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Category Theory 2015

An abstract approach to Glivenko’s theorem

Darllan Concei¸ c˜ ao Pinto (IME-USP/CAPES-PROEX) Joint work with H.L. Mariano

D.C. Pinto, H.L. Mariano (IME-USP) 1 / 32

Introduction

Index

1

Introduction

2

Preliminaries Categories of signatures and logics with strict morphism Categories of signatures and logics with flexible morphism The categories of algebrizable logics

3

Abstract Glivenko’s Theorem Institution The Institution of Lindenbaum Algebraizable Logics The Abstract Glivenko’s Theorem

4

Final Remarks and Future Works

D.C. Pinto, H.L. Mariano (IME-USP) 2 / 32

Introduction

Introduction

• 1990 decade: Rise many methods of combination of logics motivation for

categories of logics

• Methods of combinations of logics

(i) A decomposition process or analysis of logics (Ex: the ”Possible Translation Semantics”of W. Carnielli [Car]) (ii) A composition process or synthesis of logic (Ex: the ”Fibring”of D. Gabbay [Ga])

• Major concern in the study of categories of logics (CLE-UNICAMP,

IST-Lisboa): to describe condition for preservation, under the combination method, of meta-logical properties

D.C. Pinto, H.L. Mariano (IME-USP) 3 / 32

Preliminaries

Index

1

Introduction

2

Preliminaries Categories of signatures and logics with strict morphism Categories of signatures and logics with flexible morphism The categories of algebrizable logics

3

Abstract Glivenko’s Theorem Institution The Institution of Lindenbaum Algebraizable Logics The Abstract Glivenko’s Theorem

4

Final Remarks and Future Works

D.C. Pinto, H.L. Mariano (IME-USP) 4 / 32

Preliminaries Categories of signatures and logics with strict morphism

Categories of signatures and logics with strict morphism

Ss, the category of signatures(strict or simple):

– signatures (propositional, finitary): Σ = (Σn)n∈N; – formulas: F(Σ), F(Σ)[n]; – (strict) morphisms : f : Σ → Σ′; f = (fn)n∈N : (Σn)n∈N → (Σ′

n)n∈N;

– ˆ f : F(Σ) → F(Σ′).

Proposition

Ss ≃ SetN, is a finitely locally presentable category. – fp signatures ”finite support”signatures.

D.C. Pinto, H.L. Mariano (IME-USP) 5 / 32

Preliminaries Categories of signatures and logics with strict morphism

Categories of signatures and logics with strict morphism

Ss, the category of signatures(strict or simple):

– signatures (propositional, finitary): Σ = (Σn)n∈N; – formulas: F(Σ), F(Σ)[n]; – (strict) morphisms : f : Σ → Σ′; f = (fn)n∈N : (Σn)n∈N → (Σ′

n)n∈N;

– ˆ f : F(Σ) → F(Σ′).

Proposition

Ss ≃ SetN, is a finitely locally presentable category. – fp signatures ”finite support”signatures.

D.C. Pinto, H.L. Mariano (IME-USP) 5 / 32

Preliminaries Categories of signatures and logics with strict morphism

Categories of signatures and logics with strict morphism

Ls, the category of logics over Ss:

– logic l = (Σ, ⊢): Σ signature; ⊢ Tarskian consequence operator. – (strict) morphisms f : (Σ, ⊢) → (Σ′, ⊢′) f ∈ Ss(Σ, Σ′) ˆ f : F(Σ) → F(Σ′) is a (⊢, ⊢′)-translation (”continuous”): Γ ⊢ ψ ⇒ ˆ f [Γ] ⊢ ˆ f (ψ), for all Γ ∪ {ψ} ⊆ F(Σ).

Theorem

Ls is a ω-locally presentable category. – fp logics: are given by a finite set of ”axioms”and ”inference rules”, over a fp signature. The Ls does not has a good treatment of “identity problem”.

D.C. Pinto, H.L. Mariano (IME-USP) 6 / 32

Preliminaries Categories of signatures and logics with strict morphism

Categories of signatures and logics with strict morphism

Ls, the category of logics over Ss:

– logic l = (Σ, ⊢): Σ signature; ⊢ Tarskian consequence operator. – (strict) morphisms f : (Σ, ⊢) → (Σ′, ⊢′) f ∈ Ss(Σ, Σ′) ˆ f : F(Σ) → F(Σ′) is a (⊢, ⊢′)-translation (”continuous”): Γ ⊢ ψ ⇒ ˆ f [Γ] ⊢ ˆ f (ψ), for all Γ ∪ {ψ} ⊆ F(Σ).

Theorem

Ls is a ω-locally presentable category. – fp logics: are given by a finite set of ”axioms”and ”inference rules”, over a fp signature. The Ls does not has a good treatment of “identity problem”.

