ADMISSIBILITY IN A LOGICAL FRAMEWORK JEFFREY HELZNER Abstract. Theories of rational choice are often based on the assumption that the agent's choices are reducible to a preference ordering on the set of alternatives. This assumption is reflected in work from a variety of disciplines including economics [19], psychology [15], and statistics [32], as well as in philosophy where the "logic of preference" is an established topic of study [12]. However, a persistent minority of theorists have abandoned this fundamental assumption for a variety of reasons. The accounts that have emerged from this minority can be understood in terms of admissibility, a concept that does not in general reduce to preference. The purpose of this paper is to present a logic-based approach to admissibility. 1. Introduction Many, perhaps most, accounts of rationality are based on the concept of preference. The various analyses of expectation, from the expected value theory that was the target of Bernoulli's St. Petersburg example to the subjective expected utility theory of Savage [32], assume a preference relation in the form of an ordering on the set of alternatives. In many ways this basic assumption has been the most resilient in the expected utility tradition, surviving in descriptive accounts such as Kahneman and Tversky's prospect theory [15] and normative theories such as those offered by Ellsberg [7, 8] and by Gardenfors and Sahlin [10]. These accounts retain an ordering on the set of alternatives while abandoning other aspects of the expected utility tradition in order to accommodate well-known challenges, e.g., the examples of Allais [3] and of Ellsberg [7, 8]. In contrast, some theorists have relaxed the requirement that choice is reducible to an ordering on the set of alternatives. The reasons for relaxing the indicated ordering requirement range from proposals for accommodating the aforementioned challenges of Allais and Ellsberg [24], compromises in group decision making [35], models of bounded rationality [31], and general arguments within the standard framework of choice functions [38]. We will return to Sen's arguments from [38] in a later section. For now, we recall some motivating work on indeterminacy, a topic that provides what is perhaps the most familiar motivation for abandoning the standard reduction to preference. Credal indeterminacy, or uncertainty, is the most well-known form of uncertainty that is relevant to the ordering assumption of rational choice theory. While probability functions are the most familiar models of uncertainty, there is significant literature that documents objections to this use of probability functions. Early objections of this sort were expressed by Knight [18] in his discussions of unmeasurable uncertainty and Keynes [17] in his discussions concerning weight of evidence. 1 2 JEFFREY HELZNER More recent, but influential, critiques relating to the use of probability functions as models of uncertainty include those by Kyburg in [20] and by Levi in [21]. 1 2 The limitations that have been identified in connection with the use of probability functions as models of uncertainty simpliciter are even more apparent in the context of decision making under uncertainty, i.e., cases of decision making in which the decision maker does not have access to an objective probability distribution over the relevant state space. According to subjective expected utility theory, which is still the received view concerning decision making under uncertainty, the rational agent who is confronted with such a situation has a credal state that can be represented by a probability function over the relevant state space and values that can be represented by a cardinal utility function over the relevant outcome space and, moreover, that such an agent is obligated to select an alternative that maximizes expectation with respect to the indicated probability distribution and cardinal utility function [32, 25]. Classic examples, such as those offered by Ellsberg in [7, 8], have motivated some people to reject subjective expected utility theory as the proper account of rational decision making under uncertainty and, in light of this rejection, to search for other normative accounts. Prominent among such accounts is the decision rule defended by Ellsberg in [7], the decision rule defended by Gardefors and Sahlin in [10], and the decision rule defended by Levi in [21]. All three of these rules use sets of probabilities to represent uncertainty. However, there are significant differences between these rules, and some reasons for preferring the rule defended by Levi to the other two are given in [24, 34, 33]. Such reasons aside, the decision rule defended by Levi might be viewed as coming with its own conceptual price in that it is not reducible to preference. Given that other sources of indeterminacy – e.g., value conflict [23] and indeterminacy in the weighting of attributes [13] – lead to analogous challenges, and given the rather different motivations that are presented in [39], it seems reasonable to investigate departures from ordering in a more general setting. There have been several studies of this sort [2, 1, 26, 27, 28]. Set-valued choice functions provide a common foundation for these studies. If X is a set, then we write P(X) for the powerset of X and Pω(X) for the set of all finite subsets of X. C : Pω(X)→ Pω(X) is a choice function on X just in case C is a function that satisfies C(Y ) ⊆ Y for all Y ∈ Pω(X). In addition, it is usually assumed that C(Y ) is nonempty whenever Y is nonempty. Empty domains have not attracted much attention in the standard literature on choice functions. However, the issue does have some significance within the context of the current investigation. We return to this point in Section 5. 1There are, of course, numerous psychological studies that are said to show that probability functions are not descriptively adequate with respect to judgments of uncertainty, but this work, some of which is extremely interesting, is not as immediately relevant in the present context as our concerns are restricted mainly to the judgments of rational agents. 