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Neighborhood Semantics forModal Logic
Lecture 1
Eric Pacuit
ILLC, Universiteit van Amsterdamstaff.science.uva.nl/∼epacuit
August 13, 2007
Eric Pacuit: Neighborhood Semantics, Lecture 1 1
Plan for the Course
Lecture 1: Introduction, Motivation and BackgroundInformation
Lecture 2: Basic Concepts, Non-normal Modal Logics,Completeness, Decidability, Complexity,Incompleteness, Relation with Relational Semantics
Lecture 3: Advanced Topics — Topological Semantics forModal Logic, some Model Theory
Lecture 4: Advanced Topics — Topological Semantics forModal Logic, some Model Theory
Lecture 5: Neighborhood Semantics in Action: Game Logic,Coalgebra, Common Knowledge, First-Order ModalLogic
Eric Pacuit: Neighborhood Semantics, Lecture 1 2
Plan for the Course
Lecture 1: Introduction, Motivation and BackgroundInformation
Lecture 2: Basic Concepts, Non-normal Modal Logics,Completeness, Decidability, Complexity,Incompleteness, Relation with Relational Semantics
Lecture 3: Advanced Topics — Topological Semantics forModal Logic, some Model Theory
Lecture 4: Advanced Topics — Topological Semantics forModal Logic, some Model Theory
Lecture 5: Neighborhood Semantics in Action: Game Logic,Coalgebra, Common Knowledge, First-Order ModalLogic
Eric Pacuit: Neighborhood Semantics, Lecture 1 2
Plan for the Course
Lecture 1: Introduction, Motivation and BackgroundInformation
Lecture 2: Basic Concepts, Non-normal Modal Logics,Completeness, Decidability, Complexity,Incompleteness, Relation with Relational Semantics
Lecture 3: Advanced Topics — Topological Semantics forModal Logic, some Model Theory
Lecture 4: Advanced Topics — Topological Semantics forModal Logic, some Model Theory
Lecture 5: Neighborhood Semantics in Action: Game Logic,Coalgebra, Common Knowledge, First-Order ModalLogic
Eric Pacuit: Neighborhood Semantics, Lecture 1 2
Plan for the Course
Lecture 1: Introduction, Motivation and BackgroundInformation
Lecture 2: Basic Concepts, Non-normal Modal Logics,Completeness, Decidability, Complexity,Incompleteness, Relation with Relational Semantics
Lecture 3: Advanced Topics — Topological Semantics forModal Logic, some Model Theory
Lecture 4: Advanced Topics — Topological Semantics forModal Logic, some Model Theory
Lecture 5: Neighborhood Semantics in Action: Game Logic,Coalgebra, Common Knowledge, First-Order ModalLogic
Eric Pacuit: Neighborhood Semantics, Lecture 1 2
Plan for the Course
Lecture 1: Introduction, Motivation and BackgroundInformation
Lecture 2: Basic Concepts, Non-normal Modal Logics,Completeness, Decidability, Complexity,Incompleteness, Relation with Relational Semantics
Lecture 3: Advanced Topics — Topological Semantics forModal Logic, some Model Theory
Lecture 4: Advanced Topics — Topological Semantics forModal Logic, some Model Theory
Lecture 5: Neighborhood Semantics in Action: Game Logic,Coalgebra, Common Knowledge, First-Order ModalLogic
Eric Pacuit: Neighborhood Semantics, Lecture 1 2
Plan for the Course
Lecture 1: Introduction, Motivation and BackgroundInformation
Lecture 2: Basic Concepts, Non-normal Modal Logics,Completeness, Decidability, Complexity,Incompleteness, Relation with Relational Semantics
Lecture 3: Advanced Topics — Topological Semantics forModal Logic, some Model Theory
Lecture 4: Advanced Topics — Topological Semantics forModal Logic, some Model Theory
Lecture 5: Neighborhood Semantics in Action: Game Logic,Coalgebra, Common Knowledge, First-Order ModalLogic
Eric Pacuit: Neighborhood Semantics, Lecture 1 2
Plan for the Course
Course Websitestaff.science.uva.nl/∼epacuit/nbhd esslli.html
Reading Material
X Modal Logic: an Introduction, Chapters 7 - 9, by Brian Chellas
X Monotonic Modal Logics by Helle Hvid Hansen, available atwww.few.vu.nl/∼hhansen/papers/scriptie pic.pdf
X The course reader (updated version available on the website)
Concerning Modal LogicModal Logic by P. Blackburn, M. de Rijke and Y. Venema.
Eric Pacuit: Neighborhood Semantics, Lecture 1 3
Lecture 1
� Background
� Introduction
� Motivating Examples
� A Primer on Modal Logic
Eric Pacuit: Neighborhood Semantics, Lecture 1 4
Background
The Basic Modal Language: L
p | ¬ϕ | ϕ ∧ ψ | �ϕ | ♦ϕ
where p is an atomic proposition (At)
Eric Pacuit: Neighborhood Semantics, Lecture 1 5
Background
Kripke (Relational) Models
M = 〈W ,R,V 〉
I W 6= ∅I R ⊆ W ×W
I V : At → ℘(W )
Eric Pacuit: Neighborhood Semantics, Lecture 1 6
Background
Kripke (Relational) Models
M = 〈W ,R,V 〉
I W 6= ∅I R ⊆ W ×W
I V : At → ℘(W )
Eric Pacuit: Neighborhood Semantics, Lecture 1 6
Background
Truth in a Kripke Model
1. M,w |= p iff w ∈ V (p)
2. M,w |= ¬ϕ iff M,w 6|= ϕ
3. M,w |= ϕ ∧ ψ iff M,w |= ϕ and M,w |= ψ
4. M,w |= �ϕ iff for each v ∈ W , if wRv then M, v |= ϕ
5. M,w |= ♦ϕ iff there is a v ∈ W such that wRv and M, v |= ϕ
Eric Pacuit: Neighborhood Semantics, Lecture 1 7
Background
Some Validities
(M) �(ϕ ∧ ψ) → �ϕ ∧�ψ
(C) �ϕ ∧�ψ → �(ϕ ∧ ψ)
(N) �>
(K) �(ϕ→ ψ) → (�ϕ→ �ψ)
(Dual) �ϕ↔ ¬♦¬ϕ
(Nec) from ` ϕ infer ` �ϕ
(Re) from ` ϕ↔ ψ infer ` �ϕ↔ �ψ
(Mon)` ϕ→ ψ
` �ϕ→ �ψ
Eric Pacuit: Neighborhood Semantics, Lecture 1 8
Background
Some Validities
(M) �(ϕ ∧ ψ) → �ϕ ∧�ψ
(C) �ϕ ∧�ψ → �(ϕ ∧ ψ)
(N) �>
(K) �(ϕ→ ψ) → (�ϕ→ �ψ)
(Dual) �ϕ↔ ¬♦¬ϕ
(Nec) from ` ϕ infer ` �ϕ
(Re) from ` ϕ↔ ψ infer ` �ϕ↔ �ψ
(Mon)` ϕ→ ψ
` �ϕ→ �ψ
Eric Pacuit: Neighborhood Semantics, Lecture 1 8
Introduction
� Background
� Introduction
� Motivating Examples
� A Primer on Modal Logic
Eric Pacuit: Neighborhood Semantics, Lecture 1 9
Introduction
Neighborhoods in Topology
In a topology, a neighborhood of a point x is any set A containingx such that you can “wiggle” x without leaving A.
