[PPT] - Interpretability in PRA Marta Bilkova , Dick de Jongh , and Joost PowerPoint Presentation

SLIDE 1

Why and how study interpretability Proof theoretic characteristics of PRA Modal matters

Interpretability in PRA

Marta Bilkova†, Dick de Jongh∗, and Joost J. Joosten∗,

∗Institute for Logic Language and Computation

University of Amsterdam and

†Department of Logic

Charles University; Prague

14th July 2007

Marta Bilkova†, Dick de Jongh∗, and Joost J. Joosten∗, Interpretability in PRA

SLIDE 2

Why and how study interpretability Proof theoretic characteristics of PRA Modal matters Interpretations Interpretability logics

◮ We all use the notion T ⊲ S: T interprets S

Marta Bilkova†, Dick de Jongh∗, and Joost J. Joosten∗, Interpretability in PRA

SLIDE 3

Why and how study interpretability Proof theoretic characteristics of PRA Modal matters Interpretations Interpretability logics

◮ We all use the notion T ⊲ S: T interprets S ◮ T ⊲ S means (modulo some technical details)

Marta Bilkova†, Dick de Jongh∗, and Joost J. Joosten∗, Interpretability in PRA

SLIDE 4

Why and how study interpretability Proof theoretic characteristics of PRA Modal matters Interpretations Interpretability logics

◮ We all use the notion T ⊲ S: T interprets S ◮ T ⊲ S means (modulo some technical details) ◮ ∃j ∀ϕ(AxiomS(ϕ) → ∃p ProofT(p, ϕj))

Marta Bilkova†, Dick de Jongh∗, and Joost J. Joosten∗, Interpretability in PRA

SLIDE 5

Why and how study interpretability Proof theoretic characteristics of PRA Modal matters Interpretations Interpretability logics

◮ We are interested in the structural behavior of the notion of

interpretability.

Marta Bilkova†, Dick de Jongh∗, and Joost J. Joosten∗, Interpretability in PRA

SLIDE 6

Why and how study interpretability Proof theoretic characteristics of PRA Modal matters Interpretations Interpretability logics

◮ We are interested in the structural behavior of the notion of

interpretability.

◮ Interpretability can easily be formalized/arithmetized.

Marta Bilkova†, Dick de Jongh∗, and Joost J. Joosten∗, Interpretability in PRA

SLIDE 7

Why and how study interpretability Proof theoretic characteristics of PRA Modal matters Interpretations Interpretability logics

◮ We are interested in the structural behavior of the notion of

interpretability.

◮ Interpretability can easily be formalized/arithmetized. ◮ We shall consider sentential extensions of a base theory

Marta Bilkova†, Dick de Jongh∗, and Joost J. Joosten∗, Interpretability in PRA

SLIDE 8

Why and how study interpretability Proof theoretic characteristics of PRA Modal matters Interpretations Interpretability logics

◮ We are interested in the structural behavior of the notion of

interpretability.

◮ Interpretability can easily be formalized/arithmetized. ◮ We shall consider sentential extensions of a base theory ◮ ϕ ⊲T ψ stands for

Marta Bilkova†, Dick de Jongh∗, and Joost J. Joosten∗, Interpretability in PRA

SLIDE 9

Why and how study interpretability Proof theoretic characteristics of PRA Modal matters Interpretations Interpretability logics

◮ We are interested in the structural behavior of the notion of

interpretability.

◮ Interpretability can easily be formalized/arithmetized. ◮ We shall consider sentential extensions of a base theory ◮ ϕ ⊲T ψ stands for ◮ T + ϕ ⊲ T + ψ

Marta Bilkova†, Dick de Jongh∗, and Joost J. Joosten∗, Interpretability in PRA

SLIDE 10

Why and how study interpretability Proof theoretic characteristics of PRA Modal matters Interpretations Interpretability logics

◮ We are interested in the structural behavior of the notion of

interpretability.

