The logic of KM belief update is contained in the logic of AGM belief revision

The logic of KM b elief up date is con tained in the logic of A GM b elief revision Giacomo Bonanno Univ ersity of California, Davis, USA gfb onanno@ucda vis.edu Abstract F or eac h axiom of KM b elief up date we pro vide a corresp onding axiom in a modal logic containing three modal op erators: a unimo dal b elief op erator B , a bimodal conditional op erator > and the unimo dal necessit y op erator □ . W e then compare the resulting logic to the sim- ilar logic obtained from conv erting the AGM axioms of b elief revision in to mo dal axioms and show that the latter contains the former. De- noting the latter b y L AGM and the former b y L K M w e show that ev ery axiom of L K M is a theorem of L AGM . Th us A GM b elief revision can b e seen as a sp ecial case of KM b elief up date. F or the strong version of KM b elief up date we sho w that the difference b et w een L K M and L AGM can b e narro wed down to a single axiom, which deals exclu- siv ely with unsurprising information, that is, with formulas that were not initially disb elieved. Keyw ords: b elief up date, b elief revision, conditional, Kripke relation, Lewis selection function. 1 In tro duction In [4] every A GM axiom for b elief revision ([1]) was translated into a cor- resp onding axiom in a logic containing three mo dal op erators: a unimo dal b elief op erator B , a bimo dal conditional operator > and the unimodal neces- sit y (or global) op erator □ . The interpretation of B ϕ is "the agen t b eliev es ϕ ", the interpretation of ϕ > ψ is "if ϕ is (or were) the case then ψ is (or 1 w ould b e) the case" and the in terpretation of □ ϕ is " ϕ is necessarily true". Letting K b e the initial b elief set, the fact that ψ ∈ K ∗ ϕ , that is, that ψ b elongs to the revised b elief set K ∗ ϕ prompted by the informational input ϕ , is expressed in [4] by the formula B ( ϕ > ψ ) (the agen t b elieves that if ϕ w ere the case then ψ would b e the case). F or example, in [4] A GM axiom ( K ∗ 2) ( ϕ ∈ K ∗ ϕ ) is translated in to the mo dal axiom B ( ϕ > ϕ ) and A GM axiom ( K ∗ 4) (if ¬ ϕ / ∈ K then K + ϕ ⊆ K ∗ ϕ ) is translated into the mo dal axiom  ¬ B ¬ ϕ ∧ B ( ϕ → ψ )  → B ( ϕ > ψ ) . The translation w as done by means of the Kripke-Lewis seman tics put for- w ard in [3]: for ev ery AGM axiom a c haracterizing prop erty of Kripke-Lewis frames w as obtained, which was then shown to c haracterize a corresp onding mo dal-logic axiom. In this pap er we extend the analysis to the notion of b elief up date in tro- duced by Katsuno and Mendelzon (KM) in [7] b y providing, for every KM axiom, a corresponding modal axiom. W e then sho w that the modal logic – denoted b y L AGM – obtained by adding the mo dal A GM axioms to the basic logic, c ontains the logic – denoted by L K M – obtained by adding the modal KM axioms to the basic logic, in the sense that ev ery axiom of L K M is a theorem of logic L AGM . Thus w e confirm the conclusion reac hed in [10] – for the case of the strong version of KM up date – and reinforced in [3], that the notion of A GM b elief revision can be view ed as a strengthening of the notion of KM b elief up date. This conclusion b ecomes ev en more conspicuous when comparing the logic of AGM b elief revision to the logic of the str ong v ersion of KM b elief up date: in this case the difference b etw een the t wo notions reduces to a single axiom whic h deals with an item of information ϕ that is not surprising, in the sense that it is not the case that, initially , the agent b eliev ed ¬ ϕ . The pap er is structured as follows. In Section 2 w e review the KM theory of b elief up date and characterize eac h KM axiom in terms of a prop ert y of the Kripk e-Lewis frames introduced in [3]. In Section 3 w e review the mo dal logic L with the three modal op erators B , > and □ describ ed ab o ve and pro vide a mo dal axiom corresponding to eac h of the frame prop erties given in Section 2. In Section 4 w e recall the mo dal axiomatization of AGM b elief revision in tro duced in [4] and compare it to the logic of belief up date giv en in Section 3. Section 5 concludes. All the pro ofs are given in the App endix. 2 2 The KM theory of b elief up date W e consider a prop ositional logic based on a coun table set At of atomic form ulas. W e denote by Φ 0 the set of Bo olean form ulas constructed from At as follows: At ⊂ Φ 0 and if ϕ, ψ ∈ Φ 0 then ¬ ϕ and ϕ ∨ ψ b elong to Φ 0 . Define ϕ → ψ , ϕ ∧ ψ , and ϕ ↔ ψ in terms of ¬ and ∨ in the usual wa y (e.g. ϕ → ψ is a shorthand for ¬ ϕ ∨ ψ ); furthermore, ⊤ denotes a tautology and ⊥ a contradiction. Giv en a subset K of Φ 0 , its deductive closure C n ( K ) ⊆ Φ 0 is defined as follo ws: ψ ∈ C n ( K ) if and only if there exist ϕ 1 , ..., ϕ n ∈ K (with n ≥ 0 ) suc h that ( ϕ 1 ∧ ... ∧ ϕ n ) → ψ is a tautology . A set K ⊆ Φ 0 is c onsistent if C n ( K )  = Φ 0 ; it is de ductively close d if K = C n ( K ) . Giv en a set K ⊆ Φ 0 and a form ula ϕ ∈ Φ 0 , the exp ansion of K b y ϕ , denoted b y K + ϕ , is defined as follows: K + ϕ = C n ( K ∪ { ϕ } ) . Let K ⊆ Φ 0 b e a c onsistent and de ductively close d set representing the agen t’s initial b eliefs and let Ψ ⊆ Φ 0 b e a set of form ulas representing p ossible informational inputs. A b elief change function based on Ψ and K is a function ◦ : Ψ → 2 Φ 0 ( 2 Φ 0 denotes the set of subsets of Φ 0 ) that asso ciates with every form ula ϕ ∈ Ψ a set K ◦ ϕ ⊆ Φ 0 , in terpreted as the c hange in K prompted b y the input ϕ . W e follow the common practice of writing K ◦ ϕ instead of ◦ ( ϕ ) whi c h has the adv an tage of making it clear that the b elief c hange function refers to a given, fixe d , K . If Ψ  = Φ 0 then ◦ is called a p artial b elief c hange function, while if Ψ = Φ 0 then ◦ is called a ful l-domain b elief c hange function. 2.1 The KM theory of b elief up date W e consider the notion of b elief up date introduced by Katsuno and Mendel- zon (KM) in [7] and, in Section 4, w e compare it to the notion of b elief revision in tro duced by Alchourrón, Gärdenfors and Makinson (A GM) in [1]. The formalism in the tw o theories is somewhat different. In [7] a b elief state is represen ted b y a sen tence in a finite prop ositional calculus and b elief up- date is mo deled as a function o ver form ulas, while in [1] a b elief state is represen ted (as we did abov e) as a set of form ulas. Note that, while [7] allows for the p ossibility of inconsisten t initial b eliefs, follo wing [10] we take as starting p oint a consisten t b elief set. W e follo w closely the axiomatization of b elief update proposed b y [2, 9, 10], whic h has the adv an tage of making up date directly comparable to 3 revision (note, ho wev er, that [2, 9, 10] only cov er the case of "strong" up date, where axioms ( K ⋄ 6) and ( K ⋄ 7) are replaced by ( K ⋄ 9) : see Section 4 b elo w). Consider the follo wing version of the KM axioms of b elief up date based on the c onsistent and deductively closed set K (represen ting the initial b eliefs): ∀ ϕ, ψ ∈ Φ 0 , ( K ⋄ 0) K ⋄ ϕ = C n ( K ⋄ ϕ ) . ( K ⋄ 1) ϕ ∈ K ⋄ ϕ . ( K ⋄ 2) If ϕ ∈ K then K ⋄ ϕ = K . ( K ⋄ 3 a ) If ϕ is a contradiction then K ⋄ ϕ = Φ 0 . ( K ⋄ 3 b ) If ϕ is not a con tradiction then K ⋄ ϕ  = Φ 0 . ( K ⋄ 4) If ϕ ↔ ψ is a tautology then K ⋄ ϕ = K ⋄ ψ . ( K ⋄ 5) K ⋄ ( ϕ ∧ ψ ) ⊆ ( K ⋄ ϕ ) + ψ . ( K ⋄ 6 ) If ψ ∈ K ⋄ ϕ and ϕ ∈ K ⋄ ψ then K ⋄ ϕ = K ⋄ ψ . ( K ⋄ 7 ) If K is complete 1 then ( K ⋄ ϕ ) ∩ ( K ⋄ ψ ) ⊆ K ⋄ ( ϕ ∨ ψ ) . Remark 1. Katsuno and Mendelzon pr ovide an additional axiom (they name it U8), which they c al l the "disjunction rule". [2, 9, 10] tr anslate it into the fol lowing "axiom", which makes use of maximal ly c onsistent sets of formulas (MCS), also c al le d p ossible worlds. 2 Given a set of formulas Γ ⊆ Φ 0 , let J Γ K b e the set of MCS that c ontain al l the formulas in Γ . 3 The additional axiom is the fol lowing (note that J K K  = ∅ if and only if K is c onsistent): ( K ⋄ 8) If J K K  = ∅ then K ⋄ ϕ = \ w ∈ J K K ( w ⋄ ϕ ) . 