[PPT] - Arithmetic interpretation of the monadic fragment of intuitionistic PowerPoint Presentation

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Arithmetic interpretation of the monadic fragment of intuitionistic predicate logic and Casari’s formula

Julia Ilin∗ joint work with Guram Bezhanishvili† and Kristina Brantley†

∗Institute of Logic, Language and Computation, Universiteit van Amsterdam,

The Netherlands

†Department of Mathematical Sciences, New Mexico State University, USA

TACL 2017, Prague, June 2017

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Arithmetic interpretation of the monadic fragment of intuitionistic predicate logic and Casari’s formula

Arithmetic interpretation of IPC

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Arithmetic interpretation of the monadic fragment of intuitionistic predicate logic and Casari’s formula

Arithmetic interpretation of IPC

The G¨

del–McKinsey–Tarski translation t embeds IPC into S4.

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Arithmetic interpretation of the monadic fragment of intuitionistic predicate logic and Casari’s formula

Arithmetic interpretation of IPC

The G¨

del–McKinsey–Tarski translation t embeds IPC into

Grzegorczyk logic Grz.

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Arithmetic interpretation of the monadic fragment of intuitionistic predicate logic and Casari’s formula

Arithmetic interpretation of IPC

The G¨

del–McKinsey–Tarski translation t embeds IPC into

Grzegorczyk logic Grz. The splitting translation sp (replaces ϕ with ϕ ∧ ϕ) embeds Grz into the G¨

del-L¨
b logic GL.

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Arithmetic interpretation of the monadic fragment of intuitionistic predicate logic and Casari’s formula

Arithmetic interpretation of IPC

The G¨

del–McKinsey–Tarski translation t embeds IPC into

Grzegorczyk logic Grz. The splitting translation sp (replaces ϕ with ϕ ∧ ϕ) embeds Grz into the G¨

del-L¨
b logic GL.

Theorem (Grzegorczyk, Goldblatt, Boolos, Kuznetsov and Muravitsky) For every formula ϕ of IPC, IPC ⊢ ϕ iff Grz ⊢ t(ϕ) iff GL ⊢ sp(t(ϕ)).

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Arithmetic interpretation of the monadic fragment of intuitionistic predicate logic and Casari’s formula

Arithmetic interpretation of IPC

The G¨

del–McKinsey–Tarski translation t embeds IPC into

Grzegorczyk logic Grz. The splitting translation sp (replaces ϕ with ϕ ∧ ϕ) embeds Grz into the G¨

del-L¨
b logic GL.

Theorem (Grzegorczyk, Goldblatt, Boolos, Kuznetsov and Muravitsky) For every formula ϕ of IPC, IPC ⊢ ϕ iff Grz ⊢ t(ϕ) iff GL ⊢ sp(t(ϕ)). By Solovay’s theorem, GL is arithmetically complete. This provides arithmetic interpretation of IPC.

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Arithmetic interpretation of the monadic fragment of intuitionistic predicate logic and Casari’s formula

Arithmetic interpretation of IPC

The G¨

del–McKinsey–Tarski translation t embeds IPC into

Grzegorczyk logic Grz. The splitting translation sp (replaces ϕ with ϕ ∧ ϕ) embeds Grz into the G¨

del-L¨
b logic GL.

Theorem (Grzegorczyk, Goldblatt, Boolos, Kuznetsov and Muravitsky) For every formula ϕ of IPC, IPC ⊢ ϕ iff Grz ⊢ t(ϕ) iff GL ⊢ sp(t(ϕ)). By Solovay’s theorem, GL is arithmetically complete. This provides arithmetic interpretation of IPC. . The goal of this talk is to lift the above correspondences to the monadic setting as was anticipated by Esakia.

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Arithmetic interpretation of the monadic fragment of intuitionistic predicate logic and Casari’s formula

Lifting the correspondences to the full predicate setting?

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Arithmetic interpretation of the monadic fragment of intuitionistic predicate logic and Casari’s formula

Lifting the correspondences to the full predicate setting?

Let QIPC, QGrz, and QGL be the predicate extensions of IPC, Grz, and GL, respectively.

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Arithmetic interpretation of the monadic fragment of intuitionistic predicate logic and Casari’s formula

Lifting the correspondences to the full predicate setting?

Let QIPC, QGrz, and QGL be the predicate extensions of IPC, Grz, and GL, respectively. QGL is not arithmetically complete. (Montagna 1984)

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Arithmetic interpretation of the monadic fragment of intuitionistic predicate logic and Casari’s formula

Lifting the correspondences to the full predicate setting?

Let QIPC, QGrz, and QGL be the predicate extensions of IPC, Grz, and GL, respectively. QGL is not arithmetically complete. (Montagna 1984) QIPC is Kripke complete. (Kripke 1965)

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Arithmetic interpretation of the monadic fragment of intuitionistic predicate logic and Casari’s formula

Lifting the correspondences to the full predicate setting?

