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Introduction to Logic

Introduction to Logic Francesca Poggiolesi Epistemic Logic 1

April 15, 2008, Paris

Introduction to Logic Introduction to Epistemic Logic

Epistemic logic: what, when, who Epistemic Logic (EL): Logic of Knowledge and Belief

Introduction to Logic Introduction to Epistemic Logic

Epistemic logic: what, when, who Epistemic Logic (EL): Logic of Knowledge and Belief I

from the Greek word πιστ ηµη which means knowledge

Introduction to Logic Introduction to Epistemic Logic

Epistemic logic: what, when, who Epistemic Logic (EL): Logic of Knowledge and Belief I

from the Greek word πιστ ηµη which means knowledge

I

main idea: expressions like “it is known that” or “it is believed that” have systematic properties that are amenable to formal study

Introduction to Logic Introduction to Epistemic Logic

Epistemic logic: what, when, who Epistemic Logic (EL): Logic of Knowledge and Belief I

from the Greek word πιστ ηµη which means knowledge

I

main idea: expressions like “it is known that” or “it is believed that” have systematic properties that are amenable to formal study

I

EL, as we are going to treat it, i.e. within contemporary logic, is the result of the work of many philosophers and logiciens Carnap, Hintikka, Prior, von Wright - started in the early 50’s

Introduction to Logic Introduction to Epistemic Logic

Learning with an example

Let us consider the following argument:

Introduction to Logic Introduction to Epistemic Logic

Learning with an example

Let us consider the following argument: 1. When Paul lies to me, he stammers.

Introduction to Logic Introduction to Epistemic Logic

Learning with an example

Let us consider the following argument: 1. When Paul lies to me, he stammers. 2. When he stammers, I know it.

Introduction to Logic Introduction to Epistemic Logic

Learning with an example

Let us consider the following argument: 1. When Paul lies to me, he stammers. 2. When he stammers, I know it. 3. I (also) know that when Paul stammers, he is lying to me.

Introduction to Logic Introduction to Epistemic Logic

Learning with an example

Let us consider the following argument: 1. When Paul lies to me, he stammers. 2. When he stammers, I know it. 3. I (also) know that when Paul stammers, he is lying to me. 4. Therefore whenever Paul lies to me, I know that he does so.

Introduction to Logic Introduction to Epistemic Logic

Learning with an example

Let us consider the following argument: 1. When Paul lies to me, he stammers. 2. When he stammers, I know it. 3. I (also) know that when Paul stammers, he is lying to me. 4. Therefore whenever Paul lies to me, I know that he does so. Question: how do we formalise this argument?

Introduction to Logic Introduction to Epistemic Logic

Learning with an example

Let us consider the following argument: 1. When Paul lies to me, he stammers. 2. When he stammers, I know it. 3. I (also) know that when Paul stammers, he is lying to me. 4. Therefore whenever Paul lies to me, I know that he does so. Question: why is it a valid argument?

Introduction to Logic Introduction to Epistemic Logic

Remark In the example above, we have introduced the formalisation of the epistemic attitude called Knowledge.

Introduction to Logic Introduction to Epistemic Logic

Remark In the example above, we have introduced the formalisation of the epistemic attitude called Knowledge. On the other hand we have only mentioned “my” knowledge.

Introduction to Logic Introduction to Epistemic Logic

Remark In the example above, we have introduced the formalisation of the epistemic attitude called Knowledge. On the other hand we have only mentioned “my” knowledge. What happens if even Maria knows that when Paul lies to me, he stammers?

Introduction to Logic Introduction to Epistemic Logic

Remark In the example above, we have introduced the formalisation of the epistemic attitude called Knowledge. On the other hand we have only mentioned “my” knowledge. What happens if even Maria knows that when Paul lies to me, he stammers? And if John does not know this fact, but he knows something else concerning Paul? How do we formalise this?

