Uwe Scheffler
Conditionals, based on strict
entailment
Acta Universitatis Lodziensis. Folia Philosophica nr 9, 81-87
A C T A U N I V E R S I T A T I S L O D Z I E N S I S
F O L I A P H I L O S O P H I C A í). 1993
Uwe S c h effler
C O N D IT IO N A L S , B A S E D O N S T R I C T E N T A IL M E N T C o n d itio n a ls c a n be o b ta in d by several w ays:
- as a resu lt o f e m p iric a l in v estig atio n s;
- fro m o th e r c o n d itio n a l o r n o n o c o n d itio n a l p ro p o s itio n s by logical rules; - fro m d e fin itio n s o r o th e r term in o lo g ica l statem e n ts;
- p o s tu la te d ;
- fro n t sen ten ces a b o u t logical e n ta ilm e n t.
T h e la st w ay o f o b ta in in g c o n d itio n a l p ro p o s itio n s is th e m a in to p ic o f th e p a p e r w h ich d eals o n ly w ith first d eg ree c o n d itio n a ls (c o n d itio n a ls , c o n ta in in g o n ly o n e o c cu ren ce o f th e c o n d itio n a l o p e ra to r). S u ch c o n d itio n a ls a re im p o r ta n t: th ey a re logically tru e a n d a re used to d ra w co n clu sio n s fro m facts to g et facts.
E v ery tru e e n ta ilm e n t A |— В c o rre s p o n d s to a tru e c o n d itio n a l A -* B. W h a t k in d o f c o n d itio n a ls w e g e t d e p en d s o b v io u sly o n th e system o f logical e n ta ilm e n t, w hich ru les th e e n ta ilm e n ts, a n d o n th e logical rules, g o v e rn in g c o n d itio n a ls. T h e b asic sy stem is in this case th e sy stem F s o f s tric t e n ta ilm e n t c o n stru c te d b y W essel. T h e p ro p o s e d in tr o d u c tio n - ru le fo r c o n d itio n a ls allow es to use tw o im p lic a tiv e stru c tu re s: e n ta ilm e n ts a n d c o n d itio n a ls, w ith d ifferen t p ro p e rties. U sin g so m e c o n d itio n a l p rin cip les th e class o f c o n d itio n a ls c an c h an g e w hile the class o f e n ta ilm e n ts re m a in s u n ch an g e d .
T h e a lp h a b e t o f F s c o n sists o f
1) c o u n ta b le m a n y p ro p o s itio n a l v a ria b le s p, q, r, p
2) tru th -fu n c tio n a l c o n n ectiv es л (c o n ju n c tio n ), V (d isju n c tio n ), ~ (n e g atio n );
3) th e p re d ic a te o f e n ta ilm e n t |— ; 4) p a re n th e se s.
D l . A fo rm u la is a tru th -fu n c tio n a l fo rm u la , if it is c o n stru c te d b y th e usual rules w ith tru th -fu n c tio n a l c o n n ec tiv e s only.
8 2 U w e S ch efflcr
D 2. A fo rm u la is a fo rm u la o f logical c n ta ilm e n t. if it h as th e s tru c tu re A H B. a n d A a n d В a re tru th -fu n c tio n a l fo rm u la s.
T h e p o s tu la te s fo r F s a re all fo rm u la s o f logical e n ta ilm e n t h a v in g th e form o f o n e o f th e follow ing sc h e m a ta a n d m eetin g th e c o n d itio n s E l a n d E2: E l . If A I— B, th en В c o n ta in s o n ly such p ro p o s itio n a l v a ria b le s, w hich are a lso in A.
E2. If A I— B. th en A is n o t a c o n tra d ic tio n a n d В is n o t a tau to lo g y . A l . A I— - ~ A A 2. ~ ' A I— A A 3. А л В I— A A 4. А Л В І— В Λ A A 5. - (А Λ В) I A V - В A 6. ~ A V ~ В I (А Λ B) A 7. (A V В) Л С I— (А Л С) V В A 8. (А Л С ) V (В Л С ) I— (А Л В ) A C A 9. A I— А Л (В V ~ В) T h e ru les o f F s are: R l . If A I— В a n d В |— C , th en A |— C. R 2. If A I— В a n d A |— C . th en A — В л С. R 3. If A з В a n d В з A a re tau to lo g ie s, th e n С |— C [A /B ], w here C [A /B ] m ean s th a t in С all o r so m e (in c lu d in g n o o n e) o ccu ren ces o f A a rc to be rep laced by B. a n d С is n o t a c o n tra d ic tio n a n d C [A /B ] is n o t a ta u to lo g y .
