Showing papers on "Modal operator published in 2012"

First steps in synthetic guarded domain theory: step-indexing in the topos of trees

Abstract: We present the topos S of trees as a model of guarded recursion. We study the internal dependently-typed higher-order logic of S and show that S models two modal operators, on predicates and types, which serve as guards in recursive definitions of terms, predicates, and types. In particular, we show how to solve recursive type equations involving dependent types. We propose that the internal logic of S provides the right setting for the synthetic construction of abstract versions of step-indexed models of programming languages and program logics. As an example, we show how to construct a model of a programming language with higher-order store and recursive types entirely inside the internal logic of S. Moreover, we give an axiomatic categorical treatment of models of synthetic guarded domain theory and prove that, for any complete Heyting algebra A with a well-founded basis, the topos of sheaves over A forms a model of synthetic guarded domain theory, generalizing the results for S.

Model checking information flow in reactive systems

Abstract: Most analysis methods for information flow properties do not consider temporal restrictions. In practice, however, such properties rarely occur statically, but have to consider constraints such as when and under which conditions a variable has to be kept secret. In this paper, we propose a natural integration of information flow properties into linear-time temporal logics (LTL). We add a new modal operator, the hide operator, expressing that the observable behavior of a system is independent of the valuations of a secret variable. We provide a complexity analysis for the model checking problem of the resulting logic SecLTL and we identify an expressive fragment for which this question is efficiently decidable. We also show that the path based nature of the hide operator allows for seamless integration into branching time logics.

First steps in synthetic guarded domain theory: step-indexing in the topos of trees

Abstract: We present the topos S of trees as a model of guarded recursion. We study the internal dependently-typed higher-order logic of S and show that S models two modal operators, on predicates and types, which serve as guards in recursive definitions of terms, predicates, and types. In particular, we show how to solve recursive type equations involving dependent types. We propose that the internal logic of S provides the right setting for the synthetic construction of abstract versions of step-indexed models of programming languages and program logics. As an example, we show how to construct a model of a programming language with higher-order store and recursive types entirely inside the internal logic of S. Moreover, we give an axiomatic categorical treatment of models of synthetic guarded domain theory and prove that, for any complete Heyting algebra A with a well-founded basis, the topos of sheaves over A forms a model of synthetic guarded domain theory, generalizing the results for S.

Moving Arrows and Four Model Checking Results

Abstract: We study dynamic modal operators that can change the model during the evaluation of a formula. In particular, we extend the basic modal language with modalities that are able to swap, delete or add pairs of related elements of the domain, while traversing an edge of the accessibility relation. We study these languages together with the sabotage modal logic, which can arbitrarily delete edges of the model. We define a suitable notion of bisimulation for the basic modal logic extended with each of the new dynamic operators and investigate their expressive power, showing that they are all uncomparable. We also show that the complexity of their model checking problems is PSpace-complete.

Justifications, Ontology, and Conservativity.

Abstract: An ontologically transparent semantics for justifications that interprets justifications as sets of formulas they justify has been recently presented by Artemov. However, this semantics of modular models has only been studied for the case of the basic justification logic J, corresponding to the modal logic K. It has been left open how to extend and relate modular models to the already existing symbolic and epistemic semantics for justification logics with additional axioms, in particular, for logics of knowledge with factive justifications. We introduce modular models for extensions of J with any combination of the axioms (jd), (jt), (j4), (j5), and (jb), which are the explicit counterparts of standard modal axioms. After establishing soundness and completeness results, we examine the relationship of modular models to more traditional symbolic and epistemic models. This comparison yields several new semantics, including symbolic models for logics of belief with negative introspection (j5) and models for logics with the axiom (jb). Besides pure justification logics, we also consider logics with both justifications and a belief/knowledge modal operator of the same strength. In particular, we use modular models to study the conditions under which the addition of such an operator to a justification logic yields a conservative extension.

