William J. CookLogic via Topoi NCSU William J. CookLogic via Topoi. Logic via Topoi. CSC591Z: Computational Applied Logic Final Presentation

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1 Logic via Topoi CSC591Z: Computational Applied Logic Final Presentation William J. Cook Wednesday, December 8,

2 Categories 2

3 Def: A category C is a collection of objects Ob(C) and a collection of arrows Ar(C) between objects. We associate two objects to every arrow f, a source A and target B. This is denoted: f : A B. Let f : A B and g : B C be arrows, then there exists a unique arrow g f : A C. Let f : A B, g : B C, andh : C D be arrows, then h (g f) = (h g) f. For each object B there is map 1 B : B B such that for given arrows f : A B and g : B C, wehave1 B f = f and g 1 B = g. 3

4 Def: A category C is a collection of objects Ob(C) and a collection of arrows Ar(C) between objects. Let f : A B and g : B C be arrows, then there exists a unique arrow g f : A C. 4

5 Def: A category C is a collection of objects Ob(C) and a collection of arrows Ar(C) between objects. Let f : A B and g : B C be arrows, then there exists a unique arrow g f : A C. 5

6 Def: A category C is a collection of objects Ob(C) and a collection of arrows Ar(C) between objects. For each object B there is map 1 B : B B such that for given arrows f : A B and g : B C, wehave1 B f = f and g 1 B = g. 6

7 Examples Ex: SET is a category whose objects are sets and arrows are functions. Ex: The collection of all groups is a category whose arrows are group homomorphisms. Ex: The collection of all vector spaces is a category whose arrows are linear maps. Ex: The collection of all topological spaces is a category whose arrows are continuous maps. Ex: Let G be a group. Consider G. The only object is G itself, and each element of G is an arrow. We compose arrows using G s multiplication. 7

8 Topoi 8

9 Def: A topos (plural topoi or toposes) is a category E with the following properties: E is finitely complete. E is finitely co-complete. E has exponentiation. E has a subobject classifier. 9

10 Figure: A finite diagram. 10

11 Figure: A cone is an object... 11

12 Figure:...and a collection of maps... 12

13 Figure:...which are compatible. 13

14 Example: Products Figure: A discrete diagram. 14

15 Example: Products Figure: The limit (universal cone) of the discrete diagram is the product. 15

16 Figure: To get co-cones...just flip the arrows around. 16

17 Example: Co-Products Figure: The co-limit (universal co-cone) of the discrete diagram is the co-product. 17

18 Exponentiation A category E has exponentiation if (i) for every A, B Ob(E) we have that A B exists and (ii) for every pair of E-objects A and B there is an E-object B A and a E-arrow ev : B A A B such that for any E-object C and E-arrow g : C A B there is a unique E-arrow ĝ : C B A such that ev (ĝ 1 A )=g. ev is the evaluation map. Notice that the correspondence between g and ĝ gives a bijection from Hom(C A, B) tohom(c, B A ). 18

19 Exponentiation Ex: In SET given two sets A and B we can form a new set: B A = {f : A B} We have the map ev : B A A B defined by ev(f, a) =f(a). 19

20 Subobject Classifier 20

21 Subobject Classifier 21

22 Subobject Classifier 22

23 Let E be a category with a terminal object 1. A subobject classifier for E is a E-object Ω paired with an arrow :1 Ω such that for each monic f : A B there is a unique E-arrow χ f : B Ω such that A f B! χ f 1 Ω is a pull-back square. 23

24 Logic in a Topos 24

25 Consider a topos with initial object 0 and terminal object 1. We have the monic 1 1 :1 1. Notice that = χ 11 (true). We have the unique monic 0 1 :0 1. Define = χ 01 (false). Define = χ (not)., denotes the product of two arrows. Define = χ f where f =, : 1 Ω Ω (and). [, ] denotes the co-product of two arrows. Let g =[, 1 Ω, 1 Ω, ] : Ω+Ω Ω Ω. Define = χ g (or). 25

26 Let T : L 0 Hom E (1, Ω) be any function (truth assignment). Let V T : L Hom E (1, Ω) be defined as follows: V T (a) =T (a) for all a L 0. That is V T extends the truth assignment. V T (( ϕ)) = V T (ϕ) whenever ϕ L. V T ((ϕ ψ)) = V T (ϕ),v T (ψ) whenever ϕ, ψ L. V T ((ϕ ψ)) = V T (ϕ),v T (ψ) whenever ϕ, ψ L. Any function V : L Hom E (1, Ω) built up in this manner is called an E- valuation. Let ϕ L. Then ϕ is E-valid, denoted E ϕ if and only if V (ϕ) = :1 Ω for every E-valuation V. 26

27 Let ϕ, ψ, τ L. ϕ (ϕ ϕ) (ϕ ψ) (ψ ϕ) (ϕ ψ) ((ϕ τ) (ψ τ)) ((ϕ ψ) (ψ τ)) (ϕ τ) ψ (ϕ ψ) (ϕ (ϕ ψ)) ψ ϕ (ϕ ψ) (ϕ ψ) (ψ ϕ) 27

28 ((ϕ τ) (ψ τ)) ((ϕ ψ) τ) ( ϕ) (ϕ ψ) ((ϕ ψ) (ϕ ( ψ))) ( ϕ) These axioms along with the inference rule: From ϕ ψ and ϕ conclude ψ (modus ponens) make up the logical system IL (intuitionist logic). Classical logic, denoted CL, is exactly the same as IL expect that we add the following axiom: ϕ ( ϕ) (The Law of Excluded Middle) If ϕ is provable from the axioms of CL, we write CL ϕ. If ϕ is provable from the axioms of IL, we write IL ϕ. (Thus IL ϕ implies CL ϕ.) 28

29 Thm: Let E be a topos. The following are equivalent: 1. E ϕ if and only if CL ϕ for every sentence ϕ 2. E ϕ ϕ for every sentence ϕ 3. Sub(1) (the collection of subobjects of the terminal object) is a Boolean algebra. Thm: Let ϕ L(a proposition). ϕ is provable in intuitionist logic ( IL ϕ) if and only if ϕ is valid in every topos (for every topos E, wehavethate ϕ). 29