Logique dt Analyse 131-132 (1990), 311-329
STALNAKER-LEWIS CONDITIONALS:
THREE GRADES OF HOLISTIC I NV O L V ME NT
Claudio Pizzi
1. Introduction.
The aim o f the present paper is to give an analysis o f three logics o f conditionals — Lewis' system CI , Stalnaker's system C2 and a probabilistic
extension o f the latter which will be named C2Pr — in order to show that
each one o f them can be interpreted as a formal representation o f conditionals from an holistic standpoint, the differences depending on weaker or
stronger senses o f the term "holistic".
In Section 2, 3, 4 the notion o f deviance o f the conditional operator toward the material conditional and the strict conditional is introduced and
discussed with reference to each o f the above mentioned systems. Th e
notion of deviance is related to the one of collapse of explicit conditionals,
and it will be proved that the deviance increases moving from C2 to C2Pr.
In Section 5 the increasing approximation to the collapse of modalities which
may be found in the three different systems is shortly discussed from the
viewpoint o f its holistic implications.
2. Explicit and non-explicit conditionals.
System Cl has been axiomatize,d in a variety o f ways. The following axiomatization — where the capital letters stand for arbitrary wffs — has been
introduced by D.K. Lewis in 161 (p. 132):
PC. Every truth-functional tautology
A l. A 0 -0
,A2.A (
- - (A L H. A3.
A4.
1A
1 3(A) L H. B) D (A D B )
C
AS.
V (A A B ) D (A B )
(H ( ( A
.
A
A
B
)
)E
D
H
.
(
C
)B
E
-
312
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Rules:
Modus Ponens (MP) : I f H A and H A D B then B.
Conditional Necessitation (CN): I f H (B, A B2 A
A
B
) ((A
D B
C
t h e n
i
Definitions:
)
DA
A —
D
OA
( -A
eD A
0e -f
,-- LII=
A
0
A A ( A 0 - * B);
1- B
B
A
'1 A0
1
,-0- ,=
1- f,A For
=
*; sake o f simplicity, in what follows we will speak o f axioms rather
,[A, B o f axiom schemata and o f theorems rather than o f theorem schemata.
than
1; The
B
2 last two definitions are introduced by Lewis in 161 as definitions o f
1 )
=
"non-vacuous"
conditional operators. By virtue of them A 0=• B is proved
- A
D
to
be equivalent to 0 A D ( A 0 ,conditionals
e
Ithatt A D=>
i A sis not a thesis in any extension o f Cl, wh ile A
AB ) .
(-0 (
a
following
-,, Ae c uwffs
p
l are
i a theses
r i t y
B= L
1
o
f
()nt I I)o (A nD=• -B) D (A D ,a (A
B c)D - u
;• - 2)
vÀ
o
u
s
, (A
B )0 LA , 3)
,D (A
B 111=:
)
Ii B4)
( BA
)
*s ) ,D
(0 no one of the converse implications holds. No w, since we have by
-s ) D
while
A5
, A
1u D(A
= B
1c ( 0
,) ,(A A 3h A5)
B1 3 )
). 1 1
by
contraposition
we obtain
)
;I 1 D
t 1( A
E (A
l 0- i - 6)
,
B
)s ,
and
by
virtue
of the theorem
1 1
a CD
l ) (3 )
A s . 7)
D
o 1
B
s 0
)
i A
m D
p (
l A
e D
t B
)
STALNAKER-LEWI S CONDI TI ONALS
313
we further obtain
8) (
which
1 by definition of (,= equals
0
9)
A (A 0
,y
Since
A =
(
the3 above
B
result shows that, - 3 , < > = , D
B
A )
creasing
strength.
, i sD
mt p
a nl d
f o r
0
r Axiom
i e(e ls aA5tmay
i obe ncalled
s "law of conditional factuality" and symbolized
— It is commonly held that an intuitive justification for it is provided
by
CF.
o AA
f
, possible worlds semantics which is commonly associated to Cl.I n
by
d Lthe
e
D
B
particular,
Lewis
proposes to read " A B is true at a world w," as "all
FB
)
the worlds
most similar to w, at which A is true are worlds at which B is
, )
true".
) Since Lewis assumes that each world is the most similar to itself, we
B
mayDreason as follows: if A A B is true at w
,
true
i (at the world which is the most similar to w, at which A is true : so A
a
, BAB isi true
s at tw r u e
a
t
n
i
The
possible
worlds
interpretation
of
w D,
,
h
e
n
c
econditionals does not expressly
d . a reading of the conditional operator in terms of a necessary relation
suggest
i B t
t
between
the
i )
s clauses. But such an interpretation follows from the convention
proposed
by
Lewis of reading A 0—
h
, eB beathe
might
s case"that
i B".
f Since A El—
formula
be
ifeit were the case that A, it is false that it might
,i cB t e isqtouw
a read
l es (i)r "—
be
the
case
that
—
t o ( hA
1
e —
(ii)
1
B
"
.
B
u
casserting
3
a
)
s" ,ift it were
e the case that A necessarily it would be the case that
B"n
it n
t
u
i
t
i
v
ae
t
v hh
[151
Stalnaker
observes
e
l
y
lA In
a
t
.
t
e
rthat might must be intended as an operator
e
having
the whole statement and insists on the epistemic meaning
ti r h as itsi scope
t
of
it
(p.
143).