1 A b elief set K is c omplete if, for ev ery form ula ϕ ∈ Φ 0 , either ϕ ∈ K or ¬ ϕ ∈ K . 2 A set of form ulas ∆ ⊂ Φ 0 is maximally consistent if it is consistent and, furthermore, ∀ ϕ ∈ Φ 0 \ ∆ , ∆ ∪ { ϕ } is inconsisten t. Every MCS is deductively closed and complete. If K is consistent, then K is complete if and only if J K K is a singleton, where J K K denotes the set of maximally consisten t sets of form ulas that contain all the formulas in K . 3 F ollowing the common notation in the literature, w e denote b y W the set of MCS, or p ossible worlds, and b y w an elemen t of W (hence w ∈ W is a set of formulas). Thus, w ∈ J Γ K if and only if w ∈ W and Γ ⊆ w . 4 ( K ⋄ 8 ) is of a different nature than the other axioms, since it applies the up date operator not only to the initial b elief set K but also to the individual MCS contained in J K K . ( K ⋄ 8 ) can b e viewed more as a condition on the in terpretation or seman tics rather than a real axiom; it is sup erfluous in our framew ork since its role is directly captured b y the semantics describ ed b elo w; for a more detailed discussion see [3]. ( K ⋄ 0) do es not appear in the list of axioms pro vided by Katsuno and Mendelzon, since their formalism is not in terms of b elief sets. F or i ∈ { 1 , 2 , 4 , 5 , 6 , 7 } , axiom ( K ⋄ i ) is a translation of Katsuno and Mendelzon’s axiom ( U i ) (for details see [9]). The conjunction of ( K ⋄ 3 a ) and ( K ⋄ 3 b ) is the translation of Katsuno and Mendelzon’s axiom ( U 3) when atten tion is restricted to the case where the initial b elief set K is consistent. 4 The following lemma, prov ed in the App endix, shows that axiom ( K ⋄ 7) can be replaced b y the following, seemingly stronger, axiom (obtained from ( K ⋄ 7) by dropping the clause ‘if K is complete’): ( K ⋄ 7 s ) ( K ◦ ϕ ) ∩ ( K ◦ ψ ) ⊆ K ◦ ( ϕ ∨ ψ ) . Lemma 1. ( K ⋄ 7 s ) fol lows fr om ( K ⋄ 7) and ( K ⋄ 8) In order to facilitate the conv ersion to a mo dal form ula, we replace ( K ⋄ 6) with the follo wing w eaker form, which – in the presence of ( K ⋄ 0) – is equiv alen t to ( K ⋄ 6) (the role of the added clause ‘ ⊤ ∈ K ⋄ ( ϕ ∧ ψ ) ’ is explained in Remark 2): ( K ⋄ 6 w ) If ψ ∈ K ⋄ ϕ and ϕ ∈ K ⋄ ψ and ⊤ ∈ K ⋄ ( ϕ ∧ ψ ) then K ⋄ ϕ = K ⋄ ψ In view of Lemma 1 and the ab ov e observ ation that ( K ⋄ 8) is sup erfluous in our framework, we define a KM belief up date as follows. Definition 1. A KM b elief up date function , b ase d on the c onsistent and de ductively close d set K , is a ful l domain b elief change function ⋄ : Φ 0 → 2 Φ 0 that satisfies axioms ( K ⋄ 0) - ( K ⋄ 5) , ( K ⋄ 6 w ) and ( K ⋄ 7 s ) . 4 Katsuno and Mendelzon allo w for the possibility that the initial beliefs are inconsistent, in which case the conjunction of ( K ⋄ 3 a ) and ( K ⋄ 3 b ) would b e stated as follows: K ⋄ ϕ = Φ 0 if and only if either K is inconsisten t or ϕ is a con tradiction. It should b e noted that one imp ortant difference b et ween up date and revision is precisely that up dating an inconsisten t K by a consistent formula ϕ yields the inconsistent b elief set Φ 0 , while revising an inconsistent K by a consistent formula ϕ yields a cons isten t set (AGM axiom ( K ∗ 5 b ): see Section 4). 5 Katsuno and Mendelzon show that, semantically , their notion of b elief up date corresponds to partial pre-orders on the set of maximally consisten t sets of formulas. W e will, instead, make use of the Kripk e-Lewis seman tics in tro duced in [3], to which we no w turn. 2.1.1 Kripk e-Lewis seman tics Definition 2. A Kripk e-Lewis frame is a triple ⟨ S, B , f ⟩ wher e 1. S is a set of states ; subsets of S ar e c al le d ev ents . 2. B ⊆ S × S is a binary r elation on S which is serial: ∀ s ∈ S , ∃ s ′ ∈ S , such that s B s ′ ( s B s ′ is an alternative notation for ( s, s ′ ) ∈ B ). W e denote by B ( s ) the set of states that ar e r e achable fr om s by B : B ( s ) = { s ′ ∈ S : s B s ′ } . B ( s ) is interpr ete d as the set of states that, initial ly, the agent c onsiders doxastic al ly p ossible at state s . 3. f : S × (2 S \ ∅ ) → 2 S is a selection function that asso ciates with every state-event p air ( s, E ) (with E  = ∅ ) a set of states f ( s, E ) ⊆ S , interpr ete d as the set of states that ar e closest (or most similar) to s , c onditional on event E . W e require seriality of the b elief relation b ecause it ensures that the initial b eliefs at state s , represen ted b y the non-empt y set B ( s ) , are consisten t. Similarly , the requiremen t that f ( s, E ) is defined only if E  = ∅ ensures that the informational input is consisten t (how to deal with inconsisten t information is discussed in Section 3). Note the absence, in Definition 2, of the extra prop erties imposed on the selection function in [3], namely , f ( s, E ) ⊆ E (Identit y). f ( s, E )  = ∅ (Normality). if s ∈ E then s ∈ f ( s, E ) (W eak Centering). The reason why w e do not imp ose an y prop erties on the selection function is that we w ant to highligh t the role of each prop ert y in the characterization of the KM axioms. F or example, Iden tit y (lo calised to B ( s ) ) pla ys a role in the characterization of ( K ⋄ 2) but not in the characterization of the other 6 axioms, Normalit y (lo calised to B ( s ) ) is used to c haracterize axiom ( K ⋄ 3 b ) but not the other axioms. 5 A dding a v aluation to a frame yields a mo del. Thus a mo del is a tup le ⟨ S, B , f , V ⟩ where ⟨ S, B , f ⟩ is a frame and V : At → 2 S is a v aluation that assigns to every atomic form ula p ∈ At the set of states where p is true. Definition 3. Given a mo del M = ⟨ S, B , f , V ⟩ define truth of a Bo ole an formula ϕ ∈ Φ 0 at a state s ∈ S in mo del M , denote d by s | = M ϕ , as fol lows: 1. if p ∈ At then s | = M p if and only if s ∈ V ( p ) , 2. s | = M ¬ ϕ if and only if s | = M ϕ , 3. s | = M ( ϕ ∨ ψ ) if and only if s | = M ϕ or s | = M ψ (or b oth). W e denote by ∥ ϕ ∥ M the truth set of form ula ϕ in mo del M : ∥ ϕ ∥ M = { s ∈ S : s | = M ϕ } . Giv en a mo del M = ⟨ S, B , f , V ⟩ and a state s ∈ S , let K s,M = { ϕ ∈ Φ 0 : B ( s ) ⊆ ∥ ϕ ∥ M } ; thus a formula ϕ belongs to K s,M if and only if at state s the agen t b elieves ϕ , in the sense that ϕ is true at ev ery state that the agen t considers doxastically p ossible at s . W e iden tify K s,M with the agen t’s initial b eliefs at state s . It is shown in [3] that the set K s,M ⊆ Φ 0 so defined is deductiv ely closed and consisten t (the latter prop erty follo ws from serialit y of the b elief relation B ). Next, giv en a mo del M = ⟨ S, B , f , V ⟩ and a state s ∈ S , let Ψ M = { ϕ ∈ Φ 0 : ∥ ϕ ∥ M  = ∅ } 6 and define the following p artial b elief c hange function ◦ : Ψ M → 2 Φ 0 based on Ψ M and K s,M : ψ ∈ K s,M ◦ ϕ if and only if, ∀ s ′ ∈ B ( s ) , f ( s ′ , ∥ ϕ ∥ M ) ⊆ ∥ ψ ∥ M or, equiv alen tly , S s ′ ∈B ( s ) f ( s ′ , ∥ ϕ ∥ M ) ⊆ ∥ ψ ∥ M (RI) Giv en the customary in terpretation of selection functions in terms of condi- tionals, (RI) can b e interpreted as stating that ψ ∈ K s,M ◦ ϕ if and only if at state s the agen t b eliev es that "if ϕ is (were) the case then ψ is (w ould 5 In conditional logic, Iden tity corresp onds to the axiom "if ϕ then ϕ ", denoted by ϕ > ϕ , and Normalit y v alidates the axiom ( ϕ > ψ ) → ¬ ( ϕ > ¬ ψ ) for consisten t ϕ : see [8]. 6 Since, in any giv en mo del, there are form ulas ϕ such that ∥ ϕ ∥ M = ∅ (at the very least all the con tradictions), Ψ M is a prop er subset of Φ 0 . 7 b e) the case". 