permissions and obligations - IJCAI
PERMISSIONS AND OBLIGATIONS
L. T h o r n e M c C a r t y C o m p u t e r Science D e p a r t m e n t Rutgers U n i v e r s i t y New Brunswick, New Jersey
A B S T R A C T : This article describes a f o r m a l semantics f o r the deontic c o n c e p t s - - the c o n c e p t s of p e r m i s s i o n and o b l i g a t i o n — which arises naturally f r o m the representations used in artificial intelligence systems Instead of treating deontic logic as a branch of modal logic, w i t h the standard possible w o r l d s semantics, we f i r s t develop a language f o r describing actions, and we define the concepts of permission and obligation in terms of these action descriptions. Using our semantic definitions, we then derive a number of intuitively plausible inferences, and we s h o w generally that the paradoxes which are so frequently associated w i t h deontic logic do not arise in our s y s t e m *
I INTRODUCTION The representation of deontic concepts — the concepts o f p e r m i s s i o n and o b l i g a t i o n - - has not yet been seriously addressed in the artificial intelligence literature, but there are numerous application areas in which these concepts seem to be required. In our w o r k on the T A X M A N P r o j e c t [ 1 ] [ 2 ] , f o r example, w e represent the characteristics of various kinds of stocks and bonds by describing the rules of permission and obligation which are binding, at any given time, on the c o r p o r a t i o n and its securityholders In our work on the "usufructuary" provisions of the Louisiana Civil Code [ 3 ] , just recently initiated, we have encountered a similar need f o r the representation of c o m p l e x permissions and obligations Nor are these examples c o n f i n e d to legal domains In the classical w o r k on single agent planning systems, e.g., [ 4 ] , the o p e r a t o r s w h i c h change the state of the w o r l d can be interpreted as a set of "permitted" actions, but in a m o r e realistic planning environment, w i t h multiple agents, we w o u l d e x p e c t to see "obligatory" actions as well, and we w o u l d e x p e c t to see the actions of one agent p r o d u c e modifications in the rules of permission and obligation binding upon another agent. Similar observations apply to the field of computer security, see, e.g., [ 5 ] , w h e r e there has been extensive debate over the appropriate "authorization mechanisms" f o r a community of computer users. For all of these purposes, a formalization of the c o n c e p t s of permission and obligation appears to be essential Outside of the field of artificial intelligence, there exists
**This article is based upon w o r k s u p p o r t e d by Grant No. MCS-82-03591 f r o m the National Science Foundation, Washington, D.C., and by a grant f r o m the Louisiana State Law Institute, Baton Rouge, Louisiana.
an extensive literature on the deontic concepts, by logicians [ 6 ] [ 7 ] , philosophers [ 8 ] [ 9 ] , and lawyers [ 1 0 ] [ 1 1 ] [ 1 2 ] , but the attempts to formalize these concepts have generally led to paradox Since the 1 9 5 0 s , deontic logic has been treated as a branch of modal logic, with the necessity' operator replaced by the obligation" operator, O. and the "possibility" operator replaced by the "permission" operator, P Many of the theorems of modal logic turn out to be intuitively c o r r e c t under this translation For example, the dual relationship between necessity and possibility becomes a dual relationship between obligation and permission, Op = ~P~p. and this formula certainly seems plausible If it is false that you are p e r m i t t e d to do n o t - p , then you are o b l i g a t e d to do p. and vice versa The formula Op > p. which is valid in any modal system with a r e f l e x i v e accessibility relation between possible worlds, w o u l d not be plausible in a deontic logic, since people in the actual w o r l d do not always abide by their obligations, but it can be replaced by the m o r e plausible formula Op => Pp, w h i c h is valid as long as every possible w o r l d has some possible w o r l d accessible f r o m it This point was f i r s t noted by Kripke, in one of his original papers on possible w o r l d s semantics [ 1 3 ] .
