Introductory LogicPHI 120

Presentation: “Solving Proofs"

Bring the Rules Handout to lecture

Homework• Memorize the primitive rules, except ->I and

RAA

• Ex. 1.4.2 (according to these directions)

For Each Sequent, answer these two questions:1. What is the conclusion?2. How is the conclusion embedded in the

premises?

Homework I• Memorize the primitive rules

– Capable of writing the annotationm vI

– Cite how many premises make up each ruleone premise rule

– Cite what kind of premises make up each rulecan be any kind of wff (i.e., one of the disjuncts)

– Cite what sort of conclusion may be deriveda disjunction

See The Rules Handout

Except ->I and RAA

Homework I• Memorize the primitive rules

– Capable of writing the annotationm vI

– Cite how many premises make up each ruleone premise rule

– Cite what kind of premises make up each rulecan be any kind of wff (i.e., one of the disjuncts)

– Cite what sort of conclusion may be deriveda disjunction

See The Rules Handout

Except ->I and RAA

Content of Today’s Lesson

1. Proof Solving Strategy

2. The Rules

3. Doing Proofs

Expect a Learning Curve with this New Material

Homework is imperative

Study these presentations

SOLVING PROOFS“Natural Deduction”

Strategy

Key Lesson Today

(1) Read Conclusion

(2) Find Conclusion in Premises

P -> Q, Q -> R ⊢ P -> R

Valid Argument:True Premises Guarantee a True Conclusion

Ex. 1.4.2

My DirectionsConclusion

(1) What is the conclusion?

Conclusion in Premises

(2.a) Is the conclusion as a whole embedded in any premise?

If yes, where? Else…

(2.b) Where are the parts that make up the conclusion embedded in the premise(s)?

S1 – S10

2) How is the conclusion embedded in the premises?

Homework II

Conclusion in Premises• Example: S16

P -> Q, Q -> R ⊢ P -> R

Conclusion in Premises• Example: S16

P -> Q, Q -> R ⊢ P -> RC1.Conclusion:

a conditional statement

2.Conclusion in the premises: The conditional is not embedded in any premise Its antecedent “P” is the antecedent of the first premise. Its consequent “R” is the consequent of the second

premise.

Answers:

SOLVING PROOFS“Natural Deduction”

The Rules

Proofs

• Rule based system– 10 “primitive” rules

• Aim of Proofs– To derive conclusions on basis of given premises

using the primitive rules

See page 17 – “proof”

What is a Primitive Rule of Proof?

• Primitive Rules are Basic Argument Forms– simple valid argument forms

• Rule Structure– One conclusion

– Premises• Some rules employ one premise• Some rules employ two premises

Φ & Ψ ⊢ Φ

m &E Ampersand-EliminationGiven a sentence that is a conjunction, conclude either conjunct

m,n &I Ampersand-IntroductionGiven two sentences, conclude a conjunction of them.

Φ , Ψ ⊢ Φ & Ψ

Catch-22You have to memorize the rules!

1. To memorize the rules, you need to practice doing proofs.

2. To practice proofs, you need to have the rules memorized

A Solution of Sorts

"Rules to Memorize" on The Rules handout

&E ampersand elimination

vE wedge elimination

->E arrow elimination

<->E double-arrow elimination

&I ampersand introduction

vI wedge introduction

->I arrow introduction

<->I double-arrow introduction

Elimination Introduction

Make a conclusion

Break a premise

THE TEN “PRIMITIVE” RULESProofs

Elimination Rules (break a premise) Introduction Rules (make a conclusion)

* &E (ampersand Elimination) * &I (ampersand Introduction)

* vE (wedge Elimination) * vI (wedge Introduction)

* ->E (arrow Elimination) * ->I (arrow Introduction)

* <->E (double arrow Elimination) * <->I (double arrow Introduction)

A (Rule of Assumption)

RAA (Reductio ad absurdum)

The 10 Rules

Rules of Derivation

1 rule of "assumption": A

4 "elimination" rules: &E, vE, ->E, <->E

4 "introduction" rules: &I, vI, ->I, <->I

1 more rule: “RAA” (reductio ad absurdum)

= 10 rules

The 10 Rules

Rules of Derivation

1 rule of "assumption": A

4 "elimination" rules: &E, vE, ->E, <->E

4 "introduction" rules: &I, vI, ->I, <->I

1 more rule: “RAA” (reductio ad absurdum)

= 10 rules

Proofs: 1st Rule

• The most basic rule: <A> Rule of Assumption

a) Every proof begins with assumptions (i.e., basic premises)

b) You may assume any WFF at any point in a proof

Assumption Numberthe line number on which the “A” occurs.

