Max Urchs
Powerful paraconsistent logic
Acta Universitatis Lodziensis. Folia Philosophica nr 9, 109-114
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 9, 1993
M a s Urch.s
P O W E R F U L P A R A C O N S IS I E N T L O G IC
C o m in g to Ł ó d ź we h av e h a d a frien d ly d iscu ssio n in (lie tra in . Im agine th a t th e re w as a little green logician in th e o u r c o m p a rtm e n t to o . w h o w ro te d o w n a n y th in g we said. So he p ro to c o le d e.g. m y a ss e rtio n ..T o c o m e to Ł ó d ź. the tra in will go th ro u g h K u n o w ic e ” as well as th e o p p o s ite o p in io n , nam ely ..Y o u m u st be stu p id ! By no m e a n s it will tak e th a t ro u te " . A s th e little green m a n w as a lo g ician , he feels u n c o m fo rta b le : o n his sheet he w ro te d ow n a sen ten ce H to g e th e r w ith its n eg atio n non 1-І. In tw o -v a lu e d e x te n sio n a l logic o n e o f th e m m u st be false. W h en ce, a c c o rd in g to th e classical law e x falsa
(juotllihci a n y th in g follow s fro m H o r fro m non H. T h e re fo re th e little green
logician d ecided to sto p p ro lo c o llin g th e f u rth e r d iscu ssio n becau se o f its a b su rd ity .
W as he right to d o so'.’ P e rh a p s n o t. T h e q u a rre l a b o u t th e ro u te p ro d u c e d a n in c o n siste n t situ a tio n in w hich o p p o s ite claim s w ere u tte re d , th o u g h it did n o t resu lt in a so called ov erfilled s itu a tio n (th a t m e a n s in a situ a tio n in w hich a n y u tte re d sen ten ce w o u ld be a cc e p te d as tru e). T h e b e h a v io r o f all p a rtic ip a n ts in th e d iscu ssio n w as n o tw ith s ta n d in g (m o re o r less) ra tio n a l: we w o u ld be very s u rp rise d indeed if a fte r th a t a n y o f us w o u ld claim e.g. th a t b o th u tte re d sen ten ces a re tru e o r th a t K u n o w ice is th e very sam e to w n as Ł ó d ź.
In m y o p in io n , logic sh o u ld - at least to so m e e x te n t h a n d le w ith c o m p lex a rg u m e n ta tio n s o f ra tio n a l sp eak ers. Logic sh o u ld fo rm alize a p p ro p ria te frag m e n ts o f n a tu ra l la n g u a g e (e.g. th e la n g u a g e o f e m p iric a l th e o rie s) an d inv estig ate the fo rm al c o u n te rp a rts th u s o b ta in e d ra th e r th a n e la b o ra te so p h istica te d re g u la tio n s a b o u t how to use o u r la n g u a g e „ c o r re c tly " .
N a tu ra lly , it is n o t easy to specify, w h a t is a ra tio n a l sp e a k e r o r m o re gen eral: w h a t k in d s o f n a tu ra l la n g u a g e tex ts a re s u ita b le fo r logical
1 1 0 M a x U r c h s
in v e stig atio n . But never m in d , fo r the m o m e n t we ju s t su p p o se th a t som e o f th e m a re in fact in co n sisten t.
T h e re a re several a tte m p ts to form alize in c o n siste n t s itu a tio n s, th a t m ean s to c o n stru c t logical calculi w hich s u p p o rt inferences fro m in c o n siste n t sets o f p rem ises.
O n e o f th o se calculi, th e system АЬ o f so called ..discussive lo g ic", w as in tro d u c e d by S tan islaw J a ś k o w s k i1. H e n o te d a d istin c tio n b etw een tw o p ro p e rtie s o f a logical c alcu lu s, w hich a re u su ally n o t discern ed w ith in classical logic:
D ef. 1: A c alcu lu s (F O R , C n ) is called inconsistent, iff fo r so m e H є F O R : H e C n ( 0 ) & ,H є C n (0 ).