◮ Interpretability can easily be formalized/arithmetized. ◮ We shall consider sentential extensions of a base theory ◮ ϕ ⊲T ψ stands for ◮ T + ϕ ⊲ T + ψ ◮ We are interested in the interpretability logic of a theory T:

Marta Bilkova†, Dick de Jongh∗, and Joost J. Joosten∗, Interpretability in PRA

SLIDE 11

Why and how study interpretability Proof theoretic characteristics of PRA Modal matters Interpretations Interpretability logics

◮ We are interested in the structural behavior of the notion of

interpretability.

◮ Interpretability can easily be formalized/arithmetized. ◮ We shall consider sentential extensions of a base theory ◮ ϕ ⊲T ψ stands for ◮ T + ϕ ⊲ T + ψ ◮ We are interested in the interpretability logic of a theory T: ◮ The set of all model propositional logical formulas in the

language , ⊲ which are true regardless how you interpret the variables as arithmetical sentences

Marta Bilkova†, Dick de Jongh∗, and Joost J. Joosten∗, Interpretability in PRA

SLIDE 12

Why and how study interpretability Proof theoretic characteristics of PRA Modal matters Interpretations Interpretability logics

◮ We are interested in the structural behavior of the notion of

interpretability.

◮ Interpretability can easily be formalized/arithmetized. ◮ We shall consider sentential extensions of a base theory ◮ ϕ ⊲T ψ stands for ◮ T + ϕ ⊲ T + ψ ◮ We are interested in the interpretability logic of a theory T: ◮ The set of all model propositional logical formulas in the

language , ⊲ which are true regardless how you interpret the variables as arithmetical sentences

◮ Of course, reading ⊲ as ⊲T, etc.

Marta Bilkova†, Dick de Jongh∗, and Joost J. Joosten∗, Interpretability in PRA

SLIDE 13

Why and how study interpretability Proof theoretic characteristics of PRA Modal matters Interpretations Interpretability logics

◮ We are interested in the structural behavior of the notion of

interpretability.

◮ Interpretability can easily be formalized/arithmetized. ◮ We shall consider sentential extensions of a base theory ◮ ϕ ⊲T ψ stands for ◮ T + ϕ ⊲ T + ψ ◮ We are interested in the interpretability logic of a theory T: ◮ The set of all model propositional logical formulas in the

language , ⊲ which are true regardless how you interpret the variables as arithmetical sentences

◮ Of course, reading ⊲ as ⊲T, etc. ◮ Example: (ϕ ⊲ ψ) ∧ (ψ ⊲ χ) → (ϕ ⊲ χ)

Marta Bilkova†, Dick de Jongh∗, and Joost J. Joosten∗, Interpretability in PRA

SLIDE 14

Why and how study interpretability Proof theoretic characteristics of PRA Modal matters Interpretations Interpretability logics

◮ For all theories T, IL(T) contains some sort of core logic IL

Marta Bilkova†, Dick de Jongh∗, and Joost J. Joosten∗, Interpretability in PRA

SLIDE 15

Why and how study interpretability Proof theoretic characteristics of PRA Modal matters Interpretations Interpretability logics

◮ For all theories T, IL(T) contains some sort of core logic IL ◮ IL(T) is characterized by some additional axiom schemes on

top of that

Marta Bilkova†, Dick de Jongh∗, and Joost J. Joosten∗, Interpretability in PRA

SLIDE 16

Why and how study interpretability Proof theoretic characteristics of PRA Modal matters Interpretations Interpretability logics

◮ For all theories T, IL(T) contains some sort of core logic IL ◮ IL(T) is characterized by some additional axiom schemes on

top of that

◮ For example, for theories with full induction, we have that

Montagna’s Axiom holds (A ⊲ B) → ((A ∧ C) ⊲ (B ∧ C))

Marta Bilkova†, Dick de Jongh∗, and Joost J. Joosten∗, Interpretability in PRA

SLIDE 17

Why and how study interpretability Proof theoretic characteristics of PRA Modal matters Interpretations Interpretability logics