This
however does not affect the preceding analysis since "it
ss
is false that it might" is an operator which is also to be intended sensu
i e
s
composito in (I), just as the adverb "necessarily" in (ii).
oo
n
Thus would-conditionals instantiate a particular case of necessary implicaltion
y
f (they are "variably strict" conditionals, as it sometimes said) and mightaconditionals
9
instantiate the so-called relation of cotenability. Reading concditionals
) o inmthis key leads us to see that the law of conditional factuality
passerts
i any
c two true propositions are necessarily interdependent in the
i l that
as t
e
d
wa
an
yo
on
f t
314
C
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above mentioned sense.
Stalnaker's system C2 is obtaind by adding to Cl the axiom
A6 (A O H B) y ( A 111-0
which is also known as law of "conditional excluded middle" (CEM) and
by a well-known result (see Lewis [61) it can be proved to be independent
from C2.
In C2 A4, CF
0-0
B)nDd( A B ) and OE', we have (A A B) D (A ( =:. B), hence . a
10=t•
(CAB)ED M
we
have
0
A D ((A B ) D (A D B)) it follows that H A EH B)
1 a( A r
(A
D
B).
It
is
so proved that A4 follows from CF. But it can also be shown
Ae
that,
CEM, A4 implies CF. CEM in fact is equivalent to Br )given
e, l
1-h ( Ae O H
a t e
B), hence by contraposition (A A 1
n
d c of CF.
instance
113)
D
( A
1
ei
The
is then that, given CEM, A4 is logically equivD conclusion
0 we reach
3
Kn
alent
to
CF,
so
that
in
C2
we
cannot give up the latter without giving up
)A 1 1 3 ) ,
t the former. Anyone who feels puzzled by CF
also
h
i
c
h
D w
that
in C2
CF
h
.D
i
s
t
h
e
n
f
o
r c e d
s
( i
.moving
A D B logically implied by
t(Aea o also thenminimal
a
d property
m
iof having
t
A
c( fOH
a n oB(
n o lt
0
1
of Cl and C2 allows us to draw a commonly neglected
bAThe
l elanguage
o
)r-0. we m
distinction,
the
one
between non-explicit conditionals and explicit coni
0
ditionals.
o- n v ge A conditional is explicit when it is stated in conjunction with the
B
truth
d0 w or falsity of one of its clauses, and non-explicit in the contrary case
(which
)
is the case of what usually logicians call conditionals).
fDa r
,( It is not easy to draw the distinction between explicit and non-explicit
oy m
conditionals
in ordinary speech, since either their meaning or the particular
w
tA.
verbal
moods and tenses used in them often suggest that the speaker believes
h
DT
h
that
i- the clauses are true or false: so that it would be a redundancy, from the
e
h
speaker's
viewpoint, to explicit the truth or falsity of the clauses. From a
c1
a
x
a
formal
viewpoint however the distinction can be rendered fully clear. We
h
1 n o an explicit conditional to be a conjunction (5 A ( A OH B), where
idefine
c3
m
is
a for a wff in the set {A, B ,
I n a more extended sense
kto stand
o
t) s i
n
c) t
j
b. ( o
of
oI the basic laws of conditional Logic.
aS) (
i
si I A
n
inn
e
scC
h
d
ee
w
w
il
i
tl
STALNAKER-LEWI S CONDI TI ONALS
3
1
5
which for sake of simplicity will not be treated here, an explicit conditional
has the form b' A 5" A ( A 0-0 B), where 15' A 6" is a (possibly degenerate) conjunction whose first member belongs to the set {A, second
member belongs to the set {B, - IA
The
table
1
}a. following
} B
n d
w
h gives
o sa list
e of all explicit conditionals having A and
B as clauses:
1. A A ( A 0 2.
, BB A) ( A El3.
, -B- ,)
4.
A .A
The
1
( 3problem of giving a correct reading of I .- 4. will not be discussed
in these
A pages. In CI I . is equivalent to A A B A ( A Elcalls
, B(Eboth
) l afactual
n d conditionals
G o oin [31.
d mPollock
a n in [ I l l sees 2. as a correct
rendering
of
an
"even
if
"
conditional
(the
present author" has however
A
l suggested
the possibility of reading it as an "as if" conditional (see [9]). 3.
0
) .
is a full expression of what is normally called a counterfactual conditional.
-B
The importance of explicit counterfactuals can be seen in discussing the
,)
question
of the collapse of the conditional operator on the remainig three
B
operators we have taken into consideration, namely - - , 0=>, D . I t is
)
straightforward
to observe that in Cl 0- 0 does not collapse on any of the
other three operators. Let us call the following three metatheoretical statements downward collapse, upward collapse and intermediate collapse:
(DC) i - (A D B ),(UC)
1- (A 111-' B) ( A - Bi- )(A •(>= B) --, (A 111-* B)
3
(IC)
-- half
( Aof these equivalences (the one from right to left) is a thesis of
One
Cl, E
but rthe*one from left to right is not. Taking for granted that the collapse
B A) D D A cannot be proved in Cl nor in any of its extensions, C2
formula
and C2Pr, the fact that in Cl 111-. does not collapse on --), 0=t', D can be
proved in the following way:
1) From 1- (A D B) D ( A IDA),
we have H A D LIA.
, Bfrom
) which
o n e
2)
o From
b t 1a (A
i 0n - s
,iY B
- )D
(
(-I A A
3I) A
B
)
D
iA
3
t
)
fD
A o l l
(
o-) w s
1I.
A
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A and A and the one between (A v
H A D O A.
PI ZZI
- 3
A and D A we would have
3) If (IC) were a C l
- t hsuch.
be
e s iIn
s ,this case, by PC laws, also (A 0 - . B) D (A L I-. B) would be
a Cll thesis,
s o but( this wff is equivalent to CEM, which we already know to
be
- independent of Cl. End o f the proof.