D ef. 2: A c alcu lu s (F O R , C n ) is called overfilled (o r trivial) iff C n ( 0 ) = F O R .
Ja s k o w sk i's aim w as th e c o n stru c tio n o f a se n te n tia l calcu lu s w hich m eets th re e co n d itio n s: 1) w hen a p p lie d to in c o n siste n t sy stem s it w o u ld n o t alw ays en ta il th e ir triv iality , 2) it w o u ld be rich e n o u g h to e n ab le p ra c tic al inference, 3) it w o u ld have a n in tu itiv e ju stific a tio n .
T h e o ries w h ich a rc in co n sisten t b u t n o t triv ial a rc called ..p a ra c o n s is te n t" (th e n am e w as p ro p o se d by M iró Q u e sa d a , it m e a n s ..b e y o n d th e c o n sis te n t" ).
O n e c an say th a t Jaśk o w sk i n o t o n ly c o n stru c te d the first fo rm al p a ra c o n s is te n t system in h isto ry b u t th a t he m a d e av ailab le the in e ta lh eo reliea l b a c k g ro u n d to h a n d le the p h e n o m e n o n „ p a ra c o n s istc n c y ” form ally.
T h e p o in t o f his c o n stru c tio n is this: to a cc e p t a sen ten ce m ean s to claim its v alid ity , b u t w ith h id d en re strictio n s: „ so m e o n e o f th e p a rtic ip a n ts (in the d iscu ssio n ) claim s th a t H is tru e ” o r ,.H is tru e , p ro v id e d th a t the term s are used a cc o rd in g to so m e o f th e a d m issib le m e a n in g s" o r so m eth in g like th a t. In s te a d o f the u su al ,,H is true*· w e have h e n ce fo rth ,,it is p o ssib ly tru e th a t H " w ith reference to so m e c o n c e p t o f po ssib ility .
J aśk o w sk i decid ed (n o t very fo rtu n a te ly , p e rh a p s) to ta k e th e Lewis system
S 5 as m o d a l basis o f his c o n stru c tio n .
V ery ro u g h ly , his o rig in al d e fin itio n c an be re sta te d as follow s: let F O R j be the set o f all fo rm u la s b u ilt u p from a d e n u m e ra b le set o f p ro p o s itio n a l v a ria b le s by m ean s o f so m e b o o le a n c o m p le te set o f fu n c to rs a n d tw o a d d itio n a l tw o -a rg u m e n t „d isc u ssiv e ” conn ectiv es: d iscussive c o n ju c tio n &tj a n d d iscussive im p lic a tio n » j. N e x t, let / be a tra n s la tio n fro m F O R j in to the m o d a l lan g u ag e, ľ leaves p ro p o s itio n a l v a ria b le s u n c h a n g e d as well as all
1 S. J a š k o w s k i. R a c h u n e k z d a ń d la s y s te m ó w d e d u k c y jn y c h s p r z e c zn y c h , „ S tu d ia S o c ie ta tis S c ie n tia ru m T o r u n e n s is ” 1948. sec. A . n r 5. p . 5 7 -7 7 .
b o o le a n co n n ectiv es. F o r M b eein g the .S’5-possi b ílily we set: ř(H &d G ) = dr t ( H ) & M /(G ) a n d t(H » d G ) = dr M ; ( H ) » t(G ) . ( D o n 't w o rry a b o u t th e m o tiv a tio n o f th a t so m e w h a t stra n g e „ in c lin e d ” fu n c to rs, o th e r d efin itio n s a re p o ssib le.) N o w we a re ab le to define:
D 2 = d ľ {H є F O R d: M t(H ) є S5}.
