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Journal of Semantics 22: 119–128 doi:10.1093/jos/ffh030 Modality and Temporality CLEO CONDORAVDI Palo Alto Research Cen...

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Journal of Semantics 22: 119–128 doi:10.1093/jos/ffh030

Modality and Temporality CLEO CONDORAVDI Palo Alto Research Center STEFAN KAUFMANN Northwestern University

COUNTERFACTUALS Any theory of counterfactuals has to grapple with the fact that judgments about their truth or falsehood cannot be explained in terms of logical relations alone. Invariably, such judgments appear to draw on additional assumptions about non-logical dependencies between facts or propositions. This gives rise to some of the hardest problems in the theory of conditionals: What is the nature of these relations between facts or propositions? Are they all of one kind, or are different relations relevant for different counterfactuals? Can they be analysed without circular reference to counterfactuals? And just how much of this additional information needs to be incorporated into the formal semantic theory? The most prominent semantic approach to counterfactuals is ordering semantics, developed by Stalnaker (1968) and Lewis (1973b). Both authors were concerned with providing a logical theory of the special inferential relationships between counterfactuals, a purpose for which the classical Fregean material conditional is famously inadequate. Ignoring differences in detail, both rely on a model theory in which a notion of similarity between possible worlds plays a central role. For instance, the truth value of (1a) at a world at which the match was not scratched is determined by the truth value of its consequent at the most similar world(s) at which it was. Ó The Author 2005. Published by Oxford University Press. All rights reserved.

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The present collection addresses a number of issues in the semantic interpretation of modal and temporal expressions. Despite the variety the papers exhibit both in the selection of topics and the choice of formal frameworks, they are interconnected through several overarching themes that are at the centre of much ongoing research. The purpose of this brief introduction is to put the papers into context and draw the reader’s attention to some of these connections. The topics we will discuss in the remainder are: counterfactuals, causality, partiality, compositionality of conditionals, and context dependence.

120 Modality and Temporality (1) a. If that match had been scratched, it would have lighted. b. If the match had been wet and scratched, it would have lighted.

(2) If that match had been scratched, it would have been wet. The consequent of (1a) follows from the antecedent and the fact that the match was dry (ignoring certain other factors, such as the presence of oxygen). The consequent of (2) follows from the same antecedent and the fact that the match did not light. Why do most speakers consider the fact that the match was dry, but not the fact that the match did not light, relevant to their deliberations of what would have been the case if it had been scratched? Within ordering semantics, the question becomes how exactly the similarity relation ought to be specified, and whether it can be reduced to some more basic notion. The proponents of the theory have made only tentative suggestions in this regard. Lewis (1973a, 1979) proposed to use judgments about counterfactuals as empirical evidence about the way speakers assess similarity, and put forth his well-known hierarchy of ‘miracles’, which he conceived of as more or less drastic deviations from the actual course of events. Stalnaker (1968, 1984) offered his


The resulting theory correctly invalidates certain inferences involving counterfactuals, such as that from (1a) to (1b). Under the classical interpretation this inference would be valid; but clearly (1a) can be true while (1b) is false. In accounting for the logical behaviour of counterfactuals, ordering semantics was a significant step forward and must be considered an unqualified success. At the same time, many authors have voiced doubts as to whether, in itself, it really amounts to a semantic analysis, given that it delegates the most difficult questions to the unanalysed similarity relation. Thus van Fraassen (1976) noted: ‘To the question what principles govern deductive reasoning involving conditionals, Stalnaker and Lewis give exact answers. But the validity of an argument does not depend on whether its premises are true; and indeed, Stalnaker and Lewis have not notably increased our ability to decide whether particular conditionals are true or false’ (p. 266). To this day, there is little consensus on the question of how the interpretation of counterfactuals depends on the facts or, more semantically put, the truth values of atomic and truth-functional sentences. An early statement of this question is due to Goodman (1947), who saw no way of giving a non-circular logical explanation for the fact that speakers consistently judge (1a) true and (2) false.

