Recent Work inEpistemic Logic

Logic Group Colloquium

Wes HollidayPhilosophy

UC Berkeley

March 1, 2013

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 1

Outline

I Preliminaries

I Basic Epistemic Logic• Modeling Knowledge• Common Knowledge• Information Update

I Mathematical Themes• Preservation Theorems• Uniform Substitution• Relations to FOL

I Conclusion

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 2

Preliminaries

What is Epistemic Logic?

Epistemic logic provides a formal framework for modeling theknowledge of agents. Used by philosophers, theoretical computerscientists, AI researchers, game theorists, and others, epistemiclogic has become one of the main application areas for modal logic.

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 3

Preliminaries

A Very Brief History

12-15th c. Informal EL investigated by logicians of the Middle Ages.

1960s EL takes off in philosophy after Jaakko Hintikka’s Knowledge andBelief (1962) (cf. von Wright’s An Essay on Modal Logic, 1951).

1970s Extensive philosophical discussion sparked by Hintikka is reviewed inWolfgang Lenzen’s Recent Work in Epistemic Logic (1978).

1980-90s EL is “rediscovered” in theoretical computer science, leading to theclassic textbook Reasoning about Knowledge (1995) by RonaldFagin, Joseph Halpern, Yoram Moses, and Moshe Vardi.

1990s EL is discovered again in game theory by Robert Aumann (recipientof the 2005 Nobel Memorial Prize in Economics) leading to his“Interactive Epistemology I: Knowledge” (1999).

90s-now Dynamic epistemic logic is developed in CS (Jan Plaza) and logic,especially by the “Amsterdam School” around Johan van Benthem.

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 4

Preliminaries

A Very Brief History

12-15th c. Informal EL investigated by logicians of the Middle Ages.

1960s EL takes off in philosophy after Jaakko Hintikka’s Knowledge andBelief (1962) (cf. von Wright’s An Essay on Modal Logic, 1951).

1970s Extensive philosophical discussion sparked by Hintikka is reviewed inWolfgang Lenzen’s Recent Work in Epistemic Logic (1978).

1980-90s EL is “rediscovered” in theoretical computer science, leading to theclassic textbook Reasoning about Knowledge (1995) by RonaldFagin, Joseph Halpern, Yoram Moses, and Moshe Vardi.

1990s EL is discovered again in game theory by Robert Aumann (recipientof the 2005 Nobel Memorial Prize in Economics) leading to his“Interactive Epistemology I: Knowledge” (1999).

90s-now Dynamic epistemic logic is developed in CS (Jan Plaza) and logic,especially by the “Amsterdam School” around Johan van Benthem.

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 4

Preliminaries

A Very Brief History

12-15th c. Informal EL investigated by logicians of the Middle Ages.

1960s EL takes off in philosophy after Jaakko Hintikka’s Knowledge andBelief (1962) (cf. von Wright’s An Essay on Modal Logic, 1951).

1970s Extensive philosophical discussion sparked by Hintikka is reviewed inWolfgang Lenzen’s Recent Work in Epistemic Logic (1978).

1980-90s EL is “rediscovered” in theoretical computer science, leading to theclassic textbook Reasoning about Knowledge (1995) by RonaldFagin, Joseph Halpern, Yoram Moses, and Moshe Vardi.

1990s EL is discovered again in game theory by Robert Aumann (recipientof the 2005 Nobel Memorial Prize in Economics) leading to his“Interactive Epistemology I: Knowledge” (1999).

90s-now Dynamic epistemic logic is developed in CS (Jan Plaza) and logic,especially by the “Amsterdam School” around Johan van Benthem.

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 4

Preliminaries

A Very Brief History

12-15th c. Informal EL investigated by logicians of the Middle Ages.

1960s EL takes off in philosophy after Jaakko Hintikka’s Knowledge andBelief (1962) (cf. von Wright’s An Essay on Modal Logic, 1951).

1970s Extensive philosophical discussion sparked by Hintikka is reviewed inWolfgang Lenzen’s Recent Work in Epistemic Logic (1978).

1980-90s EL is “rediscovered” in theoretical computer science, leading to theclassic textbook Reasoning about Knowledge (1995) by RonaldFagin, Joseph Halpern, Yoram Moses, and Moshe Vardi.

1990s EL is discovered again in game theory by Robert Aumann (recipientof the 2005 Nobel Memorial Prize in Economics) leading to his“Interactive Epistemology I: Knowledge” (1999).

90s-now Dynamic epistemic logic is developed in CS (Jan Plaza) and logic,especially by the “Amsterdam School” around Johan van Benthem.

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 4

Preliminaries

A Very Brief History

12-15th c. Informal EL investigated by logicians of the Middle Ages.

1960s EL takes off in philosophy after Jaakko Hintikka’s Knowledge andBelief (1962) (cf. von Wright’s An Essay on Modal Logic, 1951).

1970s Extensive philosophical discussion sparked by Hintikka is reviewed inWolfgang Lenzen’s Recent Work in Epistemic Logic (1978).

1980-90s EL is “rediscovered” in theoretical computer science, leading to theclassic textbook Reasoning about Knowledge (1995) by RonaldFagin, Joseph Halpern, Yoram Moses, and Moshe Vardi.

1990s EL is discovered again in game theory by Robert Aumann (recipientof the 2005 Nobel Memorial Prize in Economics) leading to his“Interactive Epistemology I: Knowledge” (1999).

90s-now Dynamic epistemic logic is developed in CS (Jan Plaza) and logic,especially by the “Amsterdam School” around Johan van Benthem.

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 4

Preliminaries

A Very Brief History

12-15th c. Informal EL investigated by logicians of the Middle Ages.

1960s EL takes off in philosophy after Jaakko Hintikka’s Knowledge andBelief (1962) (cf. von Wright’s An Essay on Modal Logic, 1951).

1970s Extensive philosophical discussion sparked by Hintikka is reviewed inWolfgang Lenzen’s Recent Work in Epistemic Logic (1978).

1980-90s EL is “rediscovered” in theoretical computer science, leading to theclassic textbook Reasoning about Knowledge (1995) by RonaldFagin, Joseph Halpern, Yoram Moses, and Moshe Vardi.

1990s EL is discovered again in game theory by Robert Aumann (recipientof the 2005 Nobel Memorial Prize in Economics) leading to his“Interactive Epistemology I: Knowledge” (1999).

90s-now Dynamic epistemic logic is developed in CS (Jan Plaza) and logic,especially by the “Amsterdam School” around Johan van Benthem.

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 4

Preliminaries

Now I am interested in bringing epistemic logic back into contactwith philosophy, using it as a tool to formalize existing theories ofknowledge and develop new and improved ones. See, for example:

Wesley Holliday. Forthcoming.

“Epistemic Logic and Epistemology.” Handbook of Formal Philosophy.

Wesley Holliday. Forthcoming.

“Epistemic Closure and Epistemic Logic I.” Journal of Philosophical Logic.

Wesley Holliday. 2012.

“Epistemic Logic, Relevant Alternatives, and the Dynamics of Context.”

New Directions in Logic, Language, and Computation, LNCS.

Wesley Holliday and John Perry. Forthcoming.

“Roles, Rigidity, and Quantificaiton in Epistemic Logic.” Trends in Logic.

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 5

Preliminaries

Logic and Philosophy

The project of proving results about formalized philosophicaltheories has led to the development of new logical techniques.

Wesley Holliday. Forthcoming.

“Epistemic Closure and Epistemic Logic I.” Journal of Philosophical Logic.

This paper demonstrates an alternative method of provingcompleteness theorems in modal logic, without the standardHenkin-style canonical model construction, that yields results onsatisfiability in finite models and complexity almost for free.

I am working to show how this method gives simple completenessproofs for standard systems of modal logic—good for students.

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 6

Preliminaries

Logic and Philosophy

The project of proving results about formalized philosophicaltheories has led to the development of new logical techniques.

Wesley Holliday. Forthcoming.

“Epistemic Closure and Epistemic Logic I.” Journal of Philosophical Logic.

This paper demonstrates an alternative method of provingcompleteness theorems in modal logic, without the standardHenkin-style canonical model construction, that yields results onsatisfiability in finite models and complexity almost for free.

I am working to show how this method gives simple completenessproofs for standard systems of modal logic—good for students.

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 6

Preliminaries

Logic and Philosophy

The project of proving results about formalized philosophicaltheories has led to the development of new logical techniques.

Wesley Holliday. Forthcoming.

“Epistemic Closure and Epistemic Logic I.” Journal of Philosophical Logic.

This paper demonstrates an alternative method of provingcompleteness theorems in modal logic, without the standardHenkin-style canonical model construction, that yields results onsatisfiability in finite models and complexity almost for free.

I am working to show how this method gives simple completenessproofs for standard systems of modal logic—good for students.

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 6

Preliminaries

Today’s Topic

But my talk today won’t presuppose familiarity with anyphilosophical theories or standard techniques in modal logic.

Today I’ll deal with propositional epistemic logic, but first-orderepistemic logic is also philosophically and technically rich.

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 7

Preliminaries

Today’s Topic

But my talk today won’t presuppose familiarity with anyphilosophical theories or standard techniques in modal logic.

Today I’ll deal with propositional epistemic logic, but first-orderepistemic logic is also philosophically and technically rich.

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 7

Basic Epistemic Logic

Definition (Propositional Epistemic Language)

Given sets At = {p, q, r , . . . } and Agt = {a, b, c , . . . } with|Agt| = κ, the epistemic language Lκ

EL is generated by

ϕ ::= p | ¬ϕ | (ϕ ∧ ϕ) | Kaϕ.

We read Kaϕ as “agent a knows that ϕ.”

Let Kaϕ := ¬Ka¬ϕ for “ϕ is consistent with a’s knowledge.”

Given a finite set A ⊆ Agt, let

EAϕ :=∧a∈A

Kaϕ,

for “everyone in A knows that ϕ.”

Where 2 is Ka or EA, 21ϕ := 2ϕ and 2n+1ϕ := 22nϕ (n ≥ 1).

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 8

Basic Epistemic Logic

Definition (Propositional Epistemic Language)

Given sets At = {p, q, r , . . . } and Agt = {a, b, c , . . . } with|Agt| = κ, the epistemic language Lκ

EL is generated by

ϕ ::= p | ¬ϕ | (ϕ ∧ ϕ) | Kaϕ.

We read Kaϕ as “agent a knows that ϕ.”

Let Kaϕ := ¬Ka¬ϕ for “ϕ is consistent with a’s knowledge.”

Given a finite set A ⊆ Agt, let

EAϕ :=∧a∈A

Kaϕ,

for “everyone in A knows that ϕ.”

Where 2 is Ka or EA, 21ϕ := 2ϕ and 2n+1ϕ := 22nϕ (n ≥ 1).

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 8

Basic Epistemic Logic

Definition (Propositional Epistemic Language)

Given sets At = {p, q, r , . . . } and Agt = {a, b, c , . . . } with|Agt| = κ, the epistemic language Lκ

EL is generated by

ϕ ::= p | ¬ϕ | (ϕ ∧ ϕ) | Kaϕ.

We read Kaϕ as “agent a knows that ϕ.”

Let Kaϕ := ¬Ka¬ϕ for “ϕ is consistent with a’s knowledge.”

Given a finite set A ⊆ Agt, let

EAϕ :=∧a∈A

Kaϕ,

for “everyone in A knows that ϕ.”

Where 2 is Ka or EA, 21ϕ := 2ϕ and 2n+1ϕ := 22nϕ (n ≥ 1).

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 8

Basic Epistemic Logic

Definition (Propositional Epistemic Language)

Given sets At = {p, q, r , . . . } and Agt = {a, b, c , . . . } with|Agt| = κ, the epistemic language Lκ

EL is generated by

ϕ ::= p | ¬ϕ | (ϕ ∧ ϕ) | Kaϕ.

We read Kaϕ as “agent a knows that ϕ.”

Let Kaϕ := ¬Ka¬ϕ for “ϕ is consistent with a’s knowledge.”

Given a finite set A ⊆ Agt, let

EAϕ :=∧a∈A

Kaϕ,

for “everyone in A knows that ϕ.”

Where 2 is Ka or EA, 21ϕ := 2ϕ and 2n+1ϕ := 22nϕ (n ≥ 1).

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 8

Basic Epistemic Logic

Definition (Propositional Epistemic Language)

Given sets At = {p, q, r , . . . } and Agt = {a, b, c , . . . } with|Agt| = κ, the epistemic language Lκ

EL is generated by

ϕ ::= p | ¬ϕ | (ϕ ∧ ϕ) | Kaϕ.

We read Kaϕ as “agent a knows that ϕ.”

Let Kaϕ := ¬Ka¬ϕ for “ϕ is consistent with a’s knowledge.”

Given a finite set A ⊆ Agt, let

EAϕ :=∧a∈A

Kaϕ,

for “everyone in A knows that ϕ.”

Where 2 is Ka or EA, 21ϕ := 2ϕ and 2n+1ϕ := 22nϕ (n ≥ 1).

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 8

Basic Epistemic Logic

p

w1 w2

p, q

w3

p

w4

q, r

w5

a, b

a

a, b a, b

a, b a, b a, b

ab

a

b

M

Definition (Epistemic Frames and Models)

A κ-agent epistemic model is a triple M = 〈W , {Ra}a∈Agt, V 〉:

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 9

Basic Epistemic Logic

w1 w2

w3 w4 w5

a, b

a

a, b a, b

a, b a, b a, b

ab

a

b

M

Definition (Epistemic Frames and Models)

A κ-agent epistemic frame is a pair M = 〈W , {Ra}a∈Agt〉 where:

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 9

Basic Epistemic Logic

w1 w2

w3 w4 w5

a, b

a

a, b a, b

a, b a, b a, b

ab

a

b

M

Definition (Epistemic Frames and Models)

A κ-agent epistemic frame is a pair F = 〈W , {Ra}a∈Agt〉 where:

I W is a non-empty set (of “possibilities,” “scenarios,”“states,” or “worlds”);

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 9

Basic Epistemic Logic

w1 w2

w3 w4 w5

a, b

a

a, b a, b

a, b a, b a, b

ab

a

b

F

Definition (Epistemic Frames and Models)

A κ-agent epistemic frame is a pair F = 〈W , {Ra}a∈Agt〉 where:

I for each a ∈ Agt (|Agt| = κ), Ra is a binary relation on W ;Ra can be thought of as representing agent a’s uncertainty.

