**DEF**. 5.5. This paragraph does not entitle either party to any money ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ . ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ .

**DEF**. 12 guarantees that there is (at most) one root component Rr in every Gr, r > 0. Since we have infinitely many graphs in (Gr)r>0 but only finitely many processes, there is at least one process p in Rr for infinitely many r. Let r1, r2, . . . be this sequence of rounds. Moreover, let P0 = {p}, and define for each i > 0 the set Pi = Pi−1 ∪ {q : ∃q′ ∈ Pi−1 : q′ ∈ N ri }. Using induction, we will show that |Pk| “ min{n, k + 1} for k “ 0. Consequently, by the end of round rn−1 at latest, p will have causally influenced all processes in Π. Induction base k = 0: |P0| “ min{n, 1} = 1 follows immediately from P0 = {p}. Induction step k → k + 1, k “ 0: First assume that already |Pk| = n “ min{n, k + 1}; since |Pk+1| “ |Pk| = n “ min{n, k + 1}, we are done. Otherwise, consider round rk+1 and |Pk| < n: Since p is in Rrk+1 , there is a path from p to any process q, in particular, to any process q in Π \ Pk ∅. Let (v → w) be an edge on such a path, such that v ∈ Pk and

**DEF**. A quitclaim deed is a release by the grantor, or conveyor of the deed, of any interest the grantor may have in the property de scribed in the deed. Generally a quitclaim deed relieves the grantor of liability regarding the ownership of theproperty. Thus, the grantor of a quitclaim deed will not be liable to the grantee, or recipient of the deed, if a competing claim to the property is later d iscovered. A quitclaim deed is not a guarantee that the grantor has clear title to theproperty; rather it is a relinquishment of the gra ntor's rights, if any, in the property. By contrast, in a warranty deed the grantor promises that she owns the property with no cloud on the title (that is, no competing claims). The holder of a quitclaim deed receives only the interest owned by the person conveying the deed. If the grantee of a quitclaim de ed learns after accepting the deed that the grantor did not own the property, the grantee may lose theproperty to the true owner. If it turns out that the grantor had only a partial interest in the property, the quitclaim deed xxxxxx holds only that partial interest. Warranty Deed. (not typical , but sometimes seen/used in Western Mass) (def.) A warranty deed generally offers the greatest amount of protection to someone who is purchasing or receiving the title to a piece of real estate (the grantee). A warranty deed includes four basic assurances to the grantee at the time of transaction. The first warranty is that the current owner and seller of the title (the grantor) does in fact own the real estate in fee simple, which assures the grantor has absolute ownership of the property. Second, a warranty deed guarantees that the property is free from any encumbrances (anything that affects or limits the title of the property such as easements or liens) except for those specifically stated in the deed. Third, the warranty deed guarantees that the grantor of the title has the legal right to sell or transfer the property to grantee. Lastly, through the deed, the grantor promises to defend against any legal claims regarding problems with the title that arose not only during the grantor’s ownership period but also prior to that period time.

**DEF**in.sec 1.01(a)..............def.in.sec.a 2..................xxxxx.xxxx.xxx 2.01...............purch.sale.sec 2.02..................purch.price 2.03......................closing 2.03....................closing.a 2.03(a).................closing.b 2.03(c).................closing.c 2.04............closing.bal.sheet 2.04(a).......closing.bal.sheet.a 2.04(b).......closing.bal.sheet.b 2.04(c).......closing.bal.sheet.c 2.04(d).......closing.bal.sheet.d 2.05.......................adj.pp 2.05(a)..................adj.pp.a 2.05(b)..................adj.pp.b 2.06.......................all.pp

**DEF**. Let Mn+1(φ) be the proposition that is true at w just in case M0 (Mn(φ)) is G G true at w. G Def. Let ComG(φ) be the proposition that is true at w just in case Mn (φ) is true at w, for all n. Def. Let Hom[Ki(·)] be the proposition that is true at some world w just in case Ki(w) ∈ P(W) and, for every wj ∈ Ki(w), Ki(w) = Ki(wj). · ∈ Then we can show that (i) φ is common knowledge amongst G at w just in case ComG(φ) is true at w and (ii) an agent i A is third-personally introspective at a world w just in case Hom[Ki( )] holds at w.8 Now let [Ki = φ] be the proposition that is true at a world wj just in case Ki(wj) = φ. Given Uncentered Confidants, we have: 7In the case in which φ is an uncentered proposition, we can take Factivity to say: If an agent i knows φ, then φ is true. However, if φ is a centered proposition, we can’t say that such a proposition is true or false simpliciter. Instead, such a proposition will be true relative to some agents and false relative to others. Our formulation, then, is meant to apply when φ is an uncentered proposition and when φ is a centered proposition. Where φ is a centered proposition, i will be mistaken in virtue of believing φ just in case φ is false relative to i. 8See Caie [2015] for the justification of these claims. ⊆ Uncentered Representation: A group of agents G A whose epistemic states are defined over a set of possible worlds propositions are epistemic confidants at a world w just in case ComG([Ki = Ki(w)]) and ComG(Hom[Ki(·)]) hold at w, for each i ∈ G.9 In addition, we have the following theorem: Uncentered Agreement Theorem: Let F be an agreement frame and Pr(·) a probability function on P(W). If, at some w ∈ W, for each i ∈ G ⊆ A, ComG([Ki = Ki(w)]) and ComG(Hom[Ki(·)]) hold, then, for each i, j ∈ G, Pr(·|Ki(w)) = Pr(·|Kj(w)).10 ∈ ⊆ ∈ The key fact underlying this theorem is that if ComG([Ki = Ki(w)]) and ComG(Hom[Ki(·)]) are true at w, for each i G A, then it follows that, for each i, j G, Ki(w) = Kj(w). Given this, the result is obvious. Now, amongst the class of update functions, there will be some that correspond, in a natural way, to probability functions.

