SLIDE 1
the course
1st order propositional logic ↔ simple type theory λ→ 1st order predicate logic ↔ type theory with dependent types λP 2nd order propositional logic ↔ polymorphic type theory λ2
1
second order propositional logic type theory week 08 2006 04 03 0 - - PowerPoint PPT Presentation
second order propositional logic type theory week 08 2006 04 03 0 - - PowerPoint PPT Presentation
second order propositional logic type theory week 08 2006 04 03 0 the course 1st order propositional logic simple type theory 1st order predicate logic type theory with dependent types P 2nd order propositional logic
SLIDE 2
SLIDE 3
2nd order propositional logic
propositional logic
a b c . . . A → B ⊥ ⊤ ¬A A ∧ B A ∨ B
2
SLIDE 4
predicate logic
a(. . .) b(. . .) c(. . .) . . . A → B ⊥ ⊤ ¬A A ∧ B A ∨ B ∀x . A ∃ x . A x y z . . . f(. . .) g(. . .) h(. . .) . . .
3
SLIDE 5
second order propositional logic
a b c . . . A → B ⊥ ⊤ ¬A A ∧ B A ∨ B ∀a . A ∃ a . A
4
SLIDE 6
example
a → a ∀a. a → a if it’s tuesday, then it’s tuesday for every proposition, that proposition implies itself
5
SLIDE 7
the rules
introduction rules elimination rules I[x]→ E→ E⊥ I⊤ I[x]¬ E¬ I∧ El ∧ Er∧ Il ∨ Ir∨ E∨ I∀ E∀ I∃ E∃
6
SLIDE 8
propositional logic: rules for implication
implication introduction [Ax] . . . B A → B I[x]→ implication elimination . . . A → B . . . A B E→
7
SLIDE 9
propositional logic: rules for falsum and truth
falsum elimination . . . ⊥ A E⊥ truth introduction ⊤ I⊤
8
SLIDE 10
propositional logic: rules for conjunction
conjunction introduction . . . A . . . B A ∧ B I∧ conjunction elimination . . . A ∧ B A El∧ . . . A ∧ B B Er∧
9
SLIDE 11
propositional logic: rules for disjunction
disjunction introduction . . . A A ∨ B Il∨ . . . B A ∨ B Il∨ disjunction elimination . . . A ∨ B . . . A → C . . . B → C C E∨
10
SLIDE 12
2nd order propositional logic: rules for universal quantification
universal quantification introduction . . . A ∀a. A I∀ variable condition: a not a free variable in any open assumption universal quantification elimination . . . ∀a. A A[a := B] E∀
11
SLIDE 13
2nd order propositional logic: rules for existential quantification
existential quantifier introduction . . . A[a := B] ∃a. A I∃ existential quantifier elimination . . . ∃a. A . . . ∀a. (A → B) B E∃ variable condition: a not a free variable in B
12
SLIDE 14
variable conditions
- for rule I∀
check: variable does not occur in any of the available assumptions
- for rule E∃
check: variable does not occur in the conclusion
13
SLIDE 15
examples
example 1
(∀b. b) → a
14
SLIDE 16
example 2
a → ∀b. ((a → b) → b)
15
SLIDE 17
example 3
(∃b. a) → a
16
SLIDE 18
example 4
∃b.((a → b) ∨ (b → a))
17
SLIDE 19
example 5
∀a. ∀b. ((a → b) ∨ (b → a)) this needs classical logic ∀a. (a ∨ ¬a)
18
SLIDE 20
non-example 6
a → ∀a. a
19
SLIDE 21
non-example 7
(∃a. a) → a
20
SLIDE 22
higher order logic
the ‘order’ of a variable
first order
- bject
second order set of objects predicate on objects function from objects to objects third order set of second order objects predicate on predicates on objects function from second order objects to . . . etc.
21
SLIDE 23
example from 2nd order predicate logic
induction principle for natural numbers ∀a.
- a(0) → (∀m. a(m) → a(S(m))) → ∀n. a(n)
- m
1st order variable n 1st order variable 1st order constant a 2nd order variable S 2nd order constant
22
SLIDE 24
- nly predicates without arguments
quantify over predicates → 2nd order predicate logic . . . the same without terms → 2nd order propositional logic
23
SLIDE 25
impredicative encoding of inductive types
the connectives in Coq
→ hard-wired into the type theory ∀ hard-wired into the type theory ⊥ inductive type ∧ inductive type ∨ inductive type ∃ inductive type
24
SLIDE 26
inductive definition of False
Inductive False : Prop := . False_ind : ∀a. ⊥ → a the constructors are the introduction rules the induction principle is the elimination rule
25
SLIDE 27
inductive definition of and
Inductive and (a b : Prop) : Prop := conj : a → b → a ∧ b . and_ind : ∀a b c. (a → b → c) → (a ∧ b) → c the constructor is the introduction rule the induction principle gives the elimination rules
26
SLIDE 28
alternative version of conjunction elimination
conjunction elimination: alternative version . . . A ∧ B . . . A → B → C C E∧ conjunction elimination: normal version . . . A ∧ B A El∧ . . . A ∧ B B Er∧
27
SLIDE 29
inductive definition of or
Inductive or (a b : Prop) : Prop :=
- r_introl : a → a ∨ b
| or_intror : b → a ∨ b .
- r_ind : ∀a b c. (a → c) → (b → c) → (a ∨ b) → c
the constructors are the introduction rules the induction principle is the elimination rule
28
SLIDE 30
impredicative definition of False
⊥ := ∀a. a induction principle next to impredicative definition ∀a. ⊥ → a ∀a. a
29
SLIDE 31
impredicative definition of and
a ∧ b := ∀c. (a → b → c) → c induction principle next to impredicative definition ∀a b. ∀c. (a ∧ b) → (a → b → c) → c ∀c. (a → b → c) → c
30
SLIDE 32
impredicative definition of or
a ∨ b := ∀c. (a → c) → (b → c) → c induction principle next to impredicative definition ∀a b. ∀c. (a ∨ b) → (a → c) → (b → c) → c ∀c. (a → c) → (b → c) → c
31
SLIDE 33
impredicative definitions for other inductive types
impredicative definition of the booleans
∀a. a → a → a
32
SLIDE 34
impredicative definition of the natural numbers
∀a. a → (a → a) → a
33
SLIDE 35
why have inductive types as primitive then?
- one can prove less equalities
- one gets weaker induction principles
- some people don’t like impredicativity
34