SEMANTICS FOR DEONTI C LOGI C
WILLIAM H. HANSO N
1. I nt roduc t ion
In this paper I consider some deontic logics t hat are counterparts,
in a n obvious sense, of the alet hic modal logics M, t he Brouwersche
system, S4, and S5. I n section 2 these logics are specified and some of
their relat ions t o ot her modal logics, bot h alet hic and deontic, are
indicated. I n sections 3 and 4 I adapt t he semantical techniques of
Kripke [5] f or alethic modal logic to deontic logic. Specifically, section
3 contains a model-t heoret ic def init ion o f v alidit y and section 4 a
corresponding met hod o f s emantic t ableaux f o r eac h deont ic logic
under consideration. I n section 5 each deont ic logic is prov ed c onsistent and complete wi t h respect t o t he v alid f ormulas of its c orresponding mo d e l t heory , a n d t h e met hod o f s emant ic t ableaux i s
shown t o constitute a dec is ion proc edure f o r each. Modalit ies a r e
discussed i n section 6. I t is s hown t hat eac h deont ic logic has t he
same number of nonequivalent modalities as its corresponding alethic
logic.
2. Six deontic logics
We def ine six logistic systems based on an enumerably inf init e lis t
of propos it ional v ariables a n d p r i mi t i v e connectives f o r negat ion
(— ), c onjunc t ion ( A ) , and obligat ion ( D ) (
1
mulas
(wf f s ) o f these systems are as usual. The ax iom schemes and
) . Tofh inference
rules
e
w ef orl all
l - s ix
f osystems
r m eared contained i n t he f ollowing
list:
f
o
r
Al.
A2.
A3.
A4.
A5.
A D ( B DA )
( A D ( B DC) ) D ( ( A D M D ( A DC) )
(—B D
D
(A D B)
D ( A DB ) D ( D A D DB)
D A D A
(I) Familiar additional connectives are defined in terms of
in the us ual ways. I.e., A D B is defined as ( A A
-p
A- A
B
)
,
L
A
a
s
D
—
.
A
A V E
A ,
as
and EI
(
177
A6.
A7.
AB.
A9.
RI.,
R2.
D(DADA)
D (A D < > A )
DA DDDA
A D D A
i f H A and H A D B, t hen HP,.
I f H A , t hen H DA.
The s ix deontic (') logics t o be studied here are called F, D, DM, DB,
DS4, a n d DS5. They are def ined as f ollows (where R1 a n d R2 are
assumed f or all):
P is
D is
DM is
DB i s
DS4 is
DS5 is
Al
Al
Al
Al
Al
Al
—
—
—
—
—
—
A4
A5
A6
A7
A6, A8
A5, A8, A9
The names of the last f our of these deontic logics deriv e f rom t he
alethic modal logics M, t he Brouvversche system (hereaf t er ref erred
to as "B"), S4, and S5 t o whic h t hey are s imilar ('). I n fact, i f t he
axiom scheme
DA DA
(
1
)
is added t o D M (DB, DS4, DS5), t h e res ult ing system i s ident ic al
wit h M (B, S4, S5). The result of adding this scheme to F or D is also
M.
We n o w point out several relat ions bet ween t he systems jus t defined and others that have been studied or suggested in t he literature.
(a) Since none of our systems contains the ax iom scheme (1), each of
them is a result of f ollowing the suggestion of Kripk e [ 5] (p. 95) f o r
obtaining a deont ic logic . ( b) Th e systems DM, DS4, a n d DS5 are
(
logic
2 s in c e it does not c ontain A5. We s hall nevertheless do so in this paper
in )order t o have an easy way o f ref erring t o a ll the systems under consideration.
N
We
o use I T» as the name o f the system based o n A1-A4 since t his system
is fundamental
f o r (i.e., inc luded in ) a ll t he deontic and alet hic systems c ont
sidered
i n t his paper.
i
(c
$4 3
eand S5 are giv en by Lewis in 16], p.501.
