Commit d659b15d by jfschaefer

### today's stuff (semantics of FOL as view into set theory)

parent d1a1aeac
 namespace https://mathhub.info/Teaching/lbs1920 ❚ import krmt http://mathhub.info/Teaching/KRMT ❚ theory proplog : ur:?LF = prop : type ❘ # o ❙ conjunction : prop ⟶ prop ⟶ prop ❘ # 1 ∧ 2 prec 60❙ negation : prop ⟶ prop ❘ # ¬ 1 prec 80 ❙ disjunction : prop ⟶ prop ⟶ prop ❘ # 1 ∨ 2 prec 50 ❘ = [A,B] (¬ ( ¬ A ∧ ¬ B )) ❙ implication : prop ⟶ prop ⟶ prop ❘ # 1 ⇒ 2 prec 40 ❘ = [A,B] ¬ A ∨ B ❙ biimplication : prop ⟶ prop ⟶ prop ❘ # 1 ⇔ 2 prec 30 ❘ = [A, B] (A ⇒ B) ∧ (B ⇒ A) ❙ ❚ // theory TruthValues : http://mathhub.info/MitM/Foundation?DescriptionOperator = ❚ theory TruthValues : krmt:?Meta = // booleans : 𝒰 100 ❘ # ℬ ❙ // truth values ❙ booleans : type ❘ # ℬ ❙ // truth values ❙ true : ℬ ❘ # T ❙ false : ℬ ❘ # F ❙ // switch : ℬ ⟶ ℬ ❘ = [b] if b ≐ T then F else T ❙ // switch F to T and T to F ❙ switch : ℬ ⟶ ℬ ❘ = [b] if_then_else ℬ (b ≐ T) F T ❙ max : ℬ ⟶ ℬ ⟶ ℬ ❘ = [a,b] if_then_else ℬ (a ≐ T) b F ❙ // maximum element (T > F) ❙ ❚ view proplogSemantics : ?proplog -> ?TruthValues = prop = ℬ ❙ conjunction = max ❙ negation = switch ❙ ❚ theory folSyntax : ur:?LF = include ?proplog ❙ individuals : type ❘ # ι ❙ forall : (ι ⟶ o) ⟶ o ❘ # ∀ 1 ❙ // example: (∀ [x:ι] ((p x) : o)) : o ❙ exists : (ι ⟶ o) ⟶ o ❘ # ∃ 1 ❘ = [p] ¬ (∀ [x] ¬ (p x)) ❙ ❚ theory folModels : krmt:?Meta = include ☞http://mathhub.info/Teaching/KRMT?TypedSets ❙ include ?TruthValues ❙ object : type ❘ # 𝒪 ❙ domain = fullset 𝒪 ❘ # 𝒟 ❙ // set of objects ❙ set_equal : set 𝒪 ⟶ set 𝒪 ⟶ ℬ ❘ = [P1, P2] if_then_else ℬ (P1 ≐ P2) T F ❘ # 1 == 2 ❙ is_the_universe : set 𝒪 ⟶ ℬ ❘ = [P] P == 𝒟 ❙ ❚ view folSemantics : ?folSyntax -> ?folModels = include ?proplogSemantics ❙ individuals = object ❙ // forall : (object ⟶ ℬ) ⟶ ℬ ❙ forall = [P : object ⟶ ℬ] if_then_else ℬ (in ℬ F (im P 𝒟)) F T ❙ ❚ theory MLO : ur:?LF = include ?proplog ❙ box : o ⟶ o ❘ # □ 1 prec 20 ❙ diamond : o ⟶ o ❘ # ◇ 1 prec 20 ❘ = [x] ¬ □ ¬ x ❙ ❚ // theory KripkeFrames : ur:?LF = include ?TruthValues ❙ worlds : type ❘ # 𝒲 ❙ acr : 𝒲 ⟶ 𝒲 ⟶ ℬ ❘ # 1 ℛ 2 ❙ // accessibility relation ❙ allAccessible : ❚
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