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When Conditional Logic and Belief Revision Meet Substructural Logics∗

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When Conditional Logic and Belief Revision Meet Substructural Logics∗ When Conditional Logic and Belief Revision Meet Substructural Logics∗ Guillaume Aucher University of Rennes 1 INRIA Rennes, France [email protected] Abstract and non-monotonic logics [Makinson, 2005], belief revision theory [Gardenfors,¨ 1988], conditional logic [Nute and Cross, Two threads of research have been pursued in par- 2001], etc. However, a generic and general framework en- allel in logic and artificial intelligence. On the one compassing all these theories is still lacking. Instead, the cur- hand, in artificial intelligence, logic-based theories rent state of the art is such that we are left with various for- have been developed to study and formalize belief malisms which are difficult to relate formally to each other change and the so-called “common sense reason- despite numerous attempts [Makinson and Gardenfors,¨ 1989; ing”, i.e. the actual reasoning of humans. On the Aucher, 2004; Baltag and Smets, 2008], partly because they other hand, in logic, substructural logics, i.e. log- rely on different kinds of formalisms. This is problematic if ics lacking some of the structural rules of classical logic is to be viewed ultimately as a unified and unifying field logic, have been studied in depth from a theoreti- and if we want to avoid that logic goes on “riding off madly cal point of view. However, the powerful (proof- in all directions” (a metaphor used by van Benthem [2011]). theoretical) techniques and methods developed in logic have not yet been applied to artificial intel- Our objective in this article is to show that conditional logic ligence. Conditional logic and belief revision the- and belief revision can be reformulated meaningfully and nat- ory are prominent theories in artificial intelligence urally within the very general framework of substructural log- dealing with common sense reasoning. We show in ics [Restall, 2000]. More specifically, we will show that con- this article that they can both be embedded within ditional logic and belief revision theory are extensions of the the framework of substructural logics and can both well-known Lambek calculus with appropriate structural in- be seen as extensions of the Lambek calculus. This ference rules. This will allow us to compare and relate them allows us to compare and relate them to each other to each other systematically. In particular, our approach will systematically, via a natural formalization of the shed new lights on Gardenfors’¨ impossibility theorem that Ramsey test. draws attention to certain formal difficulties in defining a con- ditional connective from a revision operation, via the Ramsey test. We will also pinpoint the key to non-monotonicity and 1 Introduction we will show that it depends crucially on a constrained appli- In everyday life, the way we update and revise our beliefs cation of the (left) weakening rule. plays an important role in our representation of the surround- Other proof theoretical approaches to non-monotonic rea- ing world and therefore also in our decision making process. soning have already been proposed, notably by Bonatti This has lead researchers in artificial intelligence and com- and Olivetti [2002][1992]. However, they deal with non- puter science to develop logic-based theories that study and monotonicity at the meta-logical level by introducing specific formalize belief change and the so-called “common sense inference relations like j∼ or B. Instead of it, we will deal reasoning”. The rationale underlying the development of with non-monotonicity at the object-language level by means such theories is that it would ultimately help us understand of the substructural connective ⊃ and the introduction of ap- our everyday life reasoning and the way we update our be- propriate structural rules. liefs, and that the resulting work could subsequently lead to The article is organized as follows. In Section 2, we briefly the development of tools that could be used for example by recall elementary notions of substructural logics and we ob- artificial agents in order to act autonomously in an uncertain serve that the ternary relation can be interpreted intuitively as and changing world. a kind of update. In Section 3, we recall the basics of condi- A number of theories have been proposed to capture dif- tional logic and belief revision theory and recall how they are ferent kinds of updates and the reasoning styles that they in- formally connected. In Section 4, we show how each of them duce, using different formalisms and under various assump- can be embedded within the framework of substructural logic tions: dynamic epistemic logic [van Benthem, 2011], default that was introduced in Section 2 by adding specific structural ∗I thank Philippe Besnard for discussions. I thank three anony- inference rules. In Section 5, we discuss Gardenfors’¨ impos- mous reviewers for comments. sibility theorem. Finally, we conclude in Section 6. 