logic1 notes logic1notes chapter 2

8/12/2019 Logic1 Notes Logic1notes Chapter 2

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2/25

A 1 , A 2 , . . . , A n B A 1 , A 2 , . . . A n |= B A 1 , A 2 , . . . , A n

B

|= B B A1 , A 2 , . . . A n |= B

A 1 , A 2 , . . . , A n B

A 1 , A 2 , . . . , A n

B

( p q ) r, r |= p q

( p

q ) r

r p q


3/25

p q r ( p q ) r r p q

0 0 0 1 1 1

0 0 1 1 0 1

0 1 0 1 1 1

0 1 1 1 0 1

1 0 0 0 1 0

1 0 1 1 0 0

1 1 0 1 1 1

1 1 1 1 0 1

A1 , A 2 , . . . , A n |= B

(A1 A 2 A n ) B

A1 A 2 A n |= B

A1 , A 2 , . . . , A n |= B i i (A1 ) =i(A 2 ) = = i(An ) = 1 i(B) = 1

i i (A1 A2 An ) = 1 i (B) = 1 i(A 1 A2 An ) = 1 i(A1 ) = i(A2 ) =

i(A n ) = 1 A1 , A 2 , . . . , A n |= B A1 A2 An |= B A1 A2 An |= B

i i(A 1 A2 An ) = 1 i(B) = 0 i

i(( A1 A2 An ) B) = 0 (A 1 A2 An ) B A1

A2 An |= B (A1 A2 An ) B A 1 A2 An B

(A 1 A 2 A n ) B


4/25

p q, p r |= q r

p q r p q p r q r

0 0 0 0 1 1

0 0 1 0 1 1 0 1 0 1 1 0

0 1 1 1 1 1

1 0 0 1 1 1

1 0 1 1 0 1

1 1 0 1 1 0

1 1 1 1 0 1

i ( p ) = 0 i ( q ) = 1 i ( r ) = 0 i ( p ) = 1 i ( q ) = 1 i ( r ) = 0

p q , p r | = q r

B B

B

B B

B

B


5/25

p q

q

p

p q

q

p

p qr

s

p q r s

p

q r s

p q r s


6/25

p q r s

pq r

q (s p ) r ( t p )

q rq (s p )

r ( t p ) p

q qq (s p )q ( t p )

p

pq q

q (s p ) q ( t p )


7/25

p q s t

q q q ( s p ) q ( t p )

p

60 2 60 > 10 18


8/25

60 4

p =

q = r = s =

p (q r)

s r

( p s) q

p q rs 16

p q r s p (q r) s r ( p s) q0 0 0 0 0 1 0 0 1 1 0 1 00 0 0 1 0 1 0 1 1 1 0 1 00 0 1 0 0 1 1 0 1 0 0 1 00 0 1 1 0 1 1 1 0 0 0 1 00 1 0 0 0 1 1 0 1 1 0 1 10 1 0 1 0 1 1 1 1 1 0 1 10 1 1 0 0 1 1 0 1 0 0 1 10 1 1 1 0 1 1 1 0 0 0 1 11 0 0 0 1 0 0 0 1 1 0 1 01 0 0 1 1 0 0 1 1 1 1 0 01 0 1 0 1 1 1 0 1 0 0 1 0

1 0 1 1 1 1 1 1 0 0 1 0 01 1 0 0 1 1 1 0 1 1 0 1 11 1 0 1 1 1 1 1 1 1 1 1 11 1 1 0 1 1 1 0 1 0 0 1 11 1 1 1 1 1 1 1 0 0 1 1 1

16


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1 0

p q r s p (q r) s r ( p s) q1 1 0

( p s) q p s p

s q

p q r s p (q r) s r ( p s) q1 0 1 1 1 1 1 1 0 0

p q s

p q r s p (q r) s r ( p s) q1 0 1 1 1 0 1 1 1 1 1 0 0

s r s r r

p q r s p (q r) s r ( p s) q1 0 0 1 1 1 0 0 1 1 1 0 1 1 1 0 0

p (q r ) p q r q r

q r

p q r s p (q r) s r ( p s) q1 0 0 1 1 1 0 0 1 1 1 0 1 1 1 0 0


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p (q r )

s r

(( p s ) q )

p (q r )

s r

(( p s ) q )

p s

q

(( p s ) q )

p s p s

p (q r )

s r

(( p s ) q )

p s

q

p

s

p s p (q r ) s r q p s

q p s

s r s r

s r


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p (q r )

s r

(( p s ) q )

p s

q

p

s

s

r

s r

s r

s s

p (q r )s r

(( p s ) q )

p s

q

p

s

s

r

p (q r ) p q r

p q r

p (q r )

s r

(( p s ) q )

p s

q

p

s

s

r

p

q r

p p


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q r q r

p (q r )

s r

(( p s ) q )

p s

q

p

s

s

r

p

q r

q

r

q q r

r

p (q r )

s r

(( p s ) q )

p s

q

p

s

s

r

p

q r

q

r

p (q r ) s r ( p s ) q


13/25

A A A A

A A

16 32 64

A B

A

B

A B

A

B

A B

A

B

A B

A

B

A

B

AA

(A B )

