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Auto-Epistemic Logic. Proposed by Moore (1985) Contemplates reflection on self knowledge (auto-epistemic) Permits to talk not just about the external world, but also about the knowledge I have of it. Syntax of AEL. 1st Order Logic, plus the operator L (applied to formulas)

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Auto epistemic logic
Auto-Epistemic Logic

  • Proposed by Moore (1985)

  • Contemplates reflection on self knowledge (auto-epistemic)

  • Permits to talk not just about the external world, but also about the knowledge I have of it


Syntax of ael
Syntax of AEL

  • 1st Order Logic, plus the operator L (applied to formulas)

  • Lj signifies “I know j”

  • Examples:

    place →L place (or  L place → place)

    young (X) Lstudies (X) → studies (X)


Meaning of ael
Meaning of AEL

  • What do I know?

    • What I can derive (in all models)

  • And what do I know not?

    • What I cannot derive

  • But what can be derived depends on what I know

    • Add knowledge, then test


Semantics of ael
Semantics of AEL

  • T* is an expansion of theory T iff

    T* = Th(T{Lj : T* |= j}  {Lj : T* |≠j})

  • Assuming the inference rule j/Lj :

    T* = CnAEL(T  {Lj : T* |≠j})

  • An AEL theory is always two-valued in L, that is, for every expansion:

    j | Lj T* Lj T*


Knowledge vs belief
Knowledge vs. Belief

  • Belief is a weaker concept

    • For every formula, I know it or know it not

    • There may be formulas I do not believe in, neither their contrary

  • The Auto-Epistemic Logic of knowledge and belief (AELB), introduces also operator B j – I believe in j


Aelb example
AELB Example

  • I rent a film if I believe I’m neither going to baseball nor football games

    Bbaseball Bfootball → rent_filme

  • I don’t buy tickets if I don’t know I’m going to baseball nor know I’m going to football

     Lbaseball  Lfootball → buy_tickets

  • I’m going to football or baseball

    baseball  football

  • I should not conclude that I rent a film, but do conclude I should not buy tickets


Axioms about beliefs
Axioms about beliefs

  • Consistency Axiom

    B

  • Normality Axiom

    B(F → G) → (B F →B G)

  • Necessitation rule

    F

    B F


Minimal models
Minimal models

  • In what do I believe?

    • In that which belongs to all preferred models

  • Which are the preferred models?

    • Those that, for one same set of beliefs, have a minimal number of true things

  • A model M is minimal iff there does not exist a smaller model N, coincident with M on Bj e Lj atoms

  • When j is true in all minimal models of T, we write T |=minj


Aelb expansions
AELB expansions

  • T* is a static expansion of T iff

    T* = CnAELB(T  {Lj : T* |≠j}

     {Bj : T* |=minj})

    where CnAELB denotes closure using the axioms of AELB plus necessitation for L


The special case of aeb
The special case of AEB

  • Because of its properties, the case of theories without the knowledge operator is especially interesting

  • Then, the definition of expansion becomes:

    T* = YT(T*)

    where YT(T*) = CnAEB(T  {Bj : T* |=minj})

    and CnAEB denotes closure using the axioms of AEB


Least expansion
Least expansion

  • Theorem: Operator Y is monotonic, i.e.

    T  T1 T2→YT(T1) YT(T2)

  • Hence, there always exists a minimal expansion of T, obtainable by transfinite induction:

    • T0 = CnAEB(T)

    • Ti+1 = YT(Ti)

    • Tb = Ua < b Ta (for limit ordinals b)


Consequences
Consequences

  • Every AEB theory has at least one expansion

  • If a theory is affirmative (i.e. all clauses have at least a positive literal) then it has at least a consistent expansion

  • There is a procedure to compute the semantics


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