Cleo Condoravdi and Stefan Kaufmann 121


‘projection strategy’ as an alternative which gives more importance to epistemic considerations, explaining similarity between worlds in terms of the logic of belief update and revision. A third possible strategy would be to take certain dependencies between facts in a possible world as basic and formulate the semantic analysis of counterfactuals in terms of those. One framework in which this latter approach has been explored is premise semantics, originally proposed by Veltman (1976) and Kratzer (1981). Here the semantics of counterfactuals makes reference to premise sets, maximally consistent sets of propositions compatible with the antecedent. Premise semantics provides a way to formulate hypotheses as to which truths are given up in evaluating counterfactuals. Broadly speaking, the idea is that speakers do not view facts as mutually independent: some (but not all) facts are affected by manipulations of other facts in hypothetical reasoning. Formally, this means that some logically possible sets of premises may be irrelevant to the truth or falsehood of a given counterfactual. The question then is what determines the selection of the relevant premise sets. For Veltman (1976), this selection was driven by epistemic preferences. Kratzer (1981, 1989, 2002), on the other hand, has tried to define it in terms of assumptions about the internal structure of the world at which the counterfactual is evaluated. Two papers and one discussion note in this collection address directly the semantics of counterfactuals and the proper construction of premise sets. ‘On the Lumping Semantics of Counterfactuals’, by Makoto Kanazawa, Stefan Kaufmann and Stanley Peters, discusses Kratzer’s (1989) attempt to cast the intuition about the dependence between facts into a precise logical form and relate it to assumptions speakers appear to make about the structure of the world. Kratzer’s proposal and the underlying situation-semantic apparatus have been influential in the field, being based on intuitions that many authors find plausible. What Kanazawa et al. show is that despite its appeal, the approach is plagued by certain logical problems in its formal implementation, which are not obvious at first but lead to a number of unwelcome consequences. It is the particular formalization of lumping developed in Kratzer (1989), along with the workings of premise semantics, that leads to the triviality problems discussed in the paper. Kanazawa et al. leave open the question of whether the prima facie plausible idea can be preserved in a modified version of lumping semantics, or whether an altogether different approach is called for. This question is important, and the theory deserves that it be resolved. Kanazawa et al. show where the cracks run in the logical foundation of the most explicit formalization

122 Modality and Temporality

CAUSALITY Kratzer’s appeal to ‘lumping’ and Veltman’s notion of some facts ‘bringing others in their train’ are but two ways of placing constraints on counterfactual inferences. Another solution appeals to causal relations, which in recent years has risen to new prominence in adjacent fields, such as artificial intelligence and psychology (Ortiz, 1999a,b; Pearl, 2000), and was applied in the interpretation of conditionals by Kaufmann (2005) and


currently available. Angelika Kratzer, in her reaction paper ‘Constraining Premise Sets for Counterfactuals’, argues that by further developing the theory in directions she has indicated in more recent work, it may ultimately be possible to avoid the problems. In ‘Making Counterfactual Assumptions’, Frank Veltman gives the beginnings of a compositional analysis of counterfactuals and a new version of premise semantics for counterfactuals. Veltman takes seriously the observation, made by Tichy´ (1976), that local mismatches of facts can make for big differences in the truth of counterfactuals. Like Kratzer, he aims for a semantics of counterfactuals that pins down more precisely which facts count and which do not in evaluating the consequences of a hypothetical assumption. He makes a crucial distinction between particular facts and general laws; the latter do not depend on the particular world of evaluation. The counterfactual assumptions an agent can make are limited to those that are compatible with the laws. In Veltman’s formal system, situations assign truth values to atomic propositions, and laws complete situational bases into worlds. A basis for a world is a situation which contains all and only the basic, mutually independent facts distinguishing that world from others. A situation smaller than a basis, on the other hand, can ‘grow’ into different possible worlds. It is situations of this kind that determine the truth values of counterfactuals. A counterfactual with a false antecedent (the only case Veltman considers) is evaluated at a given world by reducing a basis of that world to a situation which admits the antecedent. This process involves the removal of some propositions, which, Veltman maintains, take others in their train: When a proposition is retracted from a world, all the independent facts that led to its truth, as well as its consequences under the laws, are retracted as well. Veltman in fact formulates the meaning of counterfactuals in terms of update conditions on belief states, whereas our informal description here is given in truth conditional terms. As Veltman notes, update distributes over worlds in this way only when the laws are fixed. Thus the reader should be alerted that our description here covers only this special case.


FROM EVENT DESCRIPTIONS AND TIME TO WORLDS Both Veltman and Hobbs make use of partial entities—situations in one case, eventualities in the other—as well as worlds, and largely abstract away from time. Veltman takes worlds and situations to be total and partial functions, respectively, from atomic formulas to truth values. Hobbs appeals to an ontology of eventualities, construed rather broadly as facts that may or may not hold in a particular world. For example, in addition to the eventuality of some basic property holding of an individual, there are negative eventualies—the eventuality of another eventuality not existing—and what we may call modal eventualities, such as the eventuality of another eventuality being hypothetical. It is fair to say that the exact nature of a model-theoretic interpretation for such ontological entities is an open question. Hobbs considers time with regard to temporal order and causal flow, but does not in general


