Lecture 4: Simple Type Theory
Teacher: Gilles Dowek
Reminder: relation
\[∃R \, ∀x, z. (x, z ∈_2 R ⟺ ∃ y \, (x = z × y))\] \[∃R; \, ε(\underbrace{R}_{\rlap{\text{propositional content of the prop. on the right}}}) ⟺ ∀x,y,z,n. \, (⋯)\]Recall that you don’t sets if you have functions: you can use characteristic functions:
Example:
\[f: \begin{cases} 4 &⟼ 4 \text{ is even} \\ 7 &⟼ 7 \text{ is even} \end{cases}\]then:
\[⊢ ε(f \, 4) \qquad (4 ∈ f)\\ ⊢ ¬ ε(f \, 7)\]Simple Type Theory
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you have an infinite number of sorts:
- $ι$: base objects
- $o$: propositional contents (what you get by applying a predicate to a term)
- $A → B$: functions from A to B
- for each pair $A,B$, application: $α_{A,B}$ of arity $⟨A → B,A,B⟩$
- a predicate symbol (the only one) $ε$ of arity $⟨o⟩$:
- for each $t$, $x_1,…,x_n ⟼ t$
- for each $A$, ${x_1,…,x_n \; \mid \; A}$
But sufficient: combinators:
\[x^A ⟼ x: A → A\\ K_{A, B} ≝ x^A, y^B ⟼ x: A → (B → A)\\ S_{A,B,C} ≝ x,y,z ⟼ (x \, z) (y \, z): (A → B → C) → (A → B) → A → C\]and then we will have axioms expressing
\[α(x ↦ x, a) = a\\ α(α(x,y ↦ x, a),b) = a\\\]Similarly:
\[\lbrace x \; \mid \; ∃ y \, ( x = 2 × y)\rbrace\\ \dot{∧} = \lbrace x,y \; \mid \; ε(x) ∧ ε(y)\rbrace\]and then:
\[ε(αα(\dot{∧},a),b) ⟺ ε(x) ∧ ε(y)\]- proposition:
the sky is blue - propositional content:
that the sky is blue - $ε$:
it_is_true(that the sky is blue)
\[∀ P, \, (P ⇒ P)\\ ∀ p \, (ε(p) ⇒ ε(p))\]
season(spring)color(4 of them)- you can’t build the proposition:
as_many(color, season)because you can’t apply predicates to predicates
To avoid binding symbols: you replace them by an operator without bound variables applied to a function: higher-order abstract syntax
\[\dot{∀}_A = \lbrace f \; \mid \; ∀x^A\, ε(\underbrace{f \, x}_{A → o})\rbrace: (A → o) → o\] \[ε(\dot{∀}_A \, f) ⟶ ∀x\, ε(f \, x)\]Term $t$, there exists $u$ st
\[(u \, y) ⟶^\ast t\]-
$t=y$: by setting $u$ to be identity: $I ≝ S \, K \, K$
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$t=x$, $x≠y$ OR $t = c$
\[u y ⟶^\ast t\]by setting $u ≝ K \, x$ or $u ≝ K \, c$
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$t= α(t_1 t_2) = (t_1 \, t_2)$:
By IH: $u_1 \, y ⟶^\ast t_1$, $u_2 \, y ⟶^\ast t_2$:
\[\underbrace{S \, u_1 \, u_2}_{≝ \, u} y ⟶ (u_1 \, y) (u_2 \, y) ⟶^\ast (t_1 \, t_2) = t\]
Proposition $A$: there exists $u$ such that
\[ε(u) ⟶^\ast A\]By induction on $A$. For example: $A ≝ A_1 ∧ A_2$. By IH: $ε(a_1) ⟶ A_1$, $ε(a_2) ⟶ A_2$:
\[ε(\dot{∧} \, a_1 \, a_2) ⟶ ε(a_1) ∧ ε(a_2)\]If $A = ε(t)$, it’s even simpler.
What’s the point? Comprehension: ex:
\[A ≝ ∃y \, (x = 2 × y)\\ ε(a) ⟶^\ast A\\ b \, x ⟶^\ast a\\ ε(b \, x) ⟶^\ast ε (a) ⟶^\ast A\]so (comprehension):
\[∃b ; \; (ε(b \, x) ⟺ A)\]Impredicative quantification:
\[∀p \, (ε(p) ⇒ ε(p))\]At first: was avoided by Russel (to avoid paradoxes like Russel’s), but his student Ramsy reintroduced it later. Martin-Löf Type Theory: predicative, on the other hand (people in Sweden at Chalmers are very reluctant to use impredicativity).
Models
All reduction rules of Simple type theory are valid in a $\lbrace 0, 1\rbrace$-valued model:
\[\underbrace{⟦ε(\dot{⊤})⟧}_{= \hat{ε} \hat{\dot{⊤}} = id(\tilde{⊤}) = 1} = \underbrace{⟦⊤⟧}_{\tilde{⊤} = 1}\]Mathematical theories
\[x=y ⟶ \dot{∀}_{ι → o} (\overline{λ} c((c \, x) \dot{⇒} (c \, y)))\]So:
\[ε(x=y) ⟶^\ast ∀c \, (ε (c \, x) ⇒ ε (c \, y))\]NB: the class $c$ is of type $ι → o$
Natural numbers:
- Peano: axioms
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Cantor: 3 = the set of all sets of cardinal 3
- modern logic (due to Frege) was invented to build Cantor numbers
- Church: Church numerals
- Von Neumann: 3 = the set $\lbrace ∅, \lbrace ∅ \rbrace, \lbrace ∅, \lbrace ∅ \rbrace\rbrace\rbrace$
Church (40’s) ⟶ his student: Henkin (50’s)
\[\begin{align*} ⟦∀y \, ε(α_{A, o} \, x \, y)⟧_φ &= \widetilde{∀} \lbrace ⟦ε(α_{A, o} \, x \, y)⟧_{φ, y ↦ a} \; \mid \; a ∈ ℳ_a \rbrace\\ &= \widetilde{∀} \lbrace \underbrace{\widetilde ε (⟦α_{A, o} \, x \, y⟧_{φ, y ↦ a})}_{= ⟦α_{A, o} \, x \, y⟧_{φ, y ↦ a}} \; \mid \; a ∈ ℳ_a \rbrace\\ & = \widetilde{∀} \lbrace ⟦α_{A, o} \, x⟧_{φ, y ↦ a}(\underbrace{⟦y⟧_{φ, y ↦ a}}_{= a}) \; \mid \; a ∈ ℳ_a \rbrace \\ & = \widetilde{∀} \lbrace ⟦α_{A, o}⟧_{φ, y ↦ a} (\underbrace{⟦x⟧_{φ, y ↦ a}}_{= φ(x)})(a) \; \mid \; a ∈ ℳ_a \rbrace \\ & = \widetilde{∀} \lbrace φ(x)(a) \; \mid \; a ∈ ℳ_a \rbrace \\ & = \widetilde{ε}\Big(\widetilde{∀} \lbrace φ(x)(a) \; \mid \; a ∈ ℳ_a \rbrace\Big)\\ & = ⟦ε(α_{A→o, o} \, \dot{∀}_A \, x)⟧_φ \end{align*}\]
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