)t
178
M
h
ia
st
g
i
it
vi
e
s
n
s
b
t
yr
ve
identical, respectively, wi t h t he systems O M+, 0S4+, a n d 0 S 5 + o f
Smiley [ 8] (
4
identical,
respectively, wi t h t he purely deont ic part s o f t he mi x e d
alethic-deontic
systems OM, OM' , and O M " o f Anders on [1]. Henc e
).
the
S mdecision
i l e procedures giv en b e l o w f o r DM, DS4, a n d DS5 apply
directly
t o t he deont ic f ragment s o f Anderson's systems. Th e o n l y
y
previously-known
dec is ion procedures f o r these f ragment s required
[
8
the
use o f procedures f or M, S4 a n d S5 o n f ormulas i n whic h t he
]
obligation
connective was expressed by Anderson's complicated def ih
a
nition.
(c ) I t is easy t o show, i n analogy wi t h t he results giv en i n
s
ssections
h 3-5, t hat t he doxastic logic o f Hint ik k a [ 3] (c hapt er 3) i s
t o t he system obt ained b y delet ing A 6 f r o m o u r DS4,
oequivalent
w
nassuming t hat t he v ariable a o f Hint ik k a' s connective B „ i s h e l d
tconstant (
5
wit h the systems called DM and DS5 by Fit c h [2]. Fitch's
hrespectively,
met hod has several features i n c ommon wi t h o u r use of
).
atree-proof
semantic tableaux.
t( d )
tF i n
ha l l y
3. Models
e,
sw
ee We giv e modellings f or the deontic systems of section 2 that closely
o
b t hos e giv en b y K r i p k e [ 5 ] f o r t h e c orres ponding alet hic
lparallel
systems
e (i.e., M, B, S4, S5). Specifically, we def ine a model structure
as
as
an
v ordered t riple (G, K, R), where K is a non-empt y set, G e K and
tr
R
is
a relat ion def ined ov er K. We also stipulate t hat G nev er bears
te
the
relat
ion R t o itself. I f no f urt her restrictions are placed on R we
et
call
t
he
model structure an F model structure. A model structure is
rh
called
a
D model structure i f it has t he f ollowing property: I f H
sa
iyt e there
then
K , is an H2 E K such t hat H
ID 1 a1 D M
H model
2 .
structure
W
e i f R is ref lex iv e ov er {I C-G
sture
}tca ( a structure
model
l
l
i f R is ref
a lex iv e and s y mmet ric ov er { K
D
-8
,
a structure
n d
if Ra is reflexive ov er ( K - G ) and transitive ov er
n G ) model
eDS4
K.
A Doamodel
structure
)m
m
,
d
D
e
B
l is called a DS5 model structure if R is transid
tive over
property: I f H
t K rand has
u thecf ollowing
ssD
IS
a (
,5 I
rtheorem
4
o f DS5. This is prov ed as TI 4 i n section 5.
ea ()
1
3
E
rter95T3) is, identicH
al wit h S4, assuming that the v ariable a o f the connective K
K
,
H
is
held
constant.
e )h
a
1
R
i a(I
H
2
,
t
d 6
179
e )iD
S
s
B
n
5
t ya
li
i t
Is
c i
o
C
a di
l neG
, 1tn
t
e
w
ie
r
c
e
and H
I RDS5 model structures are all D model structures. Henc e they all
and
H 3 the property that each member of K bears the relat ion R t o some
have
member
of K.
,
t An F (D, DM, DB, DS4, DS5) mo d e l f or a wf f A of F (D, DM, DB,
h
DS4,
DS5) i s a binary f unc t ion ( P , H ) associated wi t h a giv en
e (D, DM, DB, DS4, DS5) model structure (G, K, R). The v ariable P
F
n
ranges
o v e r wellf ormed s ubf ormulas o f A , wh i l e t h e v ariable H
ranges
ov er members of K. Th e f unc t ion 0 takes values i n t he set
H
2{T, F}.
R Assume that f or a given model 0 associated wit h a model structure
H
(G, K. R) t he value of 0 (P ,11) is specified f or all P t hat are propo:
sitional
variables of A, f or all H r K. The v alue of 0 (P, H) f o r a l l
3
subformulas
P of A can then be def ined by induc t ion as f ollows .
