This document discusses modal logics and formalisms. It defines modal logics as logics that add new logical constants like necessity (□) and possibility (◇) to classical logic. It describes how modal logics can be classified based on whether they are extended logics that add new well-formed formulas or deviant logics that interpret the usual logical constants differently. The document then focuses on modal logics, defining their language and providing details on their model theory using possible world semantics. It discusses truth in possible worlds and models. It also describes several axiomatic modal systems and the relationships between them, and examines the classes of models validated by different axioms.

1. Models and Formalisms
S. Garlatti
MR2A Informatique
2012

2. Progress
1 Introduction
2 Modal Logics
3 Model Theory
4 Axiomatic Theory
5 Main Modal Systems
6 Axioms and Class of Models
7 A Knowledge and Belief Logic
2 / 40 SG Models and Formalisms

3. Non-classical Logics
Classical logics: propositional logic, first-order logic
Non-classical logics can be classified in two main categories:
Extended logics
New logical constants are added
The set of well-formed formulas (wff) is a proper superset of the
set of well-formed formulas in classical logic.
The set of theorems generated is a proper superset of the set of
theorems generated by classical logic,
Added theorems are only the result of the new wff.
3 / 40 SG Models and Formalisms

4. Non-classical Logics
Classical logics: propositional logic, first-order logic
Non-classical logics can be classified in two main categories:
Deviant logics
The usual logical constants are kept, but with a different meaning
Only a subset of the theorems from the classical logic hold
A non-exclusive classification, a logic could be in both.
4 / 40 SG Models and Formalisms

5. Non-classical Logics
Extended Logics
Modal Logics
Tense Logics
Combination of Tense and Modality
Intensional Logic
Dynamic Logic
Deontic Logic (obligatory, permitted)
Conditional logic
...
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6. Non-classical Logics
Deviant Logics
Intuitionistic logics
Multivalued logics
Fuzzy logics
Paraconsistent logics
Probabilistic logic
...
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7. Progress
1 Introduction
2 Modal Logics
3 Model Theory
4 Axiomatic Theory
5 Main Modal Systems
6 Axioms and Class of Models
7 A Knowledge and Belief Logic
7 / 40 SG Models and Formalisms

8. Modal Logics
Traditional alethic modalities or modalities of truth
Propositional logic
New logical constants
Necessity:
Possibility: ♦
Alphabet
A set of propositional symbols S = {p1, p2, ..., pn}
A set of connectors: ¬, →, ↔, ∨, ∧
A set of modalities: , ♦
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9. The Language
Well-Formed Formulae
Let pi be a propositional symbol, (pi ∈ S), pi is an Atomic
formula
If G and H are wff
(G → H), (G ↔ H), (G ∨ H), (G ∧ H), ¬G, G and ♦G
are wff .
Let P and Q be wff
( P → ♦P)
( ♦P → ♦♦Q)
(( ♦P ∨ ♦ ♦Q) ∧ P)
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10. Progress
1 Introduction
2 Modal Logics
3 Model Theory
4 Axiomatic Theory
5 Main Modal Systems
6 Axioms and Class of Models
7 A Knowledge and Belief Logic
10 / 40 SG Models and Formalisms

