MotivationBasic Modal LogicLogic Engineering
10—Modal Logic I
CS 5209: Foundation in Logic and AI
Martin Henz and Aquinas Hobor
March 25, 2010
Generated on Thursday 25th March, 2010, 16:22
CS 5209: Foundation in Logic and AI 10—Modal Logic I 1
MotivationBasic Modal LogicLogic Engineering
1 Motivation
2 Basic Modal Logic
3 Logic Engineering
CS 5209: Foundation in Logic and AI 10—Modal Logic I 2
MotivationBasic Modal LogicLogic Engineering
1 Motivation
2 Basic Modal Logic
3 Logic Engineering
CS 5209: Foundation in Logic and AI 10—Modal Logic I 3
MotivationBasic Modal LogicLogic Engineering
Necessity
You are crime investigator and consider different suspects.
CS 5209: Foundation in Logic and AI 10—Modal Logic I 4
MotivationBasic Modal LogicLogic Engineering
Necessity
You are crime investigator and consider different suspects.Maybe the cook did it with a knife?
CS 5209: Foundation in Logic and AI 10—Modal Logic I 5
MotivationBasic Modal LogicLogic Engineering
Necessity
You are crime investigator and consider different suspects.Maybe the cook did it with a knife?Maybe the maid did it with a pistol?
CS 5209: Foundation in Logic and AI 10—Modal Logic I 6
MotivationBasic Modal LogicLogic Engineering
Necessity
You are crime investigator and consider different suspects.Maybe the cook did it with a knife?Maybe the maid did it with a pistol?
But: “The victim Ms Smith made the call before she waskilled.” is necessarily true.
CS 5209: Foundation in Logic and AI 10—Modal Logic I 7
MotivationBasic Modal LogicLogic Engineering
Necessity
You are crime investigator and consider different suspects.Maybe the cook did it with a knife?Maybe the maid did it with a pistol?
But: “The victim Ms Smith made the call before she waskilled.” is necessarily true.
“Necessarily” means in all possible scenarios (worlds)under consideration.
CS 5209: Foundation in Logic and AI 10—Modal Logic I 8
MotivationBasic Modal LogicLogic Engineering
Notions of Truth
Often, it is not enough to distinguish between “true” and“false”.
CS 5209: Foundation in Logic and AI 10—Modal Logic I 9
MotivationBasic Modal LogicLogic Engineering
Notions of Truth
Often, it is not enough to distinguish between “true” and“false”.We need to consider modalities if truth, such as:
necessity (“in all possible scenarios”)morality/law (“in acceptable/legal scenarios”)time (“forever in the future”)
CS 5209: Foundation in Logic and AI 10—Modal Logic I 10
MotivationBasic Modal LogicLogic Engineering
Notions of Truth
Often, it is not enough to distinguish between “true” and“false”.We need to consider modalities if truth, such as:
necessity (“in all possible scenarios”)morality/law (“in acceptable/legal scenarios”)time (“forever in the future”)
Modal logic constructs a framework using which modalitiescan be formalized and reasoning methods can beestablished.
CS 5209: Foundation in Logic and AI 10—Modal Logic I 11
MotivationBasic Modal LogicLogic Engineering
SyntaxSemanticsEquivalences
1 Motivation
2 Basic Modal LogicSyntaxSemanticsEquivalences
3 Logic Engineering
CS 5209: Foundation in Logic and AI 10—Modal Logic I 12
MotivationBasic Modal LogicLogic Engineering
SyntaxSemanticsEquivalences
Syntax of Basic Modal Logic
φ ::= ⊤ | ⊥ | p | (¬φ) | (φ ∧ φ)
| (φ ∨ φ) | (φ→ φ)
| (φ↔ φ)
| (�φ) | (♦φ)
CS 5209: Foundation in Logic and AI 10—Modal Logic I 13
MotivationBasic Modal LogicLogic Engineering
SyntaxSemanticsEquivalences
Pronunciation and Examples
Pronunciation
If we want to keep the meaning open, we simply say “box” and“diamond”.
CS 5209: Foundation in Logic and AI 10—Modal Logic I 14
MotivationBasic Modal LogicLogic Engineering
SyntaxSemanticsEquivalences
Pronunciation and Examples
Pronunciation
If we want to keep the meaning open, we simply say “box” and“diamond”.If we want to appeal to our intuition, we may say “necessarily”and “possibly” (or “forever in the future” and “sometime in thefuture”)
CS 5209: Foundation in Logic and AI 10—Modal Logic I 15
MotivationBasic Modal LogicLogic Engineering
SyntaxSemanticsEquivalences
Pronunciation and Examples
Pronunciation
If we want to keep the meaning open, we simply say “box” and“diamond”.If we want to appeal to our intuition, we may say “necessarily”and “possibly” (or “forever in the future” and “sometime in thefuture”)
Examples
(p ∧ ♦(p → �¬r))
CS 5209: Foundation in Logic and AI 10—Modal Logic I 16
MotivationBasic Modal LogicLogic Engineering
SyntaxSemanticsEquivalences
Pronunciation and Examples
Pronunciation
If we want to keep the meaning open, we simply say “box” and“diamond”.If we want to appeal to our intuition, we may say “necessarily”and “possibly” (or “forever in the future” and “sometime in thefuture”)
Examples
(p ∧ ♦(p → �¬r))
�((♦q ∧ ¬r) → �p)
CS 5209: Foundation in Logic and AI 10—Modal Logic I 17
MotivationBasic Modal LogicLogic Engineering
SyntaxSemanticsEquivalences
Kripke Models
Definition
A model M of basic modal logic is specified by three things:
CS 5209: Foundation in Logic and AI 10—Modal Logic I 18
MotivationBasic Modal LogicLogic Engineering
SyntaxSemanticsEquivalences
Kripke Models
Definition
A model M of basic modal logic is specified by three things:1 A set W , whose elements are called worlds;
CS 5209: Foundation in Logic and AI 10—Modal Logic I 19
MotivationBasic Modal LogicLogic Engineering
SyntaxSemanticsEquivalences
Kripke Models
Definition
A model M of basic modal logic is specified by three things:1 A set W , whose elements are called worlds;2 A relation R on W , meaning R ⊆ W × W , called the
accessibility relation;
CS 5209: Foundation in Logic and AI 10—Modal Logic I 20
MotivationBasic Modal LogicLogic Engineering
SyntaxSemanticsEquivalences
Kripke Models
Definition
A model M of basic modal logic is specified by three things:1 A set W , whose elements are called worlds;2 A relation R on W , meaning R ⊆ W × W , called the
accessibility relation;3 A function L : W → P(Atoms), called the labeling function.