Ja s k o w sk i’s idea c a n be g en eralized in to several d ire c tio n s: it is p o ssib le to use a la rg e class o f m o d a l system s, a m o n g th em even n o n -n o r m a l calculi, to o b ta in in te re stin g d iscussive system s. F u r th e r, it seem s m o re n a tu ra l to ta k e in d iscussive logic as a logical c alcu lu s (i.e. a co n se q u e n ce o p e ra tio n in a fo rm al la n g u ag e) ra th e r th a n as a set o f fo rm u la s. F o r each m o d a l logic 5 c o n ta in in g
S 3 in P a ra ko n sisten z in schw achen M o ila lka lkü len we e x p lain ed a c o n se q u e n ce
o p e ra tio n C n s in th e d iscussive la n g u a g e F O R d a n d g av e a d ire c t sem a n tic a l c h a ra c te riz a tio n fo r th e sy stem s D s th u s o b ta in e d 2.
F a c t l : 'C n S3( 0 ) = D 2
F a c t 2: V 5 V X £ F O R d V H , F є F O R d: H »dF є C n s ( X ) < = > F є C n s (X u {H})
U su a lly we in te rp re t th e fact, th a t th e d e d u c tio n th e o re m h o ld s in a system as a p ro p e rty o f th e re g a rd e d im p lic a tio n . B ut no w we g o th e o th e r w ay ro u n d : we a lre a d y k n o w , th a t th e d iscussive im p lic a tio n posesses a lo t o f p r o p erties ex p ected o f a d iscussive in feren ce. By d e d u c tio n th e o re m th e y are in d u ced to th e co n se q u e n ce re la tio n . M o re o v er, th e o rig in al system D 2 o f Jaśk o w sk i b elo n g s to the class o f system s D s o b ta in e d in th e a b o v e c o n stru c tio n .
T h e re fo re we call D s th e class o f discussive J a śk o w sk i system s. E a c h o f th e m g e n erates a class o f h ig h e r-d e g re e d iscussive system s. S u rp risin g e n o u g h : fo r n o rm a l S th e w h o le m a n ifo ld c o llap ses in to D 2. S o m e fu r th e r p ro p e rtie s o f th e n o n -n o rm a l b a se d sy stem s a re p re sen te d in P a n ik o n siste n : in schw achen
M odalka lkiilen 1.
A c o n stitu tiv e p ro p e rty o f all Jaśk o w sk i sy stem s is th e rejectio n o f A d ju n c tio n : n o n e o f th e sy stem s acc e p t the rule H , G H d H & G . T h is is e ssen tial fo r th o se system s. V ery w eak a ss u m p tio n s a b o u t th e u n d e rly in g m o d a l sy stem allo w to p ro v e b o th th e law o f ex clu d ed c o n tra d ic tio n (H & - H ) a n d C o n ju n c tiv e S p rea d H & - H »d G . T h e re fo re A d ju n c tio n w o u ld lead fro m in c o n siste n c y to triv ia lity a n d s h o u ld be ru led o u t co n se q u e n tly .
2 M . U r c h s , P a r a k o n s is ie n : in sch w a ch e n A io d a lk a /k iile n . K o n s t a n z e r B e ric h te L o g ik a n d W is s e n s c h a fts th e o r ie N r . I L 1990,
112 M a x U r c h s
But w o rse luck, th e su p e rin te n d e n t o f p a ra c o n s is te n t logic d isagrees: ..G iv e n a choice o f rejectin g o n e o r o th e r o f A d ju n c tio n a n d C o n ju n c tiv e S p read to av o id p a ra d o x e s a n d c a ta s tro p h ic s p re ad fro m a n in co n sisten cy , the rejectio n o f A d ju n c tio n is th e w ro n g c h o ic e ” 4. P riest o ffers several serious o b je c tio n s to J a s k o w sk i's c o n stru c tio n .
In his o p in io n th e d iscussive c o n se q u e n ce o p e ra tio n is to o s tro n g , it is o n ly ..h a lf-h e a rte d ly ” p a ra c o n s is te n t (b e c au se o f a cc e p tin g th e e x contradictione
qnodlibet p rin cip le) a n d th e re fo re ..to ta lly u n su ita b le as the u n d e rly in g logic o f
n aiv e set th e o ry [...] (a n d ) o f n aive sem a n tic s” 5.
