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Constructive Embeddings of Intermediate Logics

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Constructive Embeddings of Intermediate Logics Constructive embeddings of intermediate logics Sara Negri University of Helsinki Symposium on Constructivity and Computability 9-10 June 2011, Uppsala The G¨odel-Tarski-McKinsey embedding of intuitionistic logic into S4 ∗ 1. G¨odel(1933): `Int A ! `S4 A soundness ∗ 2. McKinsey & Tarski (1948): `S4 A ! `Int A faithfulness ∗ 3. Dummett & Lemmon (1959): `Int+Ax A iff `S4+Ax∗ A A modal logic M is a modal companion of a superintuitionistic ∗ logic L if `L A iff `M A . So S4 is a modal companion of Int, S4+Ax∗ is a modal companion of Int+Ax. 2. and 3. are proved semantically. We look closer at McKinsey & Tarski (1948): Proof by McKinsey & Tarski (1948) uses: I 1. Completeness of intuitionistic logic wrt Heyting algebras (Brouwerian algebras) and of S4 wrt topological Boolean algebras (closure algebras) I 2. Representation of Heyting algebras as the opens of topological Boolean algebras. I 3. The proof is indirect because of 1. and non-constructive because of 2. (Uses Stone representation of distributive lattices, in particular Zorn's lemma) The result was generalized to intermediate logics by Dummett and Lemmon (1959). No syntactic proof of faithfulness in the literature except the complex proof of the embedding of Int into S4 in Troelstra & Schwichtenberg (1996). Goal: give a direct, constructive, syntactic, and uniform proof. Background on labelled sequent systems Method for formulating systems of contraction-free sequent calculus for basic modal logic and its extensions and for systems of non-classical logics in Negri (2005). General ideas for basic modal logic K and other systems of normal modal logics. Syntax embodies the possible world semantics: I Rules for 2, 3 through meaning explanation and inversion principles. I Frame properties in the form of \non-logical rules" added to the basic sequent calculus. Extensions are modular. Relation of this approach to the extensive literature on labelled deductive systems of various forms is detailed in Negri (2011). Here explicit labelling and contraction-free systems. Sequent calculus for the basic modal logic K I Start with the calculus G3c for classical propositional logic I Enrich the language: Sequents Γ ) ∆ where Γ and ∆ consist of expressions xRy and x : A (corresponding to x A of Kripke models), with x; y; z; ::: ranging in a set W and with A any formula in the language of propositional logic extended with the modal operators of necessity and possibility, 2 and 3. I Rules for basic modal logic obtained from the inductive definition of validity in a Kripke frame. From x : 2A iff for all y; xRy implies y : A obtain the rules I If y : A can be derived for an arbitrary y accessible from x, then x : 2A can be derived xRy; Γ ) ∆; y : A R2 Γ ) ∆; x : 2A arbitrariness of y becomes the variable condition y not (free) in Γ; ∆ I If x : 2A and y is