Simple Hyperintensional Belief Revision Franz Berto F.Berto@uva.nl - - PowerPoint PPT Presentation

Simple Hyperintensional Belief Revision

Franz Berto

F.Berto@uva.nl

Bochum, 15-16 Dec 2017

• 1. Intro

Franz Berto (F.Berto@uva.nl) Simple Hyperintensional Belief Revision Bochum, 15-16 Dec 2017 2 / 34

• 1. Intro

In [Alchourrón et al., 1985] (K*1) has K ∗ φ closed under full classical logic.

Franz Berto (F.Berto@uva.nl) Simple Hyperintensional Belief Revision Bochum, 15-16 Dec 2017 3 / 34

• 1. Intro

In [Alchourrón et al., 1985] (K*1) has K ∗ φ closed under full classical logic. But, our belief states needn’t be closed under classical logic – perhaps under any kind of monotonic consequence [Jago, 2014].

Franz Berto (F.Berto@uva.nl) Simple Hyperintensional Belief Revision Bochum, 15-16 Dec 2017 3 / 34

• 1. Intro

In [Alchourrón et al., 1985] (K*1) has K ∗ φ closed under full classical logic. But, our belief states needn’t be closed under classical logic – perhaps under any kind of monotonic consequence [Jago, 2014]. (K*5) trivializes belief sets revised in the light of inconsistent information.

Franz Berto (F.Berto@uva.nl) Simple Hyperintensional Belief Revision Bochum, 15-16 Dec 2017 3 / 34

• 1. Intro

In [Alchourrón et al., 1985] (K*1) has K ∗ φ closed under full classical logic. But, our belief states needn’t be closed under classical logic – perhaps under any kind of monotonic consequence [Jago, 2014]. (K*5) trivializes belief sets revised in the light of inconsistent information. But, we shouldn’t go crazy after occasional exposure to inconsistent info.

Franz Berto (F.Berto@uva.nl) Simple Hyperintensional Belief Revision Bochum, 15-16 Dec 2017 3 / 34

• 1. Intro

In [Alchourrón et al., 1985] (K*1) has K ∗ φ closed under full classical logic. But, our belief states needn’t be closed under classical logic – perhaps under any kind of monotonic consequence [Jago, 2014]. (K*5) trivializes belief sets revised in the light of inconsistent information. But, we shouldn’t go crazy after occasional exposure to inconsistent info. (K*6) has it that, if φ and ψ are logically equivalent, then K ∗ φ = K ∗ ψ.

Franz Berto (F.Berto@uva.nl) Simple Hyperintensional Belief Revision Bochum, 15-16 Dec 2017 3 / 34

• 1. Intro

In [Alchourrón et al., 1985] (K*1) has K ∗ φ closed under full classical logic. But, our belief states needn’t be closed under classical logic – perhaps under any kind of monotonic consequence [Jago, 2014]. (K*5) trivializes belief sets revised in the light of inconsistent information. But, we shouldn’t go crazy after occasional exposure to inconsistent info. (K*6) has it that, if φ and ψ are logically equivalent, then K ∗ φ = K ∗ ψ. But, we are subject to framing effects [Kahneman and Tversky, 1984]: Lois may revise her beliefs one way when told she has 60% chances of succeeding in a task, another way when told she has has 40% chances of failing.

Franz Berto (F.Berto@uva.nl) Simple Hyperintensional Belief Revision Bochum, 15-16 Dec 2017 3 / 34

• 1. Intro

Lots of doxep logics include operators for conditional belief, Bφψ (‘Condi- tional on φ, it is believed that ψ’, or ‘It is believed that ψ after receiving the information that φ’), or dynamic belief revision, [∗φ]Bψ (‘After revision by φ, it is believed that ψ’), closely mirroring the original AGM postulates.

Franz Berto (F.Berto@uva.nl) Simple Hyperintensional Belief Revision Bochum, 15-16 Dec 2017 4 / 34

• 1. Intro

Lots of doxep logics include operators for conditional belief, Bφψ (‘Condi- tional on φ, it is believed that ψ’, or ‘It is believed that ψ after receiving the information that φ’), or dynamic belief revision, [∗φ]Bψ (‘After revision by φ, it is believed that ψ’), closely mirroring the original AGM postulates. E.g., [Spohn, 1988], [Segerberg, 1995], [Lindström and Rabinowicz, 1999], [Board, 2004], [Van Ditmarsch, 2005], [Asheim and Sövik, 2005], [Leitgeb and [van Benthem, 2007], [van Ditmarsch et al., 2007], [Baltag and Smets, 2008], [van Benthem, 2011], [Girard and Rott, 2014]

Franz Berto (F.Berto@uva.nl) Simple Hyperintensional Belief Revision Bochum, 15-16 Dec 2017 4 / 34

• 1. Intro

E.g. as for (K*6), such logics will typically satisfy: If φ ↔ ψ, then [∗φ]χ ↔ [∗ψ]χ If φ ↔ ψ, then Bφχ ↔ Bψχ Logically equivalent φ and ψ can be replaced salva veritate as indexes in [∗...] and B...: these can’t detect hyperintensional differences.

Franz Berto (F.Berto@uva.nl) Simple Hyperintensional Belief Revision Bochum, 15-16 Dec 2017 5 / 34

• 1. Intro

E.g. as for (K*6), such logics will typically satisfy: If φ ↔ ψ, then [∗φ]χ ↔ [∗ψ]χ If φ ↔ ψ, then Bφχ ↔ Bψχ Logically equivalent φ and ψ can be replaced salva veritate as indexes in [∗...] and B...: these can’t detect hyperintensional differences. But, thought is hyperintensional (framing is just a special case): Lois can wish that Superman is in love with her without wishing that Clark Kent is in love with her. We can conceive that 75 × 12 = 900 without conceiving that Fermat’s Last Theorem is true. Etc. etc.

Franz Berto (F.Berto@uva.nl) Simple Hyperintensional Belief Revision Bochum, 15-16 Dec 2017 5 / 34

• 1. Intro

In this work, I want to model an agent: whose belief revision processes are sensitive to framing effects;

Franz Berto (F.Berto@uva.nl) Simple Hyperintensional Belief Revision Bochum, 15-16 Dec 2017 6 / 34

• 1. Intro

In this work, I want to model an agent: whose belief revision processes are sensitive to framing effects; who can hold inconsistent beliefs without thereby believing everything;

Franz Berto (F.Berto@uva.nl) Simple Hyperintensional Belief Revision Bochum, 15-16 Dec 2017 6 / 34

• 1. Intro

In this work, I want to model an agent: whose belief revision processes are sensitive to framing effects; who can hold inconsistent beliefs without thereby believing everything; who is safe from various other forms of logical omniscience.

Franz Berto (F.Berto@uva.nl) Simple Hyperintensional Belief Revision Bochum, 15-16 Dec 2017 6 / 34

• 1. Intro

Start from a single, plain insight: we should take at face value the view of beliefs as (propositional) representational mental states bearing intentional- ity, that is, being about states of affairs, situations, or circumstances which make for their content.

