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 FOLIA PHILOSOPHICA 7, 1990__________________
Marek Nowak, P i o t r Rydzewski ON A GENERALIZED INFERENCE OPERATION
I . The aim of the paper is to p rese n t some form al o p e ra tio n u s e fu l fo r a f o rm a liz a tio n of the la rg e c la s s of re a s o n in g s . The a n a ly s is of v a rio u s kind s of l o g i c a l l y v a lu a b le reason ing s employed in s c ie n c e and everyd ay s i t u a t i o n s1 r e s u l t s in t h e ir e s s e n t ia l cha r a c t e r i z a t i o n :
( 1 ) The accepted prem ise must not be r e je c te d in the r e s u lt of in fe re n c e . T h e re fo re , i t g ets the s ta tu s of c o n c lu s io n .
(2 ) The enlargem ent of the in c o n s is t e n t s e t X of prem ises must not remove in c o n s is te n c y .
( 3 ) I f a p ro p o s itio n a i s in f e r r e d from the s e t X of p rem ises, i t can be a ls o concluded from e v e ry la r g e r s e t of prem ises Y i f o n ly Y U { a } i s c o n s is t e n t .
(4 ) A p ro p o s itio n at cannot be in fe r r e d from a c o n s is t e n t s e t X of prom ises, when X U { a } is in c o n s is t e n t .
I t i s s t г в i ght forw ard th a t any d e d u c tiv e reason ing possesses the above p r o p e r t ie s . For in d u c tio n , l e t us c o n s id e r the w e ll- -known sw an's example. The s e t of prem ises!
X = jswan no. i i s w h ite and i t s neck is lo n g : i * I , 2, . . . , 100} From X th e re can be in d u c t iv e ly drawn t h a t :
( a ) E ve ry swan i s w h ite . ((3) Every sw an's neck i s long.
Next prem ise: ( j ) Swan no. 101 i s grey and i t s neck i s lo n g , i s added to the s e t X. The new s e t o f prem ises X u { j } s t i l l a
l-• . .. . . . ••/. _ • V. • * \ * • . ‘
г ~i 1 e;9- K. A j d u k i e w i c z , K la s y f ik a c j a rozumowań, L in : J Ję z y k i p oznanie, t . 2, Warszawa 1965, pp. 206-225: " A r t i- f i c a l I n t e l l i g e n c e " 1980, No. 13; H. M o r t i m e r , L o g ik a in d u k c ji, Wybrane problem y, Warszawa 1982.
lows to in f e r (o ) s in c e X u j j } U Ц 1 is c o n s is t e n t , but (ot) is no more a c o n c lu s io n , s in c e ( X U y j ) U i s in c o n s is t e n t . That proves th a t (3 ) and (4 ) r e f e r to in d u c tio n .
Now we s h a ll p ro vid e another example to i l u s t r a t e non-monotonic rea so n in g . The s e t o f prem ises:
X = Ia p a tie n t has a sharp aćhe on the r ig h t s id e of h is b e lly | makes the d octo r to draw the c o n c lu s io n : (ot) the p a tie n t has a p p e n d ic it is . I f X i s e n larg ed by (p ) the p a tie n t cannot r i s e h is le g , a is s t i l l v a lid fo r (X и ) U is c o n s is t e n t . Hereby ( 3 ) . But when the next prem ise i s t h a t : (Jf) the p a tie n t had the appendix cu t o f f , then no doctor can in f e r ( a ) . N o tic e th a t
u { f } ) u j a } ie in c o n s is t e n t, thus (4 ) is o b v io u s ly f u l f i l l e d . We conclude th a t ( I ) - (4 ) c o n d itio n s are n ecessary fo r reason ing to be l o g i c a l l y v a lu a b le . A lb e it a l l the p r a c t i c a l l y used reason ing s posse3 many o th e r p r o p e r t ie s , however the fo rm a liz a tio n of the c la s s of a l l the reasonings fo r which (1 ) - (4 ) are v a lid seems to be j u s t i f i e d .