T h a t seem s n o t very d a m a g in g to d iscussive logic, b ecau se th e Ja śk o w sk i sy stem s sh a re th is p e cu liarity w ith a lot o f h o n o u ra b le logical calculi.
H o w ev er, the seco n d a rg u m e n t lo o k s really d a n g e ro u s lo r the n o n -a d ju n c tiv e a p p ro a c h to p araco n siste n cy : ..T h e o th e r side o f th is o b jectio n to d iscu rsiv e logical co n se q u e n ce is th a t it is to o w eak. T o be ex act, let Σ be a n o n -n u ll set o f z ero d egree fo rm u la s a n d let A be a first d egree fo rm u la . T h en if Σ | = (i A th e re is so m e В є Σ such th a t {B j |= а А. Т о see this, su p p o se for
m in c tio th a t th e re is no B e Σ such th a t jB] |= j A. T h e n fo r every В we can
find a m o d el A /B such th a t, fo r so m e w o rld w in М ц . В is tru e in w. w hilst for no w o rld w. A is tru e in w. Let M be th e c o lle ctio n o f all th e w o rld s in every Λ/β. T h e n M is c o u n te r-m o d e l to Σ j=a A ” ".
W h en e v e r it is p o ssib le to d e d u ce A fro m a set Σ o f prem ises. A c an be o b ta in e d fro m o n e single e lem en t o f Σ. In such a system . P riest c o n clu d es, n o th in g new will be g ain ed by c o m b in a tio n o f in fo r m a tio n s o r k n o w ledge b ases o f tw o o r m o re p a rtic ip a n ts in a d iscu ssio n . ..T h is sh o w s, th a t as a logic fo r d ra w in g in ferences in real life situ a tio n s, d iscu rsiv e logic is useless” . A n d : ..th e n o n -a d ju n c tiv e a p p ro a c h to p a ra c o n siste n c y sh o u ld be d ism issed ” 7.
P riest seco n d a rg u m e n t lo o k s q u ite o b v io u s b u t u n fo r tu n a te ly , it is no t true: th e s tru c tu re M p re sen te d a b o v e m ay be in co n siste n t, i.e. in g en eral it is n o t a m o d el. P e rh a p s, th e fo llo w in g e x am p le will suffies.
E x a m p l e : F o r a n y H . F є F O R j:
H . F »(t“tF ) b u t n e ith e r Η » d ^ F ) n o r F И л ( Н » j - i F ) .
T o com e b ack to th e first a rg u m e n t, fo r o th e r re a so n it is n o t really c o n v in c in g to o . F irst o f all, the d iscussive sy stem s respect th e discussive A d ju n c tio n :
H . G t= H G .
4 G . P r i e s t . P. R o u t l e y . ľ i i s l H is to r ic a l In tro d u ctio n : A P relim in a ry I f i s t o n o f P a ra co n sisten t ancf D ia lelftic A p p ro a ch e s, (in:] P a ra co n sisten t L o g ic . E s s a y s on th e In c o n s is te n t. M ü n c h e n 1989. p. 48.
5 G . P r i e s t . R. R o u t l e y . S y s te m s o f P a ra c o n siste n t L o g ic , [in:] P a ra co n sisten t L o g ic .... p. 160.
