Some variations of

de Jongh’s theorem 2017.07.15 Workshop in Logic and Philosophy of Mathematics

Waseda University

Naosuke Matsuda Kanagawa Univerity

Motivation and abstract

Motivation and Abstract

■Some variations of de Jongh’s theorem.

[Smorinsky1973]

■Some variations of arithmetical completeness for logic of proofs.

[Iwata-Kurahashi2017]

■Arithmetical completeness for intuitionistic Logic of Proofs.

[Artemov-Iemhoff2007, Dashkov2009]

■Search for a modal logic which captures the provability predicate in Heyting Arithmetic. [Artemov-Beklemishev2005]

Motivation and Abstract

■Some variations of de Jongh’s theorem.

[Smorinsky1973]

■Some variations of arithmetical completeness for logic of proofs.

[Iwata-Kurahashi2017]

■Arithmetical completeness for intuitionistic Logic of Proofs.

[Artemov-Iemhoff2007, Dashkov2009]

■Search for a modal logic which captures the provability predicate in Heyting Arithmetic. [Artemov-Beklemishev2005]

Preliminary

Kripke semantics

Heyting arithmetic

Notations

Heyting Arithmetic

AA : the set of Axioms of (Peano) Arithmetic

Peano Arithmetic (PA) :

Classical Logic (CL) + AA

PA ⊢ 𝛼 ⇔ AA ⊢CL 𝛼

Heyting Arithmetic (HA) :

Intuitionistic Logic (IL) + AA

HA ⊢ 𝛼 ⇔ AA ⊢IL 𝛼

An arithmetical substitution is a mapping which assigns a arithmetical sentence to a propositional variable.

Kripke semantics

Kripke model 𝒦 = ⟨𝑊,≤, {ℳ𝑤}𝑤∈𝑊⟩ :

⟨𝑊,≤⟩ is a partial order

each ℳ𝑤 is a (classical) model s.t.

• 𝑤 ≤ 𝑣 ⇒ 𝐷𝑜𝑚(ℳ𝑤) ⊆ 𝐷𝑜𝑚(ℳ𝑣)

• 𝑤 ≤ 𝑣 and ℳ𝑤 ⊨ 𝛼 ⇒ ℳ𝑣 ⊨ 𝛼 for each (ℳ𝑤-)sentence 𝛼

𝒦,𝑤 ⊩ 𝛼 :

𝒦,𝑤 ⊩ 𝛼 ⇔ ℳ𝑣 ⊨ 𝛼 if 𝛼 is atomic (ℳ𝑣-)sectence

𝒦,𝑤 ⊩ 𝛼 ∧ ∨ / → 𝛽 ⇔ 𝒦, 𝑣 ⊩ 𝛼 and (or / implies) 𝒦, 𝑣 ⊩ 𝛽 for each 𝑣 ≥ 𝑤

𝒦,𝑤 ⊩ ∀(∃)𝑥𝛼 ⇔ 𝒦, 𝑣 ⊩ 𝑥 ≔ 𝑒 𝛼,

for each (some) 𝑒 ∈ 𝐷𝑜𝑚 ℳ𝑣 , for each 𝑣 ≥ 𝑤

𝒦 ⊩ 𝛼 ⇔ 𝒦,𝑤 ⊩ 𝛼 for each 𝑤 ∈ 𝑊

Γ ⊩ 𝛼 ⇔ 𝒦 ⊩ 𝛼 for each 𝒦 s.t. 𝒦 ⊩ 𝛾 for each 𝛾 ∈ Γ

Thm [Kripke 1965]

Γ ⊢𝐼𝐿 𝛼 iff Γ ⊩ 𝛼.

Notations

𝜑 : propositional formula.

CPL : Classical Propositional Logic.

IPL : Intuitionistic Propositional Logic.

ℕ : standard PA model.

Introduction to de Jongh’s theorem

CPL vs PA

IPL vs HA

CPL vs PA

Thm1 If ⊢𝐶𝑃𝐿 𝜑, then 𝑃𝐴 ⊢ 𝜎(𝜑) for each arithmetical substitution 𝜎.

→ very easy.

Thm2 If ⊬𝐶𝑃𝐿 𝜑, then 𝑃𝐴 ⊬ 𝜎 𝜑 for some arithmetical substitution 𝜎.

Proof If ⊬𝐶𝑃𝐿 𝜑, then ℳ ⊭ 𝜑 for some model ℳ.

Let 𝜎 be the following arithmetical substitution:

𝜎 𝑝 = 0 = 0 if ℳ ⊨ 𝑝0 ≠ 0 if ℳ ⊭ 𝑝

,

then we have ℕ ⊭ 𝜑.

ℳ ⊨ 𝑝1, 𝑝2, 𝑝4

ℳ ⊭ 𝑝3

ℳ

ℕ ⊨ 0 = 0

ℕ ⊭ 0 ≠ 0

ℕ

CPL vs PA

Thm2 If ⊬𝐶𝑃𝐿 𝜑, then 𝑃𝐴 ⊬ 𝜎 𝜑 for some arithmetical substitution 𝜎.