Let QIPC, QGrz, and QGL be the predicate extensions of IPC, Grz, and GL, respectively. QGL is not arithmetically complete. (Montagna 1984) QIPC is Kripke complete. (Kripke 1965) QGL and QGrz are not Kripke complete. (Montagna 1984, Ghilardi 1991)

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Arithmetic interpretation of the monadic fragment of intuitionistic predicate logic and Casari’s formula

Lifting the correspondences to the full predicate setting?

Let QIPC, QGrz, and QGL be the predicate extensions of IPC, Grz, and GL, respectively. QGL is not arithmetically complete. (Montagna 1984) QIPC is Kripke complete. (Kripke 1965) QGL and QGrz are not Kripke complete. (Montagna 1984, Ghilardi 1991) Thus, arithmetic interpretation does not extend to the full predicate setting and a proof for the modal part of the correspondence would be essentially different than in the propositional case.

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Arithmetic interpretation of the monadic fragment of intuitionistic predicate logic and Casari’s formula

The one-variable setting (overview)

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Arithmetic interpretation of the monadic fragment of intuitionistic predicate logic and Casari’s formula

The one-variable setting (overview)

The intuitionistic bi-modal logic MIPC axiomatizes the one-variable fragment of QIPC. (Bull 1966)

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Arithmetic interpretation of the monadic fragment of intuitionistic predicate logic and Casari’s formula

The one-variable setting (overview)

The intuitionistic bi-modal logic MIPC axiomatizes the one-variable fragment of QIPC. (Bull 1966) Esakia introduced MGrz and MGL, the one variable fragments of QGrz and QGL, respectively.

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Arithmetic interpretation of the monadic fragment of intuitionistic predicate logic and Casari’s formula

The one-variable setting (overview)

The intuitionistic bi-modal logic MIPC axiomatizes the one-variable fragment of QIPC. (Bull 1966) Esakia introduced MGrz and MGL, the one variable fragments of QGrz and QGL, respectively. MIPC, MGrz, and MGL are complete with respect to finite Kripke frames (Bull, Ono, Fisher-Servi, Japaridze )

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Arithmetic interpretation of the monadic fragment of intuitionistic predicate logic and Casari’s formula

The one-variable setting (overview)

The intuitionistic bi-modal logic MIPC axiomatizes the one-variable fragment of QIPC. (Bull 1966) Esakia introduced MGrz and MGL, the one variable fragments of QGrz and QGL, respectively. MIPC, MGrz, and MGL are complete with respect to finite Kripke frames (Bull, Ono, Fisher-Servi, Japaridze ) MGL is arithmetically complete. (Japaridze 1988)

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Arithmetic interpretation of the monadic fragment of intuitionistic predicate logic and Casari’s formula

The one-variable setting (overview)

The intuitionistic bi-modal logic MIPC axiomatizes the one-variable fragment of QIPC. (Bull 1966) Esakia introduced MGrz and MGL, the one variable fragments of QGrz and QGL, respectively. MIPC, MGrz, and MGL are complete with respect to finite Kripke frames (Bull, Ono, Fisher-Servi, Japaridze ) MGL is arithmetically complete. (Japaridze 1988) The (extended) G¨

del–McKinsey–Tarski translation embeds MIPC

into MGrz. (Fischer-Servi 1977)

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Arithmetic interpretation of the monadic fragment of intuitionistic predicate logic and Casari’s formula

The one-variable setting (overview)

The intuitionistic bi-modal logic MIPC axiomatizes the one-variable fragment of QIPC. (Bull 1966) Esakia introduced MGrz and MGL, the one variable fragments of QGrz and QGL, respectively. MIPC, MGrz, and MGL are complete with respect to finite Kripke frames (Bull, Ono, Fisher-Servi, Japaridze ) MGL is arithmetically complete. (Japaridze 1988) The (extended) G¨

del–McKinsey–Tarski translation embeds MIPC

into MGrz. (Fischer-Servi 1977) However, the (extended) splitting translation does not embed MGrz into MGL.

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Arithmetic interpretation of the monadic fragment of intuitionistic predicate logic and Casari’s formula

MIPC

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Arithmetic interpretation of the monadic fragment of intuitionistic predicate logic and Casari’s formula

MIPC

MIPC = K∃,∀ + {∀p → p, ∀(p ∧ q), ↔ (∀p ∧ ∀q), ∀p → ∀∀p p → ∃p, ∃(p ∨ q) ↔ (∃p ∨ ∃q), ∃∃p → ∃p, ∃p → ∀∃p, , ∃∀p → ∀p, ∀(p → q) → (∃p → ∃q)}.