Franz Berto (F.Berto@uva.nl) Simple Hyperintensional Belief Revision Bochum, 15-16 Dec 2017 7 / 34

• 1. Intro

Start from a single, plain insight: we should take at face value the view of beliefs as (propositional) representational mental states bearing intentional- ity, that is, being about states of affairs, situations, or circumstances which make for their content. Arguably, it is precisely the aboutness of intentional states that can account for many of their hyperintensional features: as we think that 75× 12 = 900,

• ur thought is about these very integers, not about diophantine equations,

elliptical curves, or else.

Franz Berto (F.Berto@uva.nl) Simple Hyperintensional Belief Revision Bochum, 15-16 Dec 2017 7 / 34

• 2. A Hyperintensional Semantics

L with an indefinitely large set LAT of atoms p, q, r (p1, p2, ...), ¬, ∧, ∨, ≺, B, (, ). φ, ψ, χ, ..., are metavariables for formulas of L.

Franz Berto (F.Berto@uva.nl) Simple Hyperintensional Belief Revision Bochum, 15-16 Dec 2017 8 / 34

• 2. A Hyperintensional Semantics

L with an indefinitely large set LAT of atoms p, q, r (p1, p2, ...), ¬, ∧, ∨, ≺, B, (, ). φ, ψ, χ, ..., are metavariables for formulas of L. The well-formed formulas are items in LAT and, if φ and ψ are formulas: ¬φ | (φ ∧ ψ) | (φ ∨ ψ) | (φ ≺ ψ) | Bφψ

Franz Berto (F.Berto@uva.nl) Simple Hyperintensional Belief Revision Bochum, 15-16 Dec 2017 8 / 34

• 2. A Hyperintensional Semantics

L with an indefinitely large set LAT of atoms p, q, r (p1, p2, ...), ¬, ∧, ∨, ≺, B, (, ). φ, ψ, χ, ..., are metavariables for formulas of L. The well-formed formulas are items in LAT and, if φ and ψ are formulas: ¬φ | (φ ∧ ψ) | (φ ∨ ψ) | (φ ≺ ψ) | Bφψ ‘Bφψ’ = ‘Conditional on φ, the agent believes ψ’, or: ‘After revising by φ, the agent believes ψ’ (‘static’ belief revision: [Board, 2004], [Asheim and Sövik, 2005], [Bonanno, 2005], [Leitgeb and Segerberg, 2005]).

Franz Berto (F.Berto@uva.nl) Simple Hyperintensional Belief Revision Bochum, 15-16 Dec 2017 8 / 34

• 2. A Hyperintensional Semantics

A frame for L is a tuple F = W , {Rφ | φ ∈ L}, C, ⊕, c: W is a non-empty set of possible worlds; {Rφ | φ ∈ L} is a set of accessibilities, Rφ ⊆ W × W ; C is a set of contents: what the belief states are about; ⊕ is fusion on C; c : LAT → C.

Franz Berto (F.Berto@uva.nl) Simple Hyperintensional Belief Revision Bochum, 15-16 Dec 2017 9 / 34

• 2. A Hyperintensional Semantics

A frame for L is a tuple F = W , {Rφ | φ ∈ L}, C, ⊕, c: W is a non-empty set of possible worlds; {Rφ | φ ∈ L} is a set of accessibilities, Rφ ⊆ W × W ; C is a set of contents: what the belief states are about; ⊕ is fusion on C; c : LAT → C. ⊕ satisfies, for all xyz ∈ C: (Idempotence) x ⊕ x = x (Commutativity) x ⊕ y = y ⊕ x (Associativity) (x ⊕ y) ⊕ z = x ⊕ (y ⊕ z)

Franz Berto (F.Berto@uva.nl) Simple Hyperintensional Belief Revision Bochum, 15-16 Dec 2017 9 / 34

• 2. A Hyperintensional Semantics

A frame for L is a tuple F = W , {Rφ | φ ∈ L}, C, ⊕, c: W is a non-empty set of possible worlds; {Rφ | φ ∈ L} is a set of accessibilities, Rφ ⊆ W × W ; C is a set of contents: what the belief states are about; ⊕ is fusion on C; c : LAT → C. ⊕ satisfies, for all xyz ∈ C: (Idempotence) x ⊕ x = x (Commutativity) x ⊕ y = y ⊕ x (Associativity) (x ⊕ y) ⊕ z = x ⊕ (y ⊕ z) ⊕ is unrestricted: ∀xy∃z(z = x ⊕ y). C, ⊕ is a complete join semilattice.

Franz Berto (F.Berto@uva.nl) Simple Hyperintensional Belief Revision Bochum, 15-16 Dec 2017 9 / 34

• 2. A Hyperintensional Semantics

Content parthood, ≤, is: ∀xy ∈ C(x ≤ y ⇔ x ⊕ y = y).

Franz Berto (F.Berto@uva.nl) Simple Hyperintensional Belief Revision Bochum, 15-16 Dec 2017 10 / 34

• 2. A Hyperintensional Semantics

Content parthood, ≤, is: ∀xy ∈ C(x ≤ y ⇔ x ⊕ y = y). ≤ is a partial ordering – for all xyz ∈ C: (Reflexivity) x ≤ x (Antisymmetry) x ≤ y & y ≤ x ⇒ x = y (Transitivity) x ≤ y & y ≤ z ⇒ x ≤ z

Franz Berto (F.Berto@uva.nl) Simple Hyperintensional Belief Revision Bochum, 15-16 Dec 2017 10 / 34

• 2. A Hyperintensional Semantics

Content parthood, ≤, is: ∀xy ∈ C(x ≤ y ⇔ x ⊕ y = y). ≤ is a partial ordering – for all xyz ∈ C: (Reflexivity) x ≤ x (Antisymmetry) x ≤ y & y ≤ x ⇒ x = y (Transitivity) x ≤ y & y ≤ z ⇒ x ≤ z Think of all contents in C as built via fusions out of atoms: Atom(x) ⇔ ∼ ∃y(y < x). If p ∈ LAT, then c(p) ∈ {x ∈ C|Atom(x)} (a helpful idealization).

Franz Berto (F.Berto@uva.nl) Simple Hyperintensional Belief Revision Bochum, 15-16 Dec 2017 10 / 34

• 2. A Hyperintensional Semantics

c is extended to the whole L: if Atφ = {p1, ..., pn}, the set of atoms in φ, then: c(φ) = ⊕Atφ = c(p1) ⊕ ... ⊕ c(pn). A formula is about what its atoms, taken together, are about.

Franz Berto (F.Berto@uva.nl) Simple Hyperintensional Belief Revision Bochum, 15-16 Dec 2017 11 / 34

• 2. A Hyperintensional Semantics

c is extended to the whole L: if Atφ = {p1, ..., pn}, the set of atoms in φ, then: c(φ) = ⊕Atφ = c(p1) ⊕ ... ⊕ c(pn). A formula is about what its atoms, taken together, are about. Such mereology of topics tracks syntactic structure only so far, e.g.: c(φ) = c(¬¬φ) c(φ) = c(¬φ) c(φ ∧ ψ) = c(φ ∧ ψ) c(φ ∧ ψ) = c(φ) ⊕ c(ψ) = c(φ ∨ ψ)

Franz Berto (F.Berto@uva.nl) Simple Hyperintensional Belief Revision Bochum, 15-16 Dec 2017 11 / 34

• 2. A Hyperintensional Semantics

c is extended to the whole L: if Atφ = {p1, ..., pn}, the set of atoms in φ, then: c(φ) = ⊕Atφ = c(p1) ⊕ ... ⊕ c(pn). A formula is about what its atoms, taken together, are about. Such mereology of topics tracks syntactic structure only so far, e.g.: c(φ) = c(¬¬φ) c(φ) = c(¬φ) c(φ ∧ ψ) = c(φ ∧ ψ) c(φ ∧ ψ) = c(φ) ⊕ c(ψ) = c(φ ∨ ψ) (Cf. ‘awareness generated by primitive propositions’ [Schipper, 2015].)