Thm2+ There exists an arithmetical substitution 𝜎 s.t. if ⊬𝐶𝑃𝐿 𝜑, then 𝑃𝐴 ⊬ 𝜎 𝜑 .

↑

Lemma [Myhill1972]

There are Σ1-sentences 𝑆1, 𝑆2, … s.t.

if 𝑆𝑖′ ≡ 𝑆𝑖 or ¬𝑆𝑖 then 𝐴𝐴 ∪ {𝑆1

′ , 𝑆2′ , … } has a model.

𝜎 𝑝𝑖 = 𝑆𝑖 :

ℳ ⊨ 𝑝1, 𝑝2, 𝑝4

ℳ ⊭ 𝑝3

ℳ ⊭ 𝜑 𝑝1, 𝑝2, 𝑝3, 𝑝4

𝒩 ⊨ 𝑆1, 𝑆2, 𝑆4

𝒩 ⊭ 𝑆3

Σ1

𝒩 ⊭ 𝜑 𝑆1, 𝑆2, 𝑆3, 𝑆4

CPL vs PA

Q If ⊬𝐼𝑃𝐿 𝜑, then does 𝐻𝐴 ⊬ 𝜎 𝜑 hold for some 𝜎 ?

↔ Can we find the following 𝑆1, 𝑆2, … and ℒ ?

Q+ Is there an arithmetical substitution 𝜎 s.t. if ⊬𝐼𝑃𝐿 𝜑, then 𝐻𝐴 ⊬ 𝜎 𝜑 ?

ℳ ⊨ 𝑝2, 𝑝4

ℳ ⊭ 𝑝1, 𝑝3

ℳ1

𝒩 ⊨ 𝑆2, 𝑆4

𝒩 ⊭ 𝑆1, 𝑆3

𝒩1

ℳ ⊨ 𝑝1, 𝑝2, 𝑝4

ℳ ⊭ 𝑝3

ℳ2

𝒩 ⊨ 𝑆1, 𝑆2, 𝑆4

𝒩 ⊭ 𝑆3

𝒩1

𝒦 ⊮ 𝜑 𝑝1, 𝑝2, 𝑝3, 𝑝4 ℒ ⊮ 𝜑 𝑆1, 𝑆2, 𝑆3, 𝑆4

IPL vs HA

Q If ⊬𝐼𝑃𝐿 𝜑, then does 𝐻𝐴 ⊬ 𝜎 𝜑 hold for some 𝜎 ?

A Yes! There is a substitution.

(Weak version of de Jongh’s theorem / Arithmetical completeness)

Q+ Is there an arithmetical substitution 𝜎 s.t. if ⊬𝐼𝑃𝐿 𝜑, then 𝐻𝐴 ⊬ 𝜎 𝜑 ?

A Yes! There is a uniform substitution.

(de Jongh’s theorem / Uniform version of arithmetical completeness)

Π2

Some variants of de Jongh’s theorem

Proof of (weak) de Jongh’s theorem

Strong verision

de Jongh’s theorem for one

propositional variable

Proof of (weak) de Jongh’s theorem

Thm If ⊬𝐼𝑃𝐿 𝜑, then 𝐻𝐴 ⊬ 𝜎 𝜑 for some arithmetical substitution 𝜎.

Lemma [Jaskowski1936,Smorinski1973]

If ⊬𝐼𝑃𝐿 𝜑, then 𝒦 ⊮ 𝜑 for some 𝒦 based on either of the following tree:

Lemma [Myhill1972]

There are Σ1-sentences 𝑆1, 𝑆2, … s.t.

if 𝑆𝑖′ ≡ 𝑆𝑖 or ¬𝑆𝑖 then 𝐴𝐴 ∪ {𝑆1

′ , 𝑆2′ , … } has a PA model.

…

Proof of (weak) de Jongh’s theorem

Thm If ⊬𝐼𝑃𝐿 𝜑, then 𝐻𝐴 ⊬ 𝜎 𝜑 for some arithmetical substitution 𝜎.

ℳ0

ℳ12 ℳ11 ℳ13

ℳ21 ℳ22 ℳ23 ℳ24 ℳ25 ℳ26

𝒦 ⊮ 𝜑 𝑝1, 𝑝2

𝑝2

𝑝1

ℕ

ℕ ℕ ℕ

𝒯1 𝒯2 𝒯3 𝒯4 𝒯5 𝒯6

ℒ ⊮ 𝜑 𝐴1, 𝐴2

𝐴1 ≡ ¬𝑆1 ∧ ¬𝑆3 ∧ ¬𝑆4 ∧ ¬𝑆5 ∧ ¬𝑆6

𝐴2 ≡ ¬𝑆1 ∧ ¬𝑆2 ∧ ¬𝑆3 ∧ ¬𝑆4

𝒯𝑖⊨ 𝑆𝑖

but 𝒯𝑖⊭ 𝑆𝑗

Strong version

Thm If ⊬𝐼𝑃𝐿 𝜑, then 𝐻𝐴 ⊬ 𝜎 𝜑 for some arithmetical substitution 𝜎.