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Arithmetic interpretation of the monadic fragment of intuitionistic predicate logic and Casari’s formula

MIPC

MIPC = K∃,∀ + {∀p → p, ∀(p ∧ q), ↔ (∀p ∧ ∀q), ∀p → ∀∀p p → ∃p, ∃(p ∨ q) ↔ (∃p ∨ ∃q), ∃∃p → ∃p, ∃p → ∀∃p, , ∃∀p → ∀p, ∀(p → q) → (∃p → ∃q)}. An MIPC-frame is of the form F = (W , ≤, E), where

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Arithmetic interpretation of the monadic fragment of intuitionistic predicate logic and Casari’s formula

MIPC

MIPC = K∃,∀ + {∀p → p, ∀(p ∧ q), ↔ (∀p ∧ ∀q), ∀p → ∀∀p p → ∃p, ∃(p ∨ q) ↔ (∃p ∨ ∃q), ∃∃p → ∃p, ∃p → ∀∃p, , ∃∀p → ∀p, ∀(p → q) → (∃p → ∃q)}. An MIPC-frame is of the form F = (W , ≤, E), where (W , ≤) is a partial order,

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Arithmetic interpretation of the monadic fragment of intuitionistic predicate logic and Casari’s formula

MIPC

MIPC = K∃,∀ + {∀p → p, ∀(p ∧ q), ↔ (∀p ∧ ∀q), ∀p → ∀∀p p → ∃p, ∃(p ∨ q) ↔ (∃p ∨ ∃q), ∃∃p → ∃p, ∃p → ∀∃p, , ∃∀p → ∀p, ∀(p → q) → (∃p → ∃q)}. An MIPC-frame is of the form F = (W , ≤, E), where (W , ≤) is a partial order, E is an equivalence relation on W ,

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Arithmetic interpretation of the monadic fragment of intuitionistic predicate logic and Casari’s formula

MIPC

MIPC = K∃,∀ + {∀p → p, ∀(p ∧ q), ↔ (∀p ∧ ∀q), ∀p → ∀∀p p → ∃p, ∃(p ∨ q) ↔ (∃p ∨ ∃q), ∃∃p → ∃p, ∃p → ∀∃p, , ∃∀p → ∀p, ∀(p → q) → (∃p → ∃q)}. An MIPC-frame is of the form F = (W , ≤, E), where (W , ≤) is a partial order, E is an equivalence relation on W , E◦ ≤ ⊆ ≤ ◦E,

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Arithmetic interpretation of the monadic fragment of intuitionistic predicate logic and Casari’s formula

MIPC

MIPC = K∃,∀ + {∀p → p, ∀(p ∧ q), ↔ (∀p ∧ ∀q), ∀p → ∀∀p p → ∃p, ∃(p ∨ q) ↔ (∃p ∨ ∃q), ∃∃p → ∃p, ∃p → ∀∃p, , ∃∀p → ∀p, ∀(p → q) → (∃p → ∃q)}. An MIPC-frame is of the form F = (W , ≤, E), where (W , ≤) is a partial order, E is an equivalence relation on W , E◦ ≤ ⊆ ≤ ◦E, i.e. ≤ E (commutativity).

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Arithmetic interpretation of the monadic fragment of intuitionistic predicate logic and Casari’s formula

MIPC

MIPC = K∃,∀ + {∀p → p, ∀(p ∧ q), ↔ (∀p ∧ ∀q), ∀p → ∀∀p p → ∃p, ∃(p ∨ q) ↔ (∃p ∨ ∃q), ∃∃p → ∃p, ∃p → ∀∃p, , ∃∀p → ∀p, ∀(p → q) → (∃p → ∃q)}. An MIPC-frame is of the form F = (W , ≤, E), where (W , ≤) is a partial order, E is an equivalence relation on W , E◦ ≤ ⊆ ≤ ◦E, i.e. ≤ E (commutativity). If F is an MIPC-frame and v : Prop → Up≤(W ) a valuation on F,

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Arithmetic interpretation of the monadic fragment of intuitionistic predicate logic and Casari’s formula

MIPC

MIPC = K∃,∀ + {∀p → p, ∀(p ∧ q), ↔ (∀p ∧ ∀q), ∀p → ∀∀p p → ∃p, ∃(p ∨ q) ↔ (∃p ∨ ∃q), ∃∃p → ∃p, ∃p → ∀∃p, , ∃∀p → ∀p, ∀(p → q) → (∃p → ∃q)}. An MIPC-frame is of the form F = (W , ≤, E), where (W , ≤) is a partial order, E is an equivalence relation on W , E◦ ≤ ⊆ ≤ ◦E, i.e. ≤ E (commutativity). If F is an MIPC-frame and v : Prop → Up≤(W ) a valuation on F, x | = ϕ → ψ iff y | = ϕ implies x | = ψ for all x ≤ y,

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Arithmetic interpretation of the monadic fragment of intuitionistic predicate logic and Casari’s formula

MIPC

MIPC = K∃,∀ + {∀p → p, ∀(p ∧ q), ↔ (∀p ∧ ∀q), ∀p → ∀∀p p → ∃p, ∃(p ∨ q) ↔ (∃p ∨ ∃q), ∃∃p → ∃p, ∃p → ∀∃p, , ∃∀p → ∀p, ∀(p → q) → (∃p → ∃q)}. An MIPC-frame is of the form F = (W , ≤, E), where (W , ≤) is a partial order, E is an equivalence relation on W , E◦ ≤ ⊆ ≤ ◦E, i.e. ≤ E (commutativity). If F is an MIPC-frame and v : Prop → Up≤(W ) a valuation on F, x | = ϕ → ψ iff y | = ϕ implies x | = ψ for all x ≤ y, x | = ∃ϕ iff y | = ϕ for some x E y.