Franz Berto (F.Berto@uva.nl) Simple Hyperintensional Belief Revision Bochum, 15-16 Dec 2017 11 / 34

• 2. A Hyperintensional Semantics

A model M = W , {Rφ | φ ∈ L}, C, ⊕, c, is a frame + interpretation ⊆ W × LAT, relating worlds to atoms: ‘w p’ = p is true at w, ‘w p’ = ∼ w p.

Franz Berto (F.Berto@uva.nl) Simple Hyperintensional Belief Revision Bochum, 15-16 Dec 2017 12 / 34

• 2. A Hyperintensional Semantics

A model M = W , {Rφ | φ ∈ L}, C, ⊕, c, is a frame + interpretation ⊆ W × LAT, relating worlds to atoms: ‘w p’ = p is true at w, ‘w p’ = ∼ w p. is extended to all formulas of L: (S¬) w ¬φ ⇔ w φ (S∧) w φ ∧ ψ ⇔ w φ & w ψ (S∨) w φ ∨ ψ ⇔ w φ or w ψ (S≺) w φ ≺ ψ ⇔ ∀w1(w1 φ ⇒ w1 ψ) (SB) w Bφψ ⇔ ∀w1(wRφw1 ⇒ w1 ψ) & c(ψ) ≤ c(φ)

Franz Berto (F.Berto@uva.nl) Simple Hyperintensional Belief Revision Bochum, 15-16 Dec 2017 12 / 34

• 2. A Hyperintensional Semantics

A model M = W , {Rφ | φ ∈ L}, C, ⊕, c, is a frame + interpretation ⊆ W × LAT, relating worlds to atoms: ‘w p’ = p is true at w, ‘w p’ = ∼ w p. is extended to all formulas of L: (S¬) w ¬φ ⇔ w φ (S∧) w φ ∧ ψ ⇔ w φ & w ψ (S∨) w φ ∨ ψ ⇔ w φ or w ψ (S≺) w φ ≺ ψ ⇔ ∀w1(w1 φ ⇒ w1 ψ) (SB) w Bφψ ⇔ ∀w1(wRφw1 ⇒ w1 ψ) & c(ψ) ≤ c(φ) ‘wRφw1’ = ‘w1 is, from w’s viewpoint, one of the most plausible worlds where φ is true’ (plausibility coming soon!).

Franz Berto (F.Berto@uva.nl) Simple Hyperintensional Belief Revision Bochum, 15-16 Dec 2017 12 / 34

• 2. A Hyperintensional Semantics

(SB) can be expressed using set-selection functions [Lewis, 1973]. Each φ ∈ L has a fφ : W → P(W ), fφ(w) = {w1 ∈ W |wRφw1}.

Franz Berto (F.Berto@uva.nl) Simple Hyperintensional Belief Revision Bochum, 15-16 Dec 2017 13 / 34

• 2. A Hyperintensional Semantics

(SB) can be expressed using set-selection functions [Lewis, 1973]. Each φ ∈ L has a fφ : W → P(W ), fφ(w) = {w1 ∈ W |wRφw1}. If |φ| = {w ∈ W |w φ}, we can rephrase the clause for B as: (SB) w Bφψ ⇔ fφ(w) ⊆ |ψ| & c(ψ) ≤ c(φ)

Franz Berto (F.Berto@uva.nl) Simple Hyperintensional Belief Revision Bochum, 15-16 Dec 2017 13 / 34

• 2. A Hyperintensional Semantics

(SB) can be expressed using set-selection functions [Lewis, 1973]. Each φ ∈ L has a fφ : W → P(W ), fφ(w) = {w1 ∈ W |wRφw1}. If |φ| = {w ∈ W |w φ}, we can rephrase the clause for B as: (SB) w Bφψ ⇔ fφ(w) ⊆ |ψ| & c(ψ) ≤ c(φ) The two are equivalent: wRφw1 ⇔ w1 ∈ fφ(w).

Franz Berto (F.Berto@uva.nl) Simple Hyperintensional Belief Revision Bochum, 15-16 Dec 2017 13 / 34

• 2. A Hyperintensional Semantics

(SB) can be expressed using set-selection functions [Lewis, 1973]. Each φ ∈ L has a fφ : W → P(W ), fφ(w) = {w1 ∈ W |wRφw1}. If |φ| = {w ∈ W |w φ}, we can rephrase the clause for B as: (SB) w Bφψ ⇔ fφ(w) ⊆ |ψ| & c(ψ) ≤ c(φ) The two are equivalent: wRφw1 ⇔ w1 ∈ fφ(w). We impose a total similarity ordering on worlds via Lewisian spheres repre- senting subjective dispositions to revise beliefs [Grove, 1988].

Franz Berto (F.Berto@uva.nl) Simple Hyperintensional Belief Revision Bochum, 15-16 Dec 2017 13 / 34

• 2. A Hyperintensional Semantics

$ maps each w to a finite set of subsets of W , $(w) = {Sw

0 , Sw 1 , ..., Sw n }

(the spheres), with n ∈ N, such that Sw

0 ⊆ Sw 1 ⊆ ... ⊆ Sw n = W .

Franz Berto (F.Berto@uva.nl) Simple Hyperintensional Belief Revision Bochum, 15-16 Dec 2017 14 / 34

• 2. A Hyperintensional Semantics

$ maps each w to a finite set of subsets of W , $(w) = {Sw

0 , Sw 1 , ..., Sw n }

(the spheres), with n ∈ N, such that Sw

0 ⊆ Sw 1 ⊆ ... ⊆ Sw n = W .

Next, for each φ ∈ L and w ∈ W , fφ(w) is characterized thus: if |φ| = ∅, then fφ(w) = ∅. Otherwise, fφ(w) = Sw

i

∩ |φ|, where Sw

i

∈ $(w) is the smallest sphere such that Sw

i ∩ |φ| = ∅ (Limit Assumption on board).

Franz Berto (F.Berto@uva.nl) Simple Hyperintensional Belief Revision Bochum, 15-16 Dec 2017 14 / 34

• 2. A Hyperintensional Semantics

$ maps each w to a finite set of subsets of W , $(w) = {Sw

0 , Sw 1 , ..., Sw n }

(the spheres), with n ∈ N, such that Sw

0 ⊆ Sw 1 ⊆ ... ⊆ Sw n = W .

Next, for each φ ∈ L and w ∈ W , fφ(w) is characterized thus: if |φ| = ∅, then fφ(w) = ∅. Otherwise, fφ(w) = Sw

i

∩ |φ|, where Sw

i

∈ $(w) is the smallest sphere such that Sw

i ∩ |φ| = ∅ (Limit Assumption on board).

