ON TRUTH-TABLES FOR M, B, S4 and S5
Arnould BAYART
Part I
O. In tro d u ctio n
0.0. I n t h is paper use i s made o f re su lts a n d ideas t o b e
found in Godel's [ I ] f o r the formulation o f modal propositional
calculi, in Leonard's [2], in Anderson's [3] and [4] a n d in Hansons [5] f o r the construction o f truthtables, in Krip ke 's [6] and
in Hughes's and Cresswell's [7] f o r the th e o ry of models and in
Church's [8] f o r p la in proposition logic. The paper is h o we ve r
self-contained a n d demands n o p re vio u s acquaintance w i t h
these results and ideas, except o f course f o r [8].
0.1. Th e systems treated i n t h is paper a re mo d a l p ro p o sitional ca lcu li. Th e y a re the Feys- va n Wrig h t system M, th e
"Brouwerian" system B, and Lewis's systems S 4 and S 5.
0.2. Truth-tables are constructed as i n [3]. No tio n s o f " co n trol" and " a cce p ta b ility" w i l l be introduced. Th e tables co mbined wit h these two notions w i l l yie ld fo r each system under
consideration a suitable notion o f " ta u to lo g y" . I t w i l l b e o b vious that our definition of "tautology" is a decision procedure
for tautologies. The fo u r systems w i l l be formulated in Gödel
style. I t w i l l b e p ro ve d f o r each syste m th a t a fo rmu la is a
theorem i f f (i f and o n ly if ) i t is a ta u to lo g y i n th e su ita b le
sense. The theory o f models yie ld s f o r each system a suitable
notion o f " va lid it y" . I t w i l l be p ro ve d f o r each system th a t a
formula is va lid if f it is a tautology in the suitable sense.
0.3. W e t a ke negation, ma t e ria l imp lica t io n a n d necessity
as p rimitive , and we refer to object-language formulas b y wa y
of schemata in vo lvin g " — ", "---->" and " 0 " as metalogical signs
336
A
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N
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D
B A Y A RT
for the p rimit ive constants, and small Greek letters, eventually
followed b y suitable suffixes, a s metavariables ra n g in g o ve r
the class of formulas.
0.4. We use schemata like " (a A (3)" and " (a V 13)" as abbreviations f o r ' —' (a -- — p)" a n d c t - - > p y • re sp e ctive ly. W e
use also "(aa
- schemata beginning wit h a le ft parenthesis and ending wit h
In
a
we d ro p th e outmost parentheses. I n o u r
_ -rig( h3t parenthesis
) "
metalanguage
w e u se continuous co n ju n ctio n s a n d d isju n ca s
tions
a
ni n the usual wa y. W e a llo w ourselves t o consider a n y
formula
a b b as
r a degenerate continuous conjunction o r disjunction
with
a
single
member.
e v i a
t i o n
f1. Truth-tables.
o
r 1.0. Definitions. A we ll formed part of a fo rmu la a is said to
be
a subformula o f a. E ve ry fo rmu la is considered as being a
"subformula
of itself.
(
The propositional variables and the formulas of the f o rm O a
(are said to be constituents. Th e formulas o f the f o rm D a a re
a
said to be modal constituents.
—A constituent wh ich is a subformula o f a is sa id t o b e a
>
constituent o f a. I f a constituent a is a proper p a rt of a consti(
tuent 13 then a is said to be a subconstituent of 13.
3
) 1.1. Th e degree o f a fo rmu la a is defined as fo llo ws b y in A
duction
on the construction of a.
( If a is a variable, then the degree of u is O.
( If a is o f the f o rm — 13 and if the degree o f (3 is n
3 t t hofea nis n. t h e
degree
- If a is o f the fo rm (3 —> y and if the degrees o f (3 and y are n '
and
n" respectively, then the degree of a is the highest number
n) such that n n ' o r n = n".
a If a is o f the f o rm o p and if the degree o f f3 is n, then the
degree
of a is n + I .
)
)
" 1.2. W e re fe r to sets o f formulas b y wa y o f the le tte r " . "
followed
by a suitable index.
.
O N TRUTH-TABLES FO R M , B, S4 A N D S 5
3
3
7
Definition. A set .9% o f constituents is sa id t o b e adequate
iff (1) SP. is not empty, (2) .9% is fin ite and (3) e ve ry constituent
of a member o f i s a member o f .90a. I f an asc (an adequate
set o f constituents) .9% contains a ll the constituents wh ich o ccur in a formula a [wh ich occur in a n y formula o f a set .9'
formulas]
.9% is said to be an use for a [for
1
If
an
asc
.9% contains ju st a ll th e constituents wh ich o ccu r
, o f
in a formula a [wh ich occur in any fo rmu la o f a set . . r
b o f .9%f is
mulas]
o said
r - to be th e min imu m use f o r a [ f o r .9%].
If an asc .92b contains a ll the constituents wh ich occur in an
asc .90a, then .9% is said to be a subase of ,9
9
b 1.3. Theorem. Let .9% be a subasc of .9% such that .9
•just
0 n (n > 0) constituents mo re than .9%. Th e n i t is possible
to 1construct a series o f ascs .
-9 9, c o n t a i n s
90 / ,1.contains
ju st one constituent mo re than .9
9 9
91 Proof
9 r b y induction
on n.
•
• I f n — I then le t .9% be .1
a.• )i a n d o l e ' t
,9 Let
.9 n
+
1 (0 < m ) . Th e n constituents o f .9% wh ich a re
S
not
in
.9%
can
be
ordered
'D
s
u
c
h so as to f o rm a series a
'such
l1
that
foar e ve ry
t
h
t i and j (1 i < j n ) we have th a t a
not
of a
%
i,b
. i asa 9constituent
'
im
ito
e
0 i t th a t i n th e series e ve ry constituent a
sconstituents
i.,9
e c se„ odf ae degree
s
ahigher
l than
l
the degree o f a
i pT ro a
o
b
t
a
i
n
define
th
e
series
.
9
%
„
.9',,
a
s
fo llo ws. .9% i s .9%. F o r
.lt9
h
e
s
u
c
h
9every
., W ie (0 a t i h< n)
e cno n t a i n s th e constituents o f p l u s
a
%
a
isn prove
e
r n o iw that . 9 '
aWe
,ne
.9',„
and
le
m
i
s
at 13nbe any subconstituent o f a. I f a is in .9% then, as
d
.dlia
t
t s c .
•sh
9
u
f
L
e
tf
fa
m
Iie ocr in
9
e series
s
a
C
E the
o.9%
lrt,b
is
in
the
series
a,
a
, , then a,, cannot be (3 as a„ is of a degree
e
t
not
lo
we
r
than
the
degree
o f a, and so (3 is in the series a
,eh
io
a
.
.
a
n
y
h
lae
a
Iicse fo
n
s t
that
.9%,
is
an
asc.
,(n
e
sio
3
t
u
e
n
cm
o
Then
by
the
hypothesis
of the induction on n we have that the
a
ih,tb
e
s
r
n
iia
vo
n
e
s.a
9
%
fim
sd
h
e
ust.o
h
9
cu
n
co
o
,sl
3
hi(%
it(td
n
h
h.s3
s
i.a
a9
t
9
str
n
t%
338
A
R
N
O
U
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D
B A Y A RT
theorem holds f o r the series 9%, S P „ , . A s ..9
9 d'
as
theorem
„ i s
a
aholds
n fo r the
a series
s c .
9
ca 9o n
d
o
•
• A value-pair fro m an asc d '„ is a set contain r1.4.
t • Definitions.
ning
ju
st
one
o
f
the constituents o f d '
a i
"anT"sao rn"F".
d
o n e
oj A r fo w f ro m ta n asc
h , 9e
lu 9e
constituent
t ta oef . 9 r s
which
9s „ i scontain
a a.
The
table
,t so ne e t fro m an asc .9% is the set of a ll the ro ws f ro m 9
9
at We
c n oshall
dn refer
t a to
i rows
n i b y wa y of the le tte r " R" fo llo we d b y
a
and to tables b y wa y o f the le tte r " F " f o la
oh suitable
n ng l index,
y
lowed
index.
oe .j
nbyu a suitable
e s
oc t
f
I f the tables F „ and F
to 1.5.
f Definitions.
h o
the
en 1
r ascs d '
subtable
oevf i f tf eh
d' e
a
ts ,ea an rdw
ad
If
i
a
s
table
a
F
„
is
the
t
'
a
b
l
e
s table fro m the asc d '
ot r
y
[h
be
p
a
a
r
[minimum]
o
p
e
r
table
]
fo r a formula a [fo r a set o f formulas .9 '
f
r
o
m
vi
a
l
u
siff
1
,
,9
u
t
h
b
e
n
a
F
ret e s- p
s9
,1
ac cFat i i i r s
s
a
i
d
e
pu If
in
o„vse a
td
ie ' l y fo
a
i sR
.s,n If
[9a
if
i
%
tt itF
a
in
F
s.n
h
a
ha
a
g
ia
e
ensn
i,a
occurs
in R
nsnm d
t
it[dcF hg
a
o
,se
w De
e fin itio n . Let F
2[„r ib1.6.
p
stvF„.
a
a
r„a
laW ebydefine
e
aunder wh a t conditions R
ie ]a
lo
,m
ith
re
vt he
aa risab
i af ilconstituent,
ee s
o th er n a , ve rifie s a if f g t
st uIf
value-pair
- c l o sn (a,
w
p
fte
ffa
tir aT)
f i ian ne
sd fsa l ts i f ihe s ae i f f M
ish ie
sp
R
ia
pair
o nF)..t a i n s
t
h
e
en
aa rcaa (a,
via
r
If
a
is
l
o
f
u
th
e
e
f
o
rm
—13,
then
R
rsin oa
n
d
w
im
and
re i f i eat iff
s v e r i fai e s 13.
]am
m
l vi efalsifies
d. iw
h
se
i
If
a
f
is
o
f
f
the
f
o
rm (3 --> y, then a , ve rifie s u if f f a l s i f i e s (3
yu
t
s
.
9
t9
iu
a
or
f
ve
rifie
a
s
l
y,
s
and
i
.Rf i e s
to
h
o
m
9 i
i[scb
f
1
a
p
l
s
i
f
i
e
3
s
ha
b„ feb
rthse tae o
a
„
ca
h
ip
tcat h
sa f e f
o
vre
r io f
a[]h e
b
cr e]
isvn
e
s
lofpe t9w
te
a
(u
3
co
9o
r a
lo
b
a
n
cro b
„f
ifa
ru
d
p l,
n
e
fo
l
s
r[e e
F
i a
O N TRUTH-TABLES FO R M , B, S4 A N D S5
3
3
9
We sa y also th a t M, g ive s th e va lu e T (F) t o a, and th a t a
has the value T (F) fo r M
i
. 1.7. Theorem. L e t F „ be a subtable o f F
rows
l of F
a „F l e t
let
M ,
a
n
d
a M
have
n dfo r a [fo r e ve ry fo rmu la (3 in Y
F i (31 if f M b
ib
fies
e
biff tMh a t
ile
M
Proof.
ri e r svi Bf yei anr induction
via
i f i e osn the construction o f a [o f 131 lik e in
1.6.
fe
a
tp se c
a
[tl si i v e e
a
r
i
lf iy1.8.
[b
v , eDefinitions. L e t a be a variable, le t (3 be a n y fo rmu la ,
e si Y f i
rllet
a s
in a ll the constituents o f Y
esubstituting
variable
a.
Then
[ et 3h e
b
f o inr general
m u l 0a
(fta
a
min
(f imu
o
ri s said to be a (3/a-asc fro m , V
M
o
1
,3 m asc f o r fY,„. Then
itnw ia lbe
a
ne o t le t (3 be a n y formula, le t Y
hl a variable,
r,Let
a
.a
b
fro
b eem Y „,t and
h e
le t F ( 3 / a l
basc
a
n
respectively,
then
F
a
a
n
n
d
F
[es
d
b
i
s
s
a
i
d
b ca variable,
e .
le t p be any formula, le t F
ci Let
f sa be
faa
M
tL
b eeme F t t h e
P i a ,i b ohfro
stable
io
b
a
F
tfruY
l
e
e aa e b
a
yltba
,tsb
l seh t
e as M
n according
iih
v
e
r
i
f
i-ee
P
M
s
of
substituting
i
a
o s[3 f o r a in y. Th e n M
(
d
rfb
f r
o
r
tm
iu
l ie d e
t
o
l3 i ase M bsh a
ofrom
ie
table
ro
fsw
i f
cfY
ib
r sm F
a
1 a efle
e
a asnremdth
ia
o
a
h
.N
- a n ro n eo r o wwwh ic h a re Pia-rows f ro m M
„t.Fbemo
oa
e
i[tfR
also
f
happen
b
ia
,
trF
e
o
n th a t fo r some ro w M
.
f
ia
M
rows
i
n
,
fro
m
F
o
r
f
m
Mi l l
.o
w
I
t
h
w
a
w
h
m
d
. a l
M
ie
a
tology
s
t
by
h
plain
e . rproposition
e
logic and that M
i.9 k . i r .
sfira
a
i.w
e
l sth e faact th.a t some constituent b in F
e
Itn
9 fAa ii l scan
%il f ibe
f,reason
re
A
b
considered
e
c
n
a
o
a
s
o
e
n
o
a
t
s
htbh e ee
re su
r lt o f substituting (3 f o r a i n d iffe re n t
6
faF
to
„
y
and
y
i
n
F
w
n
o
isrnconstituents
L
o
w
,
to
y
and
y'.
Th
is
can
be
th e case i f f o r instance we have as
a
a
n
d
t
h
a
t
fs0
o
o
r
dir,b
m
b
constituents
e
,
i
n
F
h
i
t1
a
e
utfle
a
occur
D
s
in
s
P.
(
iHo
a g
we ve
n r fo
s r e ve ry ro w M
F
s
h
llvn
t
d
i—
be
i
a
n
i
>
unique
f
F
f
ro
w
e
M
r
a
c
e
d
e
h
ao
e
3n te1Ft h e r e
,frle )i n
a
rsit1
v
w
a
in au l d ll
u
ttye
s.n
s
e
E
y
s
h
b
l9sa
cte u l stcJ wh a
e
t(e
h
(
a
i
t
M
o
t
9n
—
M
,o
n
h
itn
i›ta
sb
f
]h
ia
s
h
te
p
f
.e
)fe
a
ito
p
T
(w
3 h /
re
a
th
r
340
A
R
N
O
U
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D
B A Y A RT
1.9. Theorem. L e t a be a variable, le t fl be a n y formula, le t
y be a n y formula, le t b be the re su lt o f substituting (I f o r a in
y, le t F
i
R
13/a-row
f ro m M
i,
ia
and
(2)
M
b e
i.n
Proof
a T ho f e(1).n B y an induction o n the construction o f y lik e in
vw
1.6
r ei fby the definition of " (
t eaand
d
Proof
-ih
v By the same kind of induction and by the definition
b e ls aof (2).