. If ( B , H) = ( C , H) = T, t h e n 0 (B A C, H) = T; o t h e r wi s e
I (B A C, H) -= F. I f ( B , H) = T, t h e n 0 (—B, I-I) = F; ot herwis e
t:1
H
m
) R H', t h e n 0 (OB, H) = T; b u t i f t here exists a n H ' s uc h t hat
H
u( R H' and 0 (B, H') = F, t hen 0 ( E B, H) = F.
s• We say t hat a wf f A is true in a model 0 associated wi t h a model
tB
structure
(G, K, R) i f 0 (A, G ) = T; f als e i f 0 (A, G) =- F. W e s ay
A is v alid i n F (D, DM, DB, DS4, DS5) i f and only i f i t is t rue
bthat
,
ein
H all its F (D, DM, DB, DS4, DS5) models. I t wi l l be s hown in section
5
e) that a wf f is v alid i n a giv en system if and only if it is prov able in
that
m
= system.
pT I nf ormally , t he modellings giv en abov e c an be ex plained as f ollows.
We take f K-G} t o be a set of «permit t ed worlds», and G t o be
h.
the
aF «real world». Th e f unc t ion 0 assigns a t rut h-v alue t o eac h
v ariable of a wf f A (and ult imat ely t o each s ubf ormula
spropositional
i
of A inc luding A itself) i n each wo r l d H. I f H1,112 6K, we int erpret
in
zHI
a R FI7 as 01-12 is permit t ed wi t h respect t o Hi » , o r «ev ery state
elof af f airs described by a proposition t hat is t rue i n 112 is permit t ed
din
l HI ». Hence, in v iew of the def init ion giv en above, «A is obligat ory
tin
y t he world H» (i.e., (T) ( E A , H) = T) is interpreted as «A is t rue i n
h,every world that is permitted wit h respect to H». The stipulation t hat
G
a
i does not bear R t o itself amounts t o saying that some states of affairs
existing in t he real wo r l d are not permitted.
tf
As
was point ed out i n section 2, t he difference bet ween o u r s ix
D
0
deontic
systems and the corresponding alethic systems (i.e., M, B, S4,
M
(
and
S5)
is t hat t he f ormer lack t he ax iom (1). A n exactly analogous
,B
D
,situation obt ains i n t h e model-t heoret ic approach. Specifically, ( 1 )
B
H
,180
'
D
)
S
=
4T
,f
o
r
e
v
e
Is equivalent to the assumption that R is reflexive ov er K. For suppose
that R is reflexive over K and that D A is true in some member of K,
call i t H. Then by the def init ion of 0 ( 0 A, H) and the ref lex iv it y of
R, i t f ollows t hat A is t rue i n H. D A D A is therefore t rue i n every
member of K, since H was chosen arbit rarily . Hence i f R is assumed
to be ref lex iv e ov er K, a deont ic int erpret at ion o f t he modellings
becomes unt enable. W e t h e n mus t adopt s omet hing l i k e Kripk e' s
alethic interpretations of his modellings f or M, B, S4, and S5 [ 5] i n
whic h K is a set of possible worlds, and «A is necessary in world H»
(i.e., (I ( DA , H) = T) i s int erpret ed as « A i s t rue i n ev ery wo r l d
(including H itself) that is possible wit h respect to H».
Consider n o w the differences among the deontic modellings t hemselves. I f no restriction is placed on R t hen we hav e the simplest of
these modellings, t he one f or F. Strictly speaking this modelling can
hardly be given a deontic interpretation since i t is possible t o specify
an F model i n whic h D AD O A is false. I ndeed a satisfactory int uitive int erpret at ion of F is dif f ic ult t o give. Since D A D A is v alid
in t he modellings f or each of the remaining systems, they may all be
properly characterized as deontic. A mo n g t hem, DM, DB, DS4, a n d
DS5 are s ignif ic ant ly dif f erent f r o m D i n t hat R i s ref lex iv e o v e r
{K-G
amounts
}
t o saying t hat ev ery proposition t rue i n a permit t ed wo r l d
is
i nalso permitted i n that world, o r that each permit t ed wo r l d is permitted
wit h respect to itself. Henc e if we t hink of a permit t ed world
e
as
a a state of affairs i n whic h ev ery t hing t hat is actual is permitted,
then
one of t he systems DM, DB, DS4, DS5 is t he deont ic logic we
c
want.