D.C. Pinto, H.L. Mariano (IME-USP) 6 / 32

Preliminaries Categories of signatures and logics with flexible morphism

Categories of signatures and logics with flexible morphism

• If Σ = (Σn)n∈N is a signature, then T(Σ) := (F(Σ)[n])n∈N is a signature too.
• h ∈ Sf (Σ, Σ′) h♯ ∈ Ss(Σ, T(Σ′)); f ∈ Ss(Σ, T(Σ′)) f ♭ ∈ Sf (Σ, Σ′).
• For each f ∈ Sf (Σ, Σ′) there is only one function ˇ

f : F(Σ) → F(Σ′)

The category Sf

The category Sf is the category of signature and flexible morphism as above. The composition in Sf is given by (f ′ • f ′′)♯ := (ˇ f ↾ ◦f ♯).

D.C. Pinto, H.L. Mariano (IME-USP) 7 / 32

Preliminaries Categories of signatures and logics with flexible morphism

Categories of signatures and logics with flexible morphism

The category Lf

• Objects: logics l = (Σ, ⊢)
• Morphisms: f : l → l′ in Lf is a flexible signature morphism f : Σ → Σ′ in Sf

such that ˇ f : F(Σ) → F(Σ′) ”preserves the consequence relation”. Due to flexible morphism, this category allows better approach to the “identity problem” of logics. Consider the flexible morphisms t : (→, ¬) − → (∨′, ¬′) such that t(→) = ¬′x ∨′ y, t(¬) = ¬′ and t′ : (∨′, ¬′) − → (→, ¬) such that t′(∨′) = ¬x → y, t′(¬′) = ¬. This pair of morphisms induce an equipollence between these presentations of classic logics [CG]. However this category does not has good categorial properties as well as logics with strict morphism.

D.C. Pinto, H.L. Mariano (IME-USP) 8 / 32

Preliminaries Categories of signatures and logics with flexible morphism

Categories of signatures and logics with flexible morphism

The category Lf

• Objects: logics l = (Σ, ⊢)
• Morphisms: f : l → l′ in Lf is a flexible signature morphism f : Σ → Σ′ in Sf

such that ˇ f : F(Σ) → F(Σ′) ”preserves the consequence relation”. Due to flexible morphism, this category allows better approach to the “identity problem” of logics. Consider the flexible morphisms t : (→, ¬) − → (∨′, ¬′) such that t(→) = ¬′x ∨′ y, t(¬) = ¬′ and t′ : (∨′, ¬′) − → (→, ¬) such that t′(∨′) = ¬x → y, t′(¬′) = ¬. This pair of morphisms induce an equipollence between these presentations of classic logics [CG]. However this category does not has good categorial properties as well as logics with strict morphism.

D.C. Pinto, H.L. Mariano (IME-USP) 8 / 32

Preliminaries Categories of signatures and logics with flexible morphism

Categories of signatures and logics with flexible morphism

The category Lf

• Objects: logics l = (Σ, ⊢)
• Morphisms: f : l → l′ in Lf is a flexible signature morphism f : Σ → Σ′ in Sf

such that ˇ f : F(Σ) → F(Σ′) ”preserves the consequence relation”. Due to flexible morphism, this category allows better approach to the “identity problem” of logics. Consider the flexible morphisms t : (→, ¬) − → (∨′, ¬′) such that t(→) = ¬′x ∨′ y, t(¬) = ¬′ and t′ : (∨′, ¬′) − → (→, ¬) such that t′(∨′) = ¬x → y, t′(¬′) = ¬. This pair of morphisms induce an equipollence between these presentations of classic logics [CG]. However this category does not has good categorial properties as well as logics with strict morphism.

D.C. Pinto, H.L. Mariano (IME-USP) 8 / 32

Preliminaries Categories of signatures and logics with flexible morphism

Other categories of logics

• QLf : ”quotient”category: f ∼ g iff ˇ

f (ϕ) ⊣′⊢ ˇ g(ϕ). The logics l and l′ are equipollent ([CG]) iff l and l′ are QLf -isomorphic.

• Lc

f ⊆ Lf : ”congruential”logics: ϕ0 ⊣⊢ ψ0, . . . , ϕn−1 ⊣⊢ ψn−1 ⇒

cn(ϕ0, . . . , ϕn−1) ⊣⊢ cn(ψ0, . . . , ψn−1). The inclusion functor Lc

f ֒

→ Lf has a left adjoint.

• QLc

f (or simply Qc f ): ”good”category of logics: represents the major part of

logics; has good categorial properties (is an accessible category complete/cocopmplete); solves the identity problem for the presentations of classical logic in terms of isomorphism; allows a good notion of algebraizable logic ([MaMe]).

D.C. Pinto, H.L. Mariano (IME-USP) 9 / 32

Preliminaries The categories of algebrizable logics

Algebrizable logics

Algebrizable logics [BP]

Let l = (Σ, ⊢) be a logic and K be a class of Σ−algebra. K is a equivalent algebraic semantics to l if ⊢ can be interpreted in | =K (semantic relation over equations) of the following form: (1) there is a finite system δi(p) ≈ ǫi(p), i = 1, ..., n of equations in a single variable p such that for all Γ ∪ {ϕ} ⊆ F(Σ) and for j < n has been:

• Γ ⊢ ϕ ⇔ {δ[γ/p] ≈ ǫ[γ/p] : γ ∈ Γ} |

=K δ[ϕ/p] ≈ ǫ[ϕ/p] where δ(p) ≈ ǫ(p) abbreviates the equation systems δi(p) ≈ ǫi(p), i = 1, ..., n.