M
of
"Pia-row".
(e
1
7
f
a
a
l
i
tsb
f a ib flhei .
a
e
o s )t
(r Co n tro
b
1 l and acceptability.
2.
i)o
y
fiw
, 2.0. Definitions. L e t a , and M
fsl i bwhee re ri oj n
case
wo t sb e in g excluded. W e define a s f o llo ws
the
CO and Cl.
va
e o relations
o
f
a
tf M
e
a
s
t
a
b
l
e
rb
if
ia i
l
FM
t
ife
C
a M
t i
h
e
if
iO
fa
ve
t M
ie
so
M
r
h C
n
vi(rd
l
2.1.
e f i f Definitions. Let M
where
e
5
ia
( eir al in jd notMbeing excluded. We shall say that
F
R
is1
b
e
i f if i
a,
iL
w i sf f a, co M
f,i erfB-controls
i o
iM
R
sie
fa n d f R
ra o
i-D
O A iff M
oS5-controls
s- a
e
) C
R
S
ca
bR
l
tys tr I C aO
zio
i
a
n
d
th
e
.p
a
lC
n
h
vM De fin itio n . F o r a ll n a tu ra l numbers n we define re cu rF
e
b 2.2.
sive
-lte
n
cl „ei ly as f,o llo ws wh a t i t is f o r a ro w M
icA
n-M-acceptable,
a
n-B-acceptable,
t a b l e n-S4-acceptable or n-S5-acceptrn
Oh
M
te troC f
able.
F
t
o
oio
M
yR „
if e
For
b
n.livr ci M. Me a
we
have that
itsve
e if M
o iso,s
i0
v
e
r
i
f
i
M
M
rA
ne s
e
rlm na
every
a
constituent
D a in F
a
iO
c
c
oiriyF sd
tie
a
w
h
e
e
h
a
v e
lfi,„f tpR t
tsn
h
a Mt
a
sfia
, ibi l e
in
f
R
A
a
ie
l tC
n
vfi,e
i+
e
r
i
sd
e ul
f1
el ss i f
i
fcslta eRai
e
ffo
.a
D
.e ni s
D
a
-.o
R
.u
t ta
ri2
M
.M On
-e
C
, ad
O N TRUTH-TABLES FO R M , B, S4 A N D S5
3
4
1
there is in F
R,
a and
a nR
iFor
n - B,
M S4
- and S5. The same definition as f o r M, replacing th e
fa a cl sc "i eM" by the prefix "B", "S4" or "S5".
prefix
fp i te as b l
ae 2.3. De fin itio n . A ro w R
able
f
a
S4-acceptable, S 5 .r i o[B-acceptable,
natural
n ut mb
[n-B-acceptable, n-S4ao ct cae pb
la b
eelr en ] gf, isi n-M-acceptable
f f
n-S5-acceptablel.
facceptable,
i
o
s
r
w
eR Remark.
s v a eIni wh
r adt yfollows we shall often formulate definitionschemata,
theorem-schemata
a n d proof-schemata, wh e re t h e
o
, t
prefix
" X " will
e occur, and which will yie ld definitions, theorems
s b
and
proofs
i
f " X " is replaced e ve rywh e re b y " M" , " B " , " 5 4 "
M
u
or a"S5".c
c
e
c
t
h p
2.4. Theorem-schema. Fo r e ve ry table F
tnumber
a t ht such
e r ethat ifi a sro w R
a
is
does not state th a t the number
ih X-acceptable.
no f a Ft u r(The
a theorem
l
taais the i same
s for tM,-B,X54- and S5).
Proof-schema.
at c c e p tThe
a set
b l ofe n + 1-X-acceptable ro ws is b y definig
tion
a
subset
of
the
set
t
h
e
n of n-X-acceptable rows. But the number
iof
R rows o f F„ being finite the successive sets o f n-X-acceptable
rows
cannot
decrease in d e fin ite ly w i t h in cre a sin g n . S o f o r
iM
c
o
some natural number t the set o f t + 1-X-acceptable ro ws w i l l
n id etn tifica l w i t h th e se t o f t-X-acceptable ro ws. I t is th e n
be
r
oseen that, f o r a ll r, i f R
easily
s
til +i r-X-acceptable.
O n t h e o t h e r h a n d f o r a l l s su ch t h a t
s t - X sa <c t cwe
(1) o f n + 1-X-acceptability th a t
e have
p t a bby condition
l e
if
t R h
e
n
if
i M
s
i s2.5. Definition-schema, T h e X -ch a ra cte ris tic n u mb e r o f a
itable
t sF
tX -w a , o f F „ we have th a t i f M
ro
a
X-acceptable.
iX
-s i s t - X a
t- hc c e p t a b l e
2.6. hTheorem-schema.
Let M
ta
e
n
c
e
Then
i
b
for
e
every
a
constituent
n
X
- Da in F
M
sc m
a
e p is
t in
a bF „ l ane X-acceptable ro w R
ceDual thcl e nc there
ia
X-controls
M ea t
,ii rsw ue c ohs9i
h and
at wvh
R
p
e
s
iti fi a
l sai f itne s
h
i
f
p
ta
.
g
ta F t
n
i a
b
a
fb . a l s i f i e s
tl
lu
e
t
re
th
a
e
lh
342
A
R
N
O
U
L
D
B A Y A RT
Proof-schema. L e t t be the X-characteristic n u mb e r o f F
ais X-acceptable and so t + 1-X-acceptable. So, b y condition (2)
•of gt +
f 1-X-acceptability, f o r e ve ry constituent E l a i n F
ia wthat
have
e if R
iro w R
being
is X-acceptable.
fi a l s t-X-acceptable
i f i e
ss u
c 2.7.
h aT h e o r e m
D
Then
tt - hh fo r ee ve ry constituent D u in T
a
tc hee m ah. a v e
naD s w
ttR
u Lverifie
hh e a ta, tand thatRR
i eich
s is X-controled b y ge
ieceptable
ii Ri rf a, lros wi fwh
veProof.
iX
f b
aDen Obvious
drea iw fhb yii the
c e fact
h s that X-co n tro l imp lie s th e re la tio n
CO,
fi-f aai and
l fby
s n2.6.
if f i e s
r
ca Xt - h . e
sa
e
a cT h ce o er e m
iol 2.8.
nl -pi rows
t ais snot
b empty.
nable
Proof-schema.
tFX sl irc eh e m a n. Let R, be a ro w of F
a
El ca in
h F t h a t
o- Ir Fsl nu
atuent
R
a
fsa e
o( oi vf re r
aa
itsuch
e
a
c ywa
n- vay ro)ew Rr
y
R
fX
ig
and
ya wit
o f degree 0 and b y associating
c,c ti bn "-oF"
n bh the
s constituents
t
i
with
faiae ln
a
i Xrceeve
s cry
t -constituent D u of degree n + 1 th e letters " T" and
le"F"
a
vnp Faseaccording
p sr ot i c as a is ve rifie d o r falsified b y the su b ro w o f R
isaw
cfdt hich
t ls i
a ll the constituents o f F
„ai- baecontains
i t w i l l f o llo w th a t R
iethan
n
a
e e
O
a t. go nf. Fro
R
hd me thgis rconstruction
in
- X
r n -+ 1-X-acceptability, b y the hypothesis o f an
facceptable.
a
ib ei rso b Ot Fo
iinduction
ifl st i r g ao nhd e fin
h
e itiorn 2.2, w e ma y suppose th a t g
ierfae ei i s l ny and
acceptable
- Xso -condition (1) fo r n + I-X-a cce p ta b ility holds.
By
stflr t the construction of M
iD
that
h rya constituent
v e
1=1 a in F„, i f R
h
vso ow feo
e r e ve
tufalsifies
ie
t i econdition
s
D(2) f oar n + 1-X-acceptability holds.
riw ff ha i l saa.i fSo
tilfs X hi e - e
R
n
R
ifite
2.9.
Remark.
X
I
f
we
limit
o u r considerations about co n tro l to
ii a st c
cfX-acceptable
o
n
t
r
o
l
ro ws, as e ve ry X-acceptable ro w is O-X-accepte
re
a
en c
s£able,
e
a
sily
seen that Cl imp lie s CO. This wo u ld a llo w us
s(sF p i t is
t
i3to
s
imp
t
lif
s
y
th
e
e
d
e fin itio n s o f S4-control a n d S5-control l i k e
"aa
llthis:
f
,
T
n
.w
a
n
,"d
h
B
d
fsui
ao
tc
lgh
sia
ir
fe
iX
e-
O N TRUTH-TABLES FO R M , B, S4 A N D S5
3
4
3
.2,S4-controls .2
i iff
M
iC shall
We
l
however avoid making use of this f a cilit y fo r reasons
which
w i l l become cle a r in the last section o f th is paper.
S
5
c Comparison between M, B, S4 and S5.
3.
o
n 3.0. T h e o r e m
t X-control
is re fle xive wit h in the set o f X-acceptable ro ws i n
r sch e ma .
Fa•
o
Proof-schema.
Th e acceptable ro ws a re 0-X-acceptable. F o r
F o r
l e vO-X-acceptable
every
ro w Sf
e r
tuent
is yw eDa inh F a v e
a
.2
ti t h aa t b ,
i,f l i f oe . 2Mr
obviously
definitions
2.1.
f FC l v
iC
e
e
r
y
vcg ae hor i f ni
O
T
s
t
.i 3.1.
e
t s - Theorem. Fo r e ve ry ta b le F
i[ Sa
D
t2
r e l a t i o n
hh5 -t eh e
acceptable]
ro
ws
inBF iac oo
o
e nr e f
Proof.
B
y
definitions
a
.tC
m
t rcro l ho e n l t r o2.1l i t is obvious th a t B-co n tro l a n d S5control
trica l wit h in th e se t o f a ll ro ws i n F
.fF
l] a o n at re symme
e
i
a
these
are
symme trica l wit h in the set o f B-acceptable
M
g
lo
i o l relations
n
.o
and
S
S5-acceptable
o
rows
respectively.
ri,s o w
vas f y e
se
I n e ve ry ta b le F „ th e re la tio n o f S4-control
tvrnm 3.2.i Theorem.
f
[S5-control]
i s tra n sitive wit h in th e se t o f S4-acceptable [S5idm e
e
h
acceptable]
ro ws in F
rs.e t
e
Proof
f
o
r
S4.
a
(
i
n
f
aI f ci n t F
9r i
ya
n
a
.9,S
then
4
we
h
w
ir.c ai m t h i anve th a t f o r e ve ry constituent D a in F
verifies
ctm
a
l s v e r i f i e s D a and i f ,2 ,
C
S
o
l oi fnh
e tDgra oethen
,verifies
ia
v
e
r
n
a.
i
f
So
i
e
we
s have
D that
a fo,r e ve ry constituent D a in F
sd
e
t
lw
o
w
i
•R
R
d
a
e
.
o
f
w
g
a
i
t
i,,2
.2
iat , i f l , l
e
th a t f o r e ve ry constituent D a i n F
-vhave
,ly(hh
then
v
a
i
f
.eS
4oe rg-i f iwe s D a , and i f v e r i f i e s D a then v e r i f i e s
rb
ai
0
vo o) ewe
Iricsy-vDa.
nr h.i at fvei thea tsf o r e ve ry constituent O a in F
n S
a
a
fiD
rX
o
l
s
et
R i
if-,h i f
iae
n
ecs
F
sce
a
D
S
et
apo
4
-ttf
ch
344
A
R
N
O
U
L
D
B A Y A RT
verifies E la then v e r i f i e s L i a and so we have ,9, Cl R
The
i
proof for S5 is the same, a first time to show that ,9, CO
". R
and
Cl R
i
C li
. Theorem. Let .9, and R
a 3.3.
have
n i bthea t a)r i f oS 5w- c so n t r o l s .9, then R, S4-controls b ) i f
.9,
f
R
a
d oS5-controls
ithen
t
h
R,
a
e
M-controls
n
b
l
e
j
a
R
a
,s ,Fd
and
) e) i f fR, S5-controls R
isRt rivia l given definitions 2.1.
iThe
R
B e -n
e at htheorem
icic .o M
Bn -t r o
Theorem.
clco s3.4.
To n
th
R r o Let
ell s R
i
b
e
a
that
a
)
i
f
R
isn n
R
R
R
i,Iid rwi s o w
e
i.cable
S
-h f R e
t o5 then
i)ia
M-acceptable
ics c e
and e) if R
n
i a
isiable.
M
i
t
s
a
S
5
b
l
p
t
a
b
l
R
m
S
a
a
e
c
c
c
c
e
e
p
t
a
F
b l eX and Y can be replaced respectively b y
e
where
fProof-schema
ie
t5
p
S5
ahh- bS4,
.l eo r b ynS5 and B, o r b y S4 and M, o r b y B and M, o r
tM
R
t „tand
by
andhnM. The
-,e
R
co T,S5on
t proofs are b y induction on definition 2.2.
If
R
ia
d
e
n
R
S
rs
o4
ls
icc)h
2.2.
, iw I of .9, is
ns n + 1-X-acceptable, then R
the
sr - no - X -of the induction R
ctio ie hypothesis
iM
dition
f-coY
rt na-e+b 1-Y-acceptability
holds fo r R
ia
e
hcsscc (1)
en p
a
l e .
lfsw iO
ia
X-acceptable
ct e yp- et then
a b fo
l er every constituent O n in F
pR
B
S
t -vc 4
.ti-h R
a
t h a t
if
XI wf en R h da v e
is
X-controled
i
s
b
y
.9,
n
and
wh
ich falsifies
+
a. B y the hypothesis
isaia
o
c
a
1
of
the
induction
R
fcbsct aa e o
n
So
condition
(2)
f o r n + 1-Y-acceptability holds f o r R
ilB
i
s
n
Y
s
c
p
t
C
ia
iea
Of c bcc e l p t a b
.lite
e .e
cee3.5.
, Theorem. There is a table F„ such that in it
R
B
a)
t
h
y re t w o ro ws .9 , a n d R , su ch t h a t R , M-co n t ro ls
shp
t re a
b
i p e
3
D
ea
)a t Rb .
3
b)
iee re a re t w o ro ws R , a n d ,R, su ch t h a t R
u
nlin at h
iR
a - c o n t r o l s
tatfd bMR
c)
t
h
,h
,h l in e re a re t w o ro ws R
a
iY
R
d -n d
e
ie eaR
n
icrsn RiRo s nu
c
h
d
te
a
h
a
t
,
r
o
l
i
B
R
A
.s-ii i n
,
d 9
'
B
d
A
sai so
c.ics O,R
n
t
r
o
eo
d
ln
,
s
s
c
o
n
F Yd
e
e
a
p -o
t
a
t ase
O N TRUTH-TABLES FO R M , B, S4 A N D S5
3
4
5
d) t h e re a re t w o ro ws R , a n d 2 , su ch t h a t R
i SR,4and
- c,R,
o does
n t rnot
o lB-control
s
j
.
Proof by table 3.9 hereafter.