11. Model Theory
In Propositional and Predicate Logics
The Interpretation function I is truth-functional: the truth or
falsity of a well-formed formula is determined by the truth or
falsity of its components.
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12. Model Theory
In Propositional and Predicate Logics
The Interpretation function I is truth-functional: the truth or
falsity of a well-formed formula is determined by the truth or
falsity of its components.
In Modal Logics, is-it the same?
P ¬P P
V F ?
F V F
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13. Model Theory
In Propositional and Predicate Logics
The Interpretation function I is truth-functional: the truth or
falsity of a well-formed formula is determined by the truth or
falsity of its components.
In Modal Logics, is-it the same?
P ¬P P
V F ?
F V F
In Modal logics, the Interpretation function I is not
truth-functional
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14. Model Theory
A standard Model is a structure M = (W, R, P) where
W is a set of possible worlds
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15. Model Theory
A standard Model is a structure M = (W, R, P) where
W is a set of possible worlds
R is a binary relation on W (R ⊆ W × W)
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16. Model Theory
A standard Model is a structure M = (W, R, P) where
W is a set of possible worlds
R is a binary relation on W (R ⊆ W × W)
R is an accessibility relationship between possible worlds.
We can write: α R β
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17. Model Theory
A standard Model is a structure M = (W, R, P) where
W is a set of possible worlds
R is a binary relation on W (R ⊆ W × W)
R is an accessibility relationship between possible worlds.
We can write: α R β
P is a mapping from natural numbers to subsets of W
(Pn ⊆ W, for each natural number n)
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18. Model Theory
A standard Model is a structure M = (W, R, P) where
W is a set of possible worlds
R is a binary relation on W (R ⊆ W × W)
R is an accessibility relationship between possible worlds.
We can write: α R β
P is a mapping from natural numbers to subsets of W
(Pn ⊆ W, for each natural number n)
P is an assignement of truth value to atomic formulae at possible
worlds.
It is a function on the set {0, 1, 2, ..., } of natural numbers such
that for each such number n, Pn is a subset of W.
It represents an assignment of sets of possible worlds to atomic
formulae (for pi ∈ S, Pi ⊆ W, ∀α such that α ∈ Pi , pi is true),
12 / 40 SG Models and Formalisms

19. Possible Worlds, Truth, Validity
Truth in a possible world, α ∈ W in M = (W, R, P)
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20. Possible Worlds, Truth, Validity
Truth in a possible world, α ∈ W in M = (W, R, P)
For propositional symbols pi , |=M
α pi iff α ∈ Pi for
n = {1, 2, 3, ..., n}
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21. Possible Worlds, Truth, Validity
Truth in a possible world, α ∈ W in M = (W, R, P)
For propositional symbols pi , |=M
α pi iff α ∈ Pi for
n = {1, 2, 3, ..., n}
|=M
α A means that A is true for the world α in the standard
model M
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22. Possible Worlds, Truth, Validity
Truth in a possible world, α ∈ W in M = (W, R, P)
For propositional symbols pi , |=M
α pi iff α ∈ Pi for
n = {1, 2, 3, ..., n}
|=M
α A means that A is true for the world α in the standard
model M
|=M
α ¬A iff not |=M
α A
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23. Possible Worlds, Truth, Validity
Truth in a possible world, α ∈ W in M = (W, R, P)
For propositional symbols pi , |=M
α pi iff α ∈ Pi for
n = {1, 2, 3, ..., n}
|=M
α A means that A is true for the world α in the standard
model M
|=M
α ¬A iff not |=M
α A
|=M
α (A ∨ B) iff either |=M
α A or |=M
α B), or both
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24. Possible Worlds, Truth, Validity
Truth in a possible world, α ∈ W in M = (W, R, P)
For propositional symbols pi , |=M
α pi iff α ∈ Pi for
n = {1, 2, 3, ..., n}
|=M
α A means that A is true for the world α in the standard
model M
|=M
α ¬A iff not |=M
α A
|=M
α (A ∨ B) iff either |=M
α A or |=M
α B), or both
|=M
α (A ∧ B) iff both |=M
α A and |=M
α B)
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25. Possible Worlds, Truth, Validity
Truth in a possible world, α ∈ W in M = (W, R, P)
|=M
α A iff for every β in M such that αRβ, |=M
β A
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26. Possible Worlds, Truth, Validity
Truth in a possible world, α ∈ W in M = (W, R, P)
|=M
α A iff for every β in M such that αRβ, |=M
β A
|=M
α ♦A iff for some β in M such that αRβ, |=M
β A
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27. Possible Worlds, Truth, Validity
Truth in a possible world, α ∈ W in M = (W, R, P)
|=M
α A iff for every β in M such that αRβ, |=M
β A
|=M
α ♦A iff for some β in M such that αRβ, |=M
β A
Truth in a Model
|=M
A iff for every world α in M, |=M
α A
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28. Possible Worlds, Truth, Validity
Truth in a possible world, α ∈ W in M = (W, R, P)
|=M
α A iff for every β in M such that αRβ, |=M
β A
|=M
α ♦A iff for some β in M such that αRβ, |=M
β A
Truth in a Model
|=M
A iff for every world α in M, |=M
α A
Validity in a class of Models
|=C A iff for every model M in C, |=M
A
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29. Progress
1 Introduction
2 Modal Logics
3 Model Theory
4 Axiomatic Theory
5 Main Modal Systems
6 Axioms and Class of Models
7 A Knowledge and Belief Logic
15 / 40 SG Models and Formalisms