A

B

(A B )

A

B

(A B )

A

B

(A B )

A

B

A

B

A

A

A B A B

A B A B

(A B ) A B A B

(A B )


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A B A B A B

A B

A

B

A B

A

B

A B

A

B

(A B )

A

B

(A B )

A

B

(A B )

A

B

A B A B A B A B A B


15/25

A , A A

A A A

A

A

A

( p q ) (r s) ( p r) (q s)

( p q ) (r s)(( p r) (q s))

p r

(q s)

q

s

p q

p

p

r

q

r s

r

p

r

s

q s p r

i( p) = 0 i(q ) = 0 i(r ) = 1 i(s) = 0

( p q ) (r s) ( p r ) (q s)

( p q) (r s) ( p r) (q s)0 1 0 1 1 0 0 0 1 1 0 0 0 0

q s p r i( p) = 1 i(q ) = 0 i(r ) = 0

i(s) = 0

( p q) (r s) ( p r) (q s)1 0 0 1 0 1 0 1 1 0 0 0 0 0


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p (q (r s ))( p (( q r ) s ))

p

(( q r ) s )

p

q (r s )

q

(q r )

q

r

s

r s

r

s

(q r )

q

r

s

p q s r r i ( p ) = 1 i ( q ) = 1 i ( s ) = 0 i ( r ) = 0 i ( r ) = 1

p ( q ( r s ) ) p ( ( q r ) s )

p (q r ) ( p q ) r

p (q r )(( p q ) r )

p q

r

p

q

p

q r

q

r


17/25

p (q r ) ( p q ) r

p (( q r ) s )(( p q ) (r s ))

p

(q r ) s

p q

(r s )

p

q

r

s

q r

q

r

s

q r q q

q

p (q r ) ( p q ) r


18/25

p (q r)(( p q ) r )

p

q r

p q

r

p

q

q

r

( p q )

r

q

p

q

q

r

p

q

q

p

(q r)

q

r

p q

r

p

q

( p q )

r

p q r

i i( p) = i(q ) = i(r ) = 1 p (q r ) ( p q ) r

p (q r) ( p q) r1 1 1 1 1 1 0 0 1 0 1

A , A A |= |=

|= A , A |=

|=

|=

|=

i i


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i i

A B

A

B

A B

A

B

A B

A

B

A B

A

B

A

B

(A B)

A

B

(A B)

A

B

(A B)

A

B

(A B)

A

B

A

B

A

A

i

A B

i (A B) = 1 i (A) = 1 i (B) = 1 i(A) = 1 A i(B) = 1

B

(A B) i((A B)) = 1 i(A B) = 0 i (A) = 0

i(B) = 0 i(A) = 1 i(B) = 1 A B (A B)

|=

|=

|=

i( p) = 1 p i(q ) = 0 q


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A B

A

B

A B

A

B

A B

A

B

A B

A

B

A

B

(A B )

A

B

(A B )

A

B

(A B )

A

B

(A B )

A

B

A

B

A

A

A B

A B A A B B A B

(A B ) A B

A B A B A B (A B )

|=

|=


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http://www.worldcat.org/oclc/318913974http://www.blackwell-synergy.com/doi/abs/10.1111/j.1755-2567.1970.tb00434.xhttp://dx.doi.org/10.1007/BF00368389http://consequently.org/writing/logical_pluralismhttp://www.amazon.com/o/asin/0691073392/consequentlyorghttp://www.amazon.com/o/asin/0691073392/consequentlyorghttp://isbn.nu/0691071926http://isbn.nu/0691071926


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http://www.flickr.com/photos/25559122@N06/6057895435/http://www.flickr.com/photos/32659528@N00/429421469http://commons.wikimedia.org/wiki/File:Barbara_partee.jpghttp://www.flickr.com/photos/andrewrusk/5598986979/http://www.calculemus.org/MathUniversalis/6/lukas.jpghttp://www.landley.net/history/mirror/pre/shannon.htmlhttp://en.wikipedia.org/wiki/Charles_Sanders_Peircehttp://commons.wikimedia.org/wiki/File:George_Boole_color.jpghttp://commons.wikimedia.org/wiki/File:George_Boole_color.jpg


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http://owpdb.mfo.de/detail?photo_id=9240http://commons.wikimedia.org/wiki/File:Gottfried_Wilhelm_von_Leibniz.jpghttp://commons.wikimedia.org/wiki/File:Gottfried_Wilhelm_von_Leibniz.jpghttp://commons.wikimedia.org/wiki/File:Thomas_Hobbes.jpeghttp://openclipart.org/detail/15413/laptop-by-mysticahttp://commons.wikimedia.org/wiki/File:Wilson_Tennis_Racquet.jpghttp://commons.wikimedia.org/wiki/File:Wilson_Tennis_Racquet.jpghttp://www.flickr.com/photos/irrational_cat/35776501


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http://commons.wikimedia.org/wiki/File:Alan_Turing.jpg

logic1 notes logic1notes chapter 2

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