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Arithmetic interpretation of the monadic fragment of intuitionistic predicate logic and Casari’s formula

MIPC

MIPC = K∃,∀ + {∀p → p, ∀(p ∧ q), ↔ (∀p ∧ ∀q), ∀p → ∀∀p p → ∃p, ∃(p ∨ q) ↔ (∃p ∨ ∃q), ∃∃p → ∃p, ∃p → ∀∃p, , ∃∀p → ∀p, ∀(p → q) → (∃p → ∃q)}. An MIPC-frame is of the form F = (W , ≤, E), where (W , ≤) is a partial order, E is an equivalence relation on W , E◦ ≤ ⊆ ≤ ◦E, i.e. ≤ E (commutativity). If F is an MIPC-frame and v : Prop → Up≤(W ) a valuation on F, x | = ϕ → ψ iff y | = ϕ implies x | = ψ for all x ≤ y, x | = ∃ϕ iff y | = ϕ for some x E y. x | = ∀ϕ iff y | = ϕ for all x Q y, where Q :=≤ ◦E.

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Arithmetic interpretation of the monadic fragment of intuitionistic predicate logic and Casari’s formula

MGrz

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Arithmetic interpretation of the monadic fragment of intuitionistic predicate logic and Casari’s formula

MGrz

MGrz = S5∀ ⊕ Grz + ∀p → ∀p

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Arithmetic interpretation of the monadic fragment of intuitionistic predicate logic and Casari’s formula

MGrz

MGrz = S5∀ ⊕ Grz + ∀p → ∀p ∃ϕ := ¬∀¬ϕ.

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Arithmetic interpretation of the monadic fragment of intuitionistic predicate logic and Casari’s formula

MGrz

MGrz = S5∀ ⊕ Grz + ∀p → ∀p ∃ϕ := ¬∀¬ϕ. An MGrz-frame is of the form F = (W , R, E), where

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Arithmetic interpretation of the monadic fragment of intuitionistic predicate logic and Casari’s formula

MGrz

MGrz = S5∀ ⊕ Grz + ∀p → ∀p ∃ϕ := ¬∀¬ϕ. An MGrz-frame is of the form F = (W , R, E), where

(W , R) is a Noetherian partial order,

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Arithmetic interpretation of the monadic fragment of intuitionistic predicate logic and Casari’s formula

MGrz

MGrz = S5∀ ⊕ Grz + ∀p → ∀p ∃ϕ := ¬∀¬ϕ. An MGrz-frame is of the form F = (W , R, E), where

(W , R) is a Noetherian partial order, E is an equivalence relation on W ,

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Arithmetic interpretation of the monadic fragment of intuitionistic predicate logic and Casari’s formula

MGrz

MGrz = S5∀ ⊕ Grz + ∀p → ∀p ∃ϕ := ¬∀¬ϕ. An MGrz-frame is of the form F = (W , R, E), where

(W , R) is a Noetherian partial order, E is an equivalence relation on W , E ◦ R ⊆ R ◦ E, i.e. R E (commutativity)

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Arithmetic interpretation of the monadic fragment of intuitionistic predicate logic and Casari’s formula

MGrz

MGrz = S5∀ ⊕ Grz + ∀p → ∀p ∃ϕ := ¬∀¬ϕ. An MGrz-frame is of the form F = (W , R, E), where

(W , R) is a Noetherian partial order, E is an equivalence relation on W , E ◦ R ⊆ R ◦ E, i.e. R E (commutativity)

If F is an MGrz-frame and v : Prop → P(W ) a valuation on F, x | = ϕ iff y | = ϕ for all xRy, x | = ∃ϕ iff y | = ϕ for some x E y, x | = ∀ϕ iff y | = ϕ for all x E y,

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Arithmetic interpretation of the monadic fragment of intuitionistic predicate logic and Casari’s formula

G¨

del–McKinsey–Tarski translation

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Arithmetic interpretation of the monadic fragment of intuitionistic predicate logic and Casari’s formula

G¨

del–McKinsey–Tarski translation

(Extended) G¨

del–McKinsey–Tarski translation t : MIPC −

→ MGrz is t(p) = p t(⊥) = ⊥ t(ϕ ◦ ψ) = t(ϕ) ◦ t(ψ)

∈ {∧, ∨}

t(ϕ → ψ) = (t(ϕ) → t(ψ)) t(∀ϕ) = ∀t(ϕ), t(∃ϕ) = ∃t(ϕ)

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Arithmetic interpretation of the monadic fragment of intuitionistic predicate logic and Casari’s formula

G¨

del–McKinsey–Tarski translation

(Extended) G¨

del–McKinsey–Tarski translation t : MIPC −

→ MGrz is t(p) = p t(⊥) = ⊥ t(ϕ ◦ ψ) = t(ϕ) ◦ t(ψ)

∈ {∧, ∨}

t(ϕ → ψ) = (t(ϕ) → t(ψ)) t(∀ϕ) = ∀t(ϕ), t(∃ϕ) = ∃t(ϕ) For every formula ϕ of MIPC, MIPC ⊢ ϕ iff MGrz ⊢ t(ϕ).