W essel pro v ed : A fo rm u la o f logical c n ta ilm e n t A — В is a th e o re m in F ' i f a n d on ly if: А з В is a ta u to lo g y , В c o n ta in s o n ly su ch v ariab les, w hich are a lso in A , A is n o t a c o n tra d ic tio n a n d В is n o t a ta u to lo g y 1.
T o get a c o n d itio n a l system w e in tro d u c e a n o n -tr u th -fu n c tio n a l c o n n ec tive -> (c o n d itio n a l o p e ra to r: if - th e n ) in to th e language:
D 3. A fo rm u la is a sen ten ce, if the fo llo w in g c o n d itio n s a rc satisfied : 1. T ru th -fu n c tio n a l fo rm u la s a re sentences. 2. I f A a n d В a rc sentences, ~ A, (А Л В), (A V В), (A -> В) a re sentences.
T h e c o n stru c tio n is c o m p le ted by th e c o n d itio n a l ax io m a n d th e c o n d itio n a l rule:
A10. I— A -► A
R 4. If A I— В a n d |— В -> С , th e n |— A -> C. F o r th e resu ltin g system F SK it is easy to show :
51. I f I— A in F SK, th e n A is a c o n d itio n a l. 52. A I— В is th e o re m in F s if a n d o n ly if A |— В is th e o re m in F SK. 53. A I— В is th e o re m in F s if a n d o n ly if |— A -» В is th e o re m in F SK. 54. I f I— A -> С a n d |— В -> С a re th e o re m s in F SK, th en I— (A V В) -> С is th e o re m in F SK. 1 C f. H . W e s s e l , L o g ik , B erlin 1984, p . 170 -1 7 3 .
C o n d itio n a ls . B ased o n S tr ie l E n ta ilm e n t 8 3
55. I f і ( Α Λ Β ) -> С is th e o re m in F SK, a n d В c o n ta in s n o v ariab les, o c cu rin g in A o r in C , th e n | A -> С is th e o re m in F SK.
■ If А л В I С is th e o re m . A ist n o t a c o n tra d ic tio n a n d С is n o t a ta u to lo g y . U n d e r th is c o n d itio n a n d b ecau se o f th e re stric tio n on В in С o c cu r o n ly such v ariab les, w hich a re also in A . Let W be a v a lu a tio n , w hich prescribes A the v alue T an d С the v alu e /·'. In an y case W c an be e x te n d ed to a v a lu a tio n W inclu d in g th e v a ria b le s o f B. th a t p re scrib es th e v alu e Г to A В to o . B ecause o f S3 a n d W essel's result m e n tio n e d a b o v e th e sen te n c e is p ro v ed . ■
56. If |— A —» В is th e o re m in F SK, th e n | (А л С ) —» В is th e o re m in F SK. w here А Л С is n o t a c o n tra d ic tio n .