Despite these positive results, there are several other modal formulae which seem counterintuitive in a deontic logic, and which cannot be so easily modified. For example, the formula f o r disjunctive permission, Pp D Pip v q), contradicts our ordinary understanding of what it means to grant permission to do p v q, but this formula is valid even in the weakest modal systems Likewise, any formula containing an iterated operator, such as OPp or POp, seems anomalous in a deontic context, and yet the various modal systems are distinguished precisely by the way in which they handle these iterated modalities. Of course, it may make sense to say that you are p e r m i t t e d to impose a particular o b l i g a t i o n upon someone else, or upon yourself, and we might conceivably w r i t e this as POp, but the inferences we w o u l d make about such statements do not c o r r e s p o n d at all to the inferences w h i c h are valid in the standard possible w o r l d semantics. Finally, even the dual relationship between permission and obligation seems problematical if we cast it into the f o r m Pp = ~ 0 ~ p If it is f a l s e that you are o b l i g a t e d to do n o t - p , i.e., if it is f a l s e that you are f o r b i d d e n to do p, does it f o l l o w that p is p e r m i t t e d ? Stringing together all of these questionable inferences, it is not surprising that we can generate a host of "deontic paradoxes," and the literature is full of them For a survey, see [ 1 4 ] and [ 1 5 ] In this paper, we will develop a formal semantics f o r
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permissions and obligations which seems to avoid these difficulties, and we will do so in a way which is entirely natural f o r an artificial intelligence system Instead of representing the deontic concepts as operators applied to p r o p o s i t i o n s , as in a standard modal logic, we will represent them as d y a d i c f o r m s which take c o n d i t i o n d e s c r i p t i o n s and action d e s c r i p t i o n s as their arguments The most important part of this representation is the use of action descriptions in the place of p r o p o s i t i o n s Instead of granting permissions and imposing obligations on the state of the w o r l d itself, we will grant permissions and impose obligations on the actions w h i c h change the state of the w o r l d This is an approach long advocated by Castaneda [ 1 5 ] , and pursued in various f o r m s by von W r i g h t [ 1 6 ] [ 1 4 ] . but to carry out this approach in full it seems necessary to establish a connection b e t w e e n the abstract d e s c r i p t i o n of an action and the c o n c r e t e changes that o c c u r in the w o r l d w h e n the action takes place This has been a major c o n c e r n of artificial intelligence research throughout its history [ 1 7 ] [ 1 8 ] [ 1 9 ] , o f course, and w e will d r a w upon this earlier w o r k in constructing our formalisms Although the actions that we actually discuss in this paper are fairly simple ones, intended to highlight the principal features of the deontic representation, the action descriptions themselves can be extended to m o r e realistic situations, in several ways We will return to this point in our concluding remarks II DEONTIC S E M A N T I C S In this section we develop a formal semantic interpretation of the deontic concepts, using a variant of the possible w o r l d s approach Our strategy p r o c e e d s in stages We start w i t h an ordinary f i r s t - o r d e r language L and a set of states S. and we use these materials to construct a new language L A in w h i c h we are able to describe actions The formulae of L 1 are evaluated w i t h respect to the states in S. as usual, but the formulae of L A are evaluated w i t h respect to sequences of states, or w o r l d s . We thus have a way of saying that an action is "true' in, or is satisfied" by, a particular w o r l d The details of these constructions are presented in Sections IIA and II.B below. N o w consider a state r w h i c h is situated at the junction b e t w e e n a "past w o r l d v and a "future" w o r l d w We assume that there exists a set P which tells us, f o r r
each past w o r l d v, all the future w o r l d s w w h i c h are "permitted." W o r k i n g exclusively w i t h this p e r m i t t e d set P , we c o n s t r u c t three expressions w h i c h tell us whether an action at r is p e r m i t t e d , forbidden, or obligatory, respectively These expressions then b e c o m e part of our deontic language L D The details of these constructions are p r e s e n t e d in Section IIC b e l o w This is not the end of the story, h o w e v e r . Since each deontic e x p r e s s i o n has a definite t r u t h value at each state in S, it turns o u t that the language L D can be embedded within our original f i r s t o r d e r language L , and thus the p r o c e s s of linguistic c o n s t r u c t i o n we have outlined here becomes fully recursive This technique enables us to represent "dynamic" permissions and obligations, i.e., permissions and obligations w h i c h change over time, w i t h o u t the use of iterated modalities
This latter point is developed in Section II.D
The principal technical d i f f i c u l t y in this development rises in connection w i t h the definition of "satisfaction" f o r