A Family of Dynamic Description Logics for Representing and Reasoning About Actions

Abstract: Description logics provide powerful languages for representing and reasoning about knowledge of static application domains. The main strength of description logics is that they offer considerable expressive power going far beyond propositional logic, while reasoning is still decidable. There is a demand to bring the power and character of description logics into the description and reasoning of dynamic application domains which are characterized by actions. In this paper, based on a combination of the propositional dynamic logic PDL, a family of description logics and an action formalism constructed over description logics, we propose a family of dynamic description logics DDL(X @) for representing and reasoning about actions, where X represents well-studied description logics ranging from the to the , and X @ denotes the extension of X with the @ constructor. The representation power of DDL(X @) is reflected in four aspects. Firstly, the static knowledge of application domains is represented as RBoxes and acyclic TBoxes of the description logic X. Secondly, the states of the world and the pre-conditions of atomic actions are described by ABox assertions of the description logic X @, and the post-conditions of atomic actions are described by primitive literals of X @. Thirdly, starting with atomic actions and ABox assertions of X @, complex actions are constructed with regular program constructors of PDL, so that various control structures on actions such as the "Sequence", "Choice", "Any-Order", "Iterate", "If-Then-Else", "Repeat-While" and "Repeat-Until" can be represented. Finally, both atomic actions and complex actions are used as modal operators for the construction of formulas, so that many properties on actions can be explicitly stated by formulas. A tableau-algorithm is provided for deciding the satisfiability of DDL(X @)-formulas; based on this algorithm, reasoning tasks such as the realizability, executability and projection of actions can be effectively carried out. As a result, DDL(X @) not only offers considerable expressive power going beyond many action formalisms which are propositional, but also provides decidable reasoning services for actions described by it.

Information completeness in Nelson algebras of rough sets induced by quasiorders

Abstract: In this paper, we give an algebraic completeness theorem for constructive logic with strong negation in terms of finite rough set-based Nelson algebras determined by quasiorders. We show how for a quasiorder $R$, its rough set-based Nelson algebra can be obtained by applying the well-known construction by Sendlewski. We prove that if the set of all $R$-closed elements, which may be viewed as the set of completely defined objects, is cofinal, then the rough set-based Nelson algebra determined by a quasiorder forms an effective lattice, that is, an algebraic model of the logic $E_0$, which is characterised by a modal operator grasping the notion of "to be classically valid". We present a necessary and sufficient condition under which a Nelson algebra is isomorphic to a rough set-based effective lattice determined by a quasiorder.

Pure type systems with corecursion on streams: from finite to infinitary normalisation

Abstract: In this paper, we use types for ensuring that programs involving streams are well-behaved. We extend pure type systems with a type constructor for streams, a modal operator next and a fixed point operator for expressing corecursion. This extension is called Pure Type Systems with Corecursion (CoPTS). The typed lambda calculus for reactive programs defined by Krishnaswami and Benton can be obtained as a CoPTS. CoPTSs allow us to study a wide range of typed lambda calculi extended with corecursion using only one framework. In particular, we study this extension for the calculus of constructions which is the underlying formal language of Coq. We use the machinery of infinitary rewriting and formalise the idea of well-behaved programs using the concept of infinitary normalisation. The set of finite and infinite terms is defined as a metric completion. We establish a precise connection between the modal operator (• A) and the metric at a syntactic level by relating a variable of type (• A) with the depth of all its occurrences in a term. This syntactic connection between the modal operator and the depth is the key to the proofs of infinitary weak and strong normalisation.

A contextual type theory with judgemental modalities for reasoning from open assumptions

Abstract: Contextual type theories are largely explored in their applications to programming languages, but less investigated for knowledge representation purposes. The combination of a constructive language with a modal extension of contexts appears crucial to explore the attractive idea of a type-theoretical calculus of provability from refutable assumptions for non-monotonic reasoning. This paper introduces such a language: the modal operators are meant to internalize two different modes of correctness, respectively with necessity as the standard notion of constructive verification and possibility as provability up to refutation of contextual conditions.

Propositions and Multiple Indexing

Abstract: It is argued that propositions cannot be the compositional semantic values of sentences (in context) simply due to issues stemming from the compositional semantics of modal operators (or modal quantifiers). In particular, the fact that the arguments for double indexing generalize to multiple indexing exposes a fundamental tension in the default philosophical conception of semantic theory. This provides further motivation for making a distinction between two sentential semantic contents—what (Dummett 1973) called “ingredient sense” and “assertoric content”.

