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.