30. Axiomatic Theory
Modal Logic KT
Axioms
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31. Axiomatic Theory
Modal Logic KT
Axioms
Necessity axiom, called T
( P → P)
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32. Axiomatic Theory
Modal Logic KT
Axioms
Necessity axiom, called T
( P → P)
Axiom K
( (P → Q) → ( P → Q))
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33. Axiomatic Theory
Modal Logic KT
Axioms
Necessity axiom, called T
( P → P)
Axiom K
( (P → Q) → ( P → Q))
Inference Rules
Necessity Rule
P P
16 / 40 SG Models and Formalisms

34. Axiomatic Theory
Modal Logic KT
Axioms
Necessity axiom, called T
( P → P)
Axiom K
( (P → Q) → ( P → Q))
Inference Rules
Necessity Rule
P P
Modus Ponens
P, (P → Q) Q
16 / 40 SG Models and Formalisms

35. KT System: Axiomatic Theory
Properties of the Modal Logic KT
Consistency: KT is Consistent
Soundness: KT is Sound
Completeness: KT is Complete
Consequences: Axioms K and T are Valid Formulae.
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36. Progress
1 Introduction
2 Modal Logics
3 Model Theory
4 Axiomatic Theory
5 Main Modal Systems
6 Axioms and Class of Models
7 A Knowledge and Belief Logic
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37. Main Modal Systems
The main modal systems are composed from a small set of axioms:
Axiom T: ( P → P)
Axiom D: ( P → ♦P)
Axiom B: (P → ♦P)
Axiom 4: ( P → P)
Axiom 5: (♦P → ♦P)
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38. Main Modal Systems
The main modal systems are composed from a small set of axioms:
Axiom T: ( P → P)
Axiom D: ( P → ♦P)
Axiom B: (P → ♦P)
Axiom 4: ( P → P)
Axiom 5: (♦P → ♦P)
From these axioms, it possible to build the following systems: KT,
KD, KB, K4, K5, K45, KB4, KD4, KTB, KT4, KT5, etc.
19 / 40 SG Models and Formalisms

39. Relations between Axiomatic Modal Systems
K
KD K4 K5 KB
KTKDB KD4KD5 K45
KTB KT4 KD45 KB4
KT5
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40. Inclusion of K Axiomatic Systems
For instance, we can demonstrate that KD is included in KT,
We have to deduce ( P → ♦P) from ( P → P)
( P → P)
(¬P → ¬ P)
(¬P → ♦¬P)
T contraposition
(P → ♦P)
( P → ♦P)
Transitivity
21 / 40 SG Models and Formalisms

41. Progress
1 Introduction
2 Modal Logics
3 Model Theory
4 Axiomatic Theory
5 Main Modal Systems
6 Axioms and Class of Models
7 A Knowledge and Belief Logic
22 / 40 SG Models and Formalisms

42. Axioms and Class of Models
Theorem: None of the Axioms D, T, B, 4 and 5 is Valid in the
class of all standards models
Proof: it is enough for each axiom to exhibit a counter-model that
falsifies it. For the axiom D: ( P → ♦P):
23 / 40 SG Models and Formalisms

