gentzen style modal logics
TRANSCRIPT
Gentzen StyleModal Logics
Andrew Parisi
Outline
Motivation
SequentCalculus
Pk
The SystemPS4
The SystemPS5
Hypersequents
The SystemPHS5
Quantification
Gentzen Style Modal Logics
Andrew Parisi
University of Connecticut
May 24, 2013
Gentzen StyleModal Logics
Andrew Parisi
Outline
Motivation
SequentCalculus
Pk
The SystemPS4
The SystemPS5
Hypersequents
The SystemPHS5
Quantification
1 Motivation
2 Sequent Calculus
3 Pk
4 The System PS4
5 The System PS5
6 Hypersequents
7 The System PHS5
8 Quantification
Gentzen StyleModal Logics
Andrew Parisi
Outline
Motivation
SequentCalculus
Pk
The SystemPS4
The SystemPS5
Hypersequents
The SystemPHS5
Quantification
Motivations
See what can be done in modal logic without mentioningpossible worlds.
Explore the Barcan formulas.
Explore cut elimination for modal systems.
See if there are any interesting overlaps between systemsof modal logic, the Barcan formulas, and Cut.
Gentzen StyleModal Logics
Andrew Parisi
Outline
Motivation
SequentCalculus
Pk
The SystemPS4
The SystemPS5
Hypersequents
The SystemPHS5
Quantification
Motivations
See what can be done in modal logic without mentioningpossible worlds.
Explore the Barcan formulas.
Explore cut elimination for modal systems.
See if there are any interesting overlaps between systemsof modal logic, the Barcan formulas, and Cut.
Gentzen StyleModal Logics
Andrew Parisi
Outline
Motivation
SequentCalculus
Pk
The SystemPS4
The SystemPS5
Hypersequents
The SystemPHS5
Quantification
Motivations
See what can be done in modal logic without mentioningpossible worlds.
Explore the Barcan formulas.
Explore cut elimination for modal systems.
See if there are any interesting overlaps between systemsof modal logic, the Barcan formulas, and Cut.
Gentzen StyleModal Logics
Andrew Parisi
Outline
Motivation
SequentCalculus
Pk
The SystemPS4
The SystemPS5
Hypersequents
The SystemPHS5
Quantification
Motivations
See what can be done in modal logic without mentioningpossible worlds.
Explore the Barcan formulas.
Explore cut elimination for modal systems.
See if there are any interesting overlaps between systemsof modal logic, the Barcan formulas, and Cut.
Gentzen StyleModal Logics
Andrew Parisi
Outline
Motivation
SequentCalculus
Pk
The SystemPS4
The SystemPS5
Hypersequents
The SystemPHS5
Quantification
Sequent Calculus
Sequents, Γ⇒ Σ, are ordered pairs of sets of sentences, Γ,Σ.
I will often write Γ ∪ Σ as ΓΣ.
A deduction is a series of sequents wherein each sequent iseither an axiom, or obtained from the previous by anapplication of a rule.
Deductions are restricted to being finite constructions.
Gentzen StyleModal Logics
Andrew Parisi
Outline
Motivation
SequentCalculus
Pk
The SystemPS4
The SystemPS5
Hypersequents
The SystemPHS5
Quantification
Sequent Calculus
Sequents, Γ⇒ Σ, are ordered pairs of sets of sentences, Γ,Σ.
I will often write Γ ∪ Σ as ΓΣ.
A deduction is a series of sequents wherein each sequent iseither an axiom, or obtained from the previous by anapplication of a rule.
Deductions are restricted to being finite constructions.
Gentzen StyleModal Logics
Andrew Parisi
Outline
Motivation
SequentCalculus
Pk
The SystemPS4
The SystemPS5
Hypersequents
The SystemPHS5
Quantification
Sequent Calculus
Sequents, Γ⇒ Σ, are ordered pairs of sets of sentences, Γ,Σ.
I will often write Γ ∪ Σ as ΓΣ.
A deduction is a series of sequents wherein each sequent iseither an axiom, or obtained from the previous by anapplication of a rule.
Deductions are restricted to being finite constructions.
Gentzen StyleModal Logics
Andrew Parisi
Outline
Motivation
SequentCalculus
Pk
The SystemPS4
The SystemPS5
Hypersequents
The SystemPHS5
Quantification
Sequent Calculus
Sequents, Γ⇒ Σ, are ordered pairs of sets of sentences, Γ,Σ.
I will often write Γ ∪ Σ as ΓΣ.
A deduction is a series of sequents wherein each sequent iseither an axiom, or obtained from the previous by anapplication of a rule.
Deductions are restricted to being finite constructions.
Gentzen StyleModal Logics
Andrew Parisi
Outline
Motivation
SequentCalculus
Pk
The SystemPS4
The SystemPS5
Hypersequents
The SystemPHS5
Quantification
Sequent Calculus (Cont.)
Most of the work in this system is done with proof rules.
There are rules for the connectives and structural rules.
Each connective has a right and a left rule, forintroduction into the right or left of a sequent.
Gentzen StyleModal Logics
Andrew Parisi
Outline
Motivation
SequentCalculus
Pk
The SystemPS4
The SystemPS5
Hypersequents
The SystemPHS5
Quantification
Sequent Calculus (Cont.)
Most of the work in this system is done with proof rules.
There are rules for the connectives and structural rules.
Each connective has a right and a left rule, forintroduction into the right or left of a sequent.
Gentzen StyleModal Logics
Andrew Parisi
Outline
Motivation
SequentCalculus
Pk
The SystemPS4
The SystemPS5
Hypersequents
The SystemPHS5
Quantification
Sequent Calculus (Cont.)
Most of the work in this system is done with proof rules.
There are rules for the connectives and structural rules.
Each connective has a right and a left rule, forintroduction into the right or left of a sequent.