(2) there is a finite system ∆j(p, q), j = 1, ..., m of two variables formulas (formed by derived binary connectives) such that for all equation ϕ ≈ ψ,

• ϕ ≈ ψ =|K|

= δ(ϕ∆ψ) ≈ ǫ(ϕ∆ψ)

ϕ∆ψ = ∆(ϕ, ψ) where ∆(ϕ, ψ) abbreviates ∆j(ϕ, ψ), j = 1, ..., m. In this case we say that a logic l is algebraizable. The set ∆(p, q), ǫ(p), δ(p) is called algebraizing pair.

D.C. Pinto, H.L. Mariano (IME-USP) 10 / 32

Preliminaries The categories of algebrizable logics

Algebrizable logics

Algebrizable logics [BP]

Let l = (Σ, ⊢) be a logic and K be a class of Σ−algebra. K is a equivalent algebraic semantics to l if ⊢ can be interpreted in | =K (semantic relation over equations) of the following form: (1) there is a finite system δi(p) ≈ ǫi(p), i = 1, ..., n of equations in a single variable p such that for all Γ ∪ {ϕ} ⊆ F(Σ) and for j < n has been:

• Γ ⊢ ϕ ⇔ {δ[γ/p] ≈ ǫ[γ/p] : γ ∈ Γ} |

=K δ[ϕ/p] ≈ ǫ[ϕ/p] where δ(p) ≈ ǫ(p) abbreviates the equation systems δi(p) ≈ ǫi(p), i = 1, ..., n.

(2) there is a finite system ∆j(p, q), j = 1, ..., m of two variables formulas (formed by derived binary connectives) such that for all equation ϕ ≈ ψ,

• ϕ ≈ ψ =|K|

= δ(ϕ∆ψ) ≈ ǫ(ϕ∆ψ)

ϕ∆ψ = ∆(ϕ, ψ) where ∆(ϕ, ψ) abbreviates ∆j(ϕ, ψ), j = 1, ..., m. In this case we say that a logic l is algebraizable. The set ∆(p, q), ǫ(p), δ(p) is called algebraizing pair.

D.C. Pinto, H.L. Mariano (IME-USP) 10 / 32

Preliminaries The categories of algebrizable logics

Algebrizable logics

Algebrizable logics [BP]

Let l = (Σ, ⊢) be a logic and K be a class of Σ−algebra. K is a equivalent algebraic semantics to l if ⊢ can be interpreted in | =K (semantic relation over equations) of the following form: (1) there is a finite system δi(p) ≈ ǫi(p), i = 1, ..., n of equations in a single variable p such that for all Γ ∪ {ϕ} ⊆ F(Σ) and for j < n has been:

• Γ ⊢ ϕ ⇔ {δ[γ/p] ≈ ǫ[γ/p] : γ ∈ Γ} |

=K δ[ϕ/p] ≈ ǫ[ϕ/p] where δ(p) ≈ ǫ(p) abbreviates the equation systems δi(p) ≈ ǫi(p), i = 1, ..., n.

(2) there is a finite system ∆j(p, q), j = 1, ..., m of two variables formulas (formed by derived binary connectives) such that for all equation ϕ ≈ ψ,

• ϕ ≈ ψ =|K|

= δ(ϕ∆ψ) ≈ ǫ(ϕ∆ψ)

ϕ∆ψ = ∆(ϕ, ψ) where ∆(ϕ, ψ) abbreviates ∆j(ϕ, ψ), j = 1, ..., m. In this case we say that a logic l is algebraizable. The set ∆(p, q), ǫ(p), δ(p) is called algebraizing pair.

D.C. Pinto, H.L. Mariano (IME-USP) 10 / 32

Preliminaries The categories of algebrizable logics

Algebrizable logics

Algebrizable logics [BP]

Let l = (Σ, ⊢) be a logic and K be a class of Σ−algebra. K is a equivalent algebraic semantics to l if ⊢ can be interpreted in | =K (semantic relation over equations) of the following form: (1) there is a finite system δi(p) ≈ ǫi(p), i = 1, ..., n of equations in a single variable p such that for all Γ ∪ {ϕ} ⊆ F(Σ) and for j < n has been:

• Γ ⊢ ϕ ⇔ {δ[γ/p] ≈ ǫ[γ/p] : γ ∈ Γ} |

=K δ[ϕ/p] ≈ ǫ[ϕ/p] where δ(p) ≈ ǫ(p) abbreviates the equation systems δi(p) ≈ ǫi(p), i = 1, ..., n.

