First-Order Logical Duality Henrik Forssell June 2008 First-Order - PowerPoint PPT Presentation

First-Order Logical Duality 1/25 First-Order Logical Duality Henrik Forssell June 2008

First-Order Logical Duality Introduction Overview 2/25 Algebra-geometry, syntax-semantics 1 Stone duality—the fact that the ‘algebraic’ category of Boolean algebras is dual to the ‘geometric’ category of Stone spaces BA op ≃ Stone has a logical interpretation as a syntax-semantics duality for classical propositional logic. 2 We present a generalization to first-order logic, which yields the propositional logical Stone duality as a special case.

First-Order Logical Duality Introduction Overview 3/25 Table of Contents 1 Introduction Stone duality—the propositional case Logical duality—the setup 2 Representation Theorem Outline of main representation result 3 Syntax-Semantics Duality The full text can be downloaded from http://folk.uio.no/jonf/

First-Order Logical Duality Introduction Stone Duality—The Propositional Case 4/25 Logical interpretation - algebras A propositional theory, T can be seen as a Boolean algebra. Definition For a propositional theory T , the Lindenbaum-Tarski algebra , L T of T consists of equivalence classes [ φ ] of formulas, where φ ∼ ψ ⇔ T ⊢ φ ↔ ψ, ordered by provability: [ φ ] ≤ [ ψ ] ⇔ T ⊢ φ → ψ. The Lindebaum-Tarski (LT) algebra of a propositional theory is a Boolean algebra. Conversely, any Boolean algebra is the LT-algebra of a classical propositional theory B ∼ = L T B .

First-Order Logical Duality Introduction Stone Duality—The Propositional Case 5/25 Logical interpretation - Stone spaces For a propositional theory T , a (2-valued) model is an assignment of formulas to the values 1 (true) and 0 (false) which preserves provability, and so can be considered to be a morphism of Boolean algebras � 2 . L T Conversely, such a morphism can be seen as a model of T . Alternatively, these morphisms can be seen as ultra-filters of L T . Therefore, the Stone space corresponding to L T can be presented as the set of ‘models’ X L T := Hom BA ( L T , 2) equipped with the ‘logical’ topology defined by basic opens U φ = { M � T M � φ } for φ a formula of T .

� � � � � � First-Order Logical Duality Introduction Stone Duality—The Propositional Case 6/25 Representing Boolean algebras as spaces of models 1 A Boolean algebra B can be recovered from its Stone space of models (or ultra-filters) X B . E.g. as follows. The map U : B → O ( X B ) defined by b �→ { f ∈ X B f ( b ) = 1 } lifts to an isomorphism of frames ˆ U , ˆ U Idl ( B ) Idl ( B ) Idl ( B ) Idl ( B ) O ( X B ) O ( X B ) O ( X B ) O ( X B ) ∼ = ����������� ����������� P U B B B B where Idl ( B ) is the ideal completion of B ; P : B → Idl ( B ) is the principal ideal embedding.

First-Order Logical Duality Introduction Stone Duality—The Propositional Case 7/25 Representing Boolean algebras as spaces of models 2 Corollary B can be recovered as the compact elements of O ( X B ) , i.e. as the compact open subsets of X B . Since X B is Stone, in particular compact and Hausdorff, that means Corollary B can be recovered as the lattice of clopen subsets of X B . The latter can be identified with the continuous functions from X B into the discrete (Stone) space 2, CL ( X B ) ∼ = Hom Stone ( X B , 2)

� � First-Order Logical Duality Introduction Stone Duality—The Propositional Case 8/25 Stone duality Sending a Boolean algebra to its Stone space of ‘models’ is (contravariantly) functorial, as is recovering a Boolean algebra as the clopens of a Stone space, and we get the familiar Stone duality: Hom Stone ( − , 2) BA op BA op ≃ Stone Stone Hom BA ( − , 2)

First-Order Logical Duality Introduction The Setup 9/25 Logical Duality - Table SYNTAX Intermediate SEMANTICS Boolean algebras Frames Stone spaces Class. B ∼ X B ∼ = L T = Hom BA ( B , 2) Prop. Idl ( B ) ∼ Logic algebraic object = space of models built from syntax O ( X B ) Bool. coh. cats Topoi Top. gpds B ≃ C T G B ⇒ X B FOL Sh( B ) top. grpd of algebraic object ≃ models and built from syntax Sh G B ( X B ) isomorphisms

First-Order Logical Duality Introduction The Setup 10/25 Syntactical categories - C T For a first-order theory T , the syntactical category C T of T has as objects formulas-in-context [ � x φ ] of T , with arrows classes of T -provably equivalent formulas-in-context � [ � | [ � x ,� y σ ] | : [ � x φ ] y ψ ] such that σ is T -provably a functional relation from φ to ψ . With T a classical f.o. theory, C T is a Boolean (coherent) category (BC). Moreover, every BC is, up to equivalence, the syntactic category of a classical f.o. theory, so that BCs represent first-order logical theories.

