The Exciting World of Natural Deduction!!!

1. The Exciting World of Natural Deduction!!! By: Dylan Kane Jordan Bradshaw Virginia Walker

2. 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

3. Background:Natural deduction proofs I’ll be back.

4. 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.

5. Natural Deduction • Natural deduction is done in step by step: • Rule • Premises • Conclusion • …

6. Logical Connectives

7. Truth Tables for Logical Connectives

8. 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.

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

10. Rules: AND/OR Rule “or E” discharges S and T.

11. Rules: IF Rule “if I” discharges S

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

13. Rules: forall (∀) • Rule “∀I” requires that “a” does not occur in S(x) or any premise on which S(a) may depend.

14. 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).

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

16. Sample proof: a tautology

17. Sample proof: a tautology A is discharged using the ->I rule.

18. Sample proof: a tautology B is discharged using the ->I rule.

19. Example using Quantifiers • “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.

20. Example using Quantifiers • 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

21. Proof using Natural Deduction

22. 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).

23. Incorrect Proof (exists E)

24. Interesting Tidbits for Further Reading • Natural Deduction book written in 1965 by Prawitz • Gallier in 1986 used Gentzen’s approach to expound the theoretical underpinning so f automated deduction.

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