CS 5209: Foundation in Logic and AI 10—Modal Logic I 21
MotivationBasic Modal LogicLogic Engineering
SyntaxSemanticsEquivalences
Who is Kripke?
How do I know I am not dreaming? Kripke asked himself thisquestion in 1952, at the age of 12.
CS 5209: Foundation in Logic and AI 10—Modal Logic I 22
MotivationBasic Modal LogicLogic Engineering
SyntaxSemanticsEquivalences
Who is Kripke?
How do I know I am not dreaming? Kripke asked himself thisquestion in 1952, at the age of 12. His father toldhim about the philosopher Descartes.
CS 5209: Foundation in Logic and AI 10—Modal Logic I 23
MotivationBasic Modal LogicLogic Engineering
SyntaxSemanticsEquivalences
Who is Kripke?
How do I know I am not dreaming? Kripke asked himself thisquestion in 1952, at the age of 12. His father toldhim about the philosopher Descartes.
Modal logic at 17 Kripke’s self-studies in philosophy and logicled him to prove a fundamental completenesstheorem on modal logic at the age of 17.
CS 5209: Foundation in Logic and AI 10—Modal Logic I 24
MotivationBasic Modal LogicLogic Engineering
SyntaxSemanticsEquivalences
Who is Kripke?
How do I know I am not dreaming? Kripke asked himself thisquestion in 1952, at the age of 12. His father toldhim about the philosopher Descartes.
Modal logic at 17 Kripke’s self-studies in philosophy and logicled him to prove a fundamental completenesstheorem on modal logic at the age of 17.
Bachelor in Mathematics from Harvard is his onlynon-honorary degree
CS 5209: Foundation in Logic and AI 10—Modal Logic I 25
MotivationBasic Modal LogicLogic Engineering
SyntaxSemanticsEquivalences
Who is Kripke?
How do I know I am not dreaming? Kripke asked himself thisquestion in 1952, at the age of 12. His father toldhim about the philosopher Descartes.
Modal logic at 17 Kripke’s self-studies in philosophy and logicled him to prove a fundamental completenesstheorem on modal logic at the age of 17.
Bachelor in Mathematics from Harvard is his onlynon-honorary degree
At Princeton Kripke taught philosophy from 1977 onwards.
CS 5209: Foundation in Logic and AI 10—Modal Logic I 26
MotivationBasic Modal LogicLogic Engineering
SyntaxSemanticsEquivalences
Who is Kripke?
How do I know I am not dreaming? Kripke asked himself thisquestion in 1952, at the age of 12. His father toldhim about the philosopher Descartes.
Modal logic at 17 Kripke’s self-studies in philosophy and logicled him to prove a fundamental completenesstheorem on modal logic at the age of 17.
Bachelor in Mathematics from Harvard is his onlynon-honorary degree
At Princeton Kripke taught philosophy from 1977 onwards.
Contributions include modal logic, naming, belief, truth, themeaning of “I”
CS 5209: Foundation in Logic and AI 10—Modal Logic I 27
MotivationBasic Modal LogicLogic Engineering
SyntaxSemanticsEquivalences
Example
W = {x1, x2, x3, x4, x5, x6}R = {(x1, x2), (x1, x3), (x2, x2), (x2, x3), (x3, x2), (x4, x5), (x5, x4), (x5, x6)}L = {(x1, {q}), (x2, {p,q}), (x3, {p}), (x4, {q}), (x5, {}), (x6, {p})}
pq
p
q
p, q
x1
x2
x3
x4
x5
x6
CS 5209: Foundation in Logic and AI 10—Modal Logic I 28
MotivationBasic Modal LogicLogic Engineering
SyntaxSemanticsEquivalences
When is a formula true in a possible world?
Definition
Let M = (W ,R,L), x ∈ W , and φ a formula in basic modallogic. We define x φ via structural induction:
CS 5209: Foundation in Logic and AI 10—Modal Logic I 29
MotivationBasic Modal LogicLogic Engineering
SyntaxSemanticsEquivalences
When is a formula true in a possible world?
Definition
Let M = (W ,R,L), x ∈ W , and φ a formula in basic modallogic. We define x φ via structural induction:
x ⊤
CS 5209: Foundation in Logic and AI 10—Modal Logic I 30
MotivationBasic Modal LogicLogic Engineering
SyntaxSemanticsEquivalences
When is a formula true in a possible world?
Definition
Let M = (W ,R,L), x ∈ W , and φ a formula in basic modallogic. We define x φ via structural induction:
x ⊤x 6 ⊥
CS 5209: Foundation in Logic and AI 10—Modal Logic I 31
MotivationBasic Modal LogicLogic Engineering
SyntaxSemanticsEquivalences
When is a formula true in a possible world?
