PHI 103 - Propositional Logic Natural Deduction (part 1)

Implication Rules valid argument forms

I. Material Implication:

A. Modus Ponens (affirming the antecedent)

B. Modus Tollens (denying the consequent)

C. Hypothetical Syllogism (the transitive property)

II. Disjunction:

A. Disjunctive Syllogism -

Propositional Logic Implication Rules

• Modus Ponens (the “way” of affirmation)

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D H D ⊃ H / D // H

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TFTF

T T T TTT F T FFF TT FTF FF FT

P1) If dogs are mammals, then they have hearts. P2) Dogs are mammals. C) Therefore, dogs have hearts.

P1) D ⊃ H P2) D ∴ H

Valid

Propositional Logic Implication Rules

• Modus Tollens (the “way” of denial)

P1) If cats are birds, then they have feathers. P2 It’s not the case that cats have feathers. C) Therefore, it’s not the case that cats are birds.

P1) C ⊃ F P2 ~ F ∴ ~ C

ValidF

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C F C ⊃ F / ~ F // ~ C

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TFTF

T T T TTT F F TF TF FT TT FF FF FT T

F FFTT

Propositional Logic Implication Rules

• Hypothetical Syllogism (the transitive property)P1) If Kato is a mammal, then he is warm-blooded. P2) If he is warm-blooded, then he has a heart. C) Therefore, If Kato is a mammal, then he has a heart.

P1) K ⊃ W P2) W ⊃ H ∴ K ⊃ H

Valid

p q r p ⊃ q / q ⊃ r // p ⊃ rT T T TTT T F TT FT TF TF TT TF FF T

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Propositional Logic Implication Rules

• Disjunctive Syllogism (the method of elimination)

P1) Either cats have feathers, or they have fur. P2) It’s not the case that cats have feathers. C) Therefore, cats have fur.

P1) C ∨ F P2) ~ C ∴ F

Valid

p q p ∨ q / ~ p // q

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TT

F

TFTF

T T T TTT F T FT FF TT FT TF FF FF T

F

Propositional Logic Implication Rules

1. (C ∨ F) ⊃ H 2. C ∨ F ∴ H

Practice

1. (K ∨ L) ⊃ W 2. W ⊃ (H ⋅ J) ∴ (K ∨ L) ⊃ (H ⋅ J)

1. D 2. D ⊃ H ∴ H

1. X ⊃ Y 2. ~ Y ∴ ~ X

MP

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MT

HS

1. ~ M ⊃ (R ⊃ S) 2. (C ⊃ K) ⊃ ~ M ∴ (C ⊃ K) ⊃ (R ⊃ S)

HS

1. ~ G ⊃ ~ (R ⊃ S) 2. ~ ~ (R ⊃ S) ∴ ~ ~ G

MT

• Demonstrating Validity (proofs in natural deduction)

1. A ⊃ B 2. C ∨ A 3. A / B

1, 3, MP4. B

1. C ⊃ B 2. A ⊃ B 3. ~ B / ~ A4. ~ A 2, 3, MT

1. S ⊃ T 2. T ⊃ U 3. R ⊃ S / R ⊃ U4. R ⊃ T 1, 3, HS5. R ⊃ U 2, 4, HS

Propositional Logic Implication Rules

• Demonstrating Validity (proofs in natural deduction)

1. A ∨ B 2. ~ C ⊃ ~ A 3. C ⊃ D 4. ~ D / B5. ~ C 3, 4, MT6. ~ A 2, 5, MP7. B 1, 6, DS

Propositional Logic Implication Rules

• Demonstrating Validity (proofs in natural deduction)1. E ⊃ (K ⊃ L) 2. F ⊃ (L ⊃ M) 3. G ∨ E 4. ~ G 5. F / (K ⊃ M)6. E 3, 4, DS7. K ⊃ L 1, 6, MP8. L ⊃ M 2, 5, MP9. K ⊃ M 7, 8, HS

Propositional Logic Implication Rules

I. Material Implication:

A. Modus Ponens (affirming the antecedent)

B. Modus Tollens (denying the consequent)

C. Hypothetical Syllogism (the transitive property)

II. Disjunction:

A. Disjunctive Syllogism B. Addition

III. Conjunction:

A. Conjunction

B. Simplification

C. Constructive Dilemma

Propositional Logic Implication Rules

• Addition (add anything you need!)

P1) Cats have feathers. C) Either cats have feathers, or they have fur.

1. C ∴ C ∨ F

Valid

p q p // p ∨ q

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TT

F

TFTF

T T TT T FTF F TTF F FF

T

Propositional Logic Implication Rules

• Conjunction (connect with the dots)

P1) Cats have fur. P2) Cats have whiskers. C) Therefore, cats have fur and cats have whiskers.

1. C 2. W ∴ C ⋅ W

ValidF

T

p q p / q // p ⋅ q

T

F

TFTF

T T TT T FFF F TFF F FF

TTFTF

Propositional Logic Implication Rules

• Simplification (dis-connect the dots)

P1) Cats have fur and cats have whiskers. C) Therefore, cats have fur.

1. F ⋅ W ∴ F

Valid

p q p ⋅ q // p

F

TT

F

TFTF

T TT TFF FFF FF

T TFTF

Propositional Logic Implication Rules

• Constructive Dilemma (concluding the consequences)

1. (M ⊃ W) ⋅ (D ⊃ F) 2. M ∨ D ∴ W ∨ F

P1) If Kato is a mammal then he is warm-blooded and if Kato is a dog then he has fur. P2) Either Kato is a mammal or he’s a dog. C) Therefore, Kato is warm-blooded, or he has fur.

Propositional Logic Implication Rules

• Constructive Dilemma (concluding the consequences)

1. (p ⊃ q) ⋅ (r ⊃ s) 2. p ∨ r ∴ q ∨ s

Propositional Logic Implication Rules

A constructive dilemma is performing two modus Ponens simultaneously.

1. (p ⊃ q) ⋅ (r ⊃ s) 2. p ∨ r ∴ q ∨ s

1. (M ⊃ W) ⋅ (D ⊃ F) 2. M ∨ D ∴ W ∨ F

CD - Valid

p q r s (p ⊃ q) ⋅ (r ⊃ s) / p ∨ r // q ∨ s

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Propositional Logic Implication Rules

Propositional Logic Implication Rules

1. p ⊃ q 2. p ∴ q

MP1. p ⊃ q 2. q ⊃ r ∴ p ⊃ r

HS1. p ⊃ q 2. ~ q ∴ ~ p

MT1. p ∨ q 2. ~ p ∴ q

DS

1. p ∴ p ∨ q

ADD1. p 2. q ∴ p ⋅ q

Conj.1. p ⋅ q ∴ p

Simp.1. (p ⊃ q) ⋅ (r ⊃ s) 2. p ∨ r ∴ q ∨ s

CD