Gentzen StyleModal Logics
Andrew Parisi
Outline
Motivation
SequentCalculus
Pk
The SystemPS4
The SystemPS5
Hypersequents
The SystemPHS5
Quantification
Propositional Logic, Pc
Axioms:
AxΓ, ϕ⇒ ϕ,Σ
L⊥Γ,⊥ ⇒ Σ
→ Rules:
Γ, ϕ⇒ Σ Γ′ ⇒ ψ,Σ′L→
ΓΓ′, ψ → ϕ⇒ ΣΣ′
Γ, ψ ⇒ ϕ,ΣR→
Γ⇒ ψ → ϕ,Σ
Gentzen StyleModal Logics
Andrew Parisi
Outline
Motivation
SequentCalculus
Pk
The SystemPS4
The SystemPS5
Hypersequents
The SystemPHS5
Quantification
Propositional Logic, Pc
Axioms:
AxΓ, ϕ⇒ ϕ,Σ
L⊥Γ,⊥ ⇒ Σ
→ Rules:
Γ, ϕ⇒ Σ Γ′ ⇒ ψ,Σ′L→
ΓΓ′, ψ → ϕ⇒ ΣΣ′
Γ, ψ ⇒ ϕ,ΣR→
Γ⇒ ψ → ϕ,Σ
Gentzen StyleModal Logics
Andrew Parisi
Outline
Motivation
SequentCalculus
Pk
The SystemPS4
The SystemPS5
Hypersequents
The SystemPHS5
Quantification
Propositional Logic, Pc
¬ Rules:
Γ⇒ ϕ,ΣL¬
Γ,¬ϕ,⇒ Σ
Γ, ϕ⇒ ΣR¬
Γ⇒ ¬ϕ,Σ
Structural Rules:
Γ⇒ ΣLWeak
Γ, ϕ⇒ ΣΓ⇒ Σ
RWeakΓ⇒ ϕ,Σ
Γ, ϕ⇒ Σ Γ′ ⇒ ϕ,Σ′Cut
ΓΓ′ ⇒ ΣΣ′
Gentzen StyleModal Logics
Andrew Parisi
Outline
Motivation
SequentCalculus
Pk
The SystemPS4
The SystemPS5
Hypersequents
The SystemPHS5
Quantification
Propositional Logic, Pc
¬ Rules:
Γ⇒ ϕ,ΣL¬
Γ,¬ϕ,⇒ Σ
Γ, ϕ⇒ ΣR¬
Γ⇒ ¬ϕ,Σ
Structural Rules:
Γ⇒ ΣLWeak
Γ, ϕ⇒ ΣΓ⇒ Σ
RWeakΓ⇒ ϕ,Σ
Γ, ϕ⇒ Σ Γ′ ⇒ ϕ,Σ′Cut
ΓΓ′ ⇒ ΣΣ′
Gentzen StyleModal Logics
Andrew Parisi
Outline
Motivation
SequentCalculus
Pk
The SystemPS4
The SystemPS5
Hypersequents
The SystemPHS5
Quantification
System Pk
Gentzen StyleModal Logics
Andrew Parisi
Outline
Motivation
SequentCalculus
Pk
The SystemPS4
The SystemPS5
Hypersequents
The SystemPHS5
Quantification
Modal Logic: Pk
Pk is the system Pc extended by the rule: �.
Γ⇒ ϕ,Σ�
�Γ⇒ �ϕ,♦Σ
Where �∆ = {�δ|δ ∈ ∆}, similarly for other connectivesand sets.
The � rule is based off of Poggiolesi [5].
Gentzen StyleModal Logics
Andrew Parisi
Outline
Motivation
SequentCalculus
Pk
The SystemPS4
The SystemPS5
Hypersequents
The SystemPHS5
Quantification
Modal Logic: Pk
Pk is the system Pc extended by the rule: �.
Γ⇒ ϕ,Σ�
�Γ⇒ �ϕ,♦Σ
Where �∆ = {�δ|δ ∈ ∆}, similarly for other connectivesand sets.
The � rule is based off of Poggiolesi [5].
Gentzen StyleModal Logics
Andrew Parisi
Outline
Motivation
SequentCalculus
Pk
The SystemPS4
The SystemPS5
Hypersequents
The SystemPHS5
Quantification
Modal Logic: Pk
Pk is the system Pc extended by the rule: �.
Γ⇒ ϕ,Σ�
�Γ⇒ �ϕ,♦Σ
Where �∆ = {�δ|δ ∈ ∆}, similarly for other connectivesand sets.
The � rule is based off of Poggiolesi [5].
Gentzen StyleModal Logics
Andrew Parisi
Outline
Motivation
SequentCalculus
Pk
The SystemPS4
The SystemPS5
Hypersequents
The SystemPHS5
Quantification
Modal Logic: Pk
Pk is the system Pc extended by the rule: �.
Γ⇒ ϕ,Σ�
�Γ⇒ �ϕ,♦Σ
Where �∆ = {�δ|δ ∈ ∆}, similarly for other connectivesand sets.
The � rule is based off of Poggiolesi [5].
Gentzen StyleModal Logics
Andrew Parisi
Outline
Motivation
SequentCalculus
Pk
The SystemPS4
The SystemPS5
Hypersequents
The SystemPHS5
Quantification
System K is the same as system Pk
The axiomatic System K licenses this rule.
Following Dosen [2] we will use a translation:
1 t(ϕ) = ϕ2 t(Γ) = t(γ1) ∧ . . . ∧ t(γn), where γi ∈ Γ.3 t(Σ) = t(σ1) ∨ . . . ∨ t(σn), where σi ∈ Σ.4 t(Γ⇒ Σ) = t(Γ)→ t(Σ).
Gentzen StyleModal Logics
Andrew Parisi
Outline
Motivation
SequentCalculus
Pk
The SystemPS4
The SystemPS5
Hypersequents
The SystemPHS5
Quantification
System K is the same as system Pk
The axiomatic System K licenses this rule.
Following Dosen [2] we will use a translation:
1 t(ϕ) = ϕ2 t(Γ) = t(γ1) ∧ . . . ∧ t(γn), where γi ∈ Γ.3 t(Σ) = t(σ1) ∨ . . . ∨ t(σn), where σi ∈ Σ.4 t(Γ⇒ Σ) = t(Γ)→ t(Σ).
Gentzen StyleModal Logics
Andrew Parisi
Outline
Motivation
SequentCalculus
Pk
The SystemPS4
The SystemPS5
Hypersequents
The SystemPHS5
Quantification
System K is the same as system Pk
The axiomatic System K licenses this rule.
Following Dosen [2] we will use a translation:
1 t(ϕ) = ϕ2 t(Γ) = t(γ1) ∧ . . . ∧ t(γn), where γi ∈ Γ.3 t(Σ) = t(σ1) ∨ . . . ∨ t(σn), where σi ∈ Σ.4 t(Γ⇒ Σ) = t(Γ)→ t(Σ).
Gentzen StyleModal Logics
Andrew Parisi
Outline
Motivation
SequentCalculus
Pk
The SystemPS4
The SystemPS5
Hypersequents
The SystemPHS5
Quantification
System K is the same system as Pk (Cont.)
Pk is the sequent calculus formulation of the axiomaticSystem K.
System K licenses the � as an inference:
t(Γ⇒ ϕ,Σ) ` t(�Γ⇒ �ϕ,♦Σ).
Gentzen StyleModal Logics
Andrew Parisi
Outline
Motivation
SequentCalculus
Pk
The SystemPS4
The SystemPS5
Hypersequents
The SystemPHS5
Quantification
System K is the same system as Pk (Cont.)
Pk is the sequent calculus formulation of the axiomaticSystem K.
System K licenses the � as an inference:
t(Γ⇒ ϕ,Σ) ` t(�Γ⇒ �ϕ,♦Σ).
Gentzen StyleModal Logics
Andrew Parisi
Outline
Motivation
SequentCalculus
Pk
The SystemPS4
The SystemPS5
Hypersequents
The SystemPHS5
Quantification
System K is the same system as Pk (Cont.)
Pk is the sequent calculus formulation of the axiomaticSystem K.
System K licenses the � as an inference:
t(Γ⇒ ϕ,Σ) ` t(�Γ⇒ �ϕ,♦Σ).
Gentzen StyleModal Logics
Andrew Parisi
Outline
Motivation
SequentCalculus
Pk
The SystemPS4
The SystemPS5
Hypersequents
The SystemPHS5
Quantification
System K is the same system as Pk (Cont.)