Thm+ If ⊬𝐼𝑃𝐿 𝜑, then 𝐻𝐴 ⊬ 𝜎 𝜑 for some arithmetical Σ1-substitution 𝜎.

Thm++ There exists an arithmetical Π2-substitution 𝜎 s.t. ⊬𝐼𝑃𝐿 𝜑 implies 𝐻𝐴 ⊢ 𝜎 𝜑 .

Thm+++ Let 𝑆1, 𝑆2, … be arbitrary Π2-sentences, independent over 𝑃𝐴 + {true Π1-sentences}, and let 𝜎 𝑝𝑖 = 𝑆𝑖. Then ⊬𝐼𝑃𝐿 𝜑 implies 𝐻𝐴 ⊢ 𝜎 𝜑 .

de Jongh’s theorem for one propositional variable

Thm

𝜑 𝑝 : propositional formula consisting of only one propositional variable 𝑝.

𝑆 : arbitrary Σ1-sentence independent over 𝑃𝐴.

⇒ ⊬𝐼𝑃𝐿 𝜑 implies 𝐻𝐴 ⊬ 𝜎 𝑆 .

Lemma [Nishimura1960]

𝜑 𝑝 : propositional formula consisting

of only one propositional variable 𝑝.

⊬𝐼𝑃𝐿 𝜑 𝑝 .

⇒ 𝒩 ⊮ 𝜑 𝑝 𝒩 :

ℳ

ℳ

ℳ

ℳ′

ℳ

ℳ

…

…

ℳ′ ⊨ 𝑝 ℳ ⊭ 𝑝

de Jongh’s theorem for one propositional variable

Thm

𝜑 𝑝 : propositional formula consisting of only one propositional variable 𝑝.

𝑆 : arbitrary Σ1-sentence independent over 𝑃𝐴.

⇒ ⊬𝐼𝑃𝐿 𝜑 implies 𝐻𝐴 ⊬ 𝜎 𝑆 .

Proof

ℳ

ℳ

ℳ

ℳ′

ℳ

ℳ

…

…

ℳ′ ⊨ 𝑝 ℳ ⊭ 𝑝

ℕ

ℕ

ℕ

𝒯

ℕ

ℕ

…

…

𝒯 ⊨ 𝑆 ℕ ⊭ 𝑆

Summary

Summary

■Some variations of de Jongh’s theorem.

[Smorinsky1973]

■Some variations of arithmetical completeness for logic of proofs.

[Iwata-Kurahashi2017]

■Arithmetical completeness for intuitionistic Logic of Proofs.

[Artemov-Iemhoff2007, Dashkov2009]

■Search for a modal logic which captures the provability predicate in Heyting Arithmetic. [Artemov-Beklemishev2005]

References

References

[Artemov-Iemhoff2007] ARTEMOV, Sergei; IEMHOFF, Rosalie. The basic intuitionistic logic of proofs. The Journal of Symbolic Logic, 2007, 72.2: 439-451.

[Artemov-Beklemishev2005] ARTEMOV, Sergei N.; BEKLEMISHEV, Lev D. Provability logic. In: Handbook of Philosophical Logic, 2nd Edition. Springer Netherlands, 2005. p. 189-360.

[Dashkov2009] DASHKOV, Evgenij. Arithmetical completeness of the intuitionistic logic of proofs. Journal of Logic and Computation, 2009, 21.4: 665-682.

[Jaskowski1936] Jaśkowski, Stanisław. Recherches sur le syst~me de la logique intuitionniste. Actes du congres international de philosophie scientifique, Philosophie des mathematiques, 1936.

[Iwata-Kurahashi2017] SOHEI, Iwata; TAISHI, Kurahashi. Arithmetical completeness theorem for Logic of proofs. Talk in the spring meeting of the Mathematical Society of Japan, 2017.

References

[Kripke 1965] KRIPKE, Saul A. Semantical analysis of intuitionistic logic I. Studies in Logic and the Foundations of Mathematics, 1965, 40: 92-130.

[Myhill 1972] MYHILL, John. An Absolutely Independent Set of Σ10-

Sentences. Mathematical Logic Quarterly, 1972, 18.7: 107-109.

[Nishimura1960] NISHIMURA, Iwao. On formulas of one variable in intuitionistic propositional calculus. The Journal of Symbolic Logic, 1960, 25.4: 327-331.

[Smorynski1973] SMORYNSKI, Craig Alan. Applications of Kripke models. In: Metamathematical investigation of intuitionistic arithmetic and analysis. Springer, Berlin, Heidelberg, 1973. p. 324-391.