43. Axioms and Class of Models
Theorem: None of the Axioms D, T, B, 4 and 5 is Valid in the
class of all standards models
Proof: it is enough for each axiom to exhibit a counter-model that
falsifies it. For the axiom D: ( P → ♦P):
1 Let M = (W, R, P) be a standard model in which W = {α},
R = ∅, Pn = ∅ for n 0
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44. Axioms and Class of Models
Theorem: None of the Axioms D, T, B, 4 and 5 is Valid in the
class of all standards models
Proof: it is enough for each axiom to exhibit a counter-model that
falsifies it. For the axiom D: ( P → ♦P):
1 Let M = (W, R, P) be a standard model in which W = {α},
R = ∅, Pn = ∅ for n 0
2 M contains one world to which no world is related and at which
every atomic formula is false.
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45. Axioms and Class of Models
Theorem: None of the Axioms D, T, B, 4 and 5 is Valid in the
class of all standards models
Proof: it is enough for each axiom to exhibit a counter-model that
falsifies it. For the axiom D: ( P → ♦P):
1 Let M = (W, R, P) be a standard model in which W = {α},
R = ∅, Pn = ∅ for n 0
2 M contains one world to which no world is related and at which
every atomic formula is false.
3 W is a set, R is a binary relation(the empty relation) and P is a
mapping from natural numbers to (empty) subsets of W
23 / 40 SG Models and Formalisms

46. Axioms and Class of Models
Theorem: None of the Axioms D, T, B, 4 and 5 is Valid in the
class of all standards models
Proof: it is enough for each axiom to exhibit a counter-model that
falsifies it. For the axiom D: ( P → ♦P):
1 Let M = (W, R, P) be a standard model in which W = {α},
R = ∅, Pn = ∅ for n 0
2 M contains one world to which no world is related and at which
every atomic formula is false.
3 W is a set, R is a binary relation(the empty relation) and P is a
mapping from natural numbers to (empty) subsets of W
4 Every β in M such that αRβ is such that |=M
β P and there is
not some β in M such that αRβ and |=M
β P
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47. Axioms and Class of Models
The necessity Axiom ( P → P) in the following Standard Model:
w1
¬p1
w2
p1
w3
p1
w4
p1R
R
R
R
R
R
RR
24 / 40 SG Models and Formalisms

48. Axioms and Class of Models
Under What Conditions, The necessity Axiom ( P → P) in
the KT system will be valid:
Hypothesis |=M
α P, we have to demonstrate |=M
α P
25 / 40 SG Models and Formalisms

49. Axioms and Class of Models
Under What Conditions, The necessity Axiom ( P → P) in
the KT system will be valid:
Hypothesis |=M
α P, we have to demonstrate |=M
α P
1 By definition |=M
α P iff for every β in M such that αRβ, |=M
β P
25 / 40 SG Models and Formalisms

50. Axioms and Class of Models
Under What Conditions, The necessity Axiom ( P → P) in
the KT system will be valid:
Hypothesis |=M
α P, we have to demonstrate |=M
α P
1 By definition |=M
α P iff for every β in M such that αRβ, |=M
β P
2 Consequently, |=M
α P iff R is reflexive, that is to say αRα.
25 / 40 SG Models and Formalisms

51. Axioms and Class of Models
Under What Conditions, The necessity Axiom ( P → P) in
the KT system will be valid:
Hypothesis |=M
α P, we have to demonstrate |=M
α P
1 By definition |=M
α P iff for every β in M such that αRβ, |=M
β P
2 Consequently, |=M
α P iff R is reflexive, that is to say αRα.
In the KT system, the necessity Axiom is valid under the following
condition:
|=C ( P → P), C is a class of models in which R is
Reflexive
25 / 40 SG Models and Formalisms

52. Axioms and Class of Models
Definition: Let M = (W, R, P) be a standard model, the
relation R, with α, β, γ ∈ W is:
26 / 40 SG Models and Formalisms