■ U se A 3 an d R l . ■
O b v io u sly A -» В is n o t a th e o re m in F SK, if A is a c o n tra d ic tio n o r if В is a ta u to lo g y . In a d ire c t sense th is system is a p a ra c o n s is te n t logic: the a p p e a ra n c e o f c o n tra d ic to ry d a ta d o e s n o t force th e system to be explosive, to derive a n y fo rm u la. T h e u n u su al restric tio n n o t to c o n clu d e fro m c o n tra d ic tio n s is a d ifferen ce b etw een relev an t a n d p a ra c o n s is te n t logics a n d F SK an d has to be ex p lain ed . In relev an t logic fro m p л — p d o e s n o t fo llow q. b u t it follow s p a n d also - p. Even if o n e s tip u la te s th a t th e re a re tru e c o n tra d ic tio n s p ro b a b ly n o t all c o n tra d ic tio n s a re tru e , th e re fo re in som e cases fro m a c o n tra d ic tio n d o es n o t follow all n o n sen se y o u w a n t (as in classical logic), b u t a little n o n sen se a n y w ay . In o rd e r to av o id a n y n o n sen se the re stric tio n o n the a n te ce d e n ts is m ad e. O n th e o th e r h a n d th e re stric tio n on the c o n se q u e n ts is u n d e rs ta n d a b le a t once: w h y w e sh o u ld c o n clu d e ta u to lo g ie s, if we a lre a d y k n o w th a t they a re ta u to lo g ie s? Such im p lic a tio n s a re o ften funny, so th e re is an old g e rm a n rule: I f th e co ck crow's on th e d u n g h ill, th e w e a th e r is c h an g in g o r it re m a in s u n c h an g e d .
T h e re tire g o o d re a so n s fo r th e restric tio n s, b u t so m etim es th ey seem to be very h a rd . S y stem atically v io la tin g th e m we c o n s tru c t w e ak e r system s.
W e s ta r t to b u ild u p several sy stem s o f c o n d itio n a l logic by a d d in g c o n d itio n a l rules. In all system s th e set o f e n ta ilm e n ts re m a in s u n c h an g e d , it is the set o f th e o re m s o f F s. T h e c o n c re te ch o ice o f rules. W'hich w'c w a n t to use. d e p en d s o f c o u rse o n p ra c tic al p u rp o se s. So it m ay, fo r e x am p le, be useful to have th e n o n -m o n o to n ic re la tio n o f e n ta ilm e n t to g e th e r u 'ith a m o n o to n ie c o n d itio n a l o p e ra to r. S u ch th in g s c an be d o n e , as we w a n t to show .
A d is a d v a n ta g e o f F SK is th e a b sen ce o f th e s u b stitu tio n rule. So it is necessary to d istin g u ish logically b etw een ( p A q ) -> p (w h a t is valid o n the base o f A 3) a n d (p л ~ p) -> p (w h a t is invalid b e ca u se o f E2). th o u g h the letter is d e riv a b le fro m th e fo rm e r by su b stitu tio n . L o g ician s w o rk in g in relev an t logic w o u ld arg u e , th a t s u b s titu tio n is a logical ru le a n d th e re fo re the set o f c o n d itio n a ls, o b ta in e d fro m sen ten ces a b o u t e n ta ilm e n ts. sh o u ld co n sist
8 4 U w e S c h e m e r
o ľ n o t o n ly the c o rre s p o n d in g c o n d itio n a ls, b u t also o t'a ll s u b stitu tio n s in such c o n d itio n a ls.
W e g e t F SK5 by a d d in g th e fo llo w in g rule to F SK:
R5. I f I A -> B. th e n |— C —* D . w h ere C -> D is th e resu lt o f su b stitu tin g p ro p o s itio n a l v a ria b le s o f A -> В by tru th -fu n c tio n a l fo rm u las.
In this system w e c a n p ro v e c o n d itio n a ls w hich d o n o t m eet the c o n d itio n E2: it is po ssib le to deriv e c o n d itio n a ls w ith c o n tra d ic to ry a n te ce d e n ts an d ta u to lo g ic a l co n se q u e n ts.
O n e o f W esscl's system s allo w s to p ro v e e n ta ilm e n ts, fulfilling the c o n d itio n E l b u t failin g to m eet restric tio n E2. H is system o f logical e n ta ilm e n t S s c a n be o b ta in e d fro m F s sim p ly by rejectin g E2. a system SSK c a n be c o n stru c te d a d d in g A 10 a n d R 4. O b v io u sly F SK5 is a system betw een F SK a n d SSK: all th e o re m s o f F SIC5 a re p ro v a b le in SSK, b u t p -> p V ~ p is th e o re m in the la tte r a n d n o t in th e fo rm e r system .
In F SK5 th e o re m s a re all c o n d itio n a ls, c o rre s p o n d in g to F s -e n ta ilm e n ts, a n d all c o n d itio n a ls b eing su b stitu tio n s in such ..in n o c e n t” fo rm u la s. Such a c o n stru c tio n is useful, if we w a n t to in tro d u c e c o u n te rfa c tu a ls w ith logically false a n te ce d e n ts in to th e system .
