A FITCH-STYLE FORMUL A TION OF CO NDI TI O NA L
LOGIC (
1
Richmond
H. THOMASON
)
The system CS of sentential conditional logic is generated by
axiom-schemes A l-A G o f [51 together wit h the rules o f modus
ponens and necessitation. I t is given an in tu itive justification in
[3] and it s connection wit h conditional p ro b a b ility is demonstrated in [4]; in [5] itse lf this system (o r rather, a quantificational extension of it) is shown to be sound and complete wit h
respect to its intended interpretation.
In th is paper we w i l l fu rth e r describe th e in fe re n tia l stru cture o f CS b y formulating it as a system o f natural deduction,
using th e fo rma t developed b y Fitch in [2] (
2that this system FCS is equivalent to CS, we w i l l use it to in )dicate
• A fh ot we th
r e rusle sh o fo FCS
w ma
i ny be
g related t o th e w a y we
reason w i t h conditionals i n e ve ryd a y situations. Th e n a tu ra l
deduction format has the further advantage of yielding a framewo rk wit h in wh ich theories o f the co n d itio n a l ma y b e cla ssified. I n particular, a ll reasonable theories o f the conditional,
including FCS, ma y be displayed as results o f adding rules to
a min ima l syste m t h a t embodies p ro p e rtie s co mmo n t o a l l
such theories.
1. Th e system FCS.
This system has D,
, and the conditional connective > as
(
ce 1Foundat ion Grants GS-1567 and GS-2517. I a m indebt ed t o R. St alnak er
for) conversations out of whic h some of the ideas f o r this res earc h grew.
T(
s imilar
t o it .
h2
e)
rT
eh
si
es
ap
ra
cp
he
lr
ep
ar
de
is
nu
gp
tp
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its p rimit ive connectives. We will regard disjunction, conjunction, a n d ma te ria l equivalence a s defined i n th e u su a l wa y.
Necessity and p o ssib ility are defined as fo llo ws.
' El A' = d f A > A •
'0•A' = d t E — A'
FCS has the usual ru le s f o r imp lica tio n and negation, and f o r
reiteration in to o rd in a ry derivations (see [6], Chapters I I I and
IV). But besides o rd in a ry derivations in FCS, we may also fo rm
strict derivations, wh ich have the fo llo win g form.
A
The in tu itive difference between such a d e riva tio n (wh ich we
will ca ll a st rict d e riva tio n in A ) and a n o rd in a ry d e riva tio n
with hypothesis A is that in the latter, we suppose th a t i t is
the case that something-or-other; in a strict derivation we suppose that it we re the case that something-or-other. Th is means
that in a strict derivation we may "bracket" o r hold in abeyance
certain portions o f o u r knowledge about o u r actual situation,
and envisage another situation in wh ich something is supposed
to hold. In an ordinary derivation we merely make a hypothesis
about the actual situation.
A ll o u r kn o wle d g e about t h e a ctu a l situ a tio n w i l l n o t i n
general obtain in other situations we may envisage. O u r formal
theory therefore mu st n o t a llo w unrestricted re ite ra tio n in t o
strict derivations, so that arguments having the fo llo win g pattern are not sanctioned as such in FCS.
A
A FI TCH-S TY LE FO RMUL A TI O N O F CO NDI TI O NA L L O G I C 3 9 9
We o b ta in a f o rma l t h e o ry o f co n d itio n a lity b y sp e cifyin g
general circumstances i n wh ich in fo rma tio n ca n be obtained
inside s t ric t d e riva tio n s. I n o t h e r wo rd s, su ch a t h e o ry i s
characterized by a set o f reiteration rules. The system FCS has
four such re ite ra tio n rules wh ich we w i l l ca ll 're it 1', ' re it 2 ,
'reit 3', and 're it 4'.