1 It 0is also straightforward to prove that any extension of Cl containing the
A
collapse
of modalities contains (UC) and (DC). So having (UC) and (DC)
v theorems in any extension of Cl is the same as having in it the collapse
as
( modalities.
of
A
Therefore, the preceding result amounts to a proof that (UC)
and
0 (DC)- are not theorems in any extension of Cl.
.
Remark.
B
Notice
)
that (A Os= B ) D (A 3also
) Bnon-theorems
)
a n do f Cl. From
(
Athe first wf f we would have by transitivity
D
D
B
)
(A E l-. B) D (A D
(D (A
(
A
3
B A
) ,- 0h
A
1 1(eA
D
3 )A
n -c e
the
of modalities.
,hence
a 3nI) .d
4
l )
(E
X collapse
=-.I( f A
.
B
B
E ) h
B
)
T
a3.
)i= Weak
:s and
r strongedeviance of conditional operators.
e
i- n- es t oc
sw
u
It
1 seems
nBnt now
l d to be a natural step to extend the notions o f downward, upoa
di) a and
t
o
ward
intermediate collapse also to explicit conditionals. We will apply
these
D d three labels in an obvious way to instances o f the metatheorem
ne
(A ( A D - . B)) ( 6 A ( A B ) , where " 0 -4
eb
"eyA
, 0=t•". q
Looking at explicit conditionals we may conjecture that in any
conditional
" s t a n logic
d s the ffollowing
o r rules" should
D hold for every value of 15:
u-A
"
3
"
lI-" , s
1o (a*) H rA D B ) -= (A El-. B)(DC) follows from H (6 A (A D B ))
-A
(6 A ( A B ) )
1,1
3 (b*)
) AH A El-. B) _= (A (w
, 3 B ) ((6UA C( A) e
B
i f o l l3 oB w) ) s
)w
. fand (b
o
(a*)
r *) may
o
be
m summarized as the thesis that the collapse o f nonexplicit
e H conditionals
u
follows from the collapse of explicit conditionals of the
O
l. A kind. To argue in favour of this conjecture we may simply show that
same
( works
itd
(
for an operator
A
which is like El-. and i n having an intermediate
0
h
E between
l
strength
- A .a n d
3a
vA B
)
D
(
e
)
T
h
e
A e f e r
rH
- n c e
eO
1
iA
3
sD
STALNAKER-LEWI S CONDI TI ONALS
3
1
7
sal" implication L I -0 defined as 0 ( A D B ), where for we h a v e H
LIA D A , 1
-toEthese
A
axioms we obtain theorems, having E -0 as the only modal operaDtor, which parallel the ones having -3 as the only modal operator (see I ll).
LIn order
I Ato simplify the argument, let us list all the possible formulas exapressing
n what we have called the downward and upward collapses of explicit
dconditionals:
H
1 DC1
1 H (A A ( A D B )) -= (A A ( A El-• B))
1 UC1 H (A A ( A El-• B)) ( A A ( A (
3A B ) )
D DC2 H (
B UC2
(
)
-' A
-D A
1 A i - ( B A ( A D B ) ( B A ( A LI-0 B))
DC3
(
( A ( B A ( A E H B)) ( B A ( A -3 B ))
A
UC3
E (D
A DC4
B
A
H (
D )UC4
-0
(
D -1
) 13
B Let
•( us
A
1
1 3hypothesize that the symbol fo r Burks' operator " EI-0" replaces
)every
- occurrence
(B
A
A
o f " 0 -0 " in the table. Then we may reason as follows
. (i)I(D
) Each o f the wffs DC1-DC4 may be easily seen to be equivalent to
Tsome
) instance o f H ô A ( A D B )) D ( A D - 0 B). The four instances o f
B
A
h A0
)(A( A D
) B ) are
a -(
n A
-11) A A ( A D B )
0
k
2)
A B A (A D B)
0
B
s
- 3)
A
)3
1
4)
(0 A)A
B
A
1
( 1
) 2) are implied by A A B , 3) and 4) by - (-A
3
1) and
) A
contraposition
3
0
1
1 AA
- of EI-0, from the latter two implications we derive H (
-IA- -B-(D
-B .
1
T h u s ,
B
using
1b1 A A
) contraposition,
0
3
y
u(B A sA ) Di (AnEI-0 gB), which equals an instance of
)A
D
the
1 same
B
theorem. Taking any truth functional tautology as an instance o f
A3 we
)B
() have, fro m the latter theorem, 1 -B D [113, hence also H A D B )
)A L1-0 B).
D (A
( (H)- Every instance o f UCI-UC4 may be seen to be equivalent to some
- 3
1 B
1 )
3 )
1
A
318
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instance of t--(6 A (A E l-. B)) D (A 3 BB))) .D (AT Bw) o
D-*
and 1 -(A A ( A L I t a truth-functional
h ( e A m
,o B )f then
taking
)
ADas
tautology,
we obtain H (B A L i B)
a
r
e
3LIB and HB L I B) D .D B respectively, which are both equivalent to the
(
F
thesis
rH LBI B
o D LmI B , which entails H A B )
( A -313). The other
tA instances,
h
e via contraposition,
m
,
two
yield (
(-A
1aI 1
result
3
which
A
is
( analogous to the preceding one (namely H L I -1LI
A 1 -31instance
) B1 ) 3 of B.
1
E As
l for
- the latter, take I,-m
A
f
o
r
B
.
a
S
i
n
c
e
H
ID
n -A ) )
Ia
LIA
D
ydA ( A I (
B)
-A
,3a
( The
b
B preceding
) ,
result concerns an implicative relation which is simp ly
1A
- nB das-having
e
3 a modal status which is intermediate between D and-3
asn
defined
1
1i it plausible
)
- to conjecture
.