3.6. Co ro lla ry. There is a table F
a) at h esreua cre ht w o tro ws
h aR t
i i aR, nand
nd R, does
R not
i S5-control
t
,,
i t h e re
b)
s a re
u tw
c o ro
h ws R , a n d R , su ch t h a t B - c o n t r o l s
R, does
t R,h and a
t not S5-control R
c)
R t hi e re a re t w o ro ws a n d s u c h t h a t R , S4-controls
, and R,
i R,
M does
- not S5-control
c
o by 3.3
n and
t 3.5.
r
o
Proof
l
s
3.7. Theorem. There is a table F
a) a
th esreuiscahro w twhhicha is tM-acceptable and not B-acceptable,
b) ith e ren is a ro w wh
i icht is M-acceptable and not S4-acceptable,
c) th e re is a ro w wh ich is B-acceptable and not S4-acceptable,
d) th e re is a ro w wh ich is S4-acceptable and not B-acceptable.
Proof by table 3.9 hereafter.
3.8. Co ro lla ry. There is a table F,, such that in it
a) th e re is a ro w wh ich is M-acceptable and not S5-acceptable,
b) th e re is a ro w wh ich is B-acceptable and not S5-acceptable,
c) th e re is a ro w which is S4-acceptable and not S5-acceptable.
Proof by 3.4 and 3.7.
3.9. L e t a be a variable. Let us consider the asc a
(u,: Da , D D e , E l — Du). Th e table F
table
a f required
r o m fort theh proof
e of 3.5. and 3.7.
a
s
c
Y
The
rows
a O-X-acceptable
i
s of F„ are the following:
RO:
(a,
T)
(
Du,
1)
(
0
Oa, T) ( 0 — Da, F).
t
h
e
• ( a , T) (D a , T) ( D a , F) — Da, 1
7 (a, T) ( D a , F) ( 0 Da, F) ( — Da, T).
R2:
). (a, T) (Da , F) ( D Du, F) ( D — Da, F).
R3:
• ( a , F) (O a , F) (L I Da, F) ( 0 — Oa, 1
R5:
(a, F) (Da , F) ( 0 Da, F) ( D O c c ,
.
).
346
A
R
N
O
U
L
D
B A Y A RT
For M we have:
RO M-controls RO and RI
• M-co n t ro ls RO, .91, R2 and R3;
R2 M-controls R2, R3, R4 and 95;
R3 M-controls O , l , a 2 , R3, R4 and R5;
R4 M-controls R 3 , R4 and R5;
15 M-controls O , l , R 2 , R3, R4 and 95;
Are M-acceptable: R i , R2, R3, R 4 and R5.
For B we have:
RO B-controls RO and RI ;
• B -co n tro ls RO, RI a n d ,93;
• B -co n tro ls R2, R3, R4 and R5 i
R3 B-controls l , R2 , R3, R4 and R5;
R4 B-controls R2, R3, R4 and R5;
• B -co n tro ls R2, R3, .94 and ,95•
Are B-acceptable: RO, R 2 , R3 and R4.
For S4 we have:
OS4-controls RO;
RI S4-controls RO and RI ;
R2 S4-controls R2 and R4
;• S4 -co n tro ls RO, RI , R2,
R4 S4-controls R2 and R4;
• S4-controls RO, 1 , R2 ,
Are S4-acceptable: RO, R2,
R3, R4 and R5
;
g23, R4 and R5.
R3, .94 and R5.
For S5 we have:
• S5 -co n tro ls RO
;• S5-controls RI ;
R2 S5-controls R2 and R4;
R3 S5-controls R3 and g 5 i
• S5-controls R2 and R4;
9 5 S5-controls R3 and R5.
Are S5-acceptable RO, R2 and R4.
(It is wo rth wh ile noticing that although ,93 is B-acceptable and
S4-acceptable, it is not S5-acceptable).
O N TRUTH-TABLES FO R M , B, S4 A N D S5
3
4
7
4. Sub tables and substitution-tables.
4.0. T h e o r e m
Let- R
c h elemt aR.
i s and
L
e have
t
a ni we
Then
that if R, does not X-control R
id Ra n d R
X-control
Proof.
i,R i g I f R, COR,
,iin a
dg
ti o
h een sn
M
-•i, dFb
n
in
o
R
t
R
, d o e s
ih
a i bhold.
o o l I f dtR
bnot
n
vteisDa
e,Cein
rhl i F,, such
e
that R
fn
ui i bvethe
d
rby
soes rdefinition
uei f i e s of "subrow", in R , verifies D a and R
tofies
D
iscO e bDa
h a r, and
e so R
holds
f
o
r
the
rw
a
ha rfCoaelonl w
i,n
s oji -d case wh e re C O M
ds owhere
e, s Cn l oR t
isi,tcase
a
, w
h
soimmediately
n
dif odoaoooo el l s ssdbyi definitions 2.1.
a
d
ln
n
f f fiodonet st d
fa
fM
o lhnael do rr e. m
cah
th
D a4.1.
RooohT
td
T
istR
M
e
. t- , t . h
nhi a te e
,u
ttB bsaach
iA
esubrow
hhe m
nel a .ofo M
abe
ftbthen
vru eLn e.0P
se
ib
i er tm
mt
ta
h
D
i.e i F
R
sTdl l h e a n Th e proofs a re b y induction o n definitions 2.2.
lProof-schema.
e aiR
sa
riIf
a
w
oRi et n
en
if, .b-i aeth
o ae
fF
rsisuch
h
X
vt R
a le
o
tbo
e
sfil sacM
a
h
o
a
r
e nw
s
n
If
R
i
s.ctw
iv e
r s rnoi fpoi g
o w
ie
a
ewe
n
io
stvehaasvefc b y t h e hypothesis o f th e in d u ctio n t h a t M
Lp
t soliet sei a i n o a
iD
t n d so n o t n + 1-X-acceptable. I f i s n o t
,eb
ftsn-X-acceptable
R
n
because co n d itio n (2 ) o f n + 1-X-acceptO f+rv 1-X-acceptable
.tia
a
tD
a
un e
a
b
ility
does
not
hold,
then there is a constituent D a in F
o
i
nite
ad al n y
F
a
that
s
R
u
c
h
d
ttfX t f.a l a
„sa
in
sbn
bwhich
s- bi i is
f lX-co n tro lle d b y R
if+
ro w M
o
ldn-X-acceptable
ia aeaeel n sd
eo
iR
w
that
h
the
i
ro
c
w
h
o
f F
s
i
f
i
c
1
s
ga g
ate
.isX
fF
a
the
a
o
w
hypothesis
f
h
l
i
s
F
c
i
h
f
of
i the
e induction
s
we have that R
e
s
i-.c - D able.
B
y
the
hypothesis
a
b
o
u
t
,
g
iC
c
b
s
c
.
e aa c
X
,O
h
B
c
ua
iO
so
d
b
y
o
4.0.
e
s
R
n
o
t
X
F
w
p aiahsp i not c- Xh - ra c c e p t -u
.f,a
be
R
a
t nda
noubo lnveb slts rir efo i l r
R
Licsfe
ta
iiae
n
o
t
y
e
w
s
n
lcta dd
a
nf ,
d
o
d
b l fch
,btsX
c
o
n
w
e
te
l ea
d
iM
lF
to
r
o
l
iea
h
a
e
p
l
fn t
,ve
e
t, s
e
iR
ob
isfefb
a
l
s
ra
h i
b
'a
.
i
e
b
siye
y ff
B
s1
iln i
348
A
R
N
O
U
L
D
B A Y A RT
ro w of A ) falsifies Da , so does R. I t fo llo ws that condition (2)
for n + 1-X-acceptability f o r R
i d o e s
n o t
i n i t. i o n
h 4.2.
o D le f d
is -a subtable of F
em
1 sch
than
F„.
L eat . us n o tice th a t o b vio u sly i f ri is o f the f o rm D y
then
, La F
ne d t F
a
Let
t Fi Rs
ia
Let
a
,t „ R
to
say
that
R
tct nano n d t a i
ia
ivd
If
b
e
Flsois ra variable then both R
ni ri
iR
If
i sufntDyd, sthen
R Ri f
re
j t iIi fia
ia
of
eF „are
- FX-controled
„ , by R
ft f ebF„
f isrswhich
a
iX
If
13
v
is
e
Dy,
r
i
f
F
then
i
e
R
y
.
o e
(b
n
sa
H
It
lle
ws
ct bet ddeia te
d ly. f ro m th is that, i f (I is Dy , Rif is
r fo
e
3
tee llo
aecimme
.i-a
selected
i so-f eFnif f fthere
is a n X-acceptable r o w i n F „ wh ich is X tyc lX
ss
controled
sn
a
.t sei i tle by
h
ucu R
i-ltd
a
n
d
X
e
d
X L c n
e
th
w
h
i
c
rlte
sa
dm
te cl eT hc et h
oer e
( 4.3.
fip
o
t fl as if bof F
t3 -h
w
e
is
aa subtable
lw
than
to
)fR
h esLme ta R,
. and M r be X-acceptable ro ws o f F
m sai ec eF.
yiro
R,
X-controls
g r. Let R
a
t
,R
t a ne d t
o L
i.rows
a
n
d
R
fib
o
t h a t
F
r FsT uR, c h
Proof
F
icsw
fe ao fo
n r tM.aI f iR
,n
ia
constituent
sb
ns - b n eD a in F „ such that R
f a
.M
a.
B
so
icd
o
v
rtbirtwe
f osri F
e
lhave
sos wthat
D there
u is no constituent E u in s u c h
.ja
n d uyenu1.7
stF
R
a
that
Da
n
is
in
d
F
R
lfX i
t
that,
tio
a
- b i f (3nfis a variable o r if (3 is D y and A , falsifies D y then
sF
o
R
,'te
e
-i1
a eaX n h-d
iA
then,
a ll the M-acceptable ro ws o f
F
fto
-h alXnas
ees -bA
f i e
scf,Dyt a
eo
s iis F„-M-selected,
C
F
vsn
e
e
r
i
l
f
e
c
ftcio l ti et du s
▪a
O
te
iR
enrif
s ie
dt y, and so again A CO
.f sev e
F
a
w
Proof
i(w
E
p u t for B. I f R, B-controls R
By
n
ih
C
u
r3
t cereasoning l i k e i n th e p ro o f f o r M w e ca n p ro ve t h a t i f
h
,id
R,
w
O
a
sm
p et nh e hn aA vC Oe ' and that i f A r a l 2, then R
.ia hCO
Proof
r S4.0 I f R R
cso
iR
d
e t c , ,f o C
cb
W
•i,th
C
S
l
4
R
o
A ,
A
tre
l hiC O
h
if
R
ithat
c,a
o
n
t
r
o
l s
A
ve
e
Ra
d
iR
C
O
%
rM
a
n
d
fS
a
l
le
ir ty
R
B
iR
y
-fio
e
so
f
. F i
ri,iM
ctT
isw e
a e hs
a,w
,-o
e
u
re
h
so s
n
h
oth
n
i
a
cn
C
O
.e
f
u
e
n
ne
vR
g
e
tIin
o
F
b
w
lA
R
rtsA
in
ra
h
iC
i.to
fn
o
X
.
o
a
kO
C
rlo
l
O N TRUTH-TABLES FO R M , B, S4 A N D S5
3
4
9
no constituent D a in F
a sDu
fies
u . cB yh the definition o f subrow we have that there is no
tconstituent
h a t D u iRn F
ibDu and
s u R
c h
(v3
ti isheD yar and
ti fM i e
isD
proof
,
fuo r CO,
. M
Dy
then,
as
ifE
if aaSll 4ss si-i f iaeR
F
csaiif ioi esnst U
nrno„ -l
a
S4-control
sE
D 4 l-u y w e h a ve t h a t a ll t h e S4-acceptable r o w s o f F
dS
F
w
a
e hl eare
t
by R
Itfsa. hich
fa ce S4-controled
l
ih
rows
e
d
,
wh
ich
are
S4-controled
by v e r i f i e y. As R
(n
saI
i n
i,3
selected
a
C
dt a r le M S 4 i,clProof
lR
r o I fl Re d
f oi osnfotrFS5.
h
b
,a
itil Sl l 5
C
clIR- y
R
sM
have
iD
v-ia
ch
S
e
o
r
4
n
that
i
t
f
r
i
if
o
M
l
s
we
vo
r
k i n d o f p ro o f w e h a ve t h a t i f C O R, a n d
iBy
.ry,,M
e
e
st hO
is C
fe saMme
i
M
iD
e
S
IS
aa
et n 4 s - o
i,ya
S
,ah c c l
l
nd
,aa
4
e
w
R
4.4.
n
p
T
d
h
e
t
e
o
r
e
m
da
C
M
-isih
n
a
at subtable
- b a l of F
lithan
Fee m a .
cfo•vei s c h
id
M
ta
subrow
o
„w
rf La hen of
R
dt the F
,C
M
ieable.
.F
theorem does n o t state that the F„-X-selected ro ws
a
ligio( F (The
n
e
n
t-R
L
of
F
e
tviR
h
b
w
C
3Xa - e
Proof-schema.
itca
sh
rviO
si eaol eincnt te dd B y in d u ctio n o n d e fin itio n 2.2. A s a
ire
,R
acceptable,
- X C
vo
a
sw M
fe.os Fi i so9ni O
n
ia
acceptable.