Definition
Let M = (W ,R,L), x ∈ W , and φ a formula in basic modallogic. We define x φ via structural induction:
x ⊤x 6 ⊥x p iff p ∈ L(x)
CS 5209: Foundation in Logic and AI 10—Modal Logic I 32
MotivationBasic Modal LogicLogic Engineering
SyntaxSemanticsEquivalences
When is a formula true in a possible world?
Definition
Let M = (W ,R,L), x ∈ W , and φ a formula in basic modallogic. We define x φ via structural induction:
x ⊤x 6 ⊥x p iff p ∈ L(x)
x ¬φ iff x 6 φ
CS 5209: Foundation in Logic and AI 10—Modal Logic I 33
MotivationBasic Modal LogicLogic Engineering
SyntaxSemanticsEquivalences
When is a formula true in a possible world?
Definition
Let M = (W ,R,L), x ∈ W , and φ a formula in basic modallogic. We define x φ via structural induction:
x ⊤x 6 ⊥x p iff p ∈ L(x)
x ¬φ iff x 6 φ
x φ ∧ ψ iff x φ and x ψ
CS 5209: Foundation in Logic and AI 10—Modal Logic I 34
MotivationBasic Modal LogicLogic Engineering
SyntaxSemanticsEquivalences
When is a formula true in a possible world?
Definition
Let M = (W ,R,L), x ∈ W , and φ a formula in basic modallogic. We define x φ via structural induction:
x ⊤x 6 ⊥x p iff p ∈ L(x)
x ¬φ iff x 6 φ
x φ ∧ ψ iff x φ and x ψ
x φ ∨ ψ iff x φ or x ψ
...
CS 5209: Foundation in Logic and AI 10—Modal Logic I 35
MotivationBasic Modal LogicLogic Engineering
SyntaxSemanticsEquivalences
When is a formula true in a possible world?
Definition (continued)
Let M = (W ,R,L), x ∈ W , and φ a formula in basic modallogic. We define x φ via structural induction:
...
x φ→ ψ iff x ψ, whenever x φ
CS 5209: Foundation in Logic and AI 10—Modal Logic I 36
MotivationBasic Modal LogicLogic Engineering
SyntaxSemanticsEquivalences
When is a formula true in a possible world?
Definition (continued)
Let M = (W ,R,L), x ∈ W , and φ a formula in basic modallogic. We define x φ via structural induction:
...
x φ→ ψ iff x ψ, whenever x φ
x φ↔ ψ iff (x φ iff x ψ)
CS 5209: Foundation in Logic and AI 10—Modal Logic I 37
MotivationBasic Modal LogicLogic Engineering
SyntaxSemanticsEquivalences
When is a formula true in a possible world?
Definition (continued)
Let M = (W ,R,L), x ∈ W , and φ a formula in basic modallogic. We define x φ via structural induction:
...
x φ→ ψ iff x ψ, whenever x φ
x φ↔ ψ iff (x φ iff x ψ)
x �φ iff for each y ∈ W with R(x , y), we have y φ
CS 5209: Foundation in Logic and AI 10—Modal Logic I 38
MotivationBasic Modal LogicLogic Engineering
SyntaxSemanticsEquivalences
When is a formula true in a possible world?
Definition (continued)
Let M = (W ,R,L), x ∈ W , and φ a formula in basic modallogic. We define x φ via structural induction:
...
x φ→ ψ iff x ψ, whenever x φ
x φ↔ ψ iff (x φ iff x ψ)
x �φ iff for each y ∈ W with R(x , y), we have y φ
x ♦φ iff there is a y ∈ W such that R(x , y) and y φ.
CS 5209: Foundation in Logic and AI 10—Modal Logic I 39
MotivationBasic Modal LogicLogic Engineering
SyntaxSemanticsEquivalences
Example
pq
p
q
p, q
x1
x2
x3
x4
x5
x6
CS 5209: Foundation in Logic and AI 10—Modal Logic I 40
MotivationBasic Modal LogicLogic Engineering
SyntaxSemanticsEquivalences
Example
pq
p
q
p, q
x1
x2
x3
x4
x5
x6
x1 q
CS 5209: Foundation in Logic and AI 10—Modal Logic I 41
MotivationBasic Modal LogicLogic Engineering
SyntaxSemanticsEquivalences
Example
pq
p
q
p, q
x1
x2
x3
x4
x5
x6
x1 q
x1 ♦q, x1 6 �q
CS 5209: Foundation in Logic and AI 10—Modal Logic I 42
MotivationBasic Modal LogicLogic Engineering
SyntaxSemanticsEquivalences
Example
pq
p
q
p, q
x1
x2
x3
x4
x5
x6
x1 q
x1 ♦q, x1 6 �q
x5 6 �p, x5 6 �q, x5 6 �p ∨ �q, x5 �(p ∨ q)
CS 5209: Foundation in Logic and AI 10—Modal Logic I 43
MotivationBasic Modal LogicLogic Engineering
SyntaxSemanticsEquivalences
Example
pq
p
q
p, q
x1
x2
x3
x4
x5
x6
x1 q
x1 ♦q, x1 6 �q
x5 6 �p, x5 6 �q, x5 6 �p ∨ �q, x5 �(p ∨ q)
x6 �φ holds for all φ, but x6 6 ♦φ
CS 5209: Foundation in Logic and AI 10—Modal Logic I 44
MotivationBasic Modal LogicLogic Engineering
SyntaxSemanticsEquivalences
Formula Schemes
Example
We said x6 �φ holds for all φ, but x6 6 ♦φ
CS 5209: Foundation in Logic and AI 10—Modal Logic I 45
MotivationBasic Modal LogicLogic Engineering
SyntaxSemanticsEquivalences
Formula Schemes
Example
We said x6 �φ holds for all φ, but x6 6 ♦φ
Notation
Greek letters denote formulas, and are not propositional atoms.