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 9

Basic Epistemic Logic

w1 w2

w3 w4 w5

a, b

a

a, b a, b

a, b a, b a, b

ab

a

b

F

Definition (Epistemic Frames and Models)

A κ-agent epistemic frame is a pair F = 〈W , {Ra}a∈Agt〉 where:

I for each a ∈ Agt (|Agt| = κ), Ra is a binary relation on W ;let’s call Ra agent a’s “epistemic accessibility” relation;wRav means that everything a knows in w is true in v .

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 9

Basic Epistemic Logic

w1 w2

w3 w4 w5

a, b

a

a, b a, b

a, b a, b a, b

ab

a

b

F

Definition (Epistemic Frames and Models)

A κ-agent epistemic frame is a pair F = 〈W , {Ra}a∈Agt〉 where:

I for each a ∈ Agt (|Agt| = κ), Ra is a binary relation on W ;let’s call Ra agent a’s “epistemic accessibility” relation;wRav means that v is compatible with a’s knowledge in w .

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 9

Basic Epistemic Logic

p

w1 w2

p, q

w3

p

w4

q, r

w5

a, b

a

a, b a, b

a, b a, b a, b

ab

a

b

M

Definition (Epistemic Model)

A κ-agent epistemic model is a triple M = 〈W , {Ra}a∈Agt, V 〉:I V : At→ ℘(w) assigns to each atomic sentence the set of

scenarios in which it holds. We say M is based on F .

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 9

Basic Epistemic Logic

Three Ideas for the Semantics

The semantics will be based on three ideas from philosophy:

1. The knowledge operator Ka can be interpreted as a restrictedquantifier ranging over those possibilities compatible with a’sknowledge (J. Hintikka, Knowledge and Belief, 1962).

2. It is common knowledge that ϕ when everyone knows that ϕ,everyone knows that everyone knows that ϕ, everyone knowsthat everyone knows that everyone knows that ϕ, etc.(inspired by D. Lewis, Convention, 1969).

3. Knowledge acquisition (information update) can beunderstood in terms of the elimination of possibilities from anagent’s initial epistemic state (R. Stalnaker, Inquiry, 1987).

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 10

Basic Epistemic Logic

Three Ideas for the Semantics

The semantics will be based on three ideas from philosophy:

1. The knowledge operator Ka can be interpreted as a restrictedquantifier ranging over those possibilities compatible with a’sknowledge (J. Hintikka, Knowledge and Belief, 1962).

2. It is common knowledge that ϕ when everyone knows that ϕ,everyone knows that everyone knows that ϕ, everyone knowsthat everyone knows that everyone knows that ϕ, etc.(inspired by D. Lewis, Convention, 1969).

3. Knowledge acquisition (information update) can beunderstood in terms of the elimination of possibilities from anagent’s initial epistemic state (R. Stalnaker, Inquiry, 1987).

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 10

Basic Epistemic Logic

Three Ideas for the Semantics

The semantics will be based on three ideas from philosophy:

1. The knowledge operator Ka can be interpreted as a restrictedquantifier ranging over those possibilities compatible with a’sknowledge (J. Hintikka, Knowledge and Belief, 1962).

2. It is common knowledge that ϕ when everyone knows that ϕ,everyone knows that everyone knows that ϕ, everyone knowsthat everyone knows that everyone knows that ϕ, etc.(inspired by D. Lewis, Convention, 1969).

3. Knowledge acquisition (information update) can beunderstood in terms of the elimination of possibilities from anagent’s initial epistemic state (R. Stalnaker, Inquiry, 1987).

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 10

Basic Epistemic Logic

Three Ideas for the Semantics

The semantics will be based on three ideas from philosophy:

1. The knowledge operator Ka can be interpreted as a restrictedquantifier ranging over those possibilities compatible with a’sknowledge (J. Hintikka, Knowledge and Belief, 1962).

2. It is common knowledge that ϕ when everyone knows that ϕ,everyone knows that everyone knows that ϕ, everyone knowsthat everyone knows that everyone knows that ϕ, etc.(inspired by D. Lewis, Convention, 1969).

3. Knowledge acquisition (information update) can beunderstood in terms of the elimination of possibilities from anagent’s initial epistemic state (R. Stalnaker, Inquiry, 1987).

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 10

Basic Epistemic Logic

Idea 1: Knowledge Operators as Restricted Quantifiers

Example (Berkeley and Copenhagen)

Let Kb stand for agent b knows that and Kc stand for agent cknows that. Suppose agent b, who lives in Berkeley, knows thatagent c lives in Copenhagen. Let r stand for ‘it’s raining inCopenhagen’. Although b doesn’t know whether it’s raining inCopenhagen, b knows that c knows whether it’s raining there:

¬(Kbr ∨Kb¬r) ∧Kb(Kc r ∨Kc¬r).

The following picture depicts a situation in which this is true,where an arrow represents compatibility with one’s knowledge:

r

w1 w2

bb, c b, c

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 11

Basic Epistemic Logic

Idea 1: Knowledge Operators as Restricted Quantifiers

Example (Berkeley and Copenhagen)

Let Kb stand for agent b knows that and Kc stand for agent cknows that. Suppose agent b, who lives in Berkeley, knows thatagent c lives in Copenhagen. Let r stand for ‘it’s raining inCopenhagen’. Although b doesn’t know whether it’s raining inCopenhagen, b knows that c knows whether it’s raining there:

¬(Kbr ∨Kb¬r) ∧Kb(Kc r ∨Kc¬r).

The following picture depicts a situation in which this is true,where an arrow represents compatibility with one’s knowledge:

r

w1 w2

bb, c b, c

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 11

Basic Epistemic Logic

Idea 1: Knowledge Operators as Restricted Quantifiers

Example (Berkeley and Copenhagen)

Let Kb stand for agent b knows that and Kc stand for agent cknows that. Suppose agent b, who lives in Berkeley, knows thatagent c lives in Copenhagen. Let r stand for ‘it’s raining inCopenhagen’. Although b doesn’t know whether it’s raining inCopenhagen, b knows that c knows whether it’s raining there:

¬(Kbr ∨Kb¬r) ∧Kb(Kc r ∨Kc¬r).

The following picture depicts a situation in which this is true,where an arrow represents compatibility with one’s knowledge:

r

w1 w2

bb, c b, c

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 11

Basic Epistemic Logic

Now suppose that agent b doesn’t know whether agent c has leftCopenhagen for a vacation. (Let v stand for ‘c has leftCopenhagen on vacation’.) Agent b knows that if c is not onvacation, then c knows whether it’s raining in Copenhagen; but ifc is on vacation, then c won’t bother to follow the weather.

Kb(¬v → (Kc r ∨Kc¬r)) ∧Kb(v → ¬(Kc r ∨Kc¬r)).

r

w1 w2

v , r

w3

v

w4

b

b bb

b, c b, c

b, cb, c

b, c

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 12

Basic Epistemic Logic

Now suppose that agent b doesn’t know whether agent c has leftCopenhagen for a vacation. (Let v stand for ‘c has leftCopenhagen on vacation’.) Agent b knows that if c is not onvacation, then c knows whether it’s raining in Copenhagen; but ifc is on vacation, then c won’t bother to follow the weather.

Kb(¬v → (Kc r ∨Kc¬r)) ∧Kb(v → ¬(Kc r ∨Kc¬r)).

r

w1 w2

v , r

w3

v

w4

b

b bb

b, c b, c

b, cb, c

b, c

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 12

Basic Epistemic Logic

Idea 1: Knowledge Operators as Restricted Quantifiers

Definition (Truth)

Given a κ-agent model M = 〈W , {Ra}a∈Agt, V 〉 with w ∈ W andϕ ∈ Lκ

EL, we define M, w � ϕ (“ϕ is true in M at w”) as follows:

M, w � p iff w ∈ V (p);M, w � ¬ϕ iff M, w 2 ϕ;M, w � (ϕ ∧ ψ) iff M, w � ϕ and M, w � ψ;

M, w � Kaϕ iff ∀v ∈ W : if wRav then M, v � ϕ;

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 13

Basic Epistemic Logic

Idea 1: Knowledge Operators as Restricted Quantifiers

Definition (Truth)

Given a κ-agent model M = 〈W , {Ra}a∈Agt, V 〉 with w ∈ W andϕ ∈ Lκ

EL, we define M, w � ϕ (“ϕ is true in M at w”) as follows:

M, w � p iff w ∈ V (p);M, w � ¬ϕ iff M, w 2 ϕ;M, w � (ϕ ∧ ψ) iff M, w � ϕ and M, w � ψ;

M, w � Kaϕ iff ∀v ∈ W : if wRav then M, v � ϕ;

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 13

Basic Epistemic Logic

Now suppose that agent b doesn’t know whether agent c has leftCopenhagen for a vacation. (Let v stand for ‘c has leftCopenhagen on vacation’.) Agent b knows that if c is not onvacation, then c knows whether it’s raining in Copenhagen; but ifc is on vacation, then c won’t bother to follow the weather.

Kb(¬v → (Kc r ∨Kc¬r)) ∧Kb(v → ¬(Kc r ∨Kc¬r)).

w1

r

w2

v

w3

v , r

w4

b

b bb

b, c b, c

b, cb, c

b, c

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 14

Basic Epistemic Logic

Idea 1: Knowledge Operators as Restricted Quantifiers

Definition (Truth and Validity)

Given a κ-agent model M = 〈W , {Ra}a∈Agt, V 〉 with w ∈ W andϕ ∈ Lκ

EL, we define M, w � ϕ as follows (other cases as before):

M, w � Kaϕ iff ∀v ∈ W : if wRav then M, v � ϕ;

We say ϕ is globally true in M iff for all w ∈ W , M, w � ϕ;

ϕ is valid on frame F iff globally true in all models based on F ;

ϕ is valid on a class C of models (resp. a class F of frames) iffglobally true in all models in C (resp. based on frames in F).

For language L, the L-theory of a class C of models, ThL(C)(resp. class F of frames) is the set of ϕ ∈ L valid on C (resp. F).

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 15

Basic Epistemic Logic

Idea 1: Knowledge Operators as Restricted Quantifiers

Definition (Truth and Validity)

Given a κ-agent model M = 〈W , {Ra}a∈Agt, V 〉 with w ∈ W andϕ ∈ Lκ

EL, we define M, w � ϕ as follows (other cases as before):

M, w � Kaϕ iff ∀v ∈ W : if wRav then M, v � ϕ;

We say ϕ is globally true in M iff for all w ∈ W , M, w � ϕ;

ϕ is valid on frame F iff globally true in all models based on F ;

ϕ is valid on a class C of models (resp. a class F of frames) iffglobally true in all models in C (resp. based on frames in F).

For language L, the L-theory of a class C of models, ThL(C)(resp. class F of frames) is the set of ϕ ∈ L valid on C (resp. F).

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 15

Basic Epistemic Logic

Idea 1: Knowledge Operators as Restricted Quantifiers

Definition (Truth and Validity)

Given a κ-agent model M = 〈W , {Ra}a∈Agt, V 〉 with w ∈ W andϕ ∈ Lκ

EL, we define M, w � ϕ as follows (other cases as before):

M, w � Kaϕ iff ∀v ∈ W : if wRav then M, v � ϕ;

We say ϕ is globally true in M iff for all w ∈ W , M, w � ϕ;

ϕ is valid on frame F iff globally true in all models based on F ;

ϕ is valid on a class C of models (resp. a class F of frames) iffglobally true in all models in C (resp. based on frames in F).

For language L, the L-theory of a class C of models, ThL(C)(resp. class F of frames) is the set of ϕ ∈ L valid on C (resp. F).

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 15

Basic Epistemic Logic

Idea 1: Knowledge Operators as Restricted Quantifiers

Definition (Truth and Validity)

Given a κ-agent model M = 〈W , {Ra}a∈Agt, V 〉 with w ∈ W andϕ ∈ Lκ

EL, we define M, w � ϕ as follows (other cases as before):

M, w � Kaϕ iff ∀v ∈ W : if wRav then M, v � ϕ;

We say ϕ is globally true in M iff for all w ∈ W , M, w � ϕ;

ϕ is valid on frame F iff globally true in all models based on F ;

ϕ is valid on a class C of models (resp. a class F of frames) iffglobally true in all models in C (resp. based on frames in F).

For language L, the L-theory of a class C of models, ThL(C)(resp. class F of frames) is the set of ϕ ∈ L valid on C (resp. F).

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 15

Basic Epistemic Logic

Idea 1: Knowledge Operators as Restricted Quantifiers

Definition (Truth and Validity)

Given a κ-agent model M = 〈W , {Ra}a∈Agt, V 〉 with w ∈ W andϕ ∈ Lκ

EL, we define M, w � ϕ as follows (other cases as before):

M, w � Kaϕ iff ∀v ∈ W : if wRav then M, v � ϕ;

We say ϕ is globally true in M iff for all w ∈ W , M, w � ϕ;

ϕ is valid on frame F iff globally true in all models based on F ;

ϕ is valid on a class C of models (resp. a class F of frames) iffglobally true in all models in C (resp. based on frames in F).

For language L, the L-theory of a class C of models, ThL(C)(resp. class F of frames) is the set of ϕ ∈ L valid on C (resp. F).

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 15

Basic Epistemic Logic

Correspondence

Proposition

For any frame F = 〈W , {Ra}a∈Agt〉:

T Kaϕ→ ϕ is valid on F iff Ra is reflexive;

4 Ka → KaKaϕ is valid on F iff Ra is transitive;

B ¬ϕ→ Ka¬Kaϕ is valid on F iff Ra is symmetric;

5 ¬Kaϕ→ Ka¬Kaϕ is valid on F iff Ra is Euclidean.