Pk deduces System K:
Proof of Nec:⇒ ϕ
� ⇒ �ϕ
Proof of Dist:
ϕ⇒ ϕ ψ ⇒ ψ
ϕ→ ψ,ϕ⇒ ψ
�(ϕ→ ψ),�ϕ⇒ �ψ
�(ϕ→ ψ)⇒ �ϕ→ �ψ
⇒ �(ϕ→ ψ)→ (�ϕ→ �ψ)
Gentzen StyleModal Logics
Andrew Parisi
Outline
Motivation
SequentCalculus
Pk
The SystemPS4
The SystemPS5
Hypersequents
The SystemPHS5
Quantification
System K is the same system as Pk (Cont.)
Pk deduces System K:
Proof of Nec:⇒ ϕ
� ⇒ �ϕProof of Dist:
ϕ⇒ ϕ ψ ⇒ ψ
ϕ→ ψ,ϕ⇒ ψ
�(ϕ→ ψ),�ϕ⇒ �ψ
�(ϕ→ ψ)⇒ �ϕ→ �ψ
⇒ �(ϕ→ ψ)→ (�ϕ→ �ψ)
Gentzen StyleModal Logics
Andrew Parisi
Outline
Motivation
SequentCalculus
Pk
The SystemPS4
The SystemPS5
Hypersequents
The SystemPHS5
Quantification
Cut Elimination
Cut Elimination Theorem:
If there is a deduction of Γ⇒ Σ in Pk , then there is adeduction of Γ⇒ Σ in Pk that does not use the Cut rule.
Gentzen StyleModal Logics
Andrew Parisi
Outline
Motivation
SequentCalculus
Pk
The SystemPS4
The SystemPS5
Hypersequents
The SystemPHS5
Quantification
Cut Elimination
Cut Elimination Theorem:
If there is a deduction of Γ⇒ Σ in Pk , then there is adeduction of Γ⇒ Σ in Pk that does not use the Cut rule.
Gentzen StyleModal Logics
Andrew Parisi
Outline
Motivation
SequentCalculus
Pk
The SystemPS4
The SystemPS5
Hypersequents
The SystemPHS5
Quantification
Cut Elimination (Cont.)
Rank of a formula:
rk(ϕ) = 0 if ϕ is atomic.rk(ϕ→ ψ) = max(rk(ϕ), rk(ψ)) + 1.rk(�ϕ) = rk(ϕ) + 1, where � ∈ {�,¬}
Rank of a deduction: The highest rank of a formula overwhich there is a cut, plus 1.
I will use `r Γ⇒ Σ to say that there is a deduction ofrank r of Γ⇒ Σ.
Gentzen StyleModal Logics
Andrew Parisi
Outline
Motivation
SequentCalculus
Pk
The SystemPS4
The SystemPS5
Hypersequents
The SystemPHS5
Quantification
Cut Elimination (Cont.)
Rank of a formula:
rk(ϕ) = 0 if ϕ is atomic.rk(ϕ→ ψ) = max(rk(ϕ), rk(ψ)) + 1.rk(�ϕ) = rk(ϕ) + 1, where � ∈ {�,¬}
Rank of a deduction: The highest rank of a formula overwhich there is a cut, plus 1.
I will use `r Γ⇒ Σ to say that there is a deduction ofrank r of Γ⇒ Σ.
Gentzen StyleModal Logics
Andrew Parisi
Outline
Motivation
SequentCalculus
Pk
The SystemPS4
The SystemPS5
Hypersequents
The SystemPHS5
Quantification
Cut Elimination (Cont.)
Rank of a formula:
rk(ϕ) = 0 if ϕ is atomic.rk(ϕ→ ψ) = max(rk(ϕ), rk(ψ)) + 1.rk(�ϕ) = rk(ϕ) + 1, where � ∈ {�,¬}
Rank of a deduction: The highest rank of a formula overwhich there is a cut, plus 1.
I will use `r Γ⇒ Σ to say that there is a deduction ofrank r of Γ⇒ Σ.
Gentzen StyleModal Logics
Andrew Parisi
Outline
Motivation
SequentCalculus
Pk
The SystemPS4
The SystemPS5
Hypersequents
The SystemPHS5
Quantification
Cut Elimination (Cont.)
Cut Elimination Theorem: If there is a Pk `r Γ⇒ Σ, thenthere is a deduction of Pk `0 Γ⇒ Σ.
Lemma 1: If Pk `r Γ, ϕ⇒ Σ and Pk `r Γ′ ⇒ ϕ,Σ′, thenPk `r ΓΓ′ ⇒ ΣΣ′.
The important case is when we have these deductions:
Γ, ψ ⇒ θ,Σ�
�Γ,�ψ ⇒ �θ,♦ΣΓ′ ⇒ ψ,Σ′
��Γ′ ⇒ �ψ,♦Σ′
Gentzen StyleModal Logics
Andrew Parisi
Outline
Motivation
SequentCalculus
Pk
The SystemPS4
The SystemPS5
Hypersequents
The SystemPHS5
Quantification
Cut Elimination (Cont.)
Cut Elimination Theorem: If there is a Pk `r Γ⇒ Σ, thenthere is a deduction of Pk `0 Γ⇒ Σ.
Lemma 1: If Pk `r Γ, ϕ⇒ Σ and Pk `r Γ′ ⇒ ϕ,Σ′, thenPk `r ΓΓ′ ⇒ ΣΣ′.
The important case is when we have these deductions:
Γ, ψ ⇒ θ,Σ�
�Γ,�ψ ⇒ �θ,♦ΣΓ′ ⇒ ψ,Σ′
��Γ′ ⇒ �ψ,♦Σ′
Gentzen StyleModal Logics
Andrew Parisi
Outline
Motivation
SequentCalculus
Pk
The SystemPS4
The SystemPS5
Hypersequents
The SystemPHS5
Quantification
Cut Elimination (Cont.)
Cut Elimination Theorem: If there is a Pk `r Γ⇒ Σ, thenthere is a deduction of Pk `0 Γ⇒ Σ.
Lemma 1: If Pk `r Γ, ϕ⇒ Σ and Pk `r Γ′ ⇒ ϕ,Σ′, thenPk `r ΓΓ′ ⇒ ΣΣ′.
The important case is when we have these deductions:
Γ, ψ ⇒ θ,Σ�
�Γ,�ψ ⇒ �θ,♦ΣΓ′ ⇒ ψ,Σ′
��Γ′ ⇒ �ψ,♦Σ′
Gentzen StyleModal Logics
Andrew Parisi
Outline
Motivation
SequentCalculus
Pk
The SystemPS4
The SystemPS5
Hypersequents
The SystemPHS5
Quantification
Lemma 1: Keeping the Cut rank the same
Pk `r Γ, ψ ⇒ θ,Σ
Pk `r Γ′ ⇒ ψ,Σ′
By hypothesis rk(ϕ) < rk(�ϕ) ≤ r .
So we have this deduction:
Γ, ψ ⇒ θ,Σ Γ′ ⇒ ψ,Σ′Cut
ΓΓ′ ⇒ θ,ΣΣ′�
�ΓΓ⇒ �θ,♦ΣΣ′
Gentzen StyleModal Logics
Andrew Parisi
Outline
Motivation
SequentCalculus
Pk
The SystemPS4
The SystemPS5
Hypersequents
The SystemPHS5
Quantification
Cut Elimination (Cont.)
