Introductory Logic PHI 120 Presentation: "Natural Deduction – Introduction“ Bring this book to lecture

Homework Handout – The Rules The Rules Homework First step in learning the rules: 1.Elimination Rules a.What are they? (Literally, can you name them?) b.Why are they called "elimination" rules? 2.Introduction Rules a.What are they? (Literally, can you name them?) b.Why are they called "introduction rules? Second step in learning the rules: 1.pp a.“primitive rules” b.“elimination” and “introduction” rules (only) Important: bring to every class!! Start learning these rules of deduction

Homework Exam 1 Being graded by your TA Will be handed back in your section (In lecture, we’re pressing on)

HOW IT WORKS “Natural Deduction” New Unit Logical Proofs Worthwhile to turn to page 40.

& (ampersand) & (ampersand)

Simple Valid Argument Forms Φ&Ψ ⊢ &E (ampersand elimination) &E (ampersand elimination)

Simple Valid Argument Forms Φ&Ψ ⊢ T Let’s presume the conjunction is true

Simple Valid Argument Forms Φ&Ψ ⊢ tTt

Φ&Ψ ⊢ Φ tTtT Φ&Ψ ⊢ Ψ tTtT &E Ampersand-Elimination Given a sentence that is a conjunction, conclude either conjunct &E Ampersand-Elimination Given a sentence that is a conjunction, conclude either conjunct

Simple Valid Argument Forms &I (ampersand introduction) &I (ampersand introduction) Φ,Ψ ⊢

Simple Valid Argument Forms Let’s presume the two sentences are true Φ,Ψ ⊢

Simple Valid Argument Forms Φ,Ψ ⊢ TT

&I Ampersand-Introduction Given two sentences, conclude a conjunction of them. &I Ampersand-Introduction Given two sentences, conclude a conjunction of them. Φ,Ψ ⊢ Φ&Ψ TTtTt Φ,Ψ ⊢ Ψ&Φ TTtTt

v (wedge) v (wedge)

Simple Valid Argument Forms ΦvΨ,~Φ ⊢ vE (wedge elimination) vE (wedge elimination) Let’s presume the two sentences are true

Simple Valid Argument Forms ΦvΨ,~Φ ⊢ T

ΦvΨ,~Φ ⊢ fTf

ΦvΨ,~Φ ⊢ fTTf

ΦvΨ,~Φ ⊢ Ψ fTtTfT ΦvΨ,~Ψ ⊢ Φ tTfTfT vE Wedge-Elimination Given a disjunction and another sentence that is the denial of one of its disjuncts, conclude the other disjunct vE Wedge-Elimination Given a disjunction and another sentence that is the denial of one of its disjuncts, conclude the other disjunct

Simple Valid Argument Forms Φ ⊢ Let’s presume this premise is true vI (wedge introduction) vI (wedge introduction)

Simple Valid Argument Forms Φ ⊢ T Let’s presume this premise is true

Simple Valid Argument Forms Φ ⊢ ΦvΨ Tt

Φ ⊢ ΦvΨ TtT Φ ⊢ ΨvΦ TTt vI Wedge-Introduction Given a sentence, conclude any disjunction having it as a disjunct. vI Wedge-Introduction Given a sentence, conclude any disjunction having it as a disjunct.

-> (arrow) -> (arrow)

Simple Valid Argument Forms Φ->Ψ,Φ ⊢ Let’s presume the two sentences are true ->E (arrow elimination) ->E (arrow elimination)

Simple Valid Argument Forms Φ->Ψ,Φ ⊢ T

Simple Valid Argument Forms Φ->Ψ,Φ ⊢ tTT

Simple Valid Argument Forms Φ->Ψ,Φ ⊢ Ψ tTtTT ->E Arrow-Elimination Given a conditional and its antecedent, conclude the consequent. ->E Arrow-Elimination Given a conditional and its antecedent, conclude the consequent.

Simple Valid Argument Forms Ψ ⊢ Let’s presume this premise is true ->I (arrow introduction) ->I (arrow introduction)

Simple Valid Argument Forms Ψ ⊢ Φ->Ψ T

Simple Valid Argument Forms Ψ ⊢ Φ->Ψ Tt

Simple Valid Argument Forms Ψ ⊢ Φ->Ψ TTt ->I Arrow-Introduction Given any sentence, conclude a conditional with it as the consequent. ->I Arrow-Introduction Given any sentence, conclude a conditional with it as the consequent.

(double-arrow) (double-arrow)

Simple Valid Argument Forms Φ Ψ ⊢ E (double-arrow elimination) E (double-arrow elimination)

Simple Valid Argument Forms Φ Ψ ⊢ T Let’s presume the biconditional is true

Simple Valid Argument Forms Φ Ψ ⊢ tTt Φ Ψ ⊢ fTf

Simple Valid Argument Forms Φ Ψ ⊢ Φ->Ψ tTttTt Φ Ψ ⊢ fTf Antecedent Consequent

Simple Valid Argument Forms Φ Ψ ⊢ Ψ->Φ tTttTt Φ Ψ ⊢ fTf Antecedent Consequent

Simple Valid Argument Forms Φ Ψ ⊢ Φ->Ψ tTttTt Φ Ψ ⊢ Φ->Ψ fTffTf AntecedentConsequent

Simple Valid Argument Forms Φ Ψ ⊢ Φ->Ψ tTttTt Φ Ψ ⊢ Ψ->Φ fTffTf E Double-Arrow Elimination Given a biconditional, conclude one or the other arrow statements E Double-Arrow Elimination Given a biconditional, conclude one or the other arrow statements AntecedentConsequent

Simple Valid Argument Forms Φ->Ψ,Ψ Φ ⊢ I (double-arrow introduction) I (double-arrow introduction) Let’s presume these premises are true

Simple Valid Argument Forms (1)Φ->Ψ,Ψ Φ ⊢ tTttTt

Simple Valid Argument Forms (2)Φ->Ψ,Ψ Φ ⊢ fTffTf

Simple Valid Argument Forms (3)Φ->Ψ,Ψ Φ ⊢ tTfT

Simple Valid Argument Forms (4)Φ->Ψ,Ψ Φ ⊢ fTtT

Simple Valid Argument Forms (4)Φ->Ψ,Ψ Φ ⊢ fTttTf

Simple Valid Argument Forms (1)Φ->Ψ,Ψ Φ ⊢ tTttTt (2)Φ->Ψ,Ψ Φ ⊢ fTffTf

Simple Valid Argument Forms (1)Φ->Ψ,Ψ Φ ⊢ Φ Ψ tTttTttTt (2)Φ->Ψ,Ψ Φ ⊢ Φ Ψ fTffTffTf 2 conditions

Simple Valid Argument Forms (1)Φ->Ψ,Ψ Φ ⊢ Ψ Φ tTttTttTt (2)Φ->Ψ,Ψ Φ ⊢ Ψ Φ fTffTffTf I Double-Arrow Introduction Given two mirror conditionals, conclude the biconditional I Double-Arrow Introduction Given two mirror conditionals, conclude the biconditional 2 conditions

Rules of Derivation &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

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

Homework Handout – The Rules The Rules Homework First step in learning the rules: 1.Elimination Rules a.What are they? (Literally, can you name them?) b.Why are they called "elimination" rules? 2.Introduction Rules a.What are they? (Literally, can you name them?) b.Why are they called "introduction rules? Second step in learning the rules: 1.pp a.“primitive rules” b.“elimination” and “introduction” rules (only) Important: bring to every class!!