Introduction to Logic Introduction to Epistemic Logic

Remark In the example above, we have introduced the formalisation of the epistemic attitude called Knowledge. On the other hand we have only mentioned “my” knowledge. What happens if even Maria knows that when Paul lies to me, he stammers? And if John does not know this fact, but he knows something else concerning Paul? How do we formalise this? In order to solve this problem, we introduce many agents 1, ..., n and we combine them with the operator K , so that we have: K1 , K2 , ..., Kn

Introduction to Logic Semantics for epistemic logic

Becoming more precise ... multi-modal propositional language

We define the propositional multi-modal language LK n in the following way:

Introduction to Logic Semantics for epistemic logic

Becoming more precise ... multi-modal propositional language

We define the propositional multi-modal language LK n in the following way: I

propositional constants: p0 , p1 , ... (AT)

Introduction to Logic Semantics for epistemic logic

Becoming more precise ... multi-modal propositional language

We define the propositional multi-modal language LK n in the following way: I

propositional constants: p0 , p1 , ... (AT)

I

agents: 1, ..., n. Let A denote the set of n agents {1, ..., n}

Introduction to Logic Semantics for epistemic logic

Becoming more precise ... multi-modal propositional language

We define the propositional multi-modal language LK n in the following way: I

propositional constants: p0 , p1 , ... (AT)

I

agents: 1, ..., n. Let A denote the set of n agents {1, ..., n}

I

connectives: ¬ and ∧

Introduction to Logic Semantics for epistemic logic

Becoming more precise ... multi-modal propositional language

We define the propositional multi-modal language LK n in the following way: I

propositional constants: p0 , p1 , ... (AT)

I

agents: 1, ..., n. Let A denote the set of n agents {1, ..., n}

I

connectives: ¬ and ∧

I

knowledge operator: Ki , for i ∈ A

Introduction to Logic Semantics for epistemic logic

Multi-modal formulas

The set of the well formed formulas of LK n (WF) is inductively defined in the following way:

Introduction to Logic Semantics for epistemic logic

Multi-modal formulas

The set of the well formed formulas of LK n (WF) is inductively defined in the following way: I

if p is a constant, then p ∈ WF

Introduction to Logic Semantics for epistemic logic

Multi-modal formulas

The set of the well formed formulas of LK n (WF) is inductively defined in the following way: I

if p is a constant, then p ∈ WF

I

if α ∈ WF, then ¬α ∈ WF

Introduction to Logic Semantics for epistemic logic

Multi-modal formulas

The set of the well formed formulas of LK n (WF) is inductively defined in the following way: I

if p is a constant, then p ∈ WF

I

if α ∈ WF, then ¬α ∈ WF

I

if α, β ∈ WF, then α ∧ β ∈ WF

Introduction to Logic Semantics for epistemic logic

Multi-modal formulas

The set of the well formed formulas of LK n (WF) is inductively defined in the following way: I

if p is a constant, then p ∈ WF

I

if α ∈ WF, then ¬α ∈ WF

I

if α, β ∈ WF, then α ∧ β ∈ WF

I

if α ∈ WF, then Ki α ∈ WF, for all i ∈ A

Introduction to Logic Semantics for epistemic logic

Epistemic operator(s)

Ki α stands for “agent i knows α”

Introduction to Logic Semantics for epistemic logic

Epistemic operator(s)

Ki α stands for “agent i knows α” Mi α ≡ ¬Ki ¬α, stands for “agent i considers α as possible”

Introduction to Logic Semantics for epistemic logic

Kripke model for multi-modal propositional logic A Kripke model for LK n is a tuple: M = (W , {Ri }i∈A , v )

Introduction to Logic Semantics for epistemic logic

Kripke model for multi-modal propositional logic A Kripke model for LK n is a tuple: M = (W , {Ri }i∈A , v )

W : non-empty set of possible words, epistemic alternatives, ....