O
R
lnB
je
fa bC uOs MBy the hypothesis of the induction M
1
able
M
i s and
- Xco-nad itio
c n
c (1)
e pf ot r -n -1- 1-X-acceptability h o ld s f o r
iirb
e
sa
,t n so
ty.v b
M
i.e
o
d
g
o
f
hrR
ae
iF
tuent
F „ wh ich is falsified b y R
•,B
ta
b
n
ir t yDaa in b
ee
ceptable
i.b
s that Mi
n
C
n
h
a4
iya
e
,i lt h. ee rroews s ui c h
Let
R
iF
X
c
o
n
t
r
o
l
s
A
O
X
e
3
.ncn
M
sC
nu
as o
iR
of
of the induction i s n-X-acceptable.
sT
a
n
M
sa
dw
tio
O
h a hypothesis
b c Bt yhi the
,iX
By
4.3
R
R
c
iM
cth
o
e
e
vn
R
l t cui - n e a
b
e t- a a D u nin b , dwh ich is in F
i•M
constituent
a
le
ih
an
n
„e hX
M
ctfin
a
p
ve
tn n
et ld s w i h f i ci h
iC
trgo oa aa
1
irh
R
e
o
l
s
s
s
sild3
X
.ia
lr t
e
itfR
variable
a
f Di y eandd M
X
,-b
R
sM
fii Fal l osr if.i (3 is
9
m
F
h
iX
v
e
r
i
f
i
e
s
b
y
-X
iC
e
a
ikf a 2
o
-e
D
'„a
,aO
criee -i . yc
X
tcrB
n
ccC
co
.S
yi s- h e e
C
e
e
co
e
o
n
d
i t i
yscp
lo
w
e
l.n
t
il1
n
o
e
n
e
M
o
p
R
B
t
(ts.cp
2
)
iitfyh t
e
fri7
o
t.ia
L
te
d
n
ro
R
asb
e
hp
ribO
lF
lter
350
A
R
N
O
U
L
D
B A Y A RT
n + 1-X-acceptability holds f o r M
iEly, th e n a s M
such
•i iI sfthatU( M3
i s
y.
iE
a L elt Ay
a
n
d
,M
subrow
- Xi bs- e of M
itX
b y 4.3 we have that M
e p t
ia c-hiscn-X-acceptable,
by
,e
1.7
i
U
X
M
c
o
n
t
r
o
a
fa cba cl eel s i f il s e s
holds
ia
.ipt bt tility
Baahyb
ln fo r d
,t-ee X,h - re
fh
sX
eyl -sl pTi hoe o r e m
ea4.5.
a
subtable
of U
fis
te
i
ste s
ci hceo
b
in
s
Ft
c
h
e
m
a
.
yie
ns d t r
).o
L l ee st t M
.of
ri LM
itfo
Proof-schema.
b
e Let n be the number of constituents in U
S
an hF
t.e
i
w
n
o
h
t
in
i
c
F„.
h The proofs are b y induction on n. I f n --- 0 the
a
a
r
n
oare
tw
nF
theorem
is
t
rivia
l. I f n 0 , we kn o w b y 1.3 that it is possible
X
e
a
n
ad
oa
h
to
co
n
stru
ct
a
se
rie s o f tables U
gia
fga d c c e p
e
n
Fa,
5
astiF
r Ua n b l e
;,a
„
justs one
u constituent
c
h
more than U
rd
igm-.1
b
ucontains
,o b U
thesis
o
f
th
e
in
d
u
ctio
n
there
is
i
n
U
rnitn
i
s
h
a
t
U
o
b
csfw t
n
M
_
,cu
B
y
t
h
e
h
y
p
o
w
X
iu
a e o
1
k
ed
ro
a
w
of
n
U
X
in-o
b
s
a
cl t
n
Mi c le p et a b l e
sa
n
c
ta
fw
b
d4.4
h
s
ra
w
cF
ti(u
ti ts hi a
e
f ot
iocn
s
s
4.6.
T
h
e
o
r
e
m
ca
h
i s
X
h
o
f
d
is
a
subtable
o
f
U
.a
ne
e u
a
c
t(b
U
fp
T
vtsF sc c h e m a .
h
o
aL
able
e ws
t t M, gand M
th
heL ro
2c.1
M
e
t
,a
and
)
r
e
s
p e c t i v e l y and M
i)p
ie
ra
s
F
ti t
a
b
tfe
ib
s
Proof-schema.
X
u
c
odhn t Le
r o nl be
s the number of constituents in U
,n
a
u
a
s
n
c
, h
h
locg
are
e
n
o
t
in
F
n
w
h
h
i
c
h
a
b
la
ah ane t
itc a
theorem
a
trice
X
h
M
t - h is t rivia l. I f n 0 le t us consider the series o f tables
ts d
n
.a
ih
rait b
FTc ht ce e
nU
d
p
n
in
r t ona o bd l
s
M
m
p
h
a
aU
+o e
are
,1fo
n
s
ka
sw
e
a t su baro
r ws bo f Mk a n d Mk' re sp e ctive ly a n d Mk X -co n tro ls
--U
b
a
Mk',
1
L
r
e
t
risu
t l
e
eM
o s
M
iaX
vsu
e
n
a
- nU
c,is
w
sM sin
u dsb
spectively.
By 4.4 .9
ih
b
o
a
o
ri u o
w
sb
cM
icfyo
M
c n d
rcftu)sX ca
s
i,e
u
4
s
u
n
d
h
- h
co
eo
,b
.pu
sw
p
a
hcfc tth c a t r e
X
ilro
5
a
n
tttM
o hh-o
te
a
c
c
e p t
t-b
yfO
.o
a
l
t
h
hIn a
a
b
l
e
,
M
n
B
e
M
R
w
m
at t
b
io
ryn
F
k<
tkr U
yn
t.if•n
o
a
os ,
M
4
.
IsM
h
w
_
L
w
l ,
in
ON TRUTH-TABLES FO R M , B, S4 A N D S5
3
5
1
trois a n d o b vio u sly M
respectively.
i a n d
a
r
e
s 4.7.
u T bh e or r eom w
la,
s -le t F
bF
o s c h e mf a .
rows
bb
M Le fero mt fromg
it,i aah nn odt X-co n tro l M
does
iM
Proof.
ela b I f eMn
i,P
in
C
F
ti hO
MA
d a
e
ae n
o
f
substituting
p f o r a in y. Th e n b y 1.9 we have th a t
i,-result
d
a
o
ti v a r i
rtM
d
se
M
, e
a sosbb pel e es
clIf
n
iu
ov de ot e s not fo ld
i M,
e
,t ioCl
,a ylt h
lfsuch
vctX
-.thea tneM
itn th o eo fnrsubstituting
et r
(3 f o r a i n Dy . Then b y the definition o f
cT
h
e
rresult
-o
irto
v
e
r
s
i
f
i
hd ( l l
(e
So
a
e
d
ih
m
b 3sM
case
i3
ce b o where
D
yn s Mt i t
tn
fa
F
iC
/a
u
e nn t
w
ith
aCl
r elM
definitions
2.1.
,vM
C
i,e
a
D
d
y
o a O
R
-n
ise
d
o
n
h
lM
wn
iee
,riF
s
4.8.
T
h
e
o
re m
rta
0
s y
d
o
e
o
n
,a
le
t
VI,
be
the
13/a-table fro m F„, le t g
vtla,
ih
8
o f
o
s
w
o
ffe
ia
ro
s
w
b
c
a
o
e
h
f
e
F
l
m
a
.
n o t
X M
f r
m
n
o
"irti1
tn
e
X-acceptable.
sF L
a
uci ec f et- p t a b l e
taProof-schema.
sw
Th e proofs a re b y induction o n definitions 2.2.
ia a e
,h
s
h
e
d
h
o
e
n
o
If
M
D
s
b
u
c
e
sia
M
ns
lo
ih
yu a dth a t M
stdsuch
iD
a
t.c svh a r (3
id
i fo r a in y. Then b y 1.9 we have that v e r i f i e s
igsubstituting
,ya
n
0
8
and
falsifies
.h
i f i l e 8, and so M
L
a t rob
civa
fa
h e
n
d
0because
i t is n o t n-X-acceptable, we have
T
o
M
M
n
a
s n o t
,e
ti +,si s1-X-acceptable
by
flhX l th
- iey hypothesis
o f the induction th a t M
tld
D
,tibh
e
If R
isM
o
cn en so
po nt ot at nbn}-l I-X-acceptable.
-e X. h
d
e
O
a
i0a ti csuand
teable
a
iierd
ceptable
i
s
n
because
o
t
co
n
d
itio
n
n
(2)
f
o
r
n
+ 1-X-acceptability does
Ir-st.a
a
c
c
e
f
p
t
8
t ( n
not
hold,
th
e
n
there
is
a
constituent
D y in F
+
1
X
a
c
tA
g 3a l
h
b
a
f,tofX
M
a
falsifies
s
u
c
D
y
h
and
t
there
h
a
is
n
t
o
n-X-acceptable
M
ro w in F
h
M
e
s-1
ifD
w
ii b i 3 f
iia
a
X-controled
e
o
ew hs i cs h b y M
/tai e
ysis
a
e
islcya
of
a
substituting
n d o
(1 f o r a i n y. Th e n f o r e ve ry n-X-acceptable
n
vh
orm
n a w
h
i
c
h
a
idt n o
sf.cre
b
slra ya l s i f i
m
e
w
.ifL
ifr f s
ta
p
e
o
fe
S
.m
fe r
trh
iyw
o
eh
ri u
e
a
b
o
eL
M
tisb
e
n
m
sT
lfra
e
yb
C
e
s0
o
th
.b
O
8
lo
te
h
Le
M
352
A
R
N
O
U
L
D
B A Y A RT
ro w .R,
, o off F
y,
F and
a b y the hypothesis o f the induction we have th a t M
n-X-acceptable.
So b y th e hypothesis about M
t
,
i, i w
ds o eMr,s and so
n b yo 4.7t M
X-control
M
iw d ho e s
n o t
iX
M
i c-i cc o n t r o l
iM
ha b ility
h for M
fsjfi dai o e s
5.
(,ln Tautologies.
a
sso t
h
lV
.i 5.0.
f so Dl e f i n i t i o n
d
sa
B
u a F t
isaid
-eu. to be
i-a
sby
c
t sacllh[be ymall
as.X-acceptablel rows of F
fra
,6 L h e t
io
.-, 5.1.
Ft D e f i n i t i o n
w
e
logy
w
P -a h [a n X
sb a
c h efom
-sfe
a
- table
er aa,. a is a F„-P-tautology [a F„-X-ta u to lo g y].
rth aalauut tht oo l leoog y
tD
5.2.
o r eum
o
g
] oaTahr e m
[a yft M
b
F
i.m
vF ll fi a fe
[a sF,c h e m a .
,IM
e
-a a
f
o
9
i 1L
ir i oe t s
ftiX
ba
7
Proof
fa
sa s fo raP. Let M
r-y-s F
i]e
which
X
a-b e is a subrow o f M
a
n
o
1
t i.a a uv td o
.e
ia
verifies
tld
a
d1.7 R, verifies a.
tTn( gr ynya. By
.ls oa
]
.rProof-schema
B
o
y
w
t
u tF Y
t
fho r eM, B, S4 and S5. Let M
yla
i9 o
h
o
oaLet
t nhM ye s X jh
ro
l,bbyeof
epfFb.
bw
stfo
o
e
F
4.1
s
M
ia
o
g
b
c
e
c
e
p
t a b l e
ta
g
w
e
h
a
nverifies
a.
By
1.7
M
b
ito
f
y
]
e
h
e
sia
b
ia va
a re rl i f i e s
it.re
s
h
e
o aw
b
tfsttb tP
i o. T h e o r e m
5.3.
a
tL
X
h
e
e
o
o
h
l
f
e
s
lF
a
h
vtthat
-sfc wFe e
rta
a
m
F
e
lh
u
r st cf hi a
a
tautology
e m[a
a [a.FL,-X-tautology] then a is a F„-P-tautology [a F
M
ca
er o
n
o
o
t
-.
ste
i
X-tautologyl
,fa
iw
p
a
fr Lrt o hed t i
cie
u
-ia
Proof
iF sF
abmhf o r P. L e t R
fcb
o
co
such
M
ia
a
e
lie
s esbMthat
b
iw
n
h
M,
s.tsa iauuverifies
ia
sn yn
a a. By 1.7 R
. i
e
td
isb
vut e br i w
s M, B, S4 and S5.
rf i eFfor
raProof-schema
h
B
T dc .o
isaF
h
.
o
b
w
yseb
h th , uf)
ta
ltb
o
e
F
b
.
rtiy-e t f „r
o
fL
e
e
h
o
w
n h
,P
o
F
fe
M
to-w ta a a
w
w l
n
a
F
iM
b
l
h
fo
(e
,.bte t s
,yM
e
s
fi2
h i
ha
.p
B
b
f
iF
u
a f
s)a
yT
e
o
a
o
fv.tv i
h
ta
tB
o
er
e
o
O N TRUTH-TABLES FO R M , B, S4 A N D S5
3
5
3
Let b e any X-acceptable ro w of Fd
•X-acceptable
B y
4 .r o 5
t h that
e r R,e is a su b ro w o f R
w .2, such
hypothesis
s
o f the theorem
i
n R, ve rifie s a . B y 1.7 M
i
i.F vBe yr ibf i te sh e a
.
a 5.4. Theorem-schema.
n
I f there is a table Fb f o r a such that a
is a F
b[an X-tautologyl.
-Proof-schema
P f o r P, M, B, S4 and S5. Let F
ttable
a a ub tf e
o r a. tLeth Fe
o
obi enhypothesis
cm l the
By
i m u o fmthe theorem a is a Fb-P-tautology [a 5 g
y
X-tautology].
So by 5.3 a is a F
aD n y
1
[But
a
-t aF b l e
a
-fFP - to a u tr o l o g y
a
F
i[ae s a .
F
b
-F 5.5. Theorem. A ll P-tautologies are M-tautologies. A ll M-taua
-tologies
s-aX Xu - are B-tautologies. A ll M-tautologies are S4-tautologies.
X
A
a rey S5-tautologies.
A l l S4-tautologies a r e
ti- llat-B-tautologies
b
u t o l o g
j .
ta
t ab
sS5-tautologies.
u
laa
et L e t F
Proof.
o
l
o
su
b
a gb ue
F
o
ftverified
ta
a b yb all M-acceptable rows of F
yF
]
o
lby
a
e ln
b rows
y
of F
,t at llehbB-acceptable
te
lable
o
3
a
.
ro
ws
4
o
f
F
a
te
h
.f.io
a
rows
s Fa
i s
f I foof
h
e
S
F
by
vng
,e
a
e hr aeri is
nfi ve
i frif
e iei dd eb y da ll S5-acceptable ro ws o f F „. I f a is
r et3.4
o
yverified
b
b
.b
a I fy yb y a ll S4-acceptable ro ws o f F
n
a
b
.a.fied
rows of F
a
e4 nS5-acceptable
y
l b
. t b.hy l all
a
iy3
a
3
.
4
a
M
I ss5ia
.ivsf 5.6.
s c
a
e c rTheorem.
e (1)p There
t is a formula which is an S5-tautology
a
.-and
not
(2) There is a formula wh ich is an S5viva f e
e an
r S4-tautology.
i
i
r
P
2tautology
i- df i and
e
e not a B-tautology. (3) There is a fo rmu la wh ich
ais
d
rs an S4-tautology and not a B-tautology. (4) There is a fo rmu la
tib
which
yb
iv
e is a B-tautology and n o t a n S4-tautology. (5 ) Th e re is
a
sa
yfr fo rmu
a
i la wh ich is a n S4-tautology and n o t a n M-ta u to lo g y.
u
ala
(6)
There
is a formula wh ich is a B-tautology and not an M-tauif
i
tF
tology.
(7)
There is a formula wh ich is an M-ta u to lo g y and not
le
d
o
cB
a
P-tautology.
b l-ld
S
a
y c4
b
o
P
a
c
e
c
c
y
g
-p
tp
t
l
a
yte
a
bb
l
l
ale
r e
urM
o
to
-w
ow
s
a
lsco
354
A
R
N
O
U
L
D
B A Y A RT
Proof. L e t a b e a va ria b le . Th e ta b le f o r a, D a , D Du a n d
—Da and the fo llo win g formulas yie ld a proof o f the theorem. (See 3.9).
For (1) and (2), — Da ---> D— Da is falsified b y R3 o f table 3.9,
but is verified by all S-5-acceptable rows.