the language L A Our initial approach is similar in spirit to the approach o f Harel [ 2 0 ] and Rosenschein [ 2 1 ] : We define a primitive action to be a relation b e t w e e n t w o states, and we define the meaning of the m o r e c o m p l e x formulae of L A by a set of recursive truth definitions on arbitrary sequences of states But the ordinary notion of s a t i s f a c t i o n in L 1 which takes into account the c o m p l e t e state of the w o r l d at a given time, is t o o imprecise f o r our purposes here, and we will supplement it w i t h a notion of s t r i c t satisfaction, w h i c h associates with each action in L A A
the specific set of changes in the w o r l d attributable to that action It turns out that this notion of strict satisfaction is absolutely essential to the construction of the deontic language LD W i t h o u t it, our definition of a rule of permission simply w o u l d not w o r k We will return to this point in Section IIIB A . State D e s c r i p t i o n s Let L 1 be a m a n y - s o r t e d f u n c t i o n - f r e e f i r s t - o r d e r language with equality, and let S be a set of states with respect to which the formulae of L 1 are evaluated We will f o l l o w the standard p r o c e d u r e s f o r specifying the syntax and the semantics of a f i r s t - o r d e r language Thus, if (Own x y) is a f o r m u l a of L 1 w i t h f r e e variables x and y, and if o is an assignment of the variables x and y to elements in the domain of interpretation of L 1 and if Own(s) is the set of tuples defining the extension of the predicate O w n f o r s € S, then we will say that (Own x y) is true m s under the assignment o" if and only if E Own(s) We will w r i t e this in general as σ,& = (Own x y), but if the assignment σ is f i x e d and clear f r o m the context, we will o f t e n omit it f r o m the notation and w r i t e s = (Own x y) Truth conditions f o r the nonatomic formulae of L will also be defined in the standard way If there are constraints in our domain of interpretation, expressible as a finite set of formulae in L 1 we will simply assume that S has been restricted in advance to include only those states in w h i c h the constraints are conjunctively satisfied To avoid any mathematical complexities, h o w e v e r , and to reveal the points of greatest importance to the representational p r o b l e m s of artificial intelligence, we will also assume, whenever it is convenient to do so, that the relevant sets are finite Thus we may assume that the predicate symbols are finite, that the domain of interpretation is finite, and so on We will attempt to r e m o v e these restrictions at a later date
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system; and we have c o n f i n e d our attention so far to a relatively simple language of actions Unfortunately, these t w o points are intimately connected The rules of inference p r o p o s e d f o r our system are c o m p l e x because they take into account the structure of the language of actions L and as we add further complexity to L A we will certainly add complexity to the rules of inference, t o o Nevertheless, we believe that the semantics of the deontic language itself, the language L D is basically c o r r e c t , and robust, and that it will remain in its present f o r m as the language L A evolves Perhaps this is even a fact of our cognitive lives that the c o n c e p t s of permission and obligation are relatively simple, and the c o m p l e x i t y arises instead f r o m our c o n c e p t of action Acknowledgments: The author wishes to thank N.S Sndharan. Ray Reiter. Alex Borgida and David Raab f o r their many helpful discussions The design of a computational version of the present deontic representation is the subject matter of a dissertation p r o p o s a l by David Raab
[14]
v o n W r i g h t . G.H., "On the Logic of N o r m s and Actions." in N e w Studies in Deontic Logic, Hilpinen, R. ed.. D Reidel, 1 9 8 1 . 3 - 3 5
[15]
Castaneda. H.-N.. "The Paradoxes of Deontic Logic The Simplest Solution to All of Them in One Fell S w o o p . ' in N e w Studies in Deontic Logic, Hilpinen, R, e d , D Reidel. 1 9 8 1 . 3 7 - 8 5
[16]
von Wright, GH An the General Theory 1968
[17]
McCarthy, J. "Programs w i t h C o m m o n Sense,' in Semantic I n f o r m a t i o n Processing, Minsky, M , e d , MIT Press. 1 9 6 8 4 0 3 - 4 1 8
[18]
Moore. Action, October
[19]
M c D e r m o t t . D "A Temporal About Processes and Plans' (1982) 1 0 1 - 1 5 5
[20]
Harel. D F i r s t Order D y n a m i c Logic. SpnngerVerlag Lecture Notes in Computer Science, Vol 6 8 , 1979
[21]
Rosenschem. S.J "Plan Synthesis A Logical P e r s p e c t i v e ' In Proceedings I J C A I - 8 1 . University o f British Columbia. August. 1 9 8 1 , 3 3 1 - 3 3 7
[22]
Barwise. J and Perry. J 'Situations and A t t i t u d e s " Journal of Philosophy. 78 11 (1981) 6 6 8 - 6 9 1
[23]
Kripke. S.A, "Semantical Analysis of Intuitionistic Logic I.,' in Formal Systems and Recursive Functions, Crossley, J.N and Dummett, M A E , e d s . N o r t h - H o l l a n d , 1965, 9 2 - 1 3 0
REFERENCES
Essay i n Deontic Logic and of A c t i o n . North-Holland,
R.C "Reasoning Technical r e p o r t 1980
About 191.