Cf. Theorem 3.5 of

Bjarni Jonsson and Alfred Tarski. 1951. “Boolean Algebras with Operators.”

In CS and game theory, it is typically assumed that every Ra is anequivalence relation. We call such models partition models.

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 16

Basic Epistemic Logic

Correspondence

Proposition

For any frame F = 〈W , {Ra}a∈Agt〉:

T Kaϕ→ ϕ is valid on F iff Ra is reflexive;

4 Ka → KaKaϕ is valid on F iff Ra is transitive;

B ¬ϕ→ Ka¬Kaϕ is valid on F iff Ra is symmetric;

5 ¬Kaϕ→ Ka¬Kaϕ is valid on F iff Ra is Euclidean.

Cf. Theorem 3.5 of

Bjarni Jonsson and Alfred Tarski. 1951. “Boolean Algebras with Operators.”

In CS and game theory, it is typically assumed that every Ra is anequivalence relation. We call such models partition models.

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 16

Basic Epistemic Logic

Correspondence

Proposition

For any frame F = 〈W , {Ra}a∈Agt〉:

T Kaϕ→ ϕ is valid on F iff Ra is reflexive;

4 Ka → KaKaϕ is valid on F iff Ra is transitive;

B ¬ϕ→ Ka¬Kaϕ is valid on F iff Ra is symmetric;

5 ¬Kaϕ→ Ka¬Kaϕ is valid on F iff Ra is Euclidean.

Cf. Theorem 3.5 of

Bjarni Jonsson and Alfred Tarski. 1951. “Boolean Algebras with Operators.”

In CS and game theory, it is typically assumed that every Ra is anequivalence relation. We call such models partition models.

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 16

Basic Epistemic Logic

Correspondence

Proposition

For any frame F = 〈W , {Ra}a∈Agt〉:

T Kaϕ→ ϕ is valid on F iff Ra is reflexive;

4 Ka → KaKaϕ is valid on F iff Ra is transitive;

B ¬ϕ→ Ka¬Kaϕ is valid on F iff Ra is symmetric;

5 ¬Kaϕ→ Ka¬Kaϕ is valid on F iff Ra is Euclidean.

Cf. Theorem 3.5 of

Bjarni Jonsson and Alfred Tarski. 1951. “Boolean Algebras with Operators.”

In CS and game theory, it is typically assumed that every Ra is anequivalence relation. We call such models partition models.

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 16

Basic Epistemic Logic

Correspondence

Proposition

For any frame F = 〈W , {Ra}a∈Agt〉:

T Kaϕ→ ϕ is valid on F iff Ra is reflexive;

4 Ka → KaKaϕ is valid on F iff Ra is transitive;

B ¬ϕ→ Ka¬Kaϕ is valid on F iff Ra is symmetric;

5 ¬Kaϕ→ Ka¬Kaϕ is valid on F iff Ra is Euclidean.

Cf. Theorem 3.5 of

Bjarni Jonsson and Alfred Tarski. 1951. “Boolean Algebras with Operators.”

In CS and game theory, it is typically assumed that every Ra is anequivalence relation. We call such models partition models.

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 16

Basic Epistemic Logic

Correspondence

Proposition

For any frame F = 〈W , {Ra}a∈Agt〉:

T Kaϕ→ ϕ is valid on F iff Ra is reflexive;

4 Ka → KaKaϕ is valid on F iff Ra is transitive;

B ¬ϕ→ Ka¬Kaϕ is valid on F iff Ra is symmetric;

5 ¬Kaϕ→ Ka¬Kaϕ is valid on F iff Ra is Euclidean.

Cf. Theorem 3.5 of

Bjarni Jonsson and Alfred Tarski. 1951. “Boolean Algebras with Operators.”

In CS and game theory, it is typically assumed that every Ra is anequivalence relation. We call such models partition models.

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 16

Basic Epistemic Logic

Complexity

Theorem (Complexity)

1. model-checking is in linear time (in the size of M and ϕ);

2. satisfiability in 1-agent models is PSPACE-complete;

3. satisfiability in 1-agent partition models is NP-complete;

4. satisfiability in 2-agent partition models is PSPACE-complete.

If we bound the nesting depth of K , satisfiability drops down toNP-complete in most cases; if we further restrict to finitely manyatomic sentences, satisfiability drops down to linear time.

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 17

Basic Epistemic Logic

Completeness

TheoremThe Lκ

EL-theory of the class of all κ-agent epistemic models isaxiomatized by the system Kκ:

I all substitutions of propositional tautologies as axioms;

I the K axiom, Ka(ϕ→ ψ)→ (Kaϕ→ Kaψ), for a ∈ Agt;

I modus ponens and necessitation, ` ϕ⇒ ` Kaϕ, for a ∈ Agt.

To axiomatize the theories of other model classes, add the axiomscorresponding to the defining properties of the class:

I for reflexive models, add the T axiom, Kaϕ→ ϕ;

I for transitive models, add the 4 axiom, Kaϕ→ KaKaϕ;

I for Euclidean models, add the 5 axiom, ¬K ϕ→ K¬K ϕa;

I etc.

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 18

Basic Epistemic Logic

Completeness

TheoremThe Lκ

EL-theory of the class of all κ-agent epistemic models isaxiomatized by the system Kκ:

I all substitutions of propositional tautologies as axioms;

I the K axiom, Ka(ϕ→ ψ)→ (Kaϕ→ Kaψ), for a ∈ Agt;

I modus ponens and necessitation, ` ϕ⇒ ` Kaϕ, for a ∈ Agt.

To axiomatize the theories of other model classes, add the axiomscorresponding to the defining properties of the class:

I for reflexive models, add the T axiom, Kaϕ→ ϕ;

I for transitive models, add the 4 axiom, Kaϕ→ KaKaϕ;

I for Euclidean models, add the 5 axiom, ¬K ϕ→ K¬K ϕa;

I etc.

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 18

Basic Epistemic Logic

Completeness

TheoremThe Lκ

EL-theory of the class of all κ-agent epistemic models isaxiomatized by the system Kκ:

I all substitutions of propositional tautologies as axioms;

I the K axiom, Ka(ϕ→ ψ)→ (Kaϕ→ Kaψ), for a ∈ Agt;

I modus ponens and necessitation, ` ϕ⇒ ` Kaϕ, for a ∈ Agt.

To axiomatize the theories of other model classes, add the axiomscorresponding to the defining properties of the class:

I for reflexive models, add the T axiom, Kaϕ→ ϕ;

I for transitive models, add the 4 axiom, Kaϕ→ KaKaϕ;

I for Euclidean models, add the 5 axiom, ¬K ϕ→ K¬K ϕa;

I etc.

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 18

Basic Epistemic Logic

Idea 2: Common Knowledge as Transitive Closure

The “everyone knows” operator is not the only group knowledgeoperator of interest. To see this, consider the following example.

Example (Consecutive Numbers (J.E. Littlewood 1953))

Two agents, b and g , are facing each other. They each see anumber on the other’s forehead. A perfectly trustworthy sourcetells them: one of you has n ∈N and the other has n + 1.

Let bm stand for “the number on b’s head is m,” and let gm standfor “the number on g ’s head is m.” The truth is: b2 ∧ g3.

What should an epistemic model representing this look like?

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 19

Basic Epistemic Logic

Idea 2: Common Knowledge as Transitive Closure

The “everyone knows” operator is not the only group knowledgeoperator of interest. To see this, consider the following example.

Example (Consecutive Numbers (J.E. Littlewood 1953))

Two agents, b and g , are facing each other. They each see anumber on the other’s forehead. A perfectly trustworthy sourcetells them: one of you has n ∈N and the other has n + 1.

Let bm stand for “the number on b’s head is m,” and let gm standfor “the number on g ’s head is m.” The truth is: b2 ∧ g3.

What should an epistemic model representing this look like?

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 19

Basic Epistemic Logic

Idea 2: Common Knowledge as Transitive Closure

The “everyone knows” operator is not the only group knowledgeoperator of interest. To see this, consider the following example.

Example (Consecutive Numbers (J.E. Littlewood 1953))

Two agents, b and g , are facing each other. They each see anumber on the other’s forehead. A perfectly trustworthy sourcetells them: one of you has n ∈N and the other has n + 1.

Let bm stand for “the number on b’s head is m,” and let gm standfor “the number on g ’s head is m.” The truth is: b2 ∧ g3.

What should an epistemic model representing this look like?

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 19

Basic Epistemic Logic

Idea 2: Common Knowledge as Transitive Closure

The “everyone knows” operator is not the only group knowledgeoperator of interest. To see this, consider the following example.

Example (Consecutive Numbers (J.E. Littlewood 1953))

Two agents, b and g , are facing each other. They each see anumber on the other’s forehead. A perfectly trustworthy sourcetells them: one of you has n ∈N and the other has n + 1.

Let bm stand for “the number on b’s head is m,” and let gm standfor “the number on g ’s head is m.” The truth is: b2 ∧ g3.

What should an epistemic model representing this look like?

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 19

Basic Epistemic Logic

Here is an epistemic model representing the example:

b0, g1 b2, g1

b2, g3 b4, g3

b4, g5 . . .

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 20

Basic Epistemic Logic

Here is an epistemic model representing the example:

b0, g1 b2, g1

b2, g3w b4, g3

b4, g5 . . .

M, w � Kbg3; M, w � Kb(b2 ∨ b4); M, w � KbKg (g1 ∨ g3 ∨ g5);M, w � KbKgKb(g1 ∨ g3 ∨ g5); and so on...

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 21

Basic Epistemic Logic

Here is an epistemic model representing the example:

b0, g1 b2, g1

b2, g3w b4, g3

b4, g5 . . .

M, w � Kbg3;

M, w � Kb(b2 ∨ b4); M, w � KbKg (g1 ∨ g3 ∨ g5);M, w � KbKgKb(g1 ∨ g3 ∨ g5); and so on...

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 21

Basic Epistemic Logic

Here is an epistemic model representing the example:

b0, g1 b2, g1

b2, g3w b4, g3

b4, g5 . . .

M, w � Kbg3; M, w � Kb(b2 ∨ b4);

M, w � KbKg (g1 ∨ g3 ∨ g5);M, w � KbKgKb(g1 ∨ g3 ∨ g5); and so on...

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 21

Basic Epistemic Logic

Here is an epistemic model representing the example:

b0, g1 b2, g1

b2, g3w b4, g3

b4, g5 . . .

M, w � Kbg3; M, w � Kb(b2 ∨ b4); M, w � KbKg (g1 ∨ g3 ∨ g5);

M, w � KbKgKb(g1 ∨ g3 ∨ g5); and so on...

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 21

Basic Epistemic Logic

Here is an epistemic model representing the example:

b0, g1 b2, g1

b2, g3w b4, g3

b4, g5 . . .

M, w � Kbg3; M, w � Kb(b2 ∨ b4); M, w � KbKg (g1 ∨ g3 ∨ g5);M, w � KbKgKb(g1 ∨ g3 ∨ g5); and so on...

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 21

Basic Epistemic Logic

Here is an epistemic model representing the example:

b0, g1 b2, g1

b2, g3w b4, g3

b4, g5 . . .

M, w � Kgg1 ∧ Kgg3;

M, w � Kg KbKgg1; and interestingly,M, w � E{b,g}¬g5 ∧ ¬E{b,g}E{b,g}¬g5. Not common knowledge.

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 22

Basic Epistemic Logic

Here is an epistemic model representing the example:

b0, g1 b2, g1

b2, g3w b4, g3

b4, g5 . . .

M, w � Kgg1 ∧ Kgg3; M, w � Kg KbKgg1;

and interestingly,M, w � E{b,g}¬g5 ∧ ¬E{b,g}E{b,g}¬g5. Not common knowledge.

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 22

Basic Epistemic Logic

Here is an epistemic model representing the example:

b0, g1 b2, g1

b2, g3w b4, g3

b4, g5 . . .

M, w � Kgg1 ∧ Kgg3; M, w � Kg KbKgg1; and interestingly,M, w � E{b,g}¬g5 ∧ ¬E{b,g}E{b,g}¬g5.

Not common knowledge.

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 22

Basic Epistemic Logic

Here is an epistemic model representing the example:

b0, g1 b2, g1

b2, g3w b4, g3

b4, g5 . . .

M, w � Kgg1 ∧ Kgg3; M, w � Kg KbKgg1; and interestingly,M, w � E{b,g}¬g5 ∧ ¬E{b,g}E{b,g}¬g5. Not common knowledge.

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 22

Basic Epistemic Logic

Definition (Basic Epistemic Languages)

Given countable sets At = {p, q, r , . . . } and Agt = {a, b, c , . . . }with |Agt| = κ, the epistemic language Lκ

EL is generated by

ϕ ::= p | ¬ϕ | (ϕ ∧ ϕ) | Kaϕ.

For the epistemic language with common knowledge, LκEL-C,

ϕ ::= p | ¬ϕ | (ϕ ∧ ϕ) | Kaϕ | CAϕ,

where A ⊆ Agt.

We read CAϕ as “it is common knowledgeamong the members of A that ϕ.”

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 23

Basic Epistemic Logic

Definition (Basic Epistemic Languages)

Given countable sets At = {p, q, r , . . . } and Agt = {a, b, c , . . . }with |Agt| = κ, the epistemic language Lκ

EL is generated by

ϕ ::= p | ¬ϕ | (ϕ ∧ ϕ) | Kaϕ.

For the epistemic language with common knowledge, LκEL-C,

ϕ ::= p | ¬ϕ | (ϕ ∧ ϕ) | Kaϕ | CAϕ,

where A ⊆ Agt. We read CAϕ as “it is common knowledgeamong the members of A that ϕ.”

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 23

Basic Epistemic Logic

Idea 2: Common Knowledge as Transitive Closure

DefinitionGiven a κ-agent model M = 〈W , {Ra}a∈Agt, V 〉 with w ∈ W andϕ ∈ Lκ

EL-C, we define M, w � ϕ as follows (other cases as before):

M, w � Kaϕ iff ∀v ∈ W : wRav implies M, v � ϕ;

M, w � CAϕ iff ∀v ∈ W : wR+A v implies M, v � ϕ,

where R+A is the transitive closure of

⋃a∈A

Ra.

