Duality for first order logic

Transcription

1 Duality for first order logic Dion Coumans Radboud University Nijmegen MLNL, May 2010

2 Outline 1 Duality in logic 2 Algebraic semantics for classical first order logic: Boolean hyperdoctrines 3 Dual notion of a Boolean hyperdoctrine: Indexed Stone spaces 4 Duality for classical first order logic 5 Future work

3 Duality in logic Logic Class of algebras Class of dual structures

4 Duality in logic CPL Boolean algebras Stone spaces

5 Duality in logic CPL Boolean algebras Stone spaces B (Uf(B), τ B ) Cl(X) X

6 Duality in logic CPL Boolean algebras Stone spaces B h C Uf(C) h 1 Uf(B) Cl(Y ) f 1 Cl(X) X f Y

7 Duality in logic CPL over a set of variables X Lindenbaum algebra of formulas over X Maps X 2 valuations

8 Duality for first order logic Classical first order logic??

9 Duality for first order logic Classical first order logic Boolean hyperdoctrines? 1 What are Boolean hyperdoctrines?

10 Duality for first order logic Classical first order logic Boolean hyperdoctrines Indexed Stone spaces 1 What are Boolean hyperdoctrines? 2 Identify the dual notion of a Boolean hyperdoctrine.

11 Algebraic semantics for first order logic We start from Signature: Σ = (f 0,..., f k 1, R 0,..., R l 1, c 0,..., c m 1 ) Set of variables: X = {x 0, x 1,...} Question: What properties does the collection of all formulas over Σ have?

12 Algebraic semantics for first order logic We start from Signature: Σ = (f 0,..., f k 1, R 0,..., R l 1, c 0,..., c m 1 ) Set of variables: X = {x 0, x 1,...} Question: What properties does the collection of all formulas over Σ have? First observation: For each n N, (F m(x 0,..., x n 1 ), ) is a Boolean algebra.

13 Algebraic semantics for first order logic [] [x 0 ] [x 0, x 1 ]...

14 Algebraic semantics for first order logic c φ(c) φ(x 0 ) [] [x 0 ] [x 0, x 1 ]... Substitutions: x 0 c φ(x 0 ) φ(c)

15 Algebraic semantics for first order logic c φ(c) φ(x 0 ) [] [x 0 ] [x 0, x 1 ]... Substitutions: x 0 c φ(x 0 ) φ(c)

16 Algebraic semantics for first order logic Contexts and substitutions form category B: Objects: natural numbers (contexts) Morphism n m: m-tuple t 0,..., t m 1 s.t. F V (t i ) {x 0,..., x n 1 } x 0 [] c [x 0 ] x 0, f(x 0 ) [x 0, x 1 ]... c, f(c)

17 Algebraic semantics for first order logic Contexts and substitutions form category B: Objects: natural numbers (contexts) Morphism n m: m-tuple t 0,..., t m 1 s.t. F V (t i ) {x 0,..., x n 1 } This category has finite products: x 0, x 1 [x 0, x 1 ] [x 0, x 1, x 2 ] x 2 [x 0 ]

18 Algebraic semantics for first order logic Contexts and substitutions form category B: Objects: natural numbers (contexts) Morphism n m: m-tuple t 0,..., t m 1 s.t. F V (t i ) {x 0,..., x n 1 } This category has finite products: x 0, x 1 x 2 [x 0, x 1 ] [x 0, x 1, x 2 ] [x 0 ] t 0, t 1, s 0 t 0, t 1 s 0 [...]