A neighborhood system of a point x is the collection ofneighborhoods of x .
J. Dugundji. Topology. 1966.
Eric Pacuit: Neighborhood Semantics, Lecture 1 10
Introduction
Neighborhoods in Modal Logic
Neighborhood Structure: 〈W ,N,V 〉
I W 6= ∅I N : W → ℘(℘(W ))
I V : At → ℘(W )
Eric Pacuit: Neighborhood Semantics, Lecture 1 11
Introduction
Some Notation
Given ϕ ∈ L and a model M, the
I proposition expressed by ϕ
I extension of ϕ
I truth set of ϕ
is
(ϕ)M = {w ∈ W | M,w |= ϕ}
Eric Pacuit: Neighborhood Semantics, Lecture 1 12
Introduction
Some Notation
Given ϕ ∈ L and a model M, the
I proposition expressed by ϕ
I extension of ϕ
I truth set of ϕ
is
(ϕ)M = {w ∈ W | M,w |= ϕ}
Eric Pacuit: Neighborhood Semantics, Lecture 1 12
Brief History
w |= �ϕ if the truth set of ϕ is a neighborhood of w
neighborhood in some topology.J. McKinsey and A. Tarski. The Algebra of Topology. 1944.
contains all the immediate neighbors in some graphS. Kripke. A Semantic Analysis of Modal Logic. 1963.
an element of some distinguished collection of setsD. Scott. Advice on Modal Logic. 1970.
R. Montague. Pragmatics. 1968.
Eric Pacuit: Neighborhood Semantics, Lecture 1 13
Brief History
w |= �ϕ if the truth set of ϕ is a neighborhood of w
What does it mean to be a neighborhood?
neighborhood in some topology.J. McKinsey and A. Tarski. The Algebra of Topology. 1944.
contains all the immediate neighbors in some graphS. Kripke. A Semantic Analysis of Modal Logic. 1963.
an element of some distinguished collection of setsD. Scott. Advice on Modal Logic. 1970.
R. Montague. Pragmatics. 1968.
Eric Pacuit: Neighborhood Semantics, Lecture 1 13
Brief History
w |= �ϕ if the truth set of ϕ is a neighborhood of w
neighborhood in some topology.J. McKinsey and A. Tarski. The Algebra of Topology. 1944.
contains all the immediate neighbors in some graphS. Kripke. A Semantic Analysis of Modal Logic. 1963.
an element of some distinguished collection of setsD. Scott. Advice on Modal Logic. 1970.
R. Montague. Pragmatics. 1968.
Eric Pacuit: Neighborhood Semantics, Lecture 1 13
Brief History
w |= �ϕ if the truth set of ϕ is a neighborhood of w
neighborhood in some topology.J. McKinsey and A. Tarski. The Algebra of Topology. 1944.
contains all the immediate neighbors in some graphS. Kripke. A Semantic Analysis of Modal Logic. 1963.
an element of some distinguished collection of setsD. Scott. Advice on Modal Logic. 1970.
R. Montague. Pragmatics. 1968.
Eric Pacuit: Neighborhood Semantics, Lecture 1 13
Brief History
w |= �ϕ if the truth set of ϕ is a neighborhood of w
neighborhood in some topology.J. McKinsey and A. Tarski. The Algebra of Topology. 1944.
contains all the immediate neighbors in some graphS. Kripke. A Semantic Analysis of Modal Logic. 1963.
an element of some distinguished collection of setsD. Scott. Advice on Modal Logic. 1970.
R. Montague. Pragmatics. 1968.
Eric Pacuit: Neighborhood Semantics, Lecture 1 13
Brief History
To see the necessity of the more general approach, wecould consider probability operators, conditionalnecessity, or, to invoke an especially perspicuous exampleof Dana Scott, the present progressive tense....Thus Nmight receive the awkward reading ‘it is being the casethat’, in the sense in which ‘it is being the case thatJones leaves’ is synonymous with ‘Jones is leaving’.
(Montague, pg. 73)
R. Montague. Pragmatics and Intentional Logic. 1970.
Eric Pacuit: Neighborhood Semantics, Lecture 1 14
Brief History
Segerberg’s Essay
K. Segerberg. An Essay on Classical Modal Logic. Uppsula Technical Report,1970.
This essay purports to deal with classical modal logic.The qualification “classical” has not yet been given anestablished meaning in connection with modallogic....Clearly one would like to reserve the label“classical” for a category of modal logics which—ifpossible—is large enough to contain all or most of thesystems which for historical or theoretical reasons havecome to be regarded as important, and which also possesa high degree of naturalness and homogeneity.
(pg. 1)
Eric Pacuit: Neighborhood Semantics, Lecture 1 15
Brief History
Segerberg’s Essay
K. Segerberg. An Essay on Classical Modal Logic. Uppsula Technical Report,1970.
This essay purports to deal with classical modal logic.The qualification “classical” has not yet been given anestablished meaning in connection with modallogic....Clearly one would like to reserve the label“classical” for a category of modal logics which—ifpossible—is large enough to contain all or most of thesystems which for historical or theoretical reasons havecome to be regarded as important, and which also possesa high degree of naturalness and homogeneity.
(pg. 1)
Eric Pacuit: Neighborhood Semantics, Lecture 1 15
Brief History
R. Goldblatt. Mathematical Modal Logic: A View of its Evolution. Handbookof the History of Logic, 2005.
Eric Pacuit: Neighborhood Semantics, Lecture 1 16
Motivating Examples
� Background
� Introduction
� Motivating Examples
� A Primer on Modal Logic
Eric Pacuit: Neighborhood Semantics, Lecture 1 17
Motivating Examples
Logics of High Probability
�ϕ means “ϕ is assigned ‘high’ probability”, where high meansabove some threshold r ∈ [0, 1].
Claim: Mon is a valid rule of inference.
Claim: C is not valid.
H. Kyburg and C.M. Teng. The Logic of Risky Knowledge. Proceedings ofWoLLIC (2002).
A. Herzig. Modal Probability, Belief, and Actions. Fundamenta Informaticae(2003).
Eric Pacuit: Neighborhood Semantics, Lecture 1 18
Motivating Examples
Logics of High Probability
�ϕ means “ϕ is assigned ‘high’ probability”, where high meansabove some threshold r ∈ [0, 1].
Claim: Mon is a valid rule of inference.
Claim: C is not valid.