7 This interpretation will be made explicit in the modal logic considered in Section 3. Remark 2. Consider an arbitr ary mo del M and state s and let ◦ : Ψ M → 2 Φ 0 b e the p artial b elief change function define d by (RI) . Then, for every c onsistent formula χ ∈ Φ 0 , ∥ χ ∥ M  = ∅ if and only if ⊤ ∈ K s,M ⋄ χ. 8 In what follo ws, when stating an axiom for a b elief change function, w e implicitly assume that it applies to every form ula in its domain . F or example, the axiom ϕ ∈ K ◦ ϕ asserts that, for all ϕ in the domain of ◦ , ϕ ∈ K ◦ ϕ . Definition 4. An axiom for b elief change functions is v alid on a frame F if, for every mo del b ase d on that fr ame and for every state s in that mo del, the p artial b elief change function define d by (RI) satisfies the axiom. An axiom is v alid on a set of frames F if it is valid on every fr ame F ∈ F . A stronger notion than v alidity is that of frame correspondence. The follo wing definition mimics the notion of frame corresp ondence in mo dal logic. Definition 5. W e say that an axiom A of b elief change functions is charac- terized b y , or corresp onds to , or c haracterizes , a pr op erty P of fr ames if the fol lowing is true: (1) axiom A is valid on the class of fr ames that satisfy pr op erty P , and (2) if a fr ame do es not satisfy pr op erty P then axiom A is not valid on that fr ame, that is, ther e is a mo del b ase d on that fr ame and a state in that mo del wher e the p artial b elief change function define d by (RI) violates axiom A . 7 Note that we allow for both the indicativ e and the sub junctive conditional. The indicativ e form (if ϕ is the case then ψ is the case) seems to b e more appropriate when the initial belief set do es not contain ¬ ϕ (that is, if the agen t initially considers ϕ p ossible), while the sub junctive form (if ϕ wer e the case then ψ would b e the case) seems to b e more appropriate when the agent initially believes ¬ ϕ . 8 Pro of. If ⊤ ∈ K s ⋄ χ then χ ∈ Ψ M and thus, by definition of Ψ M , ∥ χ ∥ M  = ∅ . Con v ersely , if ∥ χ ∥ M  = ∅ then χ ∈ Ψ M and, for ev ery s ′ ∈ B ( s ) , f ( s ′ , ∥ χ ∥ M ) is w ell- defined and, trivially , S s ′ ∈B ( s ) f ( s ′ , ∥ χ ∥ M ) ⊆ S = ∥⊤∥ , so that ⊤ ∈ K s ⋄ χ . 8 KM axiom F rame prop ert y ( K ⋄ 0) K ⋄ ϕ = C n ( K ⋄ ϕ ) No additional prop erty ( K ⋄ 1) ϕ ∈ K ⋄ ϕ ( P ∗ 2 ⋄ 1 ) ∀ s ∈ S, ∀ E ∈ 2 S \ ∅ , S s ′ ∈B ( s ) f ( s ′ , E ) ⊆ E ( K ⋄ 2) If ϕ ∈ K then K ⋄ ϕ = K ( P ⋄ 2) ∀ s ∈ S, ∀ E ∈ 2 S \ ∅ , if B ( s ) ⊆ E then, S s ′ ∈B ( s ) f ( s ′ , E ) = B ( s ) ( K ⋄ 3 b ) If ¬ ϕ is not a tautology then K ⋄ ϕ  = Φ 0 ( P ∗ 5 b ⋄ 3 b ) ∀ s ∈ S, ∀ E ∈ 2 S \ ∅ , ∃ s ′ ∈ B ( s ) suc h that f ( s ′ , E )  = ∅ ( K ⋄ 4) if ϕ ↔ ψ is a tautology then K ⋄ ϕ = K ⋄ ψ No additional prop erty ( K ⋄ 5) K ⋄ ( ϕ ∧ ψ ) ⊆ ( K ⋄ ϕ ) + ψ ( P ∗ 7 ⋄ 5 ) ∀ s ∈ S, ∀ E , F ∈ 2 S with E ∩ F  = ∅ , S s ′ ∈B ( s ) ( f ( s ′ , E ) ∩ F ) ⊆ S s ′ ∈B ( s ) f ( s ′ , E ∩ F ) ( K ⋄ 6 w ) If ψ ∈ K ⋄ ϕ and ϕ ∈ K ⋄ ψ and ⊤ ∈ K ⋄ ( ϕ ∧ ψ ) then K ⋄ ϕ = K ⋄ ψ ( P ⋄ 6 w ) ∀ s ∈ S, ∀ E , F ∈ 2 S with E ∩ F  = ∅ if S s ′ ∈B ( s ) f ( s ′ , E ) ⊆ F and S s ′ ∈B ( s ) f ( s ′ , F ) ⊆ E then S s ′ ∈B ( s ) f ( s ′ , E ) = S s ′ ∈B ( s ) f ( s ′ , F ) ( K ⋄ 7 s ) ( K ⋄ ϕ ) ∩ ( K ⋄ ψ ) ⊆ K ⋄ ( ϕ ∨ ψ ) ( P ⋄ 7 s ) ∀ s ∈ S, ∀ E , F ∈ 2 S \ ∅ S s ′ ∈B ( s ) f ( s ′ , E ∪ F ) ⊆ S s ′ ∈B ( s ) f ( s ′ , E ) ! ∪ S s ′ ∈B ( s ) f ( s ′ , F ) ! Figure 1: Semantic characterization of the KM axioms. 9 The table in Figure 1 lists, for every KM axiom, the characterizing prop- ert y of frames. 9 The pro ofs are giv en in the Appendix. The reason for the absence of axiom ( K ⋄ 3 a ) from the table in Figure 1 will b ecome clear in the next section. When a KM axiom coincides with an A GM axiom, the name of the corresp onding seman tic prop erty reflects this; for example, since KM axiom ( K ⋄ 1) coincides with AGM axiom ( K ∗ 2) , the corresp onding prop erty is denoted by ( P ∗ 2 ⋄ 1 ) . 10 3 A mo dal logic for b elief up date W e no w turn to the mo dal language considered in [4], whic h con tains three mo dal op erators: a unimo dal b elief op erator B , a bimo dal conditional op er- ator > and the unimo dal necessit y op erator □ . The interpretation of B ϕ is "the agen t b elieves ϕ ", the in terpretation of ϕ > ψ is "if ϕ is (or were) the case then ψ is (or would b e) the case" and the interpretation of □ ϕ is " ϕ is necessarily true". The set Φ of formulas in the language is defined as follows: • Φ 0 ⊆ Φ (recall that Φ 0 is the set of Bo olean form ulas built on the coun table set of atomic sentences At ), • if α , β ∈ Φ then all of the following b elong to Φ : □ α , B α , α > β and all their Bo olean com binations. W e fo cus on the basic normal logic, denoted b y L , consisting of the following axioms and rules of inference. 11 W e denote general formulas b y α , β and γ , while ϕ , ψ and χ are reserv ed for Bo ole an form ulas (e.g. in Figures 2-4). • Ev ery formula that has the form of a classical tautology is a theorem. • The consistency axiom D for B : ( D B ) B α → ¬ B ¬ α. 9 Some of the prop erties in Figure 1 are related to prop erties of the selection function discussed in the literature on conditionals ([8]). F or example, ( P ∗ 2 ⋄ 1 ) is related to Iden tity and ( P ∗ 5 b ⋄ 3 b ) to Normalit y . The remaining properties do not seem to b e related to prop erties discussed in that literature. 10 The AGM axioms are listed in Section 4. 11 W e follo w the nomenclature in [5, p.115]. 10 • The conjunction axiom C for □ , B and > : ( C □ ) □ α ∧ □ β → □ ( α ∧ β ) ( C B ) B α ∧ B β → B ( α ∧ β ) ( C > ) ( γ > α ) ∧ ( γ > β ) → ( γ > ( α ∧ β )) • The necessity-to-belief axiom: ( N B ) □ α → B α • The rule of inference Mo dus Ponens : ( M P ) α , α → β β • The rule of inference Ne c essitation for □ and > : ( N □ ) α □ α ( N > ) β α > β • The rule of inference RM for □ , B and > : ( RM □ ) α → β □ α → □ β ( RM B ) α → β B α → B β ( RM > ) α → β ( γ > α ) → ( γ > β ) Remark 3. (A) The fol lowing (which wil l b e use d in the pr o ofs) ar e the or ems of lo gic L : ( C ¬ □ ¬ ) ¬ □ ¬ ( α ∧ β ) → ¬ □ ¬ α ( C inv B ) B ( α ∧ β ) → B α ∧ B β ( K > ) ( α > β ) ∧ ( α > ( β → γ )) → ( α > γ ) ( A ∗ 1 ⋄ 0 ) B ( α > β ) ∧ B ( α > ( β → γ )) → B ( α > γ ) 11 (B) The fol lowing ar e derive d rules of infer enc e in lo gic L : ( RM ¬ □ ¬ ) α → β ¬ □ ¬ α → ¬ □ ¬ β ( N B ) α B α ( RM B > ) α → β B ( γ > α ) → B ( γ > β ) 3.1 F rame corresp ondence As seman tics for this mo dal logic w e tak e the Kripke-Lewis frames of Def- inition 2. A model based on a frame is obtained, as b efore, b y adding a v aluation V : At → 2 S . The follo wing definition expands Definition 3 b y adding v alidation rules for form ulas of the form □ α , α > β and B α . Definition 6. T ruth of a formula α ∈ Φ at state s in mo del M (denote d by s | = M α ) is define d as fol lows: 1. if p ∈ At then s | = M p if and only if s ∈ V ( p ) . 2. s | = M ¬ α if and only if s | = M α . 3. s | = M ( α ∨ β ) if and only if s | = M α or s | = M β (or b oth). 4. s | = M □ α if and only if, ∀ s ′ ∈ S , s ′ | = M α (thus s | = M ¬ □ ¬ α if and only if, for some s ′ ∈ S , s ′ | = M α , that is, ∥ α ∥ M  = ∅ ). 5. s | = M ( α > β ) if and only if, either (a) s | = M □ ¬ α (that is, ∥ α ∥ M = ∅ ), or (b) s | = M ¬ □ ¬ α (that is, ∥ α ∥ M  = ∅ ) and, for every s ′ ∈ f ( s, ∥ α ∥ M ) , s ′ | = M β (that is, f ( s, ∥ α ∥ M ) ⊆ ∥ β ∥ M ). 12 6. s | = M B α if and only if, ∀ s ′ ∈ B ( s ) , s ′ | = M α (that is, B ( s ) ⊆ ∥ α ∥ M ). The definitions of v alidit y and characterization are as in the previous section. 12 Recall that, b y definition of frame, f ( s, E ) is defined only if E  = ∅ . 12 Definition 7. A formula α ∈ Φ is v alid on a frame F if, for every mo del M b ase d on that fr ame and for every state s in that mo del, s | = M α . A formula α ∈ Φ is v alid on a set of frames F if it is valid on every fr ame F ∈ F . Definition 8. A formula α ∈ Φ is characterized by , or corresp onds to , or c haracterizes , a pr op erty P of fr ames if the fol lowing is true: (1) α is valid on the class of fr ames that satisfy pr op erty P , and (2) if a fr ame do es not satisfy pr op erty P then α is not valid on that fr ame. The table in Figure 2 sho ws, for ev ery property of frames considered in Figure 1, the mo dal form ula that corresp onds to it. When a KM axiom coincides with an AGM axiom, the name of the corresp onding mo dal formula reflects this; for example, since KM axiom ( K ⋄ 1) coincides with AGM axiom ( K ∗ 2) , the corresp onding modal form ula is denoted b y ( A ∗ 2 ⋄ 1 ) . 