Modal Logic

Question 1

Give the semantics for box and diamond

Answer 1

νw(◻A)=1 iff νx(A)=1at all worlds x such that wRx

νw(◇A)=1 iff νx(A)=1at some world x such that wRx

Question 2

What is an interpretation?

Answer 2

An interpretation is a structure ⟨W,R,ν⟩ where W is a set of worlds, R is an accessibility relation between worlds, and ν is a function assigning a truth-value to each atomic formula relative to each world

Question 3

Why interpretation?

Answer 3

Because the formula may be true/false at another world

Question 4

When is basic modal logic valid?

Answer 4

Iff every model of the premises is a model of the conclusion

Question 5

When is K a tautology?

Answer 5

⊧K A, that is,A is a tautology iff νw(A)=1 at every world of every interpretation.

Question 6

What are the Conjunction rules for K tableaux?

Answer 6

∧-rule A∧B,i ✓  ¬∧-rule ¬(A∧B),i ✓
\_\_\_\_\_\_↓ \_\_\_\_\_\_\_\_\_\_\_↙ ↘ 
\_\_\_\_\_ A,i \_\_\_\_\_\_\_\_\_\_¬A,i ¬B,i
\_\_\_\_\_ | 
\_\_\_\_\_ B,i

Question 7

What are the disjunction rules for K tableaux?

Answer 7

∨-rule A∨B,i ✓  ¬∨-rule ¬(A∨B),i ✓ 
\_\_\_\_ ↙ ↘ \_\_\_\_\_\_\_\_\_\_↓
\_\_\_\_A ,i _ B,i \_\_\_\_\_\_\_\_¬A,i 
\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ | 
\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_¬B,i

Question 8

What are the conditional rules for K tableaux?

Answer 8

⊃-rule A⊃B,i ✓  ¬⊃-rule ¬(A⊃B),i ✓ 
\_\_\_\_ ↙ ↘ \_\_\_\_\_\_\_\_\_\_\_↓
\_\_\_¬ A,i  _ B,i \_\_\_\_\_\_\_\_\_ A,i 
\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ | 
\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_¬B,i

Question 9

What are the rules for biconditional and double negation for K tableaux?

Answer 9

≡-rule A≡B,i ✓ ¬≡-rule ¬(A≡B),i ✓ DN-rule ¬¬A,i ✓
____ ↙ ↘ ________↙ ↘ ___________↓
___ A ,i __¬A,i _____ A,i __ ¬A,i _________ A,i
____| _____ | ______ | _____ |
___ B,i ___~B,i ____ ~B,i ___ B,i

Question 10

Give the K tableaux rules for diamond

Answer 10

◇-rule ◇A,i ✓  ¬◇-rule ¬◇A,i ✓ 
\_\_\_\_\_\_\_ ↓ \_\_\_\_\_\_\_\_\_\_\_\_↓ 
\_\_\_\_\_\_\_ irj \_\_\_\_\_\_\_\_\_\_◻¬A,i 
\_\_\_\_\_\_ A, j
using some new index j

Question 11

Give the K tableaux rules for box

Answer 11

◻-rule ◻A,i  ¬◻-rule ¬◻A,i ✓ 
\_\_\_\_\_\_ irj \_\_\_\_\_\_\_\_\_\_ ↓ 
\_\_\_\_\_\_ ↓ \_\_\_\_\_\_\_\_\_ ◇¬A,i
\_\_\_\_\_ A ,j 
N.B. we never check off a ◻line

Question 12

Define Irreflexive

Answer 12

For all x, ~xRx

Question 13

State non-reflexive

Answer 13

Not for all x, xRx (eg admires)

Question 14

What is symmetry?

Answer 14

xRy –> yRx

Question 15

Define Asymmetry

Answer 15

xRy –> ~yRx

Question 16

What is non-symmetry?

Answer 16

Not for all x, y, xRy –> yRx

Question 17

Explain Antisymmetry

Answer 17

(xRy ^ yRx) –> x = y (eg less than or equal to)

Basically asymmetry with some symmetry

Question 18

State Transitivity

Answer 18

(xRy ^ yRz) –> xRz

Question 19

Explain Seriality

Answer 19

For all x, VxEy xRy

For everything, there is something they are related to (eg love)

Question 20

What is denseness?

Answer 20

xRy –> Ez, xRz ^ zRy

Question 21

Define reflexive

Answer 21

wRw

Question 22

What is a Row interpretation in K?

Answer 22

An interp with Reflexive R

Question 23

What is a Tau interpretation in K?

Answer 23

An interp with Transitive R

Question 24

What is a Sigma interpretation in K?

Answer 24

An interp with Symmetrical R

Question 25

Give KRow aka

Answer 25

T

Question 26

Give KRowSig aka

Answer 26

B

Question 27

What is S4?

Answer 27

KRowTau

Question 28

What is S5?

Answer 28

KRowSigTau

Question 29

What is the process of tableaux?

Answer 29

Every time a new relation is opened:

• Check for special rule (Trans, Row etc)
• Check boxes