INTRO LOGIC

1
INTRO LOGIC

• DAY 26

Derivations in PL5
2
Exam 4 Topics

• 6 derivations in Predicate Logic
• 15 points each, plus 10 free points
• universal derivation Exercise Set C
• existential-out Exercise Set D
• negation rules Exercise Set E
• multiple quantifiers Exercise Set F
• polyadic quantifiers Exercise Set G
• polyadic quantifiers Exercise Set G

3
Example 1 (repeated from last lecture)
there is someone whom everyone Rs/ everyone Rs
someone or other
(1)
Pr
?x?yRyx
(2)
UD
? ?x?yRxy
(3)
?D (ID)
? ?yRay
(4)
As
??yRay
(5)
DD
? ?
(6)
1,
?yRyb
?O
(7)
4,
?y?Ray
??O
(8)
6,
Rab
?O
(9)
7,
?Rab
?O
(10)
8,9,
?
?I
4
Example 2

• there is a F who Rs every G/ every G is Red by
  some F or other

(1)
Pr
?x(Fx ?y(Gy ? Rxy))
(2)
? ?x(Gx ? ?y(Fy Ryx))
5
(1)
Pr
?x(Fx ?y(Gy ? Rxy))
(2)
UD
? ?x(Gx ? ?y(Fy Ryx))
(3)
CD
? Ga ? ?y(Fy Rya)
(4)
As
Ga
(5)
?D (ID)
? ?y(Fy Rya)
(6)
As
??y(Fy Rya)
(7)
DD
? ?
(9)
1,
Fb ?y(Gy ? Rby)
?O
(8)
6,
?y?(Fy Rya)
??O
(10)
9,
Fb
O
(11)
?y(Gy ? Rby)
(12)
8,
?(Fb Rba)
?O
(13)
11,
Ga ? Rba
?O
(14)
12,
Fb ? ?Rba
?O
(15)
4,13,
Rba
?O
(16)
10,14
?Rba
?O
(17)
15,16,
?
?I
6
Example 3

• there is a G who Rs no F/ every F is dis-Red
  by at least one G

(1)
Pr
?x(Gx ??y(Fy Rxy))
(2)
? ?x(Fx ? ?y(Gy ?Ryx))
7
(1)
Pr
?x(Gx ??y(Fy Rxy))
(2)
UD
? ?x(Fx ? ?y(Gy ?Ryx))
(3)
CD
? Fa ? ?y(Gy ?Rya)
(4)
As
Fa
(5)
?D (ID)
? ?y(Gy ?Rya)
(6)
As
??y(Gy ?Rya)
(7)
DD
? ?
(8)
1,
Gb ??y(Fy Rby)
?O
(9)
6,
?y?(Gy ?Rya)
??O
(10)
8,
Gb
O
(11)
??y(Fy Rby)
(12)
9,
?(Gb ?Rba)
?O
(13)
11,
?y?(Fy Rby)
??O
(14)
12,
Gb ? ??Rba
?O
(15)
13,
?(Fa Rba)
?O
(16)
15,
Fa ? ?Rba
?O
(17)
10,1,
??Rba
?O
(18)
4,16,
?Rba
?O
(19)
17,18,
?
?I
8
GOOD LUCK ON EXAM 4!
GOOD LUCK ON THE FINALS!
HAVE A GREAT SUMMER!
GO SOX!