(2) there is a finite system ∆j(p, q), j = 1, ..., m of two variables formulas (formed by derived binary connectives) such that for all equation ϕ ≈ ψ,

• ϕ ≈ ψ =|K|

= δ(ϕ∆ψ) ≈ ǫ(ϕ∆ψ)

ϕ∆ψ = ∆(ϕ, ψ) where ∆(ϕ, ψ) abbreviates ∆j(ϕ, ψ), j = 1, ..., m. In this case we say that a logic l is algebraizable. The set ∆(p, q), ǫ(p), δ(p) is called algebraizing pair.

D.C. Pinto, H.L. Mariano (IME-USP) 10 / 32

Preliminaries The categories of algebrizable logics

Example of algebrizable logics

In case of CPC and IPC the algebraizing pair ∆(p, q), ǫ(p), δ(p) is:

• 1. ∆(p, q) = {p ↔ q}
• 2. ǫ(p) = p
• 3. δ(p) = ⊤

and K is the class of Boolean algebra and Heyting algebra respectively. The logic of groups theory has as algebrizing pair ∆(p, q), ǫ(p), δ(p):

• 1. ∆(p, q) = p · q−1
• 2. δ(p) = p
• 3. ǫ(p) = e

K, in this case, is the class of groups.

D.C. Pinto, H.L. Mariano (IME-USP) 11 / 32

Preliminaries The categories of algebrizable logics

The categories of algebrizable logics

• As is the category of algebraizable logics with morphism in Ls such that

preserves algebraizing pair. In the sequence of works, [AFLM1], [AFLM2], [AFLM3] is proven that the category As is a relatively complete ω-accessible category [AR].

• Af is the category of algebraizable logics with morphisms in Lf such that

preserves algebraizing pair. Af is a subcategory of Lf , Af ֒ → Lf .

• On the category Af we have the following subcategories: Ac

f , QAf and QAc f .

• The ”Lindenbaum algebraizable”logics are logics l ∈ A such that given

formulas ϕ, ψ ∈ F(Σ), ϕ ⊣⊢ ψ ⇔⊢ ϕ∆ψ. The Lindenbaum algebraizable logics lead a subcategory of the category of algebraizable logics (j : Lind(Af ) ֒ → Af ). Lind(Af ) has a importance in the representation theory of logics. The inclusion functor Lind(Af ) ֒ → Af has a left adjoint functor L : Af → Lind(Af ).

D.C. Pinto, H.L. Mariano (IME-USP) 12 / 32

Preliminaries The categories of algebrizable logics

The categories of algebrizable logics

The following diagram represent the functors (and its adjoints) between the categories mentioned above: Af

incl

• L
• Lf

q

• c
• Qf

¯ c

• Lind(Af )

incl

• j
• Lc

f qc

• i
• Qc

f ¯ i

• Our ongoing work: (ArXiv preprints)

[MaPi1] Representation theory of logics: a categorial approach. [MaPi2] Algebraizable Logics and a functorial encoding of its morphisms.

D.C. Pinto, H.L. Mariano (IME-USP) 13 / 32

Preliminaries The categories of algebrizable logics

The categories of algebrizable logics

The main functor in this work: Given a morphism h : (Σ, ⊢) = a → a′ = (Σ′, ⊢′) in Lind(Af ). We define the following functor Σ′ − Str

h⋆

• U′
• Σ − Str

U

• Set

such that given cn ∈ Σ and M′ ∈ Σ′ − Str, ch⋆(M′)

n

= h(cn)M′. Moreover h⋆↾: QV (a′) → QV (a).

Theorem

Given h : a → a′ in Lind(Af ). h⋆↾ has a left adjoint functor Lh.

D.C. Pinto, H.L. Mariano (IME-USP) 14 / 32

Abstract Glivenko’s Theorem

Index

1

Introduction

2

Preliminaries Categories of signatures and logics with strict morphism Categories of signatures and logics with flexible morphism The categories of algebrizable logics

3

Abstract Glivenko’s Theorem Institution The Institution of Lindenbaum Algebraizable Logics The Abstract Glivenko’s Theorem

4

Final Remarks and Future Works

D.C. Pinto, H.L. Mariano (IME-USP) 15 / 32

Abstract Glivenko’s Theorem

Glivenko’s Theorem

The Glivenko’s Theorem is a well known and important result about the relationship between classical propositional logic and intuitionistic propositional logic.

Glivenko’s Theorem 1929

A formula ϕ is provable in classical propositional logic iff the formula ¬¬ϕ is provable in intuitionistic propositional logic, i.e., Γ ⊢CPC ψ ⇔ Γ ⊢IPC ¬¬ψ In this work we are trying generalize this translation to apply to different logics. To do this we use a logical-categorial framework called Institutions.