Introduction to Logic Semantics for epistemic logic

Kripke model for multi-modal propositional logic A Kripke model for LK n is a tuple: M = (W , {Ri }i∈A , v )

W : non-empty set of possible words, epistemic alternatives, .... Ri : binary relation on W , for each agent i ∈ A

Introduction to Logic Semantics for epistemic logic

Kripke model for multi-modal propositional logic A Kripke model for LK n is a tuple: M = (W , {Ri }i∈A , v )

W : non-empty set of possible words, epistemic alternatives, .... Ri : binary relation on W , for each agent i ∈ A v : AT X W → {0, 1} is a truth assignment to the propositional atoms per state

Introduction to Logic Semantics for epistemic logic

Kripke model for multi-modal propositional logic A Kripke model for LK n is a tuple: M = (W , {Ri }i∈A , v )

W : non-empty set of possible words, epistemic alternatives, .... Ri : binary relation on W , for each agent i ∈ A v : AT X W → {0, 1} is a truth assignment to the propositional atoms per state N.B. Au usual (W , {Ri }i∈A ) is a frame

Introduction to Logic Semantics for epistemic logic

Modal satisfaction

The relation of modal satisfaction, x |=M α, is defined inductively as usual.

Introduction to Logic Semantics for epistemic logic

Modal satisfaction

The relation of modal satisfaction, x |=M α, is defined inductively as usual. Let us focus our attention on: x |=M Ki α iff ∀y (xRi y → y |=M α)

Introduction to Logic Semantics for epistemic logic

Modal satisfaction

The relation of modal satisfaction, x |=M α, is defined inductively as usual. Let us focus our attention on: x |=M Ki α iff ∀y (xRi y → y |=M α) It should be read as: in each world that is compatible with what the agent i knows, it is case that α.

Introduction to Logic Semantics for epistemic logic

If it is still not clear ... Immagine the following situation:

Introduction to Logic Semantics for epistemic logic

If it is still not clear ... Immagine the following situation: A bored student in a class without a view to the outside of the building.

Introduction to Logic Semantics for epistemic logic

If it is still not clear ... Immagine the following situation: A bored student in a class without a view to the outside of the building. He wonder if it is raining.

Introduction to Logic Semantics for epistemic logic

If it is still not clear ... Immagine the following situation: A bored student in a class without a view to the outside of the building. He wonder if it is raining. Of course there are two possibilities (possible words) - he has two epistemic alternatives - one in which it rains, one in which it does not.

Introduction to Logic Semantics for epistemic logic

If it is still not clear ... Immagine the following situation: A bored student in a class without a view to the outside of the building. He wonder if it is raining. Of course there are two possibilities (possible words) - he has two epistemic alternatives - one in which it rains, one in which it does not.

Introduction to Logic Semantics for epistemic logic

If it is still not clear ... Immagine the following situation: A bored student in a class without a view to the outside of the building. He wonder if it is raining. Of course there are two possibilities (possible words) - he has two epistemic alternatives - one in which it rains, one in which it does not. On the other hand, in both these possibilities (possible worlds, epistemic alternatives) it holds that the lecture is boring.

Introduction to Logic Semantics for epistemic logic

If it is still not clear ...

Questions:

Introduction to Logic Semantics for epistemic logic

If it is still not clear ...

Questions: What does he know?

Introduction to Logic Semantics for epistemic logic

If it is still not clear ...

Questions: What does he not know?

Introduction to Logic Semantics for epistemic logic

If it is still not clear ...

Questions: What happens if he discovers that there is a storm outside?

Introduction to Logic Semantics for epistemic logic

If it is still not clear ...

Analyse the following situation:

Introduction to Logic Semantics for epistemic logic

If it is still not clear ...

Analyse the following situation: A motivated and young professor who is making a boring course

Introduction to Logic Semantics for epistemic logic

If it is still not clear ...