53. Axioms and Class of Models
Definition: Let M = (W, R, P) be a standard model, the
relation R, with α, β, γ ∈ W is:
Serial iff for every α in M there is a β in M such that α R β
26 / 40 SG Models and Formalisms

54. Axioms and Class of Models
Definition: Let M = (W, R, P) be a standard model, the
relation R, with α, β, γ ∈ W is:
Serial iff for every α in M there is a β in M such that α R β
Reflexive iff for every α in M, α R α
26 / 40 SG Models and Formalisms

55. Axioms and Class of Models
Definition: Let M = (W, R, P) be a standard model, the
relation R, with α, β, γ ∈ W is:
Serial iff for every α in M there is a β in M such that α R β
Reflexive iff for every α in M, α R α
Symmetric iff for every α and β in M, if α R β then β R α
26 / 40 SG Models and Formalisms

56. Axioms and Class of Models
Definition: Let M = (W, R, P) be a standard model, the
relation R, with α, β, γ ∈ W is:
Serial iff for every α in M there is a β in M such that α R β
Reflexive iff for every α in M, α R α
Symmetric iff for every α and β in M, if α R β then β R α
Transitive iff for every α, β and γ in M, if α R β and β R γ
then α R γ
26 / 40 SG Models and Formalisms

57. Axioms and Class of Models
Definition: Let M = (W, R, P) be a standard model, the
relation R, with α, β, γ ∈ W is:
Serial iff for every α in M there is a β in M such that α R β
Reflexive iff for every α in M, α R α
Symmetric iff for every α and β in M, if α R β then β R α
Transitive iff for every α, β and γ in M, if α R β and β R γ
then α R γ
Euclidian iff for every α, β and γ in M, if α R β and α R γ
then β R γ
26 / 40 SG Models and Formalisms

58. Axioms and Class of Models
Theorem: the following axioms are valid respectively in the
indicated class of standard models:
D: Serial
T: Reflexive
B: Symmetric
4: Transitive
5: Euclidian
27 / 40 SG Models and Formalisms

59. Axioms and Class of Models
Systems Serial Reflexive Symmetric Transitive Euclidian
KD x
KT x
KB x
K4 x
K5 x
KDB x x
KD4 x x
KD5 x x
KD45 x x
KB4 x x
KB4 x x
KTB x x
KT4 x x
KT5 x x
KT5 x x x
KT5 x x x
KT5 x x x
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60. Progress
1 Introduction
2 Modal Logics
3 Model Theory
4 Axiomatic Theory
5 Main Modal Systems
6 Axioms and Class of Models
7 A Knowledge and Belief Logic
29 / 40 SG Models and Formalisms

61. Knowledge and Belief Logic
Two modal operators, B for Belief and K for Knowledge,
Knowledge Properties for an agent λ:
Axioms
1 Axiom 1 (Distribution Axiom, K): (KλP ∧ Kλ(P → Q)) → KλQ
- Same as (Kλ(P → Q)) → (KλP → KλQ))
2 Axiom 2 (Knowledge Axiom or T): (KλP → P)
3 Axiom 3 (Positive Introspection Axiom, 4): (KλP → KλKλP)
4 Axiom 4 (Negative Introspection Axiom): (¬KλP → Kλ¬KλP)
- (¬ P → ¬ P) (♦¬P → ♦¬P)
- (♦¬P → ♦¬P) (♦Q → ♦Q) (axiom 5)
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62. Knowledge and Belief Logic
Two modal operators, B for Belief and K for Knowledge,
Knowledge Properties for an agent λ:
Inference Rules
Necessitation: P KλP
Omniscience Logic: if P Q and KλP then KλQ (axiom 1 +
necessitation rule)
The axiomatic theory correspond with the KT45 modal system in
which the accessibility relation is symmetric, reflexive and
transitive.
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63. Three-Wise-Men Problem
A king wishes to know whether his three advisors are as wise as
they claim to be.
32 / 40 SG Models and Formalisms