CS 5209: Foundation in Logic and AI 10—Modal Logic I 46
MotivationBasic Modal LogicLogic Engineering
SyntaxSemanticsEquivalences
Formula Schemes
Example
We said x6 �φ holds for all φ, but x6 6 ♦φ
Notation
Greek letters denote formulas, and are not propositional atoms.
Formula schemes
Terms where Greek letters appear instead of propositionalatoms are called formula schemes.
CS 5209: Foundation in Logic and AI 10—Modal Logic I 47
MotivationBasic Modal LogicLogic Engineering
SyntaxSemanticsEquivalences
Entailment and Equivalence
Definition
A set of formulas Γ entails a formula ψ of basic modal logic if, inany world x of any model M = (W ,R,L), whe have x ψ
whenever x φ for all φ ∈ Γ. We say Γ entails ψ and writeΓ |= ψ.
CS 5209: Foundation in Logic and AI 10—Modal Logic I 48
MotivationBasic Modal LogicLogic Engineering
SyntaxSemanticsEquivalences
Entailment and Equivalence
Definition
A set of formulas Γ entails a formula ψ of basic modal logic if, inany world x of any model M = (W ,R,L), whe have x ψ
whenever x φ for all φ ∈ Γ. We say Γ entails ψ and writeΓ |= ψ.
Equivalence
We write φ ≡ ψ if φ |= ψ and ψ |= φ.
CS 5209: Foundation in Logic and AI 10—Modal Logic I 49
MotivationBasic Modal LogicLogic Engineering
SyntaxSemanticsEquivalences
Some Equivalences
De Morgan rules: ¬�φ ≡ ♦¬φ, ¬♦φ ≡ �¬φ.
CS 5209: Foundation in Logic and AI 10—Modal Logic I 50
MotivationBasic Modal LogicLogic Engineering
SyntaxSemanticsEquivalences
Some Equivalences
De Morgan rules: ¬�φ ≡ ♦¬φ, ¬♦φ ≡ �¬φ.
Distributivity of � over ∧:
�(φ ∧ ψ) ≡ �φ ∧ �ψ
CS 5209: Foundation in Logic and AI 10—Modal Logic I 51
MotivationBasic Modal LogicLogic Engineering
SyntaxSemanticsEquivalences
Some Equivalences
De Morgan rules: ¬�φ ≡ ♦¬φ, ¬♦φ ≡ �¬φ.
Distributivity of � over ∧:
�(φ ∧ ψ) ≡ �φ ∧ �ψ
Distributivity of ♦ over ∨:
♦(φ ∨ ψ) ≡ ♦φ ∨ ♦ψ
CS 5209: Foundation in Logic and AI 10—Modal Logic I 52
MotivationBasic Modal LogicLogic Engineering
SyntaxSemanticsEquivalences
Some Equivalences
De Morgan rules: ¬�φ ≡ ♦¬φ, ¬♦φ ≡ �¬φ.
Distributivity of � over ∧:
�(φ ∧ ψ) ≡ �φ ∧ �ψ
Distributivity of ♦ over ∨:
♦(φ ∨ ψ) ≡ ♦φ ∨ ♦ψ
�⊤ ≡ ⊤, ♦⊥ ≡ ⊥
CS 5209: Foundation in Logic and AI 10—Modal Logic I 53
MotivationBasic Modal LogicLogic Engineering
SyntaxSemanticsEquivalences
Validity
Definition
A formula φ is valid if it is true in every world of every model, i.e.iff |= φ holds.
CS 5209: Foundation in Logic and AI 10—Modal Logic I 54
MotivationBasic Modal LogicLogic Engineering
SyntaxSemanticsEquivalences
Examples of Valid Formulas
All valid formulas of propositional logic
CS 5209: Foundation in Logic and AI 10—Modal Logic I 55
MotivationBasic Modal LogicLogic Engineering
SyntaxSemanticsEquivalences
Examples of Valid Formulas
All valid formulas of propositional logic
¬�φ↔ ♦¬φ
CS 5209: Foundation in Logic and AI 10—Modal Logic I 56
MotivationBasic Modal LogicLogic Engineering
SyntaxSemanticsEquivalences
Examples of Valid Formulas
All valid formulas of propositional logic
¬�φ↔ ♦¬φ�(φ ∧ ψ) ↔ �φ ∧ �ψ
CS 5209: Foundation in Logic and AI 10—Modal Logic I 57
MotivationBasic Modal LogicLogic Engineering
SyntaxSemanticsEquivalences
Examples of Valid Formulas
All valid formulas of propositional logic
¬�φ↔ ♦¬φ�(φ ∧ ψ) ↔ �φ ∧ �ψ
♦(φ ∨ ψ) ↔ ♦φ ∨ ♦ψ
CS 5209: Foundation in Logic and AI 10—Modal Logic I 58
MotivationBasic Modal LogicLogic Engineering
SyntaxSemanticsEquivalences
Examples of Valid Formulas
All valid formulas of propositional logic
¬�φ↔ ♦¬φ�(φ ∧ ψ) ↔ �φ ∧ �ψ
♦(φ ∨ ψ) ↔ ♦φ ∨ ♦ψ
Formula K : �(φ→ ψ) ∧ �φ→ �ψ.
