
d Corollary.
d
v
he 1
_Completeness of RM3 with respect to the 2 set-up 'model Aiwa'.
n 3A
tyre for RA13
c e I
2- Set-up
Model Structure for RM.,
I
- The
)
model strut ture for RAD consists of <T. K , 1 1 > „ where K is
( cset (T . T
the
(1ATTr T s and (TI T T , and k is the 3-place relation on K defined as
.. '
)follows:
8 .i t h
w
Rabe =
)
T
dr
4T
-a
1'
.T
•T
o
i.
t
sw
b
th
•
i
h
.
ec
o
Ih
r
i
COMPLETENESS PROOFS FOR 11-IE SYSTEMS RPM AND B M
9
1
This model NI ruclurc has been independently determined by Meyer,
and by Mortensen in [51. Mortensen in [51 has shown that. for all
formulae A. A is valid in this model structure i f A is valid in M
have
k 1independently shown this by a semantiK: translation method, and
the completeness proof here will add a further method.
the
v
on sentemial variables arc then given al each
element a, b of K such that, for all sentential variables p. ItTab and
sip. a) = T b ) - T. The only case where a b and RTab is
a = T• and b T . Thus, we require only vlp, T *
1for
1 all p.
) = TheTvaluations
T v are
) extended to=interprearions 1 of all formulae A
in
the
same
way
as
appears
in 141 and in [51, Chapter 4.
A
formula
A
is
true
on
valuationy
i f 1(A. T)= T.
r
,
A formula A is valid in this model structure i ff A is true on all
valuations v.
Semantic PrdAperties
a h =,,
t
The following semantic properties can easily be seen to hold. for all
a. b. c R
e KT a
b
(i) T • < T. where a < b =
(ii)
d dK laa n , zi.e.
r ʻa ebc• a.
(iii)
a
b
and
KbedaRa c d ,
a n d
(iv) Ra bc -) Kac•b• , and hence b c m
(v) Ra a a .
(vi)
By using the semantic properties (ii). WO and (iv), a i
of i dt hc eh following
i c d etwo
f itheorems
n i t i can
o nbe proved, as in [51, Chapter 4.
and in [41, p. 208:
(al F o r all formulae A , for all a . b e K, a * h and ICA.
a) • 1
- 1
(3) F u r all formulae A. B. hA B . • ) = T ( Y b K )
(b)= T -41(B. b) =
1
k .
b
)
=
T
.
2
0
KOSS T. BRADY
Truth Con&lions for A B
Using (a) and (0), the truth conditions for A B can be simplified,
as follows:
1(A B . 0 . 14ind (I(A, Te) • T c
I(A- 84 1. Ev0. )1 .
C
E K) (I (A. b ) 1 • 1 ( B . c ) T O .
( 1 Tforiall b. c eK.
=
RTsbc
( A .T
. . 4) M A
1 " and
b
) 1(I(A.
( y8 1,) . T 1(Ft. T )
)
and
(T i1(A.
3
) T
and
(ICA.
Ts.)
T )
,
e
)=
t
,)aT Tb
T
o
and
(
I
I
(
1
1
1
.
1
1
vI 1 eC .
(T
(
1D
1 1*by.)property (i) and (al above.
C( I ( A . = T
f=
3
.A
T. )
lT sby properly (i) and (a) above.
T
)
=
.
() .
A
Theorem
T 5, (Soundness). For all formulae A. %Ls A i s valid in
the 2 5ci-up
model structure for R M3_
.
I
T
t The proof follows similar lines to that in 151. Chapter 4,
Prentll
)
B
T
.>
Compkteness
Proof-Prefiminories
T
1
Thiss
(completeness proof is also modelled on the Roulley-Meyer
completeness
proof. The definitions of RM3-theory. prime RM3 )
1
theory.1
disjunctively
RM3-derivable and RM3-maximal. given in § 2.
T
are also
required
here.
Lemmas 2. 3 and 4 are also required.
.)