H. Kyburg and C.M. Teng. The Logic of Risky Knowledge. Proceedings ofWoLLIC (2002).
A. Herzig. Modal Probability, Belief, and Actions. Fundamenta Informaticae(2003).
Eric Pacuit: Neighborhood Semantics, Lecture 1 18
Motivating Examples
Logics of High Probability
�ϕ means “ϕ is assigned ‘high’ probability”, where high meansabove some threshold r ∈ [0, 1].
Claim: Mon is a valid rule of inference.
Claim: C is not valid.
H. Kyburg and C.M. Teng. The Logic of Risky Knowledge. Proceedings ofWoLLIC (2002).
A. Herzig. Modal Probability, Belief, and Actions. Fundamenta Informaticae(2003).
Eric Pacuit: Neighborhood Semantics, Lecture 1 18
Motivating Examples
Logics of High Probability
�ϕ means “ϕ is assigned ‘high’ probability”, where high meansabove some threshold r ∈ [0, 1].
Claim: Mon is a valid rule of inference.
Claim: C is not valid.
H. Kyburg and C.M. Teng. The Logic of Risky Knowledge. Proceedings ofWoLLIC (2002).
A. Herzig. Modal Probability, Belief, and Actions. Fundamenta Informaticae(2003).
Eric Pacuit: Neighborhood Semantics, Lecture 1 18
Motivating Examples
Games
�iϕ means “player i has a strategy that guarantees ϕ is true.”
Eric Pacuit: Neighborhood Semantics, Lecture 1 19
Motivating Examples
Games
�iϕ means “player i has a strategy that guarantees ϕ is true.”
As1
Bs2 B s3
p p, q p, q q
p p, q
s1 |= �Ap
p, q q
s1 |= �Ap ∧�Aq
p, q p, q
s1 |= �Ap ∧�Aq ∧ ¬�A(p ∧ q)
Eric Pacuit: Neighborhood Semantics, Lecture 1 19
Motivating Examples
Games
�iϕ means “player i has a strategy that guarantees ϕ is true.”
As1
Bs2 B s3
p p, q p, q qp p, q
s1 |= �Ap
p, q q
s1 |= �Ap ∧�Aq
p, q p, q
s1 |= �Ap ∧�Aq ∧ ¬�A(p ∧ q)
Eric Pacuit: Neighborhood Semantics, Lecture 1 19
Motivating Examples
Games
�iϕ means “player i has a strategy that guarantees ϕ is true.”
As1
Bs2 B s3
p p, q p, q q
p p, q
s1 |= �Ap
p, q q
s1 |= �Ap ∧�Aq
p, q p, q
s1 |= �Ap ∧�Aq ∧ ¬�A(p ∧ q)
Eric Pacuit: Neighborhood Semantics, Lecture 1 19
Motivating Examples
Games
�iϕ means “player i has a strategy that guarantees ϕ is true.”
As1
Bs2 B s3
p p, q p, q q
p p, q
s1 |= �Ap
p, q q
s1 |= �Ap ∧�Aq
p, q p, q
s1 |= �Ap ∧�Aq ∧ ¬�A(p ∧ q)
Eric Pacuit: Neighborhood Semantics, Lecture 1 19
Motivating Examples
Games
�iϕ means “player i has a strategy that guarantees ϕ is true.”
R. Parikh. The Logic of Games and its Applications. Annals of DiscreteMathematics (1985).
M. Pauly and R. Parikh. Game Logic — An Overview. Studia Logica (2003).
J. van Benthem. Logic and Games. Course notes (2007).
Eric Pacuit: Neighborhood Semantics, Lecture 1 19
Motivating Examples
Logic of Deduction
Let L0 ⊆ L be the set of propositional formulas.
Let Σ ⊆ L0 be the universe
Interpretation: (·)∗ : At → ℘(Σ)
I (ϕ ∨ ψ)∗ = (ϕ)∗ ∪ (ψ)∗
I (¬ϕ)∗ = Σ− (ϕ)∗
I (�ϕ)∗ = {α ∈ Σ | (ϕ)∗ ` α} (the deductive closure of ϕ)
Fact: �(ϕ→ ψ) → �ϕ→ �ψ is not valid.
P. Naumov. On modal logic of deductive closure. APAL (2005).
Eric Pacuit: Neighborhood Semantics, Lecture 1 20
Motivating Examples
Logic of Deduction
Let L0 ⊆ L be the set of propositional formulas.
Let Σ ⊆ L0 be the universe
Interpretation: (·)∗ : At → ℘(Σ)
I (ϕ ∨ ψ)∗ = (ϕ)∗ ∪ (ψ)∗
I (¬ϕ)∗ = Σ− (ϕ)∗
I (�ϕ)∗ = {α ∈ Σ | (ϕ)∗ ` α} (the deductive closure of ϕ)
Fact: �(ϕ→ ψ) → �ϕ→ �ψ is not valid.
P. Naumov. On modal logic of deductive closure. APAL (2005).
Eric Pacuit: Neighborhood Semantics, Lecture 1 20
Motivating Examples
Logic of Deduction
Let L0 ⊆ L be the set of propositional formulas.
Let Σ ⊆ L0 be the universe
Interpretation: (·)∗ : At → ℘(Σ)
I (ϕ ∨ ψ)∗ = (ϕ)∗ ∪ (ψ)∗
I (¬ϕ)∗ = Σ− (ϕ)∗
I (�ϕ)∗ = {α ∈ Σ | (ϕ)∗ ` α}
(the deductive closure of ϕ)
Fact: �(ϕ→ ψ) → �ϕ→ �ψ is not valid.
P. Naumov. On modal logic of deductive closure. APAL (2005).
Eric Pacuit: Neighborhood Semantics, Lecture 1 20
Motivating Examples
Logic of Deduction
Let L0 ⊆ L be the set of propositional formulas.
Let Σ ⊆ L0 be the universe
Interpretation: (·)∗ : At → ℘(Σ)
I (ϕ ∨ ψ)∗ = (ϕ)∗ ∪ (ψ)∗
I (¬ϕ)∗ = Σ− (ϕ)∗
I (�ϕ)∗ = {α ∈ Σ | (ϕ)∗ ` α} (the deductive closure of ϕ)
Fact: �(ϕ→ ψ) → �ϕ→ �ψ is not valid.
P. Naumov. On modal logic of deductive closure. APAL (2005).
Eric Pacuit: Neighborhood Semantics, Lecture 1 20
Motivating Examples
Logic of Deduction
Let L0 ⊆ L be the set of propositional formulas.
Let Σ ⊆ L0 be the universe
Interpretation: (·)∗ : At → ℘(Σ)
I (ϕ ∨ ψ)∗ = (ϕ)∗ ∪ (ψ)∗
I (¬ϕ)∗ = Σ− (ϕ)∗
I (�ϕ)∗ = {α ∈ Σ | (ϕ)∗ ` α} (the deductive closure of ϕ)
Fact: �(ϕ→ ψ) → �ϕ→ �ψ is not valid.