Knowledge and SRI International,
Logic f o r Reasoning C o g n i t i v e Science. 6
[I]
McCarty. L.T.. and Sndharan, N.S. "The Representation of an Evolving System of Legal Concepts I Logical Templates' In Proceedings 3 r d C S C S I - S C E 1 0 Conference. Victoria. British Columbia. 1980, 3 0 4 - 3 1 1
[2]
McCarty. LT and Sndharan. N.S "The Representation of an Evolving System of Legal Concepts II P r o t o t y p e s and D e f o r m a t i o n s " In Proceedings IJCAI-81. University of British Columbia. August. 1 9 8 1 . 2 4 6 - 5 3
[3]
DeBessonet. C G An Automated Approach to Scientific Codification' Rutgers Computer and Technology Law J o u r n a l . 9:1 (1982) 2 7 - 7 5
[24]
McCarthy, J "Circumscription A M o n o n t o n i c Reasoning" A r t i f i c i a l (1980) 2 7 - 3 9
[4]
Sacerdoti, E D A Structure f o r Plans a n d Behavior. Elsevier N o r t h - H o l l a n d , 1 9 7 7
[25]
M c D e r m o t t , D and Doyle, J " N o n - M o n o t o n i c Logic I " A r t i f i c i a l I n t e l l i g e n c e . 1 3 (1980) 4 1 - 7 2
[5]
DeMillo, R.A.. Dobkm. D P . Jones. A.K and Lipton, R.J Foundations of Secure Computation. Academic Press, 1 9 7 8
[26]
Reiter, R "A Logic f o r Default Reasoning" A r t i f i c i a l I n t e l l i g e n c e . 1 3 (1980) 8 1 - 1 3 2
[27]
[6]
von W r i g h t . G H N o r m and Routledge and Kegan Paul. 1 9 6 3
Action.
London
[7]
Rescher, N The Logic of Commands. Routledge and Kegan Paul, 1 9 6 6
London
Allen. J.F "An Interval-Based Representation of Temporal K n o w l e d g e " In P r o c e e d i n g s I J C A I - 8 1 . University of British Columbia. August. 1981. 221-226
[8]
Alchourron, C.E., and Bulygin, E N o r m a t i v e Systems. Springer Verlag, 1 9 7 1
[9]
Castaneda. H-N Thinking and Doing: P h i l o s o p h i c a l Foundations o f I n s t i t u t i o n s . D. Reidel. 1 9 7 5
[10]
H o h f e l d , W.N. "Fundamental Legal Conceptions as A p p l i e d in Judicial Reasoning: I" Vale Law J o u r n a l . 2 3 (1913) 1 6
[II]
H o h f e l d . W.N. "Fundamental Legal Conceptions a s A p p l i e d in Judicial Reasoning: I I " Yale Law J o u r n a l . 2 6 (1917) 7 1 0 .
[12]
Allen, L.E. "Formalizing Hohfeldian Analysis to Clarify the Multiple Senses of Legal Right A P o w e r f u l Lens f o r the Electronic Age." Southern C a l i f o r n i a Law R e v i e w . 4 8 (1974) 4 2 8 - .