I f , on the ot her hand, we feel t hat even permit t ed worlds may
h
have
(wit h respect t o themselves) some undesirable aspects, t hen our
o
deontic
logic is D. Among the systems DM, DB, DS4 and DS5, DM is
f
the
t simplest. Addit ional assumptions about t he relat ions among t he
various
worlds — bot h real and permit t ed — giv e t he systems DB,
h
DS4,
and DS5. I n particular, by arguments analogous t o those giv en
e
by
f Kripk e [ 5] f o r t he v arious alet hic reduc t ion ax ioms , i t c an b e
shown t hat A7 is equiv alent t o s y mmet ry of R ov er { K o
G equiv
} , t alent
h a tto t ransAit iv it8y o f R ov er K, and t hat A 9 is equiv alent
ris
to t he special c ondit ion placed on R by a DS5 model structure.
m
e
r
.
T
h
i
s
r
e
s
t
r
i
c
t
i
o
181
4. Semantic tableaux
The met hod o f semantic t ableaux has been developed f o r modal
logic by Kripk e [4], [5]. We giv e here an analogue, f or o u r deontic
systems, of what Kripk e [ 5] calls t he «S-formulation» o f t he met hod
of semantic t ableaux f o r alet hic modal logic . Familiarit y wi t h t his
method is presupposed.
The t ableaux rules f or negation and c onjunc t ion and t he rule f or
obligation on t he right are t he same, f o r all o u r deontic systems, as
those given in [5] (
for
7 each of the six systems. For F t he rule is:
) . T Yl.
h eI f D A appears on the lef t of a tableau t
r u l i e ofs each tableau t
f
o , pr 2 u st u c A
h
o
n
For
ion t o assure t hat each t ableau wi l l alway s
o bD twe
l i tadd
gh aeas t ipulat
h
t
l
e
f
t
bear the relat ion S to some other tableau:
a t i ot n
o
Yl. Iif D A appears o n t he lef t of a t ableau t
left
n
i
S of each tableau t
t
, hpt2
o
n
t u st 9u c h
.A
t hSht ea t
t
e
t
2
l
e ,i 2
For t he remaining systems t he rules dis t inguis h bet ween t he ma i n
,
.
f
t tS . t
tableau and the aux iliary tableaux of a set. We use ,
I
f
a
h
< Y 1the
nate
m »part tof o
the rule
d that
e sapplies
i g -t o a main tableau and (4Yla» t o
t
h
e
r
r
e
designate the part of the rule t hat applies t o aux iliary tableaux.
e
e
n
s
d From
i Dm
si f we have:
f
eYlm.tn rSame as YI f or D.
oI tf D A appears on the lef t of an aux iliary tableau t
e
nYla. a
u lef t o f t
t rs A o n t he
hd t
, ptc u
i ta tnhat
such
a io S n t
Notice thatn we get the effect of reflexivity ov er the aux iliary tableaux
t
2
h
e
by specifying
that A is placed on the lef t of t
e ,l in eYla
i f f
t
i
w
o
a
n
f
The rule Yl. f or DB is:
.
Ylm.t Same
ye
as
a YI f or D.
a
c
s
u
Yla, I f D h
ha A on
b the lef t of t
A b ct(1) put
l
l
t
e
a
i e tableau t2 such that t
a p peach
ut; Sa nt d
( 2 )
a r es2
(
tn2
p xu
t
o ae
the7 role o f uKripk
e's necessity connective.
a n y
t
hi2,A si f
)
t tso su n c
h
e
W
182
h
e
l 2e.t
e
l
e
f
t
f wt 2
a
o
x
i fs
t
s
o i e
s
s
f t ;
u
n
d
a h a
m
n A
e
a ou
h
x ni
e
l ti
r
a hr
e
(3) p u t A on t he lef t of t he unique aux iliary t ableau t
such
a
t hat t
a S t
Here Yla gives
us t he effect of bot h ref lex iv it y and s y mmet ry ov er
i
the aux iliary, tableaux.
