Study Guide - Introduction to Logic

STUDY GUIDES: TEST 1 | TEST 2 | TEST 3

Chapter: 1 | 4 | 5 | 6 | 7

LOGIC WORKBOOK
A COMPANION TO HURLEY
John Chiappone

7.1 NATURAL DEDUCTION IN PROPOSITIONAL LOGIC

RULES OF IMPLICATION

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MODUS PONENS (MP):

p > q

_____

Proofs Using Modus Ponens:

(1)

1. p > q

2. p / q

3. q 1, 2 mp

(2)

1. p

2. p > q / q

3. q 2, 1 mp

(3)

1. r > t

2. r / t

3. t 1, 2 mp

(4)
1. (y ≡ u) > o The main operator is > .

2. y ≡ u / o

3. o 1, 2 mp

(5)

1. x > p

2. (x > p) > r / r

3. r 2, 1 mp

(6)
1. u v y

2. (u v y) > (x v t) What is the main operator? >

3. w • s / x v t

4. x v t 2, 1 mp

(7)

1. [(o ≡ p) • x] > (s • u) What is the main operator? >

2. (o ≡ p) • x

3. ~y v t / s • u

4. s • u 1, 2 mp

(8)

1. m v ~m

2. [(m v y) v t] > [(s > p) • x]

3. (m v y) v t / (s > p) • x

4. (s > p) • x 2, 3 mp

(9)

1. p > [(x v y) v z]

2. ~r • y

3. ~u

4. p / (x v y) v z

5. (x v y) v z 1, 4 mp

MODUS TOLLENS (MT):

p > q

______

Proofs Using Modus Tollens:

(1)
1. p > q

2. ~q / ~p

3. ~p 1, 2 mt

(2)

1. ~y

2. x > y / ~x

3. ~x 2, 1 mt

(3)
1. m > ~w

2. ~~w / ~m

3. ~m 1, 2 mt

(4)
1. ~u > p

2. ~p / ~~u

3. ~~u 1, 2 mt

(5)
1. x > (o • m)

2. ~(o • m) / ~x

3. ~x 1, 2 mt

(6)
1. (p ≡ o) > u

2. ~u / ~(p ≡ o)

3. ~(p ≡ o) 1, 2 mt

(7)
1. (p ≡ o) > (o • m) What is the main operator? >

2. ~(o • m) / ~(p ≡ o)

3. ~(p ≡ o) 1, 2 mt

(8)
1. x > t

2. ~t

3. s > x / ~s

4. ~x 1, 2 mt

5. ~s 3, 4 mt

(9)
1. ~r

2. u > ~~r What is the main operator? >

3. x > p

4. p > u / ~x

5. ~u 2, 1 mt

6. ~p 4, 5 mt

7. ~x 3, 6 mt

MODUS TOLLENS (MT):

p > q

_____

MODUS PONENS (MP):

p > q

_____

Proofs Using Modus Tollens

& Modus Ponens:

(1)

1. ~x

2. p > x

3. ~p > q / q

4. ~p 2, 1 mt

5. q 3, 4 mp

(2)

1. z > q

2. ~q

3. ~z > p / p

4. ~z 1, 2 mt

5. p 3, 4 mp

(3)
1. p > ~q

2. s > q

3. p / ~s

4. ~q 1, 3 mp

5. ~s 2, 4 mt

(4)

1. x • z

2. (x • z) > r

3. r > ~y

4. q > y / ~q

5. r 2, 1 mp

6. ~y 3, 5 mp

7. ~q 4, 6 mt

(5)

1. ~t

2. x > t

3. q > x

4. ~q > r / r

5. ~x 2, 1 mt

6. ~q 3, 5 mt

7. r 4, 6 mp

(6)

1. ~m > w

2. m > z

3. z > t

4. ~t / w

5. ~z 3, 4 mt

6. ~m 2, 5 mt

7. w 1, 6 mp

Pure Hypothetical Syllogism (HS):

p > q

q > r
_____

p > r

Proofs Using Hypothetical Syllogism:

(1)

1. P > S

2. S > Q / P > Q

3. P > Q 1, 2 HS

(2)

1. D > S

2. S > T

3. T > U / S > U

4. S > U 2, 3 HS

(3)

1. (A • B) > C

2. C > D / (A • B) > D

3. (A • B) > D 1, 2 HS

(4)