Knowledge means ‘ all ', belief means ‘ most '

Abstract: We introduce a bimodal epistemic logic intended to capture knowledge as truth in all epistemically alternative states and belief as a generalized 'majority' quantifier, interpreted as truth in many (a 'majority' of the) epistemically alternative states. This doxastic interpretation is of interest in KR applications and it also has an independent philosophical and technical interest. The logic KBM comprises an S4 epistemic modal operator, a doxastic modal operator of consistent and complete belief and 'bridge' axioms which relate knowledge to belief. To capture the notion of a 'majority' we use the 'large sets' introduced independently by K. Schlechta and V. Jauregui, augmented with a requirement of completeness, which furnishes a 'weak ultrafilter' concept. We provide semantics in the form of possible-worlds frames, properly blending relational semantics with a version of general Scott-Montague (neighborhood) frames and we obtain soundness and completeness results. We examine the validity of certain epistemic principles discussed in the literature, in particular some of the 'bridge' axioms discussed by W. Lenzen and R. Stalnaker, as well as the 'paradox of the perfect believer', which is not a theorem of KBM.

Dynamic Logics of Dynamical Systems

Abstract: We survey dynamic logics for specifying and verifying properties of dynamical systems, including hybrid systems, distributed hybrid systems, and stochastic hybrid systems. A dynamic logic is a first-order modal logic with a pair of parametrized modal operators for each dynamical system to express necessary or possible properties of their transition behavior. Due to their full basis of first-order modal logic operators, dynamic logics can express a rich variety of system properties, including safety, controllability, reactivity, liveness, and quantified parametrized properties, even about relations between multiple dynamical systems. In this survey, we focus on some of the representatives of the family of differential dynamic logics, which share the ability to express properties of dynamical systems having continuous dynamics described by various forms of differential equations. We explain the dynamical system models, dynamic logics of dynamical systems, their semantics, their axiomatizations, and proof calculi for proving logical formulas about these dynamical systems. We study differential invariants, i.e., induction principles for differential equations. We survey theoretical results, including soundness and completeness and deductive power. Differential dynamic logics have been implemented in automatic and interactive theorem provers and have been used successfully to verify safety-critical applications in automotive, aviation, railway, robotics, and analogue electrical circuits.