19 Algebraic semantics for first order logic Formulas and substitutions: functor B op BA n F m(x 0,..., x n 1 ) n t 0,...,t m 1 m F m(x 0,..., x m 1 ) F m(x 0,..., x n 1 ) φ(x 0,..., x m 1 ) φ(t 0,..., t m 1 ) φ(c, f(c)) φ(x0, f(x0)) φ(x0, x1) [] c [x 0 ] x 0, f(x 0) [x 0, x 1 ]... c, f(c)

20 Algebraic semantics for first order logic Existential quantification: related to the inclusion map i φ(x 0 ) x1 ψ(x 0, x 1 ) x 0 [x 0 ] [x 0, x 1 ]

21 Algebraic semantics for first order logic Existential quantification: related to the inclusion map i φ(x 0 ) x1 ψ(x 0, x 1 ) x 0 [x 0 ] [x 0, x 1 ] x1 (ψ(x 0, x 1 )) x0 φ(x 0 ) ψ(x 0, x 1 ) x0,x 1 i(φ(x 0 ))

22 Algebraic semantics for first order logic Quantification: interaction with substitutions f(x 0 ) x1 ψ(x 0, x 1 ) f(x 0 ), x 1 [x 0 ] [x 0, x 1 ] x1 (ψ(x 0, x 1 ))[f(x 0 )/x 0 ] = x1 (ψ(f(x 0 ), x 1 )) (Beck-Chevalley)

23 Algebraic semantics for first order logic A Boolean hyperdoctrine is a functor F : B op BA s.t. 1 B is a category with finite products; 2 for all I, J B, F(π I,J ): F(I) F(I J) has a left adjoint I,J such that, for all I u K in B, F(K J) K,J F(K) F(u id) F(I J) I,J F(I) F(u) commutes.

24 Algebraic semantics for first order logic Examples of Boolean hyperdoctrines: Syntactic hyperdoctrine B = contexts and substitutions F : B op BA n F m(x 0,..., x n 1 ) Subset hyperdoctrine B = Set P : B op BA A powerset of A

25 Duality for first order logic B I BA F(I) Uf Cl Stone spaces Uf(F(I)) u J F(u) F(J) F(u) 1 Uf(F(J))

26 Duality for first order logic B I BA F(I) Uf Cl Stone spaces Uf(F(I)) u J F(u) F(J) F(u) 1 Uf(F(J)) This gives us a dual equivalence between: Functors F : B op BA F Cl G Functors G : B StSp Uf F G

27 Duality for first order logic F : B op BA G : B StSp F(π I,J ) has a left adjoint I,J for all I u K, F(K J) F(u id) F(I J) commutes. K,J I,J F(K) F(I) F(u)

28 Duality for first order logic F : B op BA F(π I,J ) has a left adjoint I,J G : B StSp G(π I,J ) is an open map for all I u K, F(K J) F(u id) F(I J) commutes. K,J I,J F(K) F(I) F(u)

29 Duality for first order logic F : B op BA F(π I,J ) has a left adjoint I,J for all I u K, G : B StSp G(π I,J ) is an open map for all I u K, F(K J) K,J F(K) z G(I J) G(π I,J ) G(I) y F(u id) F(I J) I,J F(I) F(u) G(u id) x G(K J) G(π K,J ) G(u) G(K) commutes. G(u)(x) = G(π K,J )(y) implies there exists z G(I J) s.t. G(π I,J )(z) = x G(u id)(z) = y.

30 Duality for first order logic Boolean hyperdoctrines Functors F : B op BA s.t. 1 B has finite products; 2 F(π I,J ) has a left adjoint I,J and for all I u K, Indexed Stone spaces Functors G : B StSp s.t. 1 B has finite products; 2 G(π I,J ) is an open map and for all I u K, F(K J) K,J F(K) G(I J) G(π K,J ) G(I) F(u id) F(I J) I,J F(I) F(u) G(u id) G(K J) G(π I,J ) G(u) G(K) commutes. is epicartesian.

31 Duality for first order logic Duality theorem for classical first order logic: The category of Boolean hyperdoctrines and the category of indexed Stone spaces are dually equivalent. Boolean hyperdoctrines F Cl G Indexed Stone spaces Uf F G

32 Future work Having a duality for classical first order logic we would like to: 1 Describe dual structures for non-classical first order logics. 2 Obtain information about these first order logics via studying their dual structures. 3 In particular: study the interpolation property dually.