The spheres satisfy: (C1) fφ(w) ⊆ |φ| (C2) |φ| = ∅ ⇒ fφ(w) = ∅ (C3) fφ(w) ⊆ |ψ| & fψ(w) ⊆ |φ| ⇒ fφ(w) = fψ(w) (C4) fφ(w) ∩ |ψ| = ∅ ⇒ fφ∧ψ(w) ⊆ fφ(w)

Franz Berto (F.Berto@uva.nl) Simple Hyperintensional Belief Revision Bochum, 15-16 Dec 2017 14 / 34

• 2. A Hyperintensional Semantics

Logical consequence is truth preservation at all worlds of all models based

• n $. With Σ a set of formulas:

Σ ψ ⇔ in all models M = W , {Rφ | φ ∈ L}, C, ⊕, c, and for all w ∈ W : w φ for all φ ∈ Σ ⇒ w ψ. (For single-premise entailment, {φ} ψ is φ ψ.)

Franz Berto (F.Berto@uva.nl) Simple Hyperintensional Belief Revision Bochum, 15-16 Dec 2017 15 / 34

• 2. A Hyperintensional Semantics

Logical consequence is truth preservation at all worlds of all models based

• n $. With Σ a set of formulas:

Σ ψ ⇔ in all models M = W , {Rφ | φ ∈ L}, C, ⊕, c, and for all w ∈ W : w φ for all φ ∈ Σ ⇒ w ψ. (For single-premise entailment, {φ} ψ is φ ψ.) Logical validity, φ, truth at all worlds of all models (etc.), is ∅ φ, entailment by the empty set of premises.

Franz Berto (F.Berto@uva.nl) Simple Hyperintensional Belief Revision Bochum, 15-16 Dec 2017 15 / 34

• 3. Success, Conjunction, Non-Monotonicity

C1 gives an AGM-like Success principle (as expected in static revision): (Success) Bφφ

Proof.

By C1, for any w and w1, wRφw1 ⇒ w1 φ, and of course c(φ) ≤ c(φ).

Franz Berto (F.Berto@uva.nl) Simple Hyperintensional Belief Revision Bochum, 15-16 Dec 2017 16 / 34

• 3. Success, Conjunction, Non-Monotonicity

C1 gives an AGM-like Success principle (as expected in static revision): (Success) Bφφ

Proof.

By C1, for any w and w1, wRφw1 ⇒ w1 φ, and of course c(φ) ≤ c(φ).

(Simplification) Bφ(ψ ∧ χ) Bφψ Bφ(ψ ∧ χ) Bφχ

Proof.

I do the first one (for the second, replace ψ with χ appropriately). Let w Bφ(ψ ∧ χ). By (SB), for all w1 such that wRφw1, w1 ψ ∧ χ, thus by (S∧), w1 ψ. Also, c(ψ ∧ χ) = c(ψ) ⊕ c(χ) ≤ c(φ), thus c(ψ) ≤ c(φ). Then, by (SB) again, w Bφψ.

Franz Berto (F.Berto@uva.nl) Simple Hyperintensional Belief Revision Bochum, 15-16 Dec 2017 16 / 34

• 3. Success, Conjunction, Non-Monotonicity

(Adjunction) {Bφψ, Bφχ} Bφ(ψ ∧ χ)

Proof.

Let w Bφψ and w Bφχ, that is, by (SB): for all w1 such that wRφw1, w1 ψ and w1 χ, so by (S∧) w1 φ ∧ ψ. Also, c(ψ) ≤ c(φ) and c(χ) ≤ c(φ), thus c(ψ) ⊕ c(χ) = c(ψ ∧ χ) ≤ c(φ). Then, by (SB) again, w Bφ(ψ ∧ χ).

Franz Berto (F.Berto@uva.nl) Simple Hyperintensional Belief Revision Bochum, 15-16 Dec 2017 17 / 34

• 3. Success, Conjunction, Non-Monotonicity

(Adjunction) {Bφψ, Bφχ} Bφ(ψ ∧ χ)

Proof.

Let w Bφψ and w Bφχ, that is, by (SB): for all w1 such that wRφw1, w1 ψ and w1 χ, so by (S∧) w1 φ ∧ ψ. Also, c(ψ) ≤ c(φ) and c(χ) ≤ c(φ), thus c(ψ) ⊕ c(χ) = c(ψ ∧ χ) ≤ c(φ). Then, by (SB) again, w Bφ(ψ ∧ χ).

A computer program that can determine whether φ ∧ ψ follows from some initial premises in time τ might not be able to de- termine whether ψ ∧ φ follows from those premises in time τ’ [Fagin and Halpern, 1988, p. 53, notation modified].

Franz Berto (F.Berto@uva.nl) Simple Hyperintensional Belief Revision Bochum, 15-16 Dec 2017 17 / 34

• 3. Success, Conjunction, Non-Monotonicity

(Adjunction) {Bφψ, Bφχ} Bφ(ψ ∧ χ)

Proof.

Let w Bφψ and w Bφχ, that is, by (SB): for all w1 such that wRφw1, w1 ψ and w1 χ, so by (S∧) w1 φ ∧ ψ. Also, c(ψ) ≤ c(φ) and c(χ) ≤ c(φ), thus c(ψ) ⊕ c(χ) = c(ψ ∧ χ) ≤ c(φ). Then, by (SB) again, w Bφ(ψ ∧ χ).

A computer program that can determine whether φ ∧ ψ follows from some initial premises in time τ might not be able to de- termine whether ψ ∧ φ follows from those premises in time τ’ [Fagin and Halpern, 1988, p. 53, notation modified]. But, this is no difference concerning the situation represented in the mind of an intentional agent. Representing a configuration of objects and properties making φ ∧ ψ true is the same as representing one making ψ ∧ φ true.

Franz Berto (F.Berto@uva.nl) Simple Hyperintensional Belief Revision Bochum, 15-16 Dec 2017 17 / 34

• 3. Success, Conjunction, Non-Monotonicity

The syntactic approach [...] is too fine-grained in that it considers any two sets of sentences as distinct semantic entities and, conse- quently, different belief sets. Consider, for example, the disjunction

• f φ and ψ. There is no reason to suppose that B(φ∨ψ) ≡ B(ψ∨φ)

would be valid given a syntactic understanding of B since φ ∨ ψ may be in the belief set while ψ ∨ φ may not. [But] if we consider intuitively what ‘It is believed that either φ or ψ is true.’ is saying, the order seems to be completely irrelevant [...] φ ∨ ψ is believed iff ψ ∨ φ is because these are two lexical notations for the same

• belief. [Levesque, 1984, 199-201, notation modified]

Franz Berto (F.Berto@uva.nl) Simple Hyperintensional Belief Revision Bochum, 15-16 Dec 2017 18 / 34

• 3. Success, Conjunction, Non-Monotonicity

If believing is to be taken, not just as mental symbol manipulation, but as an intentional state endowed with aboutness, then when one believes that ψ∧χ

• ne thinks about a certain scenario in which both the situation described by

ψ and the situation described by χ obtain.

Franz Berto (F.Berto@uva.nl) Simple Hyperintensional Belief Revision Bochum, 15-16 Dec 2017 19 / 34

• 3. Success, Conjunction, Non-Monotonicity

If believing is to be taken, not just as mental symbol manipulation, but as an intentional state endowed with aboutness, then when one believes that ψ∧χ

• ne thinks about a certain scenario in which both the situation described by

ψ and the situation described by χ obtain. Vice versa, when one believes that ψ and that χ, one thinks about a certain scenario, one in which ψ obtains and χ obtains as well. Then the scenario will be such that it includes both contents, and one will believe them together.