Lemma 2: We will prove that if Pk `r+1 Γ⇒ ∆ thenPk `r Γ⇒ ∆ by induction on the length of the deduction ofPk `r+1 Γ⇒ ∆.
1 The last inference in Pk `r+1 Γ⇒ ∆ is an instance of Cut:
Λ⇒ T , ϕ Λ′, ϕ⇒ T ′Cut
ΛΛ′ ⇒ TT ′
Where ΛΛ′ = Γ and TT ′ = ∆
Gentzen StyleModal Logics
Andrew Parisi
Outline
Motivation
SequentCalculus
Pk
The SystemPS4
The SystemPS5
Hypersequents
The SystemPHS5
Quantification
Cut Elimination (Cont.)
Lemma 2: We will prove that if Pk `r+1 Γ⇒ ∆ thenPk `r Γ⇒ ∆ by induction on the length of the deduction ofPk `r+1 Γ⇒ ∆.
1 The last inference in Pk `r+1 Γ⇒ ∆ is an instance of Cut:
Λ⇒ T , ϕ Λ′, ϕ⇒ T ′Cut
ΛΛ′ ⇒ TT ′
Where ΛΛ′ = Γ and TT ′ = ∆
Gentzen StyleModal Logics
Andrew Parisi
Outline
Motivation
SequentCalculus
Pk
The SystemPS4
The SystemPS5
Hypersequents
The SystemPHS5
Quantification
Cut Elimination (Cont.)
Whenever Pk ` Γ⇒ ∆, Pk `0 Γ⇒ ∆. By induction on the cutrank:
1 The base case is trivial
2 Let Pk `r Γ⇒ ∆. By Lemma 2: Pk `r−1 Γ⇒ ∆. We canthen use IH to get the result.
Gentzen StyleModal Logics
Andrew Parisi
Outline
Motivation
SequentCalculus
Pk
The SystemPS4
The SystemPS5
Hypersequents
The SystemPHS5
Quantification
Cut Elimination (Cont.)
Whenever Pk ` Γ⇒ ∆, Pk `0 Γ⇒ ∆. By induction on the cutrank:
1 The base case is trivial
2 Let Pk `r Γ⇒ ∆. By Lemma 2: Pk `r−1 Γ⇒ ∆. We canthen use IH to get the result.
Gentzen StyleModal Logics
Andrew Parisi
Outline
Motivation
SequentCalculus
Pk
The SystemPS4
The SystemPS5
Hypersequents
The SystemPHS5
Quantification
System Pk Summary
Since Pk is equivalent to the axiomatic System K, andSystem K is sound and complete for the Kripke semantics,Pk is sound and complete for the Kripke semantics.
Cut is eliminable for Pk .
Gentzen StyleModal Logics
Andrew Parisi
Outline
Motivation
SequentCalculus
Pk
The SystemPS4
The SystemPS5
Hypersequents
The SystemPHS5
Quantification
System Pk Summary
Since Pk is equivalent to the axiomatic System K, andSystem K is sound and complete for the Kripke semantics,Pk is sound and complete for the Kripke semantics.
Cut is eliminable for Pk .
Gentzen StyleModal Logics
Andrew Parisi
Outline
Motivation
SequentCalculus
Pk
The SystemPS4
The SystemPS5
Hypersequents
The SystemPHS5
Quantification
System PS4
Gentzen StyleModal Logics
Andrew Parisi
Outline
Motivation
SequentCalculus
Pk
The SystemPS4
The SystemPS5
Hypersequents
The SystemPHS5
Quantification
System PS4
Remove � from Pk , and add these two rules[4, pg. 114]:
Γ, ϕ⇒ ΣL�
Γ,�ϕ⇒ Σ
�Γ⇒ ϕ,ΣR� �Γ⇒ �ϕ,♦Σ
This is sound and complete for S4 Kripke semantics, andCut is eliminable.
Gentzen StyleModal Logics
Andrew Parisi
Outline
Motivation
SequentCalculus
Pk
The SystemPS4
The SystemPS5
Hypersequents
The SystemPHS5
Quantification
System PS4
Remove � from Pk , and add these two rules[4, pg. 114]:
Γ, ϕ⇒ ΣL�
Γ,�ϕ⇒ Σ
�Γ⇒ ϕ,ΣR� �Γ⇒ �ϕ,♦Σ
This is sound and complete for S4 Kripke semantics, andCut is eliminable.
Gentzen StyleModal Logics
Andrew Parisi
Outline
Motivation
SequentCalculus
Pk
The SystemPS4
The SystemPS5
Hypersequents
The SystemPHS5
Quantification
Features of PS4
This is an extension of Pk , it proves the � rule.
Since PS4 is an extension of Pk , we already know it provesNecessitation and Distriubution.
PS4 `⇒ �ϕ→ ��ϕ and PS4 `⇒ �ϕ→ ϕ:
Gentzen StyleModal Logics
Andrew Parisi
Outline
Motivation
SequentCalculus
Pk
The SystemPS4
The SystemPS5
Hypersequents
The SystemPHS5
Quantification
Features of PS4
This is an extension of Pk , it proves the � rule.
Since PS4 is an extension of Pk , we already know it provesNecessitation and Distriubution.
PS4 `⇒ �ϕ→ ��ϕ and PS4 `⇒ �ϕ→ ϕ:
Gentzen StyleModal Logics
Andrew Parisi
Outline
Motivation
SequentCalculus
Pk
The SystemPS4
The SystemPS5
Hypersequents
The SystemPHS5
Quantification
Features of PS4
This is an extension of Pk , it proves the � rule.
Since PS4 is an extension of Pk , we already know it provesNecessitation and Distriubution.
PS4 `⇒ �ϕ→ ��ϕ and PS4 `⇒ �ϕ→ ϕ:
Gentzen StyleModal Logics
Andrew Parisi
Outline
Motivation
SequentCalculus
Pk
The SystemPS4
The SystemPS5
Hypersequents
The SystemPHS5
Quantification
S4 Deduces PS4
S4 licenses our inferences:
L�:
Γ, ϕ⇒ Σ
Γ,�ϕ⇒ Σ
Since t(Γ, ϕ⇒ Σ) ` t(Γ,�ϕ⇒ Σ) in the axiom systemS4.
R�:
�Γ⇒ ϕ,ΣR� �Γ⇒ �ϕ,♦Σ
Since t(�Γ⇒ ϕ,Σ) ` t(�Γ⇒ �ϕ,♦Σ) in S4
Gentzen StyleModal Logics
Andrew Parisi
Outline
Motivation
SequentCalculus
Pk
The SystemPS4
The SystemPS5
Hypersequents
The SystemPHS5
Quantification
S4 Deduces PS4
S4 licenses our inferences:
L�:
Γ, ϕ⇒ Σ
Γ,�ϕ⇒ Σ
Since t(Γ, ϕ⇒ Σ) ` t(Γ,�ϕ⇒ Σ) in the axiom systemS4.