I hmakes
p
d
to
and
that a normality requirement fo r any
rA
L
sa
implicative relation intermediate between D and A
elio
property
3 sn h described
o u l d in s(a*)a and
t (b*).
i s As
f ya further remark concerning D
-dts-v( ' , usslook
let
h
ate(a*) and (b *) in the contrapositive variants : fo r every
'e
to
1 (a**) 14(A ( A D - . B ) implies 14(6 A ( A D B )) ( 6 A ( A Ell-q3))
d
A
hL
1 ()* * ) 14(A 0 b
eIA
-y3 , r
pA
ItE 1 3 )the two premises are true, (a **) and (b **) amount to saying that the
D
oSince
lconclusion
a
A
oL
( Af
o f the rules also hold. But this is very plausible at least in the
I.,k-paradigmatic
case o f first-degree conditionals, where ô stands for a truthfunctional
formula.
i, 3
w
A
- B
n
.e
Suppose
) in fact that the conclusion does not hold. Th is means that we
I i mderive
rg
W
could
p as theorems (6 A ( A D B )) D (A L I A
a
e
, l BDi ) (Ae a- n d
(
6
A
B))
t) s B A
a
(h
D A D- B o.r with A EJ-0 B may entail the modally
3
truth-functional
) .
ô with
cr) 1 u 4
a
B
stronger
tA D
u
vh
i (We
3
B
t stipulate
1
.
to call normal any operator, intermediate between D a n d ,
t(iwhich
e
3- 3 s has the property described in (a*) and (b *), and deviant any such
- A a lacking the mentioned property. The deviance can be weak and
h
h
operator
-1
re
( d AThe deviance is strong when the premise o f rule (a*) o r rule (b *)
strong.
tf1
n
tisDnot only
supposed to be a thesis o f the system under discussion but is
3
u
ch
oactually
.
such, while the conclusion is not. The deviance is weak when it is
-not
n
se
1 strong,
3
namely when the premise is not a thesis o f the system under
sc3 )
a
e
)
it-e (
g
siIh 6
a
AA
i(o
o
-) (
n
w
.t A
,Ia
lh iA
STALNAKER-LEWI S CONDI TI ONALS
3
1
9
discussion but is simply supposed to be such.
A more rigorous treatment of the notion of deviance may be given as
follows in reference to the operator 11H.
We shall say that 111(i),,some
i s value of 6 can be found which in X provides a counterexample to
(a*)(b*),
s t r o n and
g l y
(ii)d the
belongs to X.
e premise
v i aof (a*)((b*))
n
We
t shall say that EH is weakly deviant toward D (--3) in a system X if
(i)t some
o value
w ofacan be found which in X provides a counterexample to
(a*)((b*)) and (ii) the premise of (a*)((b*)) does not belong to X but to
r non
d trivial axiomatic extension of X.
some
D
We have to notice that it is possible in principle to assign a numerical
(
- the degree of weak and strong deviance, which of course is
measure
to
3
) amount of values of 6 which provide the counterexamples to
given by the
i given rules, but this possibility will not be taken into consideration in
the
n present paper.
the
a
s
y
3.sDeviance
t of El-* in Cl and C2.
e
m
The
problem
we have now to face is the following. Is 111-. a normal or
X
deviant operator in Cl and in C2 ? The answer is that it is deviant in both
i
systems, and it is interesting to see why and to which extent.
f
We may state the following theorems concerning Cl, the third of which
follows from an established result:
T i. In Cl 111-* is strongly deviant toward D
T2. In Cl Er . is weakly deviant toward -3
T3. In Cl El-. does not collapse on
The proof of the first two statements is the following:
TI D C 1 turns out to be a Cl-thesis via CF
. ( sA
uA
f f (i Ac EH
e B)i andt A A ( A
t D oB) equal A A B).
T2.
o bU Cs2 and
e UC4
r v aree not Cl-theses but, if they were added as axioms
to Cl, athis would
not yield the collapse of modalities. The proof is
t
h
t
semantic
and
is
obtained
by showing that UC2 and UC4 hold in every
b
o
t
h
Cl-model having two possible worlds, while it is well-known that the
collapse-formula A D El A is not valid in every model of this class.
320
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A Cl-model is a 3
worlds,
selection
function with the properties stated by Lewis in [6]
- p i e f is( aW
,
(p.f 58), for Cl-models, V is a standard modal value-assignment with the
additional
V
) property V(A B , w) = 1 i f and only if, for every w
that
i w
sw
u hc h e r
w
i eE
may reason as follows:
jf We
W
i( iA
s
f,1)aLet (
be
- , by Reductio false at a world w, of some Cl-model (W, f, V) such that
,w
n
o
n
W
f,A- = {w,, w
I , V(III(A D B), w, ) = O. We have to consider two possible cases:
lo=
)Ae
mdR wp
either
}r( a nw,
, t
iw
w,
o)=0,
,r y but this is incompatible with V (
sVAs B),
-n
Io1 ov7 t w) = 0 implies V(A D B, w
( ej = O. Since V(A, w) 0 , this means that f(A, w,) = f w
w
1
I
.
ym
1 A -, , w )
B
V(A
B
,
w)
=
1,
this
implies
V(B, w
t
)
i
=
0
,
IR
w
,
T
1 C
. hfl
,in
ie
o
i nV c e
w
R
idoes
B hjs ae nn d c, we s follow
( think for instance of a Cl-model such that W
A
=
)j- f (= notAnecessarily
I .,
m
e
)
,Im
= p{w,,
o) nwto r a d i c t i o n .
