Towards a Theory of Causal Implication

We present a formal theory of causal implication, which allows to reason about cause-effect relationship in a more satisfactory way than the classical calculus of proposition. In this paper only implicational subsystem is presented, but it easily extends to other propositional conjuncts and quantification. The system could serve as an underlying formalism for the development of rule-based expert systems. The classical propositional under1 yi ng formal i sm for ru commonly acknowl edged that materi a1 im~l ication is use logic is commonly acknowledged to be an le-based expert systems [4,5,15]. It is a1 so it is not a satisfactory formalism [5]. The d to represent cause-effect relationship between real world events in spite of the fact that its semantics differs quite essentially from our intuitive notion of causality. The problems connected with the use of material implication were extensively studied by logicians. The book of Anderson and Belnap [I] is a mine of information. But the motivation for most of that work (there are some exceptions such as [2], [8], [14]) came from proof theory and as a rule logical works on relevant implication are not dealing with causality. There are very many valuable ideas scattered around in philosophical literature. [16] is a very good and concise collection of some recent papers on the philosophy of causation. Several very important ideas pertaining to the development of the appropriate theory of causality were developed by A1 community. Among them works on non-monotonic logic [9,10] and logic programming [6,7] are especi a1 ly re1 evant . secondary r e l a t i o n , and " i n f e r s " f o r t he t h i r d one. We w i l l denote them by t h e s igns ->, =>, and ) correspondingly. When we use ) ( t he t u r n s t i l e ) we w i l l spec i f y what s e t o f in fe rence r u l e s we mean by p r e f i x i n g t h e name o f i t t o t h e t u r n s t i l e . For example, we can say "a, a->b (CL)b", meaning c l a s s i c a l modus ponens. When we use "A (CL)6" we mean t h a t t h e s e t o f formulas A c l a s s i c a l l y imp1 i e s B when a1 1 occurrences o f d i f f e r e n t k inds o f i m p l i c a t i o n s i n A and B are i n t e r p r e t e d as ma te r ia l i m p l i c a t i o n . Our i n t u i t i o n f o r "causes" i s based on the f o l l o w i n g premises: f a c t s do n o t cause laws; laws do n o t cause laws; laws do n o t cause f a c t s by themselves; f a c t s cause o the r f a c t s t o happen according t o the laws. I n o the r words "->" ("causes") i s a f i r s t degree f u n c t i o n which accepts o n l y atomic f a c t s . No nes t i ng o f "->" i s allowed. The negat ion o f a f a c t i s i t s absence, which can cause o the r f a c t s . So, when we r e f e r t o fac ts , we mean e i t h e r atomic f a c t s o r t h e i r negat ions. We w i l l denote a negat ion o f a as "not-a". The law o f double negat ion i s assumed throughout, and "not-not-a" i s au tomat ica l l y replaced w i t h "a". We can f o r t h e t ime being r e s t r i c t negat ion on l y t o f a c t s (so it, as w e l l as "->", i s a f i r s t degree func t i on ) because the negat ion o f a causal law i s more o r l e s s vacuous. From "a does no t cause b" no th ing much can be de r i ved about a o r b. It i s d e f i n i t e l y impossible t o d e r i v e from i t t h a t "a and n o t b" . Another s t a r t i n g p o i n t i s t h e f a c t t h a t "o r " , "and", and o the r p r o p o s i t i o n a l conjuncts are secondary t o i m p l i c a t i o n and negat ion. So, w h i l e i t i s poss ib le t o in t roduce them, we w i l l not, a t present, do i t. We are going t o use small l e t t e r s a,b,c ... x,y,z f o r atomic f a c t s and c a p i t a l l e t t e r s A,B,C.. . X , Y , Z f o r se ts o f formulas.