2In the critiques cited above, both Kyburg and Levi identify the inability of probability functions to accommodate indeterminacy as being central to their shortcomings as models of uncertainty. There is more than one way to generalize the concept of a probability function in order to provide for indeterminacy. Kyburg is among those who favor "interval-valued" probabilities, i.e., certain functions that assign an interval of values in from [0, 1], as opposed to a mere point in [0, 1], to each event (or proposition). Levi is among those who favor using sets of probability functions to represent the rational agent's credal state. It should be noted that sets of probability functions provide a framework that is more general than the interval-valued probabilities approach [11]. ADMISSIBILITY IN A LOGICAL FRAMEWORK 3 Set-valued choice functions support a general notion of admissibility : C(Y ) is interpreted as the set of alternatives that the agent would judge as admissible if it were presented with menu Y . Note that we are taking admissibility as a primitive notion, much as preference is taken in many traditional studies of rational choice. It might be suggested that preference benefits from having an operational analysis, that the concept can be unpacked in terms of behavior. However, it should be cautioned that (1) any such account must rely on some non-trivial assumptions that relate preferences to observed behavior and (2) any such account will need to distinguish 'picking' from 'choosing', to borrow the terminology of [40]. Roughly, rationality determines a set of admissible alternatives among the available alternatives. Since the remaining set of admissible alternatives may contain more than one element, 3 it is not clear how to make an appropriate, behavioral distinction between choosing a from {a, b, c} when a is uniquely admissible and choosing a from {a, b, c} when a is admissible but not uniquely so. Although it is not central to our main concerns, as we are content to take admissibility as a primitive concept, it is worth noting that ideas concerning an operational analysis of admissibility, or at least approaches to eliciting admissibility judgments, have been considered [31, 36]. Perhaps the most immediate advantage to taking admissibility rather than preference as the fundamental notion is that it provides a framework in which accounts that relax ordering can be investigated in a neutral setting. Cases in which admissibility is reducible to preference can be characterized in terms of appropriate conditions on choice functions, e.g the following conditions from Sen [37, 19]: α: For all Y,Z ∈ Pω(X), if x ∈ Y ⊆ Z and x ∈ C(Z), then x ∈ C(Y ). β: For all Y, Z ∈ Pω(X), if x, y ∈ C(Y ), Y ⊆ Z, and y ∈ C(Z), then x ∈ C(Z). These conditions are well-known to be jointly necessary and sufficient for reducing admissibility to an ordering. Despite the number of formal studies examining choice in the absence of a complete preference ranking, we are not aware of any attempts to provide a logical analysis of admissibility (i.e., attempts to provide truth conditions for statements of the form 'x is admissible from the menu consisting of y1,...,yn'). The main purpose of the present work is to offer such an analysis. The basic idea of this account is as follows: Assume a propositional language L that has been augmented by the introduction of a unary operator  and a variable-arity operator A. The intended reading of  is that φ asserts that φ is necessary. The intended reading of A is that A(φ | ψ1, ..., ψn) asserts that φ is admissible in the menu consisting of ψ1, ..., ψn. We assume that alternatives can be represented as L-formulas. Our objective is to present a semantics for L. In particular, we seek to provide truth conditions for formulas such as A(φ | ψ1, ..., ψn). Roughly, the idea is that A(φ | ψ1, ..., ψn) is true at state w just in case the alternative of moving to an otherwise unspecified state at which φ holds is admissible among the menu that, for 1 ≤ i ≤ n, includes the alternative of moving to an otherwise unspecified state at which ψi holds; here the reference to admissibility is unpacked in terms of an appropriate set-valued choice function at w. The rest of this paper is organized as follows: Section 2 and Section 3 present the relevant syntax and semantics, respectively. Formal systems are developed in 3This is certainly possible under standard theories like expected utility maximization. 4 JEFFREY HELZNER Section 4, Section 5, and Section 6. Completeness results are presented in Section 7 and Section 8, respectively. Proofs of all results are in the appendices. In Section 9 we consider some directions for future work, including the possibility of addressing challenges of the sort raised by Sen in [38]. 2. Syntax Let Ω be a countable set of atoms. The language L (over Ω) is defined by the following inductive clauses: Atoms: Ω ⊆ L Negation: If φ ∈ L, then ¬φ ∈ L Conjunction: If φ, ψ ∈ L, then (φ ∧ ψ) ∈ L Admissibility: If φ, ψ1, ..., ψn ∈ L, then A(φ | ψ1, ..., ψn) ∈ L Necessity: If φ ∈ L, then φ ∈ L Only the fourth clause is nonstandard. The intended reading of A(φ | ψ1, ..., ψn) is that φ is admissible in the menu consisting of ψ1, ..., ψn. We return to these matters in more detail in the next section. As far as additional syntactic issues, the usual conventions regarding the omitting of parentheses will be observed and the other standard connectives will be taken as abbreviations in the usual manner, e.g., φ ∨ ψ =df ¬(φ ∧ ψ), and we will take '♦', the dual of '', as an abbreviation for '¬¬'. Also, we will adopt the familiar prefix notation for iterated conjunctions and disjunctions, i.e., n∧ i=1 φi =df φ1 ∧ ... ∧ φn and n∨ i=1 φi =df φ1 ∨ ... ∨ φn where n is a positive integer greater than 1. We extend this notation to the case where n = 1 by ∧n i=1 φi =df φ1 =df ∨n i=1 φi. 3. Semantics The main objective in this section is to present a semantics for languages of the sort that are described in Section 2. In particular, we present a semantics for formulas such as A(φ | ψ1, ..., ψn). The basic idea noted above can be refined so that A(φ | ψ1, ..., ψn) is true just in case the alternative of moving to an otherwise unspecified and possible state at which φ holds is admissible among the menu that, for 1 ≤ i ≤ n, includes the alternative of moving to an otherwise unspecified and possible state at which ψi holds. That fact that the relevant notion of possibility may vary from state to state provides a way for us to distinguish the description of an alternative from its local denotation. 