Defeasible modes of inference: A preferential perspective

Abstract: Historically, approaches to defeasible reasoning have been concerned mostly with one aspect of defeasibility, viz. that of arguments, in which the focus is on normality of the premise. In this paper we are interested in another aspect of defeasibility, namely that of defeasible modes of reasoning. We do this by adopting a preferential modal semantics that we defined in previous work and which allows us to refer to the relative normality of accessible worlds. This leads us to define preferential versions of the traditional notions of knowledge, beliefs, obligations and actions, to name a few, as studied in modal logics. The resulting preferential modal logics make it possible to capture, and reason with, aspects of defeasibility heretofore beyond the reach of modal formalisms. Introduction and Motivation Defeasible reasoning, as traditionally studied in the literature on non-monotonic reasoning, has focused mostly on one aspect of defeasibility, namely that of arguments. Such is the case, for instance, in the well-known KLM approach (Kraus, Lehmann, and Magidor 1990; Lehmann and Magidor 1992), in which (propositional) defeasible consequence relations |∼ are studied. In this setting, the meaning of a defeasible statement (or a ‘conditional’, as it is sometimes referred to) of the form α |∼ β is that “all normal α-worlds are β-worlds”, leaving it open for α-worlds that are, in a sense, exceptional not to satisfy β. With the theory that has been developed around this notion it becomes possible to cope with exceptionality when performing reasoning. There are of course many other appealing and equally useful aspects of defeasibility besides that of arguments. These include notions such as typicality (Giordano et al. 2009; Booth, Meyer, and Varzinczak 2012), concerned with the most typical cases or situations (or even the most typical representatives of a class), and belief plausibility (Baltag and Smets 2008), which relates to the most plausible epistemic possibilities held by an agent, amongst others. It turns out that with KLM-style defeasible statements one cannot capture these aspects of defeasibility. This has to do partly with the syntactic restrictions imposed on |∼, namely no nesting of conditionals, but, more fundamentally, it relates to where and how the notion of normality is used in such statements. Indeed, in a KLM defeasible statement α |∼ β, the normality spotlight is somewhat put on α, as though normality was a property of the premise and not of the conclusion. Whether the β-worlds are normal or not plays no role in the reasoning that is carried out. Furthermore, normality is assumed to be a property of the premise as a whole, and not of its constituents. Technically this means one cannot refer directly to normality of a sentence in the scope of logical operators. This is also the case in recent extensions of the KLM approach to logics that are more expressive than the propositional one (Britz, Heidema, and Meyer 2008; Britz, Meyer, and Varzinczak 2011a; 2011b). In this paper we are interested in aspects of defeasibility related to the aforementioned idea of beliefs that are expressed in terms of most plausible accessible worlds. We investigate a more general notion which we refer to as defeasible modes of inference. These amount to preferential versions of the traditional notions of knowledge, beliefs, obligations and actions, to name a few, as studied in modal logics. For instance, in an action context, one may want to state that the outcome of a given action a is usually α, i.e., in the most normal situations resulting from the action’s execution, α holds. This is notably different from saying that in the most normal worlds, the result of performing the action a is always α, i.e., stating > |∼ 2aα in Britz et al.’s (2011a) modal extension to preferential reasoning. To give a more concrete example, one thing is to say that in any normal situation, a head-on collision at high speed results in a situation where there are fatalities, whereas another one is to say that in any situation, a head-on collision at high speed results in a situation in which there normally are fatalities. Here we are interested in the formalization of the latter type of statement, where it becomes important to shift the notion of normality from the antecedent of an inference to the effect of an action, and, importantly, use it in the scope of other logical constructors. The importance of defeasibility in specific modes of reasoning is also illustrated by the following example. Although one may envisage a situation where the velocity of a sub-atomic particle in a vacuum is greater than c (the speed of light in a vacuum), it is in a sense known that c is the highest possible speed. Moreover, one can derive factual consequences of this scientific theory that also will be ‘known’. This venturous version of knowledge, which patently differs from belief, provides for a more fine grained notion of knowledge that may turn out to be wrong, i.e., that is defeasible, but that is not of the same nature as suppositions or beliefs. Our proposal is not aimed at challenging the position of knowledge as indefeasible, justified true belief (Gettier 1963; Lehrer and Paxson 1969), but rather provides an extension to epistemic modal logics to allow for reasoning with a modality that we shall, argueably for lack of a more suitable term, refer to as “defeasible knowledge”. Our third example concerns obligations and weaker versions thereof. There is a subtle difference between stating that, in any normal situation, one ought to tell the truth, and stating that, in any situation, it is one’s normal duty to tell the truth. In the latter the normality of the current situation is immaterial, whereas in the former it determines the truth of the statement. Therefore, the shift in focus is again from normality of the present world, to relative normality of accessible worlds. Scenarios such as the ones depicted above require an ability to talk about the normality of effects of an action, normality of knowledge or obligations, and so on. While existing modal treatments of preferential reasoning can express preferential semantics syntactically as modalities (Boutilier 1994; Giordano et al. 2005; Britz, Heidema, and Labuschagne 2009), they do not suffice to express defeasible modes of inference as described above. In this paper we fill this gap by introducing (non-standard) modal operators allowing us to talk about relative normality in accessible worlds. With our defeasible versions of modalities, we can make statements of the form “α holds in all of the normal accessible worlds”, thereby capturing defeasibility of what is ‘expected’ in target worlds. Such a notion of defeasibility in a modality meets a variety of applications in Artificial Intelligence, ranging from reasoning about actions to deontic and epistemic reasoning. For instance, we define a defeasible-action operator allowing us to make statements of the form p∼∼paα, which we read as “α is a normal necessary effect of a”, and we define defeasible-knowledge operators with which one can state formulas such as p∼∼pAα, read as “agent A knows that normally α”. These operators are defined within the context of a general preferential modal semantics (Britz, Meyer, and Varzinczak 2011a). The relative normality of a given world in a Kripke model is determined by a preference order on states, serving as place holders for pointed Kripke models. In contrast with the plausibility models of Baltag and Smets (2008), our order on states does not define an agent’s knowledge or beliefs. Rather, it is part of the semantics of the background ontology described by the theory or knowledge base at hand. As such, it informs the meaning of defeasible actions, which can fail in their outcome, or defeasible knowledge, which may not hold in exceptional accessible worlds, in that it alters the classical semantics of these modalities. This allows for the definition of a family of modal logics in which defeasible modes of inference can be expressed, and which can be integrated with existing non-monotonic modal logics. The remainder of the present paper is structured as follows: In the next section we set up the modal notation of the paper and we recall the preferential semantics for modal logics that we shall use throughout this paper. Following that we present a logic enriched with defeasible modalities allowing for the formalization of defeasible versions of e.g. knowledge and actions, which we illustrate with examples in the following section. After a discussion of, and comparison with related work, we conclude with directions for further investigations.