13 The pro ofs of the characterizations results are given in the App endix. In axioms ( A ∗ 5 b ⋄ 3 b ) and ( A ⋄ 7 s ) the clause ¬ □ ¬ ϕ in the anteceden t ensures that, on the semantic side, ∥ ϕ ∥  = ∅ ; in particular, it rules out that ϕ is a con tradiction; similarly for the clauses ¬ □ ¬ ψ and ¬ □ ¬ ( ϕ ∧ ψ ) . The in ter- pretation of the axioms is as follows: 13 The AGM axioms are listed in Section 4. 13 ( A ∗ 2 ⋄ 1 ) the agent b elieves that if ϕ w ere the case, then ϕ w ould b e the case ( A ⋄ 2) if the agent initially believes that ϕ then she b elieves that ψ if and only if she believes that if ϕ were the case, then ψ w ould be the case ( A ∗ 5 b ⋄ 3 b ) if ϕ is not necessarily false and the agen t b elieves that if ϕ w ere the case, then ψ w ould b e the case, then the agen t do es not b eliev e that if ϕ were the case, then ψ w ould not be the case ( A ∗ 7 ⋄ 5 ) if the conjunction ϕ ∧ ψ is not necessarily false and the agen t b eliev es that if ϕ ∧ ψ w ere the case, then χ w ould b e the case then she b elieves that if ϕ w ere the case, then it w ould b e the case that either ψ is false or χ is true (that is, that ψ → χ ) ( A ⋄ 6) if ϕ ∧ ψ is not necessarily false and the agent b elieves that if ϕ were the case, then ψ w ould b e the case and that if ψ were the case, then ϕ would b e the case, then she believes that if ϕ were the case, then χ w ould b e the case if and only if she b eliev es that if ψ w ere the case, then χ would b e the case ( A ⋄ 7 s ) if neither ϕ nor ψ is necessarily false, and the agent b elieves that if ϕ w ere the case, then χ w ould b e the case and that if ψ were the case, then χ would b e the case, then she b eliev es that if ϕ ∨ ψ w ere the case, then χ w ould b e the case Figure 3 puts together Figures 1 and 2 by showing for each KM axiom the corresp onding mo dal axiom or rule of inference. Note that ( A ∗ 1 ⋄ 0 ) is a theorem of L (see Remark 3). W e denote by L K M the extension of logic L obtained by adding the mo dal axioms and rules of inference listed in Figure 3. In the next section w e define a similar extension of L , denoted b y L AGM , that captures the logic of A GM b elief revision and sho w that L K M is contained in L AGM . 14 F rame prop ert y Corresp onding mo dal formula In all the formulas, ϕ, ψ , χ are Bo olean ( P ∗ 2 ⋄ 1 ) ∀ s ∈ S, ∀ E ∈ 2 S \ ∅ , S s ′ ∈B ( s ) f ( s ′ , E ) ⊆ E ( A ∗ 2 ⋄ 1 ) B ( ϕ > ϕ ) ( P ⋄ 2) ∀ s ∈ S, ∀ E ∈ 2 S \ ∅ , if B ( s ) ⊆ E then S s ′ ∈B ( s ) f ( s ′ , E ) = B ( s ) ( A ⋄ 2) B ϕ →  B ψ ↔ B ( ϕ > ψ )  ( P ∗ 5 b ⋄ 3 b ) ∀ s ∈ S, ∀ E ∈ 2 S \ ∅ , ∃ s ′ ∈ B ( s ) suc h that f ( s ′ , E )  = ∅ ( A ∗ 5 b ⋄ 3 b ) ( ¬ □ ¬ ϕ ∧ B ( ϕ > ψ )) → ¬ B ( ϕ > ¬ ψ ) ( P ∗ 7 ⋄ 5 ) ∀ s ∈ S, ∀ E , F ∈ 2 S with E ∩ F  = ∅ , S s ′ ∈B ( s ) ( f ( s ′ , E ) ∩ F ) ⊆ S s ′ ∈B ( s ) f ( s ′ , E ∩ F ) ( A ∗ 7 ⋄ 5 ) ¬ □ ¬ ( ϕ ∧ ψ ) ∧ B (( ϕ ∧ ψ ) > χ ) → B  ϕ > ( ψ → χ )  ( P ⋄ 6 w ) ∀ s ∈ S, ∀ E , F ∈ 2 S with E ∩ F  = ∅ if S s ′ ∈B ( s ) f ( s ′ , E ) ⊆ F and S s ′ ∈B ( s ) f ( s ′ , F ) ⊆ E then S s ′ ∈B ( s ) f ( s ′ , E ) = S s ′ ∈B ( s ) f ( s ′ , F ) ( A ⋄ 6 w ) ¬ □ ¬ ( ϕ ∧ ψ ) ∧ B ( ϕ > ψ ) ∧ B ( ψ > ϕ ) → ( B ( ϕ > χ ) ↔ B ( ψ > χ )) ( P ⋄ 7 s ) ∀ s ∈ S, ∀ E , F ∈ 2 S \ ∅ S s ′ ∈B ( s ) f ( s ′ , E ∪ F ) ⊆ S s ′ ∈B ( s ) f ( s ′ , E ) ! ∪ S s ′ ∈B ( s ) f ( s ′ , F ) ! ( A ⋄ 7 s ) ¬ □ ¬ ϕ ∧ ¬ □ ¬ ψ ∧ B ( ϕ > χ ) ∧ B ( ψ > χ ) → B (( ϕ ∨ ψ ) > χ ) Figure 2: The frame prop erties of Figure 1 and the corresponding modal form ulas 15 KM axiom Mo dal axiom/Rule of inference In all the formulas, ϕ, ψ , χ are Bo olean ( K ⋄ 0) K ⋄ ϕ = C n ( K ⋄ ϕ ) ( A ∗ 1 ⋄ 0 ) B ( ϕ > ψ ) ∧ B ( ϕ > ( ψ → χ )) → B ( ϕ > χ ) ( K ⋄ 1) ϕ ∈ K ⋄ ϕ ( A ∗ 2 ⋄ 1 ) B ( ϕ > ϕ ) ( K ⋄ 2) If ϕ ∈ K then K ⋄ ϕ = K ( A ⋄ 2) B ϕ →  B ψ ↔ B ( ϕ > ψ )  ( K ⋄ 3 a ) If ¬ ϕ is a tautology then K ⋄ ϕ = Φ 0 Rule of inference ( R ∗ 5 a ⋄ 3 a ) ¬ ϕ B ( ϕ > ψ ) ( K ⋄ 3 b ) If ¬ ϕ is not a tautology then K ⋄ ϕ  = Φ 0 ( A ∗ 5 b ⋄ 3 b ) ( ¬ □ ¬ ϕ ∧ B ( ϕ > ψ )) → ¬ B ( ϕ > ¬ ψ ) ( K ⋄ 4) if ϕ ↔ ψ is a tautology then K ⋄ ϕ = K ⋄ ψ Rule of inference ( R ∗ 6 ⋄ 4 ) ϕ ↔ ψ B ( ϕ > χ ) ↔ B ( ψ > χ ) ( K ⋄ 5) K ⋄ ( ϕ ∧ ψ ) ⊆ ( K ⋄ ϕ ) + ψ ( A ∗ 7 ⋄ 5 ) ¬ □ ¬ ( ϕ ∧ ψ ) ∧ B (( ϕ ∧ ψ ) > χ ) → B  ϕ > ( ψ → χ )  ( K ⋄ 6 w ) If ψ ∈ K ⋄ ϕ and ϕ ∈ K ⋄ ψ and ⊤ ∈ K ⋄ ( ϕ ∧ ψ ) then K ⋄ ϕ = K ⋄ ψ ( A ⋄ 6 w ) ¬ □ ¬ ( ϕ ∧ ψ ) ∧ B ( ϕ > ψ ) ∧ B ( ψ > ϕ ) → ( B ( ϕ > χ ) ↔ B ( ψ > χ )) ( K ⋄ 7 s ) ( K ⋄ ϕ ) ∩ ( K ⋄ ψ ) ⊆ K ⋄ ( ϕ ∨ ψ ) ( A ⋄ 7 s ) ¬ □ ¬ ϕ ∧ ¬ □ ¬ ψ ∧ B ( ϕ > χ ) ∧ B ( ψ > χ ) → B (( ϕ ∨ ψ ) > χ ) Figure 3: The KM axioms and the corresp onding modal axioms/rules of inference 16 4 Relating KM logic to A GM logic In [4] ev ery AGM axiom of b elief revision was translated into a corresp onding mo dal axiom in a w a y similar to what w as done in the previous section for the KM axioms. Figure 4 (whic h reproduces Figure 3 in [4], with the mo dal axioms renamed to matc h the names in this pap er) shows the corresp ondence b et ween A GM axioms and modal axioms. 14 W e denote b y L AGM the extension of logic L obtained b y adding the mo dal axioms and rules of inference listed in Figure 4. According to the follo wing prop osition, which is prov ed in the App endix, AGM b elief revision can b e viewed as a strengthening of KM belief update. Prop osition 1. The lo gic L K M of KM b elief up date is c ontaine d in the lo gic L AGM of A GM b elief r evision, that is, every axiom of L K M is a the or em of L AGM . W e conclude this section by considering a stronger version of b elief up date suggested by [7]. Katsuno and Mendelzon obtain this stronger v ersion b y replacing their axioms ( U 6) and ( U 7) with a stronger axiom, whic h they call ( U 9) . 15 In our framework their axiom ( U 9) can b e translated as follows (see [2, 9, 10]): ( K ⋄ 9) If K is complete and ¬ ψ / ∈ K ⋄ ϕ then ( K ⋄ ϕ ) + ψ ⊆ K ⋄ ( ϕ ∧ ψ ) . The following Lemma is pro v ed in the Appendix. Lemma 2. The fol lowing (se emingly str onger) version of ( K ⋄ 9) (obtaine d by dr opping the clause ‘if K is c omplete’ fr om ( K ⋄ 9) ) fol lows fr om ( K ⋄ 0) , ( K ⋄ 8) and ( K ⋄ 9) : ( K ⋄ 9 s ) If ¬ ψ / ∈ K ⋄ ϕ then ( K ⋄ ϕ ) + ψ ⊆ K ⋄ ( ϕ ∧ ψ ) 14 In [3] ( K ∗ 4) was given in a weak er form, namely if ¬ ϕ / ∈ K then K ⊆ K ∗ ϕ . Ho wev er, in the presence of ( K ∗ 1) and ( K ∗ 2) , the t w o are equiv alent. That the stronger version used in Figure 4 implies the weak er version follows from the fact that K ⊆ K + { ϕ } . T o pro v e the conv erse, let ψ ∈ K + ϕ . Then since K is deductively closed, ( ϕ → ψ ) ∈ K so that, b y the weak er version, ( ϕ → ψ ) ∈ K ∗ ϕ . By ( K ∗ 2) , ϕ ∈ K ∗ ϕ and by ( K ∗ 1) K ∗ ϕ is deductively closed. Thus, since ϕ, ( ϕ → ψ ) ∈ K ∗ ϕ it follows that ψ ∈ K ∗ ϕ . 15 The authors then show that this stronger notion of b elief up date corresponds seman- tically to total pre-orders on the set of p ossible w orlds. 17 A GM axiom Mo dal axiom/Rule of Inference ( for ϕ, ψ , χ ∈ Φ 0 ) ( K ∗ 1) K ∗ ϕ = C n ( K ∗ ϕ ) ( A ∗ 1 ⋄ 0 ) B ( ϕ > ψ ) ∧ B ( ϕ > ( ψ → χ )) → B ( ϕ > χ ) ( K ∗ 2) ϕ ∈ K ∗ ϕ ( A ∗ 2 ⋄ 1 ) B ( ϕ > ϕ ) ( K ∗ 3) K ∗ ϕ ⊆ K + ϕ ( A ∗ 3)  ¬ □ ¬ ϕ ∧ B ( ϕ > ψ  → B ( ϕ → ψ ) ( K ∗ 4) if ¬ ϕ / ∈ K ∗ ϕ then K + ψ ⊆ K ∗ ϕ ( A ∗ 4)  ¬ B ¬ ϕ ∧ B ( ϕ → ψ )  → B ( ϕ > ψ ) ( K ∗ 5 a ) If ¬ ϕ is a tautology , then K ∗ ϕ = Φ 0 rule of inference: ( R ∗ 5 a ⋄ 3 a ) ¬ ϕ B ( ϕ > ψ ) ( K ∗ 5 b ) If ¬ ϕ is not a tautology then K ∗ ϕ  = Φ 0 ( A ∗ 5 b ⋄ 3 b )  ¬ □ ¬ ϕ ∧ B ( ϕ > ψ )  → ¬ B ( ϕ > ¬ ψ ) ( K ∗ 6) if ϕ ↔ ψ is a tautology then K ∗ ϕ = K ∗ ψ rule of inference: ( R ∗ 6 ⋄ 4 ) ϕ ↔ ψ B ( ϕ > χ ) ↔ B ( ψ > χ ) ( K ∗ 7) K ∗ ( ϕ ∧ ψ ) ⊆ ( K ∗ ϕ ) + ψ ( A ∗ 7) ¬ □ ¬ ( ϕ ∧ ψ ) ∧ B (( ϕ ∧ ψ ) > χ ) → B  ϕ > ( ψ → χ )  ( K ∗ 8) If ¬ ψ / ∈ K ∗ ϕ, then ( K ∗ ϕ ) + ψ ⊆ K ∗ ( ϕ ∧ ψ ) ( A ∗ 8 ⋄ 9 s ) ¬ B ( ϕ > ¬ ψ ) ∧ B ( ϕ > ( ψ → χ )) → B  ( ϕ ∧ ψ ) > ( ψ ∧ χ )  Figure 4: The corresp ondence betw een AGM axioms and their mo dal coun- terparts. 