D.C. Pinto, H.L. Mariano (IME-USP) 16 / 32

Abstract Glivenko’s Theorem Institution

Institutions

An Institution I = (Sig, Sen, Mod, | =) consist of Sig

• (Cat)op

| = Set

• 1. a category Sig, whose the objects are called signature,
• 2. a functor Sen : Sig → Set, for each signature a set whose elements are called

sentence over the signature

• 3. a functor Mod : (Sig)op → Cat, for each signature a category whose the
• bjects are called model,
• 4. a relation |

=Σ⊆ |Mod(Σ)| × Sen(Σ) for each Σ ∈ |Sig|, called Σ-satisfaction, such that for each morphism h : Σ → Σ′, the compatibility condition M′ | =Σ′ Sen(h)(φ) if and only if Mod(h)(M′) | =Σ φ holds for each M′ ∈ |Mod(Σ′)| and φ ∈ Sen(Σ)

D.C. Pinto, H.L. Mariano (IME-USP) 17 / 32

Abstract Glivenko’s Theorem Institution

Institutions

An Institution I = (Sig, Sen, Mod, | =) consist of Sig

• (Cat)op

| = Set

• 1. a category Sig, whose the objects are called signature,
• 2. a functor Sen : Sig → Set, for each signature a set whose elements are called

sentence over the signature

• 3. a functor Mod : (Sig)op → Cat, for each signature a category whose the
• bjects are called model,
• 4. a relation |

=Σ⊆ |Mod(Σ)| × Sen(Σ) for each Σ ∈ |Sig|, called Σ-satisfaction, such that for each morphism h : Σ → Σ′, the compatibility condition M′ | =Σ′ Sen(h)(φ) if and only if Mod(h)(M′) | =Σ φ holds for each M′ ∈ |Mod(Σ′)| and φ ∈ Sen(Σ)

D.C. Pinto, H.L. Mariano (IME-USP) 17 / 32

Abstract Glivenko’s Theorem Institution

Morphism of Institutions

An Institution morphism (Φ, α, β) : I → I ′ consist of Sig

տ

• Sen
• (Mod)op
• ւ
• Φ

Set Sig ′

Sen′

• Mod′op Catop
• 1. a functor Φ : Sig → Sig ′
• 2. a natural transformation α : Sen′ ◦ Φ ⇒ Sen
• 3. a natural transformation β : Mod ⇒ Mod′ ◦ Φop

such that the following compatibility condition holds: M′ | =′

Σ′ αΣ(ϕ) iff

βΣ′(M′) | =Φ(Σ′) ϕ

D.C. Pinto, H.L. Mariano (IME-USP) 18 / 32

Abstract Glivenko’s Theorem Institution

Morphism of Institutions

An Institution morphism (Φ, α, β) : I → I ′ consist of Sig

տ

• Sen
• (Mod)op
• ւ
• Φ

Set Sig ′

Sen′

• Mod′op Catop
• 1. a functor Φ : Sig → Sig ′
• 2. a natural transformation α : Sen′ ◦ Φ ⇒ Sen
• 3. a natural transformation β : Mod ⇒ Mod′ ◦ Φop

such that the following compatibility condition holds: M′ | =′

Σ′ αΣ(ϕ) iff

βΣ′(M′) | =Φ(Σ′) ϕ

D.C. Pinto, H.L. Mariano (IME-USP) 18 / 32

Abstract Glivenko’s Theorem The Institution of Lindenbaum Algebraizable Logics

The Institution of Lindenbaum Algeraizable Logics

Before define the Institution of Lindenbaum algebraizable logics, we define a notion of satisfiability of class of formulas:

Definition

Let a be algebraizable logic. Given M ∈ QV (a) and ϕ, ψ ∈ F(Σ)(X), M | =QV (a) ϕ ≈ ψ iff for every valuation v : X → M, v(ϕ) = v(ψ)

Remark

If a ∈ Lind(Af ) and [α0] = [α1], i.e. α0 ⊣⊢ α1, then: | =QV (a) δ(α0) ≈ ǫ(α0) ⇔ | =QV (a) δ(α1) ≈ ǫ(α1) .

D.C. Pinto, H.L. Mariano (IME-USP) 19 / 32

Abstract Glivenko’s Theorem The Institution of Lindenbaum Algebraizable Logics

The Institution of Lindenbaum Algebraizable Logics

Given a ∈ Lind(Af ). Ia defines the Institution of Lindenbaum algebraizable associated to a where:

• Sig a is the category whose the objects are a1 = (Σ1, ⊢1) ∈ Lind(Af ), that are

isomorphic to a in the quotient category QLind(Af ) and the morphisms are only the isomorphisms in QLind(Af ).

• Moda : (Sig a)op → Cat such that Moda(a1) = QV (a1) for all a1 ∈ |Sig| and

Moda(a1

[h]

→ a2) = (QV (a2)

h⋆ ↾

→ QV (a1)).