Franz Berto (F.Berto@uva.nl) Simple Hyperintensional Belief Revision Bochum, 15-16 Dec 2017 19 / 34

• 3. Success, Conjunction, Non-Monotonicity

Our operator is non-monotonic: Bφψ Bφ∧χψ

Proof.

Let W = {w, w1}, w Rp-accesses nothing, wRp∧rw1, w1 q, c(p) = c(q) = c(r). Then w Bpq, but w Bp∧rq.

Franz Berto (F.Berto@uva.nl) Simple Hyperintensional Belief Revision Bochum, 15-16 Dec 2017 20 / 34

• 3. Success, Conjunction, Non-Monotonicity

Our operator is non-monotonic: Bφψ Bφ∧χψ

Proof.

Let W = {w, w1}, w Rp-accesses nothing, wRp∧rw1, w1 q, c(p) = c(q) = c(r). Then w Bpq, but w Bp∧rq.

After revising your beliefs via the new information that Mike is in Strasbourg, you come to believe that he is in France. But if you are informed that Mike is in Strasbourg and that the city has been annexed by Germany, you will not come to believe that he is in France.

Franz Berto (F.Berto@uva.nl) Simple Hyperintensional Belief Revision Bochum, 15-16 Dec 2017 20 / 34

• 4. Disjunction, Indeterminacy

When, after revising by φ, one comes to believe that ψ, one does not thereby believe a disjunction between the latter and an unrelated χ. The believer need not be aware of that disconnected χ at all, or that χ might be irrelevant to the agent’s belief revision policy:

Franz Berto (F.Berto@uva.nl) Simple Hyperintensional Belief Revision Bochum, 15-16 Dec 2017 21 / 34

• 4. Disjunction, Indeterminacy

When, after revising by φ, one comes to believe that ψ, one does not thereby believe a disjunction between the latter and an unrelated χ. The believer need not be aware of that disconnected χ at all, or that χ might be irrelevant to the agent’s belief revision policy: Bφψ Bφ(ψ ∨ χ).

Proof.

Let W = {w, w1}, wRpw1, w1 q, c(p) = c(q) = c(r). Then c(q) ≤ c(p), so by (SB), w Bpq. But c(q ∨ r) = c(q) ⊕ c(r) c(p), thus w Bp(q ∨ r).

Franz Berto (F.Berto@uva.nl) Simple Hyperintensional Belief Revision Bochum, 15-16 Dec 2017 21 / 34

• 4. Disjunction, Indeterminacy

When, after revising by φ, one comes to believe that ψ, one does not thereby believe a disjunction between the latter and an unrelated χ. The believer need not be aware of that disconnected χ at all, or that χ might be irrelevant to the agent’s belief revision policy: Bφψ Bφ(ψ ∨ χ).

Proof.

Let W = {w, w1}, wRpw1, w1 q, c(p) = c(q) = c(r). Then c(q) ≤ c(p), so by (SB), w Bpq. But c(q ∨ r) = c(q) ⊕ c(r) c(p), thus w Bp(q ∨ r).

When informed that Mike is not in France, as you thought, but in New Zealand, you come to believe that Mike is in Oceania. You do not thereby automatically believe that Mike is either in Oceania or in planet Kepler-442b (you may never have heard of Kepler-442b to begin with).

Franz Berto (F.Berto@uva.nl) Simple Hyperintensional Belief Revision Bochum, 15-16 Dec 2017 21 / 34

• 4. Disjunction, Indeterminacy

Beliefs can be ‘nonprime’ due to indeterminacy in information:

Franz Berto (F.Berto@uva.nl) Simple Hyperintensional Belief Revision Bochum, 15-16 Dec 2017 22 / 34

• 4. Disjunction, Indeterminacy

Beliefs can be ‘nonprime’ due to indeterminacy in information: Bφ(ψ ∨ χ) Bφψ ∨ Bφχ

Proof.

Let W = {w, w1, w2}, wRpw1, wRpw2, w1 q but w1 r, w2 r but w2 q, c(p) = c(q) = c(r). Then by (S∨), w1 q ∨ r and w2 q ∨ r, so for all wx such that wRpwx, wx q ∨ r. Also, c(q ∨ r) = c(q) ⊕ c(r) ≤ c(p), thus by (SB), w Bp(q ∨ r). However, w Bpq and w Bpr for both q and r fail at some Rp-accessible world. Thus by (S∨), w Bpq ∨ Bpr.

Franz Berto (F.Berto@uva.nl) Simple Hyperintensional Belief Revision Bochum, 15-16 Dec 2017 22 / 34

• 4. Disjunction, Indeterminacy

Beliefs can be ‘nonprime’ due to indeterminacy in information: Bφ(ψ ∨ χ) Bφψ ∨ Bφχ

Proof.

Let W = {w, w1, w2}, wRpw1, wRpw2, w1 q but w1 r, w2 r but w2 q, c(p) = c(q) = c(r). Then by (S∨), w1 q ∨ r and w2 q ∨ r, so for all wx such that wRpwx, wx q ∨ r. Also, c(q ∨ r) = c(q) ⊕ c(r) ≤ c(p), thus by (SB), w Bp(q ∨ r). However, w Bpq and w Bpr for both q and r fail at some Rp-accessible world. Thus by (S∨), w Bpq ∨ Bpr.

Informed that Mike landed in New Zealand, you come to believe that he is either in the North Island or in the South Island. But you are not sure which

• ne. Thus, you don’t believe him to be in either rather than the other.

Franz Berto (F.Berto@uva.nl) Simple Hyperintensional Belief Revision Bochum, 15-16 Dec 2017 22 / 34

• 5. Hyperintensionality

One’s coming to believe that ψ given φ is not entailed by the corresponding strict conditional or implication: φ ≺ ψ Bφψ

Proof.

Let W = {w, w1}, wRpw1, w p, w1 q, c(p) = c(q). By Condition C1, w1 p. Now |p| ⊆ |q|, thus by (S≺), w p ≺ q. But although fp(w) ⊆ |q|, c(q) c(p), thus w Bpq.

Franz Berto (F.Berto@uva.nl) Simple Hyperintensional Belief Revision Bochum, 15-16 Dec 2017 23 / 34

• 5. Hyperintensionality

One’s coming to believe that ψ given φ is not entailed by the corresponding strict conditional or implication: φ ≺ ψ Bφψ

Proof.

Let W = {w, w1}, wRpw1, w p, w1 q, c(p) = c(q). By Condition C1, w1 p. Now |p| ⊆ |q|, thus by (S≺), w p ≺ q. But although fp(w) ⊆ |q|, c(q) c(p), thus w Bpq.

The strict conditional is ‘irrelevant’: even when all the φ-worlds are ψ-worlds (thus, all the most plausible φ-worlds are ψ-worlds), ψ may have little to do with φ. Thus, one can revise one’s beliefs by φ but fail to come to believe that ψ, although there is no way for φ to be true while ψ is not.