A > B
D
B
A
A
reit 1
r
— ( A > B)
e
A
i
>
t
2
B
13> A
13> C
A
A
reit 3
r
e
i
t
4
The rule re it 1 permits B to be reiterated into a strict derivation
in A wh ich is subordinate to an occurrence of A > B. The ru le
reit 2 permits B to be reiterated in to a st rict d e riva tio n in A
wh ich is subordinate to an occurrence o f OB . Th e ru le re it 3
permits —B to be reiterated into a strict derivation in A wh ich
is subordinate to an occurrence o f — (A > B). Fin a lly, the ru le
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re it 4 permits C to be reiterated in to a st rict d e riva tio n in A
which i s subordinate t o occurrences o f A > B, B > A, a n d
B > C.
These four reiteration rules provide the u n d e rlyin g structure
needed fo r conditional reasoning. In order to describe the logic
of th e co n d itio n a l connective w e mu st a d d t o t h is stru ctu re
rules f o r the introduction and e limin a tio n o f formulas h a vin g
the f o rm A > B. These ru le s a re exact analogues o f the co rresponding rules f o r ma te ria l imp lica tio n (').
A
A
A > B
A > B
cond in t
c
o
n
d
e lim
The ru le o f cond in t a llo ws A > B to be in fe rre d fro m a strict
derivation in A wh ich contains a step B. The ru le of cond e lim
allows B to be inferred fro m A and A > B.
2. Equivalence o f CS and FCS.
As the first step in our proof that FCS and CS are equivalent
we present derivations in FCS showing that any instance of the
axiom-schemes A l-A G of CS is derivable in FCS. These derivations w i l l also serve as illu stra tio n s o f h o w the ru le s o f FCS
can be applied. We will use the notation 'taut' b e lo w to ju st if y
a step in wh ich the conclusion can be obtained using only rules
for negation and ma te ria l implication.
(3) Th is holds t ru e f o r theories o f t he c ondit ional t hat d o not at t empt t o
embody the princ iple that the antecedent and c onc lus ion s hould be relev ant,
in t he sense discussed in I n t his case t he int roduc t ion and eliminat ion
rules f o r > mus t be adjus ted as i n [ I L
1
A FI TCH
FO RMULA TI O N OF CO NDI TI O NA L L O G I C 4 0 1
S T Y L E D(A DB )
hyp
2
_ DA
3
*
4
5
6
7
8
9
1
2
3
4
5
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
hyp
M
_ –B
A
AD B
B
DB
DA DDB
0 (A DB )D.DA DDB
D(A DB )
*
A
A> B
g
)
2, re it 2
1, re it 2
4,5, m p
3-6, co n d in t
2-7, imp in t
1-8, imp in t
hyp
hYP
A DB
B
1, re it 2
2,3, m p
2-4, co n d in t
OA
hyp
_
A> B
_
hyp
A>–B
* A
_
— A
A > — A
* - - A
_
A
— A > A
* – –A
-B
B– A
D –A
OA
– (A > – B )
( A > B) D ,--, (A > – B )
O A D . ( A > B) D --, ( A > – B)
hyp
hyp
4, t a u t
4-5, co n d in t
hyp
7, ta u t
7-8, co n d in t
hyp
3, 6, 9, re it 4
2, 6, 9, re it 4
11,12, ta u t
10-13, co n d in t
1, re it
2-15, neg in t
2-16, imp in t
1-17, imp in t
402
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2
3
4
5
6
1
2
3
4
5
6
7
8
9
10
D
H, THO M A S O N
hyp
— (A >B )
* A
3
1
N
A > (13 v C)
2
4
5
6
7
8
9
10
O
Bv C
—B
A>C
— (A >B ) ( A > C )
(A > B ) v (A > C)
( A > (13 v C)) D “ A > B ) v (A > C))
A>B
hyp
hyp
1, re it 1
2, re it 3
4, 5, ta u t
3-6, co n d in t
2-7, imp in t
8, t a u t
1-9, imp in t
hyp
A
hyp
B
A DB
(A > B )D A D B
(A > B ) A (13> A)
A>C
(A > B ) A (B > A)
A>B
B> A
* B
1, re i t
2, 3, co n d e lim
2-4, imp in t
1-4, imp in t
hyp
hyp
1, re it
3, co n j e lim
3, co n j e lim
hyp
C
2, 4, 5, re it 4
B> C
6-7, co n d in t
(A > C) D (B>C)
2-8, imp in t
YA>B) A (
1-9, imp in t
B>
A ) )
D
. these s ix axiom-schemes, CS has modus ponens and
Besides
( A as it s ru le s o f inference. I t is p la in th a t modus
necessitation
ponens>is admissible
C
in FCS. To see th a t necessitation is also
)
D
(
B
>
C
)
A FI TCH-S TY LE FO RMUL A TI O N O F CO NDI TI O NA L L O G I C 4 0 3
admissible in CS, suppose there is a d e riva tio n o f A in FCS.