◮ For all theories T, IL(T) contains some sort of core logic IL ◮ IL(T) is characterized by some additional axiom schemes on

top of that

◮ For example, for theories with full induction, we have that

Montagna’s Axiom holds (A ⊲ B) → ((A ∧ C) ⊲ (B ∧ C))

◮ It turns out that precisely ILM is, e.g. IL(PA) (Shavrukov

1988; Berarducci 1990)

Marta Bilkova†, Dick de Jongh∗, and Joost J. Joosten∗, Interpretability in PRA

SLIDE 18

Why and how study interpretability Proof theoretic characteristics of PRA Modal matters Interpretations Interpretability logics

◮ For all theories T, IL(T) contains some sort of core logic IL ◮ IL(T) is characterized by some additional axiom schemes on

top of that

◮ For example, for theories with full induction, we have that

Montagna’s Axiom holds (A ⊲ B) → ((A ∧ C) ⊲ (B ∧ C))

◮ It turns out that precisely ILM is, e.g. IL(PA) (Shavrukov

1988; Berarducci 1990)

◮ Likewise, the interpretability logic for finitely axiomatized

theories is known

Marta Bilkova†, Dick de Jongh∗, and Joost J. Joosten∗, Interpretability in PRA

SLIDE 19

Why and how study interpretability Proof theoretic characteristics of PRA Modal matters Interpretations Interpretability logics

◮ For all theories T, IL(T) contains some sort of core logic IL ◮ IL(T) is characterized by some additional axiom schemes on

top of that

◮ For example, for theories with full induction, we have that

Montagna’s Axiom holds (A ⊲ B) → ((A ∧ C) ⊲ (B ∧ C))

◮ It turns out that precisely ILM is, e.g. IL(PA) (Shavrukov

1988; Berarducci 1990)

◮ Likewise, the interpretability logic for finitely axiomatized

theories is known

◮ And no other!

Marta Bilkova†, Dick de Jongh∗, and Joost J. Joosten∗, Interpretability in PRA

SLIDE 20

Why and how study interpretability Proof theoretic characteristics of PRA Modal matters Interpretations Interpretability logics

◮ For all theories T, IL(T) contains some sort of core logic IL ◮ IL(T) is characterized by some additional axiom schemes on

top of that

◮ For example, for theories with full induction, we have that

Montagna’s Axiom holds (A ⊲ B) → ((A ∧ C) ⊲ (B ∧ C))

◮ It turns out that precisely ILM is, e.g. IL(PA) (Shavrukov

1988; Berarducci 1990)

◮ Likewise, the interpretability logic for finitely axiomatized

theories is known

◮ And no other! ◮ That’s were PRA comes in

Marta Bilkova†, Dick de Jongh∗, and Joost J. Joosten∗, Interpretability in PRA

SLIDE 21

Why and how study interpretability Proof theoretic characteristics of PRA Modal matters Beklemishev’s principle Zambella’s Principle

◮ Consider again

∃j ∀ϕ(AxiomS(ϕ) → ∃p ProofT(p, ϕj))

Marta Bilkova†, Dick de Jongh∗, and Joost J. Joosten∗, Interpretability in PRA

SLIDE 22

Why and how study interpretability Proof theoretic characteristics of PRA Modal matters Beklemishev’s principle Zambella’s Principle

◮ Consider again

∃j ∀ϕ(AxiomS(ϕ) → ∃p ProofT(p, ϕj))

◮ Certainly Σ3

Marta Bilkova†, Dick de Jongh∗, and Joost J. Joosten∗, Interpretability in PRA

SLIDE 23

Why and how study interpretability Proof theoretic characteristics of PRA Modal matters Beklemishev’s principle Zambella’s Principle

◮ Consider again

∃j ∀ϕ(AxiomS(ϕ) → ∃p ProofT(p, ϕj))

◮ Certainly Σ3 ◮ When S has finitely many axioms, then Σ1

Marta Bilkova†, Dick de Jongh∗, and Joost J. Joosten∗, Interpretability in PRA

SLIDE 24

Why and how study interpretability Proof theoretic characteristics of PRA Modal matters Beklemishev’s principle Zambella’s Principle

◮ Consider again

∃j ∀ϕ(AxiomS(ϕ) → ∃p ProofT(p, ϕj))

◮ Certainly Σ3 ◮ When S has finitely many axioms, then Σ1 ◮ When T is reflexive, then Π2. (Orey-H´

ajek).

