Modality, Si! Modal Logic, No!

This article is oriented toward the use of modality in arti cial intelligence AI An agent must reason about what it or other agents know believe want intend or owe Referentially opaque modalities are needed and must be formalized correctly Unfortunately modal logics seem too limited for many important purposes This article contains examples of uses of modality for which modal logic seems inadequate I have no proof that modal logic is inadequate so I hope modal logicians will take the examples as challenges Maybe this article will also have philosophical and mathematical logical interest Here are the main considerations Many modalities Natural language often uses several modalities in a sin gle sentence I want him to believe that I know he has lied Gab introduces a formalism for combining modalities but I don t know whether it can handle the examples mentioned in this article New Modalities Human practice sometimes introduces new modalities on an ad hoc basis The institution of owing money or the obligations the Bill of Rights imposes on the U S Government are not matters of basic logic Introducing new modalities should involve no more fuss than introducing a new predicate In particular human level AI requires that programs be able to introduce modalities when this is appropriate e g have function taking modalities as values Knowing what Pat knows Mike s telephone number is a simple exam ple In McC b this is formalized as knows pat T elephone Mike where pat stands for the person Pat Mike stands for a standard con cept of the person Mike and Telephone takes a concept of a person into a concept of his telephone number We might have telephone mike telephone mary expressing the fact that Mike and Mary have the same telephone but we won t have Telephone Mike Telephone Mary which would assert that the concept of Mike s telephone number is the same as that of Mary s telephone number This permits us to have knows pat T elephone Mary even though Pat knows Mike s telephone number which happens to be the same as Mary s The theory in McC b also includes functions from some kinds of things e g numbers or people to standard concepts of them This permits saying that Kepler did not know that the number of planets is composite while saying that Kepler knew that the number we know to be the number of planets is composite The point of this example is not mainly to advertise McC b but to advocate that a theory of knowledge must treat knowing what as well as knowing that and to illustrate some of the capabilities needed for adequately using knowing what Presumably knows pat T elephone Mike could be avoided by writing x knows pat T elephone Mike x but the required quantifying in is likely to be a nuisance Proving Non knowledge McC formalizes two puzzles whose solution requires inferring non knowledge from previously asserted non knowledge and from limitingwhat is learned when a person hears some information McC uses a variant of the Kripke accessibility relation but here it is used directly in rst order logic rather than to give semantics to a modal logic The relation is A w w person time interpreted as asserting that in world w it is possible for person that the world is w Non knowledge of a term in w is e g the color of a spot or the value of a numerical variable is expressed by saying that there is a world w in which the value of the term di ers from its value in w Lev uses a modality whose interpretation is all I know is He uses autoepistemic logic Moo a nonmonotonic modal logic This seems inadequate in general because we need to be able to express All The three wise men puzzle is as follows A certain king wishes to test his three wise men He arranges them in a circle so that they can see and hear each other and tells them that he will put a white or black spot on each of their foreheads but that at least one spot will be white In fact all three spots are white He then repeatedly asks them Do you know the color of your spot What do they answer The solution is that they answer No the rst two times the question is asked and answer Yes thereafter This is a variant form of the puzzle which avoids having wise men reason about how fast their colleagues reason Here is the Mr S and Mr P puzzle Two numbers m and n are chosen such that m n Mr S is told their sum and Mr P is told their product The following dialogue ensues Mr P I don t know the numbers Mr S I knew you didn t know I don t know either Mr P Now I know the numbers Mr S Now I know them too In view of the above dialogue what are the numbers I know about the value of x is Here s an example At one stage in Mr S and Mr P we can say that all Mr P knows about the value of the pair is their product and the fact that their sum is not the sum of two primes KPH treats the question of showing how President Bush could rea son that he didn t know whether Gorbachev was standing or sitting and how Bush could also reason that Gorbachev didn t know whether Bush was standing or sitting The treatment does not use modal logic but rather a variant of circumscription called autocircumscription proposed by Perlis Per Joint knowledge and learning In the wise men problem they learn at each stage that the others don t know the colors of their spots and in Mr S and Mr P they learn what the others have said In each case the learning is joint knowledge wherein several people knowing something jointly implies not only that each knows it but also that they know it jointly McC treats joint knowledge by introducing pseudo persons for each subset of the real knowers The pseudo person knows what the subset knows jointly The logical treatment of joint knowledge in McC makes the joint knowers S in their knowledge I don t know whether a more subtle axiomatization would avoid this McC treats learning a fact by using the time argument of the ac cessibility relation After person learns a fact p the worlds that are possible for him are those worlds that were previously possible for him and in which p holds Learning the value of a term is treated similarly Other modalities McC a treats believing and intending and McC treats introspection by robots Neither paper introduces enough for malism to provide a direct challenge to modal logic but it seems to me that the problems are even harder than those previously treated Acknowledgements This work was supported in part by DARPA ONR grant N Tom Costello provided some useful discussion Halpern and Lakemeyer in HL show that the quanti ed version of Levesque s logic is incomplete but this is a di erent complaint from the one we make here