64. Three-Wise-Men Problem
A king wishes to know whether his three advisors are as wise as
they claim to be.
Three chairs are lined up, all facing the same direction, with one
behind the other. The wise men are instructed to sit down.
32 / 40 SG Models and Formalisms

65. Three-Wise-Men Problem
A king wishes to know whether his three advisors are as wise as
they claim to be.
Three chairs are lined up, all facing the same direction, with one
behind the other. The wise men are instructed to sit down.
The wise man in the back (wise man 3) can see the backs of the
other two men.
32 / 40 SG Models and Formalisms

66. Three-Wise-Men Problem
A king wishes to know whether his three advisors are as wise as
they claim to be.
Three chairs are lined up, all facing the same direction, with one
behind the other. The wise men are instructed to sit down.
The wise man in the back (wise man 3) can see the backs of the
other two men.
The man in the middle (wise man 2) can only see the one wise
man in front of him (wise man 1).
32 / 40 SG Models and Formalisms

67. Three-Wise-Men Problem
A king wishes to know whether his three advisors are as wise as
they claim to be.
Three chairs are lined up, all facing the same direction, with one
behind the other. The wise men are instructed to sit down.
The wise man in the back (wise man 3) can see the backs of the
other two men.
The man in the middle (wise man 2) can only see the one wise
man in front of him (wise man 1).
The wise man in front (wise man 1) can see neither wise man 3
nor wise man 2.
32 / 40 SG Models and Formalisms

68. Three-Wise-Men Problem
A king wishes to know whether his three advisors are as wise as
they claim to be.
33 / 40 SG Models and Formalisms

69. Three-Wise-Men Problem
A king wishes to know whether his three advisors are as wise as
they claim to be.
The king informs the wise men that he has three cards, all of
which are either black or white, at least one of which is
white. He places one card, face up, behind each of the three
wise men.
33 / 40 SG Models and Formalisms

70. Three-Wise-Men Problem
A king wishes to know whether his three advisors are as wise as
they claim to be.
The king informs the wise men that he has three cards, all of
which are either black or white, at least one of which is
white. He places one card, face up, behind each of the three
wise men.
Each wise man must determine the color of his own card and
announce what it is as soon as he knows. The first to correctly
announce the color of his own card will be aptly rewarded. All
know that this will happen. The room is silent; then, after
several minutes, wise man 1 says: My card is white!.
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71. Three-Wise-Men Problem
A king wishes to know whether his three advisors are as wise as
they claim to be.
We assume in this puzzle that the wise men do not lie, that they
all have the same reasoning capabilities, and that they can all think
at the same speed. We then can postulate that the following
reasoning took place.
34 / 40 SG Models and Formalisms

72. Three-Wise-Men Problem
35 / 40 SG Models and Formalisms

73. Three-Wise-Men Problem
Each wise man knows there is at least one white card. If the
cards of wise man 2 and wise man 1 were black, then wise man
3 would have been able to announce immediately that his card
was white.
35 / 40 SG Models and Formalisms

74. Three-Wise-Men Problem
Each wise man knows there is at least one white card. If the
cards of wise man 2 and wise man 1 were black, then wise man
3 would have been able to announce immediately that his card
was white.
They all realize this (they are all truly wise). Since wise man 3
kept silent, either wise man 2’s card is white, or wise man 1’s is.
35 / 40 SG Models and Formalisms

75. Three-Wise-Men Problem
Each wise man knows there is at least one white card. If the
cards of wise man 2 and wise man 1 were black, then wise man
3 would have been able to announce immediately that his card
was white.
They all realize this (they are all truly wise). Since wise man 3
kept silent, either wise man 2’s card is white, or wise man 1’s is.
At this point wise man 2 would be able to determine, if wise man
1’s were black, that his card was white. They all realize this.
35 / 40 SG Models and Formalisms