P. Naumov. On modal logic of deductive closure. APAL (2005).
Eric Pacuit: Neighborhood Semantics, Lecture 1 20
Motivating Examples
Logic of Deduction
Let L0 ⊆ L be the set of propositional formulas.
Let Σ ⊆ L0 be the universe
Interpretation: (·)∗ : At → ℘(Σ)
I (ϕ ∨ ψ)∗ = (ϕ)∗ ∪ (ψ)∗
I (¬ϕ)∗ = Σ− (ϕ)∗
I (�ϕ)∗ = {α ∈ Σ | (ϕ)∗ ` α} (the deductive closure of ϕ)
Fact: �ϕ ∧�ψ → �(ϕ ∧ ψ) is not valid.
P. Naumov. On modal logic of deductive closure. APAL (2005).
Eric Pacuit: Neighborhood Semantics, Lecture 1 20
Motivating Examples
Logic of Deduction
Let L0 ⊆ L be the set of propositional formulas.
Let Σ ⊆ L0 be the universe
Interpretation: (·)∗ : At → ℘(Σ)
I (ϕ ∨ ψ)∗ = (ϕ)∗ ∪ (ψ)∗
I (¬ϕ)∗ = Σ− (ϕ)∗
I (�ϕ)∗ = {α ∈ Σ | (ϕ)∗ ` α} (the deductive closure of ϕ)
Validities: ϕ→ �ϕ,
P. Naumov. On modal logic of deductive closure. APAL (2005).
Eric Pacuit: Neighborhood Semantics, Lecture 1 20
Motivating Examples
Logic of Deduction
Let L0 ⊆ L be the set of propositional formulas.
Let Σ ⊆ L0 be the universe
Interpretation: (·)∗ : At → ℘(Σ)
I (ϕ ∨ ψ)∗ = (ϕ)∗ ∪ (ψ)∗
I (¬ϕ)∗ = Σ− (ϕ)∗
I (�ϕ)∗ = {α ∈ Σ | (ϕ)∗ ` α} (the deductive closure of ϕ)
Validities: ϕ→ �ϕ, (Mon),
P. Naumov. On modal logic of deductive closure. APAL (2005).
Eric Pacuit: Neighborhood Semantics, Lecture 1 20
Motivating Examples
Logic of Deduction
Let L0 ⊆ L be the set of propositional formulas.
Let Σ ⊆ L0 be the universe
Interpretation: (·)∗ : At → ℘(Σ)
I (ϕ ∨ ψ)∗ = (ϕ)∗ ∪ (ψ)∗
I (¬ϕ)∗ = Σ− (ϕ)∗
I (�ϕ)∗ = {α ∈ Σ | (ϕ)∗ ` α} (the deductive closure of ϕ)
Validities: ϕ→ �ϕ, (Mon), �(ϕ ∨�ϕ) → �ϕ
P. Naumov. On modal logic of deductive closure. APAL (2005).
Eric Pacuit: Neighborhood Semantics, Lecture 1 20
Motivating Examples
Deontic Logic
�ϕ mean “it is obliged that ϕ.”
1. Jones murders Smith
2. Jones ought not to murder Smith
3. If Jones murders Smith, then Jones ought to murder Smithgently
4. Jones ought to murder Smith gently
5. If Jones murders Smith gently, then Jones murders Smith.
6. If Jones ought to muder Smith gently, then Jones ought tomurder Smith
7. Jones ought to murder Smith
J. Forrester. Paradox of Gentle Murder. 1984.
L. Goble. Murder Most Gentle: The Paradox Deepens. 1991.
Eric Pacuit: Neighborhood Semantics, Lecture 1 21
Motivating Examples
Deontic Logic
�ϕ mean “it is obliged that ϕ.”
1. Jones murders Smith
2. Jones ought not to murder Smith
3. If Jones murders Smith, then Jones ought to murder Smithgently
4. Jones ought to murder Smith gently
5. If Jones murders Smith gently, then Jones murders Smith.
6. If Jones ought to muder Smith gently, then Jones ought tomurder Smith
7. Jones ought to murder Smith
J. Forrester. Paradox of Gentle Murder. 1984.
L. Goble. Murder Most Gentle: The Paradox Deepens. 1991.
Eric Pacuit: Neighborhood Semantics, Lecture 1 21
Motivating Examples
Deontic Logic
�ϕ mean “it is obliged that ϕ.”
X Jones murders Smith
2. Jones ought not to murder Smith
X If Jones murders Smith, then Jones ought to murder Smithgently
4. Jones ought to murder Smith gently
5. If Jones murders Smith gently, then Jones murders Smith.
6. If Jones ought to muder Smith gently, then Jones ought tomurder Smith
7. Jones ought to murder Smith
J. Forrester. Paradox of Gentle Murder. 1984.
L. Goble. Murder Most Gentle: The Paradox Deepens. 1991.
Eric Pacuit: Neighborhood Semantics, Lecture 1 21
Motivating Examples
Deontic Logic
�ϕ mean “it is obliged that ϕ.”
1. Jones murders Smith
2. Jones ought not to murder Smith
3. If Jones murders Smith, then Jones ought to murder Smithgently
4. Jones ought to murder Smith gently
⇒ If Jones murders Smith gently, then Jones murders Smith.
6. If Jones ought to muder Smith gently, then Jones ought tomurder Smith
7. Jones ought to murder Smith
J. Forrester. Paradox of Gentle Murder. 1984.
L. Goble. Murder Most Gentle: The Paradox Deepens. 1991.
Eric Pacuit: Neighborhood Semantics, Lecture 1 21
Motivating Examples
Deontic Logic
�ϕ mean “it is obliged that ϕ.”
1. Jones murders Smith
2. Jones ought not to murder Smith
3. If Jones murders Smith, then Jones ought to murder Smithgently
4. Jones ought to murder Smith gently
X If Jones murders Smith gently, then Jones murders Smith.
(Mon) If Jones ought to muder Smith gently, then Jones ought tomurder Smith
7. Jones ought to murder Smith
J. Forrester. Paradox of Gentle Murder. 1984.
L. Goble. Murder Most Gentle: The Paradox Deepens. 1991.
Eric Pacuit: Neighborhood Semantics, Lecture 1 21
Motivating Examples
Deontic Logic
�ϕ mean “it is obliged that ϕ.”
1. Jones murders Smith
2. Jones ought not to murder Smith
3. If Jones murders Smith, then Jones ought to murder Smithgently
X Jones ought to murder Smith gently
5. If Jones murders Smith gently, then Jones murders Smith.
X If Jones ought to muder Smith gently, then Jones ought tomurder Smith
7. Jones ought to murder Smith
J. Forrester. Paradox of Gentle Murder. 1984.
L. Goble. Murder Most Gentle: The Paradox Deepens. 1991.
Eric Pacuit: Neighborhood Semantics, Lecture 1 21
Motivating Examples
Deontic Logic
�ϕ mean “it is obliged that ϕ.”