Franz Berto (F.Berto@uva.nl) Simple Hyperintensional Belief Revision Bochum, 15-16 Dec 2017 23 / 34

• 5. Hyperintensionality

For similar reasons, [Kraus et al., 1990]’s Right Weakening fails: {Bφψ, ψ ≺ χ} Bφχ

Proof.

Let W = {w, w1}, wRpw1, w q, w1 q, w1 r, c(p) = c(q) = c(r). Then fp(w) ⊆ |q| and c(q) ≤ c(p), thus by (SB), w Bpq. Also, |q| ⊆ |r|, thus by (S≺), w q ≺ r. But although fp(w) ⊆ |r|, c(r) c(p), thus w Bpr.

Franz Berto (F.Berto@uva.nl) Simple Hyperintensional Belief Revision Bochum, 15-16 Dec 2017 24 / 34

• 5. Hyperintensionality

For similar reasons, [Kraus et al., 1990]’s Right Weakening fails: {Bφψ, ψ ≺ χ} Bφχ

Proof.

Let W = {w, w1}, wRpw1, w q, w1 q, w1 r, c(p) = c(q) = c(r). Then fp(w) ⊆ |q| and c(q) ≤ c(p), thus by (SB), w Bpq. Also, |q| ⊆ |r|, thus by (S≺), w q ≺ r. But although fp(w) ⊆ |r|, c(r) c(p), thus w Bpr.

Although all the ψ-worlds are χ-worlds, thus all the most plausible φ-worlds which are ψ-worlds are χ-worlds, ≺ can change the subject.

Franz Berto (F.Berto@uva.nl) Simple Hyperintensional Belief Revision Bochum, 15-16 Dec 2017 24 / 34

• 5. Hyperintensionality

For similar reasons, [Kraus et al., 1990]’s Right Weakening fails: {Bφψ, ψ ≺ χ} Bφχ

Proof.

Let W = {w, w1}, wRpw1, w q, w1 q, w1 r, c(p) = c(q) = c(r). Then fp(w) ⊆ |q| and c(q) ≤ c(p), thus by (SB), w Bpq. Also, |q| ⊆ |r|, thus by (S≺), w q ≺ r. But although fp(w) ⊆ |r|, c(r) c(p), thus w Bpr.

Although all the ψ-worlds are χ-worlds, thus all the most plausible φ-worlds which are ψ-worlds are χ-worlds, ≺ can change the subject. Spotting a furry animal in your garden, you come to believe that there’s a woodchuck behind home. You needn’t come to believe, too, that there’s a groundhog behind home. There’s no way for something to be a woodchuck without it being a groundhog, but you are unaware of this.

Franz Berto (F.Berto@uva.nl) Simple Hyperintensional Belief Revision Bochum, 15-16 Dec 2017 24 / 34

• 5. Hyperintensionality

Kraus’ Left Logical Equivalence fails (φ ≡ ψ =df φ ≺ ψ ∧ ψ ≺ φ): {Bφχ, φ ≡ ψ} Bψχ

Proof.

Let W = {w, w1}, wRpw1, wRqw1, w p, w q, w1 r, c(p) = c(r) = c(q). Then fp(w) ⊆ |r| and c(r) ≤ c(p), thus by (SB), w Bpr. Also, by Condition C1, w1 p and w1 q, thus |p| ⊆ |q| and |q| ⊆ |p|. Then by (S≺) and (S∧), w p ≡ q. But although fq(w) ⊆ |r|, c(r) c(q), thus w Bqr.

Franz Berto (F.Berto@uva.nl) Simple Hyperintensional Belief Revision Bochum, 15-16 Dec 2017 25 / 34

• 5. Hyperintensionality

Kraus’ Left Logical Equivalence fails (φ ≡ ψ =df φ ≺ ψ ∧ ψ ≺ φ): {Bφχ, φ ≡ ψ} Bψχ

Proof.

Let W = {w, w1}, wRpw1, wRqw1, w p, w q, w1 r, c(p) = c(r) = c(q). Then fp(w) ⊆ |r| and c(r) ≤ c(p), thus by (SB), w Bpr. Also, by Condition C1, w1 p and w1 q, thus |p| ⊆ |q| and |q| ⊆ |p|. Then by (S≺) and (S∧), w p ≡ q. But although fq(w) ⊆ |r|, c(r) c(q), thus w Bqr.

We can model framing: informed that one’s probability of making it to the short list is 1/3, one believes that one should apply for the job. But, against AGM’s (K*6), after being informed that one’s probability of failing the short list is 2/3, one does not believe that it’s worth applying. There is no way that the chances of making it are 1/3 without the chances of failing being 2/3 and vice versa, but one has been caught into a framing effect.

Franz Berto (F.Berto@uva.nl) Simple Hyperintensional Belief Revision Bochum, 15-16 Dec 2017 25 / 34

• 6. Revising by Inconsistent Information

Against AGM’s (K*5), belief revision is not trivialized by incoming inconsis- tent information: Bφ∧¬φψ

Proof.

Let W = {w}, c(p) = c(q). |p ∧ ¬p| = ∅, thus fp∧¬p(w) = ∅ ≤ |q|. However, c(q) c(p ∧ ¬p) = c(p) ⊕ c(¬p) = c(p). Thus, by (SB), w Bp∧¬pq.

Franz Berto (F.Berto@uva.nl) Simple Hyperintensional Belief Revision Bochum, 15-16 Dec 2017 26 / 34

• 6. Revising by Inconsistent Information

Against AGM’s (K*5), belief revision is not trivialized by incoming inconsis- tent information: Bφ∧¬φψ

Proof.

Let W = {w}, c(p) = c(q). |p ∧ ¬p| = ∅, thus fp∧¬p(w) = ∅ ≤ |q|. However, c(q) c(p ∧ ¬p) = c(p) ⊕ c(¬p) = c(p). Thus, by (SB), w Bp∧¬pq.

Inconsistent information may be about something, contentful: Snow is white and not white is about snow’s being white, not about grass’ being purple.

Franz Berto (F.Berto@uva.nl) Simple Hyperintensional Belief Revision Bochum, 15-16 Dec 2017 26 / 34

• 6. Revising by Inconsistent Information

Against AGM’s (K*5), belief revision is not trivialized by incoming inconsis- tent information: Bφ∧¬φψ

Proof.

Let W = {w}, c(p) = c(q). |p ∧ ¬p| = ∅, thus fp∧¬p(w) = ∅ ≤ |q|. However, c(q) c(p ∧ ¬p) = c(p) ⊕ c(¬p) = c(p). Thus, by (SB), w Bp∧¬pq.

Inconsistent information may be about something, contentful: Snow is white and not white is about snow’s being white, not about grass’ being purple. (NB we have ‘small explosion’: Bφ∧¬φ∧ψ¬ψ. A framework expanded to include non-normal or impossible worlds where a contradiction can be true would help against small detonations.)

Franz Berto (F.Berto@uva.nl) Simple Hyperintensional Belief Revision Bochum, 15-16 Dec 2017 26 / 34

• 7. Equivalents in Plausibility, Cut, Cautious Monotonicity

Important validities warranted by C3: (C3) fφ(w) ⊆ |ψ| & fψ(w) ⊆ |φ| ⇒ fφ(w) = fψ(w)

Franz Berto (F.Berto@uva.nl) Simple Hyperintensional Belief Revision Bochum, 15-16 Dec 2017 27 / 34

• 7. Equivalents in Plausibility, Cut, Cautious Monotonicity

Important validities warranted by C3: (C3) fφ(w) ⊆ |ψ| & fψ(w) ⊆ |φ| ⇒ fφ(w) = fψ(w) The Principle of Equivalents in Plausibility gives a limited recovery of (K*6): (PEP) {Bφψ, Bψφ, Bφχ} Bψχ

Proof.

