Revision by Conditionals: from Hook to Arrow

Proceedings of the 17th International Conference on Principles of Knowledge Representation and Reasoning (KR 2020) Main Track Revision by Conditionals: From Hook to Arrow Jake Chandler1 , Richard Booth2 1La Trobe University 2Cardiff University [email protected], [email protected] Abstract conditional belief. In Section 3, we outline and discuss our proposal regarding (B). Subsection 3.1 presents the basic The belief revision literature has largely focussed on the issue idea, according to which computing the result of a revi- of how to revise one’s beliefs in the light of information regarding matters of fact. Here we turn to an important but com- sion by a Ramsey Test conditional can be derived by min- paratively neglected issue: How might one extend a revision imal modification, under constraints, of the outcome of a operator to handle conditionals as input? Our approach to this revision by its corresponding material conditional. Our key question of ‘conditional revision’ is distinctive insofar as it technical contribution is presented in Subsection 3.2, where abstracts from the controversial details of how to revise by we prove that this minimal change under constraints can be factual sentences. We introduce a ‘plug and play’ method for achieved by means of a simple and familiar transformation. uniquely extending any iterated belief revision operator to the Subsection 3.3 outlines some interesting general properties conditional case. The flexibility of our approach is achieved of the proposal. These strengthen, in a plausible manner, by having the result of a conditional revision by a Ramsey the aforementioned constraints presented in (Kern-Isberner Test conditional (‘arrow’) determined by that of a plain re- 1999) and are of independent interest. Subsection 3.4 con- vision by its corresponding material conditional (‘hook’). It is shown to satisfy a number of new constraints that are of siders the upshot of pairing our proposal regarding (B) with independent interest. some well-known suggestions regarding how to tackle (A). Finally, in Section 4, we compare the suggestion made with existing work on the topic noting some important shortcom- 1 Introduction ings of the latter. We close the paper in Section 5 with a The past three decades have witnessed the development of number of questions for future research. a substantial, if inconclusive, body of work devoted to the Due to space limitations, only a couple of the more im- issue of belief revision, namely portant proofs have been provided, in Section 6. A version (A) determining the impact of a local change in belief on of the paper containing all proofs can be accessed online at both (i) the remainder of one’s prior beliefs and (ii) http://arxiv.org/abs/2006.15811. one’s prior conditional beliefs (‘Ramsey Test conditionals’). 2 Revision Surprisingly, however, very little has been done to this date The beliefs of an agent are represented by a belief state. Such on the question of conditional belief revision, that is states will be denoted by upper case Greek letters Ψ; Θ;:::. We denote by S the set of all such states. Each state de- (B) determining the impact of a local change in condi- termines a belief set, a consistent and deductively closed tional beliefs on both (i) and (ii). set of sentences, drawn from a finitely generated proposi- Furthermore, nearly all of the few proposals to tackle is- tional, truth-functional language L, equipped with the stan- sue (B), namely (Hansson 1992), (Boutilier and Goldszmidt dard connectives ⊃, ^, _, and :. We denote the belief set 1993), and (Nayak et al. 1996), have typically rested on associated with state Ψ by [Ψ]. Logical equivalence is de- somewhat contentious assumptions about how to approach noted by ≡ and the set of classical logical consequences of (A). (A noteworthy exception to this (Kern-Isberner 1999), Γ ⊆ L by Cn(Γ), with > denoting an arbitrary propositional who introduced a number of plausible general postulates tautology. The set of propositional worlds will be denoted by governing revision by conditionals whose impact on revision W and the set of models of a given sentence A by [[A]]. simpliciter remains fairly modest. More on these below.) The operation of revision ∗ returns the posterior state Ψ ∗ In this paper, we consider the prospects of providing a A that results from an adjustment of Ψ to accommodate the ‘plug and play’ solution to issue (B) that is independent of inclusion of the consistent input A in its associated belief the details of how to address (A). Its remainder is organ- set, in such a way as to maintain