For (3 ) a n d (5), D a ---> El Da i s fa lsifie d b y .21 o f ta b le 3.9,
but is verified b y all S4-acceptable rows.
For (4 ) and (6), —a —> D— Da is fa lsifie d b y R5 o f table 3.9,
but is ve rifie d b y all B-acceptable rows.
For (7), D a ---> a is falsified b y the fo llo win g not O-M-acceptable
row: (a, F) (O a , T) ( 0 Da, T) (E r— Da, T), b u t is ve rif ie d b y
all M-acceptable rows.
5.7. T h e o r e m
la,- le t y be any formula, le t b be the result of substituting f o r
a in
s cy.h Then
e m a we
. have that if y is a P-tautology [an X-tautology]
then
L 8eis at P-tautology [an X-tautology].
Proof
a for P. Let F
aF bb e e a
tthat
a aa Rb l e
fi, vi so ay, th
about
rr a ti R
iya
l a b, l e
Proof-schema f o r M, B, S4 and S5. Let F
vl(e be,Vbe
r ei f i e s
F l e
a b ethe 13/a-table
a
tfroam b
yta
tFbal and
- . le te R
F
f
o
r
y
,
1
B
rR
i ,tob l ee)ey t
lR
tR
b
1
it i(h e
, b about
iw
hypothesis
e y, .2, ve rifie s y. B y 1.9 we have that R
fi.b
e
9
fies
3
b.
r av oe rn w
i - y
X tT
w
re
h
b
o a cf c e p t a
e
a
ho
F be l e
(em
h
n
a ra 3 a/ o
va
ys w
n - ue
ng
tw
irc oy a h hb
1
a
to
e.o
t ff
e
R
w
T
r
m
hh
ifah
o
u
a
vrve
ft
e
ro
F
i
en
R
fm
b
w
i
bi
e
a
e
s
yi
b
n
h
4s
..a
d
a
8vl(
3
te
/
ta
hb
ay.r
o
tt9
w
,
h
O N TRUTH-TABLES FO R M , B, S4 A N D S5
3
5
5
PART I I
6. Pla in proposition logic.
6.0. Definitions. A n uncovered occurence o f a constituent (3
in a formula a is an occurence of 13 in a not as a part of a constituent.
An uncovered constituent o f a fo rmu la a is a constituent o f
a which has an uncovered occurence in a.
A value-assignment f o r a fo rmu la a is a monadic fu n ctio n
which takes the uncovered constituents o f a as arguments and
the letters " T" or "F" as values.
6.1 De fin itio n . Let b e a value-assignment fo r a. We define
under wh a t conditions ' r , ve rifie s o r falsifies a.
If a is a constituent (a n d so a n uncovered constituent o f
itself) then and
V 'V
i If a is of the fo rm ( 3 , then v e r i f i e s a if f 'V
-fvraei l fr ai fl si i f i e s
( 3 ,
ise iIf
a
d f o rm f3--->y, then 'VI ve rifie s a if f ' r
sf ai isn o f the
or
fe
a l ss y,i fand
i e s'V
(
3
a sivefrifie
ia
ci f af l s f i f i e s
a
li' 6.2.
r De fin itio n . A formula a is said to be a plain tautology if f
isiall value-assignments
f
f
for a verifie a.
)ifa
p'
,fp 6.3.
l L ei t F
vi'e aaedvalue-assignment
be
b re i f
f o r a. W e sa y th a t g
equivalent
i
f
f
,
f
o
r
e
ve
ry
u n co ve re d constituent (3 o f a , R
ie
a
a
e
n
s
d
a
r
e
rt
ia
v
erifies
i
f
f
'V
(sio t a - b 3
a
n
ai l e
v
e
6.4.
r
i
Theorems.
f
i
e s
Let F
d
f
o
ip
t
p
and
a
le
b
t
I
e
rʻ
a
fp
r
a
l
s
a
i( taba-equivalent.
afae bni l e Then (1) g
iflkare
is-ie[falsifies]
v
r i af .ir (2
e )sTh e fo rmu la a is a F
d f, e o
a
[vs a
l s, i ifff iti is
e asplain
]
tautology.
a
o
yre
lfa al u
sP-tautology)
Proof
oeru(1).
-ta
MPle- - t f a
e
t o lB
t oy gd ey fin itio n 6.3 a n d th e p a ra lle lism between
.id
1.6tsand
as f 6.1.
i n fd
vth(ia M
v
e
i
f
i
f iRna m lo r
osg
e
e
s b
i n i st f i e
ra
va
fitae a
b s
ifa
o
o
l re
a
ikurf w
f o
r
ea
,s"' fo
sV F
w
ftT
i a
o
ih
"u
356
A
R
N
O
U
L
D
B A Y A RT
Proof fo r (2). I f a is a F
a
rows
of F
- P - t a u twit
a
equivalent
o lho some
g y ro w o f F „. So b y (1 ) a l l value-assignments
.t
h fo rea venrifie a. I f a is a plain tautology then a is ve rifie d
by
f o r a . O b vio u sly e ve ry r o w o f F
a ba llvvalue-assignments
O
i
ais
a-equivalent
wit
h
some
value-assignment f o r Gt. S o b y (1 )
i u s ls
o
all
rows
of
F
yv
e
r
i
f
i
a
e
d
v
ve
f ifin
e itio
6.5.rr iDe
b e
y n. Let a
a
of
an
.
asc
.
9
l
ya
l
representing
9
vl , a • l a fo rmu la (th e rf ) o f M, is th e fo rmu la ( 3
1a
where
s
a
b
ei
u n e 13.
- (1
A
m
falsifies
.a tLso erhst a e
—
a
m
M
i ng n
.aim 6.6.
ci os r t d Let
dce Theorems.
i i nM, be a ro w of a table F „ and le t a be the
gb
rf
o ft eMc t
n n
iaF
a
s t
f c o sn
Proof.
a
ro p o sitio n lo g ic i t i s e a sily seen t h a t f o r
.M
ro i ot B y up la ein p n
every
ro
w
Rk
o
f
F
„ we have that a k ve rifie s a if f v e r i f i e s
ifw
T
h
s
r t
all
a
the
members
o
f
vo
e
r
i
f the continuous conjunction a, and th a t M
a n e
falsifies
a
if
f
f
a
l
s
i
f i e s one o f the members o f the continuous
k(ilfi 1e
s
conjunction
a. B y the hypothesis o f the theorem we have that
)osts
M,
ve
rifie
s
a
l l th e members o f th e continuous co n ju n ctio n a
rih
M
a
and
that
every
other ro w M
fie
continuous
a.
jvit f a l s i f conjunction
i e s
o
e
ee
a rn
Let F
o
isb 6.7.
f Theorem.
lf
a
b
e
a
the
tia
f
o
h
llo
win
e
g
fo
rmu
la s a re F
e e
logies):
t
a
b
l
e
am
e
m
b
e
s.F
(1)
w
a
i
t
h
-raaP - st a u t o l o g i e s
of
(lo
n f d
af rF a
a
V
t
i
s(2)
tnr d( a i ohs
P
;tlV
distinct
rows
of
F„
and
where
a
n
c
t
e
a
u
t
o
do
rows
of
F
iA
a
r
o
(m
a
,(3)
.r2Y w
. aa. —›(3 swh e re .a is th e r f o f some ro w M
;r(iA
Ur he such that F
,)3
a isTi anaformula
a
—
>
—13
a is the ri o f some ro w M
t(4)
a
i
h
s
ea where
w
—
e
n
e•
1
i
n
F
rit,a
a
f
b
a
l
s
n
e
d
w
h
e
r
e
h
vT
3
oa
fe
o
f
r
,eh )
it,1
a3
w h
e
r e
h n ed
-rre
sra
nd i
e
4
yn
a
og
f
t
h
e
(a
ot
fria
lth
ov,he e
r
i
f
i
re
s
a
V
e
m
1
3
.,r
u;.r
O N TRUTH-TABLES FO R M , B, S4 A N D S5
3
5
7
(5) 6 —› ( c
O L . fo
. r 13, and a
table
lV a , (3;)
venue
,w
(6) a(h a ,
e r foer 6, and a
a1
table
1 . h
ltV
falsifie
. . 6. e
3 aa f ,Fo rs (1). Eve ry ro w of F
,rV
Proofs.
oi1 vdr isju
a
the
e erfni ctio
f i en sand so ve rifie s t h e disjunction.
ta)s (2).
o
For
hn l Eve
e ry ro w of F
a antecedent
rlw
to
the
i f s
of the imp lica tio n and so falsifies th a t antecedf eand
o
,toh
ent
i ht hverifies
h
f e er the implication, o r verifies one of the members
erof
a
fm
e athem
leconsequent
sl i mf i be o
s fethe rimplication, and so ve rifie s th a t conl noand the
slrsequent
o
ru
e implication.
For
a (3 ) and
tw
o
e
hsf f(4). E ve ry ro w .2
ite(s i nso ve
and
hFrifie s a —) 6 and a --> —13. .R
R
1
according
e se r
a
rnu
e
3 v eoro i t fasi hAl
For
(5).
Eve
ry
ro
w
in
F,,
wh
i(tm
r ich falsifies 6 falsifies the antecedF
w
ic h 3e sa m n o
ent
g
vu
f i eimp
iash e or fiethe
b
rs lica
s- tio n and so ve rifie s t h e imp lica tio n . E ve ry
ro
in
>
o
t w in
r F
f
i
w
a
a
consequent
o
f
the
imp
lica
tio n and so ve rifie s th a t consequent
fF
h
a
a
l
l
s
i
s
i
f
i
hf
i
and
the
implication.
Fo
r
(6).
E ve ry ro w in F
w
h
i
fe
a i se s
co
a
a
falsifies
w
h
th
i
e
c
antecedent
h
v
e
r
o
f
i
the
f
i imp
e lica
s tio n and so ve rifie s th e
c(a
,
hrt h 3
w
(v.hm
implication.
F e r 3 i E ve ry ro w in F
a
the
members
ica
i h i c ho f the consequent of the imp lica tio n and so ve riufw
fies
that
fe
a
l s consequent
i f i e sand the implication.
s
hli
(6a
3
s
deductive system P is the system wh ich
vsa 6.8.
i nf . Th
i e e
e Derfin itio
contains
the
fo
llo
win
g
three axiom-schemata and the fo llo win g
sru i
deduction
rule.
o
n
e
fc i
AO.
a
--->
((3
—>
a).
o
f
eh s
A
ot l. (a ( P >
A2.
nh (— a ---> ( 3 ) ---> ((3—> a).
MP.
ponens for material implication.
(ea ( Modus
u
(ot
3
fF 6.9. Theorem. A fo rmu la a is a theorem o f the system P (a
P-theorem)
if f a is a P-tautology.
)ta
Proof.
(hi
cWe can consider the language, constructed from propositional
variables and the three constants f o r negation, ma te ria l
tes
Y
m
a
)e
)m
b
e
r
s
o
358
A
R
N
O
U
L
D
B A Y A RT
implication and necessity, as a language for plain propositional
logic b y considering a ll constituents, mo d a l o r not, as being
variables fo r propositions and by considering that o n ly the uncovered occurences o f the constituents are occurences o f these
variables. Such a convention is quite simila r to the convention
according to which, in a language where we have the variables
p" " etc., th e occurences o f the expressions " p "
and " p' " wit h in the va ria b le " p" " a re not considered as o ccurences o f the variables " p " and " p' ". Once we have made
such a convention we can see in Church's [8] th a t a fo rmu la a
is a P-theorem if f a is a plain tautology. The theorem then fo llows by 6.4.
7. Th e deductive systems M, B, S4 and S5.
7.0. W e g ive ourselves th e f o llo win g f iv e axiom-schemata
and the fo llo win g deduction rule.
AD. ( a ---> p) ( O a 0 ( 3 ) .
AR. D a —* a.
AS. — a —› 0 E l a
AT. D a --> 0 Oa.
AST. D a
0 D'a ,
NE. Ne c e s s it a t io n . From a to deduce Da.
(The letters " D" , " R" , " S" and " T" are suggested b y the words
"distribution", " re fle xive " , " symme trica l" and " tra n sitive " .)
The deductive systems M, B, S4 and S5 contain the fo llo win g
axiom-schemata and deduction ru le s respectively.
M. The same as P plus AD, AR and NE.
B. T h e same as M plus AS.
S4. The same as M plus AT.
S5. The same as M plus AS and AT.
To indicate that a is a theorem o f one o f these systems o r o f
some other system we writ e " I f , re lyin g o n the context to
avoid ambiguities.
7.1. Theorem. I n M, B, S4 and S5 i f 1— a-=(3, and i f 8 is th e
result of replacing an occurrence of a in y b y an occurrence o f
(3, then
ON TRUTH-TABLES FO R M . B, S4 A N D S5
3
5
9
Proof b y induction o n the construction o f y f ro m a. I f y is a
the theorem is trivia l. I f y is (1) r
,(4) OE,
o r wh e( re2r contains
)
Ea, and i f Ti is the re su lt o f replacing
an
of at in E by
- occurrence
- - >
, an occurrence of 13, then by the hypothesis
or f the induction 1--E=1. B y P we have then 1- - n ,
o
or
(E
(
3› t )= )(1-3 t), o r 1 - (t E ) . = (r, - › 'q) a n d so t h e th e o re m
is
t proved fo r cases (I), (2) and (3). Fo r case (4) we have b y P,
N E a n d A D 1 - E --) T1, - - - > E
E
rE
> 1- OE= an.
o -1› - D (T rIE [ J r- and
H
(
E
T
1
7.2. Theorem.
0 (aAp)--->
)
, I n M, a n d so i n B, S4 and S5:
( 0 a A 0(3).
Proof. (I n th is and in the f o llo win g proofs obvious steps w i l l
be omitted).
(aA(3) -÷ a (P ).
(2)1- 0 ((aA(3) -->a) (1 )(NE ).
(3)1-- ( a A(3) - › Oa (2 ) (AD) (MP).
(4)1- (aA13) ( P ) .
(5)1- ( ( a A(3) ->• 13) (4 ) (NE).
(6)1- El ((IAN 0 ( 3 (5 ) (AD) (MP).
(7) E l (aA(3) -› (ElotA EJP) (3)(6)(P).
7.3. Theorem. In M, and so in B, S4 and S5 we have:
0 (a , , A . . . A a
Proof
n by induction on n.
7.2-treats
l A a the case where n =1 .
Fornn 1 we have by 7.2:
(1)1) 0(a„A...Aa11-IAan) -* (0 (a,,A A a
By
ofEthe
n -the
- 1› hypothesis
) ( D Aa
h induction
a n ) •we have:
(2) ,1- , ( aA , .A . A. an -1) - › ( a „ A A Dun -1).
By A(1),E(2)l and
a P we have:
(3)1n 0 (a
0 - t A
A 7.4.
. l
I n M, and so in B, S4 and S5: 1 - (0 a A 013)
E. . Theorem.
La(aA(3).