Suppose w Bφψ, w Bψφ, w Bφχ. By (SB), these entail, respectively, (a) fφ(w) ⊆ |ψ| and c(ψ) ≤ c(φ), (b) fψ(w) ⊆ |φ| and c(φ) ≤ c(ψ), (c) fφ(w) ⊆ |χ| and c(χ) ≤ c(φ). From (a) and (b) we get fφ(w) = fψ(w) (by Condition C3) and c(φ) = c(ψ) (by antisymmetry of content parthood). From these and (c) we get fψ(w) ⊆ |χ| and c(χ) ≤ c(ψ). Thus by (SB) again, w Bψχ.

Franz Berto (F.Berto@uva.nl) Simple Hyperintensional Belief Revision Bochum, 15-16 Dec 2017 27 / 34

• 7. Equivalents in Plausibility, Cut, Cautious Monotonicity

‘Equivalents in plausibility’ are φ and ψ such that, when we revise by either, we come to believe the other. Then, PEP says, they can be replaced salva veritate as belief revision inputs.

Franz Berto (F.Berto@uva.nl) Simple Hyperintensional Belief Revision Bochum, 15-16 Dec 2017 28 / 34

• 7. Equivalents in Plausibility, Cut, Cautious Monotonicity

‘Equivalents in plausibility’ are φ and ψ such that, when we revise by either, we come to believe the other. Then, PEP says, they can be replaced salva veritate as belief revision inputs. Bφψ: informed that Mike is unmarried, you come to believe that he is a

• bachelor. Also, Bψφ: informed that Mike is a bachelor, you come to believe

that he is unmarried. Suppose Bφχ: informed that Mike is unmarried, you come to believe that he has no marriage allowance. Then the same happens if you are informed that he is a bachelor, Bψχ.

Franz Berto (F.Berto@uva.nl) Simple Hyperintensional Belief Revision Bochum, 15-16 Dec 2017 28 / 34

• 7. Equivalents in Plausibility, Cut, Cautious Monotonicity

C3 validates Cautious Transitivity or Cut: (CUT) {Bφψ, Bφ∧ψχ} Bφχ

Proof.

Suppose (a) w Bφψ and (b) w Bφ∧ψχ. From (a), Success, and Adjunction we get w Bφ(φ ∧ ψ), thus, by (SB), fφ(w) ⊆ |φ ∧ ψ| and c(φ ∧ ψ) ≤ c(φ). Also, w Bφ∧ψφ (from Success Bφ∧ψ(φ ∧ ψ) and Simplification). By (SB) again,fφ∧ψ(w) ⊆ |φ| and (of course) c(φ) ≤ c(φ ∧ ψ). Thus, by Condition C3 fφ(w) = fφ∧ψ(w), and c(φ ∧ ψ) = c(φ) (by antisymmetry of content parthood). Next, from (b) and (SB) again, fφ∧ψ(w) ⊆ |χ| and c(χ) ≤ c(φ ∧ ψ). Therefore, fφ∧ψ(w) = fφ(w) ⊆ |χ| and c(χ) ≤ c(φ) = c(φ ∧ ψ). Thus by (SB) again, w Bφχ.

Franz Berto (F.Berto@uva.nl) Simple Hyperintensional Belief Revision Bochum, 15-16 Dec 2017 29 / 34

• 7. Equivalents in Plausibility, Cut, Cautious Monotonicity

C3 also gives Cautious Monotonicity: (CM) {Bφψ, Bφχ} Bφ∧ψχ

Proof.

suppose (a) w Bφψ and (b) w Bφχ. From (a), Success ( Bφφ), and Adjunction, we get w Bφ(φ ∧ ψ), thus by (SB), fφ(w) ⊆ |φ ∧ ψ|. Also, w Bφ∧ψφ (from Success Bφ∧ψ(φ ∧ ψ) and Simplification), so by (SB) again, fφ∧ψ(w) ⊆ |φ|. Then, by Condition C3, fφ(w) = fφ∧ψ(w). From (b) and (SB) again, we get fφ(w) ⊆ |χ|, thus fφ∧ψ(w) ⊆ |χ|. Also, c(χ) ≤ c(φ) ⊕ c(ψ) = c(φ ∧ ψ). Thus, w Bφ∧ψχ.

Franz Berto (F.Berto@uva.nl) Simple Hyperintensional Belief Revision Bochum, 15-16 Dec 2017 30 / 34

• 7. Equivalents in Plausibility, Cut, Cautious Monotonicity

C3 also gives Cautious Monotonicity: (CM) {Bφψ, Bφχ} Bφ∧ψχ

Proof.

suppose (a) w Bφψ and (b) w Bφχ. From (a), Success ( Bφφ), and Adjunction, we get w Bφ(φ ∧ ψ), thus by (SB), fφ(w) ⊆ |φ ∧ ψ|. Also, w Bφ∧ψφ (from Success Bφ∧ψ(φ ∧ ψ) and Simplification), so by (SB) again, fφ∧ψ(w) ⊆ |φ|. Then, by Condition C3, fφ(w) = fφ∧ψ(w). From (b) and (SB) again, we get fφ(w) ⊆ |χ|, thus fφ∧ψ(w) ⊆ |χ|. Also, c(χ) ≤ c(φ) ⊕ c(ψ) = c(φ ∧ ψ). Thus, w Bφ∧ψχ.

By satisfying (1) Reflexivity-Success, (2) CUT, and (3) CM, Bφψ complies with [Gabbay, 1985]’s minimal conditions for non-monotonic entailments.

Franz Berto (F.Berto@uva.nl) Simple Hyperintensional Belief Revision Bochum, 15-16 Dec 2017 30 / 34

• 8. Further Work

Some ILLC folks (Peter Hawke, Sonja Smets, Aybüke Ozgün, Anthi Solaki) are after these: Axiomatize and prove completeness.

Franz Berto (F.Berto@uva.nl) Simple Hyperintensional Belief Revision Bochum, 15-16 Dec 2017 31 / 34

• 8. Further Work

Some ILLC folks (Peter Hawke, Sonja Smets, Aybüke Ozgün, Anthi Solaki) are after these: Axiomatize and prove completeness. Make it dynamic, as in [Baltag and Solecki, 1998], [Leitgeb and Segerberg, [Baltag and Smets, 2008], etc.

Franz Berto (F.Berto@uva.nl) Simple Hyperintensional Belief Revision Bochum, 15-16 Dec 2017 31 / 34

• 8. Further Work

Some ILLC folks (Peter Hawke, Sonja Smets, Aybüke Ozgün, Anthi Solaki) are after these: Axiomatize and prove completeness. Make it dynamic, as in [Baltag and Solecki, 1998], [Leitgeb and Segerberg, [Baltag and Smets, 2008], etc. Go first-order, model Frege’s informative identities: how Lois can (come to) believe that Clark Kent is in love with her, without (coming to) believe that Supermen is in love with her.