consistency of the resulting ised as follows. First, in Section 2, we present some standard belief set. background on problem (A), introducing along the way the The beliefs resulting from single revisions are conve- well-known notion of a Ramsey Test conditional or again niently representable by a conditional belief set [Ψ]c, which 233 Proceedings of the 17th International Conference on Principles of Knowledge Representation and Reasoning (KR 2020) Main Track ∗ 4 4 can be viewed as encoding the agent’s rules of inference over (C14) If x; y 2 [[A]] then x Ψ∗A y iff x Ψ y L in state Ψ. It is defined via the Ramsey Test: ∗ 4 4 (C24) If x; y 2 [[:A]] then x Ψ∗A y iff x Ψ y (RT) For all A; B 2 L, A) B 2 [Ψ]c iff B 2 [Ψ ∗ A] ∗ (C34) If x 2 [[A]], y 2 [[:A]] and x ≺Ψ y, then We shall call Lc the minimal extension of L that additionally x ≺Ψ∗A y ∗ 4 includes all sentences of the form A) B, with A; B 2 L. (C44) If x 2 [[A]], y 2 [[:A]] and x Ψ y, then We shall call sentences of the form A ) B ‘conditionals’ x 4Ψ∗A y and sentences of the form A ⊃ B ‘material conditionals’. We shall say that a sentence of the form A) B is consistent Importantly, while there appears to be a degree of consen- just in case A ^ B is consistent (later in the paper, we shall sus that these postulates should be strengthened, there is no explicitly disallow revisions by inconsistent conditionals). agreement as to how this should be done. Popular options Conditional belief sets are constrained by the AGM pos- include the principles respectively associated with the oper- tulates of (Alchourron,´ Gardenfors,¨ and Makinson 1985; ators of natural revision ∗N (Boutilier 1996), restrained revi- Darwiche and Pearl 1997) (henceforth ‘AGM’). Given these, sion ∗R (Booth and Meyer 2006) and lexicographic revision ∗ (Nayak, Pagnucco, and Peppas 2003), semantically de- [Ψ]c corresponds to a consistency-preserving rational con- L sequence relation, in the sense of (Lehmann and Magi- fined as follows: dor 1992). Equivalently, it is representable by a total pre- Definition 1. The operators ∗N, ∗R and ∗L are such that: order (TPO) 4Ψ of worlds, such that A ) B 2 [Ψ]c iff x 4Ψ∗ A y iff (1) x 2 min(4Ψ; [[A]]), or (2) x; y2 = min(4Ψ; A ) ⊆ B (Grove 1988; Katsuno and Mendel- N min(4Ψ; [[A]]) and x 4Ψ y zon 1991).J NoteK thatJ AK 2 [Ψ] iff >) A 2 [Ψ]c or equiva- 4 4 4 lently iff min( Ψ;W ) ⊆ [[A]]. x Ψ∗RA y iff (1) x 2 min( Ψ; [[A]]), or (2) x; y2 = Following convention, we shall call principles presented min(4Ψ; [[A]]) and either (a) x ≺Ψ y or (b) x ∼Ψ y in terms of belief sets ‘syntactic’, and call ‘semantic’ those and (x 2 [[A]] or y 2 [[:A]]) principles couched in terms of TPOs, denoting the latter by x 4 y iff (1) x 2 [[A]] and y 2 [[:A]], or (2) 4 Ψ∗LA subscripting the corresponding syntactic principle with ‘ ’. (x 2 [[A]] iff y 2 [[A]]) and x 4Ψ y. Due to space considerations and for ease of exposition, we will largely restrict our focus to a semantic perspective on The suitability of all three operators, which we will group our problem of interest. here under the heading of ‘elementary revision operators’ The AGM postulates do not entail that one’s conditional (Chandler and Booth 2019), has been called into question. beliefs are determined by one’s beliefs—in the sense that, Indeed, they assume that a state Ψ can be identified with its corresponding TPO 4Ψ and that belief revision functions if [Ψ] = [Θ], then [Ψ]c = [Θ]c—and there is widespread consensus that such determination would be unduly restric- map pairs of TPOs and sentences onto TPOs. (For this rea- tive, with (Hansson 1992) providing supporting arguments. son, we will sometimes abuse language and notation and A fortiori, one should not identify conditional beliefs with speak, for instance, of the lexicographic revision of a TPO beliefs in the corresponding material conditional. That said, rather than of a state.) But this assumption has been criti- cised as implausible, with (Booth and Chandler 2017) pro- there does remain a connection between A) B 2 [Ψ]c and A ⊃ B 2 [Ψ]. The following is well known: viding a number of counterexamples. Accordingly, (Booth and Chandler 2018; Booth and Proposition 1. Given AGM, (a) if A ) B 2 [Ψ] , then c Chandler 2020) propose a strengthening of the DP postulates A ⊃ B 2 [Ψ], but (b) the converse does not hold. that is weak enough to avoid an identification of states with Indeed, (a) is simply equivalent, given (RT), to the AGM TPOs and is consistent with the characteristic postulates of postulate of Inclusion, according to which [Ψ ∗ A] ⊆ both ∗R and ∗L (albeit not of ∗N).

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