We present a formal semantics for deontic logic based on the concept of ceteris paribus preferences. It allows to introduce notions of conditional/unconditional obligation and permission that are interpreted relative to this semantics. We show how obligations and permissions can be represented compactly using existing preference frameworks from the arti cial intelligence area.
Introduction
can compare them with an agent's preferences to understand how similar they
are [
Ceteris Paribus Deontinc Logic The ceteris paribus deontic logic - CPDL+ has the so-called universal modal operator which allows us to capture factual detachment of obligations. De nition 1. LCPDL+ (Atm) is a modal language which includes atomic propositions p; q; : : : 2 Atm, standard boolean operators and the modal operators O; P; U. The language is such that: if p 2 Atm then p 2 LCPDL+ , if '; 2 LCPDL+ then :'; '^ 2 LCPDL+ , if '; 2 LCPDL+ then O'; P'; O( j'); P( j'); U' 2 LCPDL+ :
Formulas O' and P' have to be read, respectively, \' is obligatory" and \' is permitted". Formula U' has to be read \' is universally true". Formulas O( j') and P( j') have to be read, respectively, \under condition , ' is obligatory " and \under condition , ' is permitted". The truth conditions for the formulas in the language LCPDL+ (Atm) are de ned as follows: 4 That is a binary relation on W which is re exive, transitive and complete.
A Novel Approach For a Ceteris Paribus Deontic Logic De nition 2 (Truth Conditions). Let M = (W; ) be a preference model, let w 2 W and let Atm ' = Atm n Atm(') where Atm(') is the set of atoms from Atm occurring in '. Then: { M; w j= p () p 2 w { M; w j= :' () M; w 6j= ' { M; w j= ' ^ () M; w j= ' and M; w j= { M; w j= O' () 8v; u 2 W : if M; v j= ' and v { M; w j= P' () 8v; u 2 W : if M; v j= ' and v { M; w j= U' () 8v 2 W : M; v j= ' { M; w j= O( j') () 8v; u 2 jj jjM : if M; v j= ' and v { M; w j= P( j') () 8v; u 2 jj jjM : if M; v j= ' and v Atm ' u then M; u j= ' Atm ' u then M; u j= '
Atm ' u then M; u j= ' Atm ' u then M; u j= '
In other words, O' means that, for every two possible worlds that are Atm 'indistinguishable and that disagree about the truth value of ', the world in which ' is true is better than the world in which ' is false. P' means that, for every two possible worlds that are Atm '-indistinguishable and that disagree about the truth value of ', the world in which ' is true is at least as good as the world in which ' is false. We say that the formula ' 2 LCPDL+ (Atm) is valid relative to the class of preference models P, denoted by j=P ', i , for every preference model M and for every world w in M , we have M; w j= '. We say that the formula ' 2 LCPDL+ (Atm) is satis able relative to the class of preference models i , there exists a preference model M and a world w in M , such that M; w j= '.
The proposed model has several interesting properties that we list here: { restricting our model to obligations and permissions that are stated only on atoms, then the induced preference model can be represented compactly by a CP-net; { unconditional obligation and permission do not need to be added as primitives in the language of the logic CPDL+, as they are de nable from conditional obligation and permission; { if '; are conjunctive clauses and Atm(') \ Atm( ) = ; then: j=P (O' ^
O ) ! O(' ^ ) and j=P (P' ^ P ) ! P(' ^ );
{ if ' is obligatory then it is also permitted;
{ if the condition of a conditional obligation/permission is necessarily true
then the obligation/permission is detached and becomes unconditional;
{ CPDL+ does not encounter Ross's paradox [
From Syntax Dependence to Independence
The general idea behind our ceteris paribus notion of obligation is that ' is
obligatory if and only if, the utility of a world increases in the direction by the
formula ' ceteris paribus, \all else being equal". Following Von Wright (see also
[
Conclusion and perspectives We have presented a new approach to deontic logic, based on ceteris paribus preferences, which provides a fresh foundation to the logical analysis of deontic concepts, named CPDL+. We provided a connection with knowledge representation in order to compactly represent and reason over the set of obligations and permissions using the CP-net formalism. We are currently working to develop the framework of CPDL+ in various directions, concerning both theory and applications. for ceteris paribus preferences. Journal of philosophical logic 38(1), 83{125 (2009) 0 nad ten 022 iireneng gnaaem g M n e AW dgeE legdw
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