Examples of Tautology (A → B) → ((B → C) → (A → C)) ((((A → B) → (¬C → ¬D)) → C) → E) → ((E → A) → (D → A)) ¬(A ∧ ¬A) (A ∧ ¬A) → B ((A → B) → A) → A ¬¬A → A (A → B) ↔ (¬B → ¬A) ¬(A ∧ B) ↔ (¬A ∨ ¬B) ¬(A ∨ B) ↔ (¬A ∧ ¬B)
(A → B) → ((B → C) → (A → C)) ((((A → B) → (¬C → ¬D)) → C) → E) → ((E → A) → (D → A)) ¬(A ∧ ¬A) (A ∧ ¬A) → B ((A → B) → A) → A ¬¬A → A (A → B) ↔ (¬B → ¬A) ¬(A ∧ B) ↔ (¬A ∨ ¬B) ¬(A ∨ B) ↔ (¬A ∧ ¬B) Examples of Tautology Law of Contradiction Principle of Explosion Double Negation Elimination Peirce’s Law Law of Contraposition De Morgan’s Laws Transitive Law Meredith’s Axiom
Formal System Axiom l Base premises (part of tautologies) Inference Rules l Deduces new formula from existing ones Using Axioms and Inference Rules to deduce Theorem (all possible tautologiesʣ
Styles of Formal System •Gentzen Style l Many inference rules •Natural Deduction l + No axioms •Sequent Calculus l + 1 axiom (identity) •Hilbert Style •Many axioms + 1 inference rule (modus ponens)
Hilbert Style A → B A B B → (A → B) (A → (B → C)) → ((A → B) → (A → C)) ¬¬A → A A → A ∨ B ⊥ → A B → A ∨ B (A → C) → ((B → C) → ((A ∨ B) → C)) A ∧ B → A A ∧ B → B A → (B → (A ∧ B)) (A → ¬A) → ¬A ¬A → (A → B) Modus ponens is the only available inference rule
/// `B ===> A → B` /// - Note: More formally, A ⊢ B ===> ⊢ A → B. func implyIntro(_ b: B) -> ((A) -> B) { { _ in b } } /// `A, A → B ===> B` (Modus ponens) func implyElim(_ f: (A) -> B, _ a: A) -> B { f(a) } A → B A B B A → B ⋮ [A]
/// `A, B ===> A ∧ B` func andIntro(_ a: A, _ b: B) -> (A, B) { (a, b) } /// `A ∧ B ===> A` func andElim1(_ ab: (A, B)) -> A { ab.0 } /// `A ∧ B ===> B` func andElim2(_ ab: (A, B)) -> B { ab.1 } A B A ∧ B A ∧ B A A ∧ B B
/// `A ===> A ∨ B` func orIntro1(_ a: A) -> Either { .left(a) } /// `B ===> A ∨ B` func orIntro2(_ b: B) -> Either { .right(b) } /// `A ∨ B, A → C, B → C ===> C` func orElim( _ e: Either, _ ac: (A) -> C, _ bc: (B) -> C ) -> C { switch e { case let .left(a): return ac(a) case let .right(b): return bc(b) } } A A ∨ B B A ∨ B A ∨ B C C C ⋮ [A] ⋮ [B]
struct Not { let f: (A) -> Never } /// `A ⊢ ⊥ ===> ⊢ ¬A` func notIntro(_ x: Never) -> Not { Not { _ in x } } /// `A, ¬A ===> ⊥` (NOTE: not using `fatalError`) func notElim(_ a: A, _ notA: Not) -> Never { notA.f(a) } ⊥ ¬A A ¬A ⊥ ⋮ [A]
Hilbert Style B → (A → B) (A → (B → C)) → ((A → B) → (A → C)) A → A ∨ B B → A ∨ B (A → C) → ((B → C) → ((A ∨ B) → C)) A ∧ B → A A ∧ B → B A → (B → (A ∧ B)) A → B A B Modus ponens is the only available inference rule ¬¬A → A ⊥ → A (A → ¬A) → ¬A ¬A → (A → B)
(A → C) → ((B → C) → ((A ∨ B) → C)) A → A ∨ B B → A ∨ B A ∧ B → A A ∧ B → B A → (B → (A ∧ B)) ¬¬A → A ⊥ → A (A → ¬A) → ¬A ¬A → (A → B) Hilbert Style A → B A B Modus ponens is the only available inference rule (A → (B → C)) → ((A → B) → (A → C)) A ∨ B := ¬A → B := ¬B → A A ∧ B := ¬(A → ¬B) := ¬(B → ¬A) ¬A := A → ⊥ NOTE: ∨, ∧, ¬ can be rewritten using → and ⊥ only B → (A → B)
/// `K = A → (B → A)` func k(_ a: A) -> (B) -> A { { _ in a } } /// `S = (A → (B → C)) → ((A → B) → (A → C))` func s(_ a: @escaping (C) -> (A) -> B) -> (_ b: @escaping (C) -> A) -> (C) -> B { { b in { c in a(c)(b(c)) } } }
/// `I = SKK: A → A` func i(_ a: A) -> A { let k_: (A) -> (A) -> A = k let skk: (A) -> A = s(k)(k_) return skk(a) } i("hello") // "hello" i(123) // 123