18 Definition 9. A strong b elief up date function is a ful l-domain b elief change function ⋄ : Φ 0 → 2 Φ 0 that satisfies axioms ( K ⋄ 0) - ( K ⋄ 5) and ( K ⋄ 9 s ) . Comparing the A GM axioms to the axioms of the strong version of KM up date we can see that: • KM axiom ( K ⋄ 0 ) coincides with A GM axiom ( K ∗ 1) and the corre- sp onding mo dal axiom is the following, which is a theorem of L (see Remark 3) ( A ∗ 1 ⋄ 0 ) B ( ϕ > ψ ) ∧ B ( ϕ > ( ψ → χ )) → B ( ϕ > χ ) • KM axiom ( K ⋄ 1 ) coincides with A GM axiom ( K ∗ 2) and the corre- sp onding mo dal axiom is ( A ∗ 2 ⋄ 1 ) B ( ϕ > ϕ ) • KM axiom ( K ⋄ 3 a ) coincides with AGM axiom ( K ∗ 5 a ) and it corre- sp onds to the rule of inference ( R ∗ 5 a ⋄ 3 a ) ¬ ϕ B ( ϕ > ψ ) • KM axiom ( K ⋄ 3 b ) coincides with A GM axiom ( K ∗ 5 b ) and the cor- resp onding mo dal axiom is ( A ∗ 5 b ⋄ 3 b ) ( ¬ □ ¬ ϕ ∧ B ( ϕ > ψ )) → ¬ B ( ϕ > ¬ ψ ) • KM axiom ( K ⋄ 4 ) coincides with AGM axiom ( K ∗ 6) and it corresp onds to the rule of inference ( R ∗ 6 ⋄ 4 ) ϕ ↔ ψ B ( ϕ > χ ) ↔ B ( ψ > χ ) • KM axiom ( K ⋄ 5 ) coincides with A GM axiom ( K ∗ 7) and the corre- sp onding mo dal axiom is ( A ∗ 7 ⋄ 5 ) ¬ □ ¬ ( ϕ ∧ ψ ) ∧ B (( ϕ ∧ ψ ) > χ ) → B  ϕ > ( ψ → χ )  19 • KM axiom ( K ⋄ 9 s ) coincides with A GM axiom ( K ∗ 8) and the corre- sp onding mo dal axiom is ( A ∗ 8 ⋄ 9 s ) ¬ B ( ϕ > ¬ ψ ) ∧ B ( ϕ > ( ψ → χ )) → B  ( ϕ ∧ ψ ) > ( ψ ∧ χ )  Th us the difference b etw een AGM b elief revision and the strong version of KM b elief up date is that the former con tains axioms ( K ∗ 3) and ( K ∗ 4) while the latter only requires axiom ( K ⋄ 2) . First of all, note that axiom ( A ⋄ 2) – corresponding to ( K ⋄ 2) – is a theorem of L AGM (see Lemma 4 in the App endix) and th us the lo gic L AGM c ontains also the lo gic of the str ong version of KM b elief up date . The follo wing lemma, prov ed in the Appendix, sho ws that the mo dal axiom corresp onding to A GM axiom ( K ∗ 3) is prov able in logic L K M . Lemma 3. Axiom ( A ∗ 3)  ¬ □ ¬ ϕ ∧ B ( ϕ > ψ  → B ( ϕ → ψ ) (which is the mo dal c ounterp art to A GM axiom ( K ∗ 3) ) is pr ovable in lo gic L K M . Th us, the difference b et ween AGM b elief revision and the strong version of KM b elief update reduces to one axiom. Axiom ( A ⋄ 2) B ϕ ∧ B ψ → B ( ϕ > ψ ) in the latter is replaced, in the former, b y the stronger axiom ( A ∗ 4) ¬ B ¬ ϕ ∧ B ( ϕ → ψ ) → B ( ϕ > ψ ) Axiom ( A ∗ 4) cov ers the case where the informational input ϕ is initially not disb elieve d ( ¬ ϕ / ∈ K ). Comparing the frame prop ert y ( P ⋄ 2) corresponding to KM axiom ( A ⋄ 2) (see Figure 2) with the follo wing property , which is implied by the prop erty corresp onding to AGM axiom ( A ∗ 4) 16 if B ( s ) ∩ E  = ∅ then [ s ′ ∈B ( s ) f ( s ′ , E ) ⊆ B ( s ) ∩ E 16 F or completeness, since the (standard) version of ( K ∗ 4) used here is somewhat dif- feren t from the one used in [3, 4] (see F o otnote 14) we prov e the corresp ondence results for ( K ∗ 4) and ( A ∗ 4) in Lemma 7 in the App endix. Note that Property ( P ∗ 4) given in Lemma 7 implies the ab ov e prop erty . In fact, since B ( s ) = ( B ( s ) ∩ ( S \ E )) ∪ ( B ( s ) ∩ E ) ⊆ ( S \ E ) ∪ ( B ( s ) ∩ E ) , taking F in ( P ∗ 4) to be B ( s ) ∩ E w e get the abov e property . 20 w e can see that the AGM theory requires that (letting E = ∥ ϕ ∥ ) the revised b eliefs b e concen trated on the ϕ -states in B ( s ) , while the strong version of KM up date allows the revised b eliefs to include ϕ -states outside of B ( s ) . On the other hand, there is no difference b etw een the tw o theories when the informational input ϕ is initially disb eliev ed, that is, when the agen t initially b eliev es ¬ ϕ . 5 Conclusion As remarked in [4], translating the AGM axioms of b elief revision and the KM axioms of b elief up date into mo dal formulas has the adv an tage of unifying the treatmen t of b elief, b elief revision and b elief update under the same um brella. Starting with Hintikk a’s [6] seminal con tribution, the notion of b elief has b een studied within the con text of mo dal logic, which allo ws one to express prop erties such as positive introspection of b eliefs ( B ϕ → B B ϕ ), negativ e in trosp ection of b eliefs ( ¬ B ϕ → B ¬ B ϕ ), the relationship b et w een kno wledge and b elief, etc. The mo dal AGM axioms and KM axioms listed in Figures 3 and 4 are restricted axioms in that the formulas ϕ , ψ and χ that appear in them are required to b e Boolean. Some of those axioms in volv e some nesting of the operators, such as B ( ϕ > ψ ) . Logic L op ens the do or to in vesti- gating prop erties of b elief change that go b ey ond those considered in the A GM theory and the KM theory . F or example, one can in vestigate intro- sp ection prop erties for supp ositional b eliefs: B ( ϕ > ψ ) → B B ( ϕ > ψ ) and ¬ B ( ϕ > ψ ) → B ¬ B ( ϕ > ψ ) . Other nestings of the op erators that go b eyond those considered in Figures 3 and 4 ma y also be w orth considering, suc h as B ( B ϕ > B ψ ) (the agent b elieves that if she w ere to b eliev e ϕ then she w ould b eliev e ψ ) or B  ϕ > ( ψ > χ )  (the agen t b elieves that if ϕ were the case then if ψ were the case then χ would b e the case). Perhaps some of these more complex form ulas will turn out to b e useful in characterizing the notions of iterated b elief revision or iterated belief up date. The main result of this pap er is that the mo dal logic L K M of KM b elief up date is con tained in the mo dal logic L AGM of A GM b elief revision, thus highligh ting the fact that there is no conceptual difference b etw een the t wo notions: one is merely a sp ecial case of the other. F urthermore, if one fo cuses on the strong version of KM b elief up date, then the difference b etw een the t wo theories can be narro w ed down to the differen t wa ys in whic h they deal 21 with unsurprising information (that is, with formulas ϕ that were not initially disb eliev ed), while there is no difference in ho w the t w o treat surprising information, that is, information that con tradicts the initial beliefs. A Pro ofs Lemma 1. ( K ⋄ 7 s ) fol lows fr om ( K ⋄ 7) and ( K ⋄ 8) . Pr o of. By ( K ⋄ 8) (recall the assumption that K is consistent so that J K K  = ∅ ), K ⋄ ϕ ∩ K ⋄ ψ = T w ∈ J K K ( w ⋄ ϕ ) ! ∩ T w ∈ J K K ( w ⋄ ψ ) ! = T w ∈ J K K  ( w ⋄ ϕ ) ∩ ( w ⋄ ψ )  (1) By ( K ⋄ 7) (since every MCS is complete), for every w ∈ J K K , ( w ⋄ ϕ ) ∩ ( w ⋄ ψ ) ⊆ w ⋄ ( ϕ ∨ ψ ) ; thus, \ w ∈ J K K  ( w ⋄ ϕ ) ∩ ( w ⋄ ψ )  ⊆ \ w ∈ J K K w ⋄ ( ϕ ∨ ψ ) (2) It follows from (1) and (2) that ( K ⋄ ϕ ∩ K ⋄ ψ ) ⊆ \ w ∈ J K K w ⋄ ( ϕ ∨ ψ ) (3) and by ( K ⋄ 8) \ w ∈ J K K w ⋄ ( ϕ ∨ ψ ) = K ⋄ ( ϕ ∨ ψ ) (4) Th us, b y (3) and (4), K ⋄ ϕ ∩ K ⋄ ψ ⊆ K ⋄ ( ϕ ∨ ψ ) 22 Pro ofs of the semantic c haracterizations listed in Figure 1 . The corresp ondence for - ( K ⋄ 1) is pro v ed in [4, p.4] (since ( K ⋄ 1) coincides with AGM axiom ( K ∗ 2) ). - ( K ⋄ 3 b ) is pro ved in [4, p.4] (since ( K ⋄ 3 b ) coincides with A GM axiom ( K ∗ 5 b ) ). - ( K ⋄ 5) is prov ed in Proposition 3 in [3]. 17 Th us w e only need to prov e the characterization for ( K ⋄ 2) , ( K ⋄ 6 w ) and ( K ⋄ 7 s ) , which is done in the follo wing three propositions. 18 Prop osition 2. F r ame pr op erty ( P ⋄ 2) ∀ s ∈ S, ∀ E ∈ 2 S \ ∅ , if B ( s ) ⊆ E then S s ′ ∈B ( s ) f ( s ′ , E ) = B ( s ) char acterizes the fol lowing KM axiom: ( K ⋄ 2) if ϕ ∈ K then K ⋄ ϕ = K . Pr o of. (A) Fix an arbitrary model based on a frame that satisfies Prop ert y ( P ⋄ 2) , an arbitrary state s ∈ S and let ⋄ be the partial b elief c hange function based on K s defined b y ( RI ). Let ϕ ∈ Φ 0 b e suc h that ϕ ∈ K s , that is, B ( s ) ⊆ ∥ ϕ ∥ (thus, by serialit y of B , ∥ ϕ ∥  = ∅ ). W e need to sho w that, for every formula ψ ∈ Φ 0 , ψ ∈ K ⋄ ϕ if and only if ψ ∈ K , that is, S s ′ ∈B ( s ) f ( s ′ , ∥ ϕ ∥ ) ⊆ ∥ ψ ∥ if and only if B ( s ) ⊆ ∥ ψ ∥ . This is an immediate consequence of ( P ⋄ 2) with E = ∥ ϕ ∥ ) . (B) Conv ersely , fix a frame that violates Property ( P ⋄ 2) . Then there exist s ∈ S and E ∈ 2 S suc h that B ( s ) ⊆ E (th us, b y seriality of B , E  = ∅ ) but S s ′ ∈B ( s ) f ( s ′ , E )  = B ( s ) . T wo cases are p ossible. CASE 1: S s ′ ∈B ( s ) f ( s ′ , E ) ⊆ B ( s ) . Let p, q ∈ At be atomic form ulas and 17 The characterization there w as pro ved for a differently worded prop erty , namely ∀ G ∈ 2 S , if S s ′ ∈B ( s ) f ( s ′ , E ∩ F ) ⊆ G then S s ′ ∈B ( s ) ( f ( s ′ , E ) ∩ F ) ⊆ G . It is straightfor- w ard to v erify that this prop erty is equiv alent to property ( P ∗ 7 ⋄ 5 ) in Figure 1. 