Download Natural Deduction Proof System

Natural Deduction System for
First Order Logic
Student:
Wei Lei
Instructor: W. M. Farmer
Department of Computing and Software
McMaster University, Hamilton, CA
Contents
•
•
•
•
•
Natural Deduction
Natural Deduction Proof
Rules of First Order logic
Axioms of Natural Deduction System
Reference
Natural Deduction
• Natural deduction was invented by G. Gentzen in 1934. The idea
was to have a system of derivation rules that as closely as possible
reflect the logical steps in an informal rigorous proof. For each
connective there is an introduction rule (except conjunction, which
has two) which can be seen as a definition of which conditions must
be satisfied for the proposition to be true.
• Natural Deduction tries to follow the natural style of reasoning. Most
of the proof consists of forward reasoning, i.e. deriving conclusions,
deriving new conclusions from these conclusions, etc. Occasionally
hypotheses are introduced or dropped.
• A derivation is a tree where the nodes are the rules and the leafs are
the assumptions of the derivation. The root of the tree is the
conclusion of the derivation. An assumption may be discharged by a
rule in the derivation; that an assumption A is discharged in a
derivation is denoted by [A].
Natural Deduction Proof
• Definition. A natural deduction proof is a finite tree
whose leaves are formula (over the alphabet ∧,→,⊥) and
which is built by using the rules.
• Example: p ∧ q, r |= p ∧ r
Proof:
p∧q
————∧e
p
r
————————∧i
p∧r
Rules of First Order logic
If we have a derivation of B from the assumption A, we may derive A→B and
discharge the assumption A:
[A]
→I
B
——————
A→B
Given derivations of A → B and A, we can conclude B:
A→B
A
→E
——————
B
For conjunction, the introduction rule tells us that if we have derivations of A
and B, then we can conclude A ∧ B:
A B
∧I
——————
A∧B
Rules of First Order logic
There are two elimination rules for conjunction:
A∧ B
∧E1
——————
A
A∧ B
∧E2
——————
B
We can infer a disjunction if we know one of the disjuncts. Hence there are two
introduction rules for the disjunction:
A
V I1
——————
AVB
V I2
B
——————
AVB
Rules of First Order logic
The elimination rule for disjunction is slightly more complicated to formulate. We must
make use of the fact that we know AVB but not necessarily which of A and B it is that
holds.
[A] [B]
AVB
C
C
VE
——————————
C
Rules for ¬:
A
A
┴
¬A
¬ I ————
¬ E ————
¬A
┴
Rules of First Order logic
Rules for ↔:
↔E1
↔I
A↔B
————
A→B
↔E2
A↔B
————
B→A
A→B
B→A
————
A↔B
Rules for → :
→E
A
A →B
————
B
→I
A
———
B
——————
A→B
Rules of First Order logic
Rules for ∀ :
∀E
∀x: X P(x)
——————
P[x:=t]
∀I
x is a new object of type X
——————————————
P(X)
————————————————
∀x: X P(x)
The notation P[x := t] means intelligent replacement.
Also we have rules for equality:
Equality reflexivity
Equality replacement
Axioms of Natural Deduction
System
Example of an axiomatic system of natural deduction
Name Of System: AS
•
•
•
•
•
•
•
•
•
•
Let A,B denote statements.
Undefined terms: ¬, ∧
Axiom(conjunction C): A,B; therefore A ∧ B
Axiom(simplification1 S1): A ∧ B; therefore A
Axiom(simplification2 S2): A ∧ B; therefore B
Axiom(double negation1 DN1): ¬(¬ A); therefore A
Axiom(double negation2 DN2): A; therefore ¬(¬ A)
Axiom(Conjunctive Syllogism CS): A,¬(A ∧ ¬ B); therefore B
Definition: A V B = ¬(¬ A ∧ ¬ B)
Definition: If A then B = ¬(A ∧ ¬ B)
Definition: A if ∧ only if B = (if A then B) ∧ (if B then A)
Notice that the axioms were not framed using any of the definitions.
Axioms of Natural Deduction
System
Whether or not AS is complete.
That is, the question as to whether or not AS contains enough
axioms in order for a reasoning agent to figure out all possible
natural deductions (memory limitations aside), can be raised. If the
answer is yes, then the system is said to be complete. It turns out
that AS is complete, but proving so is rather difficult.
Reference
[i] John M. Anderson and Henry W.Johnstone, Jr. Natural Deduction, The
Logical Basis of Axiom Systems. Wadsworth publishing company, inc.
Belmont, California:1962
[ii] Dirk van Dalen. Logic and Structure. Third Edition. Springer-Verlag Berlin
Heidelberg, New York:1997
[iii] Hans de Nivelle. Natural Deduction for First-Order Logic.
<www.mpi-sb.mpg.de/~nivelle/ teaching/intprooftools2001/natded.pdf>
[vi] BrainyEncyclopedia Natural deduction
<http://www.brainyencyclopedia.com/encyclopedia/n/na/natural_deduction.h
tml>