Introductory Logic PHI 120 Presentation: “Solving Proofs" Bring the Rules Handout to lecture

Homework Memorize the primitive rules, except ->I and RAA Ex (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 annotation m vI – Cite how many premises make up each rule one premise rule – Cite what kind of premises make up each rule can be any kind of wff (i.e., one of the disjuncts) – Cite what sort of conclusion may be derived a disjunction See The Rules HandoutThe Rules Except ->I and RAA

Homework I Memorize the primitive rules – Capable of writing the annotation m vI – Cite how many premises make up each rule one premise rule – Cite what kind of premises make up each rule can be any kind of wff (i.e., one of the disjuncts) – Cite what sort of conclusion may be derived a disjunction See The Rules HandoutThe Rules Except ->I and RAA

Content of Today’s Lesson 1.Proof Solving Strategy 2.The Rules 3.Doing Proofs 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”

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 Valid Argument: True Premises Guarantee a True Conclusion

Ex My Directions Conclusion (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 -> R C 1.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.

SOLVING PROOFS “Natural Deduction”

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-Elimination Given a sentence that is a conjunction, conclude either conjunct m &E Ampersand-Elimination Given a sentence that is a conjunction, conclude either conjunct m,n &I Ampersand-Introduction Given two sentences, conclude a conjunction of them. m,n &I Ampersand-Introduction Given two sentences, conclude a conjunction of them. Φ,Ψ ⊢ Φ&Ψ

Catch-22 You 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 handoutThe Rules

&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 EliminationIntroduction

THE TEN “PRIMITIVE” RULES Proofs 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: 1 st Rule The most basic rule: 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 Number the line number on which the “A” occurs. Assumption Number the 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: 2 nd – 9 th Rules – Elimination Rules – break premises – Introduction Rules – make conclusions &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”

m &Em &E Doing 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 & QA 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 & QA 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 & QA (2)

P & Q ⊢ P 1(1)P & QA (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 & QA (2)P???

P & Q ⊢ P 1(1)P & QA (2)P1 &E

P & Q ⊢ P 1(1)P & QA (2)P1 &E

P & Q ⊢ P 1(1)P & QA (2)P1 &E

P & Q ⊢ P 1(1)P & QA (2)P1 &E

P & Q ⊢ P 1(1)P & QA 1(2)P1 &E

m,n &I Doing 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)PA 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)PA 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)PA (2) A line of a proof contains four elements: (i) line number (number within parentheses)

P, Q ⊢ Q & P 1(1)PA (2)A A 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)PA (2)QA 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)

P, Q ⊢ Q & P 1(1)PA 2(2)QA 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)PA 2(2)QA (3)

P, Q ⊢ Q & P 1(1)PA 2(2)QA (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)PA 2(2)QA (3) ?, ? &I

P, Q ⊢ Q & P 1(1)PA 2(2)QA (3)Q & P ?, ? &I

P, Q ⊢ Q & P 1(1)PA 2(2)QA (3)Q & P1, 2 &I

P, Q ⊢ Q & P 1(1)PA 2(2)QA (3)Q & P1, 2 &I

P, Q ⊢ Q & P 1(1)PA 2(2)QA 1,2(3)Q & P1, 2 &I Don't forget to define the assumption set!

P, Q ⊢ Q & P 1(1)PA 2(2)QA 1, 2(3)Q & P1, 2 &I

Homework Memorize the primitive rules, except ->I and RAA Ex (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?