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Arithmetic interpretation of the monadic fragment of intuitionistic predicate logic and Casari’s formula

G¨

del–McKinsey–Tarski translation

(Extended) G¨

del–McKinsey–Tarski translation t : MIPC −

→ MGrz is t(p) = p t(⊥) = ⊥ t(ϕ ◦ ψ) = t(ϕ) ◦ t(ψ)

∈ {∧, ∨}

t(ϕ → ψ) = (t(ϕ) → t(ψ)) t(∀ϕ) = ∀t(ϕ), t(∃ϕ) = ∃t(ϕ) For every formula ϕ of MIPC, MIPC ⊢ ϕ iff MGrz ⊢ t(ϕ). If v : Prop → P(W ) is a valuation on a finite MGrz-frame F, let v(p) = {x | x | = p}.

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Arithmetic interpretation of the monadic fragment of intuitionistic predicate logic and Casari’s formula

G¨

del–McKinsey–Tarski translation

(Extended) G¨

del–McKinsey–Tarski translation t : MIPC −

→ MGrz is t(p) = p t(⊥) = ⊥ t(ϕ ◦ ψ) = t(ϕ) ◦ t(ψ)

∈ {∧, ∨}

t(ϕ → ψ) = (t(ϕ) → t(ψ)) t(∀ϕ) = ∀t(ϕ), t(∃ϕ) = ∃t(ϕ) For every formula ϕ of MIPC, MIPC ⊢ ϕ iff MGrz ⊢ t(ϕ). If v : Prop → P(W ) is a valuation on a finite MGrz-frame F, let v(p) = {x | x | = p}. Then for every ϕ of MIPC and x ∈ W , F, v, x | = ϕ iff F, v, x | = t(ϕ)

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Arithmetic interpretation of the monadic fragment of intuitionistic predicate logic and Casari’s formula

MGL

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Arithmetic interpretation of the monadic fragment of intuitionistic predicate logic and Casari’s formula

MGL

MGL = S5∀ ⊕ GL + ∀p → ∀p

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Arithmetic interpretation of the monadic fragment of intuitionistic predicate logic and Casari’s formula

MGL

MGL = S5∀ ⊕ GL + ∀p → ∀p ∃ϕ := ¬∀¬ϕ.

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Arithmetic interpretation of the monadic fragment of intuitionistic predicate logic and Casari’s formula

MGL

MGL = S5∀ ⊕ GL + ∀p → ∀p ∃ϕ := ¬∀¬ϕ. An MGL-frame is of the form F = (W , R, E), where

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Arithmetic interpretation of the monadic fragment of intuitionistic predicate logic and Casari’s formula

MGL

MGL = S5∀ ⊕ GL + ∀p → ∀p ∃ϕ := ¬∀¬ϕ. An MGL-frame is of the form F = (W , R, E), where

(W , R) is a transitive and conversely well-founded ,

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Arithmetic interpretation of the monadic fragment of intuitionistic predicate logic and Casari’s formula

MGL

MGL = S5∀ ⊕ GL + ∀p → ∀p ∃ϕ := ¬∀¬ϕ. An MGL-frame is of the form F = (W , R, E), where

(W , R) is a transitive and conversely well-founded , E is an equivalence relation on W ,

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Arithmetic interpretation of the monadic fragment of intuitionistic predicate logic and Casari’s formula

MGL

MGL = S5∀ ⊕ GL + ∀p → ∀p ∃ϕ := ¬∀¬ϕ. An MGL-frame is of the form F = (W , R, E), where

(W , R) is a transitive and conversely well-founded , E is an equivalence relation on W , E ◦ R ⊆ R ◦ E, i.e. R E (commutativity)

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Arithmetic interpretation of the monadic fragment of intuitionistic predicate logic and Casari’s formula

MGL

MGL = S5∀ ⊕ GL + ∀p → ∀p ∃ϕ := ¬∀¬ϕ. An MGL-frame is of the form F = (W , R, E), where

(W , R) is a transitive and conversely well-founded , E is an equivalence relation on W , E ◦ R ⊆ R ◦ E, i.e. R E (commutativity)

If F is an MGL-frame and v : Prop → P(W ) a valuation on F, x | = ϕ iff y | = ϕ for all xRy, x | = ∀ϕ iff y | = ϕ for all x E y,

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Arithmetic interpretation of the monadic fragment of intuitionistic predicate logic and Casari’s formula

The splitting translation from MGrz to MGL is not faithful

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Arithmetic interpretation of the monadic fragment of intuitionistic predicate logic and Casari’s formula

The splitting translation from MGrz to MGL is not faithful

Consider the following MGrz-model F ¬q q R E

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Arithmetic interpretation of the monadic fragment of intuitionistic predicate logic and Casari’s formula

The splitting translation from MGrz to MGL is not faithful

Consider the following MGrz-model F ¬q q R E F | = Ψ := ( (q → q) → q) → q, where ϕ := ∀ϕ.