1. Jones murders Smith
6 Jones ought not to murder Smith
3. If Jones murders Smith, then Jones ought to murder Smithgently
4. Jones ought to murder Smith gently
5. If Jones murders Smith gently, then Jones murders Smith.
6. If Jones ought to muder Smith gently, then Jones ought tomurder Smith
6 Jones ought to murder Smith
J. Forrester. Paradox of Gentle Murder. 1984.
L. Goble. Murder Most Gentle: The Paradox Deepens. 1991.
Eric Pacuit: Neighborhood Semantics, Lecture 1 21
Motivating Examples
Social Choice Theory
�α mean “the group accepts α.”
Eric Pacuit: Neighborhood Semantics, Lecture 1 22
Motivating Examples
Social Choice Theory
�α mean “the group accepts α.”
Note: the language is restricted so that ��α is not a wff.
Eric Pacuit: Neighborhood Semantics, Lecture 1 22
Motivating Examples
Social Choice Theory
�α mean “the group accepts α.”
Consensus: α is accepted provided everyone accepts α.
(E) �α↔ �β provided α↔ β is a tautology
(M) �(α ∧ β) → (�α ∧�β)
(C) (�α ∧�β) → (�α ∧�β)
(N) �>(D) ¬�⊥
Eric Pacuit: Neighborhood Semantics, Lecture 1 22
Motivating Examples
Social Choice Theory
�α mean “the group accepts α.”
Consensus: α is accepted provided everyone accepts α.
(E) �α↔ �β provided α↔ β is a tautology
(M) �(α ∧ β) → (�α ∧�β)
(C) (�α ∧�β) → (�α ∧�β)
(N) �>(D) ¬�⊥
Theorem The above axioms axiomatize consensus (providedn ≥ 2|At|)).
Eric Pacuit: Neighborhood Semantics, Lecture 1 22
Motivating Examples
Social Choice Theory
�α mean “the group accepts α.”
Majority: α is accepted if a majority of the agents accept α.
Eric Pacuit: Neighborhood Semantics, Lecture 1 22
Motivating Examples
Social Choice Theory
�α mean “the group accepts α.”
Majority: α is accepted if a majority of the agents accept α.
(E) �α↔ �β provided α↔ β is a tautology
(M) �(α ∧ β) → (�α ∧�β)
(S) �α→ ¬�¬α(T) ([≥]ϕ1 ∧ · · · ∧ [≥]ϕk ∧ [≤]ψ1 ∧ · · · ∧ [≤]ψk) →∧
1≤i≤k([=]ϕi ∧ [=]ψi ) where ∀v ∈ VI :|{i | v(ϕi ) = 1}| = |{i | v(ψi ) = 1}|
Theorem The above axioms axiomatize majority.
Eric Pacuit: Neighborhood Semantics, Lecture 1 22
Motivating Examples
Social Choice Theory
�α mean “the group accepts α.”
M. Pauly. Axiomatizing Collective Judgement Sets in a Minimal Logical Lan-guage. 2006.
T. Daniels. Social Choice and Logic via Simple Games. ILLC, Masters Thesis,2007.
Eric Pacuit: Neighborhood Semantics, Lecture 1 22
Motivating Examples
Other Examples
I Epistemic Logic: the logical omniscience problem.M. Vardi. On Epistemic Logic and Logical Omniscience. TARK (1986).
I Reasoning about coalitionsM. Pauly. Logic for Social Software. Ph.D. Thesis, ILLC (2001).
I Knowledge RepresentationV. Padmanabhan, G. Governatori, K. Su . Knowledge Assesment: A ModalLogic Approach. KRAQ (2007).
I Program logics: modeling concurrent programsD. Peleg. Concurrent Dynamic Logic. J. ACM (1987).
I ??????
Eric Pacuit: Neighborhood Semantics, Lecture 1 23
A Primer on Modal Logic
� Background
� Introduction
� Motivating Examples
� A Primer on Modal Logic
Eric Pacuit: Neighborhood Semantics, Lecture 1 24
A Primer on Modal Logic
Slogan 1: Modal languages are simple yet expressive languagesfor talking about relational structures.
Slogan 2: Modal languages provide an internal, local perspectiveon relational structures.
P. Blackburn, M. de Rijke and Y. Venema. Modal Logic. 2001.
Eric Pacuit: Neighborhood Semantics, Lecture 1 25
A Primer on Modal Logic
P. Blackburn and J. van Benthem. Modal Logic: A Semantics Perspective.Handbook of Modal Logic (2007).
Eric Pacuit: Neighborhood Semantics, Lecture 1 26
A Primer on Modal Logic
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A Primer on Modal Logic
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A Primer on Modal Logic
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A Primer on Modal Logic
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A Primer on Modal Logic
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A Primer on Modal Logic
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A Primer on Modal Logic
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Eric Pacuit: Neighborhood Semantics, Lecture 1 27
A Primer on Modal Logic
Notation
A Kripke frame is a pair 〈W ,R〉 where R ⊆ W ×W .
Let F = 〈W ,R〉 be a Kripke frame and M = 〈W ,R,V 〉 a modelbased on M.
ϕ is satisfiable in M if there exists w ∈ W such that M,w |= ϕ
ϕ is valid in M (M |= ϕ) if ∀w ∈ W , M,w |= ϕ
ϕ is valid on a frame F (F |= ϕ) if for all models M based on F,M |= ϕ.
Eric Pacuit: Neighborhood Semantics, Lecture 1 28
A Primer on Modal Logic
Notation
A Kripke frame is a pair 〈W ,R〉 where R ⊆ W ×W .
Let F = 〈W ,R〉 be a Kripke frame and M = 〈W ,R,V 〉 a modelbased on M.
ϕ is satisfiable in M if there exists w ∈ W such that M,w |= ϕ
ϕ is valid in M (M |= ϕ) if ∀w ∈ W , M,w |= ϕ
ϕ is valid on a frame F (F |= ϕ) if for all models M based on F,M |= ϕ.
Eric Pacuit: Neighborhood Semantics, Lecture 1 28
A Primer on Modal Logic
Notation
A Kripke frame is a pair 〈W ,R〉 where R ⊆ W ×W .
Let F = 〈W ,R〉 be a Kripke frame and M = 〈W ,R,V 〉 a modelbased on M.
ϕ is satisfiable in M if there exists w ∈ W such that M,w |= ϕ
ϕ is valid in M (M |= ϕ) if ∀w ∈ W , M,w |= ϕ
ϕ is valid on a frame F (F |= ϕ) if for all models M based on F,M |= ϕ.
Eric Pacuit: Neighborhood Semantics, Lecture 1 28
A Primer on Modal Logic
Notation
A Kripke frame is a pair 〈W ,R〉 where R ⊆ W ×W .
Let F = 〈W ,R〉 be a Kripke frame and M = 〈W ,R,V 〉 a modelbased on M.
