In earlier work, we proposed a logic that extends the Logic of General Awareness of Fagin and Halpern [1988] by allowing quantification over primitive propositions. This makes it possible to express the fact that an agent knows that there are some facts of which he is unaware. In that logic, it is not possible to model an agent who is uncertain about whether he is aware of all formulas. To overcome this problem, we keep the syntax of the earlier paper, but allow models where, with each world, a possibly different language is associated. We provide a sound and complete axiomatization for this logic and show that, under natural assumptions, the quantifier-free fragment of the logic is characterized by exactly the same axioms as the logic of Heifetz, Meier, and Schipper [2008].
Adding awareness to standard models of epistemic logic has been shown to be useful in describing many situations (see [Fagin and Halpern 1988;Heifetz, Meier, and Schipper 2006] for some examples). One of the best-known models of awareness is due to Fagin and Halpern [1988] (FH from now on). They add an awareness operator to the language, and associate with each world in a standard possible-worlds model of knowledge a set of formulas that each agent is aware of. They then say that an agent explicitly knows a formula ϕ if ϕ is true in all worlds that the agent considers possible (the traditional definition of knowledge, going back to Hintikka [1962]) and the agent is aware of ϕ.
In the economics literature, going back to the work of Modica andRustichini [1994, 1999] (MR from now on), a somewhat different approach is taken. A possibly different set L(s) of primitive propositions is associated with each world s. Intuitively, at world s, the agent is aware only of formulas that use the primitive propositions in L(s). A definition of knowledge is given in this framework, and the agent is said to be aware of ϕ if, by definition, K i ϕ ∨ K i ¬K i ϕ holds. Heifetz, Meier, andSchipper [2006, 2008] (HMS from now on), extend the ideas of MR to a multiagent setting. This extension is nontrivial, requiring lattices of state spaces, with projection functions between them. As we showed in earlier work [Halpern 2001;Halpern and Rêgo 2008], the work of MR and HMS can be seen as a special case of the FH approach, where two assumptions are made on awareness: awareness is generated by primitive propositions, that is, an agent is aware of a formula iff he is aware of all primitive propositions occurring in it, and agents know what they are aware of (so that they are aware of the same formulas in all worlds that they consider possible).
As we pointed out in [Halpern and Rêgo 2006b] (referred to as HR from now on), if awareness is generated by primitive propositions, then it is impossible for an agent to (explicitly) know that he is unaware of a specific fact. Nevertheless, an agent may well be aware that there are relevant facts that he is unaware of. For example, primary-care physicians know that specialists are aware of things that could improve a patient’s treatment that they are not aware of; investors know that investment fund companies may be aware of issues involving the financial market that could result in higher profits that they are not aware of. It thus becomes of interest to model knowledge of lack of awareness. HR does this by extending the syntax of the FH approach to allow quantification, making it possible to say that an agent knows that there exists a formula of which the agent is unaware. A complete axiomatization is provided for the resulting logic. Unfortunately, the logic has a significant problem if we assume the standard properties of knowledge and awareness: it is impossible for an agent to be uncertain about whether he is aware of all formulas.
In this paper, we deal with this problem by considering the same language as in HR (so that we can express the fact that an agent knows that he is not aware of all formulas, using quantification), but using the idea of MR that there is a different language associated with each world. As we show, this slight change makes it possible for an agent to be uncertain about whether he is aware of all formulas, while still being aware of exactly the same formulas in all worlds he considers possible. We provide a natural complete axiomatization for the resulting logic. Interestingly, knowledge in this logic acts much like explicit knowledge in the original FH framework, if we take “awareness of ϕ” to mean K i (ϕ ∨ ¬ϕ); intuitively, this is true if all the primitive propositions in ϕ are part of the language at all worlds that i considers possible. Under minimal assumptions, K i (ϕ ∨ ¬ϕ) is shown to be equivalent to K i ϕ ∨ K i ¬K i ϕ: in fact, the quantifier-free fragment of the logic that just uses the K i operator is shown to be characterized by exactly the same axioms as the HMS approach, and awareness can be defined the same way. Thus, we can capture the essence of MR and HMS approach using simple semantics and being able to reason about knowledge of lack of awareness. Board and Chung [2009] independently pointed out the problem of the HR model and proposed the solution of allowing different languages at different worlds. They also consider a model of awareness with quantification, but they use first-order modal logic, so their quantification is over domain elements. Moreover, they take awareness with respect to domain elements, not formulas; that is, agents are (un)aware of objects (i.e., domain elements), not formulas. They also allow different domains at different worlds; more precisely, they allow an agent to have a subjective view of what the set of objects is at each world. Sillari [2008] uses much the same approach as Board and Chung [2009]. That is, he has a
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