I
2. Where £ * (S , F j , . . . , Fn) is a p r o p o s itio n a l language and P ( S ) i s the power s e t o f 5 we s h a ll say th a t a fu n c tio n C: P(S)-~* —* PCS) i s a q e n e r a l i z e d i n f e r e n c e o p e r a t i o n (g . i. - o p e r a t io n fo r s h o r t) on £ i f f fo r any X, Y £ S, «. e S the fo llo w in g c o n d itio n s are s a t i s f i e d :
( i ) X £ C (X ),
( i i ) C (Y ) = S whenever X c Y and C (X ) ^ S,
( i l l ) c t e C ( X ) , X £ Y , C(Y ,
a )
/ S imply th a t o t « C ( Y ) , ( i v ) a t C (X ) whenever C (X ) / S and C (X ,« ) = S, s e t of form ulas X £ S i s c a lle d i n c o n s i s t e n t w i t h r e s p e c t t o C (С - in c o n s is t e n t ) whenever C (X ) = S.Lemma I . For any fu n c tio n C: P ( S ) — ► P ( S ) such th a t ( i i ) holds t r u e , ( i i i ) and ( i v ) are a ls o s a t i s f i e d i f and only i f f o r any X c S the fo llo w in g c o n d itio n s are e q u iv a le n t:
( . ) C (X ) / S ,
( . . ) C (X ) =
u
{ y G S: C (X U Y) i S and 3 2 S X : Y£ C ( 2 ) J .
P r o o f :
Denote fo r any X c S , U {y c S: C(X u Y ) / 5 and 3 Z s Х Г ' Y C C (Z )} = К ( X ) .
suppose th a t C (X ) / S and l e t c x e C ( X ) . So C ( X , a ) i S due to ( i v ) , thus a e K ( X ) . '
On the o th er hand assume th a t cx e K (X ). Then f o r some Y c S we have: a. e Y, C(X
U
Y) / S, 3 Z с X: Y с C (Z ). .So a e C(Z>, Z C X and accord in g to ( i i ) : C (X , a ) / S. Thus a e C ( X ) due to ( i i i ) .Now assume th a t: ( . . ) C (X ) * K (X ) and C (X ) = S fo r some X c S. So fo r any Y с S , CCX U Y) = S due to ( i i ) , thus K (X ) = 0. A c o n t r a d ic t io n .
(«=) Assume th a t ( i i ) holds tru e fo r С and the c o n d itio n s ( . ) , ( . . ) are e q u iv a le n t .
Ad ( i i i ) : suppose th a t a e C (X ), X C Y, C (Y , a ) t S . Then, accord in g to ( i i ) : C (Y ) / S, hence a ls o C (X ) i S. So C (Y ) = K (Y ) and C (X ) - K ( X ) . Hence we o b ta in th a t a e C (Z ) fo r some Z с X. Thus a 6 C (Y ) s in c e X C Y and C (Y , cx) / S.
Ad ( i v ) : le t a e C (X ) and C (X ) / S. Then we have: oc « K(X),
so due to ( i i ) : C (X ,o t ) / S . D
L e t R, T be b in a ry r e la t io n s on P ( S ) . C onsider the fo llo w in g c o n d itio n s :
(A>R R is r e f le x iv e ;
( B ) R <X, У> s ( | i f f fo r each 0 e Y: ^X, {o}> « R;
( A ) T <X, Y> « T and X U Y S ť II Y' im ply th a t <X* , Y'> с T; (A ) <X, X> e T i f f <X, S> « R;
(В У ^X., X> jť T and <X, R im ply th a t <X, T; (C ) <X, {< *[> « R, X C Y, <Y , {< *}> * T im ply th a t <Y, | a } > « R ; fo r any X, X * , Y , Y ' c S , a « S .