In other words, CAϕ is true at w iff any sequence of steps from walong any agents’ relations ends in a state where ϕ is true.

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 24

Basic Epistemic Logic

Idea 2: Common Knowledge as Transitive Closure

DefinitionGiven a κ-agent model M = 〈W , {Ra}a∈Agt, V 〉 with w ∈ W andϕ ∈ Lκ

EL-C, we define M, w � ϕ as follows (other cases as before):

M, w � Kaϕ iff ∀v ∈ W : wRav implies M, v � ϕ;

M, w � CAϕ iff ∀v ∈ W : wR+A v implies M, v � ϕ,

where R+A is the transitive closure of

⋃a∈A

Ra.

In other words, CAϕ is true at w iff any sequence of steps from walong any agents’ relations ends in a state where ϕ is true.

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 24

Basic Epistemic Logic

Example: Consecutive Numbers

b0, g1 b2, g1

b2, g3w b4, g3

b4, g5 . . .

Recall: M, w � E{b,g}¬g5 ∧ ¬E{b,g}E{b,g}¬g5.

Pattern: M, w � E 3{b,g}¬g6 ∧ ¬E 4

{b,g}¬g6.

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 25

Basic Epistemic Logic

Example: Consecutive Numbers

b0, g1 b2, g1

b2, g3w b4, g3

b4, g5 . . .

Recall: M, w � E{b,g}¬g5 ∧ ¬E{b,g}E{b,g}¬g5.

Pattern: M, w � E 3{b,g}¬g6 ∧ ¬E 4

{b,g}¬g6.

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 25

Basic Epistemic Logic

Example: Consecutive Numbers

b0, g1 b2, g1

b2, g3w b4, g3

b4, g5 . . .

Recall: M, w � E{b,g}¬g5 ∧ ¬E{b,g}E{b,g}¬g5.

Pattern: M, w � E 3{b,g}¬g6 ∧ ¬E 4

{b,g}¬g6.

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 25

Basic Epistemic Logic

Example: Consecutive Numbers

b0, g1 b2, g1

b2, g3w b4, g3

b4, g5 . . .

Surprising result: M, w � Kbg3 ∧Kgb2 ∧ ¬C{b,g}¬b1000.

It’s not common knowledge that b doesn’t have 1000 on his head!

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 26

Basic Epistemic Logic

By contrast, in our earlier example with Berkeley and Copenhagen,it is common knowledge that if c is on vacation, then he doesn’tknow whether or not it’s raining in Copenhagen:

M, w1 � C{b,c}(v → ¬(Kc r ∨Kc¬r)).

r

w1 w2

v , r

w3

v

w4

b

b bb

b, c b, c

b, cb, c

b, c

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 27

Basic Epistemic Logic

Idea 2: Common Knowledge as Transitive Closure

DefinitionGiven a κ-agent model M = 〈W , {Ra}a∈Agt, V 〉 with w ∈ W andϕ ∈ Lκ

EL-C, we define M, w � ϕ as follows (other cases as before):

M, w � CAϕ iff ∀v ∈ W : wR+A v implies M, v � ϕ,

where R+A is the transitive closure of

⋃a∈G

Ra.

ObservationGiven the infinitary character of common knowledge, it’s nosurprise that with LEL-C we lose compactness:

{EnAp | n ∈N} ∪ {¬CAp}

is finitely-satisfiable but not satisfiable.

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 28

Basic Epistemic Logic

Completeness and Complexity

TheoremThe LEL-C-theory of the class of all (resp. reflexive, preorder,partition, etc.) models is axiomatized by Kκ (resp. Tκ, S4κ, S5κ,etc.) plus the following schemas for all A ⊆ Agt:

I Fixed-Point Axiom: CAϕ↔ (ϕ ∧ EACAϕ)

I Induction Axiom: (ϕ ∧ CA(ϕ→ EAϕ))→ CAϕ

Complexity of checking satisfiability goes to EXPTIME-completewith common knowledge.

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 29

Basic Epistemic Logic

Completeness and Complexity

TheoremThe LEL-C-theory of the class of all (resp. reflexive, preorder,partition, etc.) models is axiomatized by Kκ (resp. Tκ, S4κ, S5κ,etc.) plus the following schemas for all A ⊆ Agt:

I Fixed-Point Axiom: CAϕ↔ (ϕ ∧ EACAϕ)

I Induction Axiom: (ϕ ∧ CA(ϕ→ EAϕ))→ CAϕ

Complexity of checking satisfiability goes to EXPTIME-completewith common knowledge.

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 29

Basic Epistemic Logic

Completeness and Complexity

TheoremThe LEL-C-theory of the class of all (resp. reflexive, preorder,partition, etc.) models is axiomatized by Kκ (resp. Tκ, S4κ, S5κ,etc.) plus the following schemas for all A ⊆ Agt:

I Fixed-Point Axiom: CAϕ↔ (ϕ ∧ EACAϕ)

I Induction Axiom: (ϕ ∧ CA(ϕ→ EAϕ))→ CAϕ

Complexity of checking satisfiability goes to EXPTIME-completewith common knowledge.

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 29

Basic Epistemic Logic

Idea 3: Information Update as Model Restriction

Example (Muddle Children Puzzle)

Three children—r , g , and b—have returned after playing outside.Their mother says to them, “At least one of you has mud on yourforehead. Step forward now if you know whether you are dirty.”

In fact, g and b are dirty, but no one can see his own forehead.

None of the children step forward. So their mother says again,“Step forward now if you know whether you are dirty.”

This time g and b both step forward, but r does not.

So their mother says again. “Step forward now if you knowwhether you are dirty.” Finally r steps forward.

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 30

Basic Epistemic Logic

Idea 3: Information Update as Model Restriction

Example (Muddle Children Puzzle)

Three children—r , g , and b—have returned after playing outside.Their mother says to them, “At least one of you has mud on yourforehead. Step forward now if you know whether you are dirty.”

In fact, g and b are dirty, but no one can see his own forehead.

None of the children step forward. So their mother says again,“Step forward now if you know whether you are dirty.”

This time g and b both step forward, but r does not.

So their mother says again. “Step forward now if you knowwhether you are dirty.” Finally r steps forward.

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 30

Basic Epistemic Logic

Idea 3: Information Update as Model Restriction

Example (Muddle Children Puzzle)

Three children—r , g , and b—have returned after playing outside.Their mother says to them, “At least one of you has mud on yourforehead. Step forward now if you know whether you are dirty.”

In fact, g and b are dirty, but no one can see his own forehead.

None of the children step forward. So their mother says again,“Step forward now if you know whether you are dirty.”

This time g and b both step forward, but r does not.

So their mother says again. “Step forward now if you knowwhether you are dirty.” Finally r steps forward.

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 30

Basic Epistemic Logic

Idea 3: Information Update as Model Restriction

Example (Muddle Children Puzzle)

Three children—r , g , and b—have returned after playing outside.Their mother says to them, “At least one of you has mud on yourforehead. Step forward now if you know whether you are dirty.”

In fact, g and b are dirty, but no one can see his own forehead.

None of the children step forward. So their mother says again,“Step forward now if you know whether you are dirty.”

This time g and b both step forward, but r does not.

So their mother says again. “Step forward now if you knowwhether you are dirty.” Finally r steps forward.

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 30

Basic Epistemic Logic

Idea 3: Information Update as Model Restriction

Example (Muddle Children Puzzle)

Three children—r , g , and b—have returned after playing outside.Their mother says to them, “At least one of you has mud on yourforehead. Step forward now if you know whether you are dirty.”

In fact, g and b are dirty, but no one can see his own forehead.

None of the children step forward. So their mother says again,“Step forward now if you know whether you are dirty.”

This time g and b both step forward, but r does not.

So their mother says again. “Step forward now if you knowwhether you are dirty.” Finally r steps forward.

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 30

Basic Epistemic Logic

DDD

C DD

DDC

DCD

C DC

C C CC C D

DCC

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 31

Basic Epistemic Logic

DDD

C DD

DDC

DCD

C DC

C C CC C D

DCC

“At least one of you is dirty.” Dr ∨Dg ∨Db. What happens?

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 31

Basic Epistemic Logic

DDD

C DD

DDC

DCD

C DC

C C CC C D

DCC

“At least one of you is dirty.” Dr ∨Dg ∨Db. What happens?

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 31

Basic Epistemic Logic

DDD

C DD

DDC

DCD

C DC

C C D

DCC

“At least one of you is dirty.” Dr ∨Dg ∨Db. What happens?

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 31

Basic Epistemic Logic

DDD

C DD

DDC

DCD

C DC

C C D

DCC

“If you know whether you’re dirty, step forward.”

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 31

Basic Epistemic Logic

DDD

C DD

DDC

DCD

C DC

C C D

DCC

“If you know whether you’re dirty, step forward.” No one does.¬(KrDr ∨Kr¬Dr ) ∧ ¬(KgDg ∨Kg¬Dg ) ∧ ¬(KbDb ∨Kb¬Db).

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 31

Basic Epistemic Logic

DDD

C DD

DDC

DCD

C DC

C C D

DCC

¬(KrDr ∨Kr¬Dr ) ∧ ¬(KgDg ∨Kg¬Dg ) ∧ ¬(KbDb ∨Kb¬Db).What happens after this is announced?

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 31

Basic Epistemic Logic

DDD

C DD

DDC

DCD

C DC

C C D

DCC

¬(KrDr ∨Kr¬Dr ) ∧ ¬(KgDg ∨Kg¬Dg ) ∧ ¬(KbDb ∨Kb¬Db).What happens after this? Three more possibilities are eliminated.

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 31

Basic Epistemic Logic

DDD

C DD

DDC

DCD

“If you know whether you’re dirty, step forward.”

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 31

Basic Epistemic Logic

DDD

C DD

DDC

DCD

“If you know whether you’re dirty, step forward.” Blue and green(but not red) step forward.

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 31

Basic Epistemic Logic

DDD

C DD

DDC

DCD

“If you know whether you’re dirty, step forward.” Blue and green(but not red) step forward. (KbDb ∨Kb¬Db) ∧ (KgDg ∨Kg¬Dg ).

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 31

Basic Epistemic Logic

DDD

C DD

DDC

DCD

Blue and green (but not red) step forward. (KbDb ∨ Kb¬Db) ∧(KgDg ∨Kg¬Dg ). What happens after this is announced?

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 31

Basic Epistemic Logic

DDD

C DD

DDC

DCD

(KbDb ∨Kb¬Db) ∧ (KgDg ∨Kg¬Dg ). What happens after this?The remaining non-actual possibilities are eliminated.

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 31

Basic Epistemic Logic

C DD

(KbDb ∨Kb¬Db) ∧ (KgDg ∨Kg¬Dg ). What happens after this?The remaining non-actual possibilities are eliminated.

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 31

Basic Epistemic Logic

One of the big ideas of dynamic epistemic logic is to add to ourformal language operators that can describe the kinds of modelchanges that we just saw for the Muddy Children puzzle.

Definition (Basic Epistemic Languages)

Given sets At = {p, q, r , . . . } and Agt = {a, b, c , . . . } with|Agt| = κ, the language of public announcement logic, Lκ

PAL, isgenerated by

ϕ ::= p | ¬ϕ | (ϕ ∧ ϕ) | Kaϕ | [ϕ]ϕ.

Read [ϕ]ψ as “after (every) true announcement of ϕ, ψ.”

Read 〈ϕ〉ψ := ¬[ϕ]¬ψ as “after a true announcement of ϕ, ψ.”

One can also add common knowledge CA to the language.

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 32

Basic Epistemic Logic

One of the big ideas of dynamic epistemic logic is to add to ourformal language operators that can describe the kinds of modelchanges that we just saw for the Muddy Children puzzle.

Definition (Basic Epistemic Languages)

Given sets At = {p, q, r , . . . } and Agt = {a, b, c , . . . } with|Agt| = κ, the language of public announcement logic, Lκ

PAL, isgenerated by

ϕ ::= p | ¬ϕ | (ϕ ∧ ϕ) | Kaϕ | [ϕ]ϕ.

Read [ϕ]ψ as “after (every) true announcement of ϕ, ψ.”

Read 〈ϕ〉ψ := ¬[ϕ]¬ψ as “after a true announcement of ϕ, ψ.”

One can also add common knowledge CA to the language.

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 32

Basic Epistemic Logic

One of the big ideas of dynamic epistemic logic is to add to ourformal language operators that can describe the kinds of modelchanges that we just saw for the Muddy Children puzzle.

Definition (Basic Epistemic Languages)

Given sets At = {p, q, r , . . . } and Agt = {a, b, c , . . . } with|Agt| = κ, the language of public announcement logic, Lκ

PAL, isgenerated by

ϕ ::= p | ¬ϕ | (ϕ ∧ ϕ) | Kaϕ | [ϕ]ϕ.

Read [ϕ]ψ as “after (every) true announcement of ϕ, ψ.”

Read 〈ϕ〉ψ := ¬[ϕ]¬ψ as “after a true announcement of ϕ, ψ.”

One can also add common knowledge CA to the language.

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 32

Basic Epistemic Logic

One of the big ideas of dynamic epistemic logic is to add to ourformal language operators that can describe the kinds of modelchanges that we just saw for the Muddy Children puzzle.

Definition (Basic Epistemic Languages)

Given sets At = {p, q, r , . . . } and Agt = {a, b, c , . . . } with|Agt| = κ, the language of public announcement logic, Lκ

PAL, isgenerated by

ϕ ::= p | ¬ϕ | (ϕ ∧ ϕ) | Kaϕ | [ϕ]ϕ.

Read [ϕ]ψ as “after (every) true announcement of ϕ, ψ.”

Read 〈ϕ〉ψ := ¬[ϕ]¬ψ as “after a true announcement of ϕ, ψ.”

One can also add common knowledge CA to the language.

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 32

Basic Epistemic Logic

One of the big ideas of dynamic epistemic logic is to add to ourformal language operators that can describe the kinds of modelchanges that we just saw for the Muddy Children puzzle.