R�:
�Γ⇒ ϕ,ΣR� �Γ⇒ �ϕ,♦Σ
Since t(�Γ⇒ ϕ,Σ) ` t(�Γ⇒ �ϕ,♦Σ) in S4
Gentzen StyleModal Logics
Andrew Parisi
Outline
Motivation
SequentCalculus
Pk
The SystemPS4
The SystemPS5
Hypersequents
The SystemPHS5
Quantification
S4 Deduces PS4
S4 licenses our inferences:
L�:
Γ, ϕ⇒ Σ
Γ,�ϕ⇒ Σ
Since t(Γ, ϕ⇒ Σ) ` t(Γ,�ϕ⇒ Σ) in the axiom systemS4.
R�:
�Γ⇒ ϕ,ΣR� �Γ⇒ �ϕ,♦Σ
Since t(�Γ⇒ ϕ,Σ) ` t(�Γ⇒ �ϕ,♦Σ) in S4
Gentzen StyleModal Logics
Andrew Parisi
Outline
Motivation
SequentCalculus
Pk
The SystemPS4
The SystemPS5
Hypersequents
The SystemPHS5
Quantification
S4 Deduces PS4
S4 licenses our inferences:
L�:
Γ, ϕ⇒ Σ
Γ,�ϕ⇒ Σ
Since t(Γ, ϕ⇒ Σ) ` t(Γ,�ϕ⇒ Σ) in the axiom systemS4.
R�:
�Γ⇒ ϕ,ΣR� �Γ⇒ �ϕ,♦Σ
Since t(�Γ⇒ ϕ,Σ) ` t(�Γ⇒ �ϕ,♦Σ) in S4
Gentzen StyleModal Logics
Andrew Parisi
Outline
Motivation
SequentCalculus
Pk
The SystemPS4
The SystemPS5
Hypersequents
The SystemPHS5
Quantification
S4 Deduces PS4
S4 licenses our inferences:
L�:
Γ, ϕ⇒ Σ
Γ,�ϕ⇒ Σ
Since t(Γ, ϕ⇒ Σ) ` t(Γ,�ϕ⇒ Σ) in the axiom systemS4.
R�:
�Γ⇒ ϕ,ΣR� �Γ⇒ �ϕ,♦Σ
Since t(�Γ⇒ ϕ,Σ) ` t(�Γ⇒ �ϕ,♦Σ) in S4
Gentzen StyleModal Logics
Andrew Parisi
Outline
Motivation
SequentCalculus
Pk
The SystemPS4
The SystemPS5
Hypersequents
The SystemPHS5
Quantification
S4 Deduces PS4
S4 licenses our inferences:
L�:
Γ, ϕ⇒ Σ
Γ,�ϕ⇒ Σ
Since t(Γ, ϕ⇒ Σ) ` t(Γ,�ϕ⇒ Σ) in the axiom systemS4.
R�:
�Γ⇒ ϕ,ΣR� �Γ⇒ �ϕ,♦Σ
Since t(�Γ⇒ ϕ,Σ) ` t(�Γ⇒ �ϕ,♦Σ) in S4
Gentzen StyleModal Logics
Andrew Parisi
Outline
Motivation
SequentCalculus
Pk
The SystemPS4
The SystemPS5
Hypersequents
The SystemPHS5
Quantification
Cut Elimination for PS4
Cut Elimination for PS4
The key is the reduction lemma when the formula cut overis �ϕ:
If PS4 `r Γ,�ϕ⇒ Σ and PS4 `r Γ′ ⇒ �ϕ,Σ′, thenPS4 `r ΓΓ′ ⇒ ΣΣ′.
Γ, ϕ⇒ ΣL�
Γ,�ϕ⇒ ΣΓ′ ⇒ ϕ,T
R�Γ′ ⇒ �ϕ,Σ′
where T = {σ|♦σ ∈ Σ′}.
Gentzen StyleModal Logics
Andrew Parisi
Outline
Motivation
SequentCalculus
Pk
The SystemPS4
The SystemPS5
Hypersequents
The SystemPHS5
Quantification
Cut Elimination for PS4
Cut Elimination for PS4
The key is the reduction lemma when the formula cut overis �ϕ:
If PS4 `r Γ,�ϕ⇒ Σ and PS4 `r Γ′ ⇒ �ϕ,Σ′, thenPS4 `r ΓΓ′ ⇒ ΣΣ′.
Γ, ϕ⇒ ΣL�
Γ,�ϕ⇒ ΣΓ′ ⇒ ϕ,T
R�Γ′ ⇒ �ϕ,Σ′
where T = {σ|♦σ ∈ Σ′}.
Gentzen StyleModal Logics
Andrew Parisi
Outline
Motivation
SequentCalculus
Pk
The SystemPS4
The SystemPS5
Hypersequents
The SystemPHS5
Quantification
Cut Elimination for PS4
Cut Elimination for PS4
The key is the reduction lemma when the formula cut overis �ϕ:
If PS4 `r Γ,�ϕ⇒ Σ and PS4 `r Γ′ ⇒ �ϕ,Σ′, thenPS4 `r ΓΓ′ ⇒ ΣΣ′.
Γ, ϕ⇒ ΣL�
Γ,�ϕ⇒ ΣΓ′ ⇒ ϕ,T
R�Γ′ ⇒ �ϕ,Σ′
where T = {σ|♦σ ∈ Σ′}.
Gentzen StyleModal Logics
Andrew Parisi
Outline
Motivation
SequentCalculus
Pk
The SystemPS4
The SystemPS5
Hypersequents
The SystemPHS5
Quantification
Cut Elimination for PS4
Cut Elimination for PS4
The key is the reduction lemma when the formula cut overis �ϕ:
If PS4 `r Γ,�ϕ⇒ Σ and PS4 `r Γ′ ⇒ �ϕ,Σ′, thenPS4 `r ΓΓ′ ⇒ ΣΣ′.
Γ, ϕ⇒ ΣL�
Γ,�ϕ⇒ ΣΓ′ ⇒ ϕ,T
R�Γ′ ⇒ �ϕ,Σ′
where T = {σ|♦σ ∈ Σ′}.
Gentzen StyleModal Logics
Andrew Parisi
Outline
Motivation
SequentCalculus
Pk
The SystemPS4
The SystemPS5
Hypersequents
The SystemPHS5
Quantification
Summary of PS4
This is equivalent to the axioms for S4.
Cut is eliminable for this system.
Gentzen StyleModal Logics
Andrew Parisi
Outline
Motivation
SequentCalculus
Pk
The SystemPS4
The SystemPS5
Hypersequents
The SystemPHS5
Quantification
Summary of PS4
This is equivalent to the axioms for S4.
Cut is eliminable for this system.