Then D A can be d e rive d b y repeating th is d e riva tio n inside
a strict derivation in— A as follows.
—A
A
From this, it follows that everything provable in CS is derivable in FCS. To obtain the converse result we argue as in Chapter V o f [6] th a t e ve ry derivation o f B f ro m A in FCS can be
transformed into a deduction o f B fro m A in CS. To show this,
the argument of [6] need o n ly be supplemented as follows. Let
C2
c
n
be a strict derivation occurring in an augmented derivation in
FCS, i.e. a derivation in which axioms o f CS may be introduced
at a n y point, a n d in wh ich th e ru le o f necessitation ma y be
applied to steps established without the help of any hypotheses.
We must show that C
derivation
with o u t the use o f a strict derivation.
1
>C„
c a n
b
e
o b t a i n
e d
i
n
t
h
i
s
a
u
g
m
e
n
t
e
404
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We accomplish this by erasing the line demarcating the strict
subproof and p re fixin g C
Ithe fo llo win g a rra y.
> t o
e a c h
o
f
i
t
s
s
t
e
p
s
,
C1>C1
t
h
u
C1>C2
s
o
b
t
a
i
n
i
n
g
C
1
>
C
t
It re ma in s t o b,e sh o wn th a t steps ca n b e inserted i n th is
array in such a wa y as to make an augmented derivation; th is
is accomplished b y cases according to h o w C
i wo a
the
rigsin a l jaugmented
u s t i f derivation.
i e d
If C
i i insert
we
s n t ahproof
e in CS o f CI > Ci. I f C
from
and
in CS o f C
ih cy aD
pm
o e tD hC bie, we
sy iin se
s rt a deduction
C
Im o d u s
C
>
1
C o
p deduction
a
n in eCS ofn C s
insert
o m
in CS o f A > C
i> f ra deduction
D Cf rer ito1 m
i>
by
f ro m CA
>
a
—
1
new
augmented
I
derivation
f
b y o rd in a ry reiteration. I f C, came
i
C
>
by
.n Cre
I it f 2 f ro m
C DC
id C
1
?
If
i,cC i anin saCS
tion
e of
r teC
m
e
m
1
ia
C
s
n
s
e
r
t
b
y
n eo—
d
—
>
1
from
u aci
e Cdt D
gh
t
tn
fD
>
1
elimination
g oo ; nnm from D > C
i( iro
from
/ Da ,l nCdin mD i,
C
a
—
D
i>
e
liD
3 n>
C
,(n
sumption
Ca s e
ri tcertain
deductions can be carried out in CS; e.g.
t that
o
that
>
—
(A>B)1—
A
>
—
B a n d t h a t A > B , A > - ( B D C)1— A > C.
S
d
1
D
C
a
C
n
,
CS
C
S
(d
>e C)d . u c t
iCD D
co
>
1
These
facts can be established easily using the results o f [61.
i,C
fO o n
Ia
)ia
s.