I t mus t be emphasized t hat clause (3) o f Yla
i f
can be applied
s uonlyc i f t he unique t
a s utableau.
iliary
c h h Failure
t h ato observe
t
t this restriction would give the effect
a symmetryaS
of
over all tableaux,
t
and this is certainly not what we want
in
i a deontic n
ilogic. s
a
l
sa u o x
a For DS4 nYIi isl asi follows :
Ylm.
on the lef t of a main tableau t
a
u Iaf DrxA yappears
t
ton t he
a lef t of each tableau t
t h A t h a t
, p u
2 t such
s lu cD
bno
2
left
such
et
a t hat t
Yla. Iut,if L
SI A t appears
S on t he lef
t t of an aux iliary tableau t
i
te ho ne t he
2
A
n lef t of t
, p xu
i.st2 such
tat na d
thatI t i S t2,ff
f any such t
rit2 te A
D
x
i
s
t
s
.
h
e
r
e
Notice t hat this
a
version
n s of YI gives the effect of transitivity ov er a l l
i
so
the tableauxttnof a set.
h For
e if a tableau t
a i s i napplications
subsequent
c eYI dwi l l put D A and hence ult imat ely A on
lset r oe d uf of
the
s left
u of ct .tw h
a
t
h
ao
t
t
at
.2 Finally, wefb give
l YI fSor DS5:
Ylm. Same
t
e
a asa Y l m f or DS4.
Yla. I cuf D A happears o n t he l ef t o f a n aux iliary t ableau t
a
i
,
tthen: (1)
a putb A on t he lef t of t
,
i
left
o fe eacha t ableau t2 such t hat t i S t2 i f any such t2
l2
exists;
; a nand
d (3)( put
2 D) A on the lef t of the unique tableau
w
u
u
t
ipt
In analogy ttLao the Iprevious
A cases, i t can be v erif ied t hat this version
of YI gives the
o( effect,nf or a set of tableaux, of the conditions placed on
h
R by a DS5 L
structure.
e
tmodel
h
e
Ii
We def ineA
t c los ure f o r tableaux, sets o f tableaux, systems o f t ableaux, and o
constructions
as in [5]. I t can be shown, i n analogy wit h
h
the proofs ofne Lemma 1 and Lemma 2 of [5], t hat the f ollowing result
holds (
tr
8
hm
(
).
model
8 s truc ture
ea (G, K, R) nev er bears the relat ion R t o it s elf plays a c ruc ial
i
)
n
I
183
t
o
s
r
h
a
o
u
u
x
l
i
d
l
b
i
e
a
p
r
o
The F (D, DM, DB, DS4, DS5) construction f or a wf f A is closed
if and only if A is v alid in F (D, DM, DB, DS4, DS5).
Hence we can mak e use of t he met hod o f semantic tableaux t o determine v alidit y i n t he modellings of section 3.
5. Consistency and Completeness
We n o w use t he met hod of semantic tableau t o prov e t hat a wf f
is a t heorem of one of our deontic systems i f and only i f it is v alid
in the modelling of that system. The consistency part of this result is
immediate, since it is easy t o verify t hat the appropriate construction
for each ax iom is closed and t hat the rules preserve v alidit y . I t wi l l
also be apparent to those f amiliar wit h [5] t hat it is sufficient t o prove
Case YI of t he Lemma o f section 4.2 o f [5] f o r each o f our deontic
systems i n order t o establish the completeness of each.
We begin by giv ing a number of theorems and derived rules of the
systems under consideration. Eac h o f these t heorems a n d rules i s
stated f or t he simplest system i n whic h i t holds (e.g., i f a t heorem
holds in both DB and DS5 we wi l l state it as a theorem of DB). Proofs
are given only where they are not well-k nown or obvious.
Notice t hat derived rules 1-3 and T1-T8 — all well-k nown in modal
logic — can be established in the very weak system F. Notice too that
theorems s uc h as T9 a n d T11 whos e proof s a r e t riv ial i n alet hic
modal logic hav e s imple y et nont riv ial proof s i n deont ic logic ( i n
fact, they are not prov able at all in some of our deontic systems).