18 A weak er prop erty than ( P ⋄ 2) (where in the consequen t ’ ⊆ ’ was used instead of ’ = ’) w as sufficien t in [3] to characterize ( K ⋄ 2) because in that paper the definition of frame, unlik e in this pap er, included the property of W eak Centering: if s ∈ E then s ∈ f ( s, E ) . 23 construct a model where ∥ p ∥ = E and ∥ q ∥ = B ( s ) . Then, since B ( s ) ⊆ E = ∥ p ∥ , p ∈ K s and, since B ( s ) ⊆ ∥ q ∥ , q ∈ K s but, since S s ′ ∈B ( s ) f ( s ′ , ∥ p ∥ ) ⊆ ∥ q ∥ , q / ∈ K s ⋄ p . Hence K s ⋄ p  = K s . CASE 2: B ( s ) ⊆ S s ′ ∈B ( s ) f ( s ′ , ∥ p ∥ ) . Let p, q ∈ At b e atomic form ulas and construct a mo del where ∥ p ∥ = E and ∥ q ∥ = S s ′ ∈B ( s ) f ( s ′ , E ) . Then, p ∈ K s and q ∈ K s ⋄ p but q / ∈ K s , so that K s ⋄ p  = K s . Prop osition 3. F r ame pr op erty ( P ⋄ 6 w ) ∀ s ∈ S, ∀ E , F ∈ 2 S with E ∩ F  = ∅ if S s ′ ∈B ( s ) f ( s ′ , E ) ⊆ F and S s ′ ∈B ( s ) f ( s ′ , F ) ⊆ E then S s ′ ∈B ( s ) f ( s ′ , E ) = S s ′ ∈B ( s ) f ( s ′ , F ) char acterizes the fol lowing KM axiom: ( K ⋄ 6 w ) If ψ ∈ K ⋄ ϕ and ϕ ∈ K ⋄ ψ and ⊤ ∈ K ⋄ ( ϕ ∧ ψ ) then K ⋄ ϕ = K ⋄ ψ Pr o of. Fix a frame that satisfies prop erty ( P ⋄ 6 w ) , an arbitrary mo del M based on it and an arbitrary state s ∈ S and let ⋄ b e the belief c hange function defined b y (RI). Let ϕ, ψ , ( ϕ ∧ ψ ) ∈ Φ 0 b e in the domain of ⋄ (hence ∥ ϕ ∧ ψ ∥  = ∅ and thus ∥ ϕ ∥  = ∅ and ∥ ψ ∥  = ∅ ) and supp ose that ψ ∈ K s ⋄ ϕ and ϕ ∈ K s ⋄ ψ (note that, by Remark 2, ⊤ ∈ K ⋄ ( ϕ ∧ ψ ) ). Then, S s ′ ∈B ( s ) f ( s ′ , ∥ ϕ ∥ ) ⊆ ∥ ψ ∥ and S s ′ ∈B ( s ) f ( s ′ , ∥ ψ ∥ ) ⊆ ∥ ϕ ∥ and thus, b y Prop ert y ( P ⋄ 6 w ) (with E = ∥ ϕ ∥ and F = ∥ ψ ∥ ; note that ∥ ϕ ∥ ∩ ∥ ψ ∥ = ∥ ϕ ∧ ψ ∥  = ∅ ) [ s ′ ∈B ( s ) f ( s ′ , ∥ ϕ ∥ ) = [ s ′ ∈B ( s ) f ( s ′ , ∥ ψ ∥ ) (5) It follo ws from (5) that, for every χ ∈ Φ 0 , χ ∈ K s ⋄ ϕ if and only if χ ∈ K s ⋄ ψ , that is, K s ⋄ ϕ = K s ⋄ ψ Con versely , fix a frame that violates prop erty ( P ⋄ 6 w ) . Then there exist s ∈ S and E , F ∈ 2 S suc h that E ∩ F  = ∅ , S s ′ ∈B ( s ) f ( s ′ , E ) ⊆ F and S s ′ ∈B ( s ) f ( s ′ , F ) ⊆ E but S s ′ ∈B ( s ) f ( s ′ , E )  = S s ′ ∈B ( s ) f ( s ′ , F ) T wo cases are p ossible: 24 CASE 1: S s ′ ∈B ( s ) f ( s ′ , E ) ⊈ S s ′ ∈B ( s ) f ( s ′ , F ) . CASE 2: S s ′ ∈B ( s ) f ( s ′ , F ) ⊈ S s ′ ∈B ( s ) f ( s ′ , E ) . In Case 1, let p, q , r ∈ At b e atomic sen tences and construct a model based on this frame where ∥ p ∥ = E , ∥ q ∥ = F and ∥ r ∥ = S s ′ ∈B ( s ) f ( s ′ , F ) . Then, since S s ′ ∈B ( s ) f ( s ′ , ∥ p ∥ ) ⊆ ∥ q ∥ , q ∈ K s ⋄ p and, since S s ′ ∈B ( s ) f ( s ′ , ∥ q ∥ ) ⊆ ∥ p ∥ , p ∈ K s ⋄ q and, since ∥ p ∧ q ∥  = ∅ , ⊤ ∈ K ⋄ ( p ∧ q ) (see Remark 2). F urthermore, r ∈ K s ⋄ q , but, since S s ′ ∈B ( s ) f ( s ′ , ∥ p ∥ )) ⊈ ∥ r ∥ , r / ∈ K s ⋄ p . Th us K s ⋄ p  = K s ⋄ q . In Case 2, let p, q , r ∈ At b e atomic sen tences and construct a model based on this frame where ∥ p ∥ = E , ∥ q ∥ = F and ∥ r ∥ = S s ′ ∈B ( s ) f ( s ′ , E ) . Then, q ∈ K s ⋄ p and p ∈ K s ⋄ q and r ∈ K s ⋄ p and (since ∥ p ∧ q ∥  = ∅ ) ⊤ ∈ K ⋄ ( p ∧ q ) (see Remark 2) but, since S s ′ ∈B ( s ) f ( s ′ , ∥ q ∥ )) ⊈ ∥ r ∥ , r / ∈ K s ⋄ q . Th us K s ⋄ p  = K s ⋄ q . Prop osition 4. F r ame pr op erty ( P ⋄ 7 s ) ∀ s ∈ S, ∀ E , F ∈ 2 S \ ∅ S s ′ ∈B ( s ) f ( s ′ , E ∪ F ) ⊆ S s ′ ∈B ( s ) f ( s ′ , E ) ! ∪ S s ′ ∈B ( s ) f ( s ′ , F ) ! char acterizes the fol lowing KM axiom: ( K ⋄ 7 s ) ( K ⋄ ϕ ) ∩ ( K ⋄ ψ ) ⊆ K ⋄ ( ϕ ∨ ψ ) Pr o of. Fix a frame that satisfies prop ert y ( P ⋄ 7 s ) , an arbitrary mo del based on it, an arbitrary state s ∈ S and let ⋄ b e the partial b elief change function based on K s defined b y ( R I ). Let ϕ, ψ ∈ Φ 0 b e in the domain of ⋄ (thus ∥ ϕ ∥  = ∅ and ∥ ψ ∥  = ∅ ) and fix an arbitrary χ ∈ ( K s ⋄ ϕ ) ∩ ( K s ⋄ ψ ) . Then, b y ( RI ), S s ′ ∈B ( s ) f ( s ′ , ∥ ϕ ∥ ) ⊆ ∥ χ ∥ and S s ′ ∈B ( s ) f ( s ′ , ∥ ψ ∥ ) ⊆ ∥ χ ∥ and th us, b y Prop ert y ( P ⋄ 7 s ) (with E = ∥ ϕ ∥ and F = ∥ ψ ∥ ) and the fact that ∥ ϕ ∥ ∪ ∥ ψ ∥ = ∥ ϕ ∨ ψ ∥ , S s ′ ∈B ( s ) f ( s ′ , ∥ ϕ ∨ ψ ∥ ) ⊆ ∥ χ ∥ , that is, by ( RI ), χ ∈ K s ⋄ ( ϕ ∨ ψ ) . 25 Con versely , fix a frame that violates prop ert y ( P ⋄ 7 s ) . Then there exist s ∈ S and E , F ∈ 2 S \ ∅ such that [ s ′ ∈B ( s ) f ( s ′ , E ∪ F ) ⊈ [ s ′ ∈B ( s ) f ( s ′ , E ) ∪ [ s ′ ∈B ( s ) f ( s ′ , F ) (6) Let p, q , r ∈ At b e atomic form ulas and construct a mo del where ∥ p ∥ = E , ∥ q ∥ = F and ∥ r ∥ = S s ′ ∈B ( s ) f ( s ′ , E ) ∪ S s ′ ∈B ( s ) f ( s ′ , F ) . Then, by (6) (since ∥ p ∥ ∪ ∥ q ∥ = ∥ p ∨ q ∥ ), S s ′ ∈B ( s ) f ( s ′ , ∥ p ∨ q ∥ ) ⊈ ∥ r ∥ and th us, by ( RI ), r / ∈ K s ⋄ ( p ∨ q ) . On the other hand, since S s ′ ∈B ( s ) f ( s ′ , ∥ p ∥ ) ⊆ ∥ r ∥ and S s ′ ∈B ( s ) f ( s ′ , ∥ q ∥ ) ⊆ ∥ r ∥ , r ∈ K ⋄ p and r ∈ K ⋄ q , yielding a violation of axiom ( K ⋄ 7 s ) . Pro ofs of the syntactic c haracterizations listed in Figure 2 . The characterizations of ( A ∗ 2 ⋄ 1 ) , ( A ∗ 5 b ⋄ 3 b ) and ( A ∗ 7 ⋄ 5 ) are prov ed in [4]. 19 F or ( A ⋄ 2) , ( A ⋄ 6 w ) and ( A ⋄ 7 s ) the pro ofs are given in the following three prop ositions. Prop osition 5. The mo dal formula ( A ⋄ 2) B ϕ →  B ψ ) ↔ B ( ϕ > ψ )  is char acterize d by the fol lowing pr op erty of fr ames: ( P ⋄ 2) ∀ s ∈ S, ∀ E ∈ 2 S \ ∅ , if B ( s ) ⊆ E , then S s ′ ∈B ( s ) f ( s ′ , E ) = B ( s ) Pr o of. (A) Fix an arbitrary model based on a frame that satisfies Prop ert y ( P ⋄ 2) , an arbitrary state s ∈ S and an arbitrary formula ϕ ∈ Φ 0 and supp ose that s | = B ϕ , that is, B ( s ) ⊆ ∥ ϕ ∥ (th us, b y serialit y of B , ∥ ϕ ∥  = ∅ ). W e need to sho w that, for ev ery form ula ψ ∈ Φ 0 , s | = B ψ ↔ B ( ϕ > ψ ) , that is, s | = B ψ if and only if s | = B ( ϕ > ψ ) . By Prop erty ( P ⋄ 2) (with E = ∥ ϕ ∥ ), S s ′ ∈B ( s ) f ( s ′ , ∥ ϕ ∥ ) = B ( s ) and th us, B ( s ) ⊆ ∥ ψ ∥ (that is, s | = B ψ ) if and only if S s ′ ∈B ( s ) f ( s ′ , ∥ ϕ ∥ ) ⊆ ∥ ψ ∥ (that is s | = B ( ϕ > ψ ) ). 19 Since ( K ⋄ 1) coincides with AGM axiom ( K ∗ 2) , ( K ⋄ 3 b ) coincides with A GM axiom ( K ∗ 5 b ) and ( K ⋄ 5) coincides with A GM axiom ( K ∗ 7) . 26 (B) Fix a frame that violates prop ert y ( P ⋄ 2) . Then there exist a state s ∈ S and an ev ent E ⊆ S such that (a) B ( s ) ⊆ E , (b) S s ′ ∈B ( s ) f ( s ′ , E )  = B ( s ) . T wo cases are p ossible. CASE 1: S s ′ ∈B ( s ) f ( s ′ , E ) ⊆ B ( s ) . Let p, q ∈ At b e atomic formulas and con- struct a mo del where ∥ p ∥ = E and ∥ q ∥ = B ( s ) . Then s | = B p and s | = B q but, since S s ′ ∈B ( s ) f ( s ′ , ∥ p ∥ ) ⊆ ∥ q ∥ , s | = B ( p > q ) so that s | = ( B q ↔ B ( p > q ) . Hence s | = B p → ( B q ↔ B ( p > q )) . CASE 2: B ( s ) ⊆ S s ′ ∈B ( s ) f ( s ′ , E ) . Let p, q ∈ At b e atomic form ulas and con- struct a mo del where ∥ p ∥ = E and ∥ q ∥ = S s ′ ∈B ( s ) f ( s ′ , E ) = S s ′ ∈B ( s ) f ( s ′ , ∥ p ∥ ) . Then, s | = B p and s | = B ( p > q ) but, since B ( s ) ⊆ ∥ q ∥ , s | = q so that s | = B p → ( B q ↔ B ( p > q )) . Prop osition 6. The mo dal formula ( A ⋄ 6 w ) ¬ □ ¬ ( ϕ ∧ ψ ) ∧ B ( ϕ > ψ ) ∧ B ( ψ > ϕ ) → ( B ( ϕ > χ ) ↔ B ( ψ > χ )) is char acterize d by the fol lowing pr op erty of fr ames: ( P ⋄ 6 w ) ∀ s ∈ S, ∀ E , F ∈ 2 S with E ∩ F  = ∅ , if S s ′ ∈B ( s ) f ( s ′ , E ) ⊆ F and S s ′ ∈B ( s ) f ( s ′ , F ) ⊆ E then S s ′ ∈B ( s ) f ( s ′ , E ) = S s ′ ∈B ( s ) f ( s ′ , F ) Pr o of. (A) Fix an arbitrary model based on a frame that satisfies Prop ert y ( P ⋄ 6 w ) , an arbitrary state s ∈ S and arbitrary form ulas ϕ, ψ , χ ∈ Φ 0 and supp ose that s | = ¬ □ ¬ ( ϕ ∧ ψ ) ∧ B ( ϕ > ψ ) ∧ B ( ψ > ϕ ) , that is, ∥ ϕ ∧ ψ ∥  = ∅ (and thus ∥ ϕ ∥  = ∅ and ∥ ψ ∥  = ∅ ), S s ′ ∈B ( s ) f ( s ′ , ∥ ϕ ∥ ) ⊆ ∥ ψ ∥ and S s ′ ∈B ( s ) f ( s ′ , ∥ ψ ∥ ) ⊆ ∥ ϕ ∥ . W e need to sho w that s | = B ( ϕ > χ ) ↔ B ( ψ > χ ) . By Prop ert y ( P ⋄ 6 w ) (with E = ∥ ϕ ∥ and F = ∥ ψ ∥ so that E ∩ F = ∥ ϕ ∥ ∩ ∥ ψ ∥ = ∥ ϕ ∧ ψ ∥  = ∅ ), [ s ′ ∈B ( s ) f ( s ′ , ∥ ϕ ∥ ) = [ s ′ ∈B ( s ) f ( s ′ , ∥ ψ ∥ ) (7) 27 Supp ose that s | = B ( ϕ > χ ) . Then S s ′ ∈B ( s ) f ( s ′ , ∥ ϕ ∥ ) ⊆ ∥ χ ∥ and th us, b y (7), S s ′ ∈B ( s ) f ( s ′ , ∥ ψ ∥ ) ⊆ ∥ χ ∥ , that is, s | = B ( ψ > χ ) . Con versely , sup- p ose that s | = B ( ψ > χ ) . Then S s ′ ∈B ( s ) f ( s ′ , ∥ ψ ∥ ) ⊆ ∥ χ ∥ and th us, by (7), S s ′ ∈B ( s ) f ( s ′ , ∥ ϕ ∥ ) ⊆ ∥ χ ∥ , that is, s | = B ( ϕ > χ ) . Th us, s | = B ( ϕ > χ ) ↔ B ( ψ > χ ) . (B) Conv ersely , fix a frame that violates Prop ert y ( P ⋄ 6 w ) . Then there exist s ∈ S , E , F ∈ 2 S with E ∩ F  = ∅ such that S s ′ ∈B ( s ) f ( s ′ , E ) ⊆ F and S s ′ ∈B ( s ) f ( s ′ , F ) ⊆ E but S s ′ ∈B ( s ) f ( s ′ , E )  = S s ′ ∈B ( s ) f ( s ′ , F ) . One of the follo wing t wo cases m ust hold. Case 1: S s ′ ∈B ( s ) f ( s ′ , E ) ⊆ S s ′ ∈B ( s ) f ( s ′ , F ) , or Case 2: S s ′ ∈B ( s ) f ( s ′ , F ) ⊆ S s ′ ∈B ( s ) f ( s ′ , E ) . In Case 1 let p , q and r b e atomic formulas and construct a mo del where ∥ p ∥ = E , ∥ q ∥ = F and ∥ r ∥ = S s ′ ∈B ( s ) f ( s ′ , F ) . Since ∅  = ∥ p ∥ ∩ ∥ q ∥ = ∥ p ∧ q ∥ , s | = ¬ □ ¬ ( p ∧ q ) ; furthermore, s | = B ( p > q ) ∧ B ( q > p ) ∧ B ( q > r ) but s | = B ( p > r ) . In Case 2 construct a mo del where ∥ p ∥ = E , ∥ q ∥ = F and ∥ r ∥ = S s ′ ∈B ( s ) f ( s ′ , E ) . Again, since ∅  = ∥ p ∥ ∩ ∥ q ∥ = ∥ p ∧ q ∥ , s | = ¬ □ ¬ ( p ∧ q ) ; furthermore, s | = B ( p > q ) ∧ B ( q > p ) ∧ B ( p > r ) but s | = B ( q > r ) . Prop osition 7. The mo dal formula ( A ⋄ 7 s ) ¬ □ ¬ ϕ ∧ ¬ □ ¬ ψ ∧ B ( ϕ > χ ) ∧ B ( ψ > χ ) → B (( ϕ ∨ ψ ) > χ ) is char acterize d by the fol lowing pr op erty of fr ames: ( P ⋄ 7 s ) ∀ s ∈ S, ∀ E , F ∈ 2 S \ ∅ S s ′ ∈B ( s ) f ( s ′ , E ∪ F ) ⊆ S s ′ ∈B ( s ) f ( s ′ , E ) ! ∪ S s ′ ∈B ( s ) f ( s ′ , F ) ! 28 Pr o of. Fix a frame that satisfies prop ert y ( P ⋄ 7 s ) , an arbitrary mo del based on it, an arbitrary state s ∈ S and arbitrary ϕ, ψ , χ ∈ Φ 0 and assume that s | = ¬ □ ¬ ϕ ∧ ¬ □ ¬ ψ ∧ B ( ϕ > χ ) ∧ B ( ψ > χ ) . Then ∥ ϕ ∥  = ∅ , ∥ ψ ∥  = ∅ , S s ′ ∈B ( s ) f ( s ′ , ∥ ϕ ∥ ) ⊆ ∥ χ ∥ and S s ′ ∈B ( s ) f ( s ′ , ∥ ψ ∥ ) ⊆ ∥ χ ∥ . Thus, b y prop ert y ( P ⋄ 7 s ) (with E = ∥ ϕ ∥ and F = ∥ ψ ∥ and using the fact that ∥ ϕ ∥ ∪ ∥ ψ ∥ = ∥ ϕ ∨ ψ ∥ ), S s ′ ∈B ( s ) f ( s ′ , ∥ ϕ ∨ ψ ∥ ) ⊆ ∥ χ ∥ , so that s | = B (( ϕ ∨ ψ ) > χ ) . Con versely , fix a frame that violates prop ert y ( P ⋄ 7 s ) . Then there exist s, ∈ S and E , F ∈ 2 S \ ∅ such that [ s ′ ∈B ( s ) f ( s ′ , E ∪ F ) ⊈ [ s ′ ∈B ( s ) f ( s ′ , E ) ∪ [ s ′ ∈B ( s ) f ( s ′ , F ) (8) Let p, q , r ∈ At b e atomic form ulas and construct a mo del where ∥ p ∥ = E , ∥ q ∥ = F and ∥ r ∥ = S s ′ ∈B ( s ) f ( s ′ , E ) ∪ S s ′ ∈B ( s ) f ( s ′ , F ) . Then, since ∥ p ∥  = ∅ and ∥ q ∥  = ∅ , s | = ¬ □ ¬ p ∧ ¬ □ ¬ q ; furthermore, s | = B ( p > r ) ∧ B ( q > r ) . On the other hand, since (noting that ∥ p ∥ ∪ ∥ q ∥ = ∥ p ∨ q ∥ ) S s ′ ∈B ( s ) f ( s ′ , ∥ p ∨ q ∥ ) ⊈ ∥ r ∥ , s | = B (( p ∨ q ) > r ) , yielding a violation of axiom ( A ⋄ 7 s ) . Pro of of Prop osition 1. Every axiom of L K M is a the or em of L AGM . Since some of the AGM axioms coincide with KM axioms, we only need to pro ve that ( A ⋄ 2) , ( A ⋄ 6 w ) and ( A ⋄ 7 s ) are theorems of L AGM . This is done in the follo wing three lemmas. In the pro ofs ’PL’ stands for ’Prop ositional Logic’. Lemma 4. The fol lowing mo dal KM axiom: ( A ⋄ 2) B ϕ → ( B ψ ↔ B ( ϕ > ψ )) is a the or em of L AGM . Pr o of. W e first pro ve that B ϕ → ( B ψ → B ( ϕ > ψ )) is a theorem of L AGM . 1 . B ϕ → ¬ B ¬ ϕ axiom D B 2 . ψ → ( ϕ → ψ ) tautology 3 . B ψ → B ( ϕ → ψ ) rule ( RM B ) 4 . ( B ϕ ∧ B ψ ) → ( ¬ B ¬ ϕ ∧ B ( ϕ → ψ )) 1, 3, PL 5 . ( ¬ B ¬ ϕ ∧ B ( ϕ → ψ )) → B ( ϕ > ψ ) A GM axiom ( A ∗ 4) 6 . ( B ϕ ∧ B ψ ) → B ( ϕ > ψ ) 4, 5, PL. 7 . B ϕ → ( B ψ → B ( ϕ > ψ )) 6, PL 29 Next we prov e that B ϕ → ( B ( ϕ > ψ ) → B ψ ) is a theorem of L AGM . 8 . □ ¬ ϕ → B ¬ ϕ axiom ( N B ) 9 . ¬ B ¬ ϕ → ¬ □ ¬ ϕ 8, PL 10 . B ϕ → ¬ □ ¬ ϕ 1, 9, PL 11 . ( B ϕ ∧ B ( ϕ > ψ )) → ( ¬ □ ¬ ϕ ∧ B ( ϕ > ψ )) 10, PL 12 . ( ¬ □ ¬ ϕ ∧ B ( ϕ > ψ )) → B ( ϕ → ψ ) A GM axiom ( A ∗ 3) 13 . ( B ϕ ∧ B ( ϕ > ψ )) → B ( ϕ → ψ ) 11, 12, PL 14 . ( B ϕ ∧ B ( ϕ > ψ )) → ( B ϕ ∧ B ( ϕ → ψ )) 13, PL 15 . ( B ϕ ∧ B ( ϕ → ψ )) → B ( ϕ ∧ ( ϕ → ψ )) axiom ( C B ) 16 . ϕ ∧ ( ϕ → ψ ) → ψ tautology 17 . B ( ϕ ∧ ( ϕ → ψ )) → B ψ 16, rule ( R M B ) 18 . ( B ϕ ∧ B ( ϕ > ψ )) → B ψ 14, 15, 17, PL 19 . B ϕ →  B ( ϕ > ψ ) → B ψ  18, PL Axiom ( A ⋄ 2) follo ws from lines 7 and 19. Lemma 5. The fol lowing mo dal KM axiom: ( A ⋄ 6 w ) ¬ □ ¬ ( ϕ ∧ ψ ) ∧ B ( ϕ > ψ ) ∧ B ( ψ > ϕ ) → ( B ( ϕ > χ ) ↔ B ( ψ > χ )) is a the or em of L AGM . Pr o of. First we show that the following formula is a theorem of L AGM :  ¬ □ ¬ ( ϕ ∧ ψ ) ∧ B ( ϕ > ψ )  →  B ( ϕ > χ ) → B (( ϕ ∧ ψ ) > χ )) 30 1 . ¬ □ ¬ ( ϕ ∧ ψ ) → ¬ □ ¬ ϕ ( C ¬ □ ¬ ) (see Remark 3) 2 .  ¬ □ ¬ ( ϕ ∧ ψ ) ∧ B ( ϕ > ψ )  → ( ¬ □ ¬ ϕ ∧ B ( ϕ > ψ )) 1, PL 3 . ( ¬ □ ¬ ϕ ∧ B ( ϕ > ψ )) → ¬ B ( ϕ > ¬ ψ ) axiom ( A ∗ 5 b ⋄ 3 b ) 4 .  ¬ □ ¬ ( ϕ ∧ ψ ) ∧ B ( ϕ > ψ )  → ¬ B ( ϕ > ¬ ψ ) 2, 3, PL 5 . χ → ( ψ → χ ) tautology 6 . B ( ϕ > χ ) → B ( ϕ > ( ψ → χ )) 5, rule ( R M B > ) (see Remark 3) 7 . ¬ □ ¬ ( ϕ ∧ ψ ) ∧ B ( ϕ > ψ ) ∧ B ( ϕ > χ ) → ¬ B ( ϕ > ¬ ψ ) ∧ B ( ϕ > ( ψ → χ )) 4, 6, PL 8 . ¬ B ( ϕ > ¬ ψ ) ∧ B ( ϕ > ( ψ → χ )) → B  ( ϕ ∧ ψ ) > ( ψ ∧ χ )  axiom ( A ∗ 8) 9 . ¬ □ ¬ ( ϕ ∧ ψ ) ∧ B ( ϕ > ψ ) ∧ B ( ϕ > χ ) → B  ( ϕ ∧ ψ ) > ( ψ ∧ χ )  7, 8, PL 10 . ( ψ ∧ χ ) → χ tautology 11 . B  ( ϕ ∧ ψ ) > ( ψ ∧ χ )  → B  ( ϕ ∧ ψ ) > χ  10, rule ( R M B > ) (see Remark 3) 12 . ( ¬ □ ¬ ( ϕ ∧ ψ ) ∧ B ( ϕ > ψ ) ∧ B ( ϕ > χ )) → B  ( ϕ ∧ ψ ) > χ  9, 11, PL 13 . ( ¬ □ ¬ ( ϕ ∧ ψ ) ∧ B ( ϕ > ψ )) →  B ( ϕ > χ ) → B  ( ϕ ∧ ψ ) > χ  12, PL Next, rep eating steps 1-13 with ϕ replaced by ψ and with ψ replaced by ϕ w e get 14 . ( ¬ □ ¬ ( ϕ ∧ ψ ) ∧ B ( ψ > ϕ )) →  B ( ψ > χ ) → B  ( ϕ ∧ ψ ) > χ  The next step is to sho w that the follo wing is a theorem of L AGM :  ¬ □ ¬ ( ϕ ∧ ψ ) ∧ B ( ϕ > ψ )  →  B (( ϕ ∧ ψ ) > χ ) → B ( ϕ > χ )  31 15 . ¬ □ ¬ ( ϕ ∧ ψ ) ∧ B (( ϕ ∧ ψ ) > χ ) → B  ϕ > ( ψ → χ )  axiom ( A ∗ 7) 16 . ¬ □ ¬ ( ϕ ∧ ψ ) ∧ B (( ϕ ∧ ψ ) > χ ) ∧ B ( ϕ > ψ ) →  B ( ϕ > ψ ) ∧ B  ϕ > ( ψ → χ )  15, PL 17 .  B ( ϕ > ψ ) ∧ B ( ϕ > ( ψ → χ )  → B ( ϕ > χ ) axiom ( A ∗ 1 ⋄ 0 ) 18 .  ¬ □ ¬ ( ϕ ∧ ψ ) ∧ B ( ϕ > ψ ) ∧ B (( ϕ ∧ ψ ) > χ )  → B ( ϕ > χ ) 16, 17, PL 19 .  ¬ □ ¬ ( ϕ ∧ ψ ) ∧ B ( ϕ > ψ )  →  B (( ϕ ∧ ψ ) > χ ) → B ( ϕ > χ )  18, PL Next, repeating steps 15-19 with ϕ replaced b y ψ and with ψ replaced b y ϕ w e get 20 .  ¬ □ ¬ ( ϕ ∧ ψ ) ∧ B ( ψ > ϕ )  →  B (( ϕ ∧ ψ ) > χ ) → B ( ψ > χ )  F rom 13 and 19 w e get 21 .  ¬ □ ¬ ( ϕ ∧ ψ ) ∧ B ( ϕ > ψ )  →  B (( ϕ ∧ ψ ) > χ ) ↔ B ( ϕ > χ )  and from 14 and 20 w e get 22 .  ¬ □ ¬ ( ϕ ∧ ψ ) ∧ B ( ψ > ϕ )  →  B (( ϕ ∧ ψ ) > χ ) ↔ B ( ψ > χ )  Th us, 23 .  ¬ □ ¬ ( ϕ ∧ ψ ) ∧ B ( ϕ > ψ ) ∧ B ( ψ > ϕ )  →   B (( ϕ ∧ ψ ) > χ ) ↔ B ( ϕ > χ )  ∧  B (( ϕ ∧ ψ ) > χ ) ↔ B ( ψ > χ )   21, 22, PL 24 .  