[13]
Kripke. S.A. "Semantic Analysis of Modal Logic I." Zeitschrift fur Mathematische Logik und G r u n d l a g e n d e r M a t h e m a t i k . 9 (1963) 6 7 - 9 6
The Boston
F o r m of N o n Intelligence. 13
L. T h o r n e M c C a r t y C o m p u t e r Science D e p a r t m e n t Rutgers U n i v e r s i t y New Brunswick, New Jersey
A B S T R A C T : This article describes a f o r m a l semantics f o r the deontic c o n c e p t s - - the c o n c e p t s of p e r m i s s i o n and o b l i g a t i o n — which arises naturally f r o m the representations used in artificial intelligence systems Instead of treating deontic logic as a branch of modal logic, w i t h the standard possible w o r l d s semantics, we f i r s t develop a language f o r describing actions, and we define the concepts of permission and obligation in terms of these action descriptions. Using our semantic definitions, we then derive a number of intuitively plausible inferences, and we s h o w generally that the paradoxes which are so frequently associated w i t h deontic logic do not arise in our s y s t e m *
I INTRODUCTION The representation of deontic concepts — the concepts o f p e r m i s s i o n and o b l i g a t i o n - - has not yet been seriously addressed in the artificial intelligence literature, but there are numerous application areas in which these concepts seem to be required. In our w o r k on the T A X M A N P r o j e c t [ 1 ] [ 2 ] , f o r example, w e represent the characteristics of various kinds of stocks and bonds by describing the rules of permission and obligation which are binding, at any given time, on the c o r p o r a t i o n and its securityholders In our work on the "usufructuary" provisions of the Louisiana Civil Code [ 3 ] , just recently initiated, we have encountered a similar need f o r the representation of c o m p l e x permissions and obligations Nor are these examples c o n f i n e d to legal domains In the classical w o r k on single agent planning systems, e.g., [ 4 ] , the o p e r a t o r s w h i c h change the state of the w o r l d can be interpreted as a set of "permitted" actions, but in a m o r e realistic planning environment, w i t h multiple agents, we w o u l d e x p e c t to see "obligatory" actions as well, and we w o u l d e x p e c t to see the actions of one agent p r o d u c e modifications in the rules of permission and obligation binding upon another agent. Similar observations apply to the field of computer security, see, e.g., [ 5 ] , w h e r e there has been extensive debate over the appropriate "authorization mechanisms" f o r a community of computer users. For all of these purposes, a formalization of the c o n c e p t s of permission and obligation appears to be essential Outside of the field of artificial intelligence, there exists
**This article is based upon w o r k s u p p o r t e d by Grant No. MCS-82-03591 f r o m the National Science Foundation, Washington, D.C., and by a grant f r o m the Louisiana State Law Institute, Baton Rouge, Louisiana.
an extensive literature on the deontic concepts, by logicians [ 6 ] [ 7 ] , philosophers [ 8 ] [ 9 ] , and lawyers [ 1 0 ] [ 1 1 ] [ 1 2 ] , but the attempts to formalize these concepts have generally led to paradox Since the 1 9 5 0 s , deontic logic has been treated as a branch of modal logic, with the necessity' operator replaced by the obligation" operator, O. and the "possibility" operator replaced by the "permission" operator, P Many of the theorems of modal logic turn out to be intuitively c o r r e c t under this translation For example, the dual relationship between necessity and possibility becomes a dual relationship between obligation and permission, Op = ~P~p. and this formula certainly seems plausible If it is false that you are p e r m i t t e d to do n o t - p , then you are o b l i g a t e d to do p. and vice versa The formula Op > p. which is valid in any modal system with a r e f l e x i v e accessibility relation between possible worlds, w o u l d not be plausible in a deontic logic, since people in the actual w o r l d do not always abide by their obligations, but it can be replaced by the m o r e plausible formula Op => Pp, w h i c h is valid as long as every possible w o r l d has some possible w o r l d accessible f r o m it This point was f i r s t noted by Kripke, in one of his original papers on possible w o r l d s semantics [ 1 3 ] .