Consider a prime RM3-theory T
RM3
that
L and
c osuch
n t a
i nAit nT .g for some non-theorem A of BM3. Such a
theory
T
a
l
l
L tDefine
i s hT e
e tIs t h e
o r e
m
1.
a s•b l i
1.
ah s
so
e
f has the following properties:
Lemma
6- TO
d f o 1l
b l o 1w(i) À * I n
y s : =t —
dA
4
t
r;
h
e
iT t
P
As .
r
l
i
—
m
A
i
COMPLi- r tNEss PROOFS FOR THE SYSTEMS KW AND RNA
21
(ii) T O is an RM3P1hcory.
(Ili) T c T t
Pro o f . I le n ie t h is t o the re a d e r.
For the model structure for RM3. let Inc valuation y be determined
as follows:
For all sentential variables p .
T) = T e-5p c
T
l
1
1
To satisfy the •galuation condition on v
1
.
Yip.
-)
, w Te
o y(p,
n lT)y T . This is so because Tc T. by Lemma 6,
1
a n d
r e q u i r e ,
)
Y l y is
p !Wended 10 an
The valuation
fTheorem
o 6. tinterpretationt.
r
T
.
interpretailon
I of
T
A ET
a
l
l all formulae. such that ItA,
c
I[(A,
Lp 1 ' ) T .c) A c l , " .
4
, a n d
P Proof. The proof is by induction on formulae. The proof is clear as
E
it follows along the lines of a similar proof in (.51, Chapter 4. and the
T
case for B can be dealt with as in Theorem 3,
L
. Theorem 7. (Completeness). For all formulde A, A is valid in the
•
model
structure for RM3 A .
?repo!. Let A be a non-theorem of 10P13• By Lemma 4. there is a
prime RM3•theory T
L
that A f 1
determine
a valuationy for the RM3 model stnicture. svhich is
-.
c o n t atoi an
extended
n interpretation
i n
I such that. for all fomiulae R. I(it,
1
.gIF
. – T >13 e T
in
a
1 the
B
y l KM)l model structure_
t. h he
e
T
tH reh e
n c
e eo
o
§m
r. 4. Completeness
e
m s
of 8 .
1o
l6 i A
Axiomatization
0„f, 1 w i t h of HMS
rT
B
T Primitives:
e s ) p e I ct t t, –p.
t.M
–
o
Definitions: A v B
A
4
A
3
F
t -- o l
a.
g
b Be i - d t
( A
tnf B h ) S : ( B
do r
e
s
r
w
h
iut
s
tci
o
hm
d
e
T
.
Ll
22
Ax
i uI.
r r2.
i .3.
i
• 4,
R
a
n
T. BRADY
A -*A.
A&B-*A.
A&B, 1 3 .
(A ,
1
3
5. A )& ( 1 : 1 V C )
6.
8
-A.
e -•
,i t A
7.
I( A. 'AA :- 4BA) .v
8.
(AA...A
& CB) . A -6 R9.
A
V(A
10.
A
V-43
V
(A
A
,P
12.
,. V (--A A
B 1.
/&
3 )A
I .
Rules.
C A
1.. A. B 1 h : B.
2. A. A - 4
, A , R. ( 7 - . D B , C , .
3.
4.
B I CV A. CV (A, 1 3 )--. 1 C
-• 1 3 ,
•
vTheoreon3
11.
I. A
, (A & B)&C -* A ifk(13itta
2.
3.
AA
v ,1 3
EA
l v(I3 v ( : ) , ( A v BP/C.
4.
.S. (A v 13),):1(A v C) --• A V (B&C).
6. A V A A .
7. ( A v B ) v C 8.
, A-.
A -A - &
B).
.
10.
1 - B - 4 A Ai 13)..
11.
N/ B.
. / -(Ad3c131-•-A
( B v. I C ) .
A -4
13. B, -. B VlA ( U s e Ax. 9).
14. A1 1&A--B-o. A $4.--B H ) . (Use Ax. 1 0 .
15. A v. 1 ) ) - A ) . (Use Ac. 12).
16. 1 3 )
- - 0 •-A).