4 4This is crucial to addressing the "epistemic value of the menu" problem in the manner that is discussed in Section 9. ADMISSIBILITY IN A LOGICAL FRAMEWORK 5 3.1. Frames. A frame is a tuple 〈W,R,X , {Cw}w∈W 〉 that satisfies the following requirement: F1: W is a nonempty set F2: R is a binary relation on W . F3: X is a nonempty subset of P(W ) F4: Cw is a choice function on Pω(Xw) for all w ∈W , where Xw = {Y | Y 6= ∅ and Y = Sw ∩ Z for some Z ∈ X} and where, for all w ∈W , Sw = {x | (w, x) ∈ R}. F5: X is closed under the following operations: • U 7→W − U • (U, V ) 7→ U ∩ V • U 7→ {w | Sw ⊆ U} • (U, V1, ..., Vn) 7→ {w | (Sw ∩ U) ∈ Cw({Sw ∩ V1, ..., Sw ∩ Vn})} 5 W is a set of (logically) possible states. R encodes an agent-relative notion of possibility at w. In keeping with the informal sketch of the semantics that was given earlier, the notion of possibility encoded by R is not the usual notion of epistemic possibility. Whereas the traditional notion of epistemic possibility concerns the agent's beliefs about the current state while at w, the present notion concerns the agent's beliefs about possible successor states while at w. The two notions are not mutually exclusive. However, for simplicity of exposition, and since the traditional epistemic notion is well-known, we will restrict our use of agent-relative notions of possibility to the aforementioned one involving successor states. Continuing on with the components of a frame, X is a set of propositions and each element of X determines an alternative at w by intersecting with the set of possibilities at w. Xw is the resulting set of alternatives at w, and Cw is a choice function on that set of alternatives. The closure conditions in F5 ensure that every formula is interpreted as a proposition in X (cf. Section 3.2). 3.2. Interpretations. An interpretation of L is a frame 〈W,R,X , {Cw}w∈W 〉 along with a function π from Ω to X . Given such an interpretation, π is extended to a function π∗ on L according to the following inductive clauses: Atoms: π∗(φ) = π(φ) Negation: π∗(¬φ) = W − π∗(φ) Conjunction: π∗(φ ∧ ψ) = π∗(φ) ∩ π∗(ψ) Necessity: π∗(φ) = {w | Sw ⊆ π∗(φ)} Admissibility: π∗(A(φ | ψ1, ..., ψn)) = {w | (Sw ∩ π∗(φ)) ∈ Cw({Sw ∩ π∗(ψ1), ..., Sw ∩ π∗(ψn)})} Given an interpretation I = 〈W,R,X , {Cw}w∈W , π〉, we write (I, w) |= φ just in case w ∈ π∗(φ) and write I |= φ just in case (I, w) |= φ for all w ∈W . 5Note that (U, V1, ..., Vn) ⊆ {w | Sw ∩ U 6= ∅} ∩ {w | Sw ∩ Vi 6= ∅ for all i ∈ {1, ..., n}}, since Xw is the domain of Cw 6 JEFFREY HELZNER 4. Basic axiom schemes K : (φ ∧(φ→ ψ))→ ψ C1 : A(ψ1 | ψ2, ..., ψn)→ n∧ i=1 ♦ψi C2 : n∧ i=1 ♦ψi → n∨ i=1 A(ψi | ψ1, ..., ψn) C3 : A(φ | ψ1, ..., ψn)→ n∨ i=1 (φ↔ ψi) C4 : ((φ↔ φ′) ∧ n∧ i=1 (ψi ↔ ψ′i) ∧A(φ | ψ1, ..., ψn))→ A(φ′ | ψ′1, ..., ψ′n) K is the familiar "distribution" axiom of relational semantics. C1 requires that admissibility judgments are made with respect to alternatives that are possible. C2 requires that a menu of possible alternatives has at least one admissible alternative. C3 requires that every admissible alternative is an available alternative. C4 requires that admissibility judgments are extensional. Proposition 4.1. Let I = 〈W,R,X , {Cw}w∈W , π〉 be an interpretation of L. If θ is an instance of K, C1, C2, C3 or C4, then I |= θ. 5. Additional axiom schemes 5.1. Let p be an atom of L, i.e., p ∈ Ω. We introduce the following 0-ary connective as an abbreviation: > =df p ∨ ¬p. The following axiom is mentioned in [5]. P : ♦> Recall from Section 1 that we do not rule out having an empty set of alternatives. If we were to do so, then this would amount to requiring that Sw is nonempty, which would amount to requiring that there is something that is possible at w in the relevant sense of possibility considered above; we see no reason to rule out situations that, at least from the agent's perspective, appear to be the "end of the line". P guarantees that Sw is nonempty, and, moreover, it is satisfied at w if Sw is nonempty. Proposition 5.1. Let I = 〈W,R,X , {Cw}w∈W , π〉 be an interpretation of L. If w ∈W , then w |= ♦> iff Sw 6= ∅. 5.2. The following two axiom schemes are intended as counterparts to Sen's α and β conditions as presented in Section 1: ADMISSIBILITY IN A LOGICAL FRAMEWORK 7 Cα : (A(φ | ψ1, ..., ψm, θ1, ..., θn) ∧ m∨ i=1 (φ↔ ψi))→ A(φ | ψ1, ..., ψm) Cβ : (A(φ1 | ψ1, ..., ψm) ∧A(φ2 | ψ1, ..., ψm) ∧A(φ1 | ψ1, ..., ψm, θ1, ..., θn))→ A(φ2 | ψ1, ..., ψm, θ1, ..., θn)) The following two propositions state that these axioms schemes are satisfied when the Cw satisfies the corresponding condition on choice functions: Proposition 5.2. Let I = 〈W,R,X , {Cw}w∈W , π〉 be an interpretation of L. If w ∈W and φ is an instance of Cα and Cw satisfies α, then w |= φ. Proposition 5.3. Let I = 〈W,R,X , {Cw}w∈W , π〉 be an interpretation of L. If w ∈W and φ is an instance of Cβ and Cw satisfies β, then w |= φ. 6. Systems A system consists of a collection of axioms and a set of inference rules. S is a system over L just in case (1) every axiom of S is in L and (2) L is closed under every inference rule of S. In what follows, every system that is considered is assumed to be a system over L. If S is a system, then a derivation in S is a sequence of formulas φ1, ..., φn such that each element of the sequence is either an axiom of S or follows from earlier elements of the sequence by an application of one of the inference rules for S. φ is derivable in S iff there is a derivation in S that ends with φ. We will write S ` φ just in case φ is derivable in S, and we will write S 0 φ if φ is not derivable in S. Finally, if S and S ′ are systems, then S extends S ′ just in case every axiom of S ′ is an axiom of S and every inference rule of S ′ is an inference rule of S. Let L be the system that has all tautologies in PL (see Appendix B) as axioms and has the following inference rule: φ φ→ ψ MP ψ Let C be the system that extends L by adding all instances of K, C1, C2, C3, and C4 as axioms and adds the following inference rule: φ Gen φ Let Cα be the system that extends C by adding all instances Cα. Let Cβ be the system that extends C by adding all instances Cβ . Let Cα,β be the system that extends C by adding all instances Cα and all instances of Cβ . Finally, if S is a system, then let S+ be the system that extends S by adding P. Proposition 6.1 (Soundness). Let I = 〈W,R,X , {Cw}w∈W , π〉 be an interpretation of L. The following soundness results hold: (1) System C is sound with respect to I. (2) System C+ is sound with respect to I if Sw 6= ∅ for all w ∈W . (3) System Cα is sound with respect to I if Cw satisfies α for all w ∈ W . (4) System C+α is sound with respect to I if Cw satisfies α and Sw 6= ∅ for all w ∈W . (5) System Cβ is sound with respect to I if Cw satisfies β for all w ∈W . (6) System C+β is sound with respect to I if Cw satisfies β and Sw 6= ∅ for all w ∈ W . (7) System Cα,β is sound with respect to I if Cw satisfies α and β for all w ∈ W . (8) System C+α,β is 8 JEFFREY HELZNER sound with respect to I if Cw satisfies α and β for all w ∈ W and Sw 6= ∅ for all w ∈W . 7. The canonical frame If S extends C and φ ∈ L, then let [φ]S denote the set of Γ ⊆ L such that Γ is maximally consistent with respect to S and φ ∈ Γ. Where there is no risk of confusion we will suppress the subscript and write [φ] rather than [φ]S . Let S be a system that extends C. The canonical frame for S, which we denote as IS , is defined as follows: • W is the set of all Γ ⊆ L such that Γ is maximally consistent with respect to S • (Γ,Γ′) ∈ R just in case {φ | φ ∈ Γ} ⊆ Γ′ • Y ∈ X iff there is a φ such that Y = [φ] • U ∈ CΓ({V1, ..., Vn}) iff the following conditions are satisfied: (1) U, V1, ..., Vn ∈ XΓ, (2) U = Vi for some i ∈ {1, ..., n}, (3) if φ, ψ1, ..., ψn are L formulas such that U = SΓ∩ [φ] and Vi = SΓ∩ [ψi] for all i ∈ {1, ..., n}, then A(φ | ψ1, ..., ψn) ∈ Γ. 8. Completeness Let S be a system that extends C. Let 〈W,R,X , {Cw}w∈W 〉 be the canonical frame for system S. Define π : Ω → X by π(p) = {Γ ∈ W | p ∈ Γ}. We will refer to 〈W,R,X , {Cw}w∈W , π〉 as the canonical interpretation for S. Proposition 8.1. Let IS be the canonical interpretation for S. If IS |= φ, then S ` φ. Corollary 8.2 (Basic completeness). If M is a class of interpretations that includes IS and I |= φ for all I ∈M, then S ` φ. Proposition 5.1 (in conjunction with Proposition H.1) states that if S includes P, then Sw 6= ∅ for all w in IS . Similarly, the following propositions state that if S extends Cα (resp. Cβ), then Cw satisfies α (resp. β) for all w in IS . Proposition 8.3. Let S be a system that extends Cα. Let 〈W,R,X , {Cw}w∈W 〉 be the canonical frame for S. Cw satisfies α for all w ∈W . Proposition 8.4. Let S be a system that extends Cβ. Let 〈W,R,X , {Cw}w∈W 〉 be the canonical frame for S. Cw satisfies β for all w ∈W . In light of these observations we have the following completeness results. Proposition 8.5 (Completeness). (1) System C is complete with respect to the class of all interpretations. (2) System C+ is complete with respect to the class of all interpretations such that Sw 6= ∅ for all w ∈W . (3) System Cα is complete with respect to the class of all interpretations such that Cw satisfies α for all w ∈W . (4) System C+α is complete with respect to the class of all interpretations such that Cw satisfies α and Sw 6= ∅ for all w ∈W . (5) System Cβ is complete with respect to the class of all interpretations such that Cw satisfies β for all w ∈ W . (6) System C+β is complete with respect to the class of all interpretations such that Cw satisfies β ADMISSIBILITY IN A LOGICAL FRAMEWORK 9 and Sw 6= ∅ for all w ∈W . (7) System Cα,β is complete with respect to the class of all interpretations such that Cw satisfies α and β for all w ∈ W . (8) System C+α,β is complete with respect to the class of all interpretations such that Cw satisfies α and β for all w ∈W and Sw 6= ∅ for all w ∈W . 9. Discussion Conditions such as α and β are instances of what Amartya Sen calls axioms of internal consistency of choice [38]. As Sen explains, such conditions are "'internal' to the choice function in the sense that they require correspondence between different parts of a choice function, without invoking anything outside choice (such as motivations, objectives, and substantive principles)."(495) While acknowledging the widespread use of internal consistency conditions in a variety of fields, Sen, in his well-known Presidential Address of the Econometric Society in 1984, argues that internal consistency is "essentially confused". 6 7 At the heart of Sen's critique of internal consistency are three types of examples, the second of which, according to Sen, concerns the "epistemic value of the menu". This sort of example has received attention in theoretical economics [4, 16] as well as in formal epistemology [30, 6]. The following is an instance of such an example from [38]: What is offered for choice can give us information about the underlying situation, and can thus influence our preference over the alternatives, as we see them. For example, the chooser may learn something about the person offering the choice on the basis of what he or she is offering. To illustrate, given the choice between having tea at a distant acquaintance's home (x), and not going there (y), a person who chooses to have tea (x), may nevertheless choose to go away (y), if offered – by that acquaintance – a choice over having tea (x), going away (y), and having some cocaine (z). The menu offered may provide information about the situation-in this case say something about the distant acquaintance, and this can quite reasonably affect the ranking of the alternatives x and y, and yield the pair of choices [that violate α].