SLIDE 1
✬ ✫ ✩ ✪
Natural Deduction for Classical Propositional Logic
Valentin Goranko DTU Informatics September 2010
SLIDE 2 1
✬ ✫ ✩ ✪
Natural Deduction
- ND: System for structured deduction from a set of assumptions, based
- n rules, specific to the logical connectives.
- For each logical connective: introduction rule(s) and elimination
rule(s)
- Introduction (opening) and cancelation (closing, discharge) of
assumptions. Assumptions can be re-used many times before canceled.
- Cancelation of assumptions: only when the rules allow it, but not an
- bligation.
- All open assumptions at the end of the derivation must be declared.
The less assumptions, the stronger the derivation.
Deductive systems Natural Deduction Valentin Goranko
SLIDE 3
2
✬ ✫ ✩ ✪
ND rules for the propositional connectives
Introduction rules: Elimination rules: (∧I) A, B A ∧ B (∧E) A ∧ B A , A ∧ B B (∨I) A A ∨ B , B A ∨ B (∨E) A ∨ B [A] . . . C [B] . . . C C
Deductive systems Natural Deduction Valentin Goranko
SLIDE 4
3
✬ ✫ ✩ ✪ Introduction rules: Elimination rules: (→ I) [A] . . . B A → B (→ E) A, A → B B (¬I) [A] . . . ⊥ ¬A (¬E) A, ¬A ⊥
Deductive systems Natural Deduction Valentin Goranko
SLIDE 5
4
✬ ✫ ✩ ✪
Two more ND rules
Ex falsum quodlibet: Reductio ad absurdum: (⊥) ⊥ A (RA) [¬A] . . . ⊥ A
Deductive systems Natural Deduction Valentin Goranko
SLIDE 6
5
✬ ✫ ✩ ✪
Propositional Natural Deduction: Example 1
A ∧ B ⊢ND B ∧ A : (∧I) (∧E) A ∧ B B (∧E) A ∧ B A B ∧ A
Deductive systems Natural Deduction Valentin Goranko
SLIDE 7
6
✬ ✫ ✩ ✪
Propositional Natural Deduction: Examples 2
⊢ND A → ¬¬A : (→ I) (¬I)
(¬E) [A]2, [¬A]1
⊥ ¬¬A 1 A → ¬¬A 2
Deductive systems Natural Deduction Valentin Goranko
SLIDE 8
7
✬ ✫ ✩ ✪ ⊢ND ¬¬A → A : (→ I) (RA)
(¬E) [¬¬A]2, [¬A[1
⊥ A 1 ¬¬A → A 2
Deductive systems Natural Deduction Valentin Goranko
SLIDE 9
8
✬ ✫ ✩ ✪
Propositional Natural Deduction: Examples 3
A → B ⊢ND ¬B → ¬A : [A]1, A → B B , [¬B]2 ⊥ ¬A ¬B → ¬A
2
1
Deductive systems Natural Deduction Valentin Goranko
SLIDE 10
9
✬ ✫ ✩ ✪ ¬B → ¬A ⊢ND A → B : [¬B]1, ¬B → ¬A [¬A] , [A]2 ⊥
(→I)
B A → B
2
1
Deductive systems Natural Deduction Valentin Goranko
SLIDE 11
10
✬ ✫ ✩ ✪
Propositional Natural Deduction: Example 4
A ∨ B ⊢ND ¬A → B : A ∨ B [¬A]1, [A]3 ⊥ B ¬A → B 1 [¬A]2, [B]3 ¬A → B 2 ¬A → B 3
Deductive systems Natural Deduction Valentin Goranko
SLIDE 12
11
✬ ✫ ✩ ✪
Propositional Natural Deduction: Example 5
⊢ND (A → (B → C)) → ((A ∧ B) → C) : (→ E) (∧E) [A ∧ B]1 B (→ E)
(∧E) [A ∧ B]1
A
,
[A → (B → C)]2 B → C (→ I)
(→I)
C (A ∧ B) → C
1
(A → (B → C)) → ((A ∧ B) → C) 2
Deductive systems Natural Deduction Valentin Goranko
SLIDE 13
12
✬ ✫ ✩ ✪
Propositional Deduction: a challenge!
Derive in ND the formula: p ∨ ¬p Award for the first correct entry!
Deductive systems Natural Deduction Valentin Goranko