COMPLETENESS PROOFS FOR THF. SYSTEMS RM3 AND 11N41 2 3
D r i e d Rate.
1_A--•111-1-C . C v B.
Sinee Sel-up Semantics for BN4
As for RM3. a single set-up semantics is employed to facilitate the
completeness proof. I t should he noted that the only difference
between this semantics and the corresponding one for RIO is that this
semanti
to
c be assigned to formulae_ This semantic
s f o r semantics
Dunn's
B 4 for rirst-degree
i
s
entailments in p l
differences
truth
,a lt l ho
n e being
e s in
s the
e n
t i and
a lfalsity conditon (iv) for A a n d
the
a
d a p of
t validity,
a t i o n
w sdermition
A
raluation
V
assigns
a subset of 4T. F) to each sententini %•itriable.
o
f
n e i
Symbolize
this as follows: V (p) = (T). V(p) 4 F ) . V(q) = {
t h e
V(q)
T
F etc.
} .
r %
, =4,
a The valuations V are extended l e imetpretations I of all fommhte,
inductively
as follows
s
w
(i) l ( p ) = VW),
e
(ii) T el(--A) e*F c 1(A).
l
e I l -Al T c I(A).
l
(iii) T c l(A & 13) c" T el(A) and T c l(B).
a
F cl( A&B) c
s
(Iv) T- el(A . 8 ) tlo(Tf It A) or T c I(B))
b
and
) F (Fl• l(B) or F e l(A))„
o
I'
I )
Oc A
t
(o
r
A formula A is valid
in this semantics i f T ç I(A). for all valuations
h
A
V.
F
o
H
In order toeshow that NI, and this Semantics have the same valid
f
)
formulae,
let1V,, be the M
tV, as follows:
T
4h
(1 8 )
-evFor
a l all
u asentential
t i, o n variables p.,
(
cv o r r e s p o n
ii(13) A=I T C
d
g
a i nVo(p)
V ( p) 4)
tl
o
Vm(p)
, a ni nd ,
tu
h
e
1 0F
w
1 l'T mf u
ve
a5
a
V ( ( pd )
e
V
V
t
p
tt
i a on n
(d p
)p .)
r
a)
F
u
.
fn
t
f
V
d
h
T p
(I
a
) o ,
:t
n
EV
d
74
R
O
S
S
T. BRADY
Lei I m
b&eand a s d e
tf iformula
h e
A
A is valid in .41
since
b , y„ , V
1en eixthe
fdft vahiations
1
m (eMA
an rl ,e
=
I
Lemma
e o
sx ih oar 7u. For
s t all
i formulae A, ( k A ) = T c-,141A)E• o r b) and
v h
ne .
. 1,4A) f or bl.
fo
o
r
Proof.
The
proof
fa
l
l is by induction on formulae. I leave this for the
reader.
vV a l u a t
im o n s
lipeeprem
8, For all formulae A. A is valid in the %ingle set-up
l
V
semantics
for
BN4 ct A is valid in 1
o
*
1 a
By Lemma 7. T E 1 1 0 0 c 3 i
6 1Proof,
n„
Hence,
A
is valid in this semantics for 3/44 (
m
f
- ioA t = i o r
b .
Theorem
V,
(Soundness).
A. A
A is valid in
5A
i
s
v
a
l
i d For all
i formulae
n
r
the
M msingle
. set-up semantics for N W
. u
Proofl leave this for the reader_
l
a
e
Completeness
Preq-Peelleninaries
,
u
Although
the preliminaries are similar to those required l ot the
s
completeness proof for RI D. refinements are made to the notions of
m
BN4-theory
and BN4-derivahility.
g
A
BN4-theory
a is a set of formulae closed under the following
t
closure
conditions:
h
(ii) A a and B a -"A &I I ca_
e
c
lii) 1 o (hip
5 1A - • Elt aan t l A t a
(iv)
n - ,- C V ( A - m B f a a n d C y A t a
n 1„BA L- -.
a .
