# thodg/slides 2020-05-22T11:42:51

update modal-logic

```1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
```
``````diff --git a/modal-logic/index.md b/modal-logic/index.md
index 145870b..df25f1f 100644
--- a/modal-logic/index.md
+++ b/modal-logic/index.md
@@ -1,13 +1,46 @@
# Modal logic

## 1. Notation
-¬A   "Not A"

-□A    "It is necessary that A"
+### 1.1 Not
+"Not A"

-◊A    "It is possible that A"
+¬A

-A → B "If A then B"
+### 1.2 Necessarilly
+"It is necessary that A"
+
+□A
+
+### 1.3 Possibly
+"It is possible that A"
+
+◊A
+
+### 1.4 Implies
+"If A then B"
+
+A → B
+
+### 1.5 Conjunction
+"A and B"
+
+A ∧ B
+
+### 1.6 Disjunction
+"A or B"
+
+A ∨ B
+
+### 1.7 Exclusive disjunction
+"A xor B"
+
+A ⊕ B
+
+### 1.8 Mutual implication
+"If A then B and if B then A"
+
+A ↔ B

## 2. Contruction
K is a weak logic (Saul Kripke)
@@ -21,7 +54,15 @@ A is a theorem of K → □A is a theorem of K
### 2.3 Operator ◊
◊A = ¬□¬A

-## 3. T
+## 3. Lemmas
+
+### 3.1 Necessary conjunction
+□(A ∧ B) ↔ □A ∧ □B
+
+### 3.2 Disjonction of necessities
+◻A ∨ ◻B → □(A ∨ B)
+
+## 4. T
T is K plus the following axiom :

□A → A
``````