Intuistionistic Logic

Question 1

Give semantics for A –> B

Answer 1

νw(A⊐B)=1 iff νw′(A)=0 or νw′(B)=1 for all w′ such that wRw’

Question 2

Give semantics for ~A

Answer 2

νw(⇁A)=1 iff νw′(A)=0 for all w′ such that wRw′

Question 3

Give conjunction rules for tableaux

Answer 3

+∧-rule A∧B,+i ✓ ___ −∧-rule A∧B,−i ✓
______ ↓ ______________↙ ↘
______ A ,+i _________ A,−i ___ B,−I
______ B ,+i

Question 4

Give disjunction rules for tableaux

Answer 4

+∨-rule A∨B,+i ✓ _____ −∨-rule A∨B,−i ✓
____ ↙ ↘ _______________ ↓
___ A ,+i _ B,+i _____________ A,−i
_________________________ B, -i

Question 5

Conditional rules for tab

Answer 5

\+⊐-rule A⊐B,+i \_\_\_\_\_\_\_ −⊐-rule A⊐B,−i ✓ 
 \_\_\_\_\_\_\_irj \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_↓
 \_\_\_\_\_↙ ↘ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_irj
\_\_\_ A ,−j \_\_ B,+j \_\_\_\_\_\_\_\_\_\_\_\_\_\_A,+j 
\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_B,−j

Question 6

Negation Tableaux

Answer 6

+⇁-rule ⇁A,+i _______ −⇁-rule ⇁A,−i ✓
________irj _________________↓
________↓ _________________ irj
_______ A ,−j _______________ A,+j

Question 7

Row, Tau and Heredity Tableaux

Answer 7

ρ: iri ____________ τ: irj ___________ Heredity: p,+i
_________________ jrk __________________ irj
for all i ___________ ↓ ___________________ ↓
_________________ irk __________________ p,+j

Question 8

When does a branch close?

Answer 8

A, -i

A, +i

Question 9

What is important to remember for translations in G?

Answer 9

◻Xg ⊧Kρτ Ag

P’s have boxes in front before you translate

Question 10

What is important to remember about translations and tableaux?

Answer 10

Translate before you Tableaux

Question 11

List the Translations for G

Answer 11

▸Ag = A for atomic wffs
▸(⇁A)g = ¬◻Ag
▸(A∧B)g = Ag∧Bg 
▸(A∨B)g = ◻Ag∨◻Bg 
▸(A⊐B)g = ◻Ag⊃◻Bg

Question 12

What is the G translation for atomics?

Answer 12

Ag = A for atomic wffs

Question 13

G translation for negation?

Answer 13

(⇁A)g = ¬◻Ag

Question 14

G translation for conjunction?

Answer 14

(A∧B)g = Ag∧Bg

Question 15

G translation for disjunction?

Answer 15

(A∨B)g = ◻Ag∨◻Bg

Question 16

G translation for conditional?

Answer 16

(A⊐B)g = ◻Ag⊃◻Bg

Question 17

What Logic does Godel Logic translate into?

Answer 17

S4

Question 18

What is the only difference between Classical Natural Deduction and Intuitionistic Natural Deduction?

Answer 18

DN is replaced with EFQ

Question 19

What is EFQ?

Answer 19

s1,…,sm (s) – … ⋮
s1,…,sm (u) A s EFQ

Need contradiction and then can sub-in whatever you want

Question 20

What defines Godel-Dummett Logic?

Answer 20

Intuitionistic but with constraint of Linearity

Question 21

What is Linearity?

Answer 21

▸Reflexive: xRx
▸Antisymmetric: if xRy and yRx then x=y
▸Transitive: if xRy and yRz, then xRz
▸Total: either xRy or yRx

Question 22

What is total?

Answer 22

There has to be some kind of connection between worlds

Question 23

What is the interpretation for Intuitionistic Logic?

Answer 23

{W,R,v}

Question 24

What is the interpretation for G-D Logic?

Answer 24

⟨W,R,v⟩ s.t. R is linear

Question 25

What must R be in Intuitionistic Logic?

Answer 25

Reflexive and Transitive

Question 26

What are v for countermodels?

Answer 26

If p, +i then v(p) = 1 | If p, -i then v(p) = 0

Question 27

What is Validity?

Answer 27

Truth preserving as per ``` True = 'is provable' False = 'is refutable' ```

Question 28

Describe heredity

Answer 28

If νw(A) = 1 and wRw′, then νw′(A )= 1