Despite these positive results, there are several other modal formulae which seem counterintuitive in a deontic logic, and which cannot be so easily modified. For example, the formula f o r disjunctive permission, Pp D Pip v q), contradicts our ordinary understanding of what it means to grant permission to do p v q, but this formula is valid even in the weakest modal systems Likewise, any formula containing an iterated operator, such as OPp or POp, seems anomalous in a deontic context, and yet the various modal systems are distinguished precisely by the way in which they handle these iterated modalities. Of course, it may make sense to say that you are p e r m i t t e d to impose a particular o b l i g a t i o n upon someone else, or upon yourself, and we might conceivably w r i t e this as POp, but the inferences we w o u l d make about such statements do not c o r r e s p o n d at all to the inferences w h i c h are valid in the standard possible w o r l d semantics. Finally, even the dual relationship between permission and obligation seems problematical if we cast it into the f o r m Pp = ~ 0 ~ p If it is f a l s e that you are o b l i g a t e d to do n o t - p , i.e., if it is f a l s e that you are f o r b i d d e n to do p, does it f o l l o w that p is p e r m i t t e d ? Stringing together all of these questionable inferences, it is not surprising that we can generate a host of "deontic paradoxes," and the literature is full of them For a survey, see [ 1 4 ] and [ 1 5 ] In this paper, we will develop a formal semantics f o r
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permissions and obligations which seems to avoid these difficulties, and we will do so in a way which is entirely natural f o r an artificial intelligence system Instead of representing the deontic concepts as operators applied to p r o p o s i t i o n s , as in a standard modal logic, we will represent them as d y a d i c f o r m s which take c o n d i t i o n d e s c r i p t i o n s and action d e s c r i p t i o n s as their arguments The most important part of this representation is the use of action descriptions in the place of p r o p o s i t i o n s Instead of granting permissions and imposing obligations on the state of the w o r l d itself, we will grant permissions and impose obligations on the actions w h i c h change the state of the w o r l d This is an approach long advocated by Castaneda [ 1 5 ] , and pursued in various f o r m s by von W r i g h t [ 1 6 ] [ 1 4 ] . but to carry out this approach in full it seems necessary to establish a connection b e t w e e n the abstract d e s c r i p t i o n of an action and the c o n c r e t e changes that o c c u r in the w o r l d w h e n the action takes place This has been a major c o n c e r n of artificial intelligence research throughout its history [ 1 7 ] [ 1 8 ] [ 1 9 ] , o f course, and w e will d r a w upon this earlier w o r k in constructing our formalisms Although the actions that we actually discuss in this paper are fairly simple ones, intended to highlight the principal features of the deontic representation, the action descriptions themselves can be extended to m o r e realistic situations, in several ways We will return to this point in our concluding remarks II DEONTIC S E M A N T I C S In this section we develop a formal semantic interpretation of the deontic concepts, using a variant of the possible w o r l d s approach Our strategy p r o c e e d s in stages We start w i t h an ordinary f i r s t - o r d e r language L and a set of states S. and we use these materials to construct a new language L A in w h i c h we are able to describe actions The formulae of L 1 are evaluated w i t h respect to the states in S. as usual, but the formulae of L A are evaluated w i t h respect to sequences of states, or w o r l d s . We thus have a way of saying that an action is "true' in, or is satisfied" by, a particular w o r l d The details of these constructions are presented in Sections IIA and II.B below. N o w consider a state r w h i c h is situated at the junction b e t w e e n a "past w o r l d v and a "future" w o r l d w We assume that there exists a set P which tells us, f o r r
each past w o r l d v, all the future w o r l d s w w h i c h are "permitted." W o r k i n g exclusively w i t h this p e r m i t t e d set P , we c o n s t r u c t three expressions w h i c h tell us whether an action at r is p e r m i t t e d , forbidden, or obligatory, respectively These expressions then b e c o m e part of our deontic language L D The details of these constructions are p r e s e n t e d in Section IIC b e l o w This is not the end of the story, h o w e v e r . Since each deontic e x p r e s s i o n has a definite t r u t h value at each state in S, it turns o u t that the language L D can be embedded within our original f i r s t o r d e r language L , and thus the p r o c e s s of linguistic c o n s t r u c t i o n we have outlined here becomes fully recursive This technique enables us to represent "dynamic" permissions and obligations, i.e., permissions and obligations w h i c h change over time, w i t h o u t the use of iterated modalities
This latter point is developed in Section II.D
The principal technical d i f f i c u l t y in this development rises in connection w i t h the definition of "satisfaction" f o r