A d d in g rule R 6 to F we g et F SK6: R6. If"I- A -> B. th en I В -» ~ A.
It is easy to see th a t som e fo rm u la s b ein g p ro v a b le w ith R 6 a rc v io la tin g E l . So fo rm u la s like ~ A -> ~ (А Л В) a re th e o re m s, b u t n o t A -> (A V B) (b ecau se th e re is no o rd in a ry tran sitiv ity -ru le ). In so m e c o n n e c tio n s it m ak es sense to d istin g u ish b etw een these fo rm u la s. O n e m ay arg u e , th a t A -> (A V B) m eans: 011 the base o f A it is p o ssib le to in tro d u c e in to th e d isco u rse w h a t y o u w a n t (I f roses a re red. th en ro ses a re red o r the m o o n is a green piece o f cheese); b u t ~ A -* ~ (А Л В) m ean s o n ly so m eth in g like th e „ m o n o to n ic ity o f n egative in fo r m a tio n " (I f so m e th in g is n o t the case, th e n it is n o t th e case w h a tev e r h ap p en s).
S ystem s like th e m e n tio n e d o n e m ay be used in d e o n tic logic. T h e well k n o w n principle:
F ro m A |- В follow s 0 ( A ) |- O (B )
p ro d u c e s p a ra d o x ic a l situ a tio n s in classical, re le v a n t a n d m o st o f m o d a l logics. T h e re a so n is n o t o n ly th e R o ss-p a ra d o x :
I f th e s e c re ta ry h a s to m ai! th e le tte r, s h e h a s to m a il o r to b u r n th e le tte r;
b u t a lso the po ssib le o c cu ren c e o f c o n tra d ic to ry A . O f co u rse, th e re a re c o n tra d ic to ry false n o rm a tiv e c o n te x ts, b u t th e n it is n ecessary to d ecide, w hich n o rm s o n e h as to m eet. In n o case it is in a ra tio n a l sense po ssib le to oblige s o m eo n e to g e n e ra te a c o n tra d ic to ry s itu a tio n . T h is is, by th e w ay, th e sense o f a im p o r ta n t p h ilo so p h ic al p rin c ip le in p o litical a n d social p h ilo so p h y : A ll,
C o n d ilio n a ls , B ased o n S tr ic t E n ta ilm e n t 8 5
w h at is o rd e rd , is possible. R em em b e rin g p o litical p ra c tic e it sh o u ld be ad d ed : b u t be carefu l in o rd e rin g .
C o n ce rn in g th e R o ss- p a ra d o x c o n fe r th e m e n tio n e d sen te n c e w ith
It' th e s e c re ta ry h a s to m ail th e le tte r, s h e h a s to m ail th e le tte r o r to g o to d in n e r.
B ecause th e sec re ta ry m a y first go to h av e a d in n e r a n d th e n m ail th e letter o r vice v ersa, th ere is n o th in g p a ra d o x ic a l a t all. T h e p a ra d o x in th e fa m o u s secre ta ry -e x a m p le raises u p fro m th e fact, th a t b u rn th e le tte r m ean s n o t to m ail it, a n d m ail it m ean s n o t to b u rn th e letter. T h e re fo re it is a ter- tiu m -n o n -d a tu r-c o n s tru c tio n in th e co n clu sio n o f A | - B. w h a t m ak es the m e n tio n e d d e o n tic p rin c ip le le a d in g to p a ra d o x . Such c o n stru c tio n s are explicitcly exclu d ed by E2.