ϕ is satisfiable in M if there exists w ∈ W such that M,w |= ϕ
ϕ is valid in M (M |= ϕ) if ∀w ∈ W , M,w |= ϕ
ϕ is valid on a frame F (F |= ϕ) if for all models M based on F,M |= ϕ.
Eric Pacuit: Neighborhood Semantics, Lecture 1 28
A Primer on Modal Logic
Notation
A Kripke frame is a pair 〈W ,R〉 where R ⊆ W ×W .
Let F = 〈W ,R〉 be a Kripke frame and M = 〈W ,R,V 〉 a modelbased on M.
ϕ is satisfiable in M if there exists w ∈ W such that M,w |= ϕ
ϕ is valid in M (M |= ϕ) if ∀w ∈ W , M,w |= ϕ
ϕ is valid on a frame F (F |= ϕ) if for all models M based on F,M |= ϕ.
Eric Pacuit: Neighborhood Semantics, Lecture 1 28
A Primer on Modal Logic
Definable Properties
A modal formula ϕ defines a class of frames K provided
F ∈ K iff F |= ϕ
I �ϕ→ ��ϕ defines the class of transitive frames.
I ϕ↔ �ϕ defines the class of frames consisting of isolatedreflexive points (∀x ∈ W , xRy → x = y).
I �(�ϕ→ ϕ) defines the class of secondary-reflexive frames(∀w , v ∈ W , if wRv then vRv).
Some modal formulas correspond to genuine second-orderproperties: Lob (�(�ϕ→ ϕ) → �ϕ), McKinsey (�♦ϕ→ ♦�ϕ)
Eric Pacuit: Neighborhood Semantics, Lecture 1 29
A Primer on Modal Logic
Definable Properties
A modal formula ϕ defines a class of frames K provided
F ∈ K iff F |= ϕ
X �ϕ→ ��ϕ defines the class of transitive frames.
I ϕ↔ �ϕ defines the class of frames consisting of isolatedreflexive points (∀x ∈ W , xRy → x = y).
I �(�ϕ→ ϕ) defines the class of secondary-reflexive frames(∀w , v ∈ W , if wRv then vRv).
Some modal formulas correspond to genuine second-orderproperties: Lob (�(�ϕ→ ϕ) → �ϕ), McKinsey (�♦ϕ→ ♦�ϕ)
Eric Pacuit: Neighborhood Semantics, Lecture 1 29
A Primer on Modal Logic
Definable Properties
A modal formula ϕ defines a class of frames K provided
F ∈ K iff F |= ϕ
X �ϕ→ ��ϕ defines the class of transitive frames.
X ϕ↔ �ϕ defines the class of frames consisting of isolatedreflexive points (∀x ∈ W , xRy → x = y).
I �(�ϕ→ ϕ) defines the class of secondary-reflexive frames(∀w , v ∈ W , if wRv then vRv).
Some modal formulas correspond to genuine second-orderproperties: Lob (�(�ϕ→ ϕ) → �ϕ), McKinsey (�♦ϕ→ ♦�ϕ)
Eric Pacuit: Neighborhood Semantics, Lecture 1 29
A Primer on Modal Logic
Definable Properties
A modal formula ϕ defines a class of frames K provided
F ∈ K iff F |= ϕ
X �ϕ→ ��ϕ defines the class of transitive frames.
X ϕ↔ �ϕ defines the class of frames consisting of isolatedreflexive points (∀x ∈ W , xRy → x = y).
X �(�ϕ→ ϕ) defines the class of secondary-reflexive frames(∀w , v ∈ W , if wRv then vRv).
Some modal formulas correspond to genuine second-orderproperties: Lob (�(�ϕ→ ϕ) → �ϕ), McKinsey (�♦ϕ→ ♦�ϕ)
Eric Pacuit: Neighborhood Semantics, Lecture 1 29
A Primer on Modal Logic
Definable Properties
A modal formula ϕ defines a class of frames K provided
F ∈ K iff F |= ϕ
X �ϕ→ ��ϕ defines the class of transitive frames.
X ϕ↔ �ϕ defines the class of frames consisting of isolatedreflexive points (∀x ∈ W , xRy → x = y).
X �(�ϕ→ ϕ) defines the class of secondary-reflexive frames(∀w , v ∈ W , if wRv then vRv).
Some modal formulas correspond to genuine second-orderproperties: Lob (�(�ϕ→ ϕ) → �ϕ), McKinsey (�♦ϕ→ ♦�ϕ)
Eric Pacuit: Neighborhood Semantics, Lecture 1 29
A Primer on Modal Logic
Definable Properties
A modal formula ϕ defines a class of frames K provided
F ∈ K iff F |= ϕ
X �ϕ→ ��ϕ defines the class of transitive frames.
X ϕ↔ �ϕ defines the class of frames consisting of isolatedreflexive points (∀x ∈ W , xRy → x = y).
X �(�ϕ→ ϕ) defines the class of secondary-reflexive frames(∀w , v ∈ W , if wRv then vRv).
The Sahlqvist Theorem gives an algorithm for finding a first-ordercorrespondant for certain modal formulas.
Eric Pacuit: Neighborhood Semantics, Lecture 1 29
A Primer on Modal Logic
Slogan 3: Modal logics are not isolated formal systems.
Eric Pacuit: Neighborhood Semantics, Lecture 1 30
A Primer on Modal Logic
The Standard Translation
stx : L → L1
stx : L → L1
First-order language with anappropriate signature
stx(p) = Px
stx(¬ϕ) = ¬stx(ϕ)stx(ϕ ∧ ψ) = stx(ϕ) ∧ stx(ψ)stx(�ϕ) = ∀y(xRy → sty (ϕ))stx(♦ϕ) = ∃y(xRy ∧ sty (ϕ))
Lemma For each w ∈ W , M,w |= ϕ iff M |= stx(ϕ)[x/w ].
Eric Pacuit: Neighborhood Semantics, Lecture 1 31
A Primer on Modal Logic
The Standard Translation
stx : L → L1stx : L → L1
First-order language with anappropriate signature
stx(p) = Pxstx(¬ϕ) = ¬stx(ϕ)
stx(ϕ ∧ ψ) = stx(ϕ) ∧ stx(ψ)stx(�ϕ) = ∀y(xRy → sty (ϕ))stx(♦ϕ) = ∃y(xRy ∧ sty (ϕ))
Lemma For each w ∈ W , M,w |= ϕ iff M |= stx(ϕ)[x/w ].
Eric Pacuit: Neighborhood Semantics, Lecture 1 31
A Primer on Modal Logic
The Standard Translation
stx : L → L1
stx : L → L1
First-order language with anappropriate signature
stx(p) = Pxstx(¬ϕ) = ¬stx(ϕ)stx(ϕ ∧ ψ) = stx(ϕ) ∧ stx(ψ)
stx(�ϕ) = ∀y(xRy → sty (ϕ))stx(♦ϕ) = ∃y(xRy ∧ sty (ϕ))
Lemma For each w ∈ W , M,w |= ϕ iff M |= stx(ϕ)[x/w ].
