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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

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Syntax of AEL 1st Order Logic, plus the operator L (applied to formulas) L signifies “I know ” Examples: place → L place (or L place → place) young (X) L studies (X) → studies (X)

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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

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Semantics of AEL T* is an expansion of theory T iff T* = Th(T { L : T* |= } { L : T* |≠ }) Assuming the inference rule / L : T* = Cn AEL (T { L : T* |≠ }) An AEL theory is always two-valued in L, that is, for every expansion: | L T* L T*

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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 – I believe in

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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 L baseball L football → 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

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Axioms about beliefs Consistency Axiom B Normality Axiom B (F → G) → ( B F → B G) Necessitation rule F B F

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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 B e L atoms When is true in all minimal models of T, we write T |= min

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AELB expansions T* is a static expansion of T iff T* = Cn AELB (T { L : T* |≠ } { B : T* |= min }) where Cn AELB denotes closure using the axioms of AELB plus necessitation for L

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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* = (T*) where (T*) = Cn AEB (T { B : T* |= min }) and Cn AEB denotes closure using the axioms of AEB

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Least expansion Theorem: Operator is monotonic, i.e. T T 1 T 2 → (T 1 ) (T 2 ) Hence, there always exists a minimal expansion of T, obtainable by transfinite induction: –T 0 = Cn (T) –T i+1 = (T i ) –T = U T (for limit ordinals )

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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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