C
,.adding
E
l Cll implies that Cl + UC2 is sound with rea
n V as (an axiom
to
w
.N
d)j se=l oUC2
t
i
c
e
s
(Isspect
A
class of models having no more than two possible worlds. But
, i I tohthe
sD
tjR
a
t
b
n
V
a
is a countermodel to the collapse formula A D
E
wtwo-worlds
,a )( e nmodel which
ih0
l
a
d
t-LIA
h
can
easily
be
found
by
a Reductio argument.
fsv(i n
i
V
e
2)
Let us suppose that (
1
(inA
w ghA
, at a world w, in a Cl-model in which W = w
-l,D ( a UC4 B
equals
— is false
A
i c
n
m
and
tj1
1 3wt A
w
B
,to
h
implies
ij(e
,w
)V Ar V(A, w) = 0 and V(A D B, w) = I . Since V([1(A D B), w,) =
r0,
Ic1
1 - that
0 V(A D B, = 0 , and the argument runs as in the
),)( vit1 follows
o
e
preceding
case.
End of the proof.
yB
a
)
)
=
)dt-ATo sum up, 1) and 2) prove that UC2 and UC4 do not yield the collapse
T
sD
Iaa
u
hD modalities. Since Cl-theorems are validated by models having an arof
h
e
,(
0
,bitrary
number A
of worlds, it follows that UC2 and UC4 cannot be C ln
B
ciaw
sD
V
ftheses
d
since they are refuted in Cl-models having more than two possible
e
n,
sB
i(f
worlds.
So we conclude that in Cl D idtw
m
n
)(,w i s n o t
towards
tbw
i
e
-cA
se
) t r o n g l y
yaoh
e
B
Remark.
b
u DC tI, UC2 and UC4 are the only instances of wffs in the table at
a
sndiI
w
)w
p.
317
ewhich
a f can
k be added
l
y to Cl as axioms without yielding the collapse
stica
s,fh
d
e
v
i
a
n
isn
V
R
w
h
tlpe
d
(w
i,q
c
yuiV
B
„)u
h
sln(
,Va
e
a
cA
w
(l
e
t)D
(s
STALNAKER-LEWI S CONDI TI ONALS
3
2
1
of modalities. It can he shown that from the supposition that every other wff
of the table at p. 317 is a thesis, the collapse of modalities is a straightforward consequence. The proof will be omitted since it simply rests on suitable choices o f instances of A and B. Fo r instance, in DC2 we can take B
as A , so to obtain (
-I1ALet us
A turn now to C2 and to its characteristic axiom expressing the law
(A conditional
of
A
excluded middle: (A 0 - • B) v ( A D-0 A
D 1 3 )to( derive
1
C- E ,Mfrom
) . CEM
I thet equivalence
i between
s
simple
111-0 and 0=1
. formula
the
) )
expressing
(
the intermediate collapse (IC), j. e. (A E l-' B)
,A
(A
,- n0=:.
a B).
m eThe
l y proof is simple from left to right since we have among the
I theorems
C2
A
0 A D ((A EH. B) D (A 0 -0 B)), which by permutations of
A
antecedents
gives (A 0 -0 B) D ( 0 A D (A 0 -0 B)), i. e. (A 0 -0 B) D (A
(0=:• B). In the converse direction: CE M equals (A 0 - • B) D ( A 0 - • B),
and
A since we have also 1v ( A
0 El-•
1 AB), i. e.D(A ( =t. B) D (A El-• B).
(1 Having
A - (IC) as a C2- thesis obviously yields a collapse of every explicit
1
conditional
- -0
0 -0 on the corresponding explicit conditional con0 1 1 containing
B
taining
) 0=:., ,namely
every intermediate collapse of explicit conditionals. On
I
A
)
o
) other
the
, n hand,esubjoining CEM to C l does not modify any result already
the deviance of 0 -0 toward D and toward fhreached
u inre Cl about
t
3
F
r
o
m
h
a
e
n semantic
cr viewpoint C E M expresses th e so-called "Stalnaker's A s te
h
sumption",
i.e e. the view that for every possible world w, there is only one
o
r
e
world
w
m
j TI and T3 is unaffected b y this semantic property o f C2-models. To
of
si u c
conclude,
the following metatheorems concerning C2 are provable:
s
h
t0 hT4 In C2 E1-0 is strongly deviant toward D
A
a tT5 In C2 El-' is weakly deviant toward ( w3
T6 In C2 CIi ,
) c o l l a p
4. The
=
of 0-0 toward s edeviance
s
f3 io n nC 2 P r .
(Looking
T1-T6 it is natural to ask the following question:
• •at metatheorems
•
w
is it =possible to find an extension of C2 which does not contain the collapse
) modalities and in which 111-• turns out to be not weakly but strongly
of
.
deviant
toward I3more? specifically in the following way: is it possible to find a system presertT h e
i u e s t
q
is o n
sw t
e
ra a r
i g
e
ch t o
n
322
C
L
A
U
D
I
O
PIZZI
ving the properties of Stainaker-Lewis conditionals in which I71-. collapses
on 0 =
, table at p. 317 imply the collapse o f modalities? The answer is in the
the
•,
affirmative.
We take into consideration a new system, which we shall label
D C2Pr,
as
C
whose language is a probabilistic extension of the language of C2
obtained
by adding to the language o f C2 the language o f standard real
1 ,
number
theory and a sentential operator Pr which assigns to every sentence
U
in
C the language a rational number in the interval 10, 11. The axioms o f
C2Pr
are the axioms o f a minimal f i
2
rwith
s
t
o r d fo
e r rcontingent identity (
axioms
,
fUi o r m u l a t i o n
f2
Assuming
definition
)o
e
o m
s
f L I Ao = r C, a x fi the
2
IC
A
1
7
1
*
,
lt4
place
hthe modal
e
versions Aof Kolmogorov
laws fo r Pr, i.e.
e
x
t
e
n
d
e
td
r
y
a hh ee o
d
lso
K l. 0t P
f t r ( Ae ) r I
r a
a
K2.
r
I
f
H
o
r
d A e=e B then
r Pr(A)
e = Pr(B)
e
in
K3.
