Download Jordan Bradshaw, Virginia Walker, and Dylan Kane

The Exciting World of Natural
Deduction!!!
By:
Dylan Kane
Jordan Bradshaw
Virginia Walker
Natural Deduction
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Gerhard Gentzen
Stanislaw Jaskowski
1934
Mimics the natural reasoning process,
inference rules natural to humans
Called “natural” because does not require
conversion to (unreadable) normal form
Background:
Natural deduction proofs
I’ll be back.
Natural Deduction
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Proof system for first-order logic
Designed to mimic the natural reasoning process
Process:
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Make assumptions (“A” is true)
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Letters like “A” can represent larger propositional phrases
The set of assumptions being relied on at a given step
is called the context.
Use rules to draw conclusions.
Discharge assumptions as they become no longer
necessary.
Natural Deduction
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Natural deduction is done in step by step:
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Rule
Premises
Conclusion
…
Logical Connectives
Truth Tables for Logical
Connectives
Making Conclusions
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The rules used to draw conclusions consist
mostly of the introduction (I) and elimination (E)
of these connectives.
Several of the rules serve to discharge earlier
assumptions.
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The result does not rely on the assumption being
true.
If the assumption is used by itself again somewhere
else, it must be discharged again in a step that
follows.
Introduction and Elimination
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Introduction builds the conclusion out of
the logical connective and the premises.
Elimination eliminates the logical
connective from a premise.
Rules: AND/OR
Rule “or E” discharges S and T.
Rules: IF
Rule “if I” discharges S
Rules: C
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Proof by contradiction
If by assuming S is false, you
reach a contradiction, S is true.
Discharges (not S)
Rules: forall (∀)
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Rule “∀I” requires that “a” does not occur
in S(x) or any premise on which S(a) may
depend.
Rules: exists (∃)
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Rule “∃E” requires that “a” does not occur in S(x) or T or
any assumption other than S(a) on which the derivation
of T from S(a) depends.
Rule “∃E” also discharges S(a).
Tautology
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Always true.
The proof of a tautology ultimately relies
on no assumptions.
The assumptions are discharged
throughout the proof.
Sample proof: a tautology
Sample proof: a tautology
A is discharged using the ->I rule.
Sample proof: a tautology
B is discharged using the ->I rule.
Example using Quantifiers
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“Imagine how you would convince
someone else, who didn’t know any formal
logic, of the validity of the entailment you
are trying to demonstrate.”
a.k.a. That a knowledge base entails a
sentence.
Example using Quantifiers
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Ex. We want to prove this:
{forall x (F(x) -> G(x))
forall x (G(x) -> H(x))}
|- forall x (F(x) -> H(x))
Take an arbitrary object a
Suppose a is an F
Since all Fs are Gs, a is a G
Since all Gs are Hs, a is an H
So if a is an F then a is an H
But this argument works for
any a
So all Fs are Hs
Proof using Natural Deduction
Rule exists (∃): Revisited
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Rule “∃E” requires that “a” does not occur in S(x) or T or
any assumption other than S(a) on which the derivation
of T from S(a) depends.
Rule “∃E” also discharges S(a).
Incorrect Proof (exists E)
Interesting Tidbits for Further
Reading
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Natural Deduction book written in 1965 by
Prawitz
Gallier in 1986 used Gentzen’s approach to
expound the theoretical underpinning so f
automated deduction.
Credits
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Reeves, Steve and Mike Clarke. Logic for
Computer Science. 2003.
Russell, Stuart and Peter Norvig. Artificial
Intelligence: A modern Approach. 2nd
edition. 2003