Counterpart Semantics for a Second-Order μ-Calculus

Abstract: Quantified μ-calculi combine the fix-point and modal operators of temporal logics with (existential and universal) quantifiers, and they allow for reasoning about the possible behaviour of individual components within a software system. In this paper we introduce a novel approach to the semantics of such calculi: we consider a sort of labeled transition systems called counterpart models as semantic domain, where states are algebras and transitions are defined by counterpart relations (a family of partial homomorphisms) between states. Then, formulae are interpreted over sets of state assignments (families of partial substitutions, associating formula variables to state components). Our proposal allows us to model and reason about the creation and deletion of components, as well as the merging of components. Moreover, it avoids the limitations of existing approaches, usually enforcing restrictions of the transition relation: the resulting semantics is a streamlined and intuitively appealing one, yet it is general enough to cover most of the alternative proposals we are aware of. The paper is rounded up with some considerations about expressiveness and decidability aspects.

Temporal reference in a genuinely tenseless language: The case of Hausa

Abstract: In this paper, we provide an analysis of temporality in Hausa (Chadic, Afro-Asiatic). By testing the hypothesis of covert tense (Matthewson 2006) against empirical data, we show that Hausa is genuinely tenseless in the sense that the grammar does not restrict the relation between reference time and utterance time. Rather, temporal reference is pragmatically inferred from aspectual and contextual information. We also argue that future time reference in Hausa is realized as a combination of a modal operator and a prospective aspect, thus involving the modal meaning components of intention and prediction as well as event time shifting.

Reasoning about knowledge in distributed systems using datalog

Abstract: Logic programming has been considered a viable solution for distributed computing since the Fifth Generation Computer Systems project [8]. Nowadays, this line of thought is gaining new verve, pushed by the need for new programming paradigms for addressing new emerging issues in distributed computing. We argue that a missing piece in the current state-of-the-art is the capability to express statements about the knowledge state of distributed nodes. In fact, reasoning about the knowledge state of (group of) nodes has been demonstrated to be fundamental in order to design and analyze distributed protocols [7]. To reach this goal, we designed Knowlog: Datalog¬ augmented with a set of epistemic modal operators, allowing the programmer to directly express what a node "knows" instead of low level communication details.

Denotation of syntax and metaprogramming in contextual modal type theory (CMTT)

Abstract: The modal logic S4 can be used via a Curry-Howard style correspondence to obtain a calculus. Modal (boxed) types are intuitively interpreted as ‘closed syntax of the calculus’. This -calculus is called modal type theory — this is the basic case of a more general contextual modal type theory, or CMTT. CMTT has never been given a denotational semantics in which modal types are given denotation as closed syntax. We show how this can indeed be done, with a twist. We also use the denotation to prove some properties of the system.

First Steps in Synthetic Guarded Domain Theory.

Abstract: We present the topos S of trees as a model of guarded recursion. We study the internal dependently-typed higher-order logic of S and show that S models two modal operators, on predicates and types, which serve as guards in recursive definitions of terms, predicates, and types. In particular, we show how to solve recursive type equations involving dependent types. We propose that the internal logic of S provides the right setting for the synthetic construction of abstract versions of step-indexed models of programming languages and program logics. As an example, we show how to construct a model of a programming language with higher-order store and recursive types entirely inside the internal logic of S. Moreover, we give an axiomatic categorical treatment of models of synthetic guarded domain theory and prove that, for any complete Heyting algebra A with a well-founded basis, the topos of sheaves over A forms a model of synthetic guarded domain theory, generalizing the results for S.

How Things Are Elsewhere

Abstract: When quantifiers and modal operators mingle, all sorts of troubles arise. Legend has it that after some initial confusion about how to make sense of formulas like ∃x◊Fx, the issue was finally settled by Saul Kripke, who put forward what is now known as Kripke semantics for quantified modal logic. Formulas like ∃x◊Fx are interpreted by models consisting of some “possible worlds”, each equipped with a quantifier domain, and an interpretation function that specifies which individuals satisfy which predicates relative to which worlds. Modal operators function as quantifiers over the worlds, restricted by an “accessibility” relation. ∃x◊Fx is true at a world w iff there is an individual in the domain of w that satisfies F relative to some world accessible from w.