2 Substructural Logics The semantics of substructural logics is based on the Substructural logics are a family of logics lacking some of ternary relation of the frame semantics for relevant logic orig- the structural rules of classical logic. A structural rule is a inally introduced by Routley and Meyer [1972a,b, 1973]; rule of inference which is closed under substitution of for- Routley et al. [1982]. mulas. The structural rules for classical logic are given in Definition 3 (Point set). A point set P = (P; v) is a set P Fig. 1: they are called Weakening, Contraction, Permutation together with a partial order v on P. We abusively write x 2 P and Associativity (see Definition 2 for explanations about the for x 2 P. notations used). The comma in these sequents has to be inter- The partial order v (introduced for dealing with intuition- preted as a conjunction in an antecedent and as a disjunction istic reasoning) will not be used in this article. in a consequent. While Weakening and Contraction are often Definition 4 (Model). A model is a tuple M = (P; R; I) dropped like in relevance logic and linear logic, the rule of where: Associativity is often preserved. When some of these rules •P = (P; v) is a point set; are dropped, the comma ceases to behave as a conjunction P (in the antecedent) or a disjunction (in the succedent). In that •I : P ! 2 is an interpretation function; case, the comma corresponds to other substructural connec- • R⊆P×P×P is a ternary relation on P. tives and we often introduce new punctuation marks which We abusively write x 2 M for x 2 P, and (M; x) is called a do not fulfill all these structural rules to deal with these new pointed model. substructural connectives. A model stripped out from its interpretation corresponds to Our exposition of substructural logics is based on [Restall, a frame as defined in [Restall, 2000, Def. 11.8] without truth 2000, 2006] (see also Ono [1998] for a general introduction). sets (defined in [Restall, 2000, Def. 11.7]). Truth sets are not 2.1 Syntax and Semantics needed for what concerns us here. In the sequel, P is a non-empty and finite set of propositional Definition 5 (Truth conditions). Let M be a model, x 2 M letters. and ' 2 L◦;⊃;⊂. The relation M; x ' is defined inductively as follows: Definition 1 (Language L◦;⊃;⊂). The language L◦;⊃;⊂ is de- fined inductively by the following grammar in BNF: M; x ? never ' ::= ? j p j (' ^ ') j (' ! ') M; x p iff p 2 I(x) (' ◦ ') j (' ⊃ ') j (' ⊂ ') M; x ' ^ iff M; x ' and M; x M ! M M where p ranges over P. Also, ! is material implication ; x ' iff if ; x ' then ; x whereas ⊃ and ⊂ are Lambek implications. If Con ⊆ f^; ! M; x ' ◦ iff there are y; z 2 P such that Ryzx; ; ◦; ⊃; ⊂}, then the language LCon is the language L◦;⊃;⊂ re- M; y ' and M; z stricted to the connectives of Con. The propositional lan- M; x ' ⊃ iff for all y; z 2 P where Rxyz; guage LPL is the language LCon with Con := f^; !g. if M; y ' then M; z We will use the following abbreviations: :' := ' !?, M; x ⊂ ' iff for all y; z 2 P where Ryxz > := :?, ' $ := (' ! ) ^ ( ! '), and ' _ := if M; y ' then M; z :(:'^: ). We use the following ranking of binding strength for parenthesis: :; ◦; ⊃; ⊂; ^; _; !; $. Let Con ⊆ f^; !; ◦; ⊃; ⊂}. We extend the scope of the rela- tion to also relate points to LCon–structures: Definition 2 (LCon–structure, LCon–sequent and LCon–hypersequent). Let Con ⊆ f^; !; ◦; ⊃; ⊂}. LCon– M; x X; Y iff M; x X and M; x Y structures are defined by the following grammar in BNF: M; x X ; Y iff there are y; z 2 M such that Ryzx; LCon M; y X and M; z Y SL : X ::= ' j (X; X) j (X ; X) LCon SR : Y ::= ' j (Y; Y) Let X Y be a LCon–sequent and let (M; x) be a pointed where ' ranges over LCon. Γ[X] denotes a LCon–structure model. We say that X Y is true at (M; x), written containing as substructure the LCon–structure X, and Γ[Z] de- M; x X Y, when the following holds: notes the L –structure Γ[X] where X is uniformly substi- Con M; x X Y iff if M; x X, then there is ' 2 Y tuted by the structure Z. LCon–structures are denoted U; X; Y or Z and we write ' 2 X when ' is a substructure of X. such that M; x ': A LCon–sequent is an expression of the form X Y, We say that the LCon–sequent X Y is valid, written X Y, LCon LCon Y or X where X 2 SL , Y 2 SR .A LCon– when for all pointed models (M; x), M; x X Y.
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