6 Ib id., p . 161. 7 Ib id ., p . 162.
T his is n o t d a n g e ro u s at ail b ecau se o f H & j- iH ff G . A n d seco n d , it seem s to m e th a t tru th -fu n c tio n a l A d ju n c tio n is n o t a t all so n a tu ra l as it seem s a t the first in sig th . Im ag in e once m o re th e q u a rre l in the tra in . It is q u ite n o rm a l in a n y discu ssio n a n d ra llie r h arm less fro m logical p o in t o f view, if th e re arc claim ed o p p o site sentences. But if s o m e b o d y w o u ld claim the c o n ju n c tio n o f tw o o p p o site sen ten ces s im u lta n eo u sly , he w o u ld allo w a d eep insig th in his in tellectu al c ap acities. P e rh a p s a n y such sp e a k e r sh o u ld be ex clu d ed from r a tio n a l d iscu ssio n . R u lin g o u t the tru th fu n c lio n a l A d ju n c tio n seem s p erfectly in k eep in g w ith the in tu itio n s u n d e rly in g Ja s k o w sk i's c o n stru c tio n . In o th e r w o rd s: th e re a rc s itu a tio n s in w hich ra tio n a l sp e a k e rs m a in ta in o p p o site e m p irical facts. Such s itu a tio n s sh o u ld be co n sid e re d by fo rm al logic. It is po ssib le to m o d elize th em w ith in p a ra c o n s is te n t logic. H ence it is highly e n titcled in deed to call in q u e stio n the un iv ersal valid ity o f th e classical p rin cip le e x fa ts o quodlibel.
I d o n 't know w h e th er D u n s S c o tu s really w ished to ex p ress th e p ro p e rty I have in m in d . B ut a n y h o w , it seem s n o t to be w o rth le ss to d istin g u ish
intplicalional overfillness. i.e.
H » (-.H » F)
fro m its conjunctive version H & -iH )> F. T h e hist o n e co in cid es u n d o u b te d ly w ith th e e.v contradictione quodlibet prin cip le. H en ce it m ak es sense to in d en tify th e la ir o f iniplicalional overfillness w ith the e x fa ts o qttodlihet p rinciple.
In sp ite o f th e p a rt o f th e e x fa ts o quod/ihet it is q u ite a n o th e r th in g w ith th e seco n d p rin cip le e x contradictione qttodlihet. I p erfectly a g re e w ith L u k asiew icz's c o n v ic tio n (sh a re d by Jaśk o w sk i to o ): the law o f exclu d ed
contradiction seem s to be th e k e y sto n e o f a n y ra tio n a l a r g u m e n ta tio n , it is by
all m ean s th e c rite rio n fo r ra tio n a lity . In m y o p in io n it is ex actly th e re - in the realm o f p a ra c o n s is te n t logic w h e re passes th e b o rd e rlin e b etw een th e serio u s in v e stig atio n s o f n o n-classical logics a n d m ysticism . L ogical system s, w hich v io late th e first p rin cip le m ay be in terestin g . S ystem s v io la tin g th e seco n d o ne deserve all o u r susp icio n .
H a v in g in m in d th e e n o rm o u s m e th o d o lo g ic a l p o w e r o f th e e x co ntradic
tione quodlibet prin cip le, as it w as d e m o n stra te d in L u k a sie w icz 's e ssa y 8, wve
define:
Def. 3: A system (F O R . C n ) is called (m ethodological) pow erful iff C n ( i) = F O R . (A s u su al, L d e n o te s th e lalsu m .)
I’a ra co n siste n sists in tro d u c e d th e im pressive a n d c ra fty n o tio n o f an explosive system : logical sy stem s a re e ith e r p a ra c o n s is te n t o r explosive. If you
s J. L u k a s i e w i c z . О za s a d z ie s p r z e c zn o ś c i u A r y s to te le s a , P W N . W a rs z a w a 1987: c o m p . J a ś k o w s k i . R a c h u n e k z d a ň ...
1 І 4 M a x U r c h s
th in k o f an d e d u ctiv e sy stem as o f a vehicle (e.g. a m o to r b ik e o r a sp ace s h u ttle ) w hich b rin g s y o u fro m prem ises to c o n clu sio n s, th e n it w o u ld be c alm in g to k n o w th a t it is n o t explosive. O n th e o th e r h a n d , to be a g o o d sp ace sh u ttle , it is n o t e n o u g h to be n o n -explosive: it sh o u ld be p o w e rfu l as well. N o w , we a re ab le to p re sen t J a s k o w sk i’s d iscussive c alcu lu s as th e first n o n -ex p lo siv e, b u t p o w e rfu l system o f p a ra c o n s is te n t logic.