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Arithmetic interpretation of the monadic fragment of intuitionistic predicate logic and Casari’s formula

The splitting translation from MGrz to MGL is not faithful

Consider the following MGrz-model F ¬q q R E | = q F | = Ψ := ( (q → q) → q) → q, where ϕ := ∀ϕ.

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Arithmetic interpretation of the monadic fragment of intuitionistic predicate logic and Casari’s formula

The splitting translation from MGrz to MGL is not faithful

Consider the following MGrz-model F ¬q q R E | = q | = (q → q) F | = Ψ := ( (q → q) → q) → q, where ϕ := ∀ϕ.

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Arithmetic interpretation of the monadic fragment of intuitionistic predicate logic and Casari’s formula

The splitting translation from MGrz to MGL is not faithful

Consider the following MGrz-model F ¬q q R E | = q | = (q → q) | = (q → q) F | = Ψ := ( (q → q) → q) → q, where ϕ := ∀ϕ.

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Arithmetic interpretation of the monadic fragment of intuitionistic predicate logic and Casari’s formula

The splitting translation from MGrz to MGL is not faithful

Consider the following MGrz-model F ¬q q R E | = q | = (q → q) | = (q → q) F | = Ψ := ( (q → q) → q) → q, where ϕ := ∀ϕ. But MGL | = sp(Ψ).

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Arithmetic interpretation of the monadic fragment of intuitionistic predicate logic and Casari’s formula

Adding Casari’s formula

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Arithmetic interpretation of the monadic fragment of intuitionistic predicate logic and Casari’s formula

Adding Casari’s formula

We call an E-cluster C of a frame clean iff for all y = x in C, ¬(xRy).

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Arithmetic interpretation of the monadic fragment of intuitionistic predicate logic and Casari’s formula

Adding Casari’s formula

We call an E-cluster C of a frame clean iff for all y = x in C, ¬(xRy). We call C dirty, otherwise.

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SLIDE 64

Arithmetic interpretation of the monadic fragment of intuitionistic predicate logic and Casari’s formula

Adding Casari’s formula

We call an E-cluster C of a frame clean iff for all y = x in C, ¬(xRy). We call C dirty, otherwise. Finite MGL-frames have only clean E-clusters.

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SLIDE 65

Arithmetic interpretation of the monadic fragment of intuitionistic predicate logic and Casari’s formula

Adding Casari’s formula

We call an E-cluster C of a frame clean iff for all y = x in C, ¬(xRy). We call C dirty, otherwise. Finite MGL-frames have only clean E-clusters. Consider (MCas) ∀((p → ∀p) → p) → ∀p.

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SLIDE 66

Arithmetic interpretation of the monadic fragment of intuitionistic predicate logic and Casari’s formula

Adding Casari’s formula

We call an E-cluster C of a frame clean iff for all y = x in C, ¬(xRy). We call C dirty, otherwise. Finite MGL-frames have only clean E-clusters. Consider (MCas) ∀((p → ∀p) → p) → ∀p. Lemma

1 A finite MIPC-frame validates MCas iff all its E-clusters are clean. 2 A finite MGrz-frame validates t(MCas) iff all its E-clusters are clean.

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SLIDE 67

Arithmetic interpretation of the monadic fragment of intuitionistic predicate logic and Casari’s formula

Adding Casari’s formula

We call an E-cluster C of a frame clean iff for all y = x in C, ¬(xRy). We call C dirty, otherwise. Finite MGL-frames have only clean E-clusters. Consider (MCas) ∀((p → ∀p) → p) → ∀p. Lemma

1 A finite MIPC-frame validates MCas iff all its E-clusters are clean. 2 A finite MGrz-frame validates t(MCas) iff all its E-clusters are clean.

Let M+IPC = MIPC + MCas and let M+Grz = MGrz + t(MCas).

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SLIDE 68

Arithmetic interpretation of the monadic fragment of intuitionistic predicate logic and Casari’s formula

Adding Casari’s formula

We call an E-cluster C of a frame clean iff for all y = x in C, ¬(xRy). We call C dirty, otherwise. Finite MGL-frames have only clean E-clusters. Consider (MCas) ∀((p → ∀p) → p) → ∀p. Lemma

1 A finite MIPC-frame validates MCas iff all its E-clusters are clean. 2 A finite MGrz-frame validates t(MCas) iff all its E-clusters are clean.

Let M+IPC = MIPC + MCas and let M+Grz = MGrz + t(MCas). Note that MGL ⊢ sp(t(MCas)), thus “MGL = M+GL”.

