Introductory Logic PHI 120

Presentation: “Solving Proofs" Introductory LogicPHI 120 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: • What is the conclusion? • How is the conclusion embedded in the premises?

Homework I • Memorize the primitive rules • Capable of writing the annotation mvI • 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 Except ->I and RAA See The Rules Handout

Content of Today’s Lesson • Proof Solving Strategy • The Rules • Doing Proofs

Expect a Learning Curve with this New Material Homework is imperative Study these presentations

Strategy “Natural Deduction” Solving Proofs

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 Homework II S1 – S10 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)? 2) How is the conclusion embedded in the premises?

Conclusion in Premises • Example: S16 P -> Q, Q -> R ⊢ P -> R

Conclusion in Premises • Example: S16 P -> Q, Q -> R ⊢ P -> R C • Conclusion: • a conditional statement • 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:

The Rules “Natural Deduction” Solving Proofs

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 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.

Catch-22 You have to memorize the rules! • To memorize the rules, you need to practice doing proofs. • To practice proofs, you need to have the rules memorized A Solution of Sorts "Rules to Memorize" on The Rules handout

Elimination Introduction &Eampersand elimination vEwedge elimination ->Earrow elimination <->Edouble-arrow elimination &Iampersand introduction vIwedge introduction ->Iarrow introduction <->Idouble-arrow introduction Break a premise Make a conclusion

Proofs The ten “Primitive” 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 • Every proof begins with assumptions (i.e., basic premises) • You may assume any WFF at any point in a proof Assumption Number the linenumber on which the “A” occurs.

Proofs: 2nd – 9th Rules • Elimination Rules – break premises • Introduction Rules – make conclusions &E, vE, ->E, <->E The Guts of the System &I, vI, ->I, <->I

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)

Doing Proofs “Natural Deduction” Solving Proofs

Doing Proofs The “annotation” page 18 m&E

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)

Read the sequent! P & Q ⊢ P 1 (1) P & Q A (2) P ??? "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 1 (2) P 1 &E

Doing Proofs The “annotation” m,n &I

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) 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) P A (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)

P, Q ⊢ Q & P 1 (1) P A 2 (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)

Introductory Logic PHI 120