accessible from x, then y : A y : A; x : 2A; xRy; Γ ) ∆ L2 x : 2A; xRy; Γ ) ∆ The rules for 3 are obtained similarly from the semantical explanation x : 3A iff for some y; xRy and y : A I If x : 3A, there exists y is accessible from x such that y : A xRy; y : A; Γ ) ∆ L3 x : 3A; Γ ) ∆ variable condition: y not (free) in Γ; ∆ I If y is accessible from x and y : A, then x : 3A gives the rule xRy; Γ ) ∆; x : 3A; y : A R3 xRy; Γ ) ∆; x : 3A Initial sequents: x : P; Γ ) ∆; x : P Propositional rules: x : A; x : B; Γ ) ∆ Γ ) ∆; x : A Γ ) ∆; x : B L& R& x : A&B; Γ ) ∆ Γ ) ∆; x : A&B x : A; Γ ) ∆ x : B; Γ ) ∆ Γ ) ∆; x : A; x : B L_ R_ x : A _ B; Γ ) ∆ Γ ) ∆; x : A _ B Γ ) ∆; x : A x : B; Γ ) ∆ x : A; Γ ) ∆; x : B L⊃ R⊃ x : A ⊃ B; Γ ) ∆ Γ ) ∆; x : A ⊃ B L? x :?; Γ ) ∆ Modal rules: y : A; x : 2A; xRy; Γ ) ∆ xRy; Γ ) ∆; y : A L2 R2 x : 2A; xRy; Γ ) ∆ Γ ) ∆; x : 2A xRy; y : A; Γ ) ∆ xRy; Γ ) ∆; x : 3A; y : A L3 R3 x : 3A; Γ ) ∆ xRy; Γ ) ∆; x : 3A Modal systems and frame properties I Modal logic K characterized by arbitrary frames. I Restrictions of the class of frames amount to adding certain frame properties to the calculus. I In Kripke frames for S4 the accessibility relation is reflexive 8x:xRx and transitive 8x8y8z((xRy & yRz) ⊃ xRz) I For S4:2 also directed 8xyz(xRy & xRz ⊃ 9w(yRw & zRw)) Axioms as rules (cont.) I We use the method developed in NvP (1998) and in N (2003) I Universal axioms are transformed into conjunctive normal form, that is, conjunctions of formulas of the form P1& ::: &Pm ⊃ Q1 _···_ Qn I Each conjunct is then converted into the regular rule, Q1; P1;:::; Pm; Γ ) ∆ ::: Qn; P1;:::; Pm; Γ ) ∆ Reg P1;:::; Pm; Γ ) ∆ I Other rules may be added by the closure condition. Those that are instances of contraction are admissible. Examples of universal axioms Axiom Frame property T 2A ⊃ A 8x xRx reflexivity 4 2A ⊃ 22A 8xyz(xRy & yRz ⊃ xRz) transitivity E 3A ⊃ 23A 8xyz(xRy & xRz ⊃ yRz) euclideanness B A ⊃ 23A 8xy(xRy ⊃ yRx) symmetry 3 2(2A ⊃ B) _ 2(2B ⊃ A) 8xyz(xRy & xRz ⊃ yRz _ zRy) connectedness D 2A ⊃ 3A 8x9y xRy seriality 2 32A ⊃ 23A 8xyz(xRy & xRz ⊃ 9w(yRw & zRw)) directedness Frame property Rule xRx; Γ ) ∆ T 8x xRx reflexivity Γ ) ∆ xRz; xRy; yRz; Γ ) ∆ 4 8xyz(xRy & yRz ⊃ xRz) trans. xRy; yRz; Γ ) ∆ xRz; xRy; yRz; Γ ) ∆ E 8xyz(xRy & xRz ⊃ yRz) euclid. xRy; xRz; Γ ) ∆ yRx; xRy; Γ ) ∆ B 8xy(xRy ⊃ yRx) symmetry xRy; Γ ) ∆ 3 8xyz(xRy & xRz ⊃ yRz _ zRy) yRz; xRy; xRz; Γ ) ∆ zRy; xRy; xRz; Γ ) ∆ connectedness xRy; xRz; Γ ) ∆ D 8x9y xRy seriality 2 8xyz(xRy & xRz ⊃ 9w(yRw & zRw)) Axioms as rules (cont.) I Method extended (N2003) to geometric theories, that is, theories axiomatized by formulas 8z(A ⊃ B) where A and B do not contain ⊃ or 8. I These can be reduced to conjunctions of 8z(P1& ::: &Pm ⊃ 9x1M1 _ · · · _ 9xnMn) where Mj is conjunction of atoms Qj I and turned into the geometric rule, with the yi 's not in the conclusion Q1(y1=x1); P; Γ ) ∆ ::: Qn(yn=xn); P; Γ ) ∆ GR P; Γ ) ∆ Examples of geometric implications Axiom Frame property T 2A ⊃ A 8x xRx reflexivity 4 2A ⊃ 22A 8xyz(xRy & yRz ⊃ xRz) transitivity E 3A ⊃ 23A 8xyz(xRy & xRz ⊃ yRz) euclideanness B A ⊃ 23A 8xy(xRy ⊃ yRx) symmetry 3 2(2A ⊃ B) _ 2(2B ⊃ A) 8xyz(xRy & xRz ⊃ yRz _ zRy) connectedness D 2A ⊃ 3A 8x9y xRy seriality 2 32A ⊃ 23A 8xyz(xRy & xRz ⊃ 9w(yRw & zRw)) directedness Frame property Rule xRx; Γ ) ∆ T 8x xRx reflexivity Γ ) ∆ xRz; xRy; yRz; Γ ) ∆ 4 8xyz(xRy & yRz ⊃ xRz) trans. xRy; yRz; Γ ) ∆ yRz; xRy; xRz; Γ ) ∆ E 8xyz(xRy & xRz ⊃ yRz) euclid. xRy; xRz; Γ ) ∆ yRx; xRy; Γ ) ∆ B 8xy(xRy ⊃ yRx) symmetry xRy; Γ ) ∆ yRz; xRy; xRz; Γ ) ∆ zRy; xRy; xRz; Γ ) ∆ 3 8xyz(xRy & xRz ⊃ yRz _ zRy) xRy; xRz; Γ ) ∆ xRy; Γ ) ∆ y D 8x9y xRy seriality Γ ) ∆ yRw; zRw; xRy; xRz; Γ ) ∆ w 2 8xyz(xRy & xRz ⊃ 9w(yRw & zRw)) xRy; xRz; Γ ) ∆ Structural properties of the basic system I Theorem The structural rules are admissible in extensions of G3c with regular or geometric rules satisfying the closure condition. Weakening and contraction are height-preserving admissible. All the rules are invertible. I The same structural properties hold in G3K, in addition substitution of labels is height-preserving admissible. Results Let G3K* be any extension of G3K with universal or geometric rules rules for the accessibility relation. I All the structural rules{weakening, contraction, and cut{are admissible in the system G3K*. I The characteristic axioms are derivable. I The necessitation rule is admissible. I Indirect completeness, through equivalence with the corresponding axiomatic system. I Direct completeness: proof search in the system either gives a derivation or a Kripke countemodel. I Answers to questions of undefinability through conservativity theorems. I Answers to decidability questions through algorithms of terminating proof search. Intuitionistic logic Kripke semantics for intuitionistic logic was inspired by the modal embedding. We use the semantics to get a syntactic proof of the embedding. The accessibility relation is a pre-order 8x:x 6 x (Ref) 8xyz(x 6 y& y 6 z ⊃ x 6 z) (Trans) Forcing relation as in basic modal logic except for implication: x A ⊃ B iff 8y(x 6 y & y A ⊃ y B) Gives the rules: x y; x : A ⊃ B; Γ ) ∆; y : A x y; x : A ⊃ B; y : B; Γ ) ∆ 6 6 L⊃ x 6 y; x : A ⊃ B; Γ ) ∆ x y; y : A; Γ ) ∆; y : B 6 R ⊃ Γ ) ∆; x : A ⊃ B Rule R ⊃ has the condition that y is not in the conclusion. Intuitionistic logic Initial sequents of G3K modified to have monotonicity of the forcing relation. Enough to have monotonicity with respect to atomic formulas x 6 y; x : P; Γ ) ∆; y : P Monotonicity w.r.t. arbitrary formulas admissible. The mathematical rules for the accessibility relation 6 are the rules Ref and Trans. The system G3I Initial sequents: x 6 y; x : P; Γ ) ∆; y : P Logical rules: As in G3K for &, _, ?, x y; x : A ⊃ B; Γ ) y : A; ∆; x y; x : A ⊃ B; y : B; Γ ) ∆ 6 6 L⊃ x 6 y; x : A ⊃ B; Γ ) ∆ x y; y : A; Γ ) ∆; y : B 6 R⊃ Γ ) ∆; x : A ⊃ B Order rules: x 6 x; Γ ) ∆ x 6 z; x 6 y; y 6 z; Γ ) ∆ Ref Trans Γ ) ∆ x 6 y; y 6 z; Γ ) ∆ Rule R⊃ has the condition that y must not be in Γ; ∆.
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