1. W > Q

2. S > W / S > Q

3. S > Q 2, 1 HS

(5)

1. (A v C) > S

2. D > C

3. E > (A v C) / E > S

4. E > S 2, 1 HS

(6)

1. (Q > X) > (A• N)

2. (A • N) > P / (Q > X) > P

3. (Q > X) > P 1, 2, HS

(7)

1. [(A • S) > T] > (O v B)

2. W > [(A • S) > T] / W > (O v B)

3. (O v B) 2, 1 HS

Pure Hypothetical Syllogism (HS):

p > q

q > r
_____

p > r

MODUS PONENS (MP):

p > q

_____

Proofs Using Hypothetical Syllogism and Modus Ponens:

(1)

1. P > C

2. C > L

3. P / L

4. P > L 1, 2 HS

5. L 4, 3 MP

(2)

1. A > B

2. B > C

3. (A > C) > D / D

4. A > C 1, 2 HS

5. D 3, 4 MP

(3)

1. H > P

2. P > (B > M)

3. B

4. B > H / M

5. H > (B > M) 1, 2 HS

6. H 4, 3 MP

7. B > M 5, 6 MP

8. M 7, 3 MP

(4)

1. C > T

2. T > D

3. B > C

4. B / D

5. C > D 1, 2 HS

6. C 3, 4 MP

7. D 5, 6 MP

(5)
1. F > (J > X)

2. Y > F

3. Y

4. J / X

5. F 2, 3 MP

6. J > X 1, 5 MP

5. Y > (J > X) 2, 1 HS

6. J > X 3, 5 MP

7. X 6, 4 MP

Note: A proof that takes less steps is said to be more elegant.

(6)

1. S • T

2. C

3. (S • T) > G

4. G > (C > K) / K

5. (S • T) > (C > K) 3, 4 HS

6. C > K 5, 1 MP

7. K 6, 2 MP

(7)

1. E > F

2. Y > E

3. (Y > F) > (H > S)

4. (H > S) > (I > P)

5. I / P

6. Y > F 2, 1 HS

7. (Y > F) > (I > P) 3, 4 HS

8. I > P 7, 6 MP

9. P 8, 5 MP

DISJUNCTIVE SYLLOGISM (DS):

p v q p v q

~p ~q

______ _____

q p

Proofs Using Disjunctive Syllogism:

(1)

1. ~P

2. P v Q / q

3. Q 2, 1 DS

(2)

1. s v t

2. ~s / t

3. t 1, 2 ds

(3)

1. m v u

2. ~u / m

3. m 1, 2 ds

(4)

1. (x • p) v t

2. ~t / x • p

3. x • p 1, 2 ds

(5)

1. (x v u) v t

2. ~(x v u) / t

3. t 1, 2 ds

(6)

1. y v (t > p)

2. ~(t > p) / y

3. y 1, 2 ds

(7)

1. (u v t) v (x > z)

2. ~(u v t) / x > z

3. x > z 1, 2 ds

(8)

1. (u v t) v (x > z)

2. ~(x > z) / u v t

3. u v t 1, 2 ds

(9)

1. m v ~t

2. ~m

3. t v y / y

4. ~t 1, 2 ds

5. y 3, 4 ds

(10)

1. o v ~u

2. u v r

3. ~o / r

4. ~u 1, 3 ds

5. r 2, 4 ds

(11)

1. y v m

2. t v ~y

3. ~m / t

4. y 1, 3 ds

5. t 2, 4 ds

DISJUNCTIVE SYLLOGISM (DS):

p v q p v q

~p ~q

______ _____

q p

MODUS PONENS (MP):

p > q

_____

Proofs Using Disjunctive Syllogism and Modus Ponens:

(1)
1. ~P

2. P v Q

3. Q > Z / Z

4. Q 2, 1 DS

5. Z 3,4 MP

(2)

1. D > ~X

2. D

3. X v N / N

4. ~X 1, 2 MP

5. N 3, 4 DS

(3)

1. (I • B) v (E v G)

2. ~(I • B) What is the main operator? ~

3. ~E

4. G > Y / Y

5. E v G 1, 2 DS

6. G 5, 3 DS

7. Y 4, 6 MP

(4)

1. A V C

2. ~C

3. ~B

4. A > (B v D) / B > D

5. A 1, 2 DS

6. B > D 3, 4 MP

7. D 4, 6, 3 DS

(5)