Ig o r U rb a s o b serv ed , th a t so m e system s (e.g. J o h a n s so n s m in im a l calcu lu s o r so m e o f A rru d a 's set th e o re tic a l system s) fulfil th e fo rm al c rite rio n o f p a ra co n siste n cy . th o u g h th e y a re very close to b eeing explosive. T h e re fo re he m ak es use o f a m o d ified c o n c e p t o f „ stric t p a ra c o n s is te n c y ” 0 to find o u t the system s p a ra c o n s is te n t in sp irit, n o t m erely fo rm al. H e a rg u e s 10, th a t (except in th e case o f co n clu sio n s w hich a re th e o re m s) strictly p a ra c o n s is te n t inferences from c o n tra d ic to ry prem ises satisfy the relev an t re q u ire m e n t o f s h ared v ariab les. H o w ev er, it is n o t h a rd to prove:
F a c t 3: N o D s fulfils th a t req u irem en t.
W h en ce th e Ja śk o w sk i d iscussive system s w o u ld be ru led o u t fro m p a ra c o n s is te n t logic in the stric t sense. Ja s k o w sk i’s D i is su rely n o t perfect. N ev erth eless, if so m e c rite rio n excludes n o t o n ly D?. b u t th e w hole fam ily o f Ja s k o w sk i’s d iscussive sy stem s sim ply b ecau se th ey v io la te tru th -fu n c tio n a l A d ju n c tio n , th a n th e c rite rio n s h o u ld be tried very carefu lly . So we h av e to co n clu d e, th a t e ith e r U rb a s c o n ju n c tu re is w ro n g , o r th e c o n c e p t o f stric t p a ra co n siste n cy is m islead in g . B ut p e rh a p s, tru ly p a ra c o n siste n sists c an defen d b o th . In s titu te o f L o g ic a n d T h e o r y o f S cience L eip z ig U n iv e r s ity G e rm a n y M a x U rchs O „ M O C N Y C H " L O G I K A C H P A R A K O N S Y S T E N T N Y C H W p ra c y p re z e n to w a n e są n ie k tó r e w y n ik i d o ty c z ą c e w p r o w a d z o n e j, w je d n e j z p o p rz e d n ic h p ra c a u t o r a , k lasy tzw . d y s k u s y jn y c h s y s te m ó w J a ś k o w s k ie g o . W s z c zeg ó ln o śc i z w ra c a się u w a g ę n a p a r a k o n s y s te n tn o ś ć o w y c h s y s te m ó w , b r a k re g u ły d o łą c z a n ia p ra w d z iw o ś c io w e j k o n iu n k c ji o r a z re s p e k to w a n ie re g u ły d o łą c z a n ia tzw . k o n iu n k c ji d y s k u s y jn e j w k a ż d y m z ty c h sy s te m ó w . J a k o s p e c ja ln e s y s te m y p a r a k o n s y s te n tn e s ta n o w ią o n e p u n k t w yjścia d o d y s k u s ji n a d p e w n y m i w ła s n o ś c ia m i lo g ik p a r a k o n s y s te n tn y c h . A u t o r p o d d a je k ry ty c e s ta n o w is k o P rie s la n a te m a t roli d o łą c z a n ia k o n iu n k c ji d la lo g ik p a r a k o n s y s te n tn y c h o r a z s e n s o w n o ś ć p o ję ć ścisłej p a r a k o n s y s te n - cji.
9 D . B a t e n s, P a r a c o n s is te n t E x ie n s io n a / P r o p o s itio n a l L o g ic . ..L o g iq u e e t A n a ly s e " 1980. N o . 9 0 -9 1 . p. 1 9 5 -2 3 4 .