76. Three-Wise-Men Problem
Each wise man knows there is at least one white card. If the
cards of wise man 2 and wise man 1 were black, then wise man
3 would have been able to announce immediately that his card
was white.
They all realize this (they are all truly wise). Since wise man 3
kept silent, either wise man 2’s card is white, or wise man 1’s is.
At this point wise man 2 would be able to determine, if wise man
1’s were black, that his card was white. They all realize this.
Since wise man 2 also remains silent, wise man 1 knows his card
must be white.
35 / 40 SG Models and Formalisms

77. Three-Wise-Men Problem
If we reduce the problem with two advisors A and B and have the
following axioms:
36 / 40 SG Models and Formalisms

78. Three-Wise-Men Problem
If we reduce the problem with two advisors A and B and have the
following axioms:
1 A and B know that each one can see the card of the other one
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79. Three-Wise-Men Problem
If we reduce the problem with two advisors A and B and have the
following axioms:
1 A and B know that each one can see the card of the other one
If A does not have a white card, B knows that A does not have a
white card
- (¬Blanc(A) → KB¬Blanc(A))
36 / 40 SG Models and Formalisms

80. Three-Wise-Men Problem
If we reduce the problem with two advisors A and B and have the
following axioms:
1 A and B know that each one can see the card of the other one
If A does not have a white card, B knows that A does not have a
white card
- (¬Blanc(A) → KB¬Blanc(A))
A knows that if A does not have a white card, B knows that A
does not have a white card
36 / 40 SG Models and Formalisms

81. Three-Wise-Men Problem
If we reduce the problem with two advisors A and B and have the
following axioms:
1 A and B know that each one can see the card of the other one
If A does not have a white card, B knows that A does not have a
white card
- (¬Blanc(A) → KB¬Blanc(A))
A knows that if A does not have a white card, B knows that A
does not have a white card
- KA(¬Blanc(A) → KB¬Blanc(A))
36 / 40 SG Models and Formalisms

82. Three-Wise-Men Problem
If we reduce the problem with two advisors A and B and have the
following axioms:
2 A and B know that at least one of them has a white card and
each one knows that the other knows it.
37 / 40 SG Models and Formalisms

83. Three-Wise-Men Problem
If we reduce the problem with two advisors A and B and have the
following axioms:
2 A and B know that at least one of them has a white card and
each one knows that the other knows it.
A knows that B knows that if A does not have a white card then
B has a white card
37 / 40 SG Models and Formalisms

84. Three-Wise-Men Problem
If we reduce the problem with two advisors A and B and have the
following axioms:
2 A and B know that at least one of them has a white card and
each one knows that the other knows it.
A knows that B knows that if A does not have a white card then
B has a white card
- KAKB(¬Blanc(A) → Blanc(B))
37 / 40 SG Models and Formalisms

85. Three-Wise-Men Problem
If we reduce the problem with two advisors A and B and have the
following axioms:
2 A and B know that at least one of them has a white card and
each one knows that the other knows it.
A knows that B knows that if A does not have a white card then
B has a white card
- KAKB(¬Blanc(A) → Blanc(B))
B declare that he does not say if he has a white card, then A
knows that B does not knows the color of his card.
37 / 40 SG Models and Formalisms

86. Three-Wise-Men Problem
If we reduce the problem with two advisors A and B and have the
following axioms:
2 A and B know that at least one of them has a white card and
each one knows that the other knows it.
A knows that B knows that if A does not have a white card then
B has a white card
- KAKB(¬Blanc(A) → Blanc(B))
B declare that he does not say if he has a white card, then A
knows that B does not knows the color of his card.
- KA¬KBBlanc(B)
37 / 40 SG Models and Formalisms

87. Three-Wise-Men Problem
If we can reduce the problem with two advisors and have the
following axioms:
38 / 40 SG Models and Formalisms

88. Three-Wise-Men Problem
If we can reduce the problem with two advisors and have the
following axioms:
KA(¬Blanc(A) → KB¬Blanc(A)) [1]
38 / 40 SG Models and Formalisms