W ith sim ilar resu lt it is po ssib le to a d d R 6 to F SK5 a n d SSK. T h e follo w in g rule
R 7. If | ~ A -»B , th en I- (А Л С ) -> В
a d d e d to F SK allow s to p ro v e in th e re su ltin g sy stem F SK7 c o n d itio n a ls w ith c o n tra d ic to ry a n te ce d e n ts. It is a sy stem b e tw ee n F SK a n d SSK, d iffe re n t fro m pSK5
jn
pSK7one may use jpg
m o n o to n ic c o n d itio n a l o r th e n o n -m o n o to n ic en ta ilm e n t a n d also b o th to g e th e r. T h is m ay be in te re stin g in d a ta system s, w here th e d a ta a re a rriv in g fro m d ifferen t so u rces: c o n clu sio n s w ith in the d ifferen t p o o ls sh o u ld be d ra w n w ith th e h elp o f th e m o n o to n ie c o n d itio n a l, c o n clu sio n s w ith d a ta from d ifferen t so u rc es w h o u ld be d ra w n o n th e b a se o f the e n ta ilm e n ts.T h e c o n d itio n a l in F SK67, c o n str u c te d by a d d in g R 7 to F SK6, is also a m o n o to n ie one. In th is system c o n d itio n a ls w ith ta u to lo g ic a l c o n se q u e n ts a re p ro v a b le , it is a n o th e r system betw een F SK a n d SSK, d ifferen t from F SK5.
T o g e th e r w ith F SK th e fo llo w in g ru le c o n stitu te s F SKS:
R 8. I f |— A —» B, th e n | - A -» (В V C ): w h ere В V С c o n ta in s o n ly such v a ria b le s, w hich o c c u r in A.
By R 8 c o n d itio n a ls w ith ta u to lo g ic a l c o n se q u e n ts a re d e riv a b le , th e system is n o t e q u iv a len t to o ne o f th e fo rm e r m e n tio n e d .
Let F SR be the system , c o n stru c te d by a d d in g R 5 - R8 to F SK. T h e c o n d itio n a l o p e ra to r, o c cu rin g in p ro b a b le c o n d itio n a l sentences o f this system , is n o t th e m a te ria l im p lic a tio n . T h is is sh o w n b y a n easy sentence:
57. I f |— A B, th e n th e re is a p ro p o s itio n a l v a ria b le , o c cu rin g in A a n d in B.
■ U se in d u c tio n : th e p o stu la te s h av e th e p ro p e rty , th e ru les h a n d it d o w n . ■
58. By a d d in g the tra n s itiv ily -ru le fo r c o n d itio n a ls (I f |- A -> В a n d | - В -> С , th e n I A C ) th e c o n d itio n a l o p e ra to r b eco m es m a te ria l im p lic a tio n .
8 6 U w e S c h c l ľ l c r
■ I. (В Λ ' B) V ' A |· ' A (F s)
2. I-- A -> (А л (B v ~ В)) ( F SK. R 6. T ra n s .) 3. I- (А Л (B v - В)) -* (В V ~ В) ( F SR)
4. | - А -> (В V В) (T ra n s.) ■ In o rd e r to get th e last system ol' the p a p e r we have to accept tw o a d d itio n a l rules:
R9. I f A |— В a n d В | A a n d | С -> D . th e n |- С -» D [A /B ]. RIO. If |- (A V В) -*
C,
th en |- A ->C,
if A a n d С a rc s h a rin g a c o m m o n p ro p o s itio n a l v ariab le.T h ese rules to g e th e r w ith F SR c o n stitu te th e sy stem F SR, c o n ta in in g all m ean s to c o n stru c t n o rm a l form s.
T h e ru le RIO w ith o u t re s tric tio n is o n e o f th e o fte n d iscussed rules in c o n d itio n a l logic. T h e re a re so m e c o u n te re x a m p le s a g ain st this rule, fo r instance:
K rom . . I f th e se c re ta ry h a s to w r ite a le tte r o r to g o h o m e , site w o u ld g o h o m e " fo llo w s by u n re s tr ic te d R IO ..I f th e s e c re ta ry h a s to w r ite a le tte r, s h e w o u ld g o h o m e " .
T h e re stric tio n o n R IO p re v e n ts the a p p e a ra n c e o f stlch ex am p les, fo r m ally it p re v e n ts the v a lid ity o f (p л ~ p) —► q. T h e re fo re S7 h o ld s also fo r Fs r .
S9. I f | - A -» B. th e re is a fo rm u la С su ch , th a t I- A -> С a n d | С -* В. a n d in С a re o n ly these v a ria b le s, w hich o c cu r also in A a n d in B.