Analyse the following situation: A motivated and young professor who is making a boring course He (she) does not know that the course is boring, while the student(s) does

Introduction to Logic Semantics for epistemic logic

If it is still not clear ...

Analyse the following situation: A motivated and young professor who is making a boring course He (she) does not know that the course is boring, while the student(s) does He (she) knows that the student knows if the course is boring

Introduction to Logic Semantics for epistemic logic

If it is still not clear ...

Analyse the following situation: A motivated and young professor who is making a boring course He (she) does not know that the course is boring, while the student(s) does He (she) knows that the student knows if the course is boring The student does not know that the professor does not know that the course is boring

Introduction to Logic Semantics for epistemic logic

Finally ...

We take for granted that you all know what it means for a formula to be:

Introduction to Logic Semantics for epistemic logic

Finally ...

We take for granted that you all know what it means for a formula to be: I

true in a model (global satisfaction),

Introduction to Logic Semantics for epistemic logic

Finally ...

We take for granted that you all know what it means for a formula to be: I

true in a model (global satisfaction),

I

valid in a frame (or just valid),

Introduction to Logic Semantics for epistemic logic

Finally ...

We take for granted that you all know what it means for a formula to be: I

true in a model (global satisfaction),

I

valid in a frame (or just valid),

I

valid in a class of frames.

Introduction to Logic Syntax for epistemic logic

Hilbert System Kn The Hilbert system Kn , with respect to the set of agents A = {1, ..., n} is composed of:

Introduction to Logic Syntax for epistemic logic

Hilbert System Kn The Hilbert system Kn , with respect to the set of agents A = {1, ..., n} is composed of: i all the axiom schemes of classical logic

Introduction to Logic Syntax for epistemic logic

Hilbert System Kn The Hilbert system Kn , with respect to the set of agents A = {1, ..., n} is composed of: i all the axiom schemes of classical logic ii Ki (α → β) → (Ki α → Ki β), for i = 1, ..., n (Distribution Axiom)

Introduction to Logic Syntax for epistemic logic

Hilbert System Kn The Hilbert system Kn , with respect to the set of agents A = {1, ..., n} is composed of: i all the axiom schemes of classical logic ii Ki (α → β) → (Ki α → Ki β), for i = 1, ..., n (Distribution Axiom) iii Modus Ponens

Introduction to Logic Syntax for epistemic logic

Hilbert System Kn The Hilbert system Kn , with respect to the set of agents A = {1, ..., n} is composed of: i all the axiom schemes of classical logic ii Ki (α → β) → (Ki α → Ki β), for i = 1, ..., n (Distribution Axiom) iii Modus Ponens iv Necessitation Rule: α for i = 1, ..., n Ki α

Introduction to Logic Syntax for epistemic logic

Briefly ...

We remind that:

Introduction to Logic Syntax for epistemic logic

Briefly ...

We remind that: I

the notion of derivability in a system is the usual one,

Introduction to Logic Syntax for epistemic logic

Briefly ...

We remind that: I

the notion of derivability in a system is the usual one,

I

the system Kn is valid and complete with respect to the class of all frames.

Introduction to Logic Syntax for epistemic logic

Remarks

There are at least two important remarks to make.

Introduction to Logic Syntax for epistemic logic

Remarks

1. The distribution axioms and the rule of necessitation express rather unrealistic properties for real agents. For instance, for human agents is not to be expected that their knowledge is closed under logical consequence. Indeed this would involve that humans have the disposal of an infinite body of knowledge.

Introduction to Logic Syntax for epistemic logic

Remarks

2. One could naturally ask: are we sure that with the basic system Kn we have captured all the logical properties of knowledge? The answer is negative. System Kn does not say much about knowledge. Knowledge is supposed to have additional properties; let us see them in detail.

Introduction to Logic Syntax for epistemic logic

(Possible) Logical Properties of Knowledge Let us observe the following axiom schemes:

We take for granted that you have already had (at least) a first look to: (i) these axioms, (ii) their corresponding semantic properties, (iii) the Hilbert systems that we can construct with them.