C
fm
1
n
a
ijI
,e
n
C
u
s
n
ft
ib
a
d
S
i
rf
d
cyiC e a m
n
D
o
d
C
i
rD
m
d
f>
C
1
i
n
e
D
C
>
i.ts 1
,b
1
)>
a . C
yth
•C
A FI TCH-S TY LE FO RMUL A TI O N O F CO NDI TI O NA L L O G I C 4 0 5
From these considerations i t f o llo ws th a t CS and FCS a re
equivalent systems. We can therefore regard FCS as a re fo rmulation o f conditional sentence logic.
3. FCS and in fo rma l reasoning.
In this section we w i l l use o u r reformulation o f conditional
logic t o re fin e th e account g ive n in [3] o f the relationship o f
conditional lo g ic t o reasoning i n English. Na t u ra l deduction
systems such as FCS are we ll adapted to th is purpose because
of their structure, wh ich allows fo r the positing and elimination
of hypotheses, is much closer to argumentation in natural la n guage than is that of systems such as CS. In the fo llo win g discussion we will take up in tu rn each o f the rules of FCS wh ich
helps t o determine th e properties o f th e connective > . Dis counting the rules fo r negation, wh ich characterize the classical, two -va lu e d so rt o f negation, there are s ix o f these rules:
cond in t and cond elim, and the fo u r reiteration rules.
The ru le s f o r in tro d u cin g a n d e limin a tin g fo rmu la s o f th e
sort A > B are n o t peculiar t o th is connective; f o r example,
they a re th e same i n f o rm as th e ru le s f o r ma te ria l and in tuitionistic imp lica tio n . These ru le s re fle ct v e r y deep h a b its
of reasoning wit h conditionals. Wh e n required t o establish a
conditional conclusion, one's n a tu ra l response i s t o suppose
the antecedent and then t r y t o sh o w th a t th e consequent o f
the conditional fo llo ws. Consider th e f o llo win g illu stra tio n .
" Wh y do yo u sa y th a t i f Napoleon had attacked e a rlie r a t
Waterloo, h e wo u ld have wo n ?" " We ll, suppose h e had. A s
it was, Blücher o n ly a rrive d in the n ick o f time, and i f Napoleon had attacked e a rlie r he wo u ld have had o n ly the English
to deal with . Besides, o n ly a f e w days before Wa te rlo o We llington was miles a wa y fro m his troops and t h e y we re unprepared f o r a fight. Th e Fre n ch wo u ld h a ve demoralized th e m
completely wit h an unexpected a tta ck and achieved a n easy
victo ry."
The ru le o f cond e lim can be regarded as e ith e r a wa y o f
reasoning f ro m conditional statements, o r as a w a y o f re fu ting such statements. Th e f irst aspect o f th is ru le constitutes
the core o f tru th in the v ie w o f conditionals as "inference t i-
406
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ckets"; co mmitme n t t o a conditional statement entails a conditional co mmitme n t t o it s consequent, should it s antecedent
be true. I f I cla im that I w i l l wo rk in the ya rd if it s sunny outside and then discover that it is sunny outside, my claim entails
that I w i l l wo rk in the yard. Tu rn in g th is around, we can a lways refute a conditional statement if we kn o w its antecedent
is true wh ile its consequent is false. " I f you stopped in De tro it
then o f course yo u sa w A u n t Beatrice." " No , that's n o t so. I
stopped there, b u t she was visit in g a frie n d in Cleveland and
I missed her."
The ru le o f re it 1 is , I t h in k, a n u n co n tro ve rsia l fe a tu re
of conditional reasoning; i f a conditional statement has been
posited, the supposition of its antecedent a llo ws its consequent
to be asserted. I f the b ill we re passed, i t wo u ld be declared
unconstitutional." " We l l , suppose i t we re passed. Th e n , a ccording t o wh a t y o u sa y, i t wo u ld b e declared unconstitutional..."