CS 5209: Foundation in Logic and AI 10—Modal Logic I 59
MotivationBasic Modal LogicLogic Engineering
Valid Formulas wrt ModalitiesProperties of RCorrespondence TheoryPreview: Some Modal Logics
1 Motivation
2 Basic Modal Logic
3 Logic EngineeringValid Formulas wrt ModalitiesProperties of RCorrespondence TheoryPreview: Some Modal Logics
CS 5209: Foundation in Logic and AI 10—Modal Logic I 60
MotivationBasic Modal LogicLogic Engineering
Valid Formulas wrt ModalitiesProperties of RCorrespondence TheoryPreview: Some Modal Logics
A Range of Modalities
In a particular context �φ could mean:
CS 5209: Foundation in Logic and AI 10—Modal Logic I 61
MotivationBasic Modal LogicLogic Engineering
Valid Formulas wrt ModalitiesProperties of RCorrespondence TheoryPreview: Some Modal Logics
A Range of Modalities
In a particular context �φ could mean:
It is necessarily true that φ
CS 5209: Foundation in Logic and AI 10—Modal Logic I 62
MotivationBasic Modal LogicLogic Engineering
Valid Formulas wrt ModalitiesProperties of RCorrespondence TheoryPreview: Some Modal Logics
A Range of Modalities
In a particular context �φ could mean:
It is necessarily true that φ
It will always be true that φ
CS 5209: Foundation in Logic and AI 10—Modal Logic I 63
MotivationBasic Modal LogicLogic Engineering
Valid Formulas wrt ModalitiesProperties of RCorrespondence TheoryPreview: Some Modal Logics
A Range of Modalities
In a particular context �φ could mean:
It is necessarily true that φ
It will always be true that φ
It ought to be that φ
CS 5209: Foundation in Logic and AI 10—Modal Logic I 64
MotivationBasic Modal LogicLogic Engineering
Valid Formulas wrt ModalitiesProperties of RCorrespondence TheoryPreview: Some Modal Logics
A Range of Modalities
In a particular context �φ could mean:
It is necessarily true that φ
It will always be true that φ
It ought to be that φ
Agent Q believes that φ
CS 5209: Foundation in Logic and AI 10—Modal Logic I 65
MotivationBasic Modal LogicLogic Engineering
Valid Formulas wrt ModalitiesProperties of RCorrespondence TheoryPreview: Some Modal Logics
A Range of Modalities
In a particular context �φ could mean:
It is necessarily true that φ
It will always be true that φ
It ought to be that φ
Agent Q believes that φ
Agent Q knows that φ
CS 5209: Foundation in Logic and AI 10—Modal Logic I 66
MotivationBasic Modal LogicLogic Engineering
Valid Formulas wrt ModalitiesProperties of RCorrespondence TheoryPreview: Some Modal Logics
A Range of Modalities
In a particular context �φ could mean:
It is necessarily true that φ
It will always be true that φ
It ought to be that φ
Agent Q believes that φ
Agent Q knows that φ
After any execution of program P, φ holds.
CS 5209: Foundation in Logic and AI 10—Modal Logic I 67
MotivationBasic Modal LogicLogic Engineering
Valid Formulas wrt ModalitiesProperties of RCorrespondence TheoryPreview: Some Modal Logics
A Range of Modalities
In a particular context �φ could mean:
It is necessarily true that φ
It will always be true that φ
It ought to be that φ
Agent Q believes that φ
Agent Q knows that φ
After any execution of program P, φ holds.
Since ♦φ ≡ ¬�¬φ, we can infer the meaning of ♦ in eachcontext.
CS 5209: Foundation in Logic and AI 10—Modal Logic I 68
MotivationBasic Modal LogicLogic Engineering
Valid Formulas wrt ModalitiesProperties of RCorrespondence TheoryPreview: Some Modal Logics
A Range of Modalities
From the meaning of �φ, we can conclude the meaning of ♦φ,since ♦φ ≡ ¬�¬φ:�φ ♦φ
It is necessarily true that φ
CS 5209: Foundation in Logic and AI 10—Modal Logic I 69
MotivationBasic Modal LogicLogic Engineering
Valid Formulas wrt ModalitiesProperties of RCorrespondence TheoryPreview: Some Modal Logics
A Range of Modalities
From the meaning of �φ, we can conclude the meaning of ♦φ,since ♦φ ≡ ¬�¬φ:�φ ♦φ
It is necessarily true that φ It is possibly true that φ
CS 5209: Foundation in Logic and AI 10—Modal Logic I 70
MotivationBasic Modal LogicLogic Engineering
Valid Formulas wrt ModalitiesProperties of RCorrespondence TheoryPreview: Some Modal Logics
A Range of Modalities
From the meaning of �φ, we can conclude the meaning of ♦φ,since ♦φ ≡ ¬�¬φ:�φ ♦φ
It is necessarily true that φ It is possibly true that φIt will always be true that φ
CS 5209: Foundation in Logic and AI 10—Modal Logic I 71
MotivationBasic Modal LogicLogic Engineering
Valid Formulas wrt ModalitiesProperties of RCorrespondence TheoryPreview: Some Modal Logics
A Range of Modalities
From the meaning of �φ, we can conclude the meaning of ♦φ,since ♦φ ≡ ¬�¬φ:�φ ♦φ
It is necessarily true that φ It is possibly true that φIt will always be true that φ Sometime in the future φ
CS 5209: Foundation in Logic and AI 10—Modal Logic I 72
MotivationBasic Modal LogicLogic Engineering
Valid Formulas wrt ModalitiesProperties of RCorrespondence TheoryPreview: Some Modal Logics
A Range of Modalities
From the meaning of �φ, we can conclude the meaning of ♦φ,since ♦φ ≡ ¬�¬φ:�φ ♦φ
It is necessarily true that φ It is possibly true that φIt will always be true that φ Sometime in the future φIt ought to be that φ