First-Order Logical Duality Introduction The Setup 11/25 Models Ordinary set-models of T correspond to coherent functors � Sets , C T Mod T ( Sets ) ≃ Hom Coh ( C T , Sets ) T -model isomorphisms correspond to invertible natural transformations between these coherent functors. Accordingly, the groupoid (category with all arrows invertible) of T -models and isomorphisms between them can be represented as the groupoid of coherent set-valued functors from C T with invertible natural transformations between them: In order to have sets of models and isomorphisms, lets say T (and C T ) is countable, and we only consider the countable models, i.e. those functors that take values in countable sets.

� � � � First-Order Logical Duality Introduction The Setup 12/25 Semantical groupoids For a countable Boolean coherent category B , then, we consider the groupoid i s c � G B G B × X B G B G B G B G B G B X B X B X B Id t of countable ‘models’ (coherent functors) and isomorphisms between them. We equip the sets X B and G B with topologies to make this a topological groupoid.

� � � First-Order Logical Duality Introduction The Setup 13/25 The topology on X B Definition The coherent topology on X B is the coarsest containing all sets of the form { M ∈ X B ∃ x ∈ M ( A ) . M ( f 1 )( x ) = b 1 ∧ . . . M ( f n )( x ) = b n } given by a finite span in B , � ������� A A A A � � f 1 � f n � f i � � � B 1 B 1 . . . B i B i B i B i . . . B n B n and a list b 1 , . . . , b n ∈ Sets c .

First-Order Logical Duality Introduction The Setup 14/25 The topology on G B Definition The coherent topology on G B is the coarsest such that the source and target maps G B ⇒ X B are both continuous, and containing all sets of the form U A , a �→ b = { f : M → N a ∈ M ( A ) ∧ f A ( a ) = b } given by an object A in B and a , b ∈ Sets c .

� � First-Order Logical Duality Introduction The Setup 15/25 Sheaves: Sh ( X ) For a space X , the topos of sheaves on X Sh ( X ) consists of local homeomorphisms over X f � B A A B � � � ������� � � � � a � � b X X If X is the space of objects of a topological groupoid: s G � X t the topos of equivariant sheaves , Sh G ( X ), is constructed by equipping sheaves on X with an action by G .

� � � First-Order Logical Duality Introduction The Setup 16/25 Equivariant sheaves: Sh G ( X ) Sh G ( X ) has as objects pairs � a : A → X , α � where the first component is an element of Sh ( X ) and the second component is a continuous action α G × X A A � g : y → z , d � �→ α ( g , d ) An arrow between objects � a : A → X , α � and � b : B → X , β � is an arrow f : A → B of Sh ( X ) which commutes with the actions: 1 G × f � G × X B G × X A G × X A G × X B α � β A A B B f

� � � � � � First-Order Logical Duality Introduction The Setup 17/25 The topos of coherent sheaves For a coherent category C , the topos of coherent sheaves —i.e. sheaves for the coherent, or finite epimorphic families, coverage—Sh( C ) is the ‘free topos on C ’, in the sense that coherent functors from C into a topos E correspond to geometric morphisms from E to Sh( C ): f ∗ Sh( C ) Sh( C ) Sh( C ) Sh( C ) ⊤ E E E E ����������������� ����������������� f ∗ y y F F C C C C C can be recovered, up to pretopos completion, from Sh( C ) as the coherent objects, or, if C is Boolean, as the compact decidable objects.

First-Order Logical Duality Introduction The Setup 18/25 Logical Duality - Table SYNTAX Intermediate SEMANTICS Boolean algebras Frames Stone spaces Class. B ∼ X B ∼ = L T = Hom BA ( B , 2) Prop. Idl ( B ) ∼ Logic algebraic object = space of models built from syntax O ( X B ) Bool. cats Topoi Top. gpds B ≃ C T G B ⇒ X B FOL Sh( B ) top. grpd of algebraic object ≃ models and built from syntax Sh G B ( X B ) isomorphisms

First-Order Logical Duality Representation Theorem 19/25 Stone representation theorem 1 The Stone representation theorem says that a Boolean algebra can be embedded in the lattice of subsets of a set � P ( X B ) B � � By equipping that set with a topology, on can recover B as the compact open sets. 2 Generalizing, we show that a (countable) Boolean category can be ‘embedded’ in the topos of sets over a set � Sets / X B B � � By equipping that set with a topology and introducing continuous actions, on can recover B as the compact decidable objects.