-3
n
0
(A
A
B
)
D
P
r(A
V B ) = P r(A )+P r(B )
d
t
td
h
e
h
fse these
To
i we subjoin
r
the
s following
t
axioms:
o
Pr(A A B )
m
r K4. 0 A D Pr(B/A) Pr(A)
e
K5.
P
r
(
A
)
1
n
m
e
s K6. Pr(A) 1 D L I A
w K7. P r(A A C ) 0 D (P r B/A A C = P r A 0 - • B/C) e )
lh
a
i K7 is an object-language rendering o f what is known as Generalized
w
lStalnaker Thesis — henceforth GST f r o m which the simple Stalnaker's
Thesis
Pr(A) 0 D (Pr(B/A) = Pr(A 111se
,it substituting a logical truth to C. R. (Stalnaker has proved that the logical
by
•n
dependence
e nholds
c ein fthe
o converse
r t h direction, since it is possible to derive
h B ) ) h also
S
T ST (see 1131).) According to the probabilistic notion of necessity,
veGST from
—
o
r
fle
o
l
l
o
w
s
vm(
ia
of2modal operators by using an argument of the Quine-Follesdal k ind (see Smokier 1121).
n
iIn) order to have a rigorous dis proof of A D E A in C2Pr, however, we need a semantic
which will not be treated in this paper.
A
g
nargument
r(
b
i e
possibility
with non-zero probability and necessity with max imal probability.
o
n 3s
)
tg tA
h
s rx
P
t ii
ra co
t
a
t m
e
s
n
eK
d
STALNAKER-LEWI S CONDI TI ONALS
3
2
3
which has been axiomatizekl in C2Pr, it is appropriate to see GST as a computational device such as to guarantee that every C2-thesis has the maximum
probability value, and in the light of K5 and K6, this amounts to saying that
it is logically necessary (
rise
4 to the so-called " L e wis' Trivia lity Result". A modal version o f this
theorem
in C2Pr in this form:
) • I t is derivable
i s
w e l l
k (TR)
n o( 0 (A
w A nB ) ,A 0 (A A - h 1o1 3 )w)
eD v P r
This
B , / Aopens
)
a new problem which is the following. Is it possible,
e ( rtheorem
moving
new theorems, not included in C2 , which
r
t = hfro m (TR),
a P t o prove
involve
only
truth-functional
and
conditional operators?
(
B
)
t
The
answer
is
in
the
affirmative
since we may prove the following two
G C
S)
Critical
Theses:
T
g
i
CT1:
v
eI
CT2:
A
s
D1 3
1
( (
D
(A (
( A
E
semantics
4 1 1defined by making an essential use of GST.
() 1
- I5
0 .
4
(1))n Pr(A)--- D DA
K6
S -B- ,
(2)T
(1)
) DA D ( P r (
tI A
Eh
(3)
O
(2), - ,
a
-( ,P
(4)
e rD
A D (Pr(A) 0 )
(
3
)
,
A)=0
/ A ,= Pr( P r(A
l) 10
(A
A
(5)
p
)1
E)( A A B) A 0 ( A A 1 AD
) = e 1Hypothesis
(6)
11
rn 3Pr(A
f
(5), (4)
l) A B) 0 A P r(A A - ,
7
1
(7)
a )Pr(ALI-43/B)
0
= Pr(B/ A A B)
)3
o
GST, (6)
(
1
(8)ko Pr(B/A
A B )= I
(6), P r(A A B) 0 , S T
(A
(9)fe Pr(AEH.
B/ B)=1
(
r A
DP r ( A D - 4 1 /
(10)
o
Pr(B/A 7
- f1 D
(11)
Pr(ALI-43)=Pr(A111-*B/B).PrB + P r ( A E - 4 3 /
B
43)
)
-1,1
T
B )B
= 0Probability
,
,
Calculus
)= 0
4) . P r (
( R
B
- G
S
(
)
6 i11 Pr(AEI-•13)=1.Pr(B)
1
3 )
(12)
+ 0.1
T
8
C )Pr(B/ A)=Pr(B)
3n
(13)
(
1
2
)
,
)
ST
C ( 0 (A A B) A ( A A - r(14)
(2
i 3)) D P r ( B / A )
2
-1
1
sP3result
P
1
=
1
)r ( B follows
)
The
by using Deduction Theorem, but it is possible to ex hibit another proof
without
( rp
5using
9 this
) device.
)
, rt
( ro
11
3 0
v
) u
,
e
( n
1
s
1d
)
t
a
so
fb
324
C
L
A
U
D
I
O
PI ZZI
The proof of CTI and CT2 is given in the Appendix. From it turns out
also that CT2 is a simple corollary of CTI. But, as we already know, CT !
and CT2 are equivalent to UC2 and UC4, and it has been previously proved
that they are not C2-theorems. Thus it turns out that C2Pr has the property
we were looking for: in it D-o it is strongly deviant toward —3
Since we already proved that UC2 and UC4 are refuted in some model
having more than two possible worlds, we may conclude that C2Pr is characterized by models having two possible worlds at most. I f a proposition
is conceived as a set o f possible worlds, this means that the only propositions are 0 , I w
one
D. K. Lewis draws from his Trivia lity Result (
i
6
) , The
t wresults concerning C2Pr may be summarized as follows:
)i .