Together, these three ru le s determine a min ima l basis f o r a
logical th e o ry o f th e conditional. (
4
fact
t h a t mo st lo g ica l theories o f co n d itio n a l statements d o
) E v these
satisfy
i d erules.
n c The
e
o n ly exceptions wh ich need to be taken
f
seriously
o
are,
r
I believe, Anderson and Belnap's systems E and
R
t o f hentailment
i
s and relevant implication. A kin d o f rock-boti
tom
min simu m could be obtained b y adding the restrictions o f
t[1] t o ohu r threee rules. B u t although min ima l systems o f t h is
sort ma y h a ve so me f o rma l interest, m y imme d ia te concern
is to consider enrichments o f these basic rules wh ich p ro vid e
a less impoverished logic. Th is brings us to the three remaining reiteration rules.
To ma ke cle a r the imp o rt o f the ru le o f re it 2, we should
(
rule
4 is modus
c ontaining
a ll
)
C A > A
(A > (B
a
(A > (B
l
and
l such t hat
t
h
i
s
s
y
s
t
e
m
ponens b y t ak ing t he s et S o f ax ioms t o be t he s malles t set
ins tanc es o f t h e t hree schemes.
D C)) D ((A > B) D ( A > C))
> C)) D ((A > B) D ( A > C) )
if E e S t hen D > E e S.
A FI TCH-S TY LE FO RMUL A TI O N O F CO NDI TI O NA L L O G I C 4 0 7
first say something about the content o f O A , i.e. o f —A > A .
It i s n o t e a sy t o f in d cases i n wh ich lo cu tio n s su ch a s t h is
come n a tu ra lly t o o u r lips. Perhaps th e best wa y t o d o i t is
by ta kin g a limit in g use o f the lo cu tio n 'even i f . Fo r instance,
a person wh o says " E ve n i f Hit le r had been humane i n h is
policy toward the Jews, he wo u ld have been an e vil man, even
if he had not followed an expansionist foreign p o licy he wo u ld
have been an e vil man, even i f he hadn't been e vil, he wo u ld
have been an e v il ma n " is re fu sin g in a picturesque wa y t o
acknowledge t h e p o ssib ility th a t Hit le r mig h t n o t h a ve been
evil. He cannot conceive o f a situation in wh ich Hit le r wo u ld
not be evil.
Thus, to assert something of the fo rm
>
A (or, wh a t is
equivalent on principles already established, —A > (B A —
,is to assert the absurdity o f positing the fa lsity o f A . O n th is
understanding
o f things, t h e ru le o f re it 2 sa ys t h a t wh e n
B
))
—A > A has been obtained, A must then be true in any situation whatever, no matter h o w this situation is posited. To simultaneously assert something of the form A
something
>
o f Ath e f o rma B n> Ad i s t o sa y t h a t a situ a tio n i n
d
which
e A does
n
ynot obtain cannot be conceived, and ye t that A
does n o t obtain in some situation in wh ich B is posited. I t is
this k in d o f paradoxical p o sitio n th a t is ru le d o u t b y re it 2.
The content o f the rule, then, is that if A
s
A mu >
st be truAe in e ve iry situation;
i f there is a n y possible sittuation
r uin which
e
t
h
a
— A is true, the one nin which it is posited must
be such a situation.
The ru le o f re it 3 is equivalent t o a llo win g — (A > B )
(A > which
B )
can be true, to deny a conditional o f the so rt A > B is
to
the conditional A > —B. Inferences o f this kin d are
t assert
o
common
b
in natural language. " I f yo u put yo u r we ig h t on th a t
board,
i t w i l l break." " Th a t's false: i t wo n 't." Wh a t is denied
e
here
d e is th e conditional statement th a t i f I p u t m y we ig h t o n
the
r i board, i t w i l l break. I then state the denial b y conditionally
the consequent o f the f irst conditional; i.e. I assert
v denying
e
that
if
I
put
my weight on the board, it wo n 't break. Moves o f
d
this
so
rt
occur
constantly in practice.
f
r
o
m
O
A
;
t
h
i
408
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As a mo re direct example o f the ru le re it 3, consider a questionnaire wit h th e fo llo win g lines.