A a
a n - n
Proof.
l A) a •
M
n I - a ( ( 3 -4 (a A (3)) (P ).
(2)10 (a ( ( 3 - 3 (aA(3))) . (1)(NE).
)
( 0
a
0
A .
. .
A
D
a
n
360
A
R
N
O
U
L
D
B A Y A RT
(3) D u -3 0((3 --> (Q M)) (2 )(A D)(MP ).
(4)1- ( [ 3 ( u A ( i ) ) ( 0 1 3 ( a A l i ) ) ( A D ) .
(5)1- D u - (0 t 3 ----> ( u A ( i ) ) . (3)(4)(P).
(6)1- (DOEA 0(3) ( a A ( 3 ) (5 )(P ).
7.5. Theorem. In M, and so in B, S4 and S5 we have:
Ca
Proof by induction on
n n.
7.4 treats the case where
n 1 .
)
For n 1 we have by
- -the hypothesis of the induction:
(Da
)
(2)1- (0
0 u ,,A ...A D a0
n(3)1- (A (.c. c
(
o
Da
_(4)1.( A
a
A
Io
DA u
0
In S5:
E L7.6
A
l. anTheorem.
.u .
A
A - Da - E l - Du.
Proof.
„
nA
.
D. -t--- u>
)O
(1)
Ou
1 ---> D a (A.T ).
„u F- )0- 0 Oa - Ou
((2)
. (1 )(P).
-(„ a )- (3)1O a ---> -AOa) ( 2 ) (NE).
(
1
0(4)1›0 - 0 Ou---> 0 a- Da (3 )(A D)(MP ).
a
A
(5)1A
0- Da
. .---> .0 - Dn u (A S ).
0 r (- t n - _ Oa
D
(6)1A
- (5)(4)(P).
A
u
i ) u
l
( 7.7.))Theorem. In the
n
A
A system wh ich contains the axiom-schemata
7
0
D
Aanda deduction arules o f M plus AST: 1 - - a -› - Da.
Proof.
n(•
- )
n
(1)
1
u
•
. E u - * - Ea
) (A S T).
(2)1Du
-3
(
A
R
)
.
(„1
A
.
)
A
1(3)1a- a -> •-• Du (2 )(P).
(4)
). F-. (n- - ›P - )Oct (3)(1)(P).
•. Aa 7.8.n1Theorem. In the system of 7.7: 1- Du -3 0 Du.
Proof.
_
l)
(1)1A
u•- Ou --3 0 - Oct (A S T).
t , D O u - 4 Ou (1 )(P).
(3)
D u - D u ) (2 ) (NE).
) 1- 0 ( (4)10
E
l
a -3 0 Oa (3 )(A D)(MP ).
•
(
2
)
(
3
)
(
P
)
.
O N TRUTH-TABLES FO R M , B, S4 A N D S5
3
6
1
(5)F- — 0 - 0 a — > 1 - 0 — E l a . ( A S T ) .
(6)1——1=1— Ela —> El Da (5)(4)(P).
(7)1— — Da —› — Da (A R).
(8)E-- Da ---> — D — Ela (7 )(P).
(9)1— Du— -> D a (8)(6)(P).
7.9. Theorem. S5 and the system o f 7.7 are equivalent, th a t
is they yield the same set of theorems.
Proof. By 7.6, 7.7 and 7.8.
Remark. The system o f 7.7 is the Gödel formulation o f S5. I t
should be noticed th a t in 7.6 no use is made o f AR, b u t that
use is made of AR in 7.7 and 7.8.
8. Consistency.
8.0. Theorem-schema. The formulas wh ich are axioms b y AO,
A l o r A2 are X-tautologies.
Proof. Th e fo rmu la s i n question a re p la in tautologies. S o b y
6.4 they are P-tautologies, and b y 5.5 they are X-tautologies.
8.1. Theorem-schema. M P preserves X-tautologyhood.
Proof-schema. L e t E
X-tautologies
then they are F
.
aare
bay all X-ecceptable rows of F
' „ bverified
e
-that
a
ul t eo l o g i erows
s
of F
t X a- at llba X-acceptable
alogy
.f B
a
vo
and
eny r by
i rf1d5.4
i e. an6 X-tautology.
p t. h i s
sS
sa u opo p o 1s e 3 s
ai—8.2. Theorem-schema.
s ›
The formulas wh ich are axioms b y A D
aare
a
( X-tautologies.
3n
.F d
aProof-schema.
a
Let F
I
f
-A
- ro
a
Xbw e-g t aa- u t > o oa f( 3b ) n
and
(ita11(a
l 3e D a and falsifies 1:113. Th is is impossible i f R
F
X-acceptable.
fdi i so
r Indeed if g
and
, then
b y 2.7 in F
ia
1
1X
C i sDa
a
-erifie s a w
X-controled
y M,
bX lve
(I cei cav ee pr tyb a
a
such
cn3wea R,
pndtd a b al
l[( lc( 3ro
4
.
,e
m
u
s
t
vn
B
f
i
)3 eu r t i
vtre
o
s r oi ef
nw
—
a eh
w
ib
hy
i
t›r e
0
c(fe a 3h. aD
[1
6
ie
a
-la s v
e
r
n
i— f
ys3
d
(i> e
3•
s)t1
:
1
o
h
(
3
)
b
i.
362
A
R
N
O
U
L
D
B A Y A RT
8.3. T h e o r e m
Proof-schema.
Let F,, be a table fo r Da . I f a is an X-tautology
then
s c ah lle X-acceptable
ma .
ro ws o f F „ ve rifie a. So f o r e ve ry X-acceptable
N E ro w R
X-controled
i po fr e F s b ye R, ve rifie a. So b y 2.7 e ve ry X-acceptable ro w
R
a r v e s
ia X l - l
oX 8.4.
t - a T hu e ot r e
om
fAR
are
X-tautologies.
a l c o c ge yp h
Proof-schema.
F
t soac ho
b e lm
d a
e .. L e t F „ be a table f o r Da . A ro w g
not
f ho r e D a --> a unless R
air Tofalsifie
iva
is
fv eo r sir f imeif sR
w impossible
iE
,o iu
iss lO-X-acceptable
wl Xai - ls l
a n d th e n b y d e fin itio n 2.2 i f R
eit
irfEu,R
a
c
v
e
c
n
r
e
i
p
f
d
i
t
e
a
s
w
h
i il ae .lc s i f i e
fiF
b
Isfav 8.5.
d
The formulas wh ich are axioms b y AS are Bhe n Theorem.
rw iaf e
a
e
d . and S5-tautologies.
itautologies
Proof
r B. Let F,,
iT
i be a table fo r E — Da. A ro w R
ehi er fo h
not
Da unless g
ifsis eo falsifie
f
F —a —
ia
is
impossible
f
a
l
s
i
f
i
e
if
s
g
a
g
D
ca a w xi l l
ia
n Bo d- D D a , then b y 2.7 there is a ro w i n F
iaand
h. ii sfalsifies
E
a
that
c
s
Mr
c
—
ht a
R, B-controls a n d Al
i.a
a m ueiscpsB-acceptable,
D
ib
and
l
e
so
.
ve
a
rifie
s
.
Da
.
B
y
definition
2.1 R
sr b
verifie
a.
i'IT
f
a
n
l
h
s
d
i
f
i
e
i
s
s
—
X
e y follows by 5.5.
e
,D
e
R
aForCS5
c Oathecdtheorem
ie
s
o
pa tn d
fa 8.6.b Theorem.
R
Th
e
fo
rmu
la
s
wh ic h a re a xio ms b y A T a re
l
S4-tautologies
and
S5-tautologies.
iR
m
u
s
t
e
Proof
f o r S4. Let F „ be a table f o r E D u . A ro w R
i
ii o faf lsifie
not
F Ea
-a
i is
l impossible
l
i f R, is S4-acceptable. Indeed i f R
s0 Da. wTh is
.Bi S4-acceptable
is
Dand falsifies D Da, then b y 2.7 there is a ro w
0
0 c FE c
a in
falsifies
a
u
e n p l Da
te . By definition 2.1 R
isaEu.
Cs bl
la n d
sg
For
e uS5othe theorem follows by 5.5.
R
i c ,
cv hea rn i n
o
t
vf ti e e r s i
f
i
e
D h
a a
a t
n i
d s
f Sa
l 4s
i
f
i
O N TRUTH-TABLES FO R M , B. S 4 A N D S5
3
6
3
8.7. T h e o r e m
Proof.
Fo r M: b y 8.0 — 8.4. Fo r B: b y 8.0 — 8.5. Fo r S4: b y
8.0s —
c h 8.4
e mand
a . 8.6. Fo r S5: b y 8.0 — 8.6. B y sa yin g th a t th e
deductive
A l l systems M, B, S4 a n d S5 a re consistent w e mean
that
8.7 holds.
X theorem
t h e o r
e8.8.mT hse o r e m
F ar
a es c h e m a . B y 6.6 w e see th a t R
Proof-schema.
falsified
ia X
If a fl- s bi yf an
i e X-acceptable
s
— ro w in F
an ti a By
logy.
n
8.7
d
—
a
i
is
s
not
an
n
X-theorem.
t
S
o o
a. s u t
a
d oa ln no g X —
a
tii iXa e- u ss t . o 9.
f Completeness
a c c e
a p t a b l
i 9.0.
e Theorem. Let R, be a ro w o f a table F
s of
rf
.r T h e n we have that, if £3, is not S5-acceptable, l p is an
S5-theorem.
t o
a a n d
l e t
1 1 )
h b
w
e
t
h
e 2.2.
Proof by induction on definition
e o
r a)
f If R
fD ia i s
6.9
ou n
t v --->
o D
So
we
have in S5:
ufi ta
ag(1)
n0
bn1 -d
—
'(2)
-F S
l —5> ( a A — a) ( A R ) (P).
(3)
—
„tap-e1——1p (1)(2)(P).
at(s a
b) Icf g
fnu +
a
h
ci 1-S5-acceptability
e
i s
does n o t hold, th e n M
rie
lceptable.
c p
ni sto nIn othis
t case
n we
- S
have
5 b- y athec hypothesis
o f the inducen
ha
a
b
tion:
t
P
—
t( ln e- ,
A
ta—
h
1 t+h
c) I f .92, is not n + 1-S5-acceptable although condition (1) f o r
eia
) h
1 o+e1-S5-acceptability holds, th e n co n d itio n (2) f o r n + 1-S5rn
s)tI e
S 5
acceptability
does n o t hold. S o th e re i s i n F
m
n.M
—n
- as
a
D
c
o
n s t i t u e n t
aoi- ta c
that
u
g
ntv- h
ci se
diase
- e
p t
sn
nu
r1 ra- b
ci e5
oS
X
Pl e
-S
-h
f• ia
5a
tti snc e
tchh
ea
dt h
p
a
ees ci
b r
oa
otD o
f
elrR
ue ,
364
A
R
N
O
U
L
D
B A Y A RT
So we have:
(1)Htr--> O a .
If there a re in F
P-tautology
a n o
and so a P-theorem and an S5-theorem. So we have:
r oHa.w s
(2)
a H Et u . (2)(NE).
(3)
(4)
a H (l 1 ) ( l3 ) ( P ) .
w
h
i
d) L e t us suppose we h a ve th e same case as i n c) e xce p t
c
h
that some ro ws o f F
f radistinct
l
s
the
a
f a l s i f irows
e of F
i wf h ii c eh
a(1) Hip
. — Ela.
fa
i as —>
it (x
f i e
L
(2)aH —
e
,
a
.
xi none
V
If
o f the ro ws o f F „ wh ich fa lsifie a are n-S5-acceptable,
t
h xthe ehypothesis
n
iT
V
then
by
of the induction we have:
h
ex
,w
(r
e
v
rh)(4)
3 H —a(x
n
e
:
b(5)
e
i)•a VHax (2)(4)(P).
t(6)
H El a h(5 ) (NE).
r1
i
e(7)
—
)s H ( 1 ) ( 6 ) ( P ) .
r(; 3 ) f
a
s•( e)P L e t us suppose we h a ve th e same case as i n d ) e xce p t
that
o)• . some ro ws of F „ which falsifie a are n-S5-acceptable. Then
these
ro ws a re n o t S5-controled b y g
f•
ia
g
;
i.l(2) SR 5 - c o n t r o l
—
o
b
i
f
a
lLet
x x
yrC
o
w
ir
falsifie
a and wh ich are not n-S5-acceptable.
iO
,• y
Let
m
(
ix
falsifie
a, wh ich are n-S5-acceptable and for wh ich condition (1)
p
3
,rfor S5-control by M
lLet
i) f a 1l l 1s ., be the rfs o f a ll th e t d istin ct ro ws o f F
yb
ifalsifie
g
a
c ich
h are n-S5-acceptable and fo r wh ich condition (2)
se w ha, i wh
e
i S5-control by g
for
t
b
sLet
iC f ae l l s .
h
e
tfalsifie
ll
a, wh ich are n-S5-acceptable and for which condition (3)
te
h
g S5-control by ,R
for
,h
r
e
i
ie
Let
f f a(pi l s .
e
fiaa
rs
o
n
,b
fo
l(ed p
sf
l,t(
a
o
o
4
h
b
fl
w
)
e
l
a
itrM
lt
n
flhi
h
g
steC
e
O N TRUTH-TABLES FO R M , B, S4 A N D S5
3
6
5
falsifie a, wh ich are n-S5-acceptable and fo r which condition (4)
for S5-control by R, fails. Then we have:
(1)1-4tt-> - D a .
(2). -a (y1V...Vy,1/4LIV...VittVQ1V...VNVENV...V.T,Vx1
Vx,)•
By the hypothesis of the induction and P we have:
(3 )
(4),...
- a (y,\L —V y8 V RIV • • • V iA tV @IV • • • Q u V(01 V ••• V(1
1
5
_ (
v)
- 2• can have r -0 and then we have imme d ia te ly (4). We can
(We
have
( ) s=0 , o r t = 0
11 (ocase
the
r sud -O
3, a ll m such that 1 r r n s , if M„, is the ro w o f F
, For
to4da) rw h i bc y hy„, wei have
represented
s th at there is a constituent D(3„, i n
-vFa
(
=
0
V such that a
,(t• +Pv >
u
b(5
0
vV e) )r i uf. i e s
t6
. a ll m such that 1 m t , if .9,, is the ro w o f r
)0 For
b)1
represented
a
s that there is a constituent Dt,,, in
(t 3 wn h , i bcy h11,„, wei have
F
yI-a
r
n
t(a
h
)
d
-1
sy)•
(e
7
8
h)u
y
p - laçt ; ••• 1 i-R
)For
c
a
ll
m
1 that 1 m u , i f •R,„ is the ro w o f F
oI-4
f
at l hsuch
1
a
representer
bhy) ct,„, we
i have
s
that there is a constituent DO„, in
e
-h
s wi s h f i ic 1
0
-P
tt(i such
s1V
e
sthat-,R, verifies DO(10)
;>
l i t -> D .
o
1-h
P al i t
;3
(11)
,
1
—
a
n
d
;
.