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SLIDE 69

Arithmetic interpretation of the monadic fragment of intuitionistic predicate logic and Casari’s formula

The finite model property

Theorem M+IPC has the fmp and so does M+Grz.

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SLIDE 70

Arithmetic interpretation of the monadic fragment of intuitionistic predicate logic and Casari’s formula

The finite model property

Theorem M+IPC has the fmp and so does M+Grz. The proof is via selective filtration similar to that of MIPC due to (Grefe 1998).

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SLIDE 71

Arithmetic interpretation of the monadic fragment of intuitionistic predicate logic and Casari’s formula

The finite model property

Theorem M+IPC has the fmp and so does M+Grz. The proof is via selective filtration similar to that of MIPC due to (Grefe 1998). Let us concentrate on M+IPC.

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SLIDE 72

Arithmetic interpretation of the monadic fragment of intuitionistic predicate logic and Casari’s formula

The finite model property

Theorem M+IPC has the fmp and so does M+Grz. The proof is via selective filtration similar to that of MIPC due to (Grefe 1998). Let us concentrate on M+IPC. From a descriptive refutation frame (dual of a monadic Heyting algebra) we select a finite refutation frame.

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SLIDE 73

Arithmetic interpretation of the monadic fragment of intuitionistic predicate logic and Casari’s formula

The finite model property

Theorem M+IPC has the fmp and so does M+Grz. The proof is via selective filtration similar to that of MIPC due to (Grefe 1998). Let us concentrate on M+IPC. From a descriptive refutation frame (dual of a monadic Heyting algebra) we select a finite refutation frame. Note that descriptive M+IPC-frames may have dirty clusters but the clusters in the maximum of an E-saturated clopen set are always clean.

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SLIDE 74

Arithmetic interpretation of the monadic fragment of intuitionistic predicate logic and Casari’s formula

Proof sketch

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SLIDE 75

Arithmetic interpretation of the monadic fragment of intuitionistic predicate logic and Casari’s formula

Proof sketch

Suppose F, v, x | = ϕ, where F, v is a model based on a descriptive M+IPC-frame; w.l.o.g. the E-cluster of x is clean).

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SLIDE 76

Arithmetic interpretation of the monadic fragment of intuitionistic predicate logic and Casari’s formula

Proof sketch

Suppose F, v, x | = ϕ, where F, v is a model based on a descriptive M+IPC-frame; w.l.o.g. the E-cluster of x is clean). We construct a finite frame G refuting ϕ in several rounds.

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SLIDE 77

Arithmetic interpretation of the monadic fragment of intuitionistic predicate logic and Casari’s formula

Proof sketch

Suppose F, v, x | = ϕ, where F, v is a model based on a descriptive M+IPC-frame; w.l.o.g. the E-cluster of x is clean). We construct a finite frame G refuting ϕ in several rounds. Each t ∈ G is associated with some t ∈ F.

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SLIDE 78

Arithmetic interpretation of the monadic fragment of intuitionistic predicate logic and Casari’s formula

Proof sketch

Suppose F, v, x | = ϕ, where F, v is a model based on a descriptive M+IPC-frame; w.l.o.g. the E-cluster of x is clean). We construct a finite frame G refuting ϕ in several rounds. Each t ∈ G is associated with some t ∈ F. Goal: t | = ψ iff t | = ψ for all ψ ∈ Sub(ϕ).

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SLIDE 79

Arithmetic interpretation of the monadic fragment of intuitionistic predicate logic and Casari’s formula

Proof sketch

Suppose F, v, x | = ϕ, where F, v is a model based on a descriptive M+IPC-frame; w.l.o.g. the E-cluster of x is clean). We construct a finite frame G refuting ϕ in several rounds. Each t ∈ G is associated with some t ∈ F. Goal: t | = ψ iff t | = ψ for all ψ ∈ Sub(ϕ). For t ∈ G consider the sets Σ∃(t) = {∃δ ∈ Sub(ϕ) : t ∃δ} Σ∀H(t) = {∀β ∈ Sub(ϕ) : t is maximal wrt ∀β} Σ∀V (t) = {∀γ ∈ Sub(ϕ) : t ∀γ but is not maximal wrt ∀γ} Σ→(t) = {α → σ ∈ Sub(ϕ) : t α → σ, t α}

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SLIDE 80

Arithmetic interpretation of the monadic fragment of intuitionistic predicate logic and Casari’s formula

Proof sketch (continued)

We add copies of witnesses for the formulas in the sets above, and we add points to ensure commutativity.

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SLIDE 81

Arithmetic interpretation of the monadic fragment of intuitionistic predicate logic and Casari’s formula

Proof sketch (continued)

We add copies of witnesses for the formulas in the sets above, and we add points to ensure commutativity. We only add points t to G if t is from a clean cluster.