• Sena : Sig a → Set such that Sena(a1) is the set all tuples

q = (([α0], [β0]), · · · , ([αn−1], [βn−1]); ([α], [β])) that represents quasi-equations, i.e., Eq0 ∧ ... ∧ Eqn−1 → Eq such that [αi], [βj] belongs to F(Σ1)(X)/ ⊣ ⊢, the free QV (a1)-structure on the set X, and αi = ε(ϕi), βi = δ(ϕi), for some algebraizable pair of a1, ((ε, δ), ∆). Given h : a1 → a2 which is a isomorphism in QLind(Af ). Then Sena(h) : Sena(a1) → Sena(a2) such that Sena(h)(q) = (([hα0], [hβ0]), ..., ([h(αn−1)], [h(βn−1)]), ([h(α), h(β)]))

D.C. Pinto, H.L. Mariano (IME-USP) 20 / 32

Abstract Glivenko’s Theorem The Institution of Lindenbaum Algebraizable Logics

The Institution of Lindenbaum Algebraizable Logics

Given a ∈ Lind(Af ). Ia defines the Institution of Lindenbaum algebraizable associated to a where:

• Sig a is the category whose the objects are a1 = (Σ1, ⊢1) ∈ Lind(Af ), that are

isomorphic to a in the quotient category QLind(Af ) and the morphisms are only the isomorphisms in QLind(Af ).

• Moda : (Sig a)op → Cat such that Moda(a1) = QV (a1) for all a1 ∈ |Sig| and

Moda(a1

[h]

→ a2) = (QV (a2)

h⋆ ↾

→ QV (a1)).

• Sena : Sig a → Set such that Sena(a1) is the set all tuples

q = (([α0], [β0]), · · · , ([αn−1], [βn−1]); ([α], [β])) that represents quasi-equations, i.e., Eq0 ∧ ... ∧ Eqn−1 → Eq such that [αi], [βj] belongs to F(Σ1)(X)/ ⊣ ⊢, the free QV (a1)-structure on the set X, and αi = ε(ϕi), βi = δ(ϕi), for some algebraizable pair of a1, ((ε, δ), ∆). Given h : a1 → a2 which is a isomorphism in QLind(Af ). Then Sena(h) : Sena(a1) → Sena(a2) such that Sena(h)(q) = (([hα0], [hβ0]), ..., ([h(αn−1)], [h(βn−1)]), ([h(α), h(β)]))

D.C. Pinto, H.L. Mariano (IME-USP) 20 / 32

Abstract Glivenko’s Theorem The Institution of Lindenbaum Algebraizable Logics

The Institution of Lindenbaum Algebraizable Logics

Given a ∈ Lind(Af ). Ia defines the Institution of Lindenbaum algebraizable associated to a where:

• Sig a is the category whose the objects are a1 = (Σ1, ⊢1) ∈ Lind(Af ), that are

isomorphic to a in the quotient category QLind(Af ) and the morphisms are only the isomorphisms in QLind(Af ).

• Moda : (Sig a)op → Cat such that Moda(a1) = QV (a1) for all a1 ∈ |Sig| and

Moda(a1

[h]

→ a2) = (QV (a2)

h⋆ ↾

→ QV (a1)).

• Sena : Sig a → Set such that Sena(a1) is the set all tuples

q = (([α0], [β0]), · · · , ([αn−1], [βn−1]); ([α], [β])) that represents quasi-equations, i.e., Eq0 ∧ ... ∧ Eqn−1 → Eq such that [αi], [βj] belongs to F(Σ1)(X)/ ⊣ ⊢, the free QV (a1)-structure on the set X, and αi = ε(ϕi), βi = δ(ϕi), for some algebraizable pair of a1, ((ε, δ), ∆). Given h : a1 → a2 which is a isomorphism in QLind(Af ). Then Sena(h) : Sena(a1) → Sena(a2) such that Sena(h)(q) = (([hα0], [hβ0]), ..., ([h(αn−1)], [h(βn−1)]), ([h(α), h(β)]))

D.C. Pinto, H.L. Mariano (IME-USP) 20 / 32

Abstract Glivenko’s Theorem The Institution of Lindenbaum Algebraizable Logics

The Institution of Lindenbaum Algebraizable Logics

Given a ∈ Lind(Af ). Ia defines the Institution of Lindenbaum algebraizable associated to a where:

• Sig a is the category whose the objects are a1 = (Σ1, ⊢1) ∈ Lind(Af ), that are

isomorphic to a in the quotient category QLind(Af ) and the morphisms are only the isomorphisms in QLind(Af ).

• Moda : (Sig a)op → Cat such that Moda(a1) = QV (a1) for all a1 ∈ |Sig| and

Moda(a1

[h]

→ a2) = (QV (a2)

h⋆ ↾

→ QV (a1)).