Definition (Basic Epistemic Languages)

Given sets At = {p, q, r , . . . } and Agt = {a, b, c , . . . } with|Agt| = κ, the language of public announcement logic, Lκ

PAL, isgenerated by

ϕ ::= p | ¬ϕ | (ϕ ∧ ϕ) | Kaϕ | [ϕ]ϕ.

Read [ϕ]ψ as “after (every) true announcement of ϕ, ψ.”

Read 〈ϕ〉ψ := ¬[ϕ]¬ψ as “after a true announcement of ϕ, ψ.”

One can also add common knowledge CA to the language.

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 32

Basic Epistemic Logic

Definition (Basic Epistemic Languages)

Given sets At = {p, q, r , . . . } and Agt = {a, b, c , . . . } with|Agt| = κ, the epistemic language Lκ

EL is generated by

ϕ ::= p | ¬ϕ | (ϕ ∧ ϕ) | Kaϕ.

For the epistemic language with common knowledge, LκEL-C,

ϕ ::= p | ¬ϕ | (ϕ ∧ ϕ) | Kaϕ | C ϕ.

For the language of public announcement logic, LκPAL,

ϕ ::= p | ¬ϕ | (ϕ ∧ ϕ) | Kaϕ | [ϕ]ϕ.

For public announcement with common knowledge, LκPAL-C,

ϕ ::= p | ¬ϕ | (ϕ ∧ ϕ) | Kaϕ | [ϕ]ϕ | C ϕ.

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 33

Basic Epistemic Logic

Idea 3: Information Update as Model Restriction

Definition (Truth cont.)

Given a κ-agent model M = 〈W , {Ra}a∈Agt, V 〉 with w ∈ W andϕ ∈ Lκ

PAL, we define M, w � ϕ as follows (other cases as before):

M, w � [ϕ]ψ iff M, w � ϕ implies M|ϕ, w � ψ,

where M|ϕ = 〈W|ϕ, {Ra|ϕ}a∈Agt, V|ϕ〉 is the submodel such that

W|ϕ = {v ∈ W | M, v � ϕ};∀a ∈ Agt : Ra|ϕ

= Ra ∩ (W|ϕ ×W|ϕ);

∀p ∈ At : V|ϕ(p) = V (p) ∩W|ϕ.

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 34

Basic Epistemic Logic

Public Announcement Logic

Read [ϕ]ψ as “after (every) true announcement of ϕ, ψ.”

Read 〈ϕ〉ψ := ¬[ϕ]¬ψ as “after a true announcement of ϕ, ψ.”

Let’s compare. The truth clause for the dynamic operator [ϕ] is:

I M, w � [ϕ]ψ iff M, w � ϕ implies M|ϕ, w � ψ.

So if ϕ is false, [ϕ]ψ is vacuously true. Here is the 〈ϕ〉 clause:

I M, w � 〈ϕ〉ψ iff M, w � ϕ and M|ϕ, w � ψ.

Big Idea: we evaluate [ϕ]ψ and 〈ϕ〉ψ not by looking at otherworlds in the same model, but rather by looking at a new model.

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 35

Basic Epistemic Logic

Public Announcement Logic

Read [ϕ]ψ as “after (every) true announcement of ϕ, ψ.”

Read 〈ϕ〉ψ := ¬[ϕ]¬ψ as “after a true announcement of ϕ, ψ.”

Let’s compare. The truth clause for the dynamic operator [ϕ] is:

I M, w � [ϕ]ψ iff M, w � ϕ implies M|ϕ, w � ψ.

So if ϕ is false, [ϕ]ψ is vacuously true.

Here is the 〈ϕ〉 clause:

I M, w � 〈ϕ〉ψ iff M, w � ϕ and M|ϕ, w � ψ.

Big Idea: we evaluate [ϕ]ψ and 〈ϕ〉ψ not by looking at otherworlds in the same model, but rather by looking at a new model.

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 35

Basic Epistemic Logic

Public Announcement Logic

Read [ϕ]ψ as “after (every) true announcement of ϕ, ψ.”

Read 〈ϕ〉ψ := ¬[ϕ]¬ψ as “after a true announcement of ϕ, ψ.”

Let’s compare. The truth clause for the dynamic operator [ϕ] is:

I M, w � [ϕ]ψ iff M, w � ϕ implies M|ϕ, w � ψ.

So if ϕ is false, [ϕ]ψ is vacuously true. Here is the 〈ϕ〉 clause:

I M, w � 〈ϕ〉ψ iff M, w � ϕ and M|ϕ, w � ψ.

Big Idea: we evaluate [ϕ]ψ and 〈ϕ〉ψ not by looking at otherworlds in the same model, but rather by looking at a new model.

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 35

Basic Epistemic Logic

Public Announcement Logic

Read [ϕ]ψ as “after (every) true announcement of ϕ, ψ.”

Read 〈ϕ〉ψ := ¬[ϕ]¬ψ as “after a true announcement of ϕ, ψ.”

Let’s compare. The truth clause for the dynamic operator [ϕ] is:

I M, w � [ϕ]ψ iff M, w � ϕ implies M|ϕ, w � ψ.

So if ϕ is false, [ϕ]ψ is vacuously true. Here is the 〈ϕ〉 clause:

I M, w � 〈ϕ〉ψ iff M, w � ϕ and M|ϕ, w � ψ.

Big Idea: we evaluate [ϕ]ψ and 〈ϕ〉ψ not by looking at otherworlds in the same model, but rather by looking at a new model.

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 35

Basic Epistemic Logic

DDD

C DD

DDC

DCD

C DC

C C CC C D

DCC

Where A1 := Dr ∨Dg ∨Db, check M, C C D � 〈A1〉KbDb.

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 36

Basic Epistemic Logic

DDD

C DD

DDC

DCD

C DC

C C CC C D

DCC

Where A1 := Dr ∨Dg ∨Db, check M, C C D � 〈A1〉KbDb.A1 is Mom’s announcement: “At least one of you is dirty.”

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 36

Basic Epistemic Logic

DDD

C DD

DDC

DCD

C DC

C C CC C D

DCC

Where A1 := Dr ∨Dg ∨Db, check M, C C D � 〈A1〉KbDb.

First, observe that M, C C D � Dr ∨Dg ∨Db.

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 36

Basic Epistemic Logic

DDD

C DD

DDC

DCD

C DC

C C CC C D

DCC

Where A1 := Dr ∨Dg ∨Db, check M, C C D � 〈A1〉KbDb.

Second, delete all worlds where Dr ∨Dg ∨Db is false.

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 36

Basic Epistemic Logic

DDD

C DD

DDC

DCD

C DC

C C CC C D

DCC

Where A1 := Dr ∨Dg ∨Db, check M, C C D � 〈A1〉KbDb.

Second, delete all worlds where Dr ∨Dg ∨Db is false.

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 36

Basic Epistemic Logic

DDD

C DD

DDC

DCD

C DC

C C D

DCC

Where A1 := Dr ∨Dg ∨Db, check M, C C D � 〈A1〉KbDb.

Finally, observe that M|A1, C C D � KbDb.

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 36

Basic Epistemic Logic

DDD

C DD

DDC

DCD

C DC

C C D

DCC

Where A2 :=∧

a∈{r ,g ,b}¬(KaDa ∨Ka¬Da), representing that no one

steps forward, check M|A1, C DD � 〈A2〉(KgDg ∧KbDb).

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 36

Basic Epistemic Logic

DDD

C DD

DDC

DCD

C DC

C C D

DCC

M|A1, C DD � 〈 ∧

a∈{r ,g ,b}¬(KaDa ∨Ka¬Da)〉(KgDg ∧KbDb).

First, note M|A1, C DD �

∧a∈{r ,g ,b}

¬(KaDa ∨Ka¬Da).

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 36

Basic Epistemic Logic

DDD

C DD

DDC

DCD

C DC

C C D

DCC

M|A1, C DD � 〈 ∧

a∈{r ,g ,b}¬(KaDa ∨Ka¬Da)〉(KgDg ∧KbDb).

Second, delete all worlds where∧

a∈{r ,g ,b}¬(KaDa ∨Ka¬Da) is false.

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 36

Basic Epistemic Logic

DDD

C DD

DDC

DCD

C DC

C C D

DCC

M|A1, C DD � 〈 ∧

a∈{r ,g ,b}¬(KaDa ∨Ka¬Da)〉(KgDg ∧KbDb).

Second, delete all worlds where∧

a∈{r ,g ,b}¬(KaDa ∨Ka¬Da) is false.

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 36

Basic Epistemic Logic

DDD

C DD

DDC

DCD

M|A1, C DD � 〈 ∧

a∈{r ,g ,b}¬(KaDa ∨Ka¬Da)〉(KgDg ∧KbDb).

Finally, observe M|A1|∧

a∈{r ,g ,b}¬(KaDa∨Ka¬Da), C DD � KgDg ∧KbDb.

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 36

Basic Epistemic Logic

DDD

C DD

DDC

DCD

Check M|A1|A2, C DD � 〈 ∧

a∈{g ,b}(KaDa ∨Ka¬Da)〉Kr¬Dr .

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 36

Basic Epistemic Logic

DDD

C DD

DDC

DCD

Check M|A1|A2, C DD � 〈 ∧

a∈{g ,b}(KaDa ∨Ka¬Da)〉Kr¬Dr .

This represents the info. given when g and b step forward afterMom says again, “If you know whether you’re dirty, step forward.”

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 36

Basic Epistemic Logic

DDD

C DD

DDC

DCD

Check M|A1|A2, C DD � 〈 ∧

a∈{g ,b}(KaDa ∨Ka¬Da)〉Kr¬Dr .

First, observe M|A1|A2, C DD �

∧a∈{g ,b}

(KaDa ∨Ka¬Da).

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 36

Basic Epistemic Logic

DDD

C DD

DDC

DCD

Check M|A1|A2, C DD � 〈 ∧

a∈{g ,b}(KaDa ∨Ka¬Da)〉Kr¬Dr .

Second, delete all worlds where∧

a∈{g ,b}(KaDa ∨Ka¬Da) is false.

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 36

Basic Epistemic Logic

DDD

C DD

DDC

DCD

Check M|A1|A2, C DD � 〈 ∧

a∈{g ,b}(KaDa ∨Ka¬Da)〉Kr¬Dr .

Second, delete all worlds where∧

a∈{g ,b}(KaDa ∨Ka¬Da) is false.

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 36

Basic Epistemic Logic

C DD

Check M|A1|A2, C DD � 〈 ∧

a∈{g ,b}(KaDa ∨Ka¬Da)〉Kr¬Dr .

Finally, observe M|A1|A2|∧

a∈{g ,b}(KaDa∨Ka¬Da), C DD � Kr¬Dr .

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 36

Basic Epistemic Logic

Completeness and Complexity

Theorem (PAL Axiomatization (Plaza 1989))

The LκPAL-theory of the class of all (resp. reflexive, preorder,

partition) models is axiomatized by Kω (resp. Tω, S4ω, S5ω)plus:

i. (replacement)ψ↔ χ

ϕ(ψ/p)↔ ϕ(χ/p)

ii. (atomic reduction) 〈ϕ〉p ↔ (ϕ ∧ p)

iii. (negation reduction) 〈ϕ〉¬ψ↔ (ϕ ∧ ¬〈ϕ〉ψ)iv. (conjunction reduction) 〈ϕ〉(ψ ∧ χ)↔ (〈ϕ〉ψ ∧ 〈ϕ〉χ)v. (diamond reduction) 〈ϕ〉Kaψ↔ (ϕ ∧ Ka〈ϕ〉ψ).

Complexity of satisfiability for PAL is the same as for EL.

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 37

Mathematical Themes

With the background of basic epistemic logic in place, let’s turn tomathematical themes arising in recent work in epistemic logic:

I Preservation Theorems

I Uniform Substitution

I Relations to FOL

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 38

Preservation Theorems

Preservation of Truth (in Inquiry)

Definition (Preservation Under Submodels)

A formula ϕ is preserved under submodels iff for any pointedmodel M, w and M′ ⊆M: M, w � ϕ⇒M′, w � ϕ.

Given our epistemic understanding of the models, this kind ofpreservation takes on a new meaning: once true, ϕ remains truehenceforth in inquiry, no matter what new knowledge is acquired.

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 39

Preservation Theorems

Preservation of Truth (in Inquiry)

Definition (Preservation Under Submodels)

A formula ϕ is preserved under submodels iff for any pointedmodel M, w and M′ ⊆M: M, w � ϕ⇒M′, w � ϕ.

Given our epistemic understanding of the models, this kind ofpreservation takes on a new meaning: once true, ϕ remains truehenceforth in inquiry, no matter what new knowledge is acquired.

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 39

Preservation Theorems

Definition (Preservation Under Submodels)

A formula ϕ is preserved under submodels iff for any pointedmodel M, w and M′ ⊆M: M, w � ϕ⇒M′, w � ϕ.

Definition (Universal Fragment)

The universal fragment of LκEL is generated by:

ϕ ::= p | ¬p | (ϕ ∧ ϕ) | (ϕ ∨ ϕ) | Kaϕ

for p ∈ At and a ∈ Agt.

Theorem (Modal Los-Tarski)

A formula ϕ ∈ LκEL is preserved under submodels iff it is

equivalent to a formula in the universal fragment of LκEL.

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 40

Preservation Theorems

Definition (Preservation Under Submodels)

A formula ϕ is preserved under submodels iff for any pointedmodel M, w and M′ ⊆M: M, w � ϕ⇒M′, w � ϕ.

Definition (Universal Fragment)

The universal fragment of LκEL is generated by:

ϕ ::= p | ¬p | (ϕ ∧ ϕ) | (ϕ ∨ ϕ) | Kaϕ

for p ∈ At and a ∈ Agt.

Theorem (Modal Los-Tarski)

A formula ϕ ∈ LκEL is preserved under submodels iff it is

equivalent to a formula in the universal fragment of LκEL.

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 40

Preservation Theorems

Definition (Preservation Under Submodels)

A formula ϕ is preserved under submodels iff for any pointedmodel M, w and M′ ⊆M: M, w � ϕ⇒M′, w � ϕ.