A BP
e
AV B ca-•)A
,- • 1 3 ea or 13.€11.
c
A formula
Iak l - 8 ir NW-derivable from a formula A, s
t
u i t tnc
he
n
A
B
.
i
odr t
v
aA
e
ic
s
sa
p)
r1
i1
c
m
COmPl E
FNI if 13 is Iilerivahle
from A by successive applications of the following
rules: S s
P R
O(i) OA . B
& B.
F(ii)S A 3. B. where 1 F
(iii)
1 LNA . A -.13 B .
O
(iv)
AC v A.
- C
• ( A B i ) C v B.
R,
A setTBb of. formulae is BM-derivable from a set a of farm:dal',
H
writtenEa -i.„,
B
v
B
4 b S b (n I ) . A, & &
A
set
of
formulae
a
is
BN4-maxional
a
, i f fY 1,
f o rS . .
s Lemma
o T 8. I t the set a is BN4-maximal then a is a prime BN4 E
tmheory.
eM
A
S
. Proof:
K
M
e
(i) L e t A ca. B c a a n d A & Bit& T h e n . s i nc e
)
a
/1% A
4
4BN4-nuiximality.
N
.A
D(ii) L e t E
m
0wcontradicting
ft
n
.
the BN4-masimality..
a A
8
)4
N
(iii)
- . LB
e t .A -4.B *a. A ea and 8 4 a. Since 1 4
3
q
62AR B . (A -6.B)& A A and ( A - . DI& A 01,
1 A &
1
a
A (A
,5E- —
. B)&
B , B a . nby rule
d
(iii). Then a 1 3hence
1
1 1 sBN4I-maximality.
4 4a2
4 . A c o &n t r a Bd i c t -i n. g A
•the
1
(iv)
aa Lne t Cd v(A -.13) E a. C VA ea a nd C B 1 a. Since
•
.A
&n A
and
A
& H , ( C V (A 1 1 ) ) & (C v A)
1
NM/ C '••dIA — II) and ( C V (A 1 3 )) & (C v A) l i
&
f(C
o
B v (A B-• B))& ( C A ) 7,,„ C %.• B. by rule 'iv). Then 2 0 o
2
,atradicting
a the BN4-ntaximality.
r,1
A a nand d
B a . Since 1 ,
n 4 C.or) LVe t AA v .l i c a, a
s
A4v IeaNT
4. 4n cAc o n
v
4
h
eV d
o
mality.
N
B
.
h
A
n
. A e
&
iA
V Lemma
n 9_ (Extension). Ilia e
such
,
that
b
t hb ea nn d a• is BN4-maximal,
c
B
, aÇW.
&
t
h
e
r e
B
a
s
F
i
s
1
n
i
L
,ad a
n
s
t
ce
-h
a
'
e
7e
o
f
n
A
0
f
o1
r
m
u
4c
l
a
e
7
ce
a
„
o
P
,
n
26
K
O
S
S
T. BRADY
Proof. Enumerate the formulae of liN4: A
Define
l
sets of formulae a, and b
follows
i . A :3 ,
A 3 ,
A
, *f o r = a, éb„ • =b.
0. ,
I
,
2
.(i) I f a, U fA . .
1c I uf a, rU (sA i v
r e (10
Let
a'
-=
K
e l )y a4 , and b' b
A0
b
ai By construction.
s .
. H t,e h
n ce e a U b' s e t of all formulae. We need to show:
a
n4 a , b - h, , , for All
(k)
a
a't h e n
induction on
a The proof
na , isd by
=
, ,
I
a, I
b*. by assumption.
g
,=
, E N 4 bi a nd k it a , 11 L1 ,e4t4 aU
7
ta1
(a)
Suppose
% ' 1 4 •(1, • A a, U. (A,.„) ▪ b
1Hence a, t
.a .