Gentzen StyleModal Logics
Andrew Parisi
Outline
Motivation
SequentCalculus
Pk
The SystemPS4
The SystemPS5
Hypersequents
The SystemPHS5
Quantification
System PS5
Gentzen StyleModal Logics
Andrew Parisi
Outline
Motivation
SequentCalculus
Pk
The SystemPS4
The SystemPS5
Hypersequents
The SystemPHS5
Quantification
System PS5
The System PS5 We add to Pc the following rules [4]:
1 L�
Γ, ϕ⇒ Σ
Γ,�ϕ⇒ Σ
2 R5�
�Γ⇒ ϕ,�ΣR5�
�Γ⇒ �ϕ,�Σ
Gentzen StyleModal Logics
Andrew Parisi
Outline
Motivation
SequentCalculus
Pk
The SystemPS4
The SystemPS5
Hypersequents
The SystemPHS5
Quantification
System PS5
The System PS5 We add to Pc the following rules [4]:
1 L�
Γ, ϕ⇒ Σ
Γ,�ϕ⇒ Σ
2 R5�
�Γ⇒ ϕ,�ΣR5�
�Γ⇒ �ϕ,�Σ
Gentzen StyleModal Logics
Andrew Parisi
Outline
Motivation
SequentCalculus
Pk
The SystemPS4
The SystemPS5
Hypersequents
The SystemPHS5
Quantification
PS5 extends PS4
1 PS5 is an extension of PS4
2 We already have L�
3 We can prove the R� rule from the L� and R5�.
Gentzen StyleModal Logics
Andrew Parisi
Outline
Motivation
SequentCalculus
Pk
The SystemPS4
The SystemPS5
Hypersequents
The SystemPHS5
Quantification
PS5 extends PS4
1 PS5 is an extension of PS4
2 We already have L�
3 We can prove the R� rule from the L� and R5�.
Gentzen StyleModal Logics
Andrew Parisi
Outline
Motivation
SequentCalculus
Pk
The SystemPS4
The SystemPS5
Hypersequents
The SystemPHS5
Quantification
PS5 extends PS4
1 PS5 is an extension of PS4
2 We already have L�
3 We can prove the R� rule from the L� and R5�.
Gentzen StyleModal Logics
Andrew Parisi
Outline
Motivation
SequentCalculus
Pk
The SystemPS4
The SystemPS5
Hypersequents
The SystemPHS5
Quantification
PS5 is equivalent to S5
1 PS5 is equivalent to S5.
2 This system can prove the S5 Axiom: ♦ϕ→ �♦ϕ, wherewe define ♦ϕ as ¬�¬ϕ.
3 S5 licenses our inferences:
t(Γ, ϕ⇒ Σ) ` t(Γ,�ϕ⇒ Σ)t(�Γ⇒ ϕ,�Σ) ` t(�Γ⇒ �ϕ,�Σ)
Gentzen StyleModal Logics
Andrew Parisi
Outline
Motivation
SequentCalculus
Pk
The SystemPS4
The SystemPS5
Hypersequents
The SystemPHS5
Quantification
PS5 is equivalent to S5
1 PS5 is equivalent to S5.
2 This system can prove the S5 Axiom: ♦ϕ→ �♦ϕ, wherewe define ♦ϕ as ¬�¬ϕ.
3 S5 licenses our inferences:
t(Γ, ϕ⇒ Σ) ` t(Γ,�ϕ⇒ Σ)t(�Γ⇒ ϕ,�Σ) ` t(�Γ⇒ �ϕ,�Σ)
Gentzen StyleModal Logics
Andrew Parisi
Outline
Motivation
SequentCalculus
Pk
The SystemPS4
The SystemPS5
Hypersequents
The SystemPHS5
Quantification
PS5 is equivalent to S5
1 PS5 is equivalent to S5.
2 This system can prove the S5 Axiom: ♦ϕ→ �♦ϕ, wherewe define ♦ϕ as ¬�¬ϕ.
3 S5 licenses our inferences:
t(Γ, ϕ⇒ Σ) ` t(Γ,�ϕ⇒ Σ)t(�Γ⇒ ϕ,�Σ) ` t(�Γ⇒ �ϕ,�Σ)
Gentzen StyleModal Logics
Andrew Parisi
Outline
Motivation
SequentCalculus
Pk
The SystemPS4
The SystemPS5
Hypersequents
The SystemPHS5
Quantification
Cut is not eliminable from PS5
Cut cannot be eliminated from PS5 [6].
Counterexample to Cut elimination:
Ax ϕ⇒ ϕL¬ ¬ϕ,ϕ⇒
L� �¬ϕ,ϕ⇒R¬
ϕ⇒ ¬�¬ϕ
Ax�¬ϕ⇒ �¬ϕR¬⇒ �¬ϕ,¬�¬ϕR5�⇒ �¬ϕ,�¬�¬ϕL¬¬�¬ϕ⇒ �¬�¬ϕ
Cutϕ⇒ �¬�¬ϕ
Gentzen StyleModal Logics
Andrew Parisi
Outline
Motivation
SequentCalculus
Pk
The SystemPS4
The SystemPS5
Hypersequents
The SystemPHS5
Quantification
Cut is not eliminable from PS5
Cut cannot be eliminated from PS5 [6].
Counterexample to Cut elimination:
Ax ϕ⇒ ϕL¬ ¬ϕ,ϕ⇒
L� �¬ϕ,ϕ⇒R¬
ϕ⇒ ¬�¬ϕ
Ax�¬ϕ⇒ �¬ϕR¬⇒ �¬ϕ,¬�¬ϕR5�⇒ �¬ϕ,�¬�¬ϕL¬¬�¬ϕ⇒ �¬�¬ϕ
Cutϕ⇒ �¬�¬ϕ
Gentzen StyleModal Logics
Andrew Parisi
Outline
Motivation
SequentCalculus
Pk
The SystemPS4
The SystemPS5
Hypersequents
The SystemPHS5
Quantification
Cut is not eliminable from PS5
Cut cannot be eliminated from PS5 [6].
Counterexample to Cut elimination:
Ax ϕ⇒ ϕL¬ ¬ϕ,ϕ⇒
L� �¬ϕ,ϕ⇒R¬
ϕ⇒ ¬�¬ϕ
Ax�¬ϕ⇒ �¬ϕR¬⇒ �¬ϕ,¬�¬ϕR5�⇒ �¬ϕ,�¬�¬ϕL¬¬�¬ϕ⇒ �¬�¬ϕ
Cutϕ⇒ �¬�¬ϕ
Gentzen StyleModal Logics
Andrew Parisi
Outline
Motivation
SequentCalculus
Pk
The SystemPS4
The SystemPS5
Hypersequents
The SystemPHS5
Quantification
Hypersequents
Gentzen StyleModal Logics
Andrew Parisi
Outline
Motivation
SequentCalculus
Pk
The SystemPS4
The SystemPS5
Hypersequents
The SystemPHS5
Quantification
Hypersequents
A hypersequent is a set of sequents [1]
When G and H are sequents the hypersequent composedof them is G e dH .
We can read a hypersequent, G e dH as disjunctionbetween the sequents G and H.
Gentzen StyleModal Logics
Andrew Parisi
Outline
Motivation
SequentCalculus
Pk
The SystemPS4
The SystemPS5
Hypersequents
The SystemPHS5
Quantification
Hypersequents
A hypersequent is a set of sequents [1]
When G and H are sequents the hypersequent composedof them is G e dH .
We can read a hypersequent, G e dH as disjunctionbetween the sequents G and H.
Gentzen StyleModal Logics
Andrew Parisi
Outline
Motivation
SequentCalculus
Pk
The SystemPS4
The SystemPS5
Hypersequents
The SystemPHS5
Quantification
Hypersequents
A hypersequent is a set of sequents [1]
When G and H are sequents the hypersequent composedof them is G e dH .