Definition (Universal Fragment)

The universal fragment of LκPAL-C is generated by:

ϕ ::= p | ¬p | (ϕ ∧ ϕ) | (ϕ ∨ ϕ) | Kaϕ | CAϕ | [¬ϕ]ϕ

for p ∈ At and a ∈ Agt.

Theorem (van Ditmarsch, Kooi)

A formula ϕ ∈ LκPAL-C is preserved under submodels if it is

equivalent to a formula in the universal fragment of LκPAL-C.

Open Problem 1: does ‘only if’ hold as well?

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 41

Preservation Theorems

Definition (Preservation Under Submodels)

A formula ϕ is preserved under submodels iff for any pointedmodel M, w and M′ ⊆M: M, w � ϕ⇒M′, w � ϕ.

Definition (Universal Fragment)

The universal fragment of LκPAL-C is generated by:

ϕ ::= p | ¬p | (ϕ ∧ ϕ) | (ϕ ∨ ϕ) | Kaϕ | CAϕ | [¬ϕ]ϕ

for p ∈ At and a ∈ Agt.

Theorem (van Ditmarsch, Kooi)

A formula ϕ ∈ LκPAL-C is preserved under submodels if it is

equivalent to a formula in the universal fragment of LκPAL-C.

Open Problem 1: does ‘only if’ hold as well?

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 41

Preservation Theorems

Definition (Preservation Under Submodels)

A formula ϕ is preserved under submodels iff for any pointedmodel M, w and M′ ⊆M: M, w � ϕ⇒M′, w � ϕ.

Definition (Universal Fragment)

The universal fragment of LκPAL-C is generated by:

ϕ ::= p | ¬p | (ϕ ∧ ϕ) | (ϕ ∨ ϕ) | Kaϕ | CAϕ | [¬ϕ]ϕ

for p ∈ At and a ∈ Agt.

Theorem (van Ditmarsch, Kooi)

A formula ϕ ∈ LκPAL-C is preserved under submodels if it is

equivalent to a formula in the universal fragment of LκPAL-C.

Open Problem 1: does ‘only if’ hold as well?

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 41

Preservation Theorems

Definition (Preservation Under Submodels)

A formula ϕ is preserved under submodels iff for any pointedmodel M, w and M′ ⊆M: M, w � ϕ⇒M′, w � ϕ.

Open Problem 1: Which formulas of LEL-C and LPAL-C (withcommon knowledge), syntactically characterized, are preserved?

(A place to start might be the proof of Los-Tarski for theµ-calculus (D’Agostino, 1997), of which LEL-C is a fragment.)

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 42

Preservation Theorems

Definition (Successful Formulas)

A formula ϕ is successful iff for all pointed models M, w ,

M, w � ϕ⇒M|ϕ, w � ϕ.

Successful formulas are preserved under “self-defined” submodels.

Not all formulas are successful. Some formulas are unsuccessful .

The simplest example: the Copenhagen agent c calls the Berkeleyagent b on the phone and says “You don’t know it’s raining inCopenhagen, but it’s raining in Copenhagen” (¬Kbr ∧ r).

r

w1 w2

bb, c b, c

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 43

Preservation Theorems

Definition (Successful Formulas)

A formula ϕ is successful iff for all pointed models M, w ,

M, w � ϕ⇒M|ϕ, w � ϕ.

Successful formulas are preserved under “self-defined” submodels.

Not all formulas are successful. Some formulas are unsuccessful .

The simplest example: the Copenhagen agent c calls the Berkeleyagent b on the phone and says “You don’t know it’s raining inCopenhagen, but it’s raining in Copenhagen” (¬Kbr ∧ r).

r

w1 w2

bb, c b, c

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 43

Preservation Theorems

Definition (Successful Formulas)

A formula ϕ is successful iff for all pointed models M, w ,

M, w � ϕ⇒M|ϕ, w � ϕ.

Successful formulas are preserved under “self-defined” submodels.

Not all formulas are successful. Some formulas are unsuccessful .

The simplest example: the Copenhagen agent c calls the Berkeleyagent b on the phone and says “You don’t know it’s raining inCopenhagen, but it’s raining in Copenhagen” (¬Kbr ∧ r).

r

w1 w2

bb, c b, c

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 43

Preservation Theorems

Definition (Successful Formulas)

A formula ϕ is successful iff for all pointed models M, w ,

M, w � ϕ⇒M|ϕ, w � ϕ.

Successful formulas are preserved under “self-defined” submodels.

Not all formulas are successful. Some formulas are unsuccessful .

The simplest example: the Copenhagen agent c calls the Berkeleyagent b on the phone and says “You don’t know it’s raining inCopenhagen, but it’s raining in Copenhagen” (¬Kbr ∧ r).

r

w1 w2

bb, c b, c

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 43

Preservation Theorems

Definition (Successful Formulas)

A formula ϕ is successful iff for all pointed models M, w ,

M, w � ϕ⇒M|ϕ, w � ϕ.

Successful formulas are preserved under “self-defined” submodels.

Not all formulas are successful. Some formulas are unsuccessful .

The simplest example: the Copenhagen agent c calls the Berkeleyagent b on the phone and says “You don’t know it’s raining inCopenhagen, but it’s raining in Copenhagen” (¬Kbr ∧ r).

r

w1

b, c

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 44

Preservation Theorems

Let’s go back to the first time that no students step forward. Thisis like their announcing, “We don’t know whether we are dirty.”

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 45

Preservation Theorems

DDD

C DD

DDC

DCD

C DC

C C D

DCC

M|A1, C DD � 〈 ∧

a∈{r ,g ,b}¬(KaDa ∨Ka¬Da)〉(KgDg ∧KbDb).

This is like their saying “We don’t know whether we’re dirty.”

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 46

Preservation Theorems

DDD

C DD

DDC

DCD

C DC

C C D

DCC

M|A1, C DD � 〈 ∧

a∈{r ,g ,b}¬(KaDa ∨Ka¬Da)〉(KgDg ∧KbDb).

First, note M, C DD �∧

a∈{r ,g ,b}¬(KaDa ∨Ka¬Da).

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 47

Preservation Theorems

DDD

C DD

DDC

DCD

C DC

C C D

DCC

M|A1, C DD � 〈 ∧

a∈{r ,g ,b}¬(KaDa ∨Ka¬Da)〉(KgDg ∧KbDb).

Second, delete all worlds where∧

a∈{r ,g ,b}¬(KaDa ∨Ka¬Da) is false.

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 48

Preservation Theorems

DDD

C DD

DDC

DCD

C DC

C C D

DCC

M|A1, C DD � 〈 ∧

a∈{r ,g ,b}¬(KaDa ∨Ka¬Da)〉(KgDg ∧KbDb).

Second, delete all worlds where∧

a∈{r ,g ,b}¬(KaDa ∨Ka¬Da) is false.

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 49

Preservation Theorems

DDD

C DD

DDC

DCD

M|A1, C DD � 〈 ∧

a∈{r ,g ,b}¬(KaDa ∨Ka¬Da)〉(KgDg ∧KbDb).

Finally, observe M|A1|∧

a∈{r ,g ,b}¬(KaDa∨Ka¬Da), C DD � KgDg ∧KbDb.

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 50

Preservation Theorems

DDD

C DD

DDC

DCD

Finally, observe M|A1|∧

a∈{r ,g ,b}¬(KaDa∨Ka¬Da), C DD � KgDg ∧KbDb.

Hence where ϕ :=∧

a∈{r ,g ,b}¬(KaDa ∨Ka¬Da),

M|A1, C DD � 〈ϕ〉¬ϕ.

The childrens’ true announcement that no one knows makes itfalse that no one knows! This is a so-called unsuccessful update.

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 51

Preservation Theorems

DDD

C DD

DDC

DCD

Finally, observe M|A1|∧

a∈{r ,g ,b}¬(KaDa∨Ka¬Da), C DD � KgDg ∧KbDb.

Hence where ϕ :=∧

a∈{r ,g ,b}¬(KaDa ∨Ka¬Da),

M|A1, C DD � 〈ϕ〉¬ϕ.

The childrens’ true announcement that no one knows makes itfalse that no one knows! This is a so-called unsuccessful update.

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 51

Preservation Theorems

Definition (Successful Formulas)

A formula ϕ is successful iff for all pointed models M, w ,

M, w � ϕ⇒M|ϕ, w � ϕ.

Successful formulas are preserved under “self-defined” submodels.

Not all formulas are successful. Some formulas are unsuccessful.

Example (Unsuccessful Formulas)

We’ve seen two examples so far:

I ¬Kbr ∧ r

I∧

a∈{r ,g ,b}¬(KaDa ∨Ka¬Da)

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 52

Preservation Theorems

Definition (Successful Formulas)

A formula ϕ is successful over a class of models C iff for allpointed models M, w with M in C,

M, w � ϕ⇒M|ϕ, w � ϕ.

QuestionWhich formulas of Lκ

EL, syntactically characterized, are successfulover the class of all models? The class of partition models?

Proposition (Complexity of the Success Problem)

The problem of deciding whether a formula ϕ ∈ L1EL is successful

over the class of partition models is co-NP complete.

So we cannot expect a very simple syntactic characterization.

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 53

Preservation Theorems

Definition (Successful Formulas)

A formula ϕ is successful over a class of models C iff for allpointed models M, w with M in C,

M, w � ϕ⇒M|ϕ, w � ϕ.

QuestionWhich formulas of Lκ

EL, syntactically characterized, are successfulover the class of all models? The class of partition models?

Proposition (Complexity of the Success Problem)

The problem of deciding whether a formula ϕ ∈ L1EL is successful

over the class of partition models is co-NP complete.

So we cannot expect a very simple syntactic characterization.

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 53

Preservation Theorems

Definition (Successful Formulas)

A formula ϕ is successful over a class of models C iff for allpointed models M, w with M in C,

M, w � ϕ⇒M|ϕ, w � ϕ.

QuestionWhich formulas of Lκ

EL, syntactically characterized, are successfulover the class of all models? The class of partition models?

Proposition (Complexity of the Success Problem)

The problem of deciding whether a formula ϕ ∈ L1EL is successful

over the class of partition models is co-NP complete.

So we cannot expect a very simple syntactic characterization.

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 53

Preservation Theorems

QuestionWhich formulas of Lκ

EL, syntactically characterized, are successfulover the class of all models? The class of partition models?

Q2 of van Benthem’s “Open Problems in Logical Dynamics.”

For the case κ = 1 with partition models, this was answered in:

Wesley Holliday and Thomas Icard. 2010.

“Moorean Phenomena in Epistemic Logic.” Advances in Modal Logic 8.

In essence, we show that all unsuccessful formulas involvevariations on the Moorean theme of “you don’t know it, but p.”

Open Problem 2: answer the question for κ > 1 or fornon-partition models (cf. Saraf and Sourabh for some leads).

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 54

Preservation Theorems

QuestionWhich formulas of Lκ

EL, syntactically characterized, are successfulover the class of all models? The class of partition models?

Q2 of van Benthem’s “Open Problems in Logical Dynamics.”

For the case κ = 1 with partition models, this was answered in:

Wesley Holliday and Thomas Icard. 2010.

“Moorean Phenomena in Epistemic Logic.” Advances in Modal Logic 8.

In essence, we show that all unsuccessful formulas involvevariations on the Moorean theme of “you don’t know it, but p.”

Open Problem 2: answer the question for κ > 1 or fornon-partition models (cf. Saraf and Sourabh for some leads).

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 54

Preservation Theorems

QuestionWhich formulas of Lκ

EL, syntactically characterized, are successfulover the class of all models? The class of partition models?

Q2 of van Benthem’s “Open Problems in Logical Dynamics.”

For the case κ = 1 with partition models, this was answered in:

Wesley Holliday and Thomas Icard. 2010.

“Moorean Phenomena in Epistemic Logic.” Advances in Modal Logic 8.

In essence, we show that all unsuccessful formulas involvevariations on the Moorean theme of “you don’t know it, but p.”

Open Problem 2: answer the question for κ > 1 or fornon-partition models (cf. Saraf and Sourabh for some leads).

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 54

Preservation Theorems

QuestionWhich formulas of Lκ

EL, syntactically characterized, are successfulover the class of all models? The class of partition models?

Q2 of van Benthem’s “Open Problems in Logical Dynamics.”

For the case κ = 1 with partition models, this was answered in:

Wesley Holliday and Thomas Icard. 2010.

“Moorean Phenomena in Epistemic Logic.” Advances in Modal Logic 8.

In essence, we show that all unsuccessful formulas involvevariations on the Moorean theme of “you don’t know it, but p.”

Open Problem 2: answer the question for κ > 1 or fornon-partition models (cf. Saraf and Sourabh for some leads).

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 54

Preservation Theorems

QuestionWhich formulas of Lκ

EL, syntactically characterized, are successfulover the class of all models? The class of partition models?

Q2 of van Benthem’s “Open Problems in Logical Dynamics.”

For the case κ = 1 with partition models, this was answered in:

Wesley Holliday and Thomas Icard. 2010.

“Moorean Phenomena in Epistemic Logic.” Advances in Modal Logic 8.

In essence, we show that all unsuccessful formulas involvevariations on the Moorean theme of “you don’t know it, but p.”

Open Problem 2: answer the question for κ > 1 or fornon-partition models (cf. Saraf and Sourabh for some leads).

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 54

Preservation Theorems

Other Interesting Formula Classes

Definition (Self-Refuting)

A formula ϕ is self-refuting iff for all pointed models M, w ,

M, w � ϕ⇒M|ϕ, w 2 ϕ.

Definition (Eventually Self-Refuting)

Define M|nϕ recursively by M|0ϕ =M, M|n+1ϕ =(M|nϕ

)|ϕ

.

ϕ is eventually self-refuting iff for all pointed models M, w :

M, w � ϕ⇒ ∃n :M|nϕ, w 2 ϕ.

Definition (Always Learnable)

ϕ is always learnable (for a) iff for any pointed model M, w :

M, w � ϕ⇒ ∃ψ :M|ψ, w � Kaϕ.