CS 5209: Foundation in Logic and AI 10—Modal Logic I 73
MotivationBasic Modal LogicLogic Engineering
Valid Formulas wrt ModalitiesProperties of RCorrespondence TheoryPreview: Some Modal Logics
A Range of Modalities
From the meaning of �φ, we can conclude the meaning of ♦φ,since ♦φ ≡ ¬�¬φ:�φ ♦φ
It is necessarily true that φ It is possibly true that φIt will always be true that φ Sometime in the future φIt ought to be that φ It is permitted to be that φ
CS 5209: Foundation in Logic and AI 10—Modal Logic I 74
MotivationBasic Modal LogicLogic Engineering
Valid Formulas wrt ModalitiesProperties of RCorrespondence TheoryPreview: Some Modal Logics
A Range of Modalities
From the meaning of �φ, we can conclude the meaning of ♦φ,since ♦φ ≡ ¬�¬φ:�φ ♦φ
It is necessarily true that φ It is possibly true that φIt will always be true that φ Sometime in the future φIt ought to be that φ It is permitted to be that φAgent Q believes that φ
CS 5209: Foundation in Logic and AI 10—Modal Logic I 75
MotivationBasic Modal LogicLogic Engineering
Valid Formulas wrt ModalitiesProperties of RCorrespondence TheoryPreview: Some Modal Logics
A Range of Modalities
From the meaning of �φ, we can conclude the meaning of ♦φ,since ♦φ ≡ ¬�¬φ:�φ ♦φ
It is necessarily true that φ It is possibly true that φIt will always be true that φ Sometime in the future φIt ought to be that φ It is permitted to be that φAgent Q believes that φ φ is consistent with Q’s beliefs
CS 5209: Foundation in Logic and AI 10—Modal Logic I 76
MotivationBasic Modal LogicLogic Engineering
Valid Formulas wrt ModalitiesProperties of RCorrespondence TheoryPreview: Some Modal Logics
A Range of Modalities
From the meaning of �φ, we can conclude the meaning of ♦φ,since ♦φ ≡ ¬�¬φ:�φ ♦φ
It is necessarily true that φ It is possibly true that φIt will always be true that φ Sometime in the future φIt ought to be that φ It is permitted to be that φAgent Q believes that φ φ is consistent with Q’s beliefsAgent Q knows that φ
CS 5209: Foundation in Logic and AI 10—Modal Logic I 77
MotivationBasic Modal LogicLogic Engineering
Valid Formulas wrt ModalitiesProperties of RCorrespondence TheoryPreview: Some Modal Logics
A Range of Modalities
From the meaning of �φ, we can conclude the meaning of ♦φ,since ♦φ ≡ ¬�¬φ:�φ ♦φ
It is necessarily true that φ It is possibly true that φIt will always be true that φ Sometime in the future φIt ought to be that φ It is permitted to be that φAgent Q believes that φ φ is consistent with Q’s beliefsAgent Q knows that φ For all Q knows, φ
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A Range of Modalities
From the meaning of �φ, we can conclude the meaning of ♦φ,since ♦φ ≡ ¬�¬φ:�φ ♦φ
It is necessarily true that φ It is possibly true that φIt will always be true that φ Sometime in the future φIt ought to be that φ It is permitted to be that φAgent Q believes that φ φ is consistent with Q’s beliefsAgent Q knows that φ For all Q knows, φAfter any run of P, φ holds.
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A Range of Modalities
From the meaning of �φ, we can conclude the meaning of ♦φ,since ♦φ ≡ ¬�¬φ:�φ ♦φ
It is necessarily true that φ It is possibly true that φIt will always be true that φ Sometime in the future φIt ought to be that φ It is permitted to be that φAgent Q believes that φ φ is consistent with Q’s beliefsAgent Q knows that φ For all Q knows, φAfter any run of P, φ holds. After some run of P, φ holds
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Formula Schemes that hold wrt some Modalities
�φ �φ→φ
�φ→
��φ
♦φ→
�♦φ
♦⊤ �φ→
♦φ
�φ∨�
¬φ
�(φ→ψ)
∧�φ→
�ψ
♦φ∧ ♦ψ
→♦(φ
∧ ψ)
It is necessary that φ√ √ √ √ √ × √ ×
It will always be that φ × √ × × × × √ ×It ought to be that φ × × × √ √ × √ ×Agent Q believes that φ × √ √ √ √ × √ ×Agent Q knows that φ
√ √ √ √ √ × √ ×After running P, φ × × × × × × √ ×
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Modalities lead to Interpretations of R�φ R(x , y)
It is necessarily true that φ y is possible world according to info at x
It will always be true that φ y is a future world of x
It ought to be that φ y is an acceptable world according to theinformation at x
Agent Q believes that φ y could be the actual world according toQ’s beliefs at x
Agent Q knows that φ y could be the actual world according toQ’s knowledge at x
After any execution of P, φholds
y is a possible resulting state after execu-tion of P at x
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Possible Properties of R
reflexive: for every w ∈ W , we have R(x , x).
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Possible Properties of R
reflexive: for every w ∈ W , we have R(x , x).symmetric: for every x , y ∈ W , we have R(x , y) impliesR(y , x).
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Possible Properties of R
reflexive: for every w ∈ W , we have R(x , x).symmetric: for every x , y ∈ W , we have R(x , y) impliesR(y , x).serial: for every x there is a y such that R(x , y).