I. I f the U.S. we re to adopt the A B M system CI Y e s
its milit a ry position would be strengthened. El N o
El Un d e cid e d
2. No w, suppose t h a t t h e U.S. a d o p ts t h e I : Y e s
A B M system. Wo u ld it s milit a ry position El N o
be strengthened ?
E
l
Un d e cid e d
According to the ru le o f re it 3, it wo u ld be inconsistent to answer " No " t o the f irst question and " Ye s" o r "Undecided" t o
the second. Th is p rin cip le seems t o be so cle a rly va lid a feature o f conditional reasoning th a t i t i s su rp risin g th a t so me
logicians have chosen to re je ct it.
The semantic content o f re it 3 is that, in principle, a unique
situation is selected wh e n th e antecedent o f a conditional is
posited. The va lid it y o f conditional excluded middle, (A > B )
V ( A > —B), fo llo ws fro m this fo r the same reasons that render o rd in a ry excluded mid d le , A V A , v a l i d i n cla ssica l
logic.
The rule of reit 4 permits A > C to be inferred fro m A > B,
B > A , and B C . Wi t h th is ru le , th e question whether i t
permits us to prove enough arises as we ll as the question whether i t is va lid . I n p a rticu la r, isn ' t i t a lso t ru e t h a t A > C
should f o llo w f ro m A > B and B > C alone ? A s is pointed
out in t3], th e answer to th is question is " No " ; co n d itio n a lity
is n o t transitive. Stalnaker's elegant example runs as fo llo ws.
" If J. Edgar Ho o ve r we re to d a y a communist th e n he wo u ld
be a traitor. I f J. Edgar Ho o ve r had been b o rn a Russian then
he wo u ld today be a communist. Therefore if J. Edgar Ho o ve r
had been born a Russian he wo u ld be a tra ito r."
The reason f o r this fa ilu re o f tra n sitivity is th a t in positing
that J. Edgar Ho o ve r we re a communist and positing th a t he
were b o rn a Russian, one has t wo e n tire ly different situations
in mind. I n the f irst situation he is an Ame rica n national and
would be a tra ito r in being a communist; in the second he is a
A FI TCH-S TY LE FO RMUL A TI O N O F CO NDI TI O NA L L O G I C 4 0 9
Soviet national and is a conformist in being a communist. No tice that in this case it is false that if J. Edgar Ho o ve r we re a
communist h e wo u ld h a ve been b o rn a Russian, so th a t we
lack the th ird premiss needed to show b y re it 4 that if J. Edgar
Hoover had been born a Russian he wo u ld be a tra ito r.
Wit h t h is i n mind, consider th e fo llo win g argument.
1. I f my watch cost more it wo u ld be accurate.
2. H my wa tch we re accurate i t wo u ld cost more.
3. I f my watch we re accurate I wo u ld n 't have been late.
4. Therefore i f m y wa tch co st mo re I wo u ld n 't h a ve been
late.
First, notice that, as in Stalnaker's example, step 4 doesn't fo llo w from 1 and 3; nor, fo r that matter, does it fo llo w from 2 and
3. Steps 1 and 3 ma y be true because my watch already is accurate and I wasn't late, wh ile step 4 is false because my lateness wo u ld have been caused b y m y wa tch costing more. ( I
have ju st bought th e wa tch and had o n ly enough mo n e y t o
buy it. I f it had cost more I wo u ld have had to go t o the bank
and wo u ld have been delayed.) On the other hand, steps 2 and
3 might be true because my watch isn't accurate and any watch
that is wo u ld cost more. Bu t perhaps a n y wa tch I o wn wo u ld
be inaccurate; I abuse watches te rrib ly. In that case I wo u ld be
late even if my watch cost more, and 4 wo u ld be false.