the language L A Our initial approach is similar in spirit to the approach o f Harel [ 2 0 ] and Rosenschein [ 2 1 ] : We define a primitive action to be a relation b e t w e e n t w o states, and we define the meaning of the m o r e c o m p l e x formulae of L A by a set of recursive truth definitions on arbitrary sequences of states But the ordinary notion of s a t i s f a c t i o n in L 1 which takes into account the c o m p l e t e state of the w o r l d at a given time, is t o o imprecise f o r our purposes here, and we will supplement it w i t h a notion of s t r i c t satisfaction, w h i c h associates with each action in L A A
the specific set of changes in the w o r l d attributable to that action It turns out that this notion of strict satisfaction is absolutely essential to the construction of the deontic language LD W i t h o u t it, our definition of a rule of permission simply w o u l d not w o r k We will return to this point in Section IIIB A . State D e s c r i p t i o n s Let L 1 be a m a n y - s o r t e d f u n c t i o n - f r e e f i r s t - o r d e r language with equality, and let S be a set of states with respect to which the formulae of L 1 are evaluated We will f o l l o w the standard p r o c e d u r e s f o r specifying the syntax and the semantics of a f i r s t - o r d e r language Thus, if (Own x y) is a f o r m u l a of L 1 w i t h f r e e variables x and y, and if o is an assignment of the variables x and y to elements in the domain of interpretation of L 1 and if Own(s) is the set of tuples defining the extension of the predicate O w n f o r s € S, then we will say that (Own x y) is true m s under the assignment o" if and only if E Own(s) We will w r i t e this in general as σ,& = (Own x y), but if the assignment σ is f i x e d and clear f r o m the context, we will o f t e n omit it f r o m the notation and w r i t e s = (Own x y) Truth conditions f o r the nonatomic formulae of L will also be defined in the standard way If there are constraints in our domain of interpretation, expressible as a finite set of formulae in L 1 we will simply assume that S has been restricted in advance to include only those states in w h i c h the constraints are conjunctively satisfied To avoid any mathematical complexities, h o w e v e r , and to reveal the points of greatest importance to the representational p r o b l e m s of artificial intelligence, we will also assume, whenever it is convenient to do so, that the relevant sets are finite Thus we may assume that the predicate symbols are finite, that the domain of interpretation is finite, and so on We will attempt to r e m o v e these restrictions at a later date
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system; and we have c o n f i n e d our attention so far to a relatively simple language of actions Unfortunately, these t w o points are intimately connected The rules of inference p r o p o s e d f o r our system are c o m p l e x because they take into account the structure of the language of actions L and as we add further complexity to L A we will certainly add complexity to the rules of inference, t o o Nevertheless, we believe that the semantics of the deontic language itself, the language L D is basically c o r r e c t , and robust, and that it will remain in its present f o r m as the language L A evolves Perhaps this is even a fact of our cognitive lives that the c o n c e p t s of permission and obligation are relatively simple, and the c o m p l e x i t y arises instead f r o m our c o n c e p t of action Acknowledgments: The author wishes to thank N.S Sndharan. Ray Reiter. Alex Borgida and David Raab f o r their many helpful discussions The design of a computational version of the present deontic representation is the subject matter of a dissertation p r o p o s a l by David Raab
[14]
v o n W r i g h t . G.H., "On the Logic of N o r m s and Actions." in N e w Studies in Deontic Logic, Hilpinen, R. ed.. D Reidel, 1 9 8 1 . 3 - 3 5
[15]
Castaneda. H.-N.. "The Paradoxes of Deontic Logic The Simplest Solution to All of Them in One Fell S w o o p . ' in N e w Studies in Deontic Logic, Hilpinen, R, e d , D Reidel. 1 9 8 1 . 3 7 - 8 5
[16]
von Wright, GH An the General Theory 1968
[17]
McCarthy, J. "Programs w i t h C o m m o n Sense,' in Semantic I n f o r m a t i o n Processing, Minsky, M , e d , MIT Press. 1 9 6 8 4 0 3 - 4 1 8
[18]
Moore. Action, October
[19]
M c D e r m o t t . D "A Temporal About Processes and Plans' (1982) 1 0 1 - 1 5 5
[20]
Harel. D F i r s t Order D y n a m i c Logic. SpnngerVerlag Lecture Notes in Computer Science, Vol 6 8 , 1979
[21]
Rosenschem. S.J "Plan Synthesis A Logical P e r s p e c t i v e ' In Proceedings I J C A I - 8 1 . University o f British Columbia. August. 1 9 8 1 , 3 3 1 - 3 3 7
[22]
Barwise. J and Perry. J 'Situations and A t t i t u d e s " Journal of Philosophy. 78 11 (1981) 6 6 8 - 6 9 1
[23]
Kripke. S.A, "Semantical Analysis of Intuitionistic Logic I.,' in Formal Systems and Recursive Functions, Crossley, J.N and Dummett, M A E , e d s . N o r t h - H o l l a n d , 1965, 9 2 - 1 3 0
REFERENCES
Essay i n Deontic Logic and of A c t i o n . North-Holland,
R.C "Reasoning Technical r e p o r t 1980
About 191.