2 1„ 2, „ -> D O „ .
1
a
D
f•1
>
m
O
(12)
f
1-1p
a
--->
l
s
i
;
f
i
e v s -> l a a „ (1 0 )(A T).
•t
tv For
a
ll
m
such
that
1 l m E v , if g
OM
.t
•-S D
trepresented
e we have that there is a constituent 0 4 „ in
e mS i s
•tboyhcf,,,
;•;-o
F
r
o
w
f
w
e
r•1
(
7
)o
w
y>
a
(13)
1O
X
F
h
a
v
( A S
H
-i4
e
0
s[4
(1
: ) .
f—
(h ae
vu1
t(1
h
i
c
h
if3
aw
t1
By
(4),
(6),
(8),
(I
t
),
(14)
and P we have:
-ce
4
i
s
-isv
(16)1-•
-a - - h>
)>
5
ss.e
'>
k V( -: U.
P N.1 . .VV -0 4 ).
-.-D
)1
•:
V
R
-À
1
ti P
-(v„lt B V E K I
V
.,•l • • • V D t
P
1
V
Iva 1
)-en
-rd
D
iM
->
fi
E
D
if
X
366
A
R
N
O
U
L
D
B A Y A RT
(17)1- (1 3 1 A -A f i
e A - 0 4 ) -> a (16)(P).
A
D
t
'(18)1A
t
DX,,) --> De (1.7)(NE)(AD)(MP).
A
(19)
-A 1- ( 0 [ 3
1
015„A
- DDÀ
- & A
E
By
A
1
(5),
A
(9),
0
(12), (15) and P we have:
O
(20)1-v-->
8
where 8 is the antecedent of (19).
(I 3 ,A A
A
(21)1-1p-÷
E
A . l . - . DeA (20)(19)(P).
(22)
-O
D 1- -1p (1)(21)(P).
W
D
O X , .
)\, 9.1 Theorem. Let a , be a ro w of a table F
of
l e t
l
p
-.A aa - a
. n- d
iS4-theorem.
t
h
e
. b A e
>
.D
0 r b y freplacing in the p ro o f o f 9.0 the p re fix " S 5 " b y the
Proof
T
a
prefix
" S 4 "( a n d b y in tro d u cin g th e f o llo win g simp lifica tio n s
h1
O
8
in case e).
e)t (
S4-control o f a ro w .2, b y f3, imp lie s the fo llo win g t wo condin7
ttions: M a
w
•A
•••r Tyr 0
i C O
e5
0
01 (2), (4) and (16) — (19) are shortened in accordance.
schemata
R
h)0 ((7) —
are dropped and so no use is made o f AS.
• •(9)
• 1
i
a1 (13)
3
—
(15)
E
=
I are dropped and so no use is made of AST.
a n
v)A • .t ,
d
e. 9.2.. Theorem.
Let a , be a ro w o f a table F
(
2a
tA
rf
an n d
l e t
1 1 )
) aof M
hB-theorem.
iM b d e
t
h
e
a.
M
i
b y replacing in the p ro o f o f 9.0 the p re f ix " S 5 " b y the
tProof
T
h
I" B " and b y introducing the fo llo win g simplifications i n
C
prefix
,e n ,
l
e).
icase
w
D o f a ro w a
A
fB-control
e
X ) ,R,
.i b y(1
M CO a
2tions:
h
C
o
Quon.(pi,d •••, Ty, DO
i,•••oa
a
,) l i a(2),
n
s p
(iT m
2schemata
e s(4), and (16) — (19) a re shortened in accordithe
vance.
iit, •d• h.a ,
D
e I
se
r
r
(10)
—
(12)
f% o Ol l are
o dropped and so no use is made of AT.
nC
ta
e
t
a (13)
d gare dropped and so no use is made of AST.
w
in —n(15)
oM
h
iitM o d w
ta
n
s r
.oI
S
ta
c, o oon s i
4C
,b
nD e prdX a t i
-d
io
p
ai-f, o en s
ua
br
ca
M
d u
toe
ci
ar
R
o
etid
np
IQ
p
e
p
O N TRUTH-TABLES FO R M , B, S4 A N D S5
3
6
7
9.3. Theorem. Let M
i bT eh e n awe have that, if if i s not M-acceptable, –1i) is an
rf of
M-theorem.
r o w
Proof
o b yf replacing in the p ro o f o f 9.0 th e p re f ix " S5 " b y the
prefix
a
" M " a n d b y in tro d u cin g t h e f o llo win g simp lifica tio n s
in case
t
ae). b
l
M-control
o
f
a
e
F ro w g
condition:
g
i ab y
g
i a•••,
(ph
C O
Tv, Ell,
n • • • Elt , 01%, E l k and O k
and
iM
p schemata
l i e s(2), (4) and (16) — (19) a re shortened in acdmthe
it,cordance.
h
l a r eee
.d
f (7)
l pare
l pdropped
o e d and so no use is made of AS.
t ro —o(9)
(10)
—
(12)
are
dropped and so no use is made of AT.
C
w i o i n ns . g
iu d(13)
nr (15) are
i dropped and so no use is made of AST.
, e—
a
o
q bt i u
e
n 9.4.
s
e Theorem-schema. I f a is an X-tautology then a is an X theorem.
a t
b h
Proof-schema.
Let F
o e b efalsifies
which
a
a a, then a is a P-tautology and so a P-theorem
u
and
I f some ro ws o f F „ fa lsifie a th e n these
t a an
bt X-theorem.
l e
rows
f[
oat re rn o t X-acceptable. L e t V
distinct
t
ra
b . roews o f tF „ hwh ich
e fa lsifie a. Th e n we have:
rI,(1) –f f– s> (VI V. - VIoM . (63).
f
(2)1—
–
(v
ta
h
e
h
e
(3)1—a
irt
e (1)(2)(P).
(i, By saying th a t the deductive systems M, B, S4 and S5 a re
complete
we mean that theorem 9.4 holds.
b
so
yh
n
9
o
.r10. Mo d e lso
0
w
,i 10.0. De fin itio n . Let us denote by " . 2
propositional
lo g ic w i t h negation, ma t e ria l imp lica t io n a n d
9
n 9
necessity
as
primitives.
"
t
h
e
l a n g u a g e
.F
A
mo
d
e
l
.1(
f o r 2 ' is a set o f three elements n a me ly (1) a
f
1
a o
not
o f wh ich are called " wo rld s" , (2)
l
, me mpoty set,d the aelements
a
9 dyadic relation R defined on the worlds of .11 and (3) a dyadic
function
w h i c h takes the variables o f ..2
.
92 a s f i r s t
ao r g u m e n t s
r
9
.
3
a
n
d
368
A
R
N
O
U
L
D
B A Y A RT
and the wo rld s o f d i as second arguments and wh ich takes as
values the letters " T" and "F".
We re fe r t o models and t o wo rld s b y means o f the letters
" d i " a n d " Y r" respectively, e ve n tu a lly fo llo we d b y suitable
indexes.
10.1. Definition. Let a be a fo rmu la of 2 ', le t d i
a b e the arelation
m R oandd the
e function
l
containing
a n d le t I t r
wo
i brlde o f d ai
an under what conditions " d t
aIf
. aW
is ae variable, . i t
takes
*a
d/ e "f T "i a s va lu e , a n d d i „ Y r
inr f eaIt"l s i f i e
a
.and
ib avis
f eeofrf nthe
rm H i ,r d i d r
iIf
i fu fo
i ee
ita
i
ea
d
ed
y ia„rv*kp
.e ir ti pf i el
.aie
t asnfis
If
o
f
the
o
fo
rm
(3
f d f
if"a
>
t s di t f
i
-verifie
'u yac , ry,
tldand
a
If
0[3,
dtW
F
fe
Y
i aaroisp ofn the
a
p form
l
i
dta
r 3i tf hiaet 'Wt
a
ied ch
,
iil)("e
oKs1v su
ifiva
d
i
„
R
l
K
f
Y
f
r
f
i
e
r
i
n
f
i
e
tn a
l s i f i
iefa
i fre i that
e d t,:ifr
so
d
e
t a we
"lowshave
i
f
a
l
s
i
fh
iu
va
ae
h a l axvf vle f ef ri e . y
.naritio nl . l L e t a be a fo rmu la o f 2 , a n d le t d t
stw
i'a
f(4fh o3De
a
' f fin
e i 10.2.
itmodel.
v/fld
d
e Th
i e sa y th a t a i s d i
1 dre n w
c ef . o
Y
r
a
„a
a
'ifu
Y
f
i
i
r
as 1 l / vs
n
1
ii-,e
v
,
a
iv.t
rflbi dei i efa f
10.3.
Definitions. A mo d e l is a n M-mo d e l i f f it s re la tio n R
o
ffvo
o
(e
.iu
ce r r f
reflexive.
ffiis
e
ei Ar model
y is a B-model if f its relation R is re fle xive
3
rt
ivf
.feand
symmetrical.
is a n SLI-model if f its re la tio n R is
w
r
l A model
d
a
i
o
no
reflexive
and
transitive.
A
model
is an S5-model if f its relation
4
(n
o
rfo
'r3R
d
if is reflexive, symme trica l and transitive.
,s,fe
a
l
w
The formula a is X -va lid iff for e ve ry
asp 10.4i Definition-schema.
o
f
X-model
d
t
a
the
formula
a
is dta-valid.
e
ni.
m
e
h
dy
e
a
w
.
vo
e
r
tl
h
d
a
Y
tI
d
T
ti
„o
Y
f
i,
ON TRUTH-TABLES FO R M , 13, S4 A N D S5
3
6
9
11. Truth-tables and models.
11.0. Definition-schema. A n X-ima g e o f a tru th -ta b le F
a set
a of
i sthree elements namely (1) a set, the elements o f wh ich
are called " wo rld s" , (2) a two-place re la tio n R defined on the
worlds in question and (3) a two-argument function w h i c h
takes the variables o f .2' as f irst argument and the wo rld s i n
question as second argument and wh ich takes as va lu e s th e
letters " T " and " F" , th e set o f worlds, th e re la tio n R and the
function ' r in question being such th a t there is a one-onerelation I between the X-acceptable ro ws o f F
a a n dsuch tthat
question
h (1)e i f among the wo rld s in question a n d
Y
wr o r l d s
i
n
ithen R Y / r
i question
i f f
Yr
cin
ioX - variable a in F
every
a
c oow("
r nF
e
r leat ss value if f v e r i f i e s [fa lsifie s] a.
r"T"
h
r oa nl vds es
rp
teb
g 11.1.
h ya
T h et o r e m
IsIiF - t
a
t s cp h e pm a l. B iy 2 .8 w e se e t h a t t h e X-ima g e s o f F
pProof-schema,
a
e
n F ad or because
o
omodels,
t h e se t o f X-acceptable ro ws o f F
a
er
tnempty,
oen soorthet set of wo rld s in the X-images o f F
rtd e i svand
a
a
(a
h
dbe
e yempty.
w i l l B yn3.0,o3.1t and 3.2 we see that in e ve ry X-image o f
a
2 t
e
bF
X
a n b
d
a
yand/or
-r) l
etra n sitive as needed t o be a B-model, a n S4-model o r
tItm
ianFS5-model.
o
a
krh
fo a
w
e
see
.a t11.2. T hhe o r e m
d
bee-the wo rld o f %f
m
2
sre
i mcwahR,
swcthe
hh e-ro
. o f F„, and let a be a fo rmu la such that F
pe
lioIa to
X
a
i
s
a
table
c
o
f
r
o
r
r
a.
e
Then
sa p
leo
n
L
e
t
. we have that . / t i
s i
m
verifies
(falsifies)
a.
a
o
n
d
s
c.fg g4 V e , s
bV ob
ye
tltF
t fav e rh i nf i eThe proofs are b y induction on the construction
hProof-schema.
iia
a. aI f -la is
e fX
s ai variable,
f i e ) then the theorem holds b y definitions
,eof
v(o
10.1
a
n
d
11.0.
a
o
n
e
tw i
m
a I f a is o f th e f o rm —(3, w e h a ve t h a t b y th e
en
hypothesis
o
oo g n f ee - ff th e in d u ctio n th a t d i e z r
h
liR
rir ove e l n au et
e
yw
l(i ff o a nl s i f i e )
tin
3
olfd[ F
itlo
f
f
s a
r ,
hb
ee l
rr e
oe t
f
w
370
A
R
N
O
U
L
D
B A Y A RT
R
ithis case b y definitions 1.6 and 10.1. I f a is of the fo rm 13—>y, we
vhave b y t h e hypothesis o f th e in d u ctio n t h a t . 1 4 *
e.(falsifie) i f f R
ri(falsifie)
i v e r iy f i fi f eR
ifollows
iv e r i f oi rethis
s case b y definitions 1.6 and 10.1. I f a is o f the
fform
v( ef ra0i (3
lf isaenids i f R
ro w in F
ievery
i(f fvi ea X-acceptable
rsl i sf) i i e s
verifies
c, the
h definition
i
s of "X-image" we have that for every
ea
0f( i w
(e hs(3.
3 i )By
X
wo
rld
Y
r
sty3
h
e
icn.a i onronwn tR r o l e d
(able
Y
bF
fid
dr i
r yy
iaw
.2hypothesis
a
t h i cm
o
h.9 o f t h e, in d u ctio n w e h a ve t h a t f o r e v e ry su ch
,lwo
s7ta rld
h
u laT
ht
E f t I f a is o f the fo rm 0(3 and if R
w
sicw
i. oh1 r verifie
sr(
i
f
a
l
s
i f i ethat
s there
o is pan X-acceptable
,
h
,e
ro w R
tta dsthbpy 2.7 we have
ithen
by R
ifwhich
ifo
a
hh
l i nint dist rX-controled
a
ia
de
we have that in ./It
cition
„sR
ve
i Yar onf "X-image"
w
a
that
t
h
Yr,
h
i
e
R
c
r
l
e
r
h
i
s
a
ietb
h
-tv
h h
.9,.
hypothesis
of
the
induction
we have that f a l iisw
fvyIae aeBy
or nlr the
d
r
s
i
l
f
d
i
e
s
Y
r
o t
sifie
(3.
B
y
definition
10.1
we
have
that
.1t
ip
u
c
h
s)Y
tirn f r i e • s
jB
y
X
e
tm
h
(a
u 11.3. Theorem-schema.
I f a f o rmu la a is X -v a lid th e n a is
r
o
r
h
r
e
-3ict(e
he
an
X-tautology,
0 1 3 .
d
pl s i f i e i
caie
Io
e fsna e
Proof-schema.
If
a
is
not
an X-tautology then, if F
n
o
.e
r n- id
no
a there
a,
i s isa in F t a b l e
f
o
r
n
e
o
n
dsB
.11
a
a
n
tfb
ye
iby
a -th- e one-one-relation I corresponds t o R
X
rrytrs e l
ib c cthea t p. / tC l r
have
a
o
ithat
a
ta hlisiseni ofnt i dei, va
oth
i.e
fbTa
b lid .ySo a is n o t X-va lid . Th e theorem then
a
l e
h
lm
e
n
o
n
follows
b
y
contraposition.
