The Exciting World of Natural Deduction!!!

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The Exciting World of Natural Deduction!!!

• By
• Dylan Kane
• Jordan Bradshaw
• Virginia Walker

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Natural Deduction

• 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

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BackgroundNatural deduction proofs
Ill be back.
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Natural Deduction

• Proof system for first-order logic
• Designed to mimic the natural reasoning process
• Process
• Make assumptions (A is true)
• 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.

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Natural Deduction

• Natural deduction is done in step by step
• Rule
• Premises
• Conclusion

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Logical Connectives
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Truth Tables for Logical Connectives
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Making Conclusions

• 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.
• 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.

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Introduction and Elimination

• Introduction builds the conclusion out of the
  logical connective and the premises.
• Elimination eliminates the logical connective
  from a premise.

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Rules AND/OR

• Rule or E discharges S and T.

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Rules IF

• Rule if I discharges S

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Rules C

• Proof by contradiction
• If by assuming S is false, you
• reach a contradiction, S is true.
• Discharges (not S)

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Rules forall (?)

• Rule ?I requires that a does not occur in
  S(x) or any premise on which S(a) may depend.

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Rules exists (?)

• 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).

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Tautology

• Always true.
• The proof of a tautology ultimately relies on no
  assumptions.
• The assumptions are discharged throughout the
  proof.

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Sample proof a tautology
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Sample proof a tautology
A is discharged using the -gtI rule.
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Sample proof a tautology
B is discharged using the -gtI rule.
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Example using Quantifiers

• Imagine how you would convince someone else, who
  didnt 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.

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Example using Quantifiers

• Ex. We want to prove this
• forall x (F(x) -gt G(x))
• forall x (G(x) -gt H(x))
• - forall x (F(x) -gt 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
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Proof using Natural Deduction
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Rule exists (?) Revisited

• 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).

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Incorrect Proof (exists E)
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Interesting Tidbits for Further Reading

• Natural Deduction book written in 1965 by Prawitz
• Gallier in 1986 used Gentzens approach to
  expound the theoretical underpinning so f
  automated deduction.

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Credits

• Reeves, Steve and Mike Clarke. Logic for
  Computer Science. 2003.
• Russell, Stuart and Peter Norvig. Artificial
  Intelligence A modern Approach. 2nd edition.
  2003