Knowledge and SRI International,
Logic f o r Reasoning C o g n i t i v e Science. 6
[I]
McCarty. L.T.. and Sndharan, N.S. "The Representation of an Evolving System of Legal Concepts I Logical Templates' In Proceedings 3 r d C S C S I - S C E 1 0 Conference. Victoria. British Columbia. 1980, 3 0 4 - 3 1 1
[2]
McCarty. LT and Sndharan. N.S "The Representation of an Evolving System of Legal Concepts II P r o t o t y p e s and D e f o r m a t i o n s " In Proceedings IJCAI-81. University of British Columbia. August. 1 9 8 1 . 2 4 6 - 5 3
[3]
DeBessonet. C G An Automated Approach to Scientific Codification' Rutgers Computer and Technology Law J o u r n a l . 9:1 (1982) 2 7 - 7 5
[24]
McCarthy, J "Circumscription A M o n o n t o n i c Reasoning" A r t i f i c i a l (1980) 2 7 - 3 9
[4]
Sacerdoti, E D A Structure f o r Plans a n d Behavior. Elsevier N o r t h - H o l l a n d , 1 9 7 7
[25]
M c D e r m o t t , D and Doyle, J " N o n - M o n o t o n i c Logic I " A r t i f i c i a l I n t e l l i g e n c e . 1 3 (1980) 4 1 - 7 2
[5]
DeMillo, R.A.. Dobkm. D P . Jones. A.K and Lipton, R.J Foundations of Secure Computation. Academic Press, 1 9 7 8
[26]
Reiter, R "A Logic f o r Default Reasoning" A r t i f i c i a l I n t e l l i g e n c e . 1 3 (1980) 8 1 - 1 3 2
[27]
[6]
von W r i g h t . G H N o r m and Routledge and Kegan Paul. 1 9 6 3
Action.
London
[7]
Rescher, N The Logic of Commands. Routledge and Kegan Paul, 1 9 6 6
London
Allen. J.F "An Interval-Based Representation of Temporal K n o w l e d g e " In P r o c e e d i n g s I J C A I - 8 1 . University of British Columbia. August. 1981. 221-226
[8]
Alchourron, C.E., and Bulygin, E N o r m a t i v e Systems. Springer Verlag, 1 9 7 1
[9]
Castaneda. H-N Thinking and Doing: P h i l o s o p h i c a l Foundations o f I n s t i t u t i o n s . D. Reidel. 1 9 7 5
[10]
H o h f e l d , W.N. "Fundamental Legal Conceptions as A p p l i e d in Judicial Reasoning: I" Vale Law J o u r n a l . 2 3 (1913) 1 6
[II]
H o h f e l d . W.N. "Fundamental Legal Conceptions a s A p p l i e d in Judicial Reasoning: I I " Yale Law J o u r n a l . 2 6 (1917) 7 1 0 .
[12]
Allen, L.E. "Formalizing Hohfeldian Analysis to Clarify the Multiple Senses of Legal Right A P o w e r f u l Lens f o r the Electronic Age." Southern C a l i f o r n i a Law R e v i e w . 4 8 (1974) 4 2 8 - .
[13]
Kripke. S.A. "Semantic Analysis of Modal Logic I." Zeitschrift fur Mathematische Logik und G r u n d l a g e n d e r M a t h e m a t i k . 9 (1963) 6 7 - 9 6
The Boston
F o r m of N o n Intelligence. 13