L e t CR T: P ( S ) — *• P ( S ) be a fu n c tio n d e fin e d as f o llo w s : fo r any X c S
S i f < X, X > e T
U { y b S: <X, Y> fí T and 3 Z £ X: <Z, Y > e R } ' o th , Lemma 2 . For any b in a ry r e la t io n s R, T on P ( S ) f u l f i l l i n g (B )R> ( A ) j , (А ) , ( В ) , (C ) and any X, Y c S the fo llo w in g c o n d itio n s are s a t is f i e d : (1 ) <X, Y> e T i f f CR T(X U Y) = S; (2) < P r o o f : (2 ) <X, Y> e R i f f Y C C R T( X ) . We f i r s t show th a t fo r any X С 5,
(3 )
U j y t S :
<X, Y> # T and 3Z С
X: <Z, Y> « r } / S which im p lie s ( * ) CR T(X ) » S i f f <X, X> 6 T.Suppose th a t (3 ) does not h o ld . Then a cco rd in g to ( A ) T and ( B ) R fo r some X c S , f o r any c x « S , <X, {л ]> )Г Т and <Z, ja }> e R fo r зоюе Z с X. So fo r any a « S , <X, {« }> « R due to ( C ) . Hence a cco rd in g to ( B ) R and (A ) we have: <X, X > e j . Thus a c o n t r a d ic t io n by ( A ) T .
Now, we im m ediately have ( I ) by ( A ) j and ( * ) .
To prove (2 ) assume th a t <X, Y> « R and CR T(X ) i S . Then from ( # ) : <X, X> 4 T and from ( B ) R : <X, e R fo r any oi <• Y. Thus Y с CR T (X ) due to the d e f in it io n of CR T> On the o th er s id e assume th a t ’ y с Cr t ( X ) , I f CR T(X ) = S, then from ( * ) and (A ): <X, S> e R, hence accord in g to ( Ś ) R : <X, Y>6 R. So suppose th a t ^R, T^*^ ^ Ihen <X, X> 4 T due to (# ) and fo r any ct e Y there e x is t s U c S 3uch th a t c*« U, <X, U> * T and <2, U> « R f o r some Z Ł X. Hence fo r any a « Y, <X, |a }> fŕ T due to ( A ) ? and
{ ^ J > e R from ( B ) R . T h e re fo re , accord in g to ( C ) , fo r any oíe Y, <X, {oi]> e R, thus <X, Y> e R by ( B ) R . Q’
Using lemma I and the c o n d itio n ( * ) from the proof of lemma 2 one may prove the fo llo w in g
\.j For two b in a ry r e la t io n s R, T on P ( S ) f u l f i l l i n g ( A ) R , ( B ) r , (A)^r, ( A ) , (Ö ), (C ), CR j i s a g e n e ra liz e d In fe re n c e
o p e ra tio n on S. ’ Q
Now, denote by Я the fa m ily o f a l l p a ir s <R, T> об r e la t io n s f o r which the c o n d itio n s ( A ) R , ( B ) R , <A) T , ( A ) , ( 0 ) , (C ) are s a t i s f i e d and by 2 the c la s s o f a l l g . i.- o p e r a tio n s on S.
Theorem 2. For any С « 3 th e re e x is t s a p a ir <R, T>e £ such th a t С = CR T . M oreover, the coriespúndence <R, T> — is
u nique. ’ '
P r o o f :
Using lemma I one may choose fo r g ive n g Л .- o p e ra tio n С th4 p a ir <R, T> such th at C = CR j in the following way: fo r any X, Y E S
-<X, Y> e T i f f C (X g Ý) = S,
I t i s easy v e r i f i c a t i o n th a t <R, T> « £ . F i n a l l y , i t i s obvious due to lemma 2, th a t fo r any <R1 , Tl> , <R2, T2>e J i , CRl =
= CR2 T2 im p lie s th a t Hi = R2 and T1 = T2. О
3, Each consequence on § i ’ e - 3 fu n c tio n C: P ( S ) — » P ( S ) such th a t fo r any X, У c S, X c C ( X ) , C (X ) q C (Y ) whenever X t Y and C (C (X )) с C (X ) proves to be a g .i,- o p e r a t io n , T h e re fo re the r e la t io n a l d e s c r ip tio n o f consequence o p e ra tio n is p o s s ib le . To get the a d d itio n a l c o n d itio n s fo r <R, T>*s, the fo llo w in g lemma is u s e f u l.
Lemma 3. An o p e ra tio n C: P ( S ) — > P (S ) d e fin e d fo r e v e ry X c S by the condition:
S ' i f X 4t p C (X ) =
-K (X ) i f X e i )
w ith с PCS) and K: P ( S ) — * P ( S ) being any fu n c tio n , i s a con s e q u e n t on S. i f and o n ly i f
C l) fo r any X, Y ь S, X c Y, V * ip , K (X ) / S im ply th a t X e (2 ) fo r any X * J 3 , K (X )e p whenever K (X ) / S,
(3 ) the r e s t r i c t i o n К [ s a t i s f i e s the c o n d itio n s fo r a closure o p e ra tio n .