Derived rule 1 o f F: Substitution of equivalents (subs =- )
If F-AruB, and i f D is t he result of substituting B f or one o r
more occurrences of A i n C, t hen 1–C-=-D (
F
9
Derived
) . rule 2 of F (DR2)
If 1–A DB, t hen E l A D OB.
Ti.
(
F
A
B
)
( 0 A
D OB)
role i n t his proof f o r t he system DB. Wit h o u t t his s t ipulat ion i t is easy t o
construct a DB c ountermodel t o A7. The s tipulation is apparently not needed
for the other systems, but since it is nat ural f o r deontic logic we inc lude it .
(
9
184
)
W
e
u
s
e
t
h
e
n
o
t
Derived rule 3 of F (DR3)
If A D B , t hen H O A D O B .
F
T2. 1— ( A A B )
( D A A DB)
T3. 1— <> (A VB) = (<>A V OB)
F
T4. I — E A D 171 ( A V B )
F
T5. 1 - 0 (AAB) D O A
T6. 1— ( A D B ) D [ D ( B DC) D E (ADC)1
F
T7. 1— ( A D B ) D [ D ( CDD) D D ( ( A A C) D ( B A D) ) I
F
18. ( D A A O E ) )
F
( E A A )
Proof: 1. E D (— AV(E A A)) propos it ional c alc ulus (pc )
2. < > E D O ( — A V ( E A A ) )
1
,
3. O E D A V O ( E A A ) )
2
,
4. ( D A A O E ) D O (EAA)
3
,
DR3
T3, pc
pc, def
T9. 1— E I A D D A
DM
By A4, A6.
T10. F. E ( B DC) D [ <> ( D A A B ) D O (AAC)1
DM
Proof: 1.
2.
3.
E
4.
[
5.
D
6.
(
7.
)
8.
( D A D A ) D [ ( E I A A B ) D (AAB)1
D ( DA DA ) D
D [ ( DA A B ) D (AAB)I
(<>D ( DA
A A BB) D
) O (AAB)
( B DC) D HA A B ) D (AAC)1
DA( B DC)
D
A
B E RA A B ) D (AAC)1
DI ( B C ) D [ O ( A A B ) D O ( A A C) ]
D ( B D C) D [ 0 ( E I A A B ) D ( A A C ) 1
PC
1, DR2
A6, 2, R1
Ti, 3, R1
PC
5, DR2
Ti , 6, pc
4, 7, pc
T11. D A D O D A
DS4
By A5, A8
T12. 1— ( L I A A O E ) D O (EA DA )
DS4
185
By T8, A8
T13. 1 - 0 ( D A A B ) D DA
DS5
By T5, A9
T14. 1— ( D A D A )
DS5
Proof: 1. 0 ( D
1
2. ( D A : D A ) D ( D A D A ) 1 , pc, def.
A
A 3. D ( D A D A )
A
9
,
pc, def, 2, R1
—
Case Y1 o f Kripk e's L e mma c an n o w b e prov ed f o r eac h o f o u r
A
)
deontic systems as f ollows :
D
Y1
( O DF, Case
A
A < >B e f o r e : (
,
After:
"
E I A A X A O (E
- - Justified
)l A
Aby) T8, pc.
,
D
A
A
A
O Y1 ( E
D, Case
A
X A 0
2
Subcase 1. Before: DA A X A < > E I A < > E 2 A . —
)
E
A
After: AD A A X) A < > (El A A
T
1
A
. ( E by T8, pc.
) . A .OJustified
5
A 0 E
2
Subcase
,
2 2. Before: D A A X
A
A )D A A X A 0 A
After:
p
A
A
. by
. A5,
. pc.
Justified
c
DM, Case Y l m
Same as t he t wo subcases f or D.
DM, Case Yla
Before: 0 ( D A A X A 0 E 1 A 0 E
After: 0 ( D A A A A X A 0 (E
2
Let B stand
<>E1A0E
A
)) DAA
l .A. . A.f or
AX AO
2 C stand
Let
f
or
D
A
A
X
A
0 (E
( E
lA .A. . A2by
Hence
) D, subcase
A
O 1, we have i—BDC, and by R2, 1— D ( B D C).
