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

int their proofs, and second we show how to extend this “calculus of natural deduction” to the
full language of “mathtalk”.
We will now introduce the “natural deduction” calculus for propositional logic. The calculus was
created in order to model the natural mode of reasoning e.g. in everyday mathematical practice.
This calculus was intended as a counter-approach to the well-known Hilbert style calculi, which
were mainly used as theoretical devices for studying reasoning in principle, not for modeling
particular reasoning styles.
Rather than using a minimal set of inference rules, the natural deduction calculus provides
two/three inference rules for every connective and quantifier, one “introduction rule” (an inference
rule that derives a formula with that symbol at the head) and one “elimination rule” (an inference
rule that acts on a formula with this head and derives a set of subformulae).
Calculi: Natural Deduction (ND 0 ) [Gentzen’30]
Idea: ND 0 tries to mimic human theorem proving behavior (non- minimal)
Definition 327 The ND 0 calculus has rules for the introduction and elimination of connectives
Introduction Elimination Axiom
A B
A ∧ B ∧I
[A] 1
B
A ⇒ B ⇒I1
A ∧ B
A ∧El
A ∧ B
B ∧Er
A ⇒ B A
⇒E
B
A ∨ ¬A TND
TND is used only in classical logic (otherwise constructive/intuitionistic)
c○: Michael Kohlhase 195
The most characteristic rule in the natural deduction calculus is the ⇒I rule. It corresponds to
the mathematical way of proving an implication A ⇒ B: We assume that A is true and show B
from this assumption. When we can do this we discharge (get rid of) the assumption and conclude
A ⇒ B. This mode of reasoning is called hypothetical reasoning. Note that the local hypothesis
is discharged by the rule ⇒I , i.e. it cannot be used in any other part of the proof. As the ⇒I
rules may be nested, we decorate both the rule and the corresponding assumption with a marker
(here the number 1).
Let us now consider an example of hypothetical reasoning in action.
Natural Deduction: Examples
103