General Computer Science 320201 GenCS I & II Lecture ... - Kwarc

Inference with local hypotheses
[A ∧ B] 1
B
∧Er
B ∧ A
A ∧ B ⇒ B ∧ A
[A ∧ B] 1
A ∧I
⇒I 1
∧El
c○: Michael Kohlhase 196
[A] 1
[B] 2
A ⇒I 2
B ⇒ A
⇒I 1
A ⇒ B ⇒ A
Another characteristic of the natural deduction calculus is that it has inference rules (introduction
and elimination rules) for all connectives. So we extend the set of rules from Definition 327 for
disjunction, negation and falsity.
More Rules for Natural Deduction
Definition 328 ND 0 has the following additional rules for the remaining connectives.
A
A ∨ B ∨Il
B
A ∨ B ∨Ir
[A] 1
.
F
¬A ¬I1
¬A A
F I
F
A ∨ B
[A] 1
.
C
C
¬¬A
A ¬E
F
F E
A
c○: Michael Kohlhase 197
The next step now is to extend the language of propositional logic to include the quantifiers ∀
and ∃. To do this, we will extend the language PLNQ with formulae of the form ∀x A and ∃x A,
where x is a variable and A is a formula. This system (which ist a little more involved than we
make believe now) is called “first-order logic”. 10 EdNote:10
Building on the calculus ND 0 , we define a first-order calculus for “mathtalk” by providing introduction
and elimination rules for the quantifiers.
First-Order Natural Deduction
Rules for propositional connectives just as always
Definition 329 (New Quantifier Rules) The ND extends ND 0 by the following four
rules
10 EdNote: give a forward reference
104
[B] 1
.
C
∨E 1