1. (P • Q) V (D > F) What is the main operator? V

2. ~(D > F)

3. (P • Q) > T / T

4. P • Q 1, 2 DS

5. T 3, 4 P

(6)

1. [(F v G) V (T v Y)] > (Z v F) What is the main operator? >

2. ~[(F v G) V (T v Y)]

3. ~Z / F

4. Z v F 1, 2 MP

5. F 3, 4 DS

DISJUNCTIVE SYLLOGISM (DS):

p v q p v q

~p ~q

______ _____

q p

MODUS PONENS (MP):

p > q

_____

MODUS TOLLENS (MT):

p > q

______

Proofs Using Disjunctive Syllogism,
Modus Ponens, and Modus Tollens:

(1)

1. P > ~Q

2. P

3. Q v ~W

4. C > W / ~C

5. ~Q 1, 2 MP

6. ~W 3, 5 DS

7. ~C 4, 6 MT

(2)

1. S v T

2. ~S

3. T > ~(K • F) What is the main operator? >

4. (K • F) v (R > S) / ~R

5. T 1, 2 DS

6. ~(K • F) 3, 5 MP

7. R > S 4, 5 DS

8. ~R 7, 2 MT

(3)

1. C > B

2. ~ B

3. ~C > (B v D) / D

4. ~C 1, 2 MT

5. B v D 3, 4 MP

6. D 5, 2 DS

(4)

1. (B v C) v (A v ~N)

2. (B v C) > S

3. ~S

4. ~A / ~N

5. ~(B v C) 2, 3 MT

6. A v ~N 1, 5 DS

7. ~N 6, 4 DS

(5)

1. R • X

2. C > [E > (B > D)]

3. C

4. E v L

5. ~L

6. ~D / ~B

7. E > (B > D) 2, 3 MP

8. E 4, 5 DS

9. B > D 7, 8 MP

10. ~B 9, 6 MT

MODUS PONENS (MP):

p > q

_____

MODUS TOLLENS (MT):

p > q

______

Hypothetical Syllogism (HS):

p > q

q > r
_____

p > r

DISJUNCTIVE SYLLOGISM (DS):

p v q p v q

~p ~q

______ _____

q p

Proofs Using All the Rules:

(1)

1. M > P

2. M

3. P > R / M > R

4. M > R 1, 3 HS

(2)

1. (X v D) v D

2. H

3. ~D / X

4. X v D 1, 3 DS

5. X 4, 3 DS

(3)
1. (D > S) > (T v W)

2. D > R

3. R > S / T v W

4. D > S 2, 3 HS

5. T v W 1, 4 MP

(4)
1. H > P

2. D > (P > S)

3. D / H > S

4. P > S 2, 3 MP

5. H > S 1, 4 HS

(5)

1. G > ~T

2. ~~t

3. ~G > ~F What is the main operator? >

4. F v D / D

5. ~G 1, 2 MT

6. ~F 3,5 MP

7. D 4, 6 DS

(6)

1. P > Q

2. Q > S

3. (P > S) > (D > E)

4. ~E

5. ~D > T / T

6. P > S 1, 2 HS

7. D > E 3, 6 MP

8. ~D 7, 4 MT

9. T 5, 8 MP

(7)

1. D v C

2. D > B

3. B > S

4. S > U

5. ~C / U

6. D 1, 5 DS

7. D > S 2, 3 HS

8. S 7, 6 MP

9. U 8, 4, 8 MP

(8)

1. B v W

2. B > (Y > M)

3. ~W

4. M > Q / Y > Q

5. B 1, 3 DS

6. Y > M 2, 5 MP

7. Y > Q 6, 4 HS

(9)

1. ~T

2. S > T

3. D > S

4. ~D > Z / Z

5. ~S 2, 1 MT

6. ~D 3, 5 MT

7. Z 4, 6 MP

(10)

1. (T • G) > (X v P)

2. ~P V S

3. ~S

4. T • G / X

5. ~P 2, 3 DS

6. X v P 1, 4 MP

7. X 6, 5 DS

(11)

1. A > F

2. F > (T > R)

3. R > S

4. T

5. A / S

6. A > (T > R) 1, 2 HS

7. T > R 6, 5 MP

8. T > S 7, 3 HS

9. S 8, 4 MP