Marta Bilkova†, Dick de Jongh∗, and Joost J. Joosten∗, Interpretability in PRA

SLIDE 25

Why and how study interpretability Proof theoretic characteristics of PRA Modal matters Beklemishev’s principle Zambella’s Principle

◮ Consider again

∃j ∀ϕ(AxiomS(ϕ) → ∃p ProofT(p, ϕj))

◮ Certainly Σ3 ◮ When S has finitely many axioms, then Σ1 ◮ When T is reflexive, then Π2. (Orey-H´

ajek).

◮ When T is reflexive, we have access to Montagna’s Principle:

(T ⊲ S) → ((T ∧ γ) ⊲ (S ∧ γ))

Marta Bilkova†, Dick de Jongh∗, and Joost J. Joosten∗, Interpretability in PRA

SLIDE 26

Why and how study interpretability Proof theoretic characteristics of PRA Modal matters Beklemishev’s principle Zambella’s Principle

◮ Consider again

∃j ∀ϕ(AxiomS(ϕ) → ∃p ProofT(p, ϕj))

◮ Certainly Σ3 ◮ When S has finitely many axioms, then Σ1 ◮ When T is reflexive, then Π2. (Orey-H´

ajek).

◮ When T is reflexive, we have access to Montagna’s Principle:

(T ⊲ S) → ((T ∧ γ) ⊲ (S ∧ γ))

◮ Every extension of PRA by Σ2 sentences is reflexive (Parsons,

Beklemishev, etc)

Marta Bilkova†, Dick de Jongh∗, and Joost J. Joosten∗, Interpretability in PRA

SLIDE 27

Why and how study interpretability Proof theoretic characteristics of PRA Modal matters Beklemishev’s principle Zambella’s Principle

◮ Consider again

∃j ∀ϕ(AxiomS(ϕ) → ∃p ProofT(p, ϕj))

◮ Certainly Σ3 ◮ When S has finitely many axioms, then Σ1 ◮ When T is reflexive, then Π2. (Orey-H´

ajek).

◮ When T is reflexive, we have access to Montagna’s Principle:

(T ⊲ S) → ((T ∧ γ) ⊲ (S ∧ γ))

◮ Every extension of PRA by Σ2 sentences is reflexive (Parsons,

Beklemishev, etc)

◮ (α ⊲PRA β) → ((α ∧ γ) ⊲PRA (β ∧ γ))

whenever α ∈ Σ2

Marta Bilkova†, Dick de Jongh∗, and Joost J. Joosten∗, Interpretability in PRA

SLIDE 28

Why and how study interpretability Proof theoretic characteristics of PRA Modal matters Beklemishev’s principle Zambella’s Principle

◮ B := (A ⊲ B) → (A ∧ C) ⊲ (B ∧ C)

for A ∈ ES2

Marta Bilkova†, Dick de Jongh∗, and Joost J. Joosten∗, Interpretability in PRA

SLIDE 29

Why and how study interpretability Proof theoretic characteristics of PRA Modal matters Beklemishev’s principle Zambella’s Principle

◮ B := (A ⊲ B) → (A ∧ C) ⊲ (B ∧ C)

for A ∈ ES2

◮ where

Marta Bilkova†, Dick de Jongh∗, and Joost J. Joosten∗, Interpretability in PRA

SLIDE 30

Why and how study interpretability Proof theoretic characteristics of PRA Modal matters Beklemishev’s principle Zambella’s Principle

◮ B := (A ⊲ B) → (A ∧ C) ⊲ (B ∧ C)

for A ∈ ES2

◮ where ◮

ES2 := A | ¬A | ES2 ∧ ES2 | ES2 ∨ ES2 | ¬(ES2 ⊲ A)

