Duality for first order logic

Size: px

Start display at page:

Download "Duality for first order logic"

Edgar Haynes
5 years ago
Views:

1 Duality for first order logic Dion Coumans Radboud University Nijmegen Aio-colloquium, June 2010

2 Outline 1 Introduction to logic and duality theory 2 Algebraic semantics for classical first order logic: Boolean hyperdoctrines 3 Dual notion of Boolean hyperdoctrines: Indexed Stone spaces 4 Duality for classical first order logic 5 Future work

3 Logic We doing logic we analyse mathematical reasoning Classical Propositional Logic (CPL) Propositional variables: p 0, p 1,... Build formulas using connectives:,,,, : p 0 p 1, p 0 (p 1 p 2 ), p 0,... Reasoning rules: DER φ ψ φ DER1 DER2 φ ψ φ ψ DER φ φ ψ Notation: φ ψ says ψ is derivable from φ.

4 Logic Questions Which formulas are derivable? Is it decidable wether a formula is derivable? Does the logic have the interpolation property, that is, for all formulas φ(p, q) and ψ(p, r) with φ(p, q) ψ(p, r), there exists a formula θ(p) s.t. φ(p, q) θ(p) and θ(p) ψ(p, r).

5 Algebraizing logic Relate a logic to a class of algebras: Logic Class of algebras

6 Algebraizing logic Relate a logic to a class of algebras: CPL Boolean algebras

7 Algebraizing logic Relate a logic to a class of algebras: CPL Boolean algebras A Boolean algebra is a structure A = (A,,,, 0, 1) s.t. a (b c) = (a b) c a b = b a a (a b) = a a (b c) = (a b) (a c) a a = 1 a (b c) = (a b) c a b = b a a (a b) = a a (b c) = (a b) (a c) a a = 0

8 Algebraizing logic Relate a logic to a class of algebras: CPL Boolean algebras A Boolean algebra is a structure A = (A,,,, 0, 1) s.t. a (b c) = (a b) c a b = b a a (a b) = a a (b c) = (a b) (a c) a a = 1 a (b c) = (a b) c a b = b a a (a b) = a a (b c) = (a b) (a c) a a = 0 Example: (P(X),,, () c,, X)

9 Algebraizing logic Relate a logic to a class of algebras: CPL Boolean algebras Start from a set of propositional variables P. Consider F m(p ) and define: φ ψ φ ψ and ψ φ (F m(p )/,,,, [ ], [ ]) is a Boolean algebra, where [φ] [ψ] = [φ ψ] [φ] [ψ] = [φ ψ] [φ] = [ φ] We call this the Lindenbaum algebra.

10 Algebraizing logic Interpreting a logic in a Boolean algebra: Every map f : P A (valuation) extends to a unique homomorphism f : F m(p ) A. We say φ is valid in A if, for every valuation f, f(φ) = 1. Notation: A = φ.

11 Algebraizing logic Interpreting a logic in a Boolean algebra: Every map f : P A (valuation) extends to a unique homomorphism f : F m(p ) A. We say φ is valid in A if, for every valuation f, f(φ) = 1. Notation: A = φ. Soundness and completeness theorem: For every formula φ, CP L φ = BA φ

12 Duality in logic Logic Class of algebras

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

14 Duality in logic CPL Boolean algebras

15 Duality in logic CPL Boolean algebras Stone spaces

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

17 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

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

19 Duality for first order logic Classical first order logic??

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

21 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.

22 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,...} Build terms from the variables using function symbols and constants: x 1, f 0 (x 0 ), f 2 (f 1 (x 0, f 3 (c 1 ))),... Build formulas from the terms using,,,,,, and relation symbols: R 0 (x 0 ), R 0 (x 1 ) R 1 (x 0, c 0 ), x1 R 1 (f 0 (x 0 ), x 1 ),...

23 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?

24 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.

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

26 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)

27 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)

28 Algebraic semantics for first order logic Contexts and substitutions form a 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)

29 Algebraic semantics for first order logic Contexts and substitutions form a 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 ]

30 Algebraic semantics for first order logic Contexts and substitutions form a 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 [...]

31 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)

32 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 ]

33 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 ))

34 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 ))

35 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)

36 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.

37 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

38 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))

39 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

40 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)

41 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)

42 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.

43 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.

44 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

45 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.

Canonical extension of coherent categories

Canonical extension of coherent categories Dion Coumans Radboud University Nijmegen Topology, Algebra and Categories in Logic (TACL) Marseilles, July 2011 1 / 33 Outline 1 Canonical extension of distributive