, B
b
.i mT eNpectively
h ,e n
a
)
n
s&
U
lB,&...
C , V v C„ V A
ad
( ,E
A l ,.. &. & = 13, Lind C =C, v ...vC,„ Since a, U A
.B
v
nb
, , B' of elements of a, and some disjunction C' of
at.,some
, .) . conjunction
i1, c 3.
B
d
„
.
ekmenla
o
f
h
.
H ' fb
& oC 'r . P utti ng 1 3 " = B & B • a n d
a
n
.aL
.
0
h
.
1
n
d
b, VC, 1. 3 " & A,„., C " , B " C " v
and he nc e
C"
INC'
d
e V tA
iC
B' CC
-=" can bc obtained as follows:
f 4o, 1
.1
1b
.r m 1(a) H " & A
=,
u
w l ,i h
e
r
U
I of B M. C' C " i ' ' C and, since 13 & A,„, F
th
a ByeTheorem
e
.i(,
13'
&
A
ChC 'ttr.4
. "A C". By Axiom 1. & (B'
o j& Ali
.e, & (B'
. & A,. ,) ▪ C''. By Theorem 2 of 13N4, & 13') &
hence
1
B
f
.a
n1 B ( B ' & A
1 1B •.
1
(13)
,
.a a n ,d
By Axiom
B
and, since H C v A
1 2. B&
▪n 1B y Theorem
3
of
B1'
4
4.
C
v A . , C • v (C V A,. 1
) a n d
d
11.
B"
. 1 1; i nhd e h ne n c e
(1b044
c e
,
C
C .' (y)R&B " 1 C
'vi v
,
ra A
n
V
(
„
.
Cl '"
t
C
C
,
)vh
C
i
v
B
"
s
A
y
.
d
,T
e
.h
Ier
i
.o
COMPLETENESS PROOFS POR THE sysr E m s RM1 ANo 11N4 2 7
By Theorem 3 o f 13N4. B " C . " V B" and hence. by (0 ),
13
and,
by Theorem 5 of BN4. (13 & A, „ ) arid then B" C " V (B" &
A
C
1"
since.
usi ng it, w e obtai n, by (o). C " v ( B" A
Ihence
B" 1
.1
„then
i C " . • as required
4
11 ) tB"
t. .s t
),"
C
4
C (b V
-.,
C
"
.aW
)n
C
d
V
e" The (proof is by induction on the derivation of B from A by using
C
rules
We show that A
n
v "(i)-(iv).
this
derivation.
.o
C
v
•
c a n
1
A
B
b
e
- (j) T h e yule D. E D & E is applied. C v D, C v E
w
t
y
p
c veE) and,
d
,(C vl DIsa ( C
n
by Theorem 5 of 8N4. (C D ) & (C E )
iT
,C v h&sn E) and hence C VD, C v E t C v (D451
e
fe
or , o
n
t
e.
(il) T h e MAC D -a E is applied. where D
rd
o
e
f
k
Rule
I 4of
- 13N4,1- C V D . 1 " V F. and hence C VD C " / E.
m
c
a
d
((iii)
t.
D, D. r 4 aT h e
D rule E
B
y
i6
o
T v (D10DE E
C
) e CrE iv. E.v e d
o
s
e F v D. F v (D--, Ell -4F v E is applied.
[(iv)
sh
i t sT h e rule
fh
p
2a p p l i e d
e
By Theorem
5 of BN4, C V v t 1 3 E l l ( C %IF) v (D E ) and
B
o
y.
o
n
C
V
(
F
V
D
)
1
(
C
V F) v a
fN
A
w
B
B C V (F
yv C
Hence.
v
(I) --P
v
F) v D. (C F )
4
C
t"
r mkt hence,
u
lby rule (iv), C v (F v D). By Theorem I of 8N4.
(13
E
)
,h
B
(
e
C
)(
a(C
C
i C v ( F v E).
iF I D v
"t."
.)
,
with the 'woof of Lentinii R. since 8'• Ij. C . a, b , .