Marta Bilkova†, Dick de Jongh∗, and Joost J. Joosten∗, Interpretability in PRA

SLIDE 31

Why and how study interpretability Proof theoretic characteristics of PRA Modal matters Beklemishev’s principle Zambella’s Principle

◮ If T and S are Π2 axiomatized theories with

Marta Bilkova†, Dick de Jongh∗, and Joost J. Joosten∗, Interpretability in PRA

SLIDE 32

Why and how study interpretability Proof theoretic characteristics of PRA Modal matters Beklemishev’s principle Zambella’s Principle

◮ If T and S are Π2 axiomatized theories with ◮ T ≡1 S

Marta Bilkova†, Dick de Jongh∗, and Joost J. Joosten∗, Interpretability in PRA

SLIDE 33

Why and how study interpretability Proof theoretic characteristics of PRA Modal matters Beklemishev’s principle Zambella’s Principle

◮ If T and S are Π2 axiomatized theories with ◮ T ≡1 S ◮ then, T ≡1 (T ∪ S)

Marta Bilkova†, Dick de Jongh∗, and Joost J. Joosten∗, Interpretability in PRA

SLIDE 34

Why and how study interpretability Proof theoretic characteristics of PRA Modal matters Beklemishev’s principle Zambella’s Principle

◮ If T and S are Π2 axiomatized theories with ◮ T ≡1 S ◮ then, T ≡1 (T ∪ S) ◮ So,

(α ⊲ β) ∧ (β ⊲ α) → (α ⊲ (α ∧ β)) whenever,

Marta Bilkova†, Dick de Jongh∗, and Joost J. Joosten∗, Interpretability in PRA

SLIDE 35

Why and how study interpretability Proof theoretic characteristics of PRA Modal matters Beklemishev’s principle Zambella’s Principle

◮ If T and S are Π2 axiomatized theories with ◮ T ≡1 S ◮ then, T ≡1 (T ∪ S) ◮ So,

(α ⊲ β) ∧ (β ⊲ α) → (α ⊲ (α ∧ β)) whenever,

◮ α, β ∈ Σ2, and

Marta Bilkova†, Dick de Jongh∗, and Joost J. Joosten∗, Interpretability in PRA

SLIDE 36

Why and how study interpretability Proof theoretic characteristics of PRA Modal matters Beklemishev’s principle Zambella’s Principle

◮ If T and S are Π2 axiomatized theories with ◮ T ≡1 S ◮ then, T ≡1 (T ∪ S) ◮ So,

(α ⊲ β) ∧ (β ⊲ α) → (α ⊲ (α ∧ β)) whenever,

◮ α, β ∈ Σ2, and ◮ α, β ∈ Π2.

Marta Bilkova†, Dick de Jongh∗, and Joost J. Joosten∗, Interpretability in PRA

SLIDE 37

Why and how study interpretability Proof theoretic characteristics of PRA Modal matters Beklemishev’s principle Zambella’s Principle

◮ If T and S are Π2 axiomatized theories with ◮ T ≡1 S ◮ then, T ≡1 (T ∪ S) ◮ So,

(α ⊲ β) ∧ (β ⊲ α) → (α ⊲ (α ∧ β)) whenever,

◮ α, β ∈ Σ2, and ◮ α, β ∈ Π2. ◮ In other words: α, β ∈ ∆2

Marta Bilkova†, Dick de Jongh∗, and Joost J. Joosten∗, Interpretability in PRA

SLIDE 38

Why and how study interpretability Proof theoretic characteristics of PRA Modal matters Beklemishev’s principle Zambella’s Principle

◮ Z

(A ⊲ B) ∧ (B ⊲ A) → (A ⊲ (A ∧ B)) for A and B in ED2

Marta Bilkova†, Dick de Jongh∗, and Joost J. Joosten∗, Interpretability in PRA

SLIDE 39

Why and how study interpretability Proof theoretic characteristics of PRA Modal matters Beklemishev’s principle Zambella’s Principle

