Transcript 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)
• L j signifies “I know j”
• 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{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
• 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
• 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
BF
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 |=min j
AELB expansions
• T* is a static expansion of T iff
T* = CnAELB(T  {Lj : T* |≠ j}
 {Bj : T* |=min j})
where CnAELB 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* = YT(T*)
where YT(T*) = CnAEB(T  {Bj : T* |=min j})
and CnAEB denotes closure using the axioms
of AEB
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
• 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