Eric Pacuit: Neighborhood Semantics, Lecture 1 31
A Primer on Modal Logic
The Standard Translation
stx : L → L1
stx : L → L1
First-order language with anappropriate signature
stx(p) = Pxstx(¬ϕ) = ¬stx(ϕ)stx(ϕ ∧ ψ) = stx(ϕ) ∧ stx(ψ)stx(�ϕ) = ∀y(xRy → sty (ϕ))
stx(♦ϕ) = ∃y(xRy ∧ sty (ϕ))
Lemma For each w ∈ W , M,w |= ϕ iff M |= stx(ϕ)[x/w ].
Eric Pacuit: Neighborhood Semantics, Lecture 1 31
A Primer on Modal Logic
The Standard Translation
stx : L → L1
stx : L → L1
First-order language with anappropriate signature
stx(p) = Pxstx(¬ϕ) = ¬stx(ϕ)stx(ϕ ∧ ψ) = stx(ϕ) ∧ stx(ψ)stx(�ϕ) = ∀y(xRy → sty (ϕ))stx(♦ϕ) = ∃y(xRy ∧ sty (ϕ))
Lemma For each w ∈ W , M,w |= ϕ iff M |= stx(ϕ)[x/w ].
Eric Pacuit: Neighborhood Semantics, Lecture 1 31
A Primer on Modal Logic
The Standard Translation
stx : L → L1
stx : L → L1
First-order language with anappropriate signature
stx(p) = Pxstx(¬ϕ) = ¬stx(ϕ)stx(ϕ ∧ ψ) = stx(ϕ) ∧ stx(ψ)stx(�ϕ) = ∀y(xRy → sty (ϕ))stx(♦ϕ) = ∃y(xRy ∧ sty (ϕ))
Fact: Modal logic falls in the two-variable fragment of L1.
Eric Pacuit: Neighborhood Semantics, Lecture 1 31
A Primer on Modal Logic
The Standard Translation
stx : L → L1
stx : L → L1
First-order language with anappropriate signature
stx(p) = Pxstx(¬ϕ) = ¬stx(ϕ)stx(ϕ ∧ ψ) = stx(ϕ) ∧ stx(ψ)stx(�ϕ) = ∀y(xRy → sty (ϕ))stx(♦ϕ) = ∃y(xRy ∧ sty (ϕ))
Lemma For each w ∈ W , M,w |= ϕ iff M stx(ϕ)[x/w ].
Eric Pacuit: Neighborhood Semantics, Lecture 1 31
A Primer on Modal Logic
What can we say with modal logic? What about in comparisonwith first-order logic?
Eric Pacuit: Neighborhood Semantics, Lecture 1 32
A Primer on Modal Logic
Disjoint Union
Definition Let M1 = 〈W1,R1,V1〉 and M2 = 〈W2,R2,V2〉. Thedisjoint union is the structure M1 ]M2 = 〈W ,R,V 〉 where
I W = W1 ∪W2
I R = R1 ∪ R2
I for all p ∈ At, V (p) = V1(p) ∪ V2(p)
Fact The universal modality is not definable in the basic modallanguage.
Eric Pacuit: Neighborhood Semantics, Lecture 1 33
A Primer on Modal Logic
Disjoint Union
Definition Let M1 = 〈W1,R1,V1〉 and M2 = 〈W2,R2,V2〉. Thedisjoint union is the structure M1 ]M2 = 〈W ,R,V 〉 where
I W = W1 ∪W2
I R = R1 ∪ R2
I for all p ∈ At, V (p) = V1(p) ∪ V2(p)
Lemma For each collection of Kripke structures {Mi | i ∈ I}, foreach w ∈ Wi , Mi ,w |= ϕ iff
⊎i∈I Mi ,w |= ϕ
Eric Pacuit: Neighborhood Semantics, Lecture 1 33
A Primer on Modal Logic
Disjoint Union
Definition Let M1 = 〈W1,R1,V1〉 and M2 = 〈W2,R2,V2〉. Thedisjoint union is the structure M1 ]M2 = 〈W ,R,V 〉 where
I W = W1 ∪W2
I R = R1 ∪ R2
I for all p ∈ At, V (p) = V1(p) ∪ V2(p)
Fact The universal modality is not definable in the basic modallanguage.
Eric Pacuit: Neighborhood Semantics, Lecture 1 33
A Primer on Modal Logic
Generated Submodel
Definition M′ = 〈W ′,R ′,V ′〉 is a generated submodel ofM = 〈W ,R,V 〉 provided
I W ′ ⊆ W is R-closed:for each w ′ ∈ W and v ∈ W , if wRv then v ∈ W ′.
I R ′ = R ∩W ′ ×W ′
I for all p ∈ At, V ′(p) = V (p) ∩W ′
Fact The universal modality is not definable in the basic modallanguage.
Eric Pacuit: Neighborhood Semantics, Lecture 1 34
A Primer on Modal Logic
Generated Submodel
Definition M′ = 〈W ′,R ′,V ′〉 is a generated submodel ofM = 〈W ,R,V 〉 provided
I W ′ ⊆ W is R-closed:for each w ′ ∈ W and v ∈ W , if wRv then v ∈ W ′.
I R ′ = R ∩W ′ ×W ′
I for all p ∈ At, V ′(p) = V (p) ∩W ′
Lemma If M′ is a generated submodel of M then for eachw ∈ W ′, M′,w |= ϕ iff M,w |= ϕ
Eric Pacuit: Neighborhood Semantics, Lecture 1 34
A Primer on Modal Logic
Generated Submodel
Definition M′ = 〈W ′,R ′,V ′〉 is a generated submodel ofM = 〈W ,R,V 〉 provided
I W ′ ⊆ W is R-closed:for each w ′ ∈ W and v ∈ W , if wRv then v ∈ W ′.
I R ′ = R ∩W ′ ×W ′
I for all p ∈ At, V ′(p) = V (p) ∩W ′
Fact The backwards looking modality is not definable in the basicmodal language.
Eric Pacuit: Neighborhood Semantics, Lecture 1 34
A Primer on Modal Logic
Bounded Morphism
Definition A bounded morphism between models M = 〈W ,R,V 〉and M′ = 〈W ′,R ′,V ′〉 is a function f with domain W and rangeW ′ such that:
Atomic harmony: for each p ∈ At, w ∈ V (p) iff f (w) ∈ V ′(p)
Morphism: if wRv then f (w)Rf (v)
Zag: if f (w)R ′v ′ then ∃v ∈ W such that f (v) = v ′ and wRv
Fact The universal modality is not definable in the basic modallanguage.
