Full Lambek Hyperdoctrine Full Lambek Tripos Higher-Order Categorical Substructural Logics Yoshihiro Maruyama Oxford University and Kyoto University CT 2015, Aveiro, 18 June Yoshihiro Maruyama Higher-Order Categorical Substructural Logics CT 2015, Aveiro, 18 June 1 / 8
Full Lambek Hyperdoctrine Full Lambek Tripos Overview No concept of substructural topos; no topos-style semantics for general substructural logics (fuzzy, linear, relevant, etc.) so far. In this talk I present the concept of substructural tripos, giving tripos-style semantics for a wide variety of substructural logics. Full Lambek Calculus FL is a standard system allowing us to represent various logics as axiomatic extensions incl. CL and IL . Full Lambek hyperdoctrines and triposes are introduced to give complete semantics for first-order and higher-order FL . The framework thus developed allows us to compare different categorical logics, in particular to give functorial accounts of logical translations (Gödel, Girard, GTM, Baaz, etc.). Yoshihiro Maruyama Higher-Order Categorical Substructural Logics CT 2015, Aveiro, 18 June 2 / 8
Full Lambek Hyperdoctrine Full Lambek Tripos Outline Full Lambek Hyperdoctrine 1 Full Lambek Tripos 2 Yoshihiro Maruyama Higher-Order Categorical Substructural Logics CT 2015, Aveiro, 18 June 3 / 8
Full Lambek Hyperdoctrine Full Lambek Tripos Outline Full Lambek Hyperdoctrine 1 Full Lambek Tripos 2 Yoshihiro Maruyama Higher-Order Categorical Substructural Logics CT 2015, Aveiro, 18 June 4 / 8
Overview and Background First-Order Categorical Universal Logic Higher-Order Categorical Universal Logic Typed Full Lambek Calculus TFL q has the following logical connectives: ⌦ , ^ , _ , \ , /, 1 , 0 , > , ? , 8 , 9 . Note: there are two kinds of implication connectives \ and / . In TFL q , every variable x comes with its type σ . That is, TFL q has basic types, which are denoted by letters like σ , τ , and x : σ is a formal expression meaning that a variable x is of type σ . A (type) context is a finite list of type declarations on variables: x 1 : σ 1 , ..., x n : σ n . A context is often denoted Γ . Yoshihiro Maruyama Duality, Categorical Logic, and Quantum Physics
Overview and Background First-Order Categorical Universal Logic Higher-Order Categorical Universal Logic Typed Full Lambek Calculus (cont.) Accordingly, TFL q has typed predicate symbols (aka. predicates in context) and typed function symbols (aka. function symbols in context): R ( x 1 , ..., x n ) [ x 1 : σ 1 , ..., x n : σ n ] is a formal expression meaning that R is a predicate with n variables x 1 , ..., x n of types σ 1 , ..., σ n respectively; likewise, f : τ [ x 1 : σ 1 , ..., x n : σ n ] is a formal expression meaning that f is a function symbol with n variables x 1 , ..., x n of types σ 1 , ..., σ n and with its values in τ . Then, formulae-in-context ϕ [ Γ ] and terms-in-context t : τ [ Γ ] are defined in the usual, inductive way. Yoshihiro Maruyama Duality, Categorical Logic, and Quantum Physics
Overview and Background First-Order Categorical Universal Logic Higher-Order Categorical Universal Logic Typed Full Lambek Calculus (cont.) TFL q thus has both a type structure and a logic structure, dealing with sequents-in-contexts: Φ ` ϕ [ Γ ] (or Γ | Φ ` ϕ ). There are two kinds of conjunction in TFL q : multiplicative or monoidal ⌦ and additive or cartesian ^ : Φ , ϕ , ψ , Ψ ` χ [ Γ ] Φ ` ϕ [ Γ ] Ψ ` ψ [ Γ ] Φ , ϕ ⌦ ψ , Ψ ` χ [ Γ ] ( ⌦ L ) ( ⌦ R ) Φ , Ψ ` ϕ ⌦ ψ [ Γ ] Φ , ϕ , Ψ ` χ [ Γ ] Φ , ϕ , Ψ ` χ [ Γ ] Φ , ϕ ^ ψ , Ψ ` χ [ Γ ] ( ^ L 1 ) Φ , ψ ^ ϕ , Ψ ` χ [ Γ ] ( ^ L 2 ) Φ ` ϕ [ Γ ] Φ ` ψ [ Γ ] ( ^ R ) Φ ` ϕ ^ ψ [ Γ ] The underlying type system is basically the same as that of: A. Pitts, Categorical Logic, Handbook of Logic in Comput. Sci. Yoshihiro Maruyama Duality, Categorical Logic, and Quantum Physics
Overview and Background First-Order Categorical Universal Logic Higher-Order Categorical Universal Logic Typed Full Lambek Calculus (cont.) Due to non-commutativity, there are two kinds of implication in TFL q , \ and / , which are a right adjoint of ϕ ⌦ ( - ) and a right adjoint of ( - ) ⌦ ψ respectively. Φ ` ϕ [ Γ ] Ψ 1 , ψ , Ψ 2 ` χ [ Γ ] ϕ , Φ ` ψ [ Γ ] ( \ L ) Φ ` ϕ \ ψ [ Γ ] ( \ R ) Ψ 1 , Φ , ϕ \ ψ , Ψ 2 ` χ [ Γ ] Φ ` ϕ [ Γ ] Ψ 1 , ψ , Ψ 2 ` χ [ Γ ] Φ , ϕ ` ψ [ Γ ] ( / L ) Φ ` ψ / ϕ [ Γ ] ( / R ) Ψ 1 , ψ / ϕ , Φ , Ψ 2 ` χ [ Γ ] We omit the rules for the other propositional connectives. Yoshihiro Maruyama Duality, Categorical Logic, and Quantum Physics