As these considerations show, th e ru le o f re it 4 has n o re dundant premisses; b u t are its premisses sufficient to yie ld the
conclusion ? Fo r instance, does step 4 o f th e wa tch example
really f o llo w fro m steps 1 t o 3 ? Certainly, the cases we have
considered up t o n o w d o n o t invalidate th is inference. I n the
case in wh ich my watch is accurate and I wasn't late, step 2 is
false. Indeed, th e p o in t o f the f irst case is th a t i t s false th a t
my wa tch wo u ld cost mo re if it we re accurate. S imila rly, th e
point of the second case, in wh ich my watch isn't accurate but
any wa tch I o wn wo u ld be accurate is that even if my wa tch
cost more it wo u ld not be accurate.
But a mere absence o f counterexamples does n o t show th a t
step 4 re a lly follows from steps 1 to 3: fo r this we need a general argument. To devise such an argument, we mu st re tu rn t o
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the idea th a t to posit a condition is t o imagine a situation in
which th is condition is true. Steps I t o 3 in vo lve t wo conditions, and hence determine a situation a in wh ich i t is posited
that my watch costs more and a situation t3 in which it is posited
that my watch is accurate. I f steps 1 and 2 are true, my watch
is then accurate in a and costs more in lEI. But then there is no
reason f o r a and (3 to differ, Ha vin g envisaged the situation a
in positing one co n d itio n I need n o t choose a different situation i n positing th e second condition, sin ce t h is second condition is already tru e in a; i t is tru e in a that my watch costs
more. An d if a and t3 are identical, it follows that in [I my watch
costs more, and therefore step 4, th e conclusion o f th e argument, is true.
Thus, re it 4 will be va lid if we assume that there is economy
in the choosing o f situations, i.e. if we assume that in positing
a condition we imagine a situation (3 d iffe rin g fro m a situation
a already imagined o n ly if we are forced to do so in virtu e o f
the fact that the condition is false in a. Th is ru le therefore re flects a " la w o f least e ffo rt" i n envisaging situations. Eq u a lly
well, we can regard re it 4 as re fle ctin g a n orderliness i n th e
choosing of situations; we can suppose that the choice of situations is dependent on a preferential arrangement o f situations
which can be described independently of the process of imagining. Fo r example, th e amount o f physical e n e rg y required t o
obtain the situation [3 as compared wit h that required to obtain
the situation y ma y be pertinent to such a description. I f th is
order is st rict enough t o p e rmit one a lwa ys t o speak o f th e
closest situation in wh ich a condition is true, we can then define the situation selected in positing a condition as the closest
one in which the condition is true. The va lid ity of reit 3 follows
at once fro m ths characterization o f the choosing of situations.
This argument fo r the va lid it y o f re it 4 ma y strike some as
less persuasive than o u r arguments in fa vo r o f the other rules
of FCS. Whether o r not this is so, I am reluctant to a d mit that
re it 4 represents a superficial and dispensible feature o f conditional logic. I f it weren't true that o u r selection o f situations
in positing conditions we re ru le like and o rd e rly, we wo u ld n 't
be able t o communicate e ffe ctive ly b y means o f subjunctive
A FI TCH-S TY LE FO RMUL A TI O N O F CO NDI TI O NA L L O G I C 4 1 1
conditionals and th e y wo u ld lose t h e ir o b je ctive and factual
character. O rd in a rily, w e a re prepared t o t a ke su ch co n d itionals as matter-of-fact statements about th e wo rld ; w e a re
prepared to argue th e ir tru th o r fa lsity wit h complete seriousness. Reasonable people w i l l dispute subjunctive conditionals
in debate, a n d those wh o a re le ss dispassionate ma y e ve n
come t o b lo ws o ve r them. I t is t h is seriousness w e a re p re pared t o la vish o n conditionals th a t g ive s examples su ch as
the one about Bizet and V e rd i (
5 The ru le o f re it 4 does n o t re fle ct a n y p a rticu la r system o f
) t h e i r among worlds, b u t it does re fle ct the requirement
preferences
p a these
that
r a preferences
d o x i c should
a l be systematic. I wo u ld cla im that
f
this
lre q uaire me
v n t ois ar precondition
.