V
(v
V Proceeding
C
contradicting
our
assumption.
F
C
A
B
V
Suppose a , U 8 4 4 b. T h e n a
"1)(b)
'
D
l1 #
b
v
4'tradiction.
.U
I
A
1
E
.,,o
)
Hence.
a
:
1
4
.4o
.
0
a
n
d
4t&
1 3 , d • r) b' • 04 since. iIa ii b' 0 then, for sonic A, for some
-(= By
4
B
4
A
C
I 4: a
)o
C
1 addition. a' r b
"(t") In
al
i,nIm a s
a
a
C
,V ,r eh 'q . u i
Id
C
V
A
S
r ei nd c. e ,
s
"A
v
A
4
it i
f
("
*b
n
'
,iF ,a
-.c
b
b
w
-v
1
e
.h '
IE
)
h
a
)i t
p
e
n
n
Jia
ct
h
d
.
n
2g
R
O
S
S
T. BRADY
Hence, a' is BN4-masimall. as required.
Lemma W. (Priming). Let T be the set of theorems of 8N4 and let A
be a non-theorem of BN4. Then there is a BN4-theory T' such that
T C T ', M T' and T' i prime.
Proof. To prove T $••np.4 {A}. let T ( A ) . Then. for some A
..,„3 A . e T. A . &
A .
S inc e T i s a 8 N 4 the ory ,
Ap&
, & A
m the
c four rules (i)-(iv) that can be used in establishing Brsl4-derisaof
1
1
bulky. However. A t T and hence T q
.set
TBT ' . y A t T• and T ' is, BN4-maxirnal. Hence, by
u o T '(such
A ) that
.
aLemma
ne d 8.
BM-theory.
L
m T is
m a prime
a
, Any 13N4-theory
T' which is prime and contains all BN4 iheurems
9
.
induces
. as in the semantics for 8 N4 above, on the
ft o hr a valuation
e
r Ve
sentential
as
follows:
li h e variables,
s
r .
a
T V( p) e- p 1 " .
A
e V(p) es -0 c T'.
e
T lhearear W. tInterpretationl_ The valuation V induced by 1 " is
,extended t o a n interpretation I o f a l l formulae, s uc h tha t
b
( c l (Ae o c s A r T*) and (F i: RA) c-4 -A * . T
c
a
.
u
s The proof is by induction on fommlae. and follows similar
)- Proof.
e
lines to that of Theorem 1 lise is made or Axioms 2. 3.7. S. 9 and 10
T Theorems 8.16 of BM.
and
i
s Theorem 11. (Completeness). For all formulae A, A is valid in the
d
single set-up semantics for BN4 -5 l i
o
N4 A .
s Proof Let A be a non-theorem of RN& By Lemma 10, there is a
e
prime BN4-theory Te. containing all the theorems of BN4 such that
d
Aç u
1 1such that r ERB) C) B c l . Hence T ; 1(A) and A is invalid in the
B
n
semaniics,
since the induced interpretation is an extension o f a
d
valuation
V
of the semantics.
.
eB
ry Corollary,. For all formulae A, 11,, A csA is valid in M .
eT
ah
ce
h
o
r
e
m
1
COMA-L.1EN ESS WOOFS FOR I l l E SYSTEMS It M3 AND OM
)9
§ 5. Completeness of BN4 with respect to the 2 set-tdp Pnadri
ture far RA'4
l struclbe model structure for 8 N 4 (
is the
1 set I T, T I
1
n and
s i sCP
t s T T o, and
f R is the 3-place relation on K defined as
a t )* c=oT•
follows:
) , <w i t h T
,
T K T Rabe
, =.
• • dr
i The
, tratuarioni
v on sentential variables are then given al each
(sa t R h
element
r - b of K such that, for all sentential 'variables p, R Tab and
%
e > T
.
applies.
(I w o- hr
e
r
y are extended to interpretations 1 of all formulae A
p
1 The
e vallualions
b
way as appears in (4) and in [51. Chapter 4.
