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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 Introduction 1 Preliminaries 2 Categories of signatures and logics with strict morphism Categories of signatures and logics with flexible morphism The categories of algebrizable logics Abstract Glivenko’s Theorem 3 Institution The Institution of Lindenbaum Algebraizable Logics The Abstract Glivenko’s Theorem Final Remarks and Future Works 4 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 Introduction 1 Preliminaries 2 Categories of signatures and logics with strict morphism Categories of signatures and logics with flexible morphism The categories of algebrizable logics Abstract Glivenko’s Theorem 3 Institution The Institution of Lindenbaum Algebraizable Logics The Abstract Glivenko’s Theorem Final Remarks and Future Works 4 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 S s , the category of signatures(strict or simple): – signatures (propositional, finitary): Σ = (Σ n ) n ∈ N ; – formulas: F (Σ), F (Σ)[ n ]; – (strict) morphisms : f : Σ → Σ ′ ; f = ( f n ) n ∈ N : (Σ n ) n ∈ N → (Σ ′ n ) n ∈ N ; – ˆ f : F (Σ) → F (Σ ′ ). Proposition S s ≃ Set N , 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 S s , the category of signatures(strict or simple): – signatures (propositional, finitary): Σ = (Σ n ) n ∈ N ; – formulas: F (Σ), F (Σ)[ n ]; – (strict) morphisms : f : Σ → Σ ′ ; f = ( f n ) n ∈ N : (Σ n ) n ∈ N → (Σ ′ n ) n ∈ N ; – ˆ f : F (Σ) → F (Σ ′ ). Proposition S s ≃ Set N , 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 L s , the category of logics over S s : – logic l = (Σ , ⊢ ): Σ signature; ⊢ � Tarskian consequence operator. – (strict) morphisms f : (Σ , ⊢ ) → (Σ ′ , ⊢ ′ ) f ∈ S s (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 L s is a ω -locally presentable category. – fp logics: are given by a finite set of ”axioms”and ”inference rules”, over a fp signature. The L s 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 L s , the category of logics over S s : – logic l = (Σ , ⊢ ): Σ signature; ⊢ � Tarskian consequence operator. – (strict) morphisms f : (Σ , ⊢ ) → (Σ ′ , ⊢ ′ ) f ∈ S s (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 L s is a ω -locally presentable category. – fp logics: are given by a finite set of ”axioms”and ”inference rules”, over a fp signature. The L s 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 = (Σ n ) n ∈ N is a signature, then T (Σ) := ( F (Σ)[ n ]) n ∈ N is a signature too. • h ∈ S f (Σ , Σ ′ ) � h ♯ ∈ S s (Σ , T (Σ ′ )); f ∈ S s (Σ , T (Σ ′ )) � f ♭ ∈ S f (Σ , Σ ′ ). • For each f ∈ S f (Σ , Σ ′ ) there is only one function ˇ f : F (Σ) → F (Σ ′ ) The category S f The category S f is the category of signature and flexible morphism as above. The composition in S f 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 L f - Objects: logics l = (Σ , ⊢ ) - Morphisms: f : l → l ′ in L f is a flexible signature morphism f : Σ → Σ ′ in S f 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. → ( ∨ ′ , ¬ ′ ) such that Consider the flexible morphisms t : ( → , ¬ ) − 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 L f - Objects: logics l = (Σ , ⊢ ) - Morphisms: f : l → l ′ in L f is a flexible signature morphism f : Σ → Σ ′ in S f 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. → ( ∨ ′ , ¬ ′ ) such that Consider the flexible morphisms t : ( → , ¬ ) − 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 L f - Objects: logics l = (Σ , ⊢ ) - Morphisms: f : l → l ′ in L f is a flexible signature morphism f : Σ → Σ ′ in S f 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. → ( ∨ ′ , ¬ ′ ) such that Consider the flexible morphisms t : ( → , ¬ ) − 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 • Q L f : ”quotient”category: f ∼ g iff ˇ f ( ϕ ) ⊣ ′ ⊢ ˇ g ( ϕ ). The logics l and l ′ are equipollent ([ CG ]) iff l and l ′ are Q L f -isomorphic. • L c f ⊆ L f : ”congruential”logics: ϕ 0 ⊣⊢ ψ 0 , . . . , ϕ n − 1 ⊣⊢ ψ n − 1 ⇒ c n ( ϕ 0 , . . . , ϕ n − 1 ) ⊣⊢ c n ( ψ 0 , . . . , ψ n − 1 ). The inclusion functor L c f ֒ → L f has a left adjoint. • Q L c f (or simply Q c 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