D
D
( E
A
A
)
2
From
this by T10 and RI we get 1 - 0 ( D A A B ) D ( A A C ) , whic h
A
.
.
.
)
DM
A A )
is
the
result
we
want.
A
.
.
.
(lc) B y «before» and «after» we mean the relev ant part of the characteristic
formula o f t he alternativ e set o f tableaux i n question, before and af t er t he
application o f Yl.
186
DB, Case Y l m
Same as the t wo subcases f or D.
DB, Case Y l a
Before: 0 L X A 0 ( DA A Y A 0 E 1 A < > E 2 A • • • ) 1
After: O P C A A A O ( E I A A A A Y A < > (E1AA) A ( E
2
Justified
A as
A )follows:
A•••)]
1. [ 0 ( E A A Y A O E I A < > E 2 A • • • )
(ELAAAAYA<> ( E
Case Yla of DM,
1 R2
2. E (X ) C)
c
,
R2
A A )p
A
3. I I [ 0 ( D A A Y A
0
E
1
A
0
E
2
A
.
.
.
)
D
O
D
A
]
T5,
R2
( E
4. L i ( D A D A )
A
7
,
pc, subs = , def
2
5. D [ 0 ( D A A Y A O E
A A )
I (Before) D (After)
6.
I
,
2, 5, T7, Ti ,
A
.
.
.
)
A0E
2 DS4, Case Y l m]
A. . . )
Subcase
D L Before:
A
] EI AAXA<>E1A0E2A. –
3
After:
, E I A A X A 0 A A E A ) A ( E 2 A DA ) A
4
Justified
,
b y T12, pc.
T
6
,
R 2. Before: ID A A X
Subcase
After: E f A A X A 0 D A
Justified by T11, pc.
DS-4, Case Yla
Before: 0 ( E I A A X A 0 E 1 A 0 E 2 A . . . )
After: 0 ( E I A A A A X A O (E
Analogous
l A to DM,
D ACase
) Yla, except that T12 is used in place of TB.
A
O
(
E
2DS5, Case Y l mA
D CaseAY l m f or
) DS4.
Same as
A
.
.
.
)
DS5, Case Y l a
Before: X A 0 ( D A A Y A 0 E
After:
1
X A E I A A 0 ( D A A A A Y A 0 (E
1A 0 E
A
2 D A )
A. . . )
(E
2
A
D
A
)
A
—
)
187
Justified by Case Yla f or DS4, T13, pc. This depends on T14 (i.e., on
the fact that A6 can be deriv ed as a t heorem of DS5).
All remaining steps i n t he completeness proof c an be established
exactly as in [51. Hence we have the f ollowing metatheorem:
A wf f A is a t heorem of F (D, DM, DB, DS4, DS5) i f and only
if it is v alid in F (D, DM, DB, DS4, DS5).
It should also be point ed out that the met hod of semantic tableaux
can b e us ed t o giv e dec is ion procedures f o r eac h o f o u r deont ic
systems. This f ollows easily f or F, D, DM, and DB, since each of the
tableaux rules f o r these systems eliminat es a connective. Th e rules
f or DS4 and DS5 do not all hav e t his property, but constructions i n
these systems can also be made to terminate by adopting a procedure
s imilar to that of [51 f or eliminat ing repetitive tableaux.
The relations between our six deontic systems and the corresponding alet hic system are expressed by t he f ollowing diagram:
S5
S
055
4
D
S4
D
DB
D
M
M
Here an arrow means t hat t he system f rom whic h i t originates contains a l l t he theorems o f t he system t o wh i c h i t points, a n d more
besides. I f it is not possible t o mov e f r om one system t o anot her by
f ollowing arrows , t hen t he lat t er system contains theorems t hat are
not contained i n the f ormer.