Overview and Background First-Order Categorical Universal Logic Higher-Order Categorical Universal Logic Typed Full Lambek Calculus (cont.) Finally, we have the following rules for quantifiers 8 and 9 , in which type contexts explicitly change; notice that type contexts do not change in the rest of the rules presented above. Φ 1 , ϕ , Φ 2 ` ψ [ x : σ , Γ ] Φ ` ϕ [ x : σ , Γ ] Φ 1 , 8 x ϕ , Φ 2 ` ψ [ x : σ , Γ ] ( 8 L ) ( 8 R ) Φ ` 8 x ϕ [ Γ ] Φ 1 , ϕ , Φ 2 ` ψ [ x : σ , Γ ] Φ ` ϕ [ x : σ , Γ ] ( 9 L ) Φ ` 9 x ϕ [ x : σ , Γ ] ( 9 R ) Φ 1 , 9 x ϕ , Φ 2 ` ψ [ Γ ] As usual, there are eigenvariable conditions on the rules above. The other two rules do not have eigenvariable conditions, and this is why contexts do not change in them. Yoshihiro Maruyama Duality, Categorical Logic, and Quantum Physics
Overview and Background First-Order Categorical Universal Logic Higher-Order Categorical Universal Logic Typed Full Lambek Calculus (cont.) The following sequents-in-context are deducible in TFL q : ϕ ⌦ ( 9 x ψ ) ` 9 x ( ϕ ⌦ ψ ) [ Γ ] and 9 x ( ϕ ⌦ ψ ) ` ϕ ⌦ ( 9 x ψ ) [ Γ ] . ( 9 x ψ ) ⌦ ϕ ` 9 x ( ψ ⌦ ϕ ) [ Γ ] and 9 x ( ψ ⌦ ϕ ) ` ( 9 x ψ ) ⌦ ϕ [ Γ ] . where it is supposed that ϕ does not contain x as a free variable, and Γ contains type declarations on those free variables that appear in ϕ and 9 x ψ . Yoshihiro Maruyama Duality, Categorical Logic, and Quantum Physics
Overview and Background First-Order Categorical Universal Logic Higher-Order Categorical Universal Logic Full Lambek Algebra ( A , ⌦ , ^ , _ , \ , /, 1 , 0 , > , ? ) is called an FL algebra iff ( A , ⌦ , 1 ) is a monoid; 0 is an element of A ; ( A , ^ , _ , > , ? ) is a bounded lattice; for any a 2 A , a \ ( - ) : A ! A is a right adjoint of a ⌦ ( - ) : a ⌦ b c iff b a \ c ; for any b 2 A , ( - ) / b : A ! A is a right adjoint of ( - ) ⌦ b : a ⌦ b c iff a c / b . FL denotes the category of FL algebras and homomorphisms. This is propositional. How to extend all this for predicate logic? Alg. Log.: A propositional logic is a single algebra A . Cat. Log.: A predicate logic is a fibred algebra ( A C ) C ∈ C ; the base cat. C accounts for the underlying type theory. Yoshihiro Maruyama Duality, Categorical Logic, and Quantum Physics
Overview and Background First-Order Categorical Universal Logic Higher-Order Categorical Universal Logic Full Lambek Hyperdoctrine An FL hyperdoctrine is an FL -valued presheaf P : C op → FL such that C is a cat. with finite products, and for any projection π : X × Y → Y , P ( π ) : P ( Y ) → P ( X × Y ) has a right adjoint ∀ π : P ( X × Y ) → P ( Y ) with the corresponding Beck-Chevalley condition ∀ π P ( X × Y ) P ( Y ) - P ( X × f ) P ( f ) ? ? P ( X × Z ) P ( Z ) - ∀ π 0 (This means ( ∀ x ϕ )[ t / y ] = ∀ x ( ϕ [ t / y ]) in the syntactic hypdoc.) Yoshihiro Maruyama Duality, Categorical Logic, and Quantum Physics
Overview and Background First-Order Categorical Universal Logic Higher-Order Categorical Universal Logic Full Lambek Hyperdoctrine (def. cont’d) Furthermore, for any projection π : X × Y → Y in C , P ( π ) : P ( Y ) → P ( X × Y ) has a left adjoint ∃ π : P ( X × Y ) → P ( Y ) with the corresponding Beck-Chevalley condition: ∃ π P ( X × Y ) P ( Y ) - P ( X × f ) P ( f ) ? ? P ( X × Z ) P ( Z ) - ∃ π 0 This comes from Lawvere’s idea of quantifiers as adjoints. Equality, comprehension, and the like can be treated as well. Yoshihiro Maruyama Duality, Categorical Logic, and Quantum Physics
Overview and Background First-Order Categorical Universal Logic Higher-Order Categorical Universal Logic Full Lambek Hyperdoctrine (def. cont’d) Furthermore, the Frobenius Reciprocity conditions hold: for any π : X × Y → Y in C , any a ∈ P ( Y ) , and any b ∈ P ( X × Y ) , a ⊗ ( ∃ π b ) = ∃ π ( P ( π )( a ) ⊗ b ) ( ∃ π b ) ⊗ a = ∃ π ( b ⊗ P ( π )( a )) . For an axiomatic extension FL X of FL , an FL X hyperdoctrine is defined by restricting the value cat. FL into FL X . An FL (resp. FL X ) hypdoc. is also called a fibred FL (resp. FL X ) algebra. In full generality we conceive of universally algebraised hyperdoctrines P : C op → Alg ( T ) (or P : C op → V ). Yoshihiro Maruyama Duality, Categorical Logic, and Quantum Physics
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