o f th e in te rsu b je ctive use
of subjunctive conditionals. Communication wo u ld break down
if the Bizet and V e rd i phenomenon we re the ru le ra th e r than
the exception. Furthermore, I wo u ld ma in ta in t h a t i t sh o u ld
be possible t o describe the conventions used b y a p a rticu la r
community in selecting situations, and I believe that this would
be a p h ilo so p h ica lly useful a n d illu min a t in g ta sk. Ho we ve r,
this enterprise should be distinguished fro m the mo re general
task o f exhibiting logical fo rm and p ro vid in g a semantic theory o f va lid ity.
One unpleasant consequence does f o llo w f ro m o u r account
of situation-choosing as systematic; i t appears on this account
to b e impossible t o ima g in e one situ a tio n wit h o u t somehow
imagining th e m a ll. E v e ry situ a tio n mu st b e chosen i n t h e
light of a complete system in wh ich a ll possible situations are
arranged in some preferential order.
I d o f e e l t h a t t h is re q u ire me n t i s t o o strong, a n d t h a t i t
needs t o be modified b y introducing the mo re modest n o tio n
of a p a rt ia l ru le o f selecting situations. Su ch a p a rt ia l ru le
would select situations w i t h re g a rd t o so me conditions, b u t
not necessarily fo r all. Wi t h respect to a p a rtia l ru le o f selection, a su b ju n ctive co n d itio n a l h a vin g a n antecedent co n d i(5) O n e man says I f Biz et and V e rd i had been c ompatriots , Biz et wo u ld
have been I t alian. " Th e ot her says "No . V e r d i wo u ld hav e been Frenc h. "
The ex ample is due t o Quine.
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tion not fa llin g under the ru le wo u ld not in general be tru e o r
false. F o r instance, i f w e h a ve n o t ma d e a d e cisio n a b o u t
which situation is intended as the one in wh ich Bizet and Ve rd i
are compatriots, then i t is neither tru e n o r false th a t i f Bizet
and V e rd i we re compatriots th e n Bize t wo u ld b e Ita lia n . I n
van Fraassen's technique o f supervaluations, discussed in [7],
there exists a semantic technique o f p ro vid in g f o r truth-value
gaps such as these. Th e application o f th is technique t o th e
subjunctive conditional is le f t fo r a fu tu re paper. Wh e n mo d ified in this way, the crite rio n of orderliness in the selection o f
situations becomes a plausible as we ll as a necessary feature
of the lo g ic o f conditionals.
Yale Un ive rsit y
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BIBLIOGRAPHY
I ll ANDERSON, A . and N. BEINAP, Jr., T h e pure c alc ulus of ent ailment . " The
journal o f s y mbolic logic , 27 (1962), pp. 19-52.
[2] FITCH, F., Sy mbolic logic . N e w York , 1952.
[3] STALNAKER, R. , " A t h e o ry o f c ondit ionals . " St udies i n lo g ic a l t heory ,
N. Rescher, ed., Ox ford, 1968, pp. 98-112.
[4] STALNAKER, R. , "P ro b a b ilit y a n d c ondit ionals . " P hilos ophy o f s c ienc e,
37 (1970), pp. 64-80.
[5] STALNAKER, R. a n d R. THOMASON, " A s emant ic analy s is o f c ondit ional
logic ." Theoria, 3 6 (1970), pp. 23-42.
[6] THOMASON, R., Sy mbolic logic : an int roduc t ion. N e w York , 1970.
[7] VAN FRAASSEN, B., "S ingular terms , t rut h-v alue gaps, and f ree logic . " Th e
journal o f philos ophy , 63 (1966), pp. 481-495.