We can read a hypersequent, G e dH as disjunctionbetween the sequents G and H.
Gentzen StyleModal Logics
Andrew Parisi
Outline
Motivation
SequentCalculus
Pk
The SystemPS4
The SystemPS5
Hypersequents
The SystemPHS5
Quantification
Hypersequents (Cont.)
Some examples of rules:
G e d Γ⇒ ϕ,Σ e dH G ′ e d Γ′, ψ ⇒ Σ′ e dH ′L→
G e dG ′ e d ΓΓ′, ϕ→ ψ ⇒ ΣΣ′ e dH e dH ′
G e d Γ⇒ ϕ,Σ e dHL¬
G e d Γ,¬ϕ⇒ Σ e dH
Gentzen StyleModal Logics
Andrew Parisi
Outline
Motivation
SequentCalculus
Pk
The SystemPS4
The SystemPS5
Hypersequents
The SystemPHS5
Quantification
Hypersequents (Cont.)
Some examples of rules:
G e d Γ⇒ ϕ,Σ e dH G ′ e d Γ′, ψ ⇒ Σ′ e dH ′L→
G e dG ′ e d ΓΓ′, ϕ→ ψ ⇒ ΣΣ′ e dH e dH ′
G e d Γ⇒ ϕ,Σ e dHL¬
G e d Γ,¬ϕ⇒ Σ e dH
Gentzen StyleModal Logics
Andrew Parisi
Outline
Motivation
SequentCalculus
Pk
The SystemPS4
The SystemPS5
Hypersequents
The SystemPHS5
Quantification
Hypersequents (Cont.)
New rules:
G e dG e dHContract e
G e dHG e dH
Weak eG e dG ′ e dH
G e dHExchange e
H e dG
Gentzen StyleModal Logics
Andrew Parisi
Outline
Motivation
SequentCalculus
Pk
The SystemPS4
The SystemPS5
Hypersequents
The SystemPHS5
Quantification
New Rules (Cont.)
G e d ΓΓ′ ⇒ ΣΣ′ e dHS
G e d Γ⇒ Σ e d Γ′ ⇒ Σ′ e dH
G e d Γ⇒ ϕ,Σ e dH G ′ e d Γ′, ϕ⇒ Σ′ e dH ′Cut
G e dG ′ e d ΓΓ′ ⇒ ΣΣ′ e dH e dH ′
Gentzen StyleModal Logics
Andrew Parisi
Outline
Motivation
SequentCalculus
Pk
The SystemPS4
The SystemPS5
Hypersequents
The SystemPHS5
Quantification
The Sytem PHS5
1 PS4 within a sequent.
2 PHS5 proves the S5 Axiom
Gentzen StyleModal Logics
Andrew Parisi
Outline
Motivation
SequentCalculus
Pk
The SystemPS4
The SystemPS5
Hypersequents
The SystemPHS5
Quantification
The Sytem PHS5
1 PS4 within a sequent.
2 PHS5 proves the S5 Axiom
Gentzen StyleModal Logics
Andrew Parisi
Outline
Motivation
SequentCalculus
Pk
The SystemPS4
The SystemPS5
Hypersequents
The SystemPHS5
Quantification
PHS5 is equivalent to S5 (Cont.)
S5 licenses PHS5 inferences:
A hypersequent, G e d Γ⇒ Σ e dH is represented in S5as: t(G ) ∨ t(Γ⇒ Σ) ∨ t(H) [1].
New Cases:
G e dG e dHContract e
G e dHG e dH
Weak eG e dG ′ e dH
G e dHExchange e
H e dG
Gentzen StyleModal Logics
Andrew Parisi
Outline
Motivation
SequentCalculus
Pk
The SystemPS4
The SystemPS5
Hypersequents
The SystemPHS5
Quantification
PHS5 is equivalent to S5 (Cont.)
S5 licenses PHS5 inferences:
A hypersequent, G e d Γ⇒ Σ e dH is represented in S5as: t(G ) ∨ t(Γ⇒ Σ) ∨ t(H) [1].
New Cases:
G e dG e dHContract e
G e dHG e dH
Weak eG e dG ′ e dH
G e dHExchange e
H e dG
Gentzen StyleModal Logics
Andrew Parisi
Outline
Motivation
SequentCalculus
Pk
The SystemPS4
The SystemPS5
Hypersequents
The SystemPHS5
Quantification
PHS5 is equivalent to S5 (Cont.)
S5 licenses PHS5 inferences:
A hypersequent, G e d Γ⇒ Σ e dH is represented in S5as: t(G ) ∨ t(Γ⇒ Σ) ∨ t(H) [1].
New Cases:
G e dG e dHContract e
G e dHG e dH
Weak eG e dG ′ e dH
G e dHExchange e
H e dG
Gentzen StyleModal Logics
Andrew Parisi
Outline
Motivation
SequentCalculus
Pk
The SystemPS4
The SystemPS5
Hypersequents
The SystemPHS5
Quantification
PHS5 is equivalent to S5 (Cont.)
S5 lisences Splitting:
G e d ΓΓ′ ⇒ ΣΣ′ e dHS
G e d Γ⇒ Σ e d Γ′ ⇒ Σ′ e dH
A closer look:
�p ⇒ pS
�p ⇒ed⇒ p
Gentzen StyleModal Logics
Andrew Parisi
Outline
Motivation
SequentCalculus
Pk
The SystemPS4
The SystemPS5
Hypersequents
The SystemPHS5
Quantification
PHS5 is equivalent to S5 (Cont.)
S5 lisences Splitting:
G e d ΓΓ′ ⇒ ΣΣ′ e dHS
G e d Γ⇒ Σ e d Γ′ ⇒ Σ′ e dHA closer look:
�p ⇒ pS
�p ⇒ed⇒ p
Gentzen StyleModal Logics
Andrew Parisi
Outline
Motivation
SequentCalculus
Pk
The SystemPS4
The SystemPS5
Hypersequents
The SystemPHS5
Quantification
Cut is Eliminable for PHS5
Cut Elimination for PHS5
The important lemma is Reduction: IfPHS5 `r G e d Γ, ϕ⇒ Σ e dH andPHS5 `r G ′ e d Γ′ ⇒ ϕ,Σ e dH ′ thenPHS5 `r G e dG ′ e d ΓΓ′ ⇒ ΣΣ′ e dH e dH ′.
Gentzen StyleModal Logics
Andrew Parisi
Outline
Motivation
SequentCalculus
Pk
The SystemPS4
The SystemPS5
Hypersequents
The SystemPHS5
Quantification
Cut Free ♦ϕ→ �♦ϕ, S in action
PHS5 `0 ϕ→ �¬�¬ϕ [1]:
ϕ⇒ �¬�¬ϕ,ϕL¬ ¬ϕ,ϕ⇒ �¬�¬ϕ
L� �¬ϕ,ϕ⇒ �¬�¬ϕS
ϕ⇒ �¬�¬ϕ e d�¬ϕ⇒R¬
ϕ⇒ �¬�¬ϕ e d⇒ ¬�¬ϕR�
ϕ⇒ �¬�¬ϕ e d⇒ �¬�¬ϕWeak i
ϕ⇒ �¬�¬ϕ e dϕ⇒ �¬�¬ϕContraction e
ϕ⇒ �¬�¬ϕR→ ⇒ ϕ→ �¬�¬ϕ
Gentzen StyleModal Logics
Andrew Parisi
Outline
Motivation
SequentCalculus
Pk
The SystemPS4
The SystemPS5
Hypersequents
The SystemPHS5
Quantification
Adding Quantification
We can add the sequent rules for quantification.