◮ Z

(A ⊲ B) ∧ (B ⊲ A) → (A ⊲ (A ∧ B)) for A and B in ED2

◮

ED2 := A | ¬ED2 | ED2 ∧ ED2 | ED2 ∨ ED2

Marta Bilkova†, Dick de Jongh∗, and Joost J. Joosten∗, Interpretability in PRA

SLIDE 40

Why and how study interpretability Proof theoretic characteristics of PRA Modal matters Beklemishev’s principle Zambella’s Principle

◮ Z

(A ⊲ B) ∧ (B ⊲ A) → (A ⊲ (A ∧ B)) for A and B in ED2

◮

ED2 := A | ¬ED2 | ED2 ∧ ED2 | ED2 ∨ ED2

◮ Is this all?

Marta Bilkova†, Dick de Jongh∗, and Joost J. Joosten∗, Interpretability in PRA

SLIDE 41

Why and how study interpretability Proof theoretic characteristics of PRA Modal matters The basics Frame conditions

The logic IL

L1: (A → B) → (A → B) L2: A → A L3: (A → A) → A J1: (A → B) → A ⊲ B J2: (A ⊲ B) ∧ (B ⊲ C) → A ⊲ C J3: (A ⊲ C) ∧ (B ⊲ C) → A ∨ B ⊲ C J4: A ⊲ B → (♦A → ♦B) J5: ♦A ⊲ A

Marta Bilkova†, Dick de Jongh∗, and Joost J. Joosten∗, Interpretability in PRA

SLIDE 42

Why and how study interpretability Proof theoretic characteristics of PRA Modal matters The basics Frame conditions

◮ A Veltman frame F = W , R, S,

R ⊆ W × W , Sw ⊆ W × W for each w ∈ W .

Marta Bilkova†, Dick de Jongh∗, and Joost J. Joosten∗, Interpretability in PRA

SLIDE 43

Why and how study interpretability Proof theoretic characteristics of PRA Modal matters The basics Frame conditions

◮ ◮ A Veltman frame F = W , R, S,

R ⊆ W × W , Sw ⊆ W × W for each w ∈ W .

◮ R is conversely well-founded and transitive

Marta Bilkova†, Dick de Jongh∗, and Joost J. Joosten∗, Interpretability in PRA

SLIDE 44

Why and how study interpretability Proof theoretic characteristics of PRA Modal matters The basics Frame conditions

◮ ◮ A Veltman frame F = W , R, S,

R ⊆ W × W , Sw ⊆ W × W for each w ∈ W .

◮ R is conversely well-founded and transitive ◮ ySxz → xRy ∧ xRz

Marta Bilkova†, Dick de Jongh∗, and Joost J. Joosten∗, Interpretability in PRA

SLIDE 45

Why and how study interpretability Proof theoretic characteristics of PRA Modal matters The basics Frame conditions

◮ ◮ A Veltman frame F = W , R, S,

R ⊆ W × W , Sw ⊆ W × W for each w ∈ W .

◮ R is conversely well-founded and transitive ◮ ySxz → xRy ∧ xRz ◮ xRyRz → ySxz

Marta Bilkova†, Dick de Jongh∗, and Joost J. Joosten∗, Interpretability in PRA

SLIDE 46

Why and how study interpretability Proof theoretic characteristics of PRA Modal matters The basics Frame conditions

◮ ◮ A Veltman frame F = W , R, S,

R ⊆ W × W , Sw ⊆ W × W for each w ∈ W .

◮ R is conversely well-founded and transitive ◮ ySxz → xRy ∧ xRz ◮ xRyRz → ySxz ◮ Sx is transitive and reflexive for each x

Marta Bilkova†, Dick de Jongh∗, and Joost J. Joosten∗, Interpretability in PRA

SLIDE 47

Why and how study interpretability Proof theoretic characteristics of PRA Modal matters The basics Frame conditions

◮ ◮ A Veltman frame F = W , R, S,

R ⊆ W × W , Sw ⊆ W × W for each w ∈ W .