Pro o f by easy v e r i f i c a t i o n . °
Theorem A g . i .- o p e ra tio n CR T on 5, fo r <R, T> e Ä is a consequence o p e ra tio n i f and o n ly i f the fo llo w in g c o n d itio n s are s a t is f i e d fo r any X, Y, Z C S:
( C ) R <Z, Y> e R whenever <X, Y> « R and X G, Z,
(0 )д R is t r a n s i t i v e , „
-( В / <X, X> 4 T and <X, Y> e R im ply th a t <X, Y> * T. E r q o f :
(-*) According to lemma 2, the c o n d itio n s ( C ) R , C0)R , ( 0 / follow im m ediately from the ussumtion th a t j i s a consequence o p e ra
t io n . ’
( f ) Assume th a t <R, T> c X f u l f i l s the c o n d itio n s ( C ) R , ( 0 ) R , ( B ) ' . Due to lemma 3 i t i s s u f f i c i e n t to show th a t the c o n d itio n s ( 1 ) , ( 2 J , ( 3 ) from i t h old tru e fo r
P • {X t S : <X, X> i T } and (u s in g (CR )> K (X ) « и { ¥ C S : <X, Y> e R - f } , X C S.
N o tice th a t fo r any X c S , К ( X ) / S ( c f . the proof of lemma
2
).So the c o n d itio n (1 ) fo llo w s im m ediately from (A )y . To prove ( 2 ) assume th a t X « J) , th a t i s <X, X> i T. N o tic » th a t fo r any ot e K (X ), <X, | a } > i S by ( B ) R , so a ls o from ( B ) R : <X, K (X )> c R,
thus <X, K (X )> ft T by ( B ) ' , 30 from (A ) j : < K (X ), K ( X ) > * T i . e . К ( Х ) б £ .
N a t u r a lly , fo r any Х « Р , X с K (X ) due to ( A ) R .
To prove the m o n o to n ic a lity of k [ %) assume th a t <У, Y> t T, X C Y and ot « K (X ). Hence <X, {<*}> * R by and t h e re fo r e , accordin g to ( C ) R we have: <Y, { “ }> c R- Moreover, due to (8) ' , <Y, |a }> 4 T, thus a e K ( Y ) .
To the end assume th a t <X, X> ? T and об e K ( K ( X ) ) . Then < K (X ){o t}> ^ t and < K(X), {a t} > e R due to ( A ) T and ( B ) R . S in ce X Q K (X ), so accord in g to ( A ) y : < X , { a } > £ T. Moreover, from ( B ) R we have: <X, K (X )> £ R which to g e th e r w ith < K (X), { t t } > с К, lead s by ( 0 ) R to the c o n clu sio n th a t <X, { a } > c R. Thus c r e K ( X ) . О
U n iv e r s it y of Łódź Poland
Marek 'lowak, P i o t r Rydzewski O UOGÓLNIONEJ OPERACJI INFERENCJI
Celem p racy j e s t a n a liz a form alna .pewnych ogólnych w łasn o ści c h a ra k te ry z u ją c y c h wnioskowania. Autorzy tw ie rd z ę , że p o siad a n ie tych w łasn o ści j e s t warunkiem koniecznym, aby wnioskowanie b yło lo g ic z n ie w artościo w e. Wprowadzają aksjom atycznie p o ję c ie "u o g ó l n io n e j o p e r a c ji i n f e r e n c j i " , k tó re fo rm a ln ie ujmuje wnioskowanie mające owe cechy. N astępnie re p re z e n tu ją uogólnioną o p e ra c ję i n f e r e n c j i przy uZyciu r e l a c j i b in a rn ych o k reślo n ych na podzbiorach ję z y k a . Podają również taką re p re z e n ta c ję d la lo g ic z n e j o p e r a c ji konsekw encji (każda lo g icz n a o p e ra cja konsekw encji j e s t u o g ó ln io ną o p e ra cją i n f e r e n c j i ) .
