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) L  signifies “I know  ” Examples: place → L place (or  L place →  place) young (X)   L  studies (X) → studies (X)

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 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*

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 

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

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

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 

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

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

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  )

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