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Possible Properties of R
reflexive: for every w ∈ W , we have R(x , x).symmetric: for every x , y ∈ W , we have R(x , y) impliesR(y , x).serial: for every x there is a y such that R(x , y).transitive: for every x , y , z ∈ W , we have R(x , y) andR(y , z) imply R(x , z).
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Possible Properties of R
reflexive: for every w ∈ W , we have R(x , x).symmetric: for every x , y ∈ W , we have R(x , y) impliesR(y , x).serial: for every x there is a y such that R(x , y).transitive: for every x , y , z ∈ W , we have R(x , y) andR(y , z) imply R(x , z).Euclidean: for every x , y , z ∈ W with R(x , y) and R(x , z),we have R(y , z).
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Possible Properties of R
reflexive: for every w ∈ W , we have R(x , x).symmetric: for every x , y ∈ W , we have R(x , y) impliesR(y , x).serial: for every x there is a y such that R(x , y).transitive: for every x , y , z ∈ W , we have R(x , y) andR(y , z) imply R(x , z).Euclidean: for every x , y , z ∈ W with R(x , y) and R(x , z),we have R(y , z).functional: for each x there is a unique y such that R(x , y).
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Possible Properties of R
reflexive: for every w ∈ W , we have R(x , x).symmetric: for every x , y ∈ W , we have R(x , y) impliesR(y , x).serial: for every x there is a y such that R(x , y).transitive: for every x , y , z ∈ W , we have R(x , y) andR(y , z) imply R(x , z).Euclidean: for every x , y , z ∈ W with R(x , y) and R(x , z),we have R(y , z).functional: for each x there is a unique y such that R(x , y).linear: for every x , y , z ∈ W with R(x , y) and R(x , z), wehave R(y , z) or y = z or R(z, y).
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Possible Properties of R
reflexive: for every w ∈ W , we have R(x , x).symmetric: for every x , y ∈ W , we have R(x , y) impliesR(y , x).serial: for every x there is a y such that R(x , y).transitive: for every x , y , z ∈ W , we have R(x , y) andR(y , z) imply R(x , z).Euclidean: for every x , y , z ∈ W with R(x , y) and R(x , z),we have R(y , z).functional: for each x there is a unique y such that R(x , y).linear: for every x , y , z ∈ W with R(x , y) and R(x , z), wehave R(y , z) or y = z or R(z, y).total: for every x , y ∈ W , we have R(x , y) and R(y , x).
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Possible Properties of R
reflexive: for every w ∈ W , we have R(x , x).symmetric: for every x , y ∈ W , we have R(x , y) impliesR(y , x).serial: for every x there is a y such that R(x , y).transitive: for every x , y , z ∈ W , we have R(x , y) andR(y , z) imply R(x , z).Euclidean: for every x , y , z ∈ W with R(x , y) and R(x , z),we have R(y , z).functional: for each x there is a unique y such that R(x , y).linear: for every x , y , z ∈ W with R(x , y) and R(x , z), wehave R(y , z) or y = z or R(z, y).total: for every x , y ∈ W , we have R(x , y) and R(y , x).equivalence: reflexive, symmetric and transitive.
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Example
Consider the modality in which �φ means“it ought to be that φ”.
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Example
Consider the modality in which �φ means“it ought to be that φ”.
Should R be reflexive?
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Example
Consider the modality in which �φ means“it ought to be that φ”.
Should R be reflexive?
Should R be serial?
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Necessarily true and Reflexivity
Guess
R is reflexive if and only if �φ→ φ is valid.
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Motivation
We would like to establish that some formulas holdwhenever R has a particular property.
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Motivation
We would like to establish that some formulas holdwhenever R has a particular property.
Ignore L, and only consider the (W ,R) part of a model,called frame.
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Motivation
We would like to establish that some formulas holdwhenever R has a particular property.
Ignore L, and only consider the (W ,R) part of a model,called frame.
Establish formula schemes based on properties of frames.
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Reflexivity and Transitivity
Theorem 1
Let F = (W ,R) be a frame. The following statements areequivalent:
R is reflexive;
F satisfies �φ→ φ;
F satisfies �p → p for any atom p
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Reflexivity and Transitivity
Theorem 1
Let F = (W ,R) be a frame. The following statements areequivalent:
R is reflexive;
F satisfies �φ→ φ;
F satisfies �p → p for any atom p
Theorem 2
The following statements are equivalent:
R is transitive;
F satisfies �φ→ ��φ;
F satisfies �p → ��p for any atom p
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Proof of Theorem 1
Let F = (W ,R) be a frame. The following statements areequivalent:
1 R is reflexive;2 F satisfies �φ→ φ;3 F satisfies �p → p for any atom p
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Proof of Theorem 1
Let F = (W ,R) be a frame. The following statements areequivalent:
1 R is reflexive;2 F satisfies �φ→ φ;3 F satisfies �p → p for any atom p
1 ⇒ 2: Let R be reflexive.
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Proof of Theorem 1
Let F = (W ,R) be a frame. The following statements areequivalent:
1 R is reflexive;2 F satisfies �φ→ φ;3 F satisfies �p → p for any atom p
1 ⇒ 2: Let R be reflexive. Let L be any labeling function;M = (W ,R,L).
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Proof of Theorem 1
Let F = (W ,R) be a frame. The following statements areequivalent:
1 R is reflexive;2 F satisfies �φ→ φ;3 F satisfies �p → p for any atom p
1 ⇒ 2: Let R be reflexive. Let L be any labeling function;M = (W ,R,L). Need to show for any x :x �φ→ φ
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Proof of Theorem 1
Let F = (W ,R) be a frame. The following statements areequivalent:
1 R is reflexive;2 F satisfies �φ→ φ;3 F satisfies �p → p for any atom p
1 ⇒ 2: Let R be reflexive. Let L be any labeling function;M = (W ,R,L). Need to show for any x :x �φ→ φ Suppose x �φ.