(502) The alleged violation of α in the given example is clear: the agent judges x to be uniquely admissible from the menu {x, y} but judges y to be uniquely admissible from the menu {x, y, z}. If we want to count these judgments as rational, then it seems we have to abandon α as a basic principle of rationality – here we are putting aside other arguments for abandoning α, in particular the arguments from indeterminacy that were discussed earlier. One response, which Sen anticipates, suggests that the x from {x, y} is not the one that is offered in {x, y, z}: It is, of course, true that the chooser has different information even about x (i.e., having tea with the acquaintance) when the acquaintance gives him the choice of having cocaine with him, and it can certainly be argued that in the "intentional" (as opposed to "extensional") sense the alternative x is no longer the same. But an "intentional" definition of alternatives would be, in general, 6Sen mentions that "Internal consistency of choice has been a central concept in demand theory, social choice theory, decision theory, behavioral economics, and related fields."(495) 7Sen's Presidential Address was published as [38]. 10 JEFFREY HELZNER quite hopeless in invoking inter-menu consistency, especially when (as in this case) the intentional characterization changes precisely with the alternatives available for choice (i.e. with the menus offered).(502) Sen's point is easy enough to appreciate. However, if inter-menu consistency conditions are translated into the object language that was presented in Section 2, then the framework that has been presented seems to provide a way to preserve the rationality of the choices in Sen's example without trivializing inter-menu consistency. Let us consider Sen's example more closely. It is suggested that the example demonstrates a rational violation of α. Recall that the judgments of admissibility which are to be constrained by α are purely synchronic. Of course such judgments may change over time as a result of changes to antecedent states such as the agent's relevant beliefs or desires – e.g., consider how a subjective expected utility maximizer's judgments of admissibility change as a result of changes in that agent's subjective probability or cardinal utility functions – but the judgments encoded by a set-valued choice functions are taken to be suppositional. In Sen's example it is implicit that the agent regards the alternative of having tea at the distant acquaintance's home as uniquely admissible from the menu that consists of (x) the alternative of having tea at the distant acquaintance's house and (y) the alternative of being away from the acquaintance. Let φ assert that the agent has tea at the distant acquaintance's home. Let ψ assert that the agent is away from the acquaintance. Letting w denote the current state, and working within the proposed framework, the example suggests that w |= ¬A(ψ | φ, ψ). Similarly, Sen's remarks concerning the three-way choice suggest that w |= A(ψ | φ, ψ, θ), where θ asserts that the agent has cocaine with the the acquaintance. According to the relevant semantic clause in Section 3 there must be a w′ ∈ W such that wRw′ and w′ |= θ. That is, since θ is a genuine alternative for the agent at w, it is required that at w the agent regards some θ state as a possible successor. The agent's recognition of such possibilities when considering the two-way choice would seem to undermine the plausibility of the rational agent judging φ as admissible in the pair and, as a result, renders doubtful claim that w |= ¬A(ψ | φ, ψ). In general, allowing for the "epistemic value of the menu" requires a framework that allows for changes of epistemic state, something that exceeds the standard interpretation of set-valued choice functions. The proposed framework, which uses an indexed set of choice functions in the semantic account, does allow for such changes. Let w0 be a state in which the agent does not regard any θ states as possible successors. Let w1 be a state in which the agent does regard some θ states as possible successors. According to the semantics, w0 will not satisfy A(ψ | φ, ψ, θ), since θ will not even pick out an alternative at w0. Moreover, interpreting the example to imply that w0 |= ¬A(ψ | ψ, φ) and w1 |= A(ψ | φ, ψ, θ) is obviously compatible with Cw0 and Cw1 both satisfying α. Such a story is compatible with a frame that satisfies Cα, which, in light of the completeness results presented presented above, is a non-trivial condition on frames. Finally, another direction for future work, apart from further investigation into the examples raised by Sen, concerns the interaction between the admissibility and traditional epistemic modalities. As noted in Section 3.1, the accessibility relation R used in the semantic frames above is not given the standard epistemic interpretation. While we have avoided the introduction of a second accessibility ADMISSIBILITY IN A LOGICAL FRAMEWORK 11 relation in the present work in order to focus on the semantics of admissibility, we do plan to introduce a belief operator in future work. Combining the present semantics of admissibility with a belief operator would seem to offer a novel way of pursuing the program of taking belief change to be a species of rational choice [22, 29]. With the present framework augmented by the addition of a belief operator B, the admissibility of Bφi in menu {Bφ1, ..., Bφn} would be represented by the truth of A(Bφi | Bφ1, ..., Bφn). This approach to a rational-choice gloss on belief change seems natural enough in the present framework and worthy of further investigation. Rott has remarked that Sen's examples concerning the epistemic value of the menu are even more problematic within the context of belief change [30]. It is reasonable to suspect that something like the above analysis of Sen's example might extend to the context of belief revision through the suggested representation of theoretical choice problems. Technical Appendices Appendix A. Results from Section 3.2 Proposition A.1. π∗(φ) ∈ X for all φ ∈ L. Proof. Straightforward induction on L.  Appendix B. Inclusion of propositional logic Let ΩL be the set consisting of Ω and all formulas in L of the form A(φ | ψ1, ..., ψn) and all formulas in L of the form φ. Let PL be the propositional language over ΩL. That is, PL is defined inductively by the following clauses: Atoms: ΩL ⊆ PL Negation: If φ ∈ PL, then ¬φ ∈ PL Conjunction: If φ, ψ ∈ L, then (φ ∧ ψ) ∈ PL Proposition B.1. PL ⊆ L Proof. Trivial proof by induction on the structure of PL.  