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 55

Preservation Theorems

Other Interesting Formula Classes

Definition (Self-Refuting)

A formula ϕ is self-refuting iff for all pointed models M, w ,

M, w � ϕ⇒M|ϕ, w 2 ϕ.

Definition (Eventually Self-Refuting)

Define M|nϕ recursively by M|0ϕ =M, M|n+1ϕ =(M|nϕ

)|ϕ

.

ϕ is eventually self-refuting iff for all pointed models M, w :

M, w � ϕ⇒ ∃n :M|nϕ, w 2 ϕ.

Definition (Always Learnable)

ϕ is always learnable (for a) iff for any pointed model M, w :

M, w � ϕ⇒ ∃ψ :M|ψ, w � Kaϕ.

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 55

Preservation Theorems

Other Interesting Formula Classes

Definition (Self-Refuting)

A formula ϕ is self-refuting iff for all pointed models M, w ,

M, w � ϕ⇒M|ϕ, w 2 ϕ.

Definition (Eventually Self-Refuting)

Define M|nϕ recursively by M|0ϕ =M, M|n+1ϕ =(M|nϕ

)|ϕ

.

ϕ is eventually self-refuting iff for all pointed models M, w :

M, w � ϕ⇒ ∃n :M|nϕ, w 2 ϕ.

Definition (Always Learnable)

ϕ is always learnable (for a) iff for any pointed model M, w :

M, w � ϕ⇒ ∃ψ :M|ψ, w � Kaϕ.

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 55

Uniform Substitution

Substitution Closure

A striking feature of epistemic logic with operators for informationupdate is that it is not closed under uniform substitution: e.g.,[p]p is valid, but [¬Kbr ∧ r ]([¬Kbr ∧ r) is not valid, as we saw.

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 56

Uniform Substitution

Definition (Substitution Core & Schematic Validity)

A substitution is any function σ : At→ L; and we define(·)σ : L → L as the extension such that (ϕ)σ is obtained from ϕ byreplacing, for all p ∈ At(ϕ), each occurrence of p in ϕ by σ(p).

The substitution core of logic L is the set

{ϕ ∈ LL | (ϕ)σ ∈ L for all substitutions σ}.

Formulas in the substitution core are schematically valid.

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 57

Uniform Substitution

Definition (Substitution Core & Schematic Validity)

A substitution is any function σ : At→ L; and we define(·)σ : L → L as the extension such that (ϕ)σ is obtained from ϕ byreplacing, for all p ∈ At(ϕ), each occurrence of p in ϕ by σ(p).

The substitution core of logic L is the set

{ϕ ∈ LL | (ϕ)σ ∈ L for all substitutions σ}.

Formulas in the substitution core are schematically valid.

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 57

Uniform Substitution

Example (Valid but not Schematically Valid)

Where C is the class of all models, formulas that are in ThLκPAL

(C )but are not in the substitution core of ThLκ

PAL(C ) include (κ ≥ 1):

[p]p Kap → [p]Kap[p]Kap Kap → [p](p → Kap)[p](p → Kap) Ka(p → q)→ (〈q〉Kar → 〈p〉Kar)[p ∧ ¬Kap]¬(p ∧ ¬Kap) (〈p〉Kar ∧ 〈q〉Kar)→ 〈p ∨ q〉Kar .

So what are the uniform principles of information dynamics?

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 58

Uniform Substitution

Example (Valid but not Schematically Valid)

Where C is the class of all models, formulas that are in ThLκPAL

(C )but are not in the substitution core of ThLκ

PAL(C ) include (κ ≥ 1):

[p]p Kap → [p]Kap[p]Kap Kap → [p](p → Kap)[p](p → Kap) Ka(p → q)→ (〈q〉Kar → 〈p〉Kar)[p ∧ ¬Kap]¬(p ∧ ¬Kap) (〈p〉Kar ∧ 〈q〉Kar)→ 〈p ∨ q〉Kar .

So what are the uniform principles of information dynamics?

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 58

Uniform Substitution

QuestionIs the substitution core of PAL axiomatizable?

Q1 in van Benthem’s “Open Problems in Logical Dynamics.”

The answer is given in:

Wesley Holliday, Tomohiro Hoshi, and Thomas Icard. 2012.

“A Uniform Logic of Information Dynamics.” Advances in Modal Logic 9.

Theorem (Axiomatization)

For the class C of all (resp. reflexive, preorder, partition) ω-agentmodels, the substitution core of ThLω

PAL(C ) is axiomatizable.

Upshot: we have a logic for reasoning about information change inwhich the atomic sentences p can be genuine propositionalvariables, standing in for any propositions, even epistemic ones.

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 59

Uniform Substitution

QuestionIs the substitution core of PAL axiomatizable?

Q1 in van Benthem’s “Open Problems in Logical Dynamics.”

The answer is given in:

Wesley Holliday, Tomohiro Hoshi, and Thomas Icard. 2012.

“A Uniform Logic of Information Dynamics.” Advances in Modal Logic 9.

Theorem (Axiomatization)

For the class C of all (resp. reflexive, preorder, partition) ω-agentmodels, the substitution core of ThLω

PAL(C ) is axiomatizable.

Upshot: we have a logic for reasoning about information change inwhich the atomic sentences p can be genuine propositionalvariables, standing in for any propositions, even epistemic ones.

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 59

Uniform Substitution

QuestionIs the substitution core of PAL axiomatizable?

Q1 in van Benthem’s “Open Problems in Logical Dynamics.”

The answer is given in:

Wesley Holliday, Tomohiro Hoshi, and Thomas Icard. 2012.

“A Uniform Logic of Information Dynamics.” Advances in Modal Logic 9.

Theorem (Axiomatization)

For the class C of all (resp. reflexive, preorder, partition) ω-agentmodels, the substitution core of ThLω

PAL(C ) is axiomatizable.

Upshot: we have a logic for reasoning about information change inwhich the atomic sentences p can be genuine propositionalvariables, standing in for any propositions, even epistemic ones.

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 59

Uniform Substitution

The axiomatization is obtained by adding to the normal modallogic Kω (or Tω, S4ω, S5ω depending on the model class):

1. (uniformity)ϕ

(ϕ)σ for any substitution σ

2. (necessitation)ϕ

[p]ϕ

3. (extensionality)ϕ↔ ψ

〈ϕ〉p ↔ 〈ψ〉p4. (distribution) [p](q → r)→ ([p]q → [p]r)

5. (p-seriality) p → 〈p〉>6. (truthfulness) 〈p〉> → p

7. (>-reflexivity) p → 〈>〉p8. (functionality) 〈p〉q → [p]q

9. (pa-commutativity) 〈p〉Kaq → Ka〈p〉q10. (ap-commutativity) Ka〈p〉q → [p]Kaq

11. (composition) 〈p〉〈q〉r ↔ 〈〈p〉q〉r

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 60

Uniform Substitution

The axiomatization is obtained by adding to the normal modallogic Kω (or Tω, S4ω, S5ω depending on the model class):

1. (uniformity)ϕ

(ϕ)σ for any substitution σ

2. (necessitation)ϕ

[p]ϕ

3. (extensionality)ϕ↔ ψ

〈ϕ〉p ↔ 〈ψ〉p4. (distribution) [p](q → r)→ ([p]q → [p]r)

5. (p-seriality) p → 〈p〉>6. (truthfulness) 〈p〉> → p

7. (>-reflexivity) p → 〈>〉p8. (functionality) 〈p〉q → [p]q

9. (pa-commutativity) 〈p〉Kaq → Ka〈p〉q10. (ap-commutativity) Ka〈p〉q → [p]Kaq

11. (composition) 〈p〉〈q〉r ↔ 〈〈p〉q〉r

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 60

Uniform Substitution

The axiomatization is obtained by adding to the normal modallogic Kω (or Tω, S4ω, S5ω depending on the model class):

1. (uniformity)ϕ

(ϕ)σ for any substitution σ

2. (necessitation)ϕ

[p]ϕ

3. (extensionality)ϕ↔ ψ

〈ϕ〉p ↔ 〈ψ〉p4. (distribution) [p](q → r)→ ([p]q → [p]r)

5. (p-seriality) p → 〈p〉>6. (truthfulness) 〈p〉> → p

7. (>-reflexivity) p → 〈>〉p8. (functionality) 〈p〉q → [p]q

9. (pa-commutativity) 〈p〉Kaq → Ka〈p〉q10. (ap-commutativity) Ka〈p〉q → [p]Kaq

11. (composition) 〈p〉〈q〉r ↔ 〈〈p〉q〉r

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 60

Uniform Substitution

The axiomatization is obtained by adding to the normal modallogic Kω (or Tω, S4ω, S5ω depending on the model class):

1. (uniformity)ϕ

(ϕ)σ for any substitution σ

2. (necessitation)ϕ

[p]ϕ

3. (extensionality)ϕ↔ ψ

〈ϕ〉p ↔ 〈ψ〉p

4. (distribution) [p](q → r)→ ([p]q → [p]r)

5. (p-seriality) p → 〈p〉>6. (truthfulness) 〈p〉> → p

7. (>-reflexivity) p → 〈>〉p8. (functionality) 〈p〉q → [p]q

9. (pa-commutativity) 〈p〉Kaq → Ka〈p〉q10. (ap-commutativity) Ka〈p〉q → [p]Kaq

11. (composition) 〈p〉〈q〉r ↔ 〈〈p〉q〉r

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 60

Uniform Substitution

The axiomatization is obtained by adding to the normal modallogic Kω (or Tω, S4ω, S5ω depending on the model class):

1. (uniformity)ϕ

(ϕ)σ for any substitution σ

2. (necessitation)ϕ

[p]ϕ

3. (extensionality)ϕ↔ ψ

〈ϕ〉p ↔ 〈ψ〉p4. (distribution) [p](q → r)→ ([p]q → [p]r)

5. (p-seriality) p → 〈p〉>6. (truthfulness) 〈p〉> → p

7. (>-reflexivity) p → 〈>〉p8. (functionality) 〈p〉q → [p]q

9. (pa-commutativity) 〈p〉Kaq → Ka〈p〉q10. (ap-commutativity) Ka〈p〉q → [p]Kaq

11. (composition) 〈p〉〈q〉r ↔ 〈〈p〉q〉r

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 60

Uniform Substitution

The axiomatization is obtained by adding to the normal modallogic Kω (or Tω, S4ω, S5ω depending on the model class):

1. (uniformity)ϕ

(ϕ)σ for any substitution σ

2. (necessitation)ϕ

[p]ϕ

3. (extensionality)ϕ↔ ψ

〈ϕ〉p ↔ 〈ψ〉p4. (distribution) [p](q → r)→ ([p]q → [p]r)

5. (p-seriality) p → 〈p〉>

6. (truthfulness) 〈p〉> → p

7. (>-reflexivity) p → 〈>〉p8. (functionality) 〈p〉q → [p]q

9. (pa-commutativity) 〈p〉Kaq → Ka〈p〉q10. (ap-commutativity) Ka〈p〉q → [p]Kaq

11. (composition) 〈p〉〈q〉r ↔ 〈〈p〉q〉r

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 60

Uniform Substitution

The axiomatization is obtained by adding to the normal modallogic Kω (or Tω, S4ω, S5ω depending on the model class):

1. (uniformity)ϕ

(ϕ)σ for any substitution σ

2. (necessitation)ϕ

[p]ϕ

3. (extensionality)ϕ↔ ψ

〈ϕ〉p ↔ 〈ψ〉p4. (distribution) [p](q → r)→ ([p]q → [p]r)

5. (p-seriality) p → 〈p〉>6. (truthfulness) 〈p〉> → p

7. (>-reflexivity) p → 〈>〉p8. (functionality) 〈p〉q → [p]q

9. (pa-commutativity) 〈p〉Kaq → Ka〈p〉q10. (ap-commutativity) Ka〈p〉q → [p]Kaq

11. (composition) 〈p〉〈q〉r ↔ 〈〈p〉q〉r

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 60

Uniform Substitution

The axiomatization is obtained by adding to the normal modallogic Kω (or Tω, S4ω, S5ω depending on the model class):

1. (uniformity)ϕ

(ϕ)σ for any substitution σ

2. (necessitation)ϕ

[p]ϕ

3. (extensionality)ϕ↔ ψ

〈ϕ〉p ↔ 〈ψ〉p4. (distribution) [p](q → r)→ ([p]q → [p]r)

5. (p-seriality) p → 〈p〉>6. (truthfulness) 〈p〉> → p

7. (>-reflexivity) p → 〈>〉p

8. (functionality) 〈p〉q → [p]q

9. (pa-commutativity) 〈p〉Kaq → Ka〈p〉q10. (ap-commutativity) Ka〈p〉q → [p]Kaq

11. (composition) 〈p〉〈q〉r ↔ 〈〈p〉q〉r

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 60

Uniform Substitution

The axiomatization is obtained by adding to the normal modallogic Kω (or Tω, S4ω, S5ω depending on the model class):

1. (uniformity)ϕ

(ϕ)σ for any substitution σ

2. (necessitation)ϕ

[p]ϕ

3. (extensionality)ϕ↔ ψ

〈ϕ〉p ↔ 〈ψ〉p4. (distribution) [p](q → r)→ ([p]q → [p]r)

5. (p-seriality) p → 〈p〉>6. (truthfulness) 〈p〉> → p

7. (>-reflexivity) p → 〈>〉p8. (functionality) 〈p〉q → [p]q

9. (pa-commutativity) 〈p〉Kaq → Ka〈p〉q10. (ap-commutativity) Ka〈p〉q → [p]Kaq

11. (composition) 〈p〉〈q〉r ↔ 〈〈p〉q〉r

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 60

Uniform Substitution

The axiomatization is obtained by adding to the normal modallogic Kω (or Tω, S4ω, S5ω depending on the model class):

1. (uniformity)ϕ

(ϕ)σ for any substitution σ

2. (necessitation)ϕ

[p]ϕ

3. (extensionality)ϕ↔ ψ

〈ϕ〉p ↔ 〈ψ〉p4. (distribution) [p](q → r)→ ([p]q → [p]r)