In C2Pr D-. is strongly deviant toward D
l , T7.
I w
t T8. In C2Pr D-. is strongly deviant toward —3
, T9.
w In C2Pr D-. collapses on 0=:.
i
) In. C2Pr, D C ! , UC1 , UC2 are theorems, wh ile nothing change in the
T
already
h mentioned proof that the remaining wffs o f the table o f p. 317,
added
i
s as axioms, yield the collapse of modalities.
c o n
c l u s
5.
conditionals as holistic conditionals.
i Stalnaker-Lewis
o n
i
s is useful to recall that the theorems which in Cl. C2, C2Pr respectively
It
allow
us to prove the collapses mentioned in the receding sections are the
i
following:
n
a
g
r CF:
e (A A B ) D (A B )
e CEM:
m (A B ) v ( A 111-. 1
1
CT
e
nI:3 ) A D ((A B ) D (A —
t 3 B ) )
w it is straightforward to see that they are essential to the relevant concluand
i
sions.
The intuitive meaning o f each one of these wffs may be seen to be
t expression of an holistic view of the relations holding among propositions
an
h
t
h(
incompatible
sentences, then we may call it a triv ial language. We have proved this theorem:
e6
any
) language having a universal probability conditional is a t riv ial language ([7], I). 3 0 0 ).
I
n
h
i
s
w
o
r
STALNAKER-LEWI S CONDI TI ONALS
3
2
5
of suitably defined classes. I f we agree with the idea that conditional imp lication is a kind of necessary implication, CF asserts that any true proposition has a necessary dependence on any other true proposition. CEM yields
what we called Intermediate Collapse (of E -. on ) and asserts that if
any proposition A is (non-vacuously) cotenable with another proposition B,
B has a necessary dependence on A. Bas van Fraassen in [161 sees Stalnaker's logic as a logic suitable to "hidden variable theories" ("Stalnaker
models are Lewis models with hidden variables." 1161, p. 176): and since
Ei-0 has no temporal connotation, a possible interpretation of van Fraassen's
result is that Stalnaker's logic presupposes an holistic conception o f events
which are described as cotenable.
As regards CT1, it must be noticed that it implies via CEM
CT1'
D
(
which
1 (asserts
A
that every proposition B cotenable with a false statement A
E I -implied
0
is strictly
by it. Notice that moving from CT1' we can perform the
following
- , steps:
1 3 )
D
(1)
D
(
-(
(2)
D
( 0 (A A - ,
A () A- D
1
B
(3)
( A
[-(4)
IE
A7 (l 1 -- . .
3
D
- - ,
) be
) read b y saying that every B consistent with any false A is
(4)I(BA
may
)
(A
0
1
) it. CT2 gives also rise to a meaningful result via the
logically
A
implied
by
following
)0
proof:
(,B
)(A
T
) (1)
A
-C 1 1
A
(2)
T
1
3
B 1 B D ( 0 (A A B ) D (A El-. B))
(3)
( 2 ) ,
CE M
1
D
)
3
' may be read by saying that every A consistent with any true B con()3
(3)D
)
(D
ditionally
implies it. We cannot have a theorem analogous to (3) having-3
D
A
(
0
in place
of EH. since instantiating A by any truth-functional tautology would
Ato (B A 0 B) D B , hence to the collapse o f modalities.
lead (B
)- may conjecture that ro w (4) o f the first proof and ro w (3 ) o f the
A
We
- proof, even if they do not imply the collapse of modalities, together
D
3
A
second
-Bthe strongest approximation to hegelian holism which is expressible
state(3
A
)1
-(
1
1
3
3
B
)
),
D
-)P
(
-C
326
C
L
A
U
D
I
O
PI ZZI
in a logical calculus preserving modal distinctions.
The three Stalnaker-Lewis systems discussed in the preceding pages are
interesting contributions to the formalization o f reasoning moving from
holistic presupposition. It is clear however that anyone not willing to cornmitt himself to such presuppositions can accept neither the three formulas
CF, CEM, CT I nor the approximations to collapse which are provable by
their means. The anti-holist has then to choose between working with weakened axiomatic bases of the Stalnaker-Lewis family or turning to an alternative conception of conditionals such as to avoid any trivialization of both
explicit and non-explicit conditionals (
7
).
ACKNOWLEDGEMENTS
The author wishes to thank Peter Gardenfors and Wlodzimierz Rabinowicz
for useful comments to a preceding version of the present paper.
APPENDIX
To reach theorems CT I and CT2 we first prove:
L i ((A —
3 B )
vin the following way:
( A
—
(I ) 3(2)
1 (A 111-• - -1
( AB- )
D
1(B (1 A
3(3)
) )(A )—
B
)
D
D (A —
3
(4)
(((5)
B B
3
((A) —
(A
(3
) - ((A EH* B) D (A B ) )
D
A
A 1( 3 )
—
(D
1
0yA
1 (
(3
-B
1
([A 1
•)-A
0( 1
tigations
on conditionals: see for instance Kvart
B
-)0 7
—
holistic
)
account of conditionals see Pizzi [IO].