Proposition B.2. Let I = 〈W,R,X , {Cw}w∈W , π〉 be an interpretation of L. Define the function vw : ΩL → {T, F} by vw(φ) = T iff (I, w) |= φ. Let v∗w : PL → {T, F} be the canonical extension of vw to all of PL. For all φ ∈ PL, v∗w(φ) = T iff (I, w) |= φ. Proof. Trivial proof by induction on the structure of PL.  Corollary B.3. If θ is a tautology in PL and I = 〈W,X , C, π〉 is an interpretation of L, then I |= θ. Proof. Suppose that θ is a tautology in PL. Hence, v ∗ w(θ) = T for all w ∈W . Thus, by Proposition B.2, it follows that (I, w) |= θ for all w ∈W . Hence, I |= θ.  12 JEFFREY HELZNER Appendix C. Results from Section 4 Proof of Proposition 4.1. The result follows from the lemmas in this section.  Lemma C.1. Let I = 〈W,R,X , {Cw}w∈W , π〉 be an interpretation of L. If θ is an instance of K, then I |= θ. Proof. Standard. Suppose that w |= φ ∧ (φ → ψ). Hence, w |= φ and w |= (φ→ ψ). Suppose that v ∈ Sw. It follows that v |= φ→ ψ. Thus, if v |= φ, then v |= ψ. Since v ∈ Sw, it follows that v |= φ. Hence, v |= ψ. Thus, v |= ψ for all v ∈ Sw, and we have shown that w |= ψ.  Lemma C.2. Let I = 〈W,R,X , {Cw}w∈W , π〉 be an interpretation of L. If w ∈W and φ ∈ L, then Sw ∩ π∗(φ) 6= ∅ iff w ∈ π∗(♦φ). Proof. The result follows easily from the fact that ♦φ =df ¬¬φ.  Lemma C.3. Let I = 〈W,R,X , {Cw}w∈W , π〉 be an interpretation of L. If θ is an instance of C1, then I |= θ. Proof. Suppose that w |= A(ψ1 | ψ2, ..., ψn). From the F4 condition on frames and the admissibility clause for interpretations it follows that Sw ∩ π∗(ψi) 6= ∅ for all i ∈ {1, ..., n}. Thus, by Lemma C.2, w |= ♦ψi for all i ∈ {1, ..., n}, and it follows that w |= ∧n i=1 ♦ψi.  Lemma C.4. Let I = 〈W,R,X , {Cw}w∈W , π〉 be an interpretation of L. If θ is an instance of C2, then I |= θ. Proof. Suppose that w |= ∧n i=1 ♦ψi. Hence, w |= ♦ψi for all i ∈ {1, ..., n}. From Lemma C.2, it follows that Sw ∩ π∗(ψi) 6= ∅ for all i ∈ {1, ..., n}. Hence, by Proposition A.1, Sw ∩ π∗(ψi) ∈ Xw for all i ∈ {1, ..., n}. Since Cw is a choice function, there is a j ∈ {1, ..., n} such that Sw ∩ π∗(ψj) is an element of Cw({Sw ∩ π∗(ψ1), ..., Sw ∩ π∗(ψn)}). Hence, there is a j ∈ {1, ..., n} such that w |= A(ψj | ψ1, ..., ψn). It follows that w |= ∨n i=1A(ψi | ψ1, ..., ψn).  Lemma C.5. Let I = 〈W,R,X , {Cw}w∈W , π〉 be an interpretation of L. If w ∈W and φ, ψ ∈ L, then w ∈ π∗((φ↔ ψ)) iff Sw ∩ π∗(φ) = Sw ∩ π∗(ψ). Proof. The result follows easily from the fact that φ↔ ψ =df (φ∧ψ)∨(¬φ∧¬ψ).  Lemma C.6. Let I = 〈W,R,X , {Cw}w∈W , π〉 be an interpretation of L. If θ is an instance of C3, then I |= θ. Proof. Suppose that w |= A(φ | ψ1, ..., ψn). It follows that Sw ∩ π∗(φ) ∈ Cw(Sw ∩ ψ1, ..., Sw ∩ ψn). Thus, Sw ∩ π∗(φ) = Sw ∩ π∗(ψj) for some j ∈ {1, ..., n}. Hence, from Lemma C.5, w |= (φ ↔ ψj) for some j ∈ {1, ..., n}. It follows that w |=∨n i=1 (φ↔ ψi).  Lemma C.7. Let I = 〈W,R,X , {Cw}w∈W , π〉 be an interpretation of L. If θ is an instance of C4, then I |= θ. Proof. Suppose that w |= (φ↔ φ′)∧ ∧n i=1 (ψi ↔ ψ′i)∧A(φ | ψ1, ..., ψn). Hence, w |= (φ ↔ φ′), w |= ∧n i=1 (ψi ↔ ψ′i), and w |= A(φ | ψ1, ..., ψn). From the last of these it follows that Sw ∩ π∗(φ) 6= ∅, Sw ∩ π∗(ψi) 6= ∅ for all i ∈ {1, ..., n}, and Sw ∩π∗(φ) is in Cw(Sw ∩π∗(ψ1), ..., Sw ∩π∗(ψn)). Since w |= (φ↔ φ′), it follows from Lemma C.5 that Sw ∩ π∗(φ) = Sw ∩ π∗(φ′). Likewise, since w |= (ψi ↔ ψ′i), ADMISSIBILITY IN A LOGICAL FRAMEWORK 13 it follows from Lemma C.5 that Sw ∩ π∗(ψi) = Sw ∩ π∗(ψ′i) for all i ∈ {1, ..., n}. Thus, Sw ∩ π∗(φ′) 6= ∅, Sw ∩ π∗(ψ′i) 6= ∅ for all i ∈ {1, ..., n}, and Sw ∩ π∗(φ′) is in Cw(Sw ∩ π∗(ψ′1), ..., Sw ∩ π∗(ψ′n)). It follows that w |= A(φ′ | ψ′1, ..., ψ′n).  Appendix D. Results from Section 5 Proof of Proposition 5.1. Trivial  Proof of Proposition 5.2. Assume that w ∈ W and φ is an instance of Cα and Cw satisfies α. Suppose that w |= A(φ | ψ1, ..., ψm, θ1, ..., θn) ∧ ∨m i=1 (φ↔ ψi). Thus, (1) w |= A(φ | ψ1, ..., ψm, θ1, ..., θn) and (2) w |= ∨m i=1 (φ ↔ ψi). From (1) it follows that Sw∩π∗(φ) is in Cw({Sw∩π∗(ψ1), ..., Sw∩π∗(ψm), Sw∩π∗(θ1), ..., Sw∩ π∗(θn)}). From (2) it follows that w |= (φ ↔ ψi) for some i ∈ {1, ...,m}. Thus, from Lemma C.5, it follows that Sw ∩ π∗(φ) = Sw ∩ π∗(ψi) for some i ∈ {1, ...,m}. Thus, since Cw satisfies α, it follows that Sw∩π∗(φ) is in Cw({Sw∩π∗(ψ1), ..., Sw∩ π∗(ψm)}). Finally, from (1), Sw ∩ π∗(φ) 6= ∅ and Sw ∩ π∗(ψi) 6= ∅ for all i ∈ {1, ...,m}. Hence, w |= A(φ | ψ1, ..., ψm).  Proof of 5.3. Assume that w ∈ W and φ is an instance of Cβ and Cw satisfies β. Suppose that w |= A(φ1 | ψ1, ..., ψm)∧A(φ2 | ψ1, ..., ψm)∧A(φ1 | ψ1, ..., ψm, θ1, ..., θn). Thus, (1) w |= A(φ1 | ψ1, ..., ψm), (2) w |= A(φ2 | ψ1, ..., ψm), and (3) w |= A(φ1 | ψ1, ..., ψm, θ1, ..., θn). From (1) it follows that Sw ∩ π∗(φ1) is in Cw({Sw ∩ π∗(ψ1), ..., Sw ∩ π∗(ψn)}). From (2) it follows that Sw ∩ π∗(φ2) is in Cw({Sw ∩ π∗(ψ1), ..., Sw ∩ π∗(ψn)}). From (3) it follows that Sw ∩ π∗(φ1) is in Cw({Sw ∩ π∗(ψ1), ..., Sw ∩ π∗(ψm), Sw ∩ π∗(θ1), ..., Sw ∩ π∗(θn)}). Hence, since Cw satisfies β, it follows that Sw ∩ π∗(φ2) is in Cw({Sw ∩ π∗(ψ1), ..., Sw ∩ π∗(ψm), Sw ∩ π∗(θ1), ..., Sw ∩ π∗(θn)}). Finally, from (2), Sw ∩ π∗(φ2) 6= ∅ and π∗(ψi) 6= ∅ for all i ∈ {1, ...,m}, and, from (3), Sw ∩ π∗(θi) 6= ∅ for all i ∈ {1, ..., n}. Hence, w |= A(φ2 | ψ1, ..., ψm, θ1, ..., θn).  Appendix E. Results from Section 6 Proof of Proposition 6.1. Straightforward inductive proof using the soundness of the two inference rules and the soundness of the various axioms, as established in Propositions 4.1, 5.1, 5.2, and 5.3.  