The 10 Rules

Rules of Derivation

1 rule of "assumption": A

4 "elimination" rules: &E, vE, ->E, <->E

4 "introduction" rules: &I, vI, ->I, <->I

1 more rule: “RAA” (reductio ad absurdum)

= 10 rules

The 10 Rules

Rules of Derivation

1 rule of "assumption": A

4 "elimination" rules: &E, vE, ->E, <->E

4 "introduction" rules: &I, vI, ->I, <->I

1 more rule: “RAA” (reductio ad absurdum)

= 10 rules

Proofs: 2nd – 9th Rules

–Elimination Rules – break premises

–Introduction Rules – make conclusions

The Guts of the System

&I, vI, ->I, <->I

&E, vE, ->E, <->E

The 10 Rules

Rules of Derivation

1 rule of "assumption": A

4 "elimination" rules: &E, vE, ->E, <->E

4 "introduction" rules: &I, vI, ->I, <->I

1 more rule: “RAA” (reductio ad absurdum)

= 10 rules

The 10 Rules

Rules of Derivation

1 rule of "assumption": A

4 "elimination" rules: &E, vE, ->E, <->E

4 "introduction" rules: &I, vI, ->I, <->I

1 more rule: “RAA” (reductio ad absurdum)

= 10 rules

(later)

SOLVING PROOFS“Natural Deduction”

Doing Proofs

m &EDoing Proofs

The “annotation”

page 18

P & Q P⊢

A line of a proof contains four elements: (i) line number (number within parentheses) (ii) annotation (at the very right) (iii) sentence derived (next to line number) (iv) assumption set (number to very left)

P & Q P⊢(1)

A line of a proof contains four elements: (i) line number (number within parentheses) (ii) annotation (at the very right) (iii) sentence derived (next to line number) (iv) assumption set (number to very left)

P & Q P⊢(1) A

A line of a proof contains four elements: (i) line number (number within parentheses) (ii) annotation (at the very right) (iii) sentence derived (next to line number) (iv) assumption set (number to very left)

P & Q P⊢(1) P & Q A

A line of a proof contains four elements: (i) line number (number within parentheses) (ii) annotation (at the very right) (iii) sentence derived (next to line number) (iv) assumption set (number to very left)

P & Q P⊢1 (1) P & Q A

A line of a proof contains four elements: (i) line number (number within parentheses) (ii) annotation (at the very right) (iii) sentence derived (next to line number) (iv) assumption set (number to very left)

P & Q P⊢1 (1) P & Q A

(2)

P & Q P ⊢1 (1) P & Q A

(2) P ???

Read the sequent!

"P" is embedded in the premise.

We will have to break it out of the conjunction. Hence &E.

P & Q P⊢1 (1) P & Q A

(2) P ???

P & Q P⊢1 (1) P & Q A

(2) P 1 &E

P & Q P⊢1 (1) P & Q A

(2) P 1 &E

P & Q P⊢1 (1) P & Q A

(2) P 1 &E

P & Q P⊢1 (1) P & Q A

(2) P 1 &E

P & Q P⊢1 (1) P & Q A1 (2) P 1 &E

m,n &IDoing Proofs

The “annotation”

P, Q Q & P⊢

A line of a proof contains four elements: (i) line number (number within parentheses) (ii) annotation (at the very right) (iii) sentence derived (next to line number) (iv) assumption set (number to very left)

P, Q Q & P⊢(1)

A line of a proof contains four elements: (i) line number (number within parentheses) (ii) annotation (at the very right) (iii) sentence derived (next to line number) (iv) assumption set (number to very left)

P, Q Q & P⊢(1) A

A line of a proof contains four elements: (i) line number (number within parentheses) (ii) annotation (at the very right) (iii) sentence derived (next to line number) (iv) assumption set (number to very left)

P, Q Q & P⊢(1) P A

A line of a proof contains four elements: (i) line number (number within parentheses) (ii) annotation (at the very right) (iii) sentence derived (next to line number) (iv) assumption set (number to very left)

P, Q Q & P⊢1 (1) P A

A line of a proof contains four elements: (i) line number (number within parentheses) (ii) annotation (at the very right) (iii) sentence derived (next to line number) (iv) assumption set (number to very left)

P, Q Q & P⊢1 (1) P A

(2)A line of a proof contains four elements: (i) line number (number within parentheses)

P, Q Q & P⊢1 (1) P A

(2) AA line of a proof contains four elements: (i) line number (number within parentheses) (ii) annotation (at the very right)

P, Q Q & P⊢1 (1) P A

(2) Q AA line of a proof contains four elements: (i) line number (number within parentheses) (ii) annotation (at the very right) (iii) sentence derived (next to line number)

P, Q Q & P⊢1 (1) P A2 (2) Q A

A line of a proof contains four elements: (i) line number (number within parentheses) (ii) annotation (at the very right) (iii) sentence derived (next to line number) (iv) assumption set (number to very left)

P, Q Q & P⊢1 (1) P A2 (2) Q A

(3)

P, Q Q & P⊢1 (1) P A2 (2) Q A

(3) ???

Read the sequent!

"P & Q" is not embedded in any premise.

We will have to make the conjunction. Hence &I

P, Q Q & P⊢1 (1) P A2 (2) Q A

(3) ?, ? &I

P, Q Q & P⊢1 (1) P A2 (2) Q A

(3) Q & P ?, ? &I

P, Q Q & P⊢1 (1) P A2 (2) Q A

(3) Q & P 1, 2 &I

P, Q Q & P⊢1 (1) P A2 (2) Q A

(3) Q & P 1, 2 &I

P, Q Q & P⊢1 (1) P A2 (2) Q A1,2 (3) Q & P 1, 2 &I

Don't forget to define the assumption set!

P, Q Q & P⊢1 (1) P A2 (2) Q A1, 2 (3) Q & P 1, 2 &I

Homework• Memorize the primitive rules, except ->I and

RAA

• Ex. 1.4.2 (according to these directions)

For Each Sequent, answer these two questions:1. What is the conclusion?2. How is the conclusion embedded in the

premises?