((A → B) → A) → A A ∨ ¬A ¬¬A → A ¬(A ∧ B) → (¬A ∨ ¬B) (A → B) → (¬A ∨ B) (¬B → ¬A) → (A → B) Law of Excluded Middle Double Negation Elimination (One of) Law of Contraposition (One of) De Morgan’s Law Material Implication Peirce’s Law NOTE: Converse is tautology Tautologies that can’t be implemented in Swift
/// `((A → B) → A) → A` func peirceLaw() -> (((A) -> B) -> A) -> A { { (aba: ((A) -> B) -> A) in let ab: (A) -> B = { fatalError("Can't impl ") }() let a = aba(ab) return a } }
Curry-Howard Correspondence Intuitionistic Logic Swift Programming Proposition Type Proof Program → Introduction Function / Closure Modus Ponens Function Application Axiom Global Variable Assumption Free Variable Discharged Assumption Bound Variable
Kripke Semantics • Kripke Frame l Ordered set of possible world e.g. → → • Kripke Model l Assigning inheriting truth value for every possible world on the Frame • Tautology in Intuitionistic Logic … For any frame and for any valuation (= for any model), formula is always true "͕ಘΒΕΔͱɺ #ࣗಈతʹಘΒΕΔ
Classical Logic ⊂ Intuitionistic Logic Minimal Intuitionistic Classical A ∨ ¬A ⊥ → A → , ∧ , ∨ , ¬, ⊥ Formulas using A → A ∨ B A ∧ B → A Law of Contradiction Law of Excluded Middle ⋮ NOTE: Doesn’t mean Intuitionistic Logic owns Law of Excluded Middle ”directly” (See next slide)
((A → B) → A) → A A ∨ ¬A ¬¬A → A ¬(A ∧ B) → (¬A ∨ ¬B) (A → B) → (¬A ∨ B) (¬B → ¬A) → (A → B) Law of Excluded Middle Double Negation Elimination (One of) Law of Contraposition (One of) De Morgan’s Law Material Implication Peirce’s Law NOTE: Converse is tautology Tautologies that can’t be implemented in Swift
¬¬(((A → B) → A) → A) ¬¬(A ∨ ¬A) ¬¬(¬¬A → A) ¬¬(¬(A ∧ B) → (¬A ∨ ¬B)) ¬¬((A → B) → (¬A ∨ B)) ¬¬((¬B → ¬A) → (A → B)) Tautologies in Intuitionistic Logic Adding double-negation will hold in Intuitionistic Logic!
/// `¬¬(A ∨ ¬A)` (Law of excluded middle in Intuitionistic Logic) func excludedMiddle_IL() -> Not>>> { Not>>> { notEither in let notA: Not = Not { a in notEither.f(.left(a)) } let either: Either> = .right(notA) return notEither.f(either) } }
/// `¬¬¬(A ∧ B) ===> ¬¬(¬A ∨ ¬B)` (De-Morgan's Law in IL) func deMorgan4_IL(_ notNotNotAB: Not>>) -> Not, Not>>> { Not, Not>>> { (notEither: Not, Not>>) in let notNotAB = Not> { notAB in let notB = Not { b in let notA = Not { a in notAB.f((a, b)) } let either: Either, Not> = .left(notA) return notEither.f(either) } let either: Either, Not> = .right(notB) return notEither.f(either) } return notNotNotAB.f(notNotAB) } }
/// `¬¬((A → B) → A) → ¬¬A` (Peirce's Law in IL) func peirceLaw_IL() -> (Not B) -> A>>) -> Not> { { notNotF in Not> { notA in let notABA = Not<((A) -> B) -> A> { aba in let ab: (A) -> B = { a in absurd(notElim(a, notA)) } let a = aba(ab) return notA.f(a) } return notNotF.f(notABA) } } }
Peirce’s Law → Call with Current Continuation ¬¬(A → B) ¬¬A → ¬¬B ((A → ¬¬B) → ¬¬A) → ¬¬A Here, are used to gain ¬¬(A → B) A → ¬¬B ¬¬(((A → B) → A) → A) Peirce’s Law
Recap •Syntax and Semantics •Tautology as “always true” formula •Formal System: Axioms and Inference Rules •Intuitionistic Logic 㱻 Swift Programming (Curry-Howard) •Writing Law of Excluded Middle in Swift (Glivenko) •Learning Logic will help understand Swift’s type (system)
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