5. (p-seriality) p → 〈p〉>6. (truthfulness) 〈p〉> → p

7. (>-reflexivity) p → 〈>〉p8. (functionality) 〈p〉q → [p]q

9. (pa-commutativity) 〈p〉Kaq → Ka〈p〉q

10. (ap-commutativity) Ka〈p〉q → [p]Kaq

11. (composition) 〈p〉〈q〉r ↔ 〈〈p〉q〉r

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 60

Uniform Substitution

The axiomatization is obtained by adding to the normal modallogic Kω (or Tω, S4ω, S5ω depending on the model class):

1. (uniformity)ϕ

(ϕ)σ for any substitution σ

2. (necessitation)ϕ

[p]ϕ

3. (extensionality)ϕ↔ ψ

〈ϕ〉p ↔ 〈ψ〉p4. (distribution) [p](q → r)→ ([p]q → [p]r)

5. (p-seriality) p → 〈p〉>6. (truthfulness) 〈p〉> → p

7. (>-reflexivity) p → 〈>〉p8. (functionality) 〈p〉q → [p]q

9. (pa-commutativity) 〈p〉Kaq → Ka〈p〉q10. (ap-commutativity) Ka〈p〉q → [p]Kaq

11. (composition) 〈p〉〈q〉r ↔ 〈〈p〉q〉r

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 60

Uniform Substitution

The axiomatization is obtained by adding to the normal modallogic Kω (or Tω, S4ω, S5ω depending on the model class):

1. (uniformity)ϕ

(ϕ)σ for any substitution σ

2. (necessitation)ϕ

[p]ϕ

3. (extensionality)ϕ↔ ψ

〈ϕ〉p ↔ 〈ψ〉p4. (distribution) [p](q → r)→ ([p]q → [p]r)

5. (p-seriality) p → 〈p〉>6. (truthfulness) 〈p〉> → p

7. (>-reflexivity) p → 〈>〉p8. (functionality) 〈p〉q → [p]q

9. (pa-commutativity) 〈p〉Kaq → Ka〈p〉q10. (ap-commutativity) Ka〈p〉q → [p]Kaq

11. (composition) 〈p〉〈q〉r ↔ 〈〈p〉q〉r

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 60

Uniform Substitution

Theorem (Axiomatization)

For the class C of all (resp. reflexive, preorder, partition) ω-agentmodels, the substitution core of ThLω

PAL(C ) is axiomatizable.

Wesley Holliday, Tomohiro Hoshi, and Thomas Icard. 2012.

“A Uniform Logic of Information Dynamics.” Advances in Modal Logic 9.

Theorem (Decidability)

For the class C of all (resp. reflexive, preorder, partition) ω-agentmodels, the substitution core of ThLω

PAL(C ) is decidable.

Wesley Holliday, Tomohiro Hoshi, and Thomas Icard. Forthcoming.

“Information Dynamics and Uniform Substitution.” Synthese.

Wesley Holliday, Tomohiro Hoshi, and Thomas Icard. 2011.

“Schematic Validity in Dynamic Epistemic Logic: Decidability.” LNAI.

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 61

Uniform Substitution

Theorem (Axiomatization)

For the class C of all (resp. reflexive, preorder, partition) ω-agentmodels, the substitution core of ThLω

PAL(C ) is axiomatizable.

Wesley Holliday, Tomohiro Hoshi, and Thomas Icard. 2012.

“A Uniform Logic of Information Dynamics.” Advances in Modal Logic 9.

Theorem (Decidability)

For the class C of all (resp. reflexive, preorder, partition) ω-agentmodels, the substitution core of ThLω

PAL(C ) is decidable.

Wesley Holliday, Tomohiro Hoshi, and Thomas Icard. Forthcoming.

“Information Dynamics and Uniform Substitution.” Synthese.

Wesley Holliday, Tomohiro Hoshi, and Thomas Icard. 2011.

“Schematic Validity in Dynamic Epistemic Logic: Decidability.” LNAI.

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 61

Uniform Substitution

Theorem (Axiomatization)

For the class C of all (resp. reflexive, preorder, partition) ω-agentmodels, the substitution core of ThLω

PAL(C ) is axiomatizable.

Wesley Holliday, Tomohiro Hoshi, and Thomas Icard. 2012.

“A Uniform Logic of Information Dynamics.” Advances in Modal Logic 9.

Open Problem 3: Does this extend to ThLωPAL-C(C )? Is there a

Uniform Logic of Information Dynamics & Common Knowledge?

(The obstacle is that given the failure of compactness, we mustuse a finite version of the Henkin-style canonical modelconstruction to prove completeness; but the use of the standardFisher-Ladner closure to do so proves difficult in this case...)

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 62

Uniform Substitution

Theorem (Axiomatization)

For the class C of all (resp. reflexive, preorder, partition) ω-agentmodels, the substitution core of ThLω

PAL(C ) is axiomatizable.

Wesley Holliday, Tomohiro Hoshi, and Thomas Icard. 2012.

“A Uniform Logic of Information Dynamics.” Advances in Modal Logic 9.

Open Problem 3: Does this extend to ThLωPAL-C(C )? Is there a

Uniform Logic of Information Dynamics & Common Knowledge?

(The obstacle is that given the failure of compactness, we mustuse a finite version of the Henkin-style canonical modelconstruction to prove completeness; but the use of the standardFisher-Ladner closure to do so proves difficult in this case...)

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 62

Relations to First-Order Logic

Transferring the Questions to FOL

In “Open Problems in Logical Dynamics,” van Benthem poses thequestions we have been discussing for FOL as well.

Open Problem 4: Characterize the formulas ϕ of FOL withexactly one free variable x such that for any FO structure A andvariable assignment s, if A, s � ϕ, then Aϕ, s � ϕ, where Aϕ is thesubstructure of A with domain {d ∈ |A| | A, s[x :=d ] � ϕ}.

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 63

Relations to First-Order Logic

Transferring the Questions to FOL

In “Open Problems in Logical Dynamics,” van Benthem poses thequestions we have been discussing for FOL as well.

Open Problem 5: Extend the language of FOL with relativizationoperators (·)ϕ for all ϕ with exactly one free variable x , such that

A, s � (ψ)ϕ ⇔ if ∀y ∈ FV (ψ)A, s[x :=s(y )] � ϕ, then Aϕ, s � ψ,

where FV (ψ) is the set of free variables in ψ.

What is the complete logic of FOL with relativization?

Since this logic is not closed under uniform substitution (e.g.,Px → (Px)Qz is valid, but (Px ∧ ∃y¬Py)→ (Px ∧ ∃y¬Py)Qz

is not), is the substitution core of this logic axiomatizable?

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 64

Relations to First-Order Logic

Transferring the Questions to FOL

In “Open Problems in Logical Dynamics,” van Benthem poses thequestions we have been discussing for FOL as well.

Open Problem 5: Extend the language of FOL with relativizationoperators (·)ϕ for all ϕ with exactly one free variable x , such that

A, s � (ψ)ϕ ⇔ if ∀y ∈ FV (ψ)A, s[x :=s(y )] � ϕ, then Aϕ, s � ψ,

where FV (ψ) is the set of free variables in ψ.

What is the complete logic of FOL with relativization?

Since this logic is not closed under uniform substitution (e.g.,Px → (Px)Qz is valid, but (Px ∧ ∃y¬Py)→ (Px ∧ ∃y¬Py)Qz

is not), is the substitution core of this logic axiomatizable?

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 64

Relations to First-Order Logic

Translating EL into FOL

Can we transfer our results for modal logic to fragments of FOL?

Definition (Propositional Epistemic Language)

Given sets At = {p, q, r , . . . } and Agt = {a, b, c , . . . } with|Agt| = κ, the epistemic language Lκ

EL is generated by

ϕ ::= p | ¬ϕ | (ϕ ∧ ϕ) | Kaϕ.

Consider a first-order language that contains, for every p ∈ At anda ∈ Agt, a one-place predicate P and a two-place predicate Ra.

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 65

Relations to First-Order Logic

Translating EL into FOL

Can we transfer our results for modal logic to fragments of FOL?

Definition (Propositional Epistemic Language)

Given sets At = {p, q, r , . . . } and Agt = {a, b, c , . . . } with|Agt| = κ, the epistemic language Lκ

EL is generated by

ϕ ::= p | ¬ϕ | (ϕ ∧ ϕ) | Kaϕ.

Consider a first-order language that contains, for every p ∈ At anda ∈ Agt, a one-place predicate P and a two-place predicate Ra.

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 65

Relations to First-Order Logic

Translating EL into FOL

Definition (Standard Translation)

Fix two variables x and y and define functions STx and STy :

I STx (p) = Px ;

I STx (¬ϕ) = ¬STx (ϕ);

I STx (ϕ ∧ ψ) = STx (ϕ) ∧ STx (ψ);

I STx (Kaϕ) = ∀y(Raxy → STy (ϕ))

I STy (p) = Py ;

I STy (¬ϕ) = ¬STy (ϕ);

I STy (ϕ ∧ ψ) = STy (ϕ) ∧ STy (ψ);

I STy (Kaϕ) = ∀y(Raxy → STx (ϕ))

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 66

Relations to First-Order Logic

Translating EL into FOL

Definition (Standard Translation)

Fix two variables x and y and define functions STx and STy :

I STx (p) = Px ;

I STx (¬ϕ) = ¬STx (ϕ);

I STx (ϕ ∧ ψ) = STx (ϕ) ∧ STx (ψ);

I STx (Kaϕ) = ∀y(Raxy → STy (ϕ));

I and similarly for STy .

Any modal model A can be viewed as a first-order structure Asuch that for any ϕ ∈ Lκ

EL, variable x , and assignment s,

A, w � ϕ iff A, s[x :=w ] � STx (ϕ)

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 67

Relations to First-Order Logic

Under the translation, the modal language lives in the two-variablefragment of FOL. It also lives in the “guarded” fragment of FOL,since all the quantifiers have guards. The satisfiability/validityproblem for formulas in both of these fragment is decidable.

Van Benthem’s Characterization Theorem gives a semanticcharacterization of the FOL formulas (with one free variable) thatare equivalent to the standard translation of some modal formula:they are the FOL formulas that are invariant for bisimulation.

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 68

Relations to First-Order Logic

Under the translation, the modal language lives in the two-variablefragment of FOL. It also lives in the “guarded” fragment of FOL,since all the quantifiers have guards. The satisfiability/validityproblem for formulas in both of these fragment is decidable.

Van Benthem’s Characterization Theorem gives a semanticcharacterization of the FOL formulas (with one free variable) thatare equivalent to the standard translation of some modal formula:they are the FOL formulas that are invariant for bisimulation.

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 68

Relations to First-Order Logic

Monadic First-Order Logic

In the other direction, any formula ϕ of Monadic First-Order Logicwith only x free (and not in the scope of any quantifier) can betranslated into a modal formula ϕ′ such that for any MFOstructure A and assignment s, when we view A as a single-agentepistemic partition model A (with Ra the universal relation),

A, s � ϕ iff A, s(x) � ϕ′.

Since we have solved the success problem and substitution coreproblem for single-agent epistemic logic over partition models, wecan transfer these results to the modal-like fragment of MFOL.

But it is a long way from here to answers for full FOL.

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 69

Relations to First-Order Logic

Monadic First-Order Logic

In the other direction, any formula ϕ of Monadic First-Order Logicwith only x free (and not in the scope of any quantifier) can betranslated into a modal formula ϕ′ such that for any MFOstructure A and assignment s, when we view A as a single-agentepistemic partition model A (with Ra the universal relation),

A, s � ϕ iff A, s(x) � ϕ′.

Since we have solved the success problem and substitution coreproblem for single-agent epistemic logic over partition models, wecan transfer these results to the modal-like fragment of MFOL.

But it is a long way from here to answers for full FOL.

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 69

Relations to First-Order Logic

Monadic First-Order Logic

In the other direction, any formula ϕ of Monadic First-Order Logicwith only x free (and not in the scope of any quantifier) can betranslated into a modal formula ϕ′ such that for any MFOstructure A and assignment s, when we view A as a single-agentepistemic partition model A (with Ra the universal relation),

A, s � ϕ iff A, s(x) � ϕ′.

Since we have solved the success problem and substitution coreproblem for single-agent epistemic logic over partition models, wecan transfer these results to the modal-like fragment of MFOL.

But it is a long way from here to answers for full FOL.

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 69

Conclusion

Directions from Here

Going beyond the basic epistemic logic we have covered today:

I Quantified epistemic logic (Ka∃xSpy(x) vs. ∃xKaSpy(x));

I “Softer” belief revision instead of “hard” knowledge update;

I Modeling private communication, eavesdropping, etc.;

I Epistemic logic and probability (qualitative and quantitative);

I Philosophical applications of EL, mentioned at the beginning;

I and more...

Two sources for learning about recents research directions are:

I Advances in Modal Logic, biennial conference and book series.

I European Summer School for Logic, Language, and Information.

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 70

Conclusion

Directions from Here

Going beyond the basic epistemic logic we have covered today:

I Quantified epistemic logic (Ka∃xSpy(x) vs. ∃xKaSpy(x));

I “Softer” belief revision instead of “hard” knowledge update;

I Modeling private communication, eavesdropping, etc.;

I Epistemic logic and probability (qualitative and quantitative);

I Philosophical applications of EL, mentioned at the beginning;

I and more...

Two sources for learning about recents research directions are:

I Advances in Modal Logic, biennial conference and book series.

I European Summer School for Logic, Language, and Information.

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 70

Thank you!

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 71

Bonus Puzzle

Sum and Product

A says to S and P: “I have picked x , y ∈N such that 1 < x < yand x + y ≤ 100. Shortly I will whisper their sum s = x + y to Sonly, and their product p = x · y to P only.” A acts accordingly.The following conversation between S and P then takes place:

1. P: “I don’t know the numbers.”

2. S: “I knew that you didn’t know them.”

3. P: “Now I know the numbers.”

4. S: “Now I know them too.”

Question: what are x and y? (There is a unique answer.)

Wes Holliday: Recent Work in Epistemic Logic, Logic Group Colloquium 72