Γ⇒ ϕ(x),ΣR∀
Γ⇒ ∀xϕ(x),Σ
Γ, ϕ(t)⇒ ΣL∀
Γ,∀xϕ(x)⇒ Σ
Where x is an eigenvariable, and t is any term.
Let ∃xϕ be defined as ¬∀x¬ϕ.
Gentzen StyleModal Logics
Andrew Parisi
Outline
Motivation
SequentCalculus
Pk
The SystemPS4
The SystemPS5
Hypersequents
The SystemPHS5
Quantification
Adding Quantification
We can add the sequent rules for quantification.
Γ⇒ ϕ(x),ΣR∀
Γ⇒ ∀xϕ(x),Σ
Γ, ϕ(t)⇒ ΣL∀
Γ,∀xϕ(x)⇒ Σ
Where x is an eigenvariable, and t is any term.
Let ∃xϕ be defined as ¬∀x¬ϕ.
Gentzen StyleModal Logics
Andrew Parisi
Outline
Motivation
SequentCalculus
Pk
The SystemPS4
The SystemPS5
Hypersequents
The SystemPHS5
Quantification
Adding Quantification
We can add the sequent rules for quantification.
Γ⇒ ϕ(x),ΣR∀
Γ⇒ ∀xϕ(x),Σ
Γ, ϕ(t)⇒ ΣL∀
Γ,∀xϕ(x)⇒ Σ
Where x is an eigenvariable, and t is any term.
Let ∃xϕ be defined as ¬∀x¬ϕ.
Gentzen StyleModal Logics
Andrew Parisi
Outline
Motivation
SequentCalculus
Pk
The SystemPS4
The SystemPS5
Hypersequents
The SystemPHS5
Quantification
Barcan Formulas
BF: ∀x�ϕ→ �∀xϕ
CBF: �∀xϕ→ ∀x�ϕOur sequent systems prove only CBF [3].
PS4 can’t prove BF:
?ϕ(t)⇒ ∀xϕ(x)
L��ϕ(t)⇒ ∀xϕ(x)
R��ϕ(t)⇒ �∀xϕ(x)
L∀ ∀x�ϕ(x)⇒ �∀xϕ(x)
Gentzen StyleModal Logics
Andrew Parisi
Outline
Motivation
SequentCalculus
Pk
The SystemPS4
The SystemPS5
Hypersequents
The SystemPHS5
Quantification
Barcan Formulas
BF: ∀x�ϕ→ �∀xϕCBF: �∀xϕ→ ∀x�ϕ
Our sequent systems prove only CBF [3].
PS4 can’t prove BF:
?ϕ(t)⇒ ∀xϕ(x)
L��ϕ(t)⇒ ∀xϕ(x)
R��ϕ(t)⇒ �∀xϕ(x)
L∀ ∀x�ϕ(x)⇒ �∀xϕ(x)
Gentzen StyleModal Logics
Andrew Parisi
Outline
Motivation
SequentCalculus
Pk
The SystemPS4
The SystemPS5
Hypersequents
The SystemPHS5
Quantification
Barcan Formulas
BF: ∀x�ϕ→ �∀xϕCBF: �∀xϕ→ ∀x�ϕOur sequent systems prove only CBF [3].
PS4 can’t prove BF:
?ϕ(t)⇒ ∀xϕ(x)
L��ϕ(t)⇒ ∀xϕ(x)
R��ϕ(t)⇒ �∀xϕ(x)
L∀ ∀x�ϕ(x)⇒ �∀xϕ(x)
Gentzen StyleModal Logics
Andrew Parisi
Outline
Motivation
SequentCalculus
Pk
The SystemPS4
The SystemPS5
Hypersequents
The SystemPHS5
Quantification
Barcan Formulas
BF: ∀x�ϕ→ �∀xϕCBF: �∀xϕ→ ∀x�ϕOur sequent systems prove only CBF [3].
PS4 can’t prove BF:
?ϕ(t)⇒ ∀xϕ(x)
L��ϕ(t)⇒ ∀xϕ(x)
R��ϕ(t)⇒ �∀xϕ(x)
L∀ ∀x�ϕ(x)⇒ �∀xϕ(x)
Gentzen StyleModal Logics
Andrew Parisi
Outline
Motivation
SequentCalculus
Pk
The SystemPS4
The SystemPS5
Hypersequents
The SystemPHS5
Quantification
PHS5 proves BF
PHS5 proves BF:
ϕ(a)⇒ �∀xϕ(x), ϕ(a)L�
�ϕ(a)⇒ �∀xϕ(x), ϕ(a)S
�ϕ(a)⇒ �∀xϕ(x) e d⇒ ϕ(a)L∀ ∀x�ϕ(x)⇒ �∀xϕ(x) e d⇒ ϕ(a)
R∀ ∀x�ϕ(x)⇒ �∀xϕ(x) e d⇒ ∀xϕ(x)R� ∀x�ϕ(x)⇒ �∀xϕ(x) e d⇒ �∀xϕ(x)
Weak i ∀x�ϕ(x)⇒ �∀xϕ(x) e d ∀x�ϕ(x)⇒ �∀xϕ(x)
∀x�ϕ(x)⇒ �∀xϕ(x)
Gentzen StyleModal Logics
Andrew Parisi
Outline
Motivation
SequentCalculus
Pk
The SystemPS4
The SystemPS5
Hypersequents
The SystemPHS5
Quantification
PHS5, BF, and Cut
Conjecture: There is a close relation between Cut, BF andaxiom B. Hughes and Cresswell [3] point out that BF is invalidwithout Axiom B. Perhaps Cut/Hypersequent Calculus is alsorequired.
Gentzen StyleModal Logics
Andrew Parisi
Outline
Motivation
SequentCalculus
Pk
The SystemPS4
The SystemPS5
Hypersequents
The SystemPHS5
Quantification
Arnon Avron.The Method of Hypersequents in the Proof Theory ofPropositional Non-Classical Logics, volume Logic: FromFoundations to Applications.Oxford University Press, New York, 1993.
Kosta Dosen.Sequent-systems for modal logics.The Journal of Symbolic Logic, 50(1), 1985.
G Hughes and M Cresswell.A New Introduction to Modal Logic.Taylor and Francis, 2012.
M. Ohnishi and K. Matsumoto.Gentzen method in modal calculi.Osaka Mathematical Journal, 9:113–130, 1957.
Gentzen StyleModal Logics
Andrew Parisi
Outline
Motivation
SequentCalculus
Pk
The SystemPS4
The SystemPS5
Hypersequents
The SystemPHS5
Quantification
Francesca Poggiolesi.Gentzen Calculi for Modal Propositional Logic.Springer, 2010.
Phiniki Stouppa.A deep inference system for the modal logic s5.Studia Logica: An International Journal for SymbolicLogic, 85(2):199–214, March 2007.