B (( ϕ ∧ ψ ) > χ ) ↔ B ( ϕ > χ )  ∧  B (( ϕ ∧ ψ ) > χ ) ↔ B ( ψ > χ )  → ( B ( ϕ > χ ) ↔ B ( ψ > χ )) tautology 25 . ¬ □ ¬ ( ϕ ∧ ψ ) ∧ B ( ϕ > ψ ) ∧ B ( ψ > ϕ ) →  B ( ϕ > χ ) ↔ B ( ψ > χ )  23, 24, PL; this is ( A ⋄ 6) 32 Lemma 6. The fol lowing mo dal KM axiom: ( A ⋄ 7 s ) ¬ □ ¬ ϕ ∧ ¬ □ ¬ ψ ∧ B ( ϕ > χ ) ∧ B ( ψ > χ ) → B (( ϕ ∨ ψ ) > χ ) is a the or em of L AGM . Pr o of. First we prov e that the following formula is a theorem of L AGM : ¬ □ ¬ ϕ ∧ B ( ϕ > χ ) → B  ( ϕ ∨ ψ ) > ( ϕ → χ )  1 . ϕ ↔  ( ϕ ∨ ψ ) ∧ ϕ  tautology 2 . B ( ϕ > χ ) ↔ B  (( ϕ ∨ ψ ) ∧ ϕ ) > χ  1, rule ( R ∗ 6 ⋄ 4 ) 3 . B ( ϕ > χ ) → B  (( ϕ ∨ ψ ) ∧ ϕ ) > χ  2, PL 4 . ϕ →  ( ϕ ∨ ψ ) ∧ ϕ  tautology 5 . ¬ □ ¬ ϕ → ¬ □ ¬ (( ϕ ∨ ψ ) ∧ ϕ ) 1, rule ( R M ¬ □ ¬ ) ( see Remark 3 ) 6 . ¬ □ ¬ ϕ ∧ B ( ϕ > χ ) → ¬ □ ¬ (( ϕ ∨ ψ ) ∧ ϕ ) ∧ B  (( ϕ ∨ ψ ) ∧ ϕ ) > χ  3 , 5 , P L 7 . ¬ □ ¬ (( ϕ ∨ ψ ) ∧ ϕ ) ∧ B  (( ϕ ∨ ψ ) ∧ ϕ ) > χ  → B  ( ϕ ∨ ψ ) > ( ϕ → χ )  axiom ( A ∗ 7 ⋄ 5 ) 8 . ¬ □ ¬ ϕ ∧ B ( ϕ > χ ) → B  ( ϕ ∨ ψ ) > ( ϕ → χ )  6, 7, PL Rep eating steps 1-8 with ϕ replaced by ψ and ψ by ϕ w e get 9 . ¬ □ ¬ ψ ∧ B ( ψ > χ ) → B  ( ϕ ∨ ψ ) > ( ψ → χ )  Th us, 33 10 . ¬ □ ¬ ϕ ∧ B ( ϕ > χ ) ∧ ¬ □ ¬ ψ ∧ B ( ψ > χ ) → B  ( ϕ ∨ ψ ) > ( ϕ → χ )  ∧ B  ( ϕ ∨ ψ ) > ( ψ → χ )  8, 9, PL 11 . B  ( ϕ ∨ ψ ) > ( ϕ → χ )  ∧ B  ( ϕ ∨ ψ ) > ( ψ → χ )  → B  ( ϕ ∨ ψ ) > ( ϕ → χ ∧ ψ → χ )  axiom ( C B ) 12 . ( ϕ → χ ) ∧ ( ψ → χ ) → (( ϕ ∨ ψ ) → χ ) tautology 13 . B  ( ϕ ∨ ψ ) > ( ϕ → χ ∧ ψ → χ )  → B  ( ϕ ∨ ψ ) > (( ϕ ∨ ψ ) → χ )  12, rule ( R M B > ) (see Remark 3) 14 . ¬ □ ¬ ϕ ∧ B ( ϕ > χ ) ∧ ¬ □ ¬ ψ ∧ B ( ψ > χ ) → B  ( ϕ ∨ ψ ) > (( ϕ ∨ ψ ) → χ )  10, 11, 13, PL 15 . B  ( ϕ ∨ ψ ) > ( ϕ ∨ ψ )  axiom ( A ∗ 2 ⋄ 1 ) 16 . B  ( ϕ ∨ ψ ) > (( ϕ ∨ ψ ) → χ )  → B  ( ϕ ∨ ψ ) > ( ϕ ∨ ψ )  ∧ B  ( ϕ ∨ ψ ) > (( ϕ ∨ ψ ) → χ )  15, PL 17 . B  ( ϕ ∨ ψ ) > ( ϕ ∨ ψ )  ∧ B  ( ϕ ∨ ψ ) > (( ϕ ∨ ψ ) → χ )  → B  ( ϕ ∨ ψ ) > χ  axiom ( A ∗ 1 ⋄ 0 ) ( with ϕ and ψ replaced by ϕ ∨ ψ ) 18 . B  ( ϕ ∨ ψ ) > (( ϕ ∨ ψ ) → χ )  → B  ( ϕ ∨ ψ ) > χ  16, 17, PL 19 . ¬ □ ¬ ϕ ∧ B ( ϕ > χ ) ∧ ¬ □ ¬ ϕ ∧ B ( ψ > χ ) → B  ( ϕ ∨ ψ ) > χ  14, 18, PL this is ( A ⋄ 7 s ) Lemma 2 . ( K ⋄ 9 s ) fol lows fr om ( K ⋄ 0) , ( K ⋄ 8) and ( K ⋄ 9) . Pr o of. Fix arbitrary ϕ, ψ ∈ Φ 0 . By ( K ⋄ 8) (recall that J K K = { w ∈ W : K ⊆ w } ) K ⋄ ϕ = \ w ∈ J K K ( w ⋄ ϕ ) (9) Let A = { w ∈ J K K : ¬ ψ / ∈ w ⋄ ϕ } and B = { w ∈ J K K : ¬ ψ ∈ w ⋄ ϕ } (clearly J K K = A ∪ B ). Assume that ¬ ϕ / ∈ K ⋄ ϕ . Then A  = ∅ . First note that ∀ w ∈ B , ( w ⋄ ϕ ) + ψ = Φ 0 ; th us, \ w ∈ B  ( w ⋄ ϕ ) + ψ  = Φ 0 (10) 34 It follows from (10) that \ w ∈ J K K (( w ⋄ ϕ ) + ψ ) = \ w ∈ A (( w ⋄ ϕ ) + ψ ) ∩ Φ 0 = \ w ∈ A (( w ⋄ ϕ ) + ψ ) (11) By ( K ⋄ 9) (since eac h w is complete), ∀ w ∈ A, ( w ⋄ ϕ ) + ψ ⊆ w ⋄ ( ϕ ∧ ψ ) (12) Next, note that, by ( K ⋄ 0) (whic h ensures that K ⋄ ϕ = C n ( K ⋄ ϕ ) and w ⋄ ϕ = C n ( w ⋄ ϕ ) ) and (9), 20 ( K ⋄ ϕ ) + ψ =   \ w ∈ J K K ( w ⋄ ϕ )   + ψ = \ w ∈ J K K ( w ⋄ ϕ + ψ ) (13) It follows from (11), (12) and (13) that ( K ⋄ ϕ ) + ψ ⊆ \ w ∈ J K K ( w ⋄ ( ϕ ∧ ψ )) (14) and, by ( K ⋄ 8) , T w ∈ J K K ( w ⋄ ( ϕ ∧ ψ )) = K ⋄ ( ϕ ∧ ψ ) . Lemma 3 . ( A ∗ 3)  ¬ □ ¬ ϕ ∧ B ( ϕ > ψ  → B ( ϕ → ψ ) is pr ovable in lo gic L K M . Pr o of. Recall that ( A ⋄ 2) is the following axiom, for χ, ξ ∈ Φ 0 , B χ →  B ξ ↔ B ( χ > ξ )  ; in line 1 we take the instance of ( A ⋄ 2) with χ = ϕ ∨ ¬ ϕ and ξ = ϕ → ψ . Recall also that ( A ⋄ 5) is the follo wing axiom, for χ, ξ , θ ∈ Φ 0 ,  ¬ □ ¬ ( χ ∧ ξ ) ∧ B (( χ ∧ ξ ) > θ )  → B ( χ > ( ξ → θ )) ; in line 5 w e tak e the instance of ( A ⋄ 5) with χ = ϕ ∨ ¬ ϕ and ξ = ϕ and θ = ψ . 20 Since K ⋄ ϕ is deductively closed, for every χ ∈ Φ 0 , χ ∈ K ⋄ ϕ + ψ if and only if ( ψ → χ ) ∈ K ⋄ ϕ . Since, ∀ w ∈ W , w ⋄ ϕ is deductiv ely closed, T w ∈ J K K ( w ⋄ ϕ ) is deductiv ely closed and thus χ ∈  T w ∈ J K K ( w ⋄ ϕ )  + ψ if and only if ( ψ → χ ) ∈ T w ∈ J K K ( w ⋄ ϕ ) if and only if χ ∈ T w ∈ J K K ( w ⋄ ϕ + ψ ) . 35 1 . B ( ϕ ∨ ¬ ϕ ) →  B ( ϕ → ψ ) ↔ B (( ϕ ∨ ¬ ϕ ) > ( ϕ → ψ ))  axiom ( A ⋄ 2) 2 . ϕ ∨ ¬ ϕ tautology 3 . B ( ϕ ∨ ¬ ϕ ) 2, rule ( N B ) 4 . B ( ϕ → ψ ) ↔ B (( ϕ ∨ ¬ ϕ ) > ( ϕ → ψ )) 1, 3, rule ( M P ) 5 . ¬ □ ¬ (( ϕ ∨ ¬ ϕ ) ∧ ϕ ) ∧ B (( ϕ ∨ ¬ ϕ ) ∧ ϕ ) > ψ ) → B (( ϕ ∧ ¬ ϕ ) > ( ϕ → ψ )) axiom ( A ⋄ 5) 6 . ϕ → ( ϕ ∨ ¬ ϕ ) ∧ ϕ tautology 7 . ¬ □ ¬ ϕ → ¬ □ ¬ (( ϕ ∨ ¬ ϕ ) ∧ ϕ ) 6, rule ( R M ¬ □ ¬ ) see Remark 3 8 . B ( ϕ > ψ ) → B (( ϕ ∨ ¬ ϕ ) ∧ ϕ ) > ψ ) 6, rule ( R M B > ) see Remark 3 9 . ¬ □ ¬ ϕ ∧ B ( ϕ > ψ ) → ¬ □ ¬ (( ϕ ∨ ¬ ϕ ) ∧ ϕ ) ∧ B (( ϕ ∨ ¬ ϕ ) ∧ ϕ ) > ψ ) 7, 8 PL 10 . ¬ □ ¬ ϕ ∧ B ( ϕ > ψ ) → B (( ϕ ∧ ¬ ϕ ) > ( ϕ → ψ )) 9, 5, PL 11 . ¬ □ ¬ ϕ ∧ B ( ϕ > ψ ) → B ( ϕ → ψ ) 10, 4, PL W e conclude by pro ving the seman tic corresp ondence for A GM axiom ( K ∗ 4) and its modal coun terpart ( A ∗ 4) . Lemma 7. The fol lowing pr op erty of fr ames: ( P ∗ 4) ∀ s ∈ S, ∀ E , F ∈ 2 S , if B ( s ) ∩ E  = ∅ and B ( s ) ⊆ ( S \ E ) ∪ F then S s ′ ∈B ( s ) f ( s ′ , E ) ⊆ F char acterizes the AGM axiom ( K ∗ 4) If ¬ ϕ / ∈ K then K + ϕ ⊆ K ∗ ϕ and its mo dal c ounterp art ( A ∗ 4) ¬ B ¬ ϕ ∧ B ( ϕ → ψ ) → B ( ϕ > ψ ) Pr o of. (A) First we pro v e that ( P ∗ 4) characterizes ( K ∗ 4) . Fix an arbitrary mo del based on a frame that satisfies Prop erty ( P ∗ 4) , an arbitrary state s and an arbitrary form ula ϕ and suppose that ¬ ϕ / ∈ K s , that is, B ( s ) ∩ ∥ ϕ ∥  = ∅ . Let ψ ∈ K s + ϕ ; then, since K s is deductiv ely closed, ϕ → ψ ∈ K s , that is, B ( s ) ⊆ ∥ ϕ → ψ ∥ = ( S \ ∥ ϕ ∥ ) ∪ ∥ ψ ∥ . Then b y Property ( P ∗ 4) (with E = ∥ ϕ ∥ and F = ∥ ψ ∥ ) S s ′ ∈B ( s ) f ( s ′ , ∥ ϕ ∥ ) ⊆ ∥ ψ ∥ , that is, ψ ∈ K ⋄ ϕ . 36 Con versely , fix a frame that violates Prop ert y ( P ∗ 4) . Then there exist s ∈ S and E , F ∈ 2 S suc h that B ( s ) ∩ E  = ∅ and B ( s ) ⊆ ( S \ E ) ∪ F but S s ′ ∈B ( s ) f ( s ′ , E ) ⊆ F . Let p, q ∈ At b e atomic sentences and construct a mo del based on this f rame where ∥ p ∥ = E and ∥ q ∥ = F . Then, since B ( s ) ∩ ∥ p ∥  = ∅ , ¬ p / ∈ K s and, since B ( s ) ⊆ ( S \ ∥ p ∥ ) ∪ ∥ q ∥ = ∥ p → q ∥ , ( p → q ) ∈ K s , that is q ∈ K s + p ; ho wev er, since S s ′ ∈B ( s ) f ( s ′ , ∥ p ∥ ) ⊆ ∥ q ∥ , q / ∈ K ⋄ p . (B) Next we pro v e that ( P ∗ 4) c haracterizes ( A ∗ 4) . Fix an arbitrary mo del based on a frame that satisfies Prop erty ( P ∗ 4) , an arbitrary state s and arbitrary form ulas ϕ and ψ and supp ose that s | = ¬ B ¬ ϕ ∧ B ( ϕ → ψ ) . W e need to sho w that s | = B ( ϕ > ψ ) . Since s | = ¬ B ¬ ϕ , B ( s ) ∩ ∥ ϕ ∥  = ∅ . Since s | = B ( ϕ → ψ ) , B ( s ) ⊆ ∥ ϕ → ψ ∥ = ( S \ ∥ ϕ ∥ ) ∪ ∥ ψ ∥ . Thus, b y Prop erty ( P ∗ 4) (with E = ∥ ϕ ∥ ) and F = ∥ ψ ∥ ) , S s ′ ∈B ( s ) f ( s ′ , ∥ ϕ ∥ ) ⊆ ∥ ψ ∥ , that is , s | = B ( ϕ > ψ ) . Con versely , fix a frame that violates Prop ert y ( P ∗ 4) . Then there exist s ∈ S and E , F ∈ 2 S suc h that B ( s ) ∩ E  = ∅ and B ( s ) ⊆ ( S \ E ) ∪ F but S s ′ ∈B ( s ) f ( s ′ , E ) ⊆ F . Let p, q ∈ At b e atomic sentences and construct a mo del based on this frame where ∥ p ∥ = E and ∥ q ∥ = F . Then, since B ( s ) ∩ ∥ p ∥  = ∅ , s | = ¬ B ¬ p and, since B ( s ) ⊆ ( S \ ∥ p ∥ ) ∪ ∥ q ∥ = ∥ p → q ∥ , s | = B ( p → q ) ; ho w ever, since S s ′ ∈B ( s ) f ( s ′ , ∥ p ∥ ) ⊆ ∥ q ∥ , s | = B ( p > q ) . References [1] Alchourr ón, C., Gärdenfors, P., and Makinson, D. On the logic of theory c hange: partial meet contraction and revision functions. The Journal of Symb olic L o gic 50 (1985), 510–530. [2] Ara v anis, T. On the consistency b etw een b elief revision and b elief up date. Journal of A rtificial Intel ligenc e R ese ar ch 82 (2025), 1743– 1771. [3] Bonanno, G. A Kripk e-Lewis semantics for b elief up date and b elief revision. A rtificial Intel ligenc e 339 (2025). 37 [4] Bonanno, G. A mo dal-logic translation of the AGM axioms for b elief revision. The Eur op e an Journal on Artificial Intel ligenc e 0 , 0 (2025), 1–14. [5] Chellas, B. Mo dal lo gic: an intr o duction . Cam bridge Universit y Press, 1984. [6] Hintikka, J. Know le dge and b elief: an intr o duction to the lo gic of the two notions . Cornell Univ ersity Press, Cornell, 1962. [7] Ka tsuno, H., and Mendelzon, A. O. On the difference b et ween up dating a knowledge base and revising it. In Pr o c e e dings of the Se c- ond International Confer enc e on Principles of Know le dge R epr esenta- tion and R e asoning (San F rancisco, 1991), KR’91, Morgan Kaufmann Publishers Inc., pp. 387–394. [8] Lewis, D. Completeness and decidability of three logics of counterfac- tual conditionals. The oria 37 , 1 (1971), 74–85. [9] Pepp as, P. Belief change and r e asoning ab out action: an axiomatic appr o ach to mo del ling dynamic worlds and the c onne ction to the lo gic of the ory change . PhD thesis, Universit y of Sydney , 1993. [10] Pepp as, P., Na y ak, A., P agnucco, M., F oo, N. Y., Kwok, R., and Pr okopenk o, M. Revision vs. up date: taking a closer lo ok. In ECAI ’96: 12th Eur op e an Confer enc e on Artificial Intel ligenc e (1996), W. W ahlster, Ed., John Wiley & Sons, Ltd, pp. 95–99. 38