◮ R is conversely well-founded and transitive ◮ ySxz → xRy ∧ xRz ◮ xRyRz → ySxz ◮ Sx is transitive and reflexive for each x

Marta Bilkova†, Dick de Jongh∗, and Joost J. Joosten∗, Interpretability in PRA

SLIDE 48

Why and how study interpretability Proof theoretic characteristics of PRA Modal matters The basics Frame conditions

A model M = W , R, S, , ⊆ W × Prop

◮ ◮ w ⊥

Marta Bilkova†, Dick de Jongh∗, and Joost J. Joosten∗, Interpretability in PRA

SLIDE 49

Why and how study interpretability Proof theoretic characteristics of PRA Modal matters The basics Frame conditions

A model M = W , R, S, , ⊆ W × Prop

◮ w ⊥ ◮ w A → B iff w A or w B

Marta Bilkova†, Dick de Jongh∗, and Joost J. Joosten∗, Interpretability in PRA

SLIDE 50

Why and how study interpretability Proof theoretic characteristics of PRA Modal matters The basics Frame conditions

A model M = W , R, S, , ⊆ W × Prop

◮ w ⊥ ◮ w A → B iff w A or w B ◮ w A iff ∀v (wRv ⇒ v A)

Marta Bilkova†, Dick de Jongh∗, and Joost J. Joosten∗, Interpretability in PRA

SLIDE 51

Why and how study interpretability Proof theoretic characteristics of PRA Modal matters The basics Frame conditions

A model M = W , R, S, , ⊆ W × Prop

◮ w ⊥ ◮ w A → B iff w A or w B ◮ w A iff ∀v (wRv ⇒ v A) ◮ w A ⊲ B iff ∀u (wRu ∧ u A ⇒ ∃v(uSwv B))

Marta Bilkova†, Dick de Jongh∗, and Joost J. Joosten∗, Interpretability in PRA

SLIDE 52

Why and how study interpretability Proof theoretic characteristics of PRA Modal matters The basics Frame conditions

◮ Montagna has a nice frame condition

(A ⊲ B) → ((A ∧ C) ⊲ (B ∧ C))

Marta Bilkova†, Dick de Jongh∗, and Joost J. Joosten∗, Interpretability in PRA

SLIDE 53

Why and how study interpretability Proof theoretic characteristics of PRA Modal matters The basics Frame conditions

◮ Montagna has a nice frame condition

(A ⊲ B) → ((A ∧ C) ⊲ (B ∧ C))

◮ Beklemishev is somewhat similar

Marta Bilkova†, Dick de Jongh∗, and Joost J. Joosten∗, Interpretability in PRA

SLIDE 54

Why and how study interpretability Proof theoretic characteristics of PRA Modal matters The basics Frame conditions

A B-simulation on a frame is a binary relation S for which the following holds.

1. S(x, x′) → x↑ = x′↑
2. S(x, x′) & xRy → ∃y′(ySxy′ ∧ S(y, y′) ∧ y′Sx′↑ ⊆ ySx↑)

F | = CB if and only if there is a B-simulation S on F such that for all x and y, xRy → ∃y′(ySxy′ ∧ S(y, y′) ∧ ∀d, e (y′SxdRe → yRd)).

Marta Bilkova†, Dick de Jongh∗, and Joost J. Joosten∗, Interpretability in PRA

SLIDE 55

Why and how study interpretability Proof theoretic characteristics of PRA Modal matters The basics Frame conditions

◮

ES0

2

:= ED2 ESn+1

2

:= ESn

2 | ESn+1 2

∧ ESn+1

2

| ESn+1

2

∨ ESn+1

2

| ¬(ESn

2 ⊲ Form)

Marta Bilkova†, Dick de Jongh∗, and Joost J. Joosten∗, Interpretability in PRA

SLIDE 56

Why and how study interpretability Proof theoretic characteristics of PRA Modal matters The basics Frame conditions

◮

ES0

2

:= ED2 ESn+1

2

:= ESn

2 | ESn+1 2

∧ ESn+1

2

| ESn+1

2

∨ ESn+1

2

| ¬(ESn

2 ⊲ Form) ◮

S0(b, u) := b↑=u↑ Sn+1(b, u) := Sn(b, u)∧ ∀c (bRc → ∃c′ (uRc′ ∧ Sn(c, c′)∧ cSbc′ ∧ c′Su↑ ⊆ cSb↑))

