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# Auto-Epistemic Logic - PowerPoint PPT Presentation

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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## PowerPoint Slideshow about 'Auto-Epistemic Logic' - colman

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Presentation Transcript

• 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

• 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)

• 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

• 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*

• 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

• 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

• Consistency Axiom

B

• Normality Axiom

B(F → G) → (B F →B G)

• Necessitation rule

F

B F

• 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

• 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

• 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

• 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)

• 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