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

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Auto-Epistemic Logic

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  1. 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

  2. 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)

  3. 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

  4. 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*

  5. 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

  6. 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

  7. Axioms about beliefs • Consistency Axiom B • Normality Axiom B(F → G) → (B F →B G) • Necessitation rule F B F

  8. 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

  9. 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

  10. 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

  11. 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)

  12. 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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