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Proof of Theorem 1
Let F = (W ,R) be a frame. The following statements areequivalent:
1 R is reflexive;2 F satisfies �φ→ φ;3 F satisfies �p → p for any atom p
1 ⇒ 2: Let R be reflexive. Let L be any labeling function;M = (W ,R,L). Need to show for any x :x �φ→ φ Suppose x �φ.Since R is reflexive, we have x φ.
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Proof of Theorem 1
Let F = (W ,R) be a frame. The following statements areequivalent:
1 R is reflexive;2 F satisfies �φ→ φ;3 F satisfies �p → p for any atom p
1 ⇒ 2: Let R be reflexive. Let L be any labeling function;M = (W ,R,L). Need to show for any x :x �φ→ φ Suppose x �φ.Since R is reflexive, we have x φ.Using the semantics of →: x �φ→ φ
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Proof of Theorem 1
Let F = (W ,R) be a frame. The following statements areequivalent:
1 R is reflexive;2 F satisfies �φ→ φ;3 F satisfies �p → p for any atom p
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Proof of Theorem 1
Let F = (W ,R) be a frame. The following statements areequivalent:
1 R is reflexive;2 F satisfies �φ→ φ;3 F satisfies �p → p for any atom p
2 ⇒ 3: Just set φ to be p
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Proof of Theorem 1
Let F = (W ,R) be a frame. The following statements areequivalent:
1 R is reflexive;2 F satisfies �φ→ φ;3 F satisfies �p → p for any atom p
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Proof of Theorem 1
Let F = (W ,R) be a frame. The following statements areequivalent:
1 R is reflexive;2 F satisfies �φ→ φ;3 F satisfies �p → p for any atom p
3 ⇒ 1: Suppose the frame satisfies �p → p.
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Proof of Theorem 1
Let F = (W ,R) be a frame. The following statements areequivalent:
1 R is reflexive;2 F satisfies �φ→ φ;3 F satisfies �p → p for any atom p
3 ⇒ 1: Suppose the frame satisfies �p → p.Take any world x from W .
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Proof of Theorem 1
Let F = (W ,R) be a frame. The following statements areequivalent:
1 R is reflexive;2 F satisfies �φ→ φ;3 F satisfies �p → p for any atom p
3 ⇒ 1: Suppose the frame satisfies �p → p.Take any world x from W .Choose a labeling function L such that p 6∈ L(x),but p ∈ L(y) for all y with y 6= x
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Proof of Theorem 1
Let F = (W ,R) be a frame. The following statements areequivalent:
1 R is reflexive;2 F satisfies �φ→ φ;3 F satisfies �p → p for any atom p
3 ⇒ 1: Suppose the frame satisfies �p → p.Take any world x from W .Choose a labeling function L such that p 6∈ L(x),but p ∈ L(y) for all y with y 6= xProof by contradiction: Assume (x , x) 6∈ R.
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Proof of Theorem 1
Let F = (W ,R) be a frame. The following statements areequivalent:
1 R is reflexive;2 F satisfies �φ→ φ;3 F satisfies �p → p for any atom p
3 ⇒ 1: Suppose the frame satisfies �p → p.Take any world x from W .Choose a labeling function L such that p 6∈ L(x),but p ∈ L(y) for all y with y 6= xProof by contradiction: Assume (x , x) 6∈ R. Thenwe would have x �p, but not x p.Contradiction!
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Formula Schemes and Properties of R
name formula scheme property of RT �φ→ φ reflexiveB φ→ �♦φ symmetricD �φ→ ♦φ serial4 �φ→ ��φ transitive5 ♦φ→ �♦φ Euclidean
�φ↔ ♦φ functional�(φ ∧ �φ→ ψ) ∨ �(ψ ∧ �ψ → φ) linear
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Which Formula Schemes to Choose?
Definition
Let L be a set of formula schemes and Γ ∪ {ψ} a set offormulas of basic modal logic.
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Which Formula Schemes to Choose?
Definition
Let L be a set of formula schemes and Γ ∪ {ψ} a set offormulas of basic modal logic.
A set of formula schemes is said to be closed iff it containsall substitution instances of its elements.
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Which Formula Schemes to Choose?
Definition
Let L be a set of formula schemes and Γ ∪ {ψ} a set offormulas of basic modal logic.
A set of formula schemes is said to be closed iff it containsall substitution instances of its elements.
Let Lc be the smallest closed superset of L.
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Which Formula Schemes to Choose?
Definition
Let L be a set of formula schemes and Γ ∪ {ψ} a set offormulas of basic modal logic.
A set of formula schemes is said to be closed iff it containsall substitution instances of its elements.
Let Lc be the smallest closed superset of L.
Γ entails ψ in L iff Γ ∪ Lc semantically entails ψ. We sayΓ |=L ψ.
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Examples of Modal Logics: K
K is the weakest modal logic, L = ∅.
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Examples of Modal Logics: KT45
L = {T ,4,5}
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Examples of Modal Logics: KT45
L = {T ,4,5}
Used for reasoning about knowledge.
T: Truth: agent Q only knows true things.
4: Positive introspection: If Q knows something, he knowsthat he knows it.
5: Negative introspection: If Q doesn’t know something, heknows that he doesn’t know it.
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Next Week
Examples of Modal Logic
Natural deduction in modal logic
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