
## References

Ben-David, S. and Y. Gafni, 1989: All we believe fails in impossible worlds. *Manuscript*.

Fagin, R., J. Y. Halpern, Y. Moses, and M. Y. Vardi, 1995: *Reasoning About Knowledge*. The MIT
Press.

Halpern, J. and G. Lakemeyer, 2001: Multi-agent only knowing. *Journal of Logic and Computation*,
11(1), 41.

Halpern, J. and Y. Moses, 1985: Towards a theory of knowledge and ignorance: preliminary report.

Logics and models of concurrent systems 459–476.

Halpern, J. Y., 1993: Reasoning about only knowing with many agents. In *AAAI*, 655-661.

Halpern, J. Y. and G. Lakemeyer, 1995: Levesque's axiomatization of only knowing is incomplete.

Artif. Intell., 74(2), 381–387.

Hughes, G. E. and M. J. Cresswell, 1972: *An introduction to modal logic*. Methuen London,, x, 388 p.

Hughes, G. E. and M. J. Cresswell, 1984: *A companion to modal logic*. Routledge.

Lakemeyer, G., 1993: All they know: A study in multi-agent autoepistemic reasoning. In *IJCAI-93*.

Lakemeyer, G. and H. J. Levesque, 2005: Only-knowing: taking it beyond autoepistemic reasoning.

In *AAAI'05*, AAAI Press, 633–638.

Levesque, H. and G. Lakemeyer, 2001: *The logic of knowledge bases*. The MIT Press. Levesque, H. J., 1990: All I know: a study in autoepistemic logic. *Artif. Intell.*, 42(2-3), 263–309.

Moore, R. C., 1985: Semantical considerations on nonmonotonic logic. *Artif. Intell.*, 25(1), 75–94.

Rosati, R., 2000: On the decidability and complexity of reasoning about only knowing. *Artif. Intell.*,
116(1-2), 193–215.

Waaler, A., 2004: Consistency proofs for systems of multi-agent only knowing. In Advances in Modal Logic, 347-366.

Waaler, A. and B. Solhaug, 2005: Semantics for multi-agent only knowing: extended abstract. In TARK, 109-125.

## 9 Appendix

Lemma 6.1. *For all* α **∈ ON L**+
n t, AX+
n t⊢ α iff |= α.

Proof: The proof is via induction. Using Theorems 4.15 and 4.18 as the base cases in the induction, there is one additional step on the structure of formulas.

Soundness: The base case holds for formulas α **∈ ON L**tn for AXtn
. Suppose now if AXtn ⊢ α, then AX+
n t⊢ Val(α). But if AXtn ⊢ α then (by induction hypothesis) at all models e ka
, e j b
, w |= α, and so by the definition at all models e ka
, e j b
, w |= Val(α) or |= Val(α).

Completeness. For the base case, we know that if for all models e ka
, e j b
, w |= α then AXtn ⊢ α.

Suppose |= Val(α), then by definition, for all models e ka
, e j b
, w |= α iff (by hypothesis) AXtn ⊢ α. So, AX+
n t⊢ Val(α).