.in
f the
K usame
e n
A
formula
ac
t ) i A is true dm voluatum v iff 1(A. T) T .
A is valid in this model structure in' A is true tin All
)o A formula
na
valuations
v..
=o
n
T
n
d Properties
-Semontic
K
(
1s
a
a e:, b R T a b
1u
T
(c
.
The
semantic properties can easily he seen to hold, for all a.
p
h following
b. cc K' :
.t
o
b
h
r(I) a içb 4-sa = h.
)a
Rabc R a e l ts .
i
T
t
b
.
•
I ruth CumlitlinLs
for A
S
T
i
al(A — B. n n
n
and
1 ill A, T
c
d
1
-1(A T s ) - T c4 NA. T) - r 1 (B. . T*0 - T. since a b
e ) a = cT and h 1 • • •
)( 1 -( A .
R
T ) = 1 "
T
'- - U S
a Theorem 12_ (Soundness). For all formulae A. I
the 2 set-up
.5 1 model
1 1 3Astructure ifor HMI.
s
h mo A
T
, l si d
v
a
i
n
c
)T .
•
s
i
)
)
n
T
i
c
i
i
e
h
I
.
t
t
T
30
R
O
S
S
T. BRADY
Proof
.
T h
Compleieness
Proof-Prirliminariet
e
p
rThe definitions or BN44heoly. prime BM-theory. BM-derivable
and
o BM-maximal. given In § 4. are also required here_ Lemmas 8. 9
and
o 10
f are also required.
Consider
a prime BN4-theory TL containing all the theorems ot
f
8N4
and
such
that Ast T
o
Ll
theory
T
, lDefine
L
fi o
s r TO as follows :
s oso t m e
e
nw
a
bo l TO
ni -=-,
it
t Lonma
s
sh ee o11,r T e m
(A l
A Is
d
(
—
oC
b
h af s
i ' (ii)
. T O is closed under Adjunction and provable BN4-impli1
y
tm h A
3 e t T eNis closed under the first two closure conditions of 4
cation.
i.e.
1
T
4
L
fi o . l l
BN4-theory.
})
So
e
l w Aiu n
.
cg
m
a
Proof.e 1hleave this to the reader.
ap
m
o
rFor rthe
T model structure for 1344. let the valuation v he determined
ap
e r
as
lfollows:
L
1 tiFori alle sentential variables p . v (p, T ) = T p r T
4
0s
n
v :T
a4 n
T ed l . sThe
i valuation
p .
condition on y applies.
. e
—
s
Theorem
A 13. (Interpretation). The valuation y is extended to an
t
interpretation
I of all formulae. such that 14A. T) T C4 A E T . and
l
o l's) T c-> A e T e .
t
h
Proof.
The proof is by induction on formulae. The proof is clear as
a
it follows along similar lines to a proof in 151. Chapter 4. The case for
A—,
t 15 can be dealt ssith as in Theorem 3.
i
n
Theorem N. 4Completeness). Ear all formulae A. A is valid in the
model
1 structure for BN4 A .
5
1
.
C
h
a
p
t
e
r
COMPLI-JENESS M O M S FOR THE SYSTEMS MM) AND BN4
3 1
Proof. Let A be a non-theorem of BN4. By Lemma 10. there is a
prime BN4-theory T . containing all the theorems of BN4 and such
that Ai TL. By Theorem 13. T
determine
a valuation v for the BN4 model structure which is
IL
extended
to
I such tha t
. t o g e tan
h interpretation
e r
,w )f=oiT r4 t ah l l ' F
T
,fL Hence,
o r m1(A,
u T1
l aF efor this model, and A is invalid in the 8 N4
model
B
1
, a
* 1 3 Fstructure.11
s1
11
(d
T
1e
1
.