6. Deont ic modalit ies
Since each of the systems F, D, and DM is contained in M, it f ollows
that each has an inf init e number of nonequiv alent modalities. Simi-
188
larly, t hat t he number of nonequiv alent modalit ies i n DS4 (DS5) is
no s maller t han f ourt een (s ix ) f ollows f r o m t he f ac t t hat S4 (S5)
contains DS4 (DS5). The decision procedures f or DS4 and DS5 can be
used t o s how t hat a l l t he reduc t ion equivalences o f S4 and S5 are
theorems o f t he c orres ponding deont ic systems. I n part ic ular, t h e
converse of A8 and
0 - 0 - 0 - 0 A = D • - - D A
are theorems of DS4, and t he converse of A9 is a t heorem of DS5.
Hence t here are no more t han fourteen (six) non-equiv alent modalities i n DS4 (DS5).
Since DB is contained i n B. it f ollows that DB has at least as many
nonequivalent modalit ies as B. Th a t t he n u mb e r o f nonequiv alent
modalities i n B is inf init e c an be s hown as follows. Cons ider wf f s of
the f orm
DAD D
m
where A
D ,„stands f or a string of n El's. By means of Kripke's semantictableaux decision procedure f or B (giv en in [51) it can be s hown that,
in general, a wf f of this f orm is not a t heorem of B unless n?-m. As
an example, let n = 2 and m -= 3. The construction is t hen:
D EJA
DA
A
DA
A
A
Ell1ADDEDA
DODA
0 DA
DA
A
Obviously, whenev er n < m t h e l e f t sides o f t h e t ableaux i n t h e
construction wi l l alway s be «behind» t he right sides, s o that closure
wi l l not occur. Hence, i n general, a wf f of the f orm
189
DAm 0
is not a t heorem of B unless n m . Fr o m this i t f ollows easily t hat
the number of nonequivalent modalit ies i n B (and hence i n DB) i s
infinite.
It s hould be point ed out t hat an exactly analogous argument c an
be used to show that M has inf init ely many nonequivalent modalities.
This res ult was originally established b y Sobocinski [ 9] , us ing a n
inf init e ma t r i x o f Mc Kins ey [ 7] . Th e argument f r o m s emant ic t ableaux is s impler and, s inc e i t is based o n a n int uit iv ely plaus ible
interpretation of M, more nat ural than the argument f rom McKinsey's
matrix.
UNI VAC
William H. HANSON
Division of Sperry Rand Corporat ion
St-Paul, Minnesota.
REFERENCES
[1] A l a n Ross ANDERSON, «The f ormal analysis of normativ e systems, Tec hnic al Report No. 2, Contract SAR/Nonr-609(16), Office of Nav al Research,
Group Psychology Branch, Ne w Haven, 1956.
[21 Frederic B. FITCH, «Tree proofs i n mo d a l logic » (abs trac t), f ort hc oming
in The J ournal of Sy mbolic Logic .
[31 J aak k o HINTIKKA, Knowledge and belief; A n introduc tion t o t he logic o f
the t wo notions , Ithaca, 1962.
[41 Saul A . KRIPKE, « A completeness theorem i n modal logic ', Th e J ournal
of Sy mbolic Logic , X X I V (1959), 1-14.
[5]
,
«Semantical analy s is o f mo d a l lo g ic I », Zeit s c hrif t f ü r mat hematisclie Logik u n d G rundlagen d e r Mathematik , I X (1963), 67-96.
[6] C. I . LEWIS a n d C. H. LANGFORD, Sy mbolic Logic , 2 n d ed., N e w Y ork ,
1959.
171 J . C. C. MCKINSEY, ' P r o o f t hat t here a re inf init ely m a n y modalit ies i n
Lewis's system S2», Th e J ournal o f Sy mbolic Logic , V (1940), 110-112.
181 Timo t h y SMILEY, «Relativ e nec es s ity , Th e J ournal o f Sy mbolic Logic ,
XXVI I I (1963), 113-134.
[9] Boles law SOROCINSKI, «Note on a modal system of Feys-von Wright », The
J ournal o f Comput ing Systems, I (1953), 171-178.
[10] G eorg H. VON WRIGHT, A n Essay i n Mo d a l Logic , Ams terdam, 1951.
190