Eric Pacuit: Neighborhood Semantics, Lecture 1 35
A Primer on Modal Logic
Bounded Morphism
Definition A bounded morphism between models M = 〈W ,R,V 〉and M′ = 〈W ′,R ′,V ′〉 is a function f with domain W and rangeW ′ such that:
Atomic harmony: for each p ∈ At, w ∈ V (p) iff f (w) ∈ V ′(p)
Morphism: if wRv then f (w)Rf (v)
Zag: if f (w)R ′v ′ then ∃v ∈ W such that f (v) = v ′ and wRv
Lemma If M′ is a bounded morphic image of M then for eachw ∈ W , M,w |= ϕ iff M′, f (w) |= ϕ
Eric Pacuit: Neighborhood Semantics, Lecture 1 35
A Primer on Modal Logic
Bounded Morphism
Definition A bounded morphism between models M = 〈W ,R,V 〉and M′ = 〈W ′,R ′,V ′〉 is a function f with domain W and rangeW ′ such that:
Atomic harmony: for each p ∈ At, w ∈ V (p) iff f (w) ∈ V ′(p)
Morphism: if wRv then f (w)Rf (v)
Zag: if f (w)R ′v ′ then ∃v ∈ W such that f (v) = v ′ and wRv
Fact Counting modalities are not definable in the basic modallanguage (eg., ♦1ϕ iff ϕ is true in more than 1 accessible world).
Eric Pacuit: Neighborhood Semantics, Lecture 1 35
A Primer on Modal Logic
Bisimulation
A bisimulation between M = 〈W ,R,V 〉and M′ = 〈W ′,R ′,V ′〉 is anon-empty binary relation Z ⊆ W ×W ′ such that whenever wZw ′:
Atomic harmony: for each p ∈ At, w ∈ V (p) iff w ′ ∈ V ′(p)
Zig: if wRv , then ∃v ′ ∈ W ′ such that vZv ′ and w ′R ′v ′
Zag: if w ′R ′v ′ then ∃v ∈ W such that vZv ′ and wRv
Fact The universal modality is not definable in the
Eric Pacuit: Neighborhood Semantics, Lecture 1 36
A Primer on Modal Logic
Bisimulation
A bisimulation between M = 〈W ,R,V 〉and M′ = 〈W ′,R ′,V ′〉 is anon-empty binary relation Z ⊆ W ×W ′ such that whenever wZw ′:
Atomic harmony: for each p ∈ At, w ∈ V (p) iff w ′ ∈ V ′(p)
Zig: if wRv , then ∃v ′ ∈ W ′ such that vZv ′ and w ′R ′v ′
Zag: if w ′R ′v ′ then ∃v ∈ W such that vZv ′ and wRv
We write M,w ↔ M′,w ′ if there is a Z such that wZw ′.
Eric Pacuit: Neighborhood Semantics, Lecture 1 36
A Primer on Modal Logic
Bisimulation
A bisimulation between M = 〈W ,R,V 〉and M′ = 〈W ′,R ′,V ′〉 is anon-empty binary relation Z ⊆ W ×W ′ such that whenever wZw ′:
Atomic harmony: for each p ∈ At, w ∈ V (p) iff w ′ ∈ V ′(p)
Zig: if wRv , then ∃v ′ ∈ W ′ such that vZv ′ and w ′R ′v ′
Zag: if w ′R ′v ′ then ∃v ∈ W such that vZv ′ and wRv
We write M,w ! M′,w ′ iff ∀ϕ ∈ L, M,w |= ϕ iff M′,w ′ |= ϕ.
Eric Pacuit: Neighborhood Semantics, Lecture 1 36
A Primer on Modal Logic
Bisimulation
A bisimulation between M = 〈W ,R,V 〉and M′ = 〈W ′,R ′,V ′〉 is anon-empty binary relation Z ⊆ W ×W ′ such that whenever wZw ′:
Atomic harmony: for each p ∈ At, w ∈ V (p) iff w ′ ∈ V ′(p)
Zig: if wRv , then ∃v ′ ∈ W ′ such that vZv ′ and w ′R ′v ′
Zag: if w ′R ′v ′ then ∃v ∈ W such that vZv ′ and wRv
Lemma If M,w ↔ M′,w ′ then M,w ! M′,w ′.
Eric Pacuit: Neighborhood Semantics, Lecture 1 36
A Primer on Modal Logic
Bisimulation
A bisimulation between M = 〈W ,R,V 〉and M′ = 〈W ′,R ′,V ′〉 is anon-empty binary relation Z ⊆ W ×W ′ such that whenever wZw ′:
Atomic harmony: for each p ∈ At, w ∈ V (p) iff w ′ ∈ V ′(p)
Zig: if wRv , then ∃v ′ ∈ W ′ such that vZv ′ and w ′R ′v ′
Zag: if w ′R ′v ′ then ∃v ∈ W such that vZv ′ and wRv
Lemma On finite frames, if M,w ! M′,w ′ then M,w ↔ M′,w ′.
Eric Pacuit: Neighborhood Semantics, Lecture 1 36
A Primer on Modal Logic
The Van Benthem Characterization Theorem
Modal logic is the bisimulation invariant fragment of first-orderlogic.
Eric Pacuit: Neighborhood Semantics, Lecture 1 37
A Primer on Modal Logic
The Van Benthem Characterization Theorem
For any first-order formula ϕ(x), TFAE:
1. ϕ(x) is invariant for bisimulation
2. ϕ(x) is equivalent to the standard translation of a basic modalformula.
Eric Pacuit: Neighborhood Semantics, Lecture 1 37
A Primer on Modal Logic
The Goldblatt-Thomason Theorem
An elementary class of frames K is modally definiable iff
it is closedunder disjoint unions, bounded morphic images, generatedsubframes, and reflects ultrafilter extensions.
Eric Pacuit: Neighborhood Semantics, Lecture 1 38
A Primer on Modal Logic
The Goldblatt-Thomason Theorem
An elementary class of frames K is modally definiable iff it is closedunder disjoint unions,
bounded morphic images, generatedsubframes, and reflects ultrafilter extensions.
Eric Pacuit: Neighborhood Semantics, Lecture 1 38
A Primer on Modal Logic
The Goldblatt-Thomason Theorem
An elementary class of frames K is modally definiable iff it is closedunder disjoint unions, bounded morphic images,
generatedsubframes, and reflects ultrafilter extensions.
Eric Pacuit: Neighborhood Semantics, Lecture 1 38
A Primer on Modal Logic
The Goldblatt-Thomason Theorem
An elementary class of frames K is modally definiable iff it is closedunder disjoint unions, bounded morphic images, generatedsubframes,
and reflects ultrafilter extensions.
Eric Pacuit: Neighborhood Semantics, Lecture 1 38
A Primer on Modal Logic
The Goldblatt-Thomason Theorem
An elementary class of frames K is modally definiable iff it is closedunder disjoint unions, bounded morphic images, generatedsubframes, and reflects ultrafilter extensions.
Eric Pacuit: Neighborhood Semantics, Lecture 1 38
A Primer on Modal Logic
End of lecture 1.
Eric Pacuit: Neighborhood Semantics, Lecture 1 39