• Sena : Sig a → Set such that Sena(a1) is the set all tuples

q = (([α0], [β0]), · · · , ([αn−1], [βn−1]); ([α], [β])) that represents quasi-equations, i.e., Eq0 ∧ ... ∧ Eqn−1 → Eq such that [αi], [βj] belongs to F(Σ1)(X)/ ⊣ ⊢, the free QV (a1)-structure on the set X, and αi = ε(ϕi), βi = δ(ϕi), for some algebraizable pair of a1, ((ε, δ), ∆). Given h : a1 → a2 which is a isomorphism in QLind(Af ). Then Sena(h) : Sena(a1) → Sena(a2) such that Sena(h)(q) = (([hα0], [hβ0]), ..., ([h(αn−1)], [h(βn−1)]), ([h(α), h(β)]))

D.C. Pinto, H.L. Mariano (IME-USP) 20 / 32

Abstract Glivenko’s Theorem The Institution of Lindenbaum Algebraizable Logics

The Institution of Lindenbaum Algebraizable Logics

Given a′ ∈ Sig a, M′ ∈ QV (a′) and q′ ∈ Sena(a′), we say that M′ | =a

a′ q′ when

M′ | =QV (a′) [α′

i] ≈ [β′ i ] ∀ i = 0, ..., n − 1

then M′ | =QV (a′) [α′] ≈ [β′] Let h : a1 → a2 ∈ Sig a (then h is a isomorphism in QLind(Af )), M2 ∈ QV (a2) and q1 ∈ Senaa(a1) then M2 | =a

a2 Sen(h)(q1) ⇔ Mod(h)(M2) |

=a

a1 q1

In this way, Ia is a Institution.

D.C. Pinto, H.L. Mariano (IME-USP) 21 / 32

Abstract Glivenko’s Theorem The Institution of Lindenbaum Algebraizable Logics

The Institution of Lindenbaum Algebraizable Logics

Given a′ ∈ Sig a, M′ ∈ QV (a′) and q′ ∈ Sena(a′), we say that M′ | =a

a′ q′ when

M′ | =QV (a′) [α′

i] ≈ [β′ i ] ∀ i = 0, ..., n − 1

then M′ | =QV (a′) [α′] ≈ [β′] Let h : a1 → a2 ∈ Sig a (then h is a isomorphism in QLind(Af )), M2 ∈ QV (a2) and q1 ∈ Senaa(a1) then M2 | =a

a2 Sen(h)(q1) ⇔ Mod(h)(M2) |

=a

a1 q1

In this way, Ia is a Institution.

D.C. Pinto, H.L. Mariano (IME-USP) 21 / 32

Abstract Glivenko’s Theorem The Abstract Glivenko’s Theorem

Glivenko’s context

Definition

A Glivenko’s context is a pair G = (h : a → a′, ρ) where h ∈ Lind(Af )(a, a′) and ¯ ρ : h⋆↾ ◦Lh ⇒ Id is a natural transformation that is a section of the unit ¯ ∂ : Id ⇒ h⋆↾ ◦Lh).

First example

inclusion : IPL → CPL, ∂ is given by ¬¬.

Remark

If G = (h : a → a′, ρ) is a Glivenko’s context then h is a ”dense”morphism (∀ψ′∃ψ, ψ′ ⊣⊢ ˇ h(ψ)). However not every dense morphisms induces a institution morphism MG.

D.C. Pinto, H.L. Mariano (IME-USP) 22 / 32

Abstract Glivenko’s Theorem The Abstract Glivenko’s Theorem

Glivenko’s context

Theorem

Each G = (h : a → a′, ρ) Glivenko’s context induces some institutions morphism MG : Ia → Ia′.

D.C. Pinto, H.L. Mariano (IME-USP) 23 / 32

Abstract Glivenko’s Theorem The Abstract Glivenko’s Theorem

Abstract Glivenko’s theorem

Corollary

For each Glivenko’s context G = (h : a → a′, ρ), is associated an abstract Glivenko’s theorem between a and a′ i.e; given Γ ∪ {ϕ} ⊆ F(X) then ρX∂X[Γ] ⊢ ρX∂X(ϕ) ⇔ ˇ h[Γ] ⊢′ ˇ h(ϕ) Sketch of proof Due to a and a′ are Lindenbaum algebraizable, is enough to show that {¯ ρX ¯ ∂X[ε(ψ)] ≈ ¯ ρX ¯ ∂X[δ(ψ)], ψ ∈ Γ} | =QV (a) ¯ ρX ¯ ∂X[ε(ϕ)] ≈ ¯ ρX ¯ ∂X[δ(ϕ)]

• {[ˇ

h(ε(ψ))] ≈ [ˇ h(δ(ψ))], ψ ∈ Γ} | =QV (a′) [ˇ h(ε(ϕ))] ≈ [ˇ h(δ(ϕ))] Consider Γ = {ψ0, ..., ψn−1} and q′ = (([ˇ h(ε(ψ0))], [ˇ h(δ(ψ0))]), ..., ([ˇ h(ε(ψn−1))], [ˇ h(δ(ψn−1))])([ˇ h(ε(ϕ))], [ˇ h(δ(ϕ))])) Observe that ¯ ∂X[ϕ] = [ˇ h(ϕ)] for all ϕ ∈ F(Σ)(X) and given M′ ∈ QV (a′) there is M ∈ QV (a) such that LhM ∼ = M′. With this, it is enough to show that for every M ∈ QV (a), M | =a ¯ ρXq′ ⇔ LhM | =a′ q′ And this last equivalence is consequence of the theorem above.