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SLIDE 82

Arithmetic interpretation of the monadic fragment of intuitionistic predicate logic and Casari’s formula

Proof sketch (continued)

We add copies of witnesses for the formulas in the sets above, and we add points to ensure commutativity. We only add points t to G if t is from a clean cluster. Problematic case: Finding the right witnesses for formulas in Σ→(t). Here we may introduce R-arrows in G coming from original Q-arrows. (Here our proof differs from that of Grefe.)

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SLIDE 83

Arithmetic interpretation of the monadic fragment of intuitionistic predicate logic and Casari’s formula

Arithmetic interpretation of M+IPC

Theorem For every formula ϕ of MIPC, M+IPC ⊢ ϕ iff M+Grz ⊢ t(ϕ) iff MGL ⊢ sp(t(ϕ)).

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SLIDE 84

Arithmetic interpretation of the monadic fragment of intuitionistic predicate logic and Casari’s formula

Arithmetic interpretation of M+IPC

Theorem For every formula ϕ of MIPC, M+IPC ⊢ ϕ iff M+Grz ⊢ t(ϕ) iff MGL ⊢ sp(t(ϕ)). Since Solovay’s Theorem extends to MGL, we get an arithmetic interpretation of M+IPC.

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SLIDE 85

Arithmetic interpretation of the monadic fragment of intuitionistic predicate logic and Casari’s formula

Arithmetic interpretation of M+IPC

Theorem For every formula ϕ of MIPC, M+IPC ⊢ ϕ iff M+Grz ⊢ t(ϕ) iff MGL ⊢ sp(t(ϕ)). Since Solovay’s Theorem extends to MGL, we get an arithmetic interpretation of M+IPC.

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SLIDE 86

Arithmetic interpretation of the monadic fragment of intuitionistic predicate logic and Casari’s formula

One variable fragments of predicate logics

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SLIDE 87

Arithmetic interpretation of the monadic fragment of intuitionistic predicate logic and Casari’s formula

One variable fragments of predicate logics

For a formula ϕ of MIPC define a translation Ψ to QIPC by

Ψ(p) = P(x), for each prop. letter p and a unary predicate P(x), Ψ(ϕ ◦ ψ) = Ψ(ϕ) ◦ Ψ(ψ) for ◦ ∈ {∧, ∨, →} Ψ(∀ϕ) = ∀xΨ(ϕ), Ψ(∃ϕ) = ∃xΨ(ϕ),

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SLIDE 88

Arithmetic interpretation of the monadic fragment of intuitionistic predicate logic and Casari’s formula

One variable fragments of predicate logics

For a formula ϕ of MIPC define a translation Ψ to QIPC by

Ψ(p) = P(x), for each prop. letter p and a unary predicate P(x), Ψ(ϕ ◦ ψ) = Ψ(ϕ) ◦ Ψ(ψ) for ◦ ∈ {∧, ∨, →} Ψ(∀ϕ) = ∀xΨ(ϕ), Ψ(∃ϕ) = ∃xΨ(ϕ),

Let MIPC ⊆ L be an intuitionistic bi-modal logic and let QIPC ⊆ S an intuitionistic predicate logic. L is the one-variable fragment of S iff for all ϕ of MIPC L ⊢ ϕ iff S ⊢ Ψ(ϕ)

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SLIDE 89

Arithmetic interpretation of the monadic fragment of intuitionistic predicate logic and Casari’s formula

M+IPC is the one-variable fragment of Q+IPC

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SLIDE 90

Arithmetic interpretation of the monadic fragment of intuitionistic predicate logic and Casari’s formula

M+IPC is the one-variable fragment of Q+IPC

(Ono and Suzuki 1988) identify a criterion to detect whether L is the one-variable fragment of S.

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SLIDE 91

Arithmetic interpretation of the monadic fragment of intuitionistic predicate logic and Casari’s formula

M+IPC is the one-variable fragment of Q+IPC

(Ono and Suzuki 1988) identify a criterion to detect whether L is the one-variable fragment of S. Let (Cas) ∀x[(P(x) → ∀xP(x)) → ∀xP(x)] → ∀xP(x).

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SLIDE 92

Arithmetic interpretation of the monadic fragment of intuitionistic predicate logic and Casari’s formula

M+IPC is the one-variable fragment of Q+IPC

(Ono and Suzuki 1988) identify a criterion to detect whether L is the one-variable fragment of S. Let (Cas) ∀x[(P(x) → ∀xP(x)) → ∀xP(x)] → ∀xP(x). Let Q+IPC = QIPC + Cas.

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SLIDE 93

Arithmetic interpretation of the monadic fragment of intuitionistic predicate logic and Casari’s formula

M+IPC is the one-variable fragment of Q+IPC

(Ono and Suzuki 1988) identify a criterion to detect whether L is the one-variable fragment of S. Let (Cas) ∀x[(P(x) → ∀xP(x)) → ∀xP(x)] → ∀xP(x). Let Q+IPC = QIPC + Cas. Corollary M+IPC is the one-variable fragment of Q+IPC.

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SLIDE 94

Arithmetic interpretation of the monadic fragment of intuitionistic predicate logic and Casari’s formula

Thank you!

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