The Square of Opposition in Orthomodular Logic

Abstract: In Aristotelian logic, categorical propositions are divided in Universal Affirmative, Universal Negative, Particular Affirmative and Particular Negative. Possible relations between two of the mentioned type of propositions are encoded in the square of opposition. The square expresses the essential properties of monadic first order quantification which, in an algebraic approach, may be represented taking into account monadic Boolean algebras. More precisely, quantifiers are considered as modal operators acting on a Boolean algebra and the square of opposition is represented by relations between certain terms of the language in which the algebraic structure is formulated. This representation is sometimes called the modal square of opposition. Several generalizations of the monadic first order logic can be obtained by changing the underlying Boolean structure by another one giving rise to new possible interpretations of the square.

An approach to extraction of linguistic recommendation rules: application of modal conditionals grounding

Abstract: An approach to linguistic summarization of distributed databases is considered. It is assumed that summarizations are produced for the case of incomplete access to existing data. To cope with the problem the stored data are processed partially (sampled). In consequence summarizations become equivalent to the natural language modal conditionals with modal operators of knowledge, belief and possibility. To capture this case of knowledge processing an original theory for grounding of modal languages is applied. Simple implementation scenarios and related computational techniques are suggested to illustrate a possible utilization of this model of linguistic summarization.

Internal Models for Coalgebraic Modal Logics

Abstract: We present ongoing work into the systematic study of the use of dual adjunctions in coalgebraic modal logic. We introduce a category of internal models for a modal logic. These are constructed from syntax, and yield a generalised notion of canonical model. Further, expressivity of a modal logic is shown to be characterised by factorisation of its models via internal models and the existence of cospans of internal models.

Knowlog: a declarative language for reasoning about knowledge in distributed systems

Abstract: In the last few years, researchers started to investigate how recursive queries and deductive languages can be applied to find solutions to the new emerging trends in distributed computing. We conjecture that a missing piece in the current state-of-the-art in logic programming is the capability to express statements about the knowledge state of distributed nodes. In fact, reasoning about the state of remote nodes is fundamental in distributed contexts in order to design and analyze protocols behavior. To reach this goal, we leveraged Datalog¬ with an epistemic modal operator, allowing the programmer to directly express nodes' state of knowledge instead of low level communication details. To support the effectiveness of our proposal, we introduce, as example, the declarative implementation of the two phase commit protocol.

The Genesis of Hi-Worlds: Towards a Principle-Based Possible World Semantics

Abstract: A Leibnizian semantics proposed by Becker in 1952 for the modal operators has recently been reviewed in Copeland’s paper The Genesis of Possible World Semantics (Copeland in J Philos Logic 31:99–137, 2002), with a remark that “neither the binary relation nor the idea of proving completeness was present in Becker’s work”. In light of Frege’s celebrated Sense-Determines-Reference principle, we find, however, that it is Becker’s semantics, rather than Kripke’s semantics, that has captured the true spirit of Frege’s semantic program. Furthermore, for Kripke’s possible world semantics to fit in Frege’s framework of senses, worlds and referents, it will have to be thoroughly reformulated. By introducing the notion of a hi-world into the picture, we manage to keep the key ingredients of Becker’s semantics intact, while at the same time solve a fatal problem that used to shadow Becker’s original semantics—it had not been able to make sense of inhomogeneous modality. The resulting generalized Beckerian semantics provides, in effect, a Beckerian analysis of the Kripkean possible worlds. It reveals the subtle hierarchical internal structure of a Kripkean world that has not been discovered before.

Indispensability arguments and instrumental nominalism

Abstract: In the philosophy of mathematics, indispensability arguments aim to show that we are justified in believing that mathematical objects exist on the grounds that we make indispensable reference to such objects in our best scientific theories (Quine, 1981a; Putnam, 1979a) and in our everyday reasoning (Ketland, 2005). I wish to defend a particular objection to such arguments called instrumental nominalism. Existing formulations of this objection are either insufficiently precise or themselves make reference to mathematical objects or possible worlds. I show how to formulate the position precisely without making any such reference. To do so, it is necessary to supplement the standard modal operators with two new operators that allow us to shift the locus of evaluation for a subformula. I motivate this move and give a semantics for the new operators.