D.C. Pinto, H.L. Mariano (IME-USP) 24 / 32

Abstract Glivenko’s Theorem The Abstract Glivenko’s Theorem

Abstract Glivenko’s theorem

Corollary

For each Glivenko’s context G = (h : a → a′, ρ), is associated an abstract Glivenko’s theorem between a and a′ i.e; given Γ ∪ {ϕ} ⊆ F(X) then ρX∂X[Γ] ⊢ ρX∂X(ϕ) ⇔ ˇ h[Γ] ⊢′ ˇ h(ϕ) Sketch of proof Due to a and a′ are Lindenbaum algebraizable, is enough to show that {¯ ρX ¯ ∂X[ε(ψ)] ≈ ¯ ρX ¯ ∂X[δ(ψ)], ψ ∈ Γ} | =QV (a) ¯ ρX ¯ ∂X[ε(ϕ)] ≈ ¯ ρX ¯ ∂X[δ(ϕ)]

• {[ˇ

h(ε(ψ))] ≈ [ˇ h(δ(ψ))], ψ ∈ Γ} | =QV (a′) [ˇ h(ε(ϕ))] ≈ [ˇ h(δ(ϕ))] Consider Γ = {ψ0, ..., ψn−1} and q′ = (([ˇ h(ε(ψ0))], [ˇ h(δ(ψ0))]), ..., ([ˇ h(ε(ψn−1))], [ˇ h(δ(ψn−1))])([ˇ h(ε(ϕ))], [ˇ h(δ(ϕ))])) Observe that ¯ ∂X[ϕ] = [ˇ h(ϕ)] for all ϕ ∈ F(Σ)(X) and given M′ ∈ QV (a′) there is M ∈ QV (a) such that LhM ∼ = M′. With this, it is enough to show that for every M ∈ QV (a), M | =a ¯ ρXq′ ⇔ LhM | =a′ q′ And this last equivalence is consequence of the theorem above.

D.C. Pinto, H.L. Mariano (IME-USP) 24 / 32

Final Remarks and Future Works

Index

1

Introduction

2

Preliminaries Categories of signatures and logics with strict morphism Categories of signatures and logics with flexible morphism The categories of algebrizable logics

3

Abstract Glivenko’s Theorem Institution The Institution of Lindenbaum Algebraizable Logics The Abstract Glivenko’s Theorem

4

Final Remarks and Future Works

D.C. Pinto, H.L. Mariano (IME-USP) 25 / 32

Final Remarks and Future Works

Final Remarks and Future Works

• Describe the exact dependence of the morphism of institution on the

Glivenko’s context.

• Apply the powerful methods and results of Institution Theory to the study of

propositional logics.

• Search for generalizations...

D.C. Pinto, H.L. Mariano (IME-USP) 26 / 32

Final Remarks and Future Works

Bibliography

[AFLM1]P. Arndt, R. A. Freire, O. O. Luciano, H. L. Mariano, Fibring and Sheaves, Proceedings of IICAI-05, Special Session at the 2nd Indian International Conference on Artificial Intelligence, Pune, India, (2005), 1679-1698. [AFLM2]P. Arndt, R. A. Freire, O. O. Luciano, H. L. Mariano, On the category of algebraizable logics, CLE e-Prints 6(1) (2006), 24 pages. [AFLM3]P. Arndt, R. A. Freire, O. O. Luciano, H. L. Mariano, A global glance on categories in Logic, Logica Universalis 1 (2007), 3-39. [AR]J. Ad´ amek, J. Rosick´ y, Locally Presentable and Accessible Categories, Lecture Notes Series of the LMS 189, Cambridge University Press, Cambridge, Great Britain, 1994. [Bez]J.Y. B´ eziau, From Consequence Operator to Universal Logic: A Survey

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Theory of Logic (J.-Y. Beziau, ed.), Birkhaeuser, Basel, 2005.

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Final Remarks and Future Works

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Final Remarks and Future Works

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[Con]M. E. Coniglio, The Meta-Fibring environment: Preservation of meta-properties by fibring, CLE e-Prints 5(4) (2005), 36 p´ aginas. http://www.cle.unicamp.br/e-prints/. [Cze]J. Czelakowski, Protoalgebraic logic, Trends in Logic, Studia Logica Library, Kluwer Academic Publishers, 2001. [CC1]W. A. Carnielli, M. E. Coniglio, A categorial approach to the combination of logics, Manuscrito 22 (1999), 64-94. [CC2]W. A. Carnielli, M. E. Coniglio, Transfers between logics and their applications, Studia Logica 72 (2002); CLE e-Prints 1(4) (2001), 31 pages. [CC3]W. A. Carnielli, M. E. Coniglio, Combining Logics, Stanford Encyclopedia of Philosophy,

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Final Remarks and Future Works

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Universal Construction, Handbook of Philosophical Logic 13 (2005) (editors:

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Final Remarks and Future Works

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D.C. Pinto, H.L. Mariano (IME-USP) 31 / 32

Final Remarks and Future Works

THANK YOU.

D.C. Pinto, H.L. Mariano (IME-USP) 32 / 32