The Exciting World of The Exciting World of Natural Deduction!!!Natural Deduction!!!

By: By:

Dylan KaneDylan Kane

Jordan BradshawJordan Bradshaw

Virginia WalkerVirginia Walker

Natural DeductionNatural Deduction

Gerhard Gentzen Gerhard Gentzen Stanislaw JaskowskiStanislaw Jaskowski 19341934 Mimics the natural reasoning process, Mimics the natural reasoning process,

inference rules natural to humansinference rules natural to humans Called “natural” because does not Called “natural” because does not

require conversion to (unreadable) require conversion to (unreadable) normal formnormal form

Background:Background:Natural deduction proofsNatural deduction proofs

I’ll be back.

Natural DeductionNatural Deduction

Proof system for first-order logicProof system for first-order logic Designed to mimic the natural reasoning Designed to mimic the natural reasoning

processprocess Process:Process:

Make assumptions (“A” is true)Make assumptions (“A” is true) Letters like “A” can represent larger propositional phrasesLetters like “A” can represent larger propositional phrases The set of assumptions being relied on at a given The set of assumptions being relied on at a given

step is called the context.step is called the context. Use rules to draw conclusions.Use rules to draw conclusions. Discharge assumptions as they become no longer Discharge assumptions as they become no longer

necessary.necessary.

Natural DeductionNatural Deduction

Natural deduction is done in step by Natural deduction is done in step by step:step: RuleRule PremisesPremises ConclusionConclusion ……

Logical ConnectivesLogical Connectives

Truth Tables for Logical Truth Tables for Logical Connectives Connectives

Making ConclusionsMaking Conclusions

The rules used to draw conclusions consist The rules used to draw conclusions consist mostly of the introduction (I) and mostly of the introduction (I) and elimination (E) of these connectives.elimination (E) of these connectives.

Several of the rules serve to discharge Several of the rules serve to discharge earlier assumptions.earlier assumptions. The result does not rely on the assumption The result does not rely on the assumption

being true.being true. If the assumption is used by itself again If the assumption is used by itself again

somewhere else, it must be discharged again somewhere else, it must be discharged again in a step that follows.in a step that follows.

Introduction and EliminationIntroduction and Elimination

Introduction builds the conclusion out Introduction builds the conclusion out of the logical connective and the of the logical connective and the premises. premises.

Elimination eliminates the logical Elimination eliminates the logical connective from a premise. connective from a premise.

Rules: AND/ORRules: AND/OR

Rule “or E” discharges S and T.Rule “or E” discharges S and T.

Rules: IFRules: IF

Rule “if I” discharges SRule “if I” discharges S

Rules: CRules: C

Proof by contradictionProof by contradiction If by assuming S is false, youIf by assuming S is false, you

reach a contradiction, S is true.reach a contradiction, S is true. Discharges (not S)Discharges (not S)

Rules: forall (∀)Rules: forall (∀)

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

Rules: exists (∃)Rules: exists (∃)

Rule “∃E” requires that “a” does not occur in S(x) Rule “∃E” requires that “a” does not occur in S(x) or T or any assumption other than S(a) on which or T or any assumption other than S(a) on which the derivation of T from S(a) depends.the derivation of T from S(a) depends.

Rule “∃E” also discharges S(a).Rule “∃E” also discharges S(a).

TautologyTautology

Always true. Always true. The proof of a tautology ultimately The proof of a tautology ultimately

relies on no assumptions. relies on no assumptions. The assumptions are discharged The assumptions are discharged

throughout the proof. throughout the proof.

Sample proof: a tautologySample proof: a tautology

Sample proof: a tautologySample proof: a tautology

A is discharged using the ->I rule.

Sample proof: a tautologySample proof: a tautology

B is discharged using the ->I rule.

Example using QuantifiersExample using Quantifiers

““Imagine how you would convince Imagine how you would convince someone else, who didn’t know any someone else, who didn’t know any formal logic, of the validity of the formal logic, of the validity of the entailment you are trying to entailment you are trying to demonstrate.”demonstrate.”

a.k.a. That a knowledge base entails a.k.a. That a knowledge base entails a sentence. a sentence.

Example using QuantifiersExample using Quantifiers

Ex. We want to prove this:Ex. We want to prove this: {forall x (F(x) -> G(x)){forall x (F(x) -> G(x)) forall x (G(x) -> H(x))} forall x (G(x) -> H(x))}

|- forall x (F(x) -> H(x))|- forall x (F(x) -> H(x))

Take an arbitrary object aSuppose a is an FSince all Fs are Gs, a is a GSince all Gs are Hs, a is an HSo if a is an F then a is an HBut this argument works for

any aSo all Fs are Hs

Proof using Natural Proof using Natural DeductionDeduction

Rule exists (∃): RevisitedRule exists (∃): Revisited

Rule “∃E” requires that “a” does not occur in S(x) Rule “∃E” requires that “a” does not occur in S(x) or T or any assumption other than S(a) on which or T or any assumption other than S(a) on which the derivation of T from S(a) depends.the derivation of T from S(a) depends.

Rule “∃E” also discharges S(a).Rule “∃E” also discharges S(a).

Incorrect Proof (exists E)Incorrect Proof (exists E)

Interesting Tidbits for Interesting Tidbits for Further ReadingFurther Reading

Natural Deduction book written in Natural Deduction book written in 1965 by Prawitz1965 by Prawitz

Gallier in 1986 used Gentzen’s Gallier in 1986 used Gentzen’s approach to expound the theoretical approach to expound the theoretical underpinning so f automated underpinning so f automated deduction. deduction.

CreditsCredits

Reeves, Steve and Mike Clarke. Reeves, Steve and Mike Clarke. Logic for Computer ScienceLogic for Computer Science. 2003.. 2003.

Russell, Stuart and Peter Norvig. Russell, Stuart and Peter Norvig. Artificial Intelligence: A modern Artificial Intelligence: A modern Approach. Approach. 22ndnd edition. 2003 edition. 2003