)3
*BI
B
D
)B
)t
(D
)Di
A
(s
)(D
—
(At
A
o
3-—
Ab
B
3
-e
3
[5] and
1 (1),
A D ((A L I ---*
1 -D
1A
11 3 )
(D 1- (A —
(2),
( A
3A
BD
) (13 D A)
A
E
D
1 A(4 )
(D
(3),
[-B 1 ] ) 3 )
44
3
)
Nute )[81 F o r an alternative anti,
STALNAKER-LEWI S CONDI TI ONALS
327
We also need the following theorems (L2 and L3)
L2 ((A E l-. B) A - ,
(A 3
which
Bmay) be
) proved in the following way:
D
(I ) ( 0r ( A (A BA) A ( A A P
P r
H 0 (A A B )D 0 A
11 1 31) ) ) D ( A
0
0
0(2)
((A B ((P
(3)
= ) A P(P Dr R)) D (R V Q )
P
C
B
3
- ) y )Q
B 1((A
=(4)
1
3 )111-0 B) V ,
1T
(V A
- 3r
(A 3B
(,
) )
3
S Let(us call
1V
A
B Tfo)r-sake o f simplicity C the wf f (A 111-0 B) D ( A -3 B ) and
,let us rnotice
B
P
( that
A 1 L1i equals
)
B
)
3 )
) i ' ((AP El- 0 B)
=
r D
L
A -1 (A -3 B )) D ( 0 ( A A B ) A 0 ( A A v1 B ) ) :
B
(
Then
we have:
(P
2
(
)r
,
>
( L3
A
AC
3
1
D
)0 proof is the- following:
D
The
(,,
0 ) ( 0 ( A A (A
L
P
(I
( A B )) A O (A A - ,
(riBA Pr(A
A A BB) A
) )r
o Dw
(
I
)
o f prec. proof, A A B /B
)(2)
B
P ( r0 ( A
( AA BB) A 0 ( A A ( A A = 1 3 Pr(A
D
1
) -) AD0B )
P( r 1 ( A
)
,
H 0 (A A (3)
C
2
(P ( A
1A
1E
A3H) (A
D ) A - B ))
, ( A 0 - * B)
r
A
(4)
( (( 0 A(A A B
, ) AA 0 ( A A - , B
)
(B
(5)) )( ( A )D)0 -0 B)
)P( A) r - ,( A
AA -- 3 •
=
(0
> B
)
L i ' , L2, (4), PC, TR
1 )P ) r ( AA
A
=
B
Thus
)
B
we are ableD to prove the two lemmas L4 and L5:
A
D
),P
B
r X B 2(
)
L4
,)T Cv (B D A) A
=
( PrI3r 0 3 D (P
P
(1)
B rB =)P r(A A B )) D P rB /P rB =P r(A A B)/PrB
(
A
)
x ( x = y D y/ x = x/ x)
A O B D (Pr(13)=Pr(A A B ) D P r(B A ) = 1) ( 1 ) , K4, K7 (ST)
(2)
B
)
328
C
L
A
U
D
I
O
PIZZI
(3) 0 B D (P r(B )=P r(A A B ) D (3 D A ))
(2), 1 - D A D A
(4) B D (B D A)
(5)
1 0(4), L3
1
BC
D
L5 ( (PC V (1)
L4
1
1
(2)
3 V -r 1 C
(1)
3
V
1
3
(1 1
(3)
1
C
y
(B
D
D
B
)
y
(
A
(2)
D
yA
3 ) D - . (A B ) ) D (A ( A 111-> B))) V ((A E l-. B) D
-D
(4)((A
D
= A DB(ADD - . B)) y (
1
-A
(3), A LI-.13/B
)P
-y1
'
r
C
A
)
(5)((A 111-0 B) D (A -(3
(I A) ) D V 0 AB
(4), C2
1
(A (-10 0
(6)((A
AB A) )D (A (5), A vA A
A
' ) )
)B B
B
3
y
yD
)Hence
A
(
T i simply follows by the following steps:
A
D
-( D
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REFERENCES
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pp. 363-382
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4, 1975, pp. 133-154
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W
STALNAKER-LEWI S CONDI TI ONALS
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[3] G o o d ma n , N., "The Problem of Counterfactual Conditionals" ,Journal
o f Philosophy, 44, 1947, pp. 113- 128.
[4] Ka lis h , D. - Montague, R. Logic. Techniques of Formal Reasoning,
Harcourt, Brace 8z. World Inc., N. Y . 1964.
[5] K v a r t , I., A Theory of Counterfactuals, Hackett Pu. Co., Indianapolis,
1986
[6] L e w i s , D. K., Counterfactuals, Blackwell, Oxford, 1973
[7] L e w i s , D. K., "Probability of Conditionals and Conditional Probability" , Philosophical Review, 3, 1976, pp. 297-315
[8] N u t e , D. Topics in Conditional Logic, Reidel, Dordrecht, 1981.
[9] P i z z i , C. " S in ce ', 'Even I f ' , ' A s I f " in M. L . Da lla Chiara ed.,
Italian Studies in the Philosophy of Science, Reidel, Dordrecht, 1981,
pp. 73-87
1101 Pizzi, C. Dalla logica della rilevanza alla logica condizionale, E. U.
Roma, 1987
[11] Po llo ck, J., Subjunctive Reasoning, Reidel, Dordrecht, 1976
[12] Smo kie r, H . " Th e collapse o f Modal Distinctions in Probabilistic
Contexts", Theoria, 45, 1979, pp. 1-7
[13] Stalnaker, R. "Letter to B. C. van Fraassen", in W. L . Harper and
C. H. Hooker(eds.), Foundations of Probability Theory, Statistical Inference and Statistical Theories of Science, Reidel, Dordrecht, 1976,
I, pp. 302-206
[14] Stainaker, R. "Probability and Conditionals", Philosophy o f Science,
37, 1970, pp. 68-80
[15] Stalnaker, R., Inquiry, MI T Press, 1984
[16] V a n Fraassen, B. "Hidden Variables in Conditional Logic", Theoria,
40, 1974, pp. 176-190