Many-Valued Logics

Question 1

What is validity for MV Logics?

Answer 1

Iff it is impossible to have designated P’s and an undesignated Conc

Question 2

What is designated in K3?

Answer 2

{1}

Question 3

What are the values for K3 negation?

Answer 3

1 if ν(A) = 0
n if ν(A) = n
0 if ν(A) = 1

Question 4

What is the conditional in K3?

Answer 4

A –> B := ~AvB

Translate this before doing tableaux!

Question 5

What is important to remember about translations (eg conditional)?

Answer 5

Do it before you start the Tableaux

Question 6

What is the difference between the semantics of L3 and K3?

Answer 6

L3 has primitive conditional (–> filled in)

Question 7

Give the TT for L3 –>

Answer 7

Write it down and that

Question 8

What are designated values in LP?

Answer 8

{1,b}

Question 9

Give values of negation for LP

Answer 9

1 if ν(A) = 0
b if ν(A) = b
0 if ν(A) = 1

Question 10

What is the difference between the semantics of LP and RM3?

Answer 10

RM3 has primitive conditional (–>)

Question 11

Do TT for RM3 conditional

Answer 11

Aye

Question 12

What Tableaux we be using?

Answer 12

K3 and LP

Question 13

Conjunction for Tab

Answer 13

+∧rule A∧B,+✓
______ A,+
______ B,+

−∧rule A∧B,−✓
_____/ __ \
____A,− __B,−

Question 14

Disjunction for Tab

Answer 14

+∨rule A∨B,+✓
_____/ __ \
____A,+ __B,+

−∨rule A∨B,−✓
______ A,−
______ B,−

Question 15

Negated Conjunction for Tab

Answer 15

+¬∧rule ¬(A∧B),+✓
_________ |
_______¬A∨¬B,+

−¬∧rule ¬(A∧B),−✓
_________ |
_______¬A∨¬B,−

Question 16

Negated Disjunction for Tab

Answer 16

+¬∨rule ¬(A∨B),+✓
_________ |
_______¬A∧¬B,+

−¬∨rule ¬(A∨B),−✓
_________ |
_______¬A∧¬B,−

Question 17

What are the DN rules for Tableaux?

Answer 17

+¬¬rule ¬¬A,+✓
________ |
________A,+

−¬¬rule ¬¬A,−✓
________ |
________A,−

Question 18

When does a branch close in K3?

Answer 18

1. If a branch contains both A,+ and A,− for some sentence A
2. If a branch contains both A,+ and ¬A,+ for some sentence A

Question 19

What are the countermodel values for K3?

Answer 19

1. For every atomic or negated atomic such that A,+ is on the branch, assign ν(A) = 1
2. If A,− and ¬A,− are on the branch, set ν(A) = n
3. If A,− is on the branch but its negative force negation is not on the branch, set ν(A) = 0

Question 20

How does a branch close in LP?

Answer 20

1. If a branch contains contains both A,+ and A,− for some sentence A
2. If a branch contains both A,− and ¬A,− for some sentence A

Question 21

What are the countermodel values for LP?

Answer 21

1. Where A,+ is on the branch but its positive-force negation is not on the branch, set ν(A) = 1
2. Where A ,+ and ¬A,+ are on the branch, set ν(A) = b
3. For every atomic or negated atomic such that A,− is on the branch, assign ν(A) = 0