1
a
1. o .
2
w
e
r
e
d
e
IF
te
n
S
o
w
d
fta 11.4
in
Give n a table F
ho
tX h Definition.
e
R
b
n
Y//',a o-i f d t
a
o
ie
r-i
e
yso
tnrow
b , in oa f Fm o d e l
.
iw
h
e
,tX(falsifies)
R
o
a
n
a
iff
.1
4
1
r
1
4
,
sm
i e cl
„hr1
i u
v ec rhi nf i e d
-asw
a
h 11.5.
i s i f i Let R
a
0
ts( a
fhata alTheorem.
a
ea
g
n
foF i nab e l t h e
n
tye w)
o
r
l
d
t.cX
e
sIa r ei - p f r e s e
fhcta ho .
d
1
oo n et o i ns g
im
setw
ra t
e
faf r e o
d
o
e
t
e
v
p
oo
lF
.t w
w
h
h
e
r
trd
a
Lh i
h
a
e
y
e-ia
ee n
im
vcr eo n
n
a
t t
h
cfew
sp rt i
sdo e
h
te us e
otu
l t
a
vhrn
e nt
O N TRUTH-TABLES FO R M , B, S4 A N D S5
3
7
1
that F
a iff v e r i f i e (f a lsif ie ) a.
Proof.
By an induction on the construction of a like in 1.6.
i s
a
t 11.6.
a
Theorems. G ive n th e ta b le F „ and th e mo d e l 4 4 , le t
b and
l
gf,
b e respectively the representing ro ws in F
e o f 1V t h e
worlds
a
if r a n d
Y
.o Cl g r.
Proof
fo.r (1). Fo r e ve ry constituent D a in F
ifr
a
gt,
e s Dhu a
then
v bey 11.4 td i h
a
t
•1 ve
/w rifie
lit1
verifie
f a and b y 11.5 it fo llo ws that R
.
i,h
Proof
TY .r f o r (2). Fo r e ve ry constituent D a in F „ we have th a t i f
M
•ie
s
a
.
h v ee r r i if fi ei e
T
iS
ty
o
f
R,
fo
r
every
wo
rld
lit
,
- in .//t
D
ao .
M
e
h
n
viB
h s Vu cy h C t h aO t
nthat
e
M
e
iw R . /0f1 e .r
that
1
S
1.
w
in
follows
that
R
rd
h
Y
1
V
a
v
i
b
l
r e i
e
w
C
ie
,hi ' I l t
i•O
fa
h•i- 11.7. T h e o r e m
.1
in
vvva (e- r i f i e
a
s,n sthe
1
c hwo
em
rldas. Ylf
e
veof
d
itD
,,
an
a
nrai d v eandn if YV
s(e
vd G
eX-model
a
nfo
l2
iY
Proof
,hr ' M. eB y 11.6 fo r any model and so fo r a ll M-models, i f
D
b fRt KiI V
th
d
R
t
e
. /M, CO M
u
y t f ah e bn
e
aio
)h
sa l /i t fo
tia
,1
ehr B.e B yn the symme try o f R if Yr, R Y r
tProof
iftM
o
It
M
1
M
F.lloawsn bdy 11.6 that if 'Yr
h
,0 fo
i,IM
h YM
e
o rn - gV
iV
.t sso
X
. „R
a
(,eand
ia
Proof
B
f
o
i
t rI f oR is tra n sitive then, b y 11.6 we h a ve th a t i f
n
1 ao hn rMS4.
n
vcT
i1
-n
yr,
R
Ylt
C
M
b
, ns- t n h Re n
d
)e.•le
R
Proof
iircw
to S5.
lsf od cn for
n CtByr the
o Olsymme try and the tra n sitivity of R we have
d
y1
by
sl s trh a t i f Y f f
b
e
o ot 11.6
fi1•rM
1
Cl
an
ie
R
t1
h
a
ainf d Md
tr h e Y
fh
V
7
i
ir.n
vh
e
e e
.iM
,e t C T
,g
11.8.
hhl h ee o rne m
ijR
tea
e
4
v m
a
n
iC
s,sd,M
a
t ddO
te1 (oiR
J1
p
i,r i ess c h e omla .
la
(C
able.
M i Ov e n
„C
b
if1
e
in
y G
fO
.cilM
t
hi
e
s ,
1
a,d
d
S
5
i9
e
b
t'ilT
tV
u t2 a
tiV
,,ic lc oe n
h
fs
a Ft r no l
M
e
vjR
i,i,h
.
id
a
e
svw as
vifn
g
M
b
w
le
o a
e
sin
id
yrtee
r ni
t,M
tirsh
il d,
C
h
ifd t.
h
fa
372
A
R
N
O
U
L
D
B A Y A RT
Proof-schema b y a n in d u ctio n o n d e fin itio n 2.2. I n X-mo d e ls
R is re fle xive and so R Y r
iamounts t o th e same as sa yin g th a t M
the
s y 0 -1 X1 -o.f th
.i iBhypothesis
6 e in d u ctio n R
iw
condition
a ic sce en p(1)
- tXf a
o -r bn l+e1-X-acceptability
.
holds f o r M
ia
B c a
c e yvp Da
t a
e ic h is falsified by M
hconstituent
e inb Fl wh
tion
.a
i w
F 11.4
o
e nr thhaeta.d1vv/ ee r y b
y
M
si1
d ea wo
f rld
oi Int t i M
, 1 Vi
C
O
a,
o f the induction 2,• is n-X-acceptable and
tM
i bf y,athe
l shypothesis
i f i e
by
11.7
M
,iD s a u c . h
a
holds
ib
BbXility
t- h
hy a for
w
i M c
ce
hid o etn t f i
rt.n o11.9.
li1s Theorem-schema.
I f a formula a is an X-tautology then
t i o
a
ihn is .X-valid.
V
I f a is n o t X -va lid th e n th e re is a mo d e l M b
W
eProof-schema.
1 R. 0
a wo
S
rand
.
a
1 rld Y r
it i nn f ohr a and le t a , b e th e representing r o w i n F
o
etable
wo
M
od
b fY rr t h e
cpa
e rld
in
ceptable.
s
u
re dI cr Soh a is not a F
iieti n t b
a
logy.
/ atheorem then fo llo ws b y contraposition.
h.The
-Iists X
t o-i t ar u t o l o g y
in
a
fei No
12.
l d
systems.
iant reflexive
s„(nsn ,i of
B
n
t
e
I n a[71 considerations
can be found about systems wh ich
2
ti 12.0.
f o
can
b
e
ca
lle
d
n o t re f le xive " because about t h e mo d e ls f o r
y)ia l
a
sn
1
X
these
i t is n o t stipulated th a t th e re la tio n R should
fn.
i1-systems
f
t.o
be
re
5
a
fle
xive
u
.
W
te shoa ll c a ll these systems ' ' M°" , " B °" , "S4°",
L
g
i
e
-R
S5°
",
M
'
",
B
' ", S 4 " and " S5' ". I t w i l l h o we ve r appear
re a
,n
that
ot
.S5' is id e n tica l w i t h S5. Th e wh o le preceding t h e o ry is
f+
easily
a
F
w L adapted to these eight systems once the fo llo win g basic
are made.
l1
a se
iadaptations
i-nb f t
iX
e Definitions. A l l models a s defined i n 10.0 a re M° -mo
e 12.1.
F
Mo d e ls wh e re th e re la tio n R is symme trica l a re 1P-mos-adels.
a
dels.
Models where the relation R is tra n sitive are SV-models.
a
a
o
Models
wh e re t h e re la t io n R i s symme t rica l a n d t ra n sit ive
a
cf
are
S5('-models.
n
c1
d
e
1
b
p
t
yti
1
-•
1
•
.B
8
y
M
1
i
ON TRUTH-TABLES FO R M , 13, S4 A N D S5
3
7
3
12.2. Definitions. T h e d e d u ctive syste ms M° , B °, S 4 ° a n d
S5° contain the fo llo win g axiom-schemata and deduction-rules
respectively.
M°. The same as P plus AD and NE.
B°. The same as M° plus AS.
S4°. The same as M° plus AT.
S5°. The same as M° plus AS and AT.
Remark. I n S5° we have 1—— Du —> D O a because n o use
is made o f A R in theorem 7.6. Bu t as i n theorems 7.7 and 7.8
use is made o f A R, S5° cannot be defined a s co n ta in in g th e
same postulates as M° plus AST.
12.3. Definitions. M% co n t ro l, 13%control, S 4 °-co n trol a n d
S5°
-and S5-control respectively.
con
t r o12.4.
l De fin itio n
a replaced
b y " M°" , "B°", "S4°" or "S5°".
r In
F , a ll ro ws are 0
s cehve
e ry
m atable
,
eF„-i is n + 1-X°-acceptable if f (1) R
for
vehryn constituent
E a in F
ii X
i esG
w
i- Xc ° -h
a
dthere
a -tcwciseehinp Fht aea b vl ee
tetrols
a aphac Rcn n
a p ttea b
dle.
re
inin
( Remark.
A
f- X f i°2 - Ax s w)e d o n o t stipulate a n y mo re th a t f o r e v e ry
a
a r"in
c c e,o pX
t0-X"-acceptable
wr o w R
g
fcid
t ai°w
aabneymo
ll e " sre th
i a t,f b ei twe
e e n X°-acceptable ro ws, C l imp lie s
say
saR
rCO.
m ic So
u no
s use
t n can be made here o f the simp lifie d d e fin itio n o f
D
io a a
a e S5-control considered in 2.9.
lS4-control
h
v and
fw a
w
R
,
n
i
De
iclR s12.5.
ofin itio n . A mo d e l v i t
b
ft,w ia
wo
rld
i
s
i
n a./11„
n there
M is a wo rld Y F
e
s .i snM'-models
d t t where the relation R is symme trica l are B
hie
dels.
M'-mo
a
s o uadd eecls lwh
h e re the re la tio n R is tra n sitive are S 4
M
cu - .m
.tn
R is symme trica l and transi.c i -hM'-models
m of aft where the
Y relation
r
-dels.
tive
are
S5'-models.
-cih
m
o
f
o
r
R
n
o
easily rseen that in S5'-models R is re fle xive and
e
v I t is e
ot Remark.
so
S5'-models
a
re S5-models. A s i n S5-models R is re fle xive ,
nh y
are obviously S5'-models.
a
tS5-models
tr
oR
li
,X
°
B
-cc
oo
n
374
A
R
N
O
U
L
D
B A Y A RT
12.6, Definitions. W e g iv e ourselves t h e f o llo win g a xio mschema:
AC. El —a D u .
(the letter " C
- i ssystems M', B', S4 a n d S5' contain the f o llo win g a xio mtive
schemata
s u g g and
e deduction rules respectively.
M'.
The
same
as P plus AD, A C and NE.
s t e d
B'.
The
same
as
M' plus AS.
b
S4'.
y The same as M' plus AT.
S5'.
The
t
h same as M' plus AS and AT.
e Remarks. a ) Fo r the same reason as in 12.2, S5' cannot be
w
defined
oa s containing t h e same postulates a s M r p lu s A S T.
b)
—
Du .
r In S5
d we have 1—
In
" S5'
c weo have 1— De a as follows.
n
t— —
r D — Du (A S ).
(2)
o 1—l — " Da —› — 0 Du (A C ).
(3)
) 1—. — u ---> — D u (1)(2)(P).
(4)
T 1— D u ---> a (3 )(P).
(5)
( A T ) .
h — >
(6)1—
Du
a
(5)(4)(P).
e
So
d S5 and S5' yie ld the same theorems.
e
d 12.7. Definitions. M'-co n tro l, B
control
are identical wit h M-co n tro l, B-control, S4-control and
u .
S5-control
c
o
n
t rrespectively.
ol, S t l ' c
- c o n t r o l
a
12.8.
Definition-schema,
n
d
in wh ich the p re fix " X' " can be re placed
by
S
5 " M' ", " B' ", " S4' - o r " S5' ".
In
. every table F„ a ll rows are 0 - X
A
.- ro w R
ceptable,
i- ai cnc e(2)
p tf oar be lve
e ry
. constituent D e is F
a F w e D e hthere
falsifies
a is
v ine F
t „a
a
R
h n a n t- X ' - i
f
is
R iicn cF
ia
e
,
p
t
a
b
l
e
s
R
a
r n
X
o
w
ia
's Remarks.
u a) fo
c r the same reason as i n 12.4 S4'-control and
+n
.n
S5'-control
-h 1- - X cannot receive the simp lifie d d e fin itio n considered
in
X
' - h
ct .2.9.
a
c
o t
cn ae c c
p
t et ap t
b
r al eb l
ro e
o
l i
w
s f
Rf
i (
O N TRUTH-TABLES FO R M , B, S4 A N D S5
3
7
5
b) B y an induction on definition 2.2 one can prove that in every
table F
duction
o n d e fin itio n 12.8 one can p ro ve th a t in e ve ry ta b le
a
a l all
l S5-acceptable rows are S5'-acceptable.
S 5 '
- 12.9. Fin a l remark. The technique proposed in this paper can
be
the technique o f "clean truth-tables".
a ccalled
c
e p t
a b l
e
REFERENCES
r
o 1KURT G6DEL, E lne I nt erpret at ion des int uit ionis t is c hen Aus s agenk alk tils ,
w Ergebnisse eines mathematis c hen Kolloquiums , He f t 4, p p . 39-40 (1933).
[2]
s HENRY S. LEONARD, Two-v alued truth-tables f or modal functions, Struc ture,
a method a n d meaning, Essays i n h o n o r o f He n r y M . Sheffer, p p . 42-67,
New-York , 1951.
r
[3] ALAN Ross ANDERSON, I mprov ed dec is ion proc edures f o r Lewis 's c alc ulus
e S4 and v on Wrig h t 's c alc ulus M, J.S.L., V o l. 19 (1954), pp. 201-214.
S
[4] ALAN Ross ANDERSON, Correc t ion t o a paper o n modal logic , V o l .
5 20 (1955), p. 150.
-151 WILLIAM H . HANSON, O n s o me a lle d g e d dec is ion proc edures f o r S 4,
J.S.L., V ol. 31 (1966) pp. 641-643.
a[6] SAUL A. KRIPICE, Semant ic al analy s is o f mo d a l l o g i c I . N o r m a l m o d a l
c propos itional c alc uli. Zeit s c hrif t f ü r mathentatis c he L o g ik u n d G ru n d c lagen der Malhematik , Vol. 9, (1963), pp. 67-96.
e(7) G . E. HUGHES and M. J. CRESWELL, A n int roduc t ion t o modal logic , London, 1968.
p
18] ALONZO CHURCH, I nt roduc t ion t o mat hemat ic al logic , V o l. 1., Princ eton,
t 1956.
a
b
l
e
.
B
y
a
n
i
n
-