■In Fs r a re all m ean s to c o n s tru c t fo r a n y fo rm u la th e c o rre s p o n d in g fo rm u la in e x te n d ed d isju n ctiv e n o rm a l fo rm (a d isju n ctiv e n o rm a l fo rm such, th a t fo r all o c cu rin g v a ria b le s h o ld s: th ey o c cu r - w ith o r w ith o u t n eg atio n - in all e le m en ta ry co n ju n c tio n s). B ecause o f R 9 it is sufficien t to sh o w S9 for fo rm u les in ex te n d ed n o rm a l form .
L et A a n d В be fo rm u la s in c x ta n d e d d isju n ctiv e n o rm a l fo rm a n d Ι
Α -> В. Let С be th e result o f e ra sin g in A all p ro p o s itio n a l v ariab les, w hich d o n o t o c cu r in В. С exists b ecau se o f S7.
F o r all e le m en ta ry c o n ju n c tio n s A; o f A th e re is a n e le m en ta ry c o n ju n c tio n Cj o f С su ch , th a t fo r th e sets o f o c c u rin g a to m ic fo rm u la s {Aj} a n d |C j] h o ld s {CjJ ε [A j]. F o r these Aj a n d Cj th e c o n d itio n a l Aj -> Cj is p ro v a b le b ecau se o f A 3, a n d so is |~ Aj -> С fo r all Aj. By S4 follow s | - A -* C.
Since I-
A
-► B. fo r allAj
b e ca u se o f RIO is valid | -Aj
-* B. A n yAj
is a c o n ju n c tio nCj
лD;,
a n dDj
d o e s n o t sh a re v a ria b le s w ithC;
a n d B; th e re fo re (by S5) fo llo w s ICj
-> B. Since th is h o ld s fo r allCj,
| - С -» В is valid because o f S4. ■C o n d itio n a ls . B ased o n S tric t E n ta ilm e n t 8 7
I f |·· A -» В a n d A is n o t a c o n tr a d i c t i o n , th e n th e re is a fo r m u la С s u c h , th a t A С a n d С -» В . a n d C' c o n t a in s o n ly v a r ia b le s , o c c u rin g in B.
F SR is n o t e q u iv a le n t to the relev an t system F D E o f first d eg ree en ta ilm e n t. In all m e n tio n e d sy stem s 1— (( — p q) p) -> q (th e у-p rin cip le) is valid, in F D E n o t. In F D E we h av e u n re stric te d tra n sitiv ity , S8 sh o w s, th a t F SR to g e th e r w ith tra n s itiv ity co llap ses to a system o f m a te ria l im p lic a tio n .
Fs r is n o t e q u iv a len t to th e first d egree frag m e n t o f th e system SI o f s tric t im p lic a tio n . T h e so called p a ra d o x e s o f stric t im p lic a tio n a re n o t p ro v a b le in F S R . b u t it is easy to see, th a t F SR is a su b sy stem o f SI.
H u m b o ld -U n iv c r s ity , B erlin G e r m a n y
Uwe Scluj'fler
O K R E S Y W A R U N K O W E O P A R T E N A Ś C I S Ł Y M „ E N T A I L M E N T '
W a r ty k u le rozw aż;» się o k re s y w a ru n k o w e o p a r te n a s y s te m ie ś cisłeg o e n ta ilm e n t F s s k o n s tru o w a n y m p rz e z W e s s e la . P o p rz e z u z u p e łn ie n ie a k s jo m a ty k i i re g u ł in re re n c ji F s o trz y m u je się s y s te m F SK. p o s ia d a ją c y d w ie s t r u k t u r y im p lik a c y jn e . ty p u : e n ta ilm e n t i o k re s u w a ru n k o w e g o . K o n s e k w e n c ją d a ls z e j m o d y fik a c ji s y s te m u F SK p o p rz e z w p r o w a d z e n ie d o d a tk o w y c h re g u ł in fe re n c ji, s y s te m y F SK5. F SK6 i F SK7. je s t z m ia n a o d p o w ie d n ic h k la s o k re s ó w w a r u n k o w y c h b ez z m ia n y e n ta ilm e n t.