Introduction to Logic Syntax for epistemic logic

(Possible) Logical Properties of Knowledge Let us observe the following axiom schemes: T Ki α → α, for i = 1, ..., n Known facts are true

We take for granted that you have already had (at least) a first look to: (i) these axioms, (ii) their corresponding semantic properties, (iii) the Hilbert systems that we can construct with them.

Introduction to Logic Syntax for epistemic logic

(Possible) Logical Properties of Knowledge Let us observe the following axiom schemes: T Ki α → α, for i = 1, ..., n Known facts are true 4 Ki α → Ki Ki α, for i = 1, ..., n An agent knows that he knows something

We take for granted that you have already had (at least) a first look to: (i) these axioms, (ii) their corresponding semantic properties, (iii) the Hilbert systems that we can construct with them.

Introduction to Logic Syntax for epistemic logic

(Possible) Logical Properties of Knowledge Let us observe the following axiom schemes: T Ki α → α, for i = 1, ..., n Known facts are true 4 Ki α → Ki Ki α, for i = 1, ..., n An agent knows that he knows something 5 ¬Ki α → Ki ¬Ki α, for i = 1, ..., n An agent knows that he does not know something We take for granted that you have already had (at least) a first look to: (i) these axioms, (ii) their corresponding semantic properties, (iii) the Hilbert systems that we can construct with them.

Introduction to Logic Syntax for epistemic logic

Chose the best!

An interesting question could now be: which is the ‘best’ system to capture knowledge?

Introduction to Logic Syntax for epistemic logic

Chose the best!

An interesting question could now be: which is the ‘best’ system to capture knowledge? The answer depends on the philosopher, e.g.:

Introduction to Logic Syntax for epistemic logic

Chose the best!

An interesting question could now be: which is the ‘best’ system to capture knowledge? The answer depends on the philosopher, e.g.: I

Hintikka has argued for S4,

Introduction to Logic Syntax for epistemic logic

Chose the best!

An interesting question could now be: which is the ‘best’ system to capture knowledge? The answer depends on the philosopher, e.g.: I

Hintikka has argued for S4,

I

Van Hoek, Fagin and al. have argued for S5.

Introduction to Logic Towards Common Knowledge

To resume

We started describing (the logic of) knowledge by using a single agent.

Introduction to Logic Towards Common Knowledge

To resume

It was easy to realise that this approach is too narrow: what happens if there are n knower agents?

Introduction to Logic Towards Common Knowledge

To resume

The situation was easily solved within the introduction of: (i) a set of n agents that ‘interact’ with the knowledge operator; (ii) n different accessibility relations, one for each agent.

Introduction to Logic Towards Common Knowledge

To resume

This way we can express things that was not possible to formulate before, e.g. Paul knows that Maria knows that when he is lying to her, he stammers.

Introduction to Logic Towards Common Knowledge

To resume

Question: is it possible to go further? For example, is it possible to formalise the fact that everybody knows something and everybody knows that everybody knows something, and that.... and so on?

Introduction to Logic Towards Common Knowledge

To resume

Question: is it possible to go further? For example, is it possible to formalise the fact that everybody knows something and everybody knows that everybody knows something, and that.... and so on? The answer is positive and takes the name of the logic of common knowledge. lt represent the beginning of the next course ....

Introduction to Logic Towards Common Knowledge

References

I

Stanford Encyclopedia of Philosophy, entry “Epistemic Logic”, http://plato.stanford.edu/entries/logic-epistemic/

I

Fagin, R., Halpern, Y. J., Moses, Y. and Vardi, Y. M Reasoning about Knowledge, MIT press, 1995.

I

Blackburn, P., de Rijke, M. et Venema, Y. Modal Logic, Cambridge University Press, 2001.