Appendix F. Maximally consistent sets If S is a system and φ ∈ L, then φ is consistent (with respect to S) just in case S 0 ¬φ. A finite set {φ1, ..., φn} ⊆ L is consistent iff ∧n i=1 φi is consistent. An infinite set Γ ⊆ L is consistent just in case every finite subset of Γ is consistent. We will say that a formula (or set of formulas) is inconsistent iff it is not consistent. A set Γ ⊆ L is maximally consistent just in case (1) it is consistent and (2) Γ∪{φ} is inconsistent for all φ ∈ L− Γ. The following two lemmas are standard results that apply to all systems in which classical, propositional reasoning can be conducted. Proofs are readily available, e.g. Lemma 1.4.3 in [14] or Lemma 3.1.2 in [9]. Lemma F.1. If S is a system that extends L and Γ is consistent with respect to S, then there is a set Γ′ such that Γ ⊆ Γ′ and Γ′ is maximally consistent. Lemma F.2. If S is a system that extends L and Γ is maximally consistent with respect to S, then the following hold for all φ, ψ ∈ L: • φ ∈ Γ or ¬φ ∈ Γ; 14 JEFFREY HELZNER • φ ∧ ψ ∈ Γ iff φ ∈ Γ and ψ ∈ Γ; • if φ ∈ Γ and φ→ ψ ∈ Γ, then ψ ∈ Γ; • if S ` φ, then φ ∈ Γ. Corollary F.3. If S is a system that extends L and Γ is maximally consistent with respect to S, then ∨n i=1 θi ∈ Γ just in case θi ∈ Γ for some i ∈ {1, ..., n}. Appendix G. Results from Section 7 Note that it is not immediate that IS satisfies the frame conditions. In particular, it is not clear that F4 and F5 are satisfied. We now turn our attention to this issue and show that IS does indeed satisfy the frame conditions. Lemma G.1. Let S be a system that extends C. Let 〈W,R,X , {Cw}w∈W 〉 be the canonical frame for S. If SΓ ⊆ [φ], then φ ∈ Γ. Proof. P.21 in van der Hoek [14]. Note that this uses Gen and the K schema.  Lemma G.2. Let S be a system that extends C. Let 〈W,R,X , {Cw}w∈W 〉 be the canonical frame for S. For all Γ ∈W , SΓ ∩ [φ] 6= ∅ iff ♦φ ∈ Γ. Proof. Suppose that SΓ ∩ [φ] 6= ∅. Hence, there is a Γ′ ∈ SΓ such that φ ∈ Γ′. If ♦φ /∈ Γ, then it follows from Lemma F.2 that ¬φ ∈ Γ. Hence, ¬φ ∈ Γ′, contradicting the assumption that Γ′ is consistent. Suppose that ♦φ ∈ Γ but that there is no Γ′ ∈ SΓ such that φ ∈ Γ′. It follows that ¬φ ∈ Γ′ for all Γ′ ∈ SΓ. Hence, by Lemma G.1, ¬φ ∈ Γ, contradicting the assumption Γ is consistent.  Lemma G.3. Let S be a system that extends C. Let 〈W,R,X , {Cw}w∈W 〉 be the canonical frame for S. Assume that Γ ∈ W . It follows that (φ ↔ ψ) ∈ Γ iff SΓ ∩ [φ] = SΓ ∩ [ψ]. Proof. Assume that Γ ∈ W and (φ ↔ ψ) ∈ Γ. It follows that φ ↔ ψ ∈ Γ′ for all Γ′ ∈ SΓ. Hence, (φ∧ψ)∨ (¬φ∧¬ψ) ∈ Γ′ for all Γ′ ∈ SΓ. Thus, by Corollary F.3, if Γ′ ∈ SΓ, then φ ∧ ψ ∈ Γ′ or ¬φ ∧ ¬ψ ∈ Γ′. From Lemma F.2, it follows that for all Γ′ ∈ SΓ, either φ, ψ ∈ Γ of φ, ψ /∈ Γ. The desired conclusion follows immediately. Assume that Γ ∈W and SΓ ∩ [φ] = SΓ ∩ [ψ]. It follows that, for all Γ′ ∈ SΓ, either φ, ψ ∈ Γ′ or ¬φ,¬ψ ∈ Γ′. Hence, by classical, propositional reasoning, if Γ′ ∈ SΓ, then (φ∧ψ)∨ (¬φ∧¬ψ) ∈ Γ′. Thus, φ↔ ψ ∈ Γ′ for all Γ′ ∈ SΓ. From Lemma G.1, it follows that (φ↔ ψ) ∈ Γ.  Proposition G.4. Let S be a system that extends C. Let 〈W,R,X , {Cw}w∈W 〉 be the canonical frame for S. CΓ is a choice function on Pω(XΓ) for each Γ ∈W . Proof. We must show that if V1, ..., Vn ∈ XΓ, where n ≥ 1, then there is a U such that U ∈ CΓ({V1, ..., Vn}). Suppose that V1, ..., Vn ∈ XΓ. It follows that, for all i ∈ {1, ..., n}, Vi 6= ∅, and it follows that there are L-formulas φ1, ..., φn such that Vi = SΓ ∩ [φi] for all i ∈ {1, ..., n}. By Lemma G.2, ♦φi ∈ Γ for all i ∈ {1, ..., n}. Hence, by Lemma F.2, ∧n i=1 ♦φi ∈ Γ. Since, by Lemma F.2, Γ contain all instances of C2, it follows that Γ contains ∧n i=1 ♦φi → ∨n i=1A(φi | φ1, ..., φn). From Lemma F.2, Γ is closed under MP. Hence, Γ contains ∨n i=1A(φi | φ1, ..., φn). Thus, by Corollary F.3, A(φj | φ1, ..., φn) ∈ Γ for some j ∈ {1, ..., n}. We claim that Vj ∈ CΓ({V1, ..., Vn}). By assumption, Conditions (1) and (2) in the specification of CΓ are satisfied. Now, suppose that φ ′ 1, ..., φ ′ n are L-formulas such that Vi = SΓ∩[φ′i] for all i ∈ {1, ..., n}. From Lemma G.3, it follows that (φi ↔ φ′i) ∈ Γ for all ADMISSIBILITY IN A LOGICAL FRAMEWORK 15 i ∈ {1, ..., n}. By Lemma F.2, Γ contains all instances of C4. In particular, Γ contains the following formula: ((φj ↔ φ′j) ∧ n∧ i=1 (φi ↔ φ′i) ∧A(φj | φ1, ..., φn))→ A(φ′j | φ′1, ..., φ′n). Since A(φj | φ1, ..., φn) ∈ Γ and (φi ↔ φ′i) ∈ Γ for i ∈ {1, ..., n}, it follows from Lemma F.2 that (φj ↔ φ′j) ∧ ∧n i=1 (φi ↔ φ′i) ∧ A(φj | φ1, ..., φn) ∈ Γ. By Lemma F.2, Γ is closed under MP. Thus, Γ contains A(φ′j | φ′1, ..., φ′n), and we have shown that Condition (3) in the specification of CΓ is satisfied.  Proposition G.5. If S is a system that extends C and 〈W,R,X , {Cw}w∈W 〉 is the canonical frame for S, then the following hold for all φ, ψ1, ..., ψn ∈ L. • [¬φ] = W − [φ] • [φ ∧ ψ] = [φ] ∩ [ψ] • [φ] = {Γ | SΓ ⊆ [φ]} • [A(φ | ψ1, ..., ψn)] = {Γ | SΓ ∩ [φ] ∈ CΓ({SΓ ∩ [ψ1], ..., SΓ ∩ [ψn]})} Proof. Suppose that Γ ∈ [¬φ]. Since Γ is consistent, it follows that Γ /∈ [φ]. Hence, Γ ∈W−[φ]. Conversely, suppose that Γ ∈W−[φ]. Hence, φ /∈ Γ. From Lemma F.2, it follows that ¬φ ∈ Γ. Thus, Γ ∈ [¬φ]. Suppose that Γ ∈ [φ∧ψ]. Hence, φ∧ψ ∈ Γ. From Lemma F.2, it follows that φ, ψ ∈ Γ. Thus, Γ ∈ [φ]∩ [ψ]. Conversely, suppose that Γ ∈ [φ] ∩ [ψ]. It follows that φ, ψ ∈ Γ. Thus, by Lemma F.2, φ ∧ ψ ∈ Γ. Hence, Γ ∈ [φ ∧ ψ]. Suppose that Γ ∈ [φ]. Thus, φ ∈ Γ, and it follows immediately that SΓ ⊆ [φ]. Conversely, suppose that SΓ ⊆ [φ]. From Lemma G.1, it follows that φ ∈ Γ. Thus, Γ ∈ [φ]. Suppose that Γ ∈ [A(φ | ψ1, ..., ψn)]. Hence, A(φ | ψ1, ..., ψn) ∈ Γ. By Lemma F.2, Γ contains all instances of C1. In particular, Γ contains A(φ | ψ1, ..., ψn) → (♦φ ∧ ∧n i=1 ♦ψi). By Lemma F.2, Γ is closed under MP. Thus, Γ contains ♦φ ∧ ∧n i=1 ♦ψi. By Lemma F.2, ♦φ ∈ Γ and ♦ψi ∈ Γ for all i ∈ {1, ..., n}. Thus, by Lemma G.2, SΓ ∩ [φ] 6= ∅ and SΓ ∩ [ψi] 6= ∅ for all i ∈ {1, ..., n}. We have verified the first condition in the specification of CΓ. By Lemma F.2, Γ contains all instances of C3. In particular, Γ contains A(φ | ψ1, ..., ψn) → ∨n i=1 (φ ↔ ψi). By Lemma F.2, Γ is closed under MP. Thus, Γ contains ∨n i=1 (φ ↔ ψi). By Corollary F.3, there is a j ∈ {1, ..., n} such that (φ ↔ ψj) ∈ Γ. Thus, by Lemma G.3, there is a j ∈ {1, ..., n} such that SΓ ∩ [φ] = SΓ ∩ [ψj ]. We have verified the second condition in the specification of CΓ. Suppose that φ ′, ψ′1, ..., ψ ′ n are L-formulas such that SΓ ∩ [φ] = SΓ ∩ [φ′] and SΓ ∩ [ψi] = SΓ ∩ [ψ′i] for all i ∈ {1, ..., n}. It follows from Lemma G.3 that (φ↔ φ′) ∈ Γ and (ψi ↔ ψ′i) ∈ Γ for all i ∈ {1, ..., n}. Thus, by Lemma F.2, Γ contains the formula (φ ↔ φ′) ∧ ∧n i=1 (ψi ↔ ψ′i) ∧ A(φ | ψ1, ..., ψn). By Lemma F.2, Γ contains all instances of C4. In particular, Γ contains ((φ↔ φ′) ∧ ∧n i=1 (ψi ↔ ψ′i)∧A(φ | ψ1, ..., ψn))→ A(φ′ | ψ′1, ..., ψ′n). By Lemma F.2, Γ is closed under MP. Thus, Γ contains A(φ′ | ψ′1, ..., ψ′n). We have verified the third condition in the specification of CΓ. Hence, SΓ ∩ [φ] ∈ CΓ({SΓ ∩ [ψ1], ..., SΓ ∩ [ψn]}). Conversely, suppose that SΓ ∩ [φ] ∈ CΓ({SΓ ∩ [ψ1], ..., SΓ ∩ [ψn]}). It follows immediately from the third condition in the specification of CΓ that A(φ | ψ1, ..., ψn) ∈ Γ.  