**DEF**. We say that an agent a is first-personally G-introspective just in case, if a’s epistemic state is K, then a first-personally knows that it is commonly known in G that their epistemic state is characterized by K. Consider now the following claim:22 22A few points that are perhaps worth highlighting. First, Group Introspection only provides a necessary condition on a group of agents being epistemic confidants. The proponent of Group Introspection will, I take it, accept that the conditions imposed by Centered Confidants are also necessary for any group of agents to be epistemic confidants. They will, however, deny that these conditions are sufficient. Second, insofar as one is attracted to Group Introspection, there are further conditions involving first- personal higher-order knowledge of one’s own and others epistemic states that one may naturally be inclined to require of a group of epistemic confidants. For example, given Group Introspection, it is also natural to require of a group of epistemic confidants that each a ∈ G be such that, if their epistemic state is characterized G z G z x by K, then, for any n iterations of M0 Ka , Ka({˙z : (M0 Ka )n{˙x : Ka (˙x) = K}}) holds. That is, if a set of agents G are epistemic confidants, then if some member a’s epistemic state is characterized by K, then a will know, in a first-personal way, that each member of G knows that they know that their epistemic state is characterized by K, and a will know, in a first-personal way, that each member of G knows that they know that each member of G knows that they know that their epistemic state is characterized by K, and so on.

**DEF**students-NOM where 0-xxxxxxxx-XX.XX ‘The students where are (they) studying?’ (Xxxxxxxx, 1993, 243) Xxxxxxxxxxx, 0000, 28), (8).

**DEF**. Let M0 (φ) be the proposition that is true at w just in case, for every i ∈ G, Ki(w) ⊆ φ.

**DEF**. Let ComG(φ) be the proposition that is true at ˙q just in case Mn (φ) is true at ˙q, for all n. Def. Let Hom[Ki(·)] be the proposition that is true at some centered world ˙q just in case Ki(˙q) ∈ P(C) and, for every ˙z ∈ Ki(˙q), Ki(˙q) = Ki(˙z). · ∈ Then we can show that (i) φ is common knowledge amongst G at wz just in case ComG(φ) is true at ˙z and (ii) an agent i A is third-personally introspective at wz just in case Hom[Ki( )] holds at ˙z.13 In addition, we can also represent the property of being first-personally introspective in a centered agreement frame. Def. Let CHom[Ki(·)] be the proposition that is true at some centered world ˙q just in case Ki(˙q) ∈ P(C) and Ki(˙q) = Kaz (˙z), for all ˙z ∈ Ki(˙q). Then we can show that an agent i ∈ A is first-personally introspective at wz just in case CHom[Ki(·)] holds at ˙z.14 Now let [Ki = φ] be the proposition that is true at a centered world ˙z just in case Ki(˙z) = φ. Given Centered Confidants, we have: ⊆ Centered Representation: A group of agents G A whose epistemic states are defined over a set of centered worlds propositions are epistemic confidants at a world wz just in case ComG([Ki = Ki(˙z)]), ComG(Hom[Ki(·)]) and ComG(CHom[Ki(·)]) hold at ˙z, for each i ∈ G. In addition, we have the following theorem: Centered Agreement Theorem: Let C be a centered agreement frame. Then there exists some probability function Pr(·) defined over P(C), such that, for any ˙q ∈ C, if every i ∈ G ⊆ A is such that the following propositions are true at ˙q: ComG([Ki = Ki(˙q)]), ComG(Hom[Ki(·)]), ComG(CHom[Ki(·)]), then, for each uncentered proposition ψ and each i, j ∈ G, Pr(ψ|Ki(˙q)) = Pr(ψ|Kj(˙q)).15 P P Given Centered Confidants, then, we can see that Permissible Agreement is compatible Strong Centered Bayesianism. For, given Centered Confidants, we have Centered Representation. And, given Centered Representation, it follows from the Centered Agreement Theorem that, given an algebra (C), there is some update function that is determined by a probability function over this algebra, that ensures that any epistemic confidants, with epistemic and xxxxxx states defined over (C), who update in line with this function will assign the same credence to every uncentered proposition over which their credal states are defined. In addition, we can also see that, given Centered Confidants, Permissible Agreement is compatible with Strong Centered Anti-Bayesianism. For, let pr(·) be an update function that 13Again, see Caie [2015] for t...