89. Three-Wise-Men Problem
If we can reduce the problem with two advisors and have the
following axioms:
KA(¬Blanc(A) → KB¬Blanc(A)) [1]
KAKB(¬Blanc(A) → Blanc(B)) [2]
KA¬KBBlanc(B) [3]
38 / 40 SG Models and Formalisms

90. Three-Wise-Men Problem
If we can reduce the problem with two advisors and have the
following axioms:
KA(¬Blanc(A) → KB¬Blanc(A)) [1]
KAKB(¬Blanc(A) → Blanc(B)) [2]
KA¬KBBlanc(B) [3]
38 / 40 SG Models and Formalisms

91. Three-Wise-Men Problem
If we can reduce the problem with two advisors and have the
following axioms:
KA(¬Blanc(A) → KB¬Blanc(A)) [1]
KAKB(¬Blanc(A) → Blanc(B)) [2]
KA¬KBBlanc(B) [3]
[1] (KA(¬Blanc(A) → KB¬Blanc(A)) (KλP → P) [Axiom2]
[4] (¬Blanc(A) → KB¬Blanc(A))
38 / 40 SG Models and Formalisms

92. Three-Wise-Men Problem
If we can reduce the problem with two advisors and have the
following axioms:
KA(¬Blanc(A) → KB¬Blanc(A)) [1]
KAKB(¬Blanc(A) → Blanc(B)) [2]
KA¬KBBlanc(B) [3]
[1] (KA(¬Blanc(A) → KB¬Blanc(A)) (KλP → P) [Axiom2]
[4] (¬Blanc(A) → KB¬Blanc(A))
[2] KAKB(¬Blanc(A) → Blanc(B)) (KλP → P) [Axiom2]
[5] (KB(¬Blanc(A) → Blanc(B))
(Kλ(P → Q) → (KλP → KλQ)) [Axiom1]
[6] (KB¬Blanc(A) → KBBlanc(B))
38 / 40 SG Models and Formalisms

93. Three-Wise-Men Problem
Omniscience
Contraposition
Transitivity
[4] (¬Blanc(A) → KB¬Blanc(A))
(KB¬Blanc(A) → KBBlanc(B)) [6]
[7] (¬Blanc(A) → KBBlanc(B))
[8] (¬KBBlanc(B) → Blanc(A))
[1] [2] [4] [5]
[9] (KA(¬KBBlanc(B) → Blanc(A)))
(Kλ(P → Q) → (KλP → KλQ)) [Axiom1]
(KA¬KBBlanc(A) → KABlanc(A))
KA¬KBBlanc(B) [3]
[11] KABlanc(A)
39 / 40 SG Models and Formalisms

94. Knowledge and Belief Logic
Two modal operators, B for Belief and K for Knowledge, Belief
Properties for an agent λ:
Axioms
40 / 40 SG Models and Formalisms

95. Knowledge and Belief Logic
Two modal operators, B for Belief and K for Knowledge, Belief
Properties for an agent λ:
Axioms
An agent cannot belief a contradiction: ¬Bλ(False)
40 / 40 SG Models and Formalisms

96. Knowledge and Belief Logic
Two modal operators, B for Belief and K for Knowledge, Belief
Properties for an agent λ:
Axioms
An agent cannot belief a contradiction: ¬Bλ(False)
Positive Introspection Axiom (axiom 4): (BλP → BλBλP)
an agent believes what he believes
40 / 40 SG Models and Formalisms

97. Knowledge and Belief Logic
Two modal operators, B for Belief and K for Knowledge, Belief
Properties for an agent λ:
Axioms
An agent cannot belief a contradiction: ¬Bλ(False)
Positive Introspection Axiom (axiom 4): (BλP → BλBλP)
an agent believes what he believes
(BλBλP → BλP) (converse of the previous axiom)
(BλBαP → BλP)
an agent can belief what another agent believes
40 / 40 SG Models and Formalisms