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

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Presentation Transcript

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)
- Lj signifies “I know j”
- Examples:
place →L place (or L place → place)

young (X) Lstudies (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
Bbaseball Bfootball → 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

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 Bj e Lj atoms
- When j is true in all minimal models of T, we write T |=minj

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

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

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

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