f
i
n
e
)L
d
,a
b
o
v § 6. Concluding Rralari,
e
,
A similar treatment can be given for LAikasiewices 3-valued logic as
was given for RM3 and BN1 above. This logic is represented by the
following matrix set L
L
t n
:
& II n r
tot
l't 1 1 n n r r
•
n
f ff f
s
i ff
It can be shown that any formulan
A is valid in L
n
nn
n
•1
n
s
-
t
n i
t n
t t
j
following
single set-up semantics. The single set-up semantics for 1 .
is the
i f f same
A as that
i for
s 1013.
v excepr
a l ithat
d valuations V assign a proper
3
subArt
of
variable. The valuations V are
i
n T , F . tot eachhsentential
e
extended to interpretations 1 using the same T- and F- conditions as
for RM3. The valuations correspond as follows:
V
V
m
(•
Vm(p)
f 44T f 1/(p) and F %
(pp
4
Using
a completeness proof similar to that for BN4, L
).
axiomatized
I c 4a n as 1.3:
b BN4
e + A & - A A 6: - A — B. where A it --A
--•• A &t1 - A '
both
AC
.
)and -A , for any formula A. and thus the valuati011 V induced
•by I " is3
walways a proper subset of (T. F .
6 There
B T
n is also a 2 set-up model structure for L3, which is the same a%
e n s *c u
r e sV
4
t h (T
a t p
t
t
h)
i
a
e
l
n
p
d
(
n
1
p
m
32
R
O
S
S
T. BRADY
lhe one for lits14 except that Rabc is defined as ta T or b T s or
c T s ) anti la T ' or (1, T and C T s ) ) . The valuations v must
satisfy the c onditk
-T
with
on:
1 respect
F o r to this model structure. in a similar manner to that for
a1 l l
sT Ite isnalso
i eworth
n i noting that a completeness proof for 1013 can be
carried
i. a l oui in the SWIM manner as for RN& resulting in the axiomiati&Ilion
vT a ofr RSI3
i as
a BN4 + A y —A.
b
h l e s
Ross T. BRADY
La
p
e Train. University
and
.a
AuTtredian
National University
vx
(i
p
.o
NOTES
Im
)a I') Tio% name it amen because the system contains the belie tystent k cdf 151.
at
Chapter
4, and has a characteristic 41-sulurd matrix set_ one of the values being '11.
*i
tepresenung
notha truth nor falsity.
T
z et I aelinwaktige help newt Ft. K. Mullis in getting the amornauration two this foam
ya et The idea of allowing hoth atues Î rani F o be assigned to formulae Ci!etne% from
I. M. D i . 1m a Dunn also allows ricithei I MiK F to bc assigned to formulae.
it e t 1 clinoinior itii% model stnxture for 0 N4 by using the Mortensen method oir
p
i
construed's;
model structurel from mania lets. ris described in 151.
.o4'1 1 acknowledge help from a referee oirLogique ct Analyw in putting thit paper in a
suitable
form for Fioblicatiors.
n
L
i
a
REFERENCES
b
1o11 A R ANDEHOel and N
D
v B Neve-olii)•••
1 . 1
NVol,I I. Princel011. I975.
121
A
P ,butts . 41intainue &mammies for First-Degree Entaibmeas Anil Vedettes'
e J.M.
Studies
Jc e Trots%
. . • Phdosophicsi
. E
1)1
G.
P
,
RI
V
M,
o
l
•Sun
.
of
2
Logic
9
of
renidox•. presented at & K U Logic illnittal
na t a i l m e n
Ill R. ROLTILIgY
( 1 9 7alai
6 It
) k, , MEYER. 'Ti l e Semantics of Entailment-. in I L Leblanc
tn .
-Truth.
p
c
. Syntax and Medal i t-. pp. 199-243.
h
e
PJ
b R. Roomy
1 4 9and- R.K.
1 Mksea
4 4 vrai.. •Relevimi Logics tad their Itivekt.• NUhog
L
o
v in Itat2)
ireatheolesog
,
c
e
s
o
fh
o
R
e
l
e
vw
a
n
c
en
.
2t
n
.o
1
b
e