Marta Bilkova†, Dick de Jongh∗, and Joost J. Joosten∗, Interpretability in PRA

SLIDE 57

Why and how study interpretability Proof theoretic characteristics of PRA Modal matters The basics Frame conditions

◮

ES0

2

:= ED2 ESn+1

2

:= ESn

2 | ESn+1 2

∧ ESn+1

2

| ESn+1

2

∨ ESn+1

2

| ¬(ESn

2 ⊲ Form) ◮

S0(b, u) := b↑=u↑ Sn+1(b, u) := Sn(b, u)∧ ∀c (bRc → ∃c′ (uRc′ ∧ Sn(c, c′)∧ cSbc′ ∧ c′Su↑ ⊆ cSb↑))

◮ For every i we define the frame condition Ci to be

∀ a, b (aRb → ∃u (bSau ∧ Si(b, u) ∧ ∀ d, e (uSadRe → bRe))).

Marta Bilkova†, Dick de Jongh∗, and Joost J. Joosten∗, Interpretability in PRA

SLIDE 58

Why and how study interpretability Proof theoretic characteristics of PRA Modal matters The basics Frame conditions

◮

ES0

2

:= ED2 ESn+1

2

:= ESn

2 | ESn+1 2

∧ ESn+1

2

| ESn+1

2

∨ ESn+1

2

| ¬(ESn

2 ⊲ Form) ◮

S0(b, u) := b↑=u↑ Sn+1(b, u) := Sn(b, u)∧ ∀c (bRc → ∃c′ (uRc′ ∧ Sn(c, c′)∧ cSbc′ ∧ c′Su↑ ⊆ cSb↑))

◮ For every i we define the frame condition Ci to be

∀ a, b (aRb → ∃u (bSau ∧ Si(b, u) ∧ ∀ d, e (uSadRe → bRe))).

Marta Bilkova†, Dick de Jongh∗, and Joost J. Joosten∗, Interpretability in PRA

SLIDE 59

Why and how study interpretability Proof theoretic characteristics of PRA Modal matters The basics Frame conditions

◮

ES0

2

:= ED2 ESn+1

2

:= ESn

2 | ESn+1 2

∧ ESn+1

2

| ESn+1

2

∨ ESn+1

2

| ¬(ESn

2 ⊲ Form) ◮

S0(b, u) := b↑=u↑ Sn+1(b, u) := Sn(b, u)∧ ∀c (bRc → ∃c′ (uRc′ ∧ Sn(c, c′)∧ cSbc′ ∧ c′Su↑ ⊆ cSb↑))

◮ For every i we define the frame condition Ci to be

∀ a, b (aRb → ∃u (bSau ∧ Si(b, u) ∧ ∀ d, e (uSadRe → bRe))).

◮ Theorem

A finite frame F validates all instances of Beklemishev’s principle if and only if ∀i F | = Ci.

Marta Bilkova†, Dick de Jongh∗, and Joost J. Joosten∗, Interpretability in PRA

SLIDE 60

Why and how study interpretability Proof theoretic characteristics of PRA Modal matters The basics Frame conditions

◮ B ⊢ Z

Marta Bilkova†, Dick de Jongh∗, and Joost J. Joosten∗, Interpretability in PRA

SLIDE 61

Why and how study interpretability Proof theoretic characteristics of PRA Modal matters The basics Frame conditions

◮ B ⊢ Z ◮ B |

= Z

Marta Bilkova†, Dick de Jongh∗, and Joost J. Joosten∗, Interpretability in PRA

SLIDE 62

Why and how study interpretability Proof theoretic characteristics of PRA Modal matters The basics Frame conditions

◮ B ⊢ Z ◮ B |

= Z

◮ Frame condition Zambella?

Marta Bilkova†, Dick de Jongh∗, and Joost J. Joosten∗, Interpretability in PRA