Proposition G.6. Let S be a system that extends C. Let 〈W,R,X , {Cw}w∈W 〉 be the canonical frame for S. X satisfies the closure requirements of F5. Proof. Follows easily from Proposition G.5.  16 JEFFREY HELZNER Appendix H. Results from Section 8 Proposition H.1. For all φ ∈ L, π∗(φ) = [φ]. Proof. The result follows from Proposition G.5 via a straightforward induction on L. Atom: If φ is an atom, then the result follows immediately. Negation: Assume that the result holds for φ. From the negation clause for interpretations, π∗(¬φ) = W − π∗(φ). Thus, from the inductive hypothesis, π∗(¬φ) = W − [φ]. Since, by Proposition G.5, [¬φ] = W − [φ], it follows that π∗(¬φ) = [¬φ]. Conjunction: Assume that the result holds for φ and ψ. From the negation clause for interpretations, π∗(φ ∧ ψ) = π∗(φ) ∩ π∗(ψ). Thus, from the inductive hypothesis, π∗(φ ∧ ψ) = [φ] ∩ [ψ]. Since, by Proposition G.5, [φ ∧ ψ] = [φ] ∩ [ψ], it follows that π∗(φ ∧ ψ) = [φ ∧ ψ]. Necessity : Assume that the result holds for φ. From the necessity clause for interpretations, π∗(φ) = {Γ ∈ W | SΓ ⊆ π∗(φ)}. Thus, from the inductive hypothesis, π∗(φ) = {Γ ∈ W | SΓ ⊆ [φ]}. Since, by Proposition G.5, [φ] = {Γ ∈W | SΓ ⊆ [φ]}, it follows that π∗(φ) = [φ]. Admissibility : Assume that the result holds for φ and ψ1, ..., ψn. From the admissibility clause for interpretations, π∗(A(φ | ψ1, ..., ψn)) is equal to {Γ | SΓ ∩ π∗(φ)) ∈ CΓ({SΓ ∩ π∗(ψ1), ..., SΓ ∩ π∗(ψn)})} which, by the inductive hypothesis, is equal to {Γ | SΓ ∩ [φ] ∈ CΓ({SΓ ∩ [ψ1], ..., SΓ ∩ [ψn]})}. which, by Proposition G.5, is equal to [A(φ | ψ1, ..., ψn)].  Proof of 8.1. We prove the contrapositive. Suppose that S 0 φ. It follows that S 0 ¬¬φ, since S extends L. Hence, ¬φ is consistent with respect to S. Let IS = 〈W, {Rw}w∈W ,X , {Cw}w∈W , π〉 be the canonical interpretation for S. By Lemma F.1 there is a Γ ∈ W such that ¬φ ∈ Γ. Hence, Γ ∈ [¬φ]. By Proposition H.1, π∗(¬φ) = [¬φ]. Thus, Γ ∈ π∗(¬φ), i.e. (IS ,Γ) |= ¬φ. Thus, it is not the case that IS |= φ.  Proof of Proposition 8.3. Suppose that Γ ∈W and that Y1, ..., Ym, Z1, ..., Zn ∈ XΓ. Hence, there are L-formulas φ1, ..., φm such that SΓ ∩ [φi] = Yi 6= ∅ for all i ∈ {1, ...,m} and L-formulas ψ1, ..., ψn such that SΓ∩[ψi] = Zi 6= ∅ for all i ∈ {1, ..., n}. Assume that Yj ∈ CΓ({Y1, ..., Ym, Z1, ..., Zn}), where j ∈ {1, ...,m}. It follows from the specification of CΓ that A(φj | φ1, ..., φm, ψ1, ..., ψn) is in Γ. Since S extends Cα it follows from Lemma F.2 that Γ contains all instances of Cα. In particular, Γ contains the following formula: (A(φj | φ1, ..., φm, ψ1, ..., ψn) ∧ m∨ i=1 (φj ↔ φi))→ A(φj | φ1, ..., φm). By Lemma G.3, it is clear that (φj ↔ φj) ∈ Γ. Hence, by Lemma F.3, ∨m i=1 (φj ↔ φi) is in Γ. Thus, by Lemma F.2, A(φj | φ1, ..., φm, ψ1, ..., ψn) ∧ m∨ i=1 (φj ↔ φi) is in Γ. Since, by Lemma F.2, Γ is closed under MP, it follows that A(φj | φ1, ..., φm) is in Γ. Now we must show that Yj ∈ CΓ(Y1, ..., Ym). Of the three conditions in the specification of CΓ, all but Condition (3) follow immediately from what has already been assumed. Thus, suppose that φ′1, ..., φ ′ m are L-formulas ADMISSIBILITY IN A LOGICAL FRAMEWORK 17 such that SΓ ∩ [φ′i] = Yi for all i ∈ {1, ...,m}. It follows from Lemma G.3 that (φi ↔ φ′i) ∈ Γ for all i ∈ {1, ...,m}. Since S extends C it follows from Lemma F.2 that Γ contains all instances of C4. In particular, Γ contains the following formula: ((φj ↔ φ′j) ∧ m∧ i=1 (φi ↔ φ′i) ∧A(φj | φ1, ..., φm))→ A(φ′j | φ′1, ..., φ′m). We have shown that Γ contains each of the conjuncts from the antecedent of this formula. Hence, by Lemma F.2, Γ contains (φj ↔ φ′j) ∧ ∧m i=1 (φi ↔ φ′i) ∧ A(φj | φ1, ..., φm). By Lemma F.2, Γ is closed under MP. Thus, Γ contains A(φ′j | φ′1, ..., φ ′ m). This completes the verification of Condition (3) in the specification of CΓ. Thus, Yj ∈ C{Γ(Y1, ..., Ym}).  Proof of 8.4. Suppose that Γ ∈ W and that Y1, ..., Ym, Z1, ..., Zn ∈ XΓ. Hence, there are L-formulas φ1, ..., φm such that SΓ ∩ [φi] = Yi 6= ∅ for all i ∈ {1, ...,m} and L-formulas ψ1, ..., ψn such that SΓ∩ [ψi] = Zi 6= ∅ for all i ∈ {1, ..., n}. Assume that Yj , Yk ∈ CΓ({Y1, ..., Ym}) and Yj ∈ CΓ({Y1, ..., Ym, Z1, ..., Zn}) , where j, k ∈ {1, ...,m}. It follows from the specification of CΓ that Γ contains A(φj | φ1, ..., φm), A(φk | φ1, ..., φm), and A(φj | φ1, ..., φm, ψ1, ..., ψn). Hence, by Lemma F.2, A(φj | φ1, ..., φm) ∧A(φk | φ1, ..., φm) ∧A(φj | φ1, ..., φm, ψ1, ..., ψn) is contained in Γ. Since S extends Cβ it follows from Lemma F.2 that Γ contains all instances of Cα. In particular, Γ contains the following formula: (A(φj | φ1, ..., φm) ∧A(φk | φ1, ..., φm) ∧A(φj | φ1, ..., φm, ψ1, ..., ψn))→ A(φk | φ1, ..., φm, ψ1, ..., ψn)). By Lemma F.2, Γ is closed under MP. Thus, Γ containsA(φk | φ1, ..., φm, ψ1, ..., ψn). Now we must show that Yk ∈ CΓ({Y1, ..., Ym, Z1, ..., Zn}). Of the three conditions in the specification of CΓ, all but Condition (3) follow immediately from what has already been assumed. Thus, suppose that φ′1, ..., φ ′ m are L-formulas such that SΓ ∩ [φ′i] = Yi for all i ∈ {1, ...,m}, and suppose that ψ′1, ..., ψ′n are L-formulas such that SΓ ∩ [ψ′i] = Zi for all i ∈ {1, ..., n}. It follows from Lemma G.3 that (φi ↔ φ′i) ∈ Γ for all i ∈ {1, ...,m} and (ψi ↔ ψ′i) ∈ Γ for all i ∈ {1, ..., n}. Thus, by Lemma F.2, Γ contains (φk ↔ φ′k) ∧ m∧ i=1 (φi ↔ φ′i) ∧ n∧ i=1 (ψi ↔ ψ′i) ∧A(φk | φ1, ..., φm, ψ1, ..., ψn). Since S extends C it follows from Lemma F.2 that Γ contains all instances of C4. In particular, Γ contains the following formula: ((φk ↔ φ′k) ∧ m∧ i=1 (φi ↔ φ′i) ∧ n∧ i=1 (ψi ↔ ψ′i) ∧A(φk | φ1, ..., φm, ψ1, ..., ψn))→ A(φ′k | φ′1, ..., φ′m, ψ′1, ..., ψ′n). By Lemma F.2, Γ is closed under MP. Hence, Γ containsA(φ′k | φ′1, ..., φ′m, ψ′1, ..., ψ′n). This completes the verification of Condition (3). Yk ∈ CΓ({Y1, ..., Ym, Z1, ..., Zn}).  18 JEFFREY HELZNER References 1. 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