Franz Berto (F.Berto@uva.nl) Simple Hyperintensional Belief Revision Bochum, 15-16 Dec 2017 31 / 34

Thanks!

Franz Berto (F.Berto@uva.nl) Simple Hyperintensional Belief Revision Bochum, 15-16 Dec 2017 32 / 34

Appendix: (K*7) and (K*8)

A peculiar asymmetry related to the AGM (K*7) and (K*8). A natural counterpart of (K*7) (see [Board, 2004], p. 55) fails: {¬Bφ¬ψ, Bφ∧ψχ} Bφ(ψ ≺ χ)

Proof.

Let W = {w}, fp(w) = ∅, fp∧q = ∅, c(p) = c(q) = c(r). Then by (SB), w Bp¬q because c(¬q) = c(q) c(p), so w ¬Bp¬q; and w Bp∧qr, because (trivially) fp∧q(w) ⊆ |r|, and c(r) = c(q) ≤ c(p) ⊕ c(q) = c(p ∧ q). However, w Bp(q ≺ r), because c(q ≺ r) = c(q) ⊕ c(r) = c(q) c(p).

Franz Berto (F.Berto@uva.nl) Simple Hyperintensional Belief Revision Bochum, 15-16 Dec 2017 33 / 34

Appendix: (K*7) and (K*8)

But a natural counterpart of (K*8) (see [Board, 2004], Ibid), obtained by flipping premise and conclusion, holds: {¬Bφ¬ψ, Bφ(ψ ≺ χ)} Bφ∧ψχ

Proof.

Suppose (a) w ¬Bφ¬ψ and (b) w Bφ(ψ ≺ χ). By (a) and (S¬), w Bφ¬ψ, that is: either fφ(w) |¬ψ|, that is, fφ(w) ∩ |ψ| = ∅, or c(¬ψ) = c(ψ) c(φ). But it can’t be the latter, because by (b) and (SB), c(ψ ≺ χ) = c(ψ) ⊕ c(χ) ≤ c(φ), thus in particular c(ψ) ≤ c(φ); so it must be the former. Applying Condition C5 to it, fφ∧ψ(w) ⊆ fφ(w). By Condition C1, fφ∧ψ(w) ⊆ |φ ∧ ψ|, so by (S∧), fφ∧ψ(w) ⊆ |ψ|. By (b) and (SB) again, fφ(w) ⊆ |ψ ≺ χ|. Putting things together: fφ∧ψ(w) ⊆ fφ(w) ⊆ |ψ ≺ χ|, so fφ∧ψ(w) ⊆ |ψ ≺ χ|; and since fφ∧ψ(w) ⊆ |ψ|, then by modus ponens fφ∧ψ(w) ⊆ |χ|. Also, by (b) again, c(ψ) ⊕ c(χ) ≤ c(φ) ≤ c(φ ∧ ψ), thus c(χ) ≤ c(φ ∧ ψ). Thus, by (SB), w Bφ∧ψχ.

Franz Berto (F.Berto@uva.nl) Simple Hyperintensional Belief Revision Bochum, 15-16 Dec 2017 34 / 34

Alchourrón, C., Gärdenfors, P., and Makinson, D. (1985). On the logic of theory change: Partial meet functions for contraction and revision. Journal of Symbolic Logic, 50:510–30. Asheim, G. and Sövik, Y. (2005). Preference-based belief operators. Mathematical Social Sciences, 50:61–82. Baltag, A. and Smets, S. (2008). A qualitative theory of dynamic interactive belief revision. In Bonanno, G., van der Hoek, W., and Wooldridge, M., editors, Logic and the Foundations of Game and Decision Theory, pages 9–58. Ams- terdam University Press, Amsterdam. Baltag, A., M. L. and Solecki, S. (1998). The logic of public announcements, common knowledge, and private suspicions. In Gilboa, I., editor, Proceedings of TARK 98, pages 43–56. Morgan and Kaufmann, Evanston, IL.

Franz Berto (F.Berto@uva.nl) Simple Hyperintensional Belief Revision Bochum, 15-16 Dec 2017 34 / 34

Board, O. (2004). Dynamic interactive epistemology. Games and Economic Behaviour, 49:49–80. Bonanno, G. (2005). A simple modal logic for belief revision. Synthèse, 147:193–228. Fagin, R. and Halpern, J. (1988). Belief, awareness and limited reasoning. Artificial Intelligence, 34:39–76. Gabbay, D. (1985). Theoretical Foundations for Non-Monotonic Reasoning. Springer, Berlin. Girard, P. and Rott, H. (2014). Belief revision and dynamic logic. In Baltag, A. and Smets, S., editors, Johan van Benthem on Logic and Information Dynamics, pages 203–33. Springer, Dordrecht.

Franz Berto (F.Berto@uva.nl) Simple Hyperintensional Belief Revision Bochum, 15-16 Dec 2017 34 / 34

Grove, A. (1988). Two modellings for theory change. Journal of Philosophical Logic, 17:157–170. Jago, M. (2014). The Impossible. An Essay on Hyperintensionality. Oxford University Press, Oxford. Kahneman, D. and Tversky, A. (1984). Choices, values, and frames. American Psychologist, 39:341–50. Kraus, S., Lehmann, D., and Magidor, M. (1990). Nonmonotonic reasoning, preferential models and cumulative logics. Artificial Intelligence, 44:167–207. Leitgeb, H. and Segerberg, K. (2005). Dynamic doxastic logic: Why, how, and where to? Synthèse, 155:167–90. Levesque, H. (1984).

Franz Berto (F.Berto@uva.nl) Simple Hyperintensional Belief Revision Bochum, 15-16 Dec 2017 34 / 34

A logic of implicit and explicit belief. National Conference on AI, AAAI-84:198–202. Lewis, D. (1973). Counterfactuals. Blackwell, Oxford. Lindström, S. and Rabinowicz, W. (1999). Ddl unlimited: Dynamic doxastic logic for introspective agents. Erkenntnis, 50:353–85. Schipper, B. (2015). Awareness. In van Ditmarsch, H., Halpern, J., van der Hoek, W., and Kooi, B., edi- tors, Handbook of Epistemic Logic, pages 79–146. College Publications, London. Segerberg, K. (1995). Belief revision from the point of view of doxastic logic. Bulletin of the IGPL, 3:535–53. Spohn, W. (1988).

Franz Berto (F.Berto@uva.nl) Simple Hyperintensional Belief Revision Bochum, 15-16 Dec 2017 34 / 34

Ordinal conditional functions: a dynamic theory of epistemic states. In Hrper, L. and Skyrms, B., editors, Causation in Decision, Belief Change, and Statistics, volume 2, pages 105–34. Kluwer, Dordrecht. van Benthem, J. (2007). Dynamic logic for belief revision. Journal of Applied Non-Classical Logic, 17:129–55. van Benthem, J. (2011). Logical Dynamics of Information and Interaction. Cambridge University Press, Cambridge. Van Ditmarsch, H. (2005). Prolegomena to dynamic logic for belief revision. Synthèse, 147:229–75. van Ditmarsch, H., van der Hoek, W., and Kooi, B. (2007). Dynamic Epistemic Logic. Springer, Dordrecht.

Franz Berto (F.Berto@uva.nl) Simple Hyperintensional Belief Revision Bochum, 15-16 Dec 2017 34 / 34