others. Here again, we face unresolved foundational questions: What are the relata of causal relations? What are their logical properties? How do they enter into speakers’ reasoning about particular sentences? How are they utilized in the absence of full knowledge about the relevant facts? The importance of an adequate notion of causality is by no means restricted to counterfactuals. Jerry Hobbs, in the paper ‘Toward a Useful Concept of Causality for Lexical Semantics’, starts out by noting that it is required for the analysis of a wide variety of expressions, and moves on to propose an account of its logical properties that is independent of any particular linguistic application. Central to the proposal is the notion of a causal complex, a collection of eventualities which in their totality are responsible for the effect, and none of which is irrelevant to the occurrence of the effect. Sceptical about the possibility of giving a complete definition of the concept, Hobbs’ goal is to identify general conditions on causal complexes that support linguistically relevant inferences. Using techniques from non-monotonic logic, Hobbs weakens inferences which go against the direction of causality to ensure that they do not lead to counterintuitive consequences in causal reasoning. Along the way, he develops a number of auxiliary notions that should prove independently interesting, especially that of a closest world, which is similar but not equivalent to that of the Stalnaker/Lewis theory of counterfactuals. This raises new questions, such as what exactly the relationship is and whether Hobbs’ notion of closeness may offer a causal explication of Stalnaker’s.

124 Modality and Temporality

A COMPOSITIONAL SEMANTICS FOR INDICATIVE CONDITIONALS It has been widely accepted since the work of Lewis (1975) and Kratzer (1979) that the semantic contribution of if-clauses is to restrict the domain of an overt or covert modal operator with scope over the consequent clause. The question remains, however, how this restriction comes about in the process of compositional interpretation. Von Fintel (1994) proposed that if-clauses may act as modifiers of consequent clauses, but left open the details of a compositional analysis. Another largely unaddressed issue regarding the meaning of indicative conditionals is the interpretation of Present and Past tenses in their antecedents and consequents (a notable exception is Crouch 1993). The variety of temporal readings and the semantic interdependence between the tenses in antecedent and consequent are illustrated by (3) and (4). (3) a. b. c.

If he comes out smiling, the interview went well. If he came out smiling, the interview went well. If he went in smiling, the interview will go well.


address change through time, i.e. the fact that both an eventuality and its negation can be realized in a world at different times. In the paper ‘Schedules in a Temporal Interpretation of Modals’, Tim Fernando approaches temporal matters constructively, treating eventuality descriptions rather than worlds as primitives. He formulates both worlds and eventualities as relations between time and eventuality descriptions—relations he calls schedules. Schedules ground eventuality descriptions in time, and, as such, amount to temporal realizations of eventuality descriptions. Insofar as eventuality descriptions can be understood as intensional notions and schedules as extensional notions, Fernando’s formation of schedules from eventuality descriptions reverses the Montagovian tradition of deriving intensions from extensions (by abstracting over a world parameter). Of particular interest in the paper is the fact that schedules satisfy not just atomic eventuality descriptions, but also descriptions with temporal and modal operators. Fernando argues that satisfaction need not rest on worlds, even for modal formulas involving epistemic or historical alternatives, and offers a reformulation of the temporal interpretation of modals proposed in Condoravdi (2002). The paper generalizes the notion of eventive, stative and temporal properties by defining a satisfaction predicate (‘forcing’) that is persistent relative to a partial order on schedules. Persistence then allows one to reconstruct worlds from certain so-called generic sets of schedules.


(4)

a.

If he is at the interview (now/when we call him), he will be late for the meeting. b. If he is at the interview, the interviews are on schedule.

CONTEXT DEPENDENCE AND DYNAMIC INTERPRETATION Moving above the level of individual sentences, we face the inextricable context dependence of modal and conditional expressions. Not only is their interpretation determined and constrained by a variety of contextual parameters, but they in turn operate on the context, affecting the interpretation of subsequent utterances. The cross-sentential dependencies that result from such interactions are


Note, for instance, that the Past tense in the consequent of (3a) indicates backshifting from a future time, wherease those in the antecedents of (3b,c) and in the consequent of (3b), on their most natural interpretations, indicate backshifting from the time of utterance. Similarly, the Present tense in the antecedent of (3a) calls for a forwardshifted interpretation, while that in the antecedent and consequent of (4b) indicates overlap with the time of utterance and that in the antecedent of (4a) is compatible with either interpretation. In ‘Conditional Truth and Future Reference’, Stefan Kaufmann proposes a compositional semantics for indicative conditionals which brings together the modal and temporal elements of their interpretation. He treats if-clauses as modifiers of the consequent, with the desired effect of restricting the modal base associated with the latter. Regarding temporal interpretation, he makes the (at first sight striking) claim that the tenses in the antecedent and consequent of indicative conditionals receive the same interpetation as in isolation, and demonstrates that this assumption helps explain a number of otherwise puzzling facts. He shows how the same basic meaning can give rise to predictive and nonpredictive, metaphysical and doxastic readings, depending on contextual parameters and the choice of modal base for the consequent, and explains why non-predictive readings tend to be associated with doxastic modality. The main technical innovation of Kaufmann’s paper is to take apart the various parameters involved in the interpretation of a modal and then have them enter the process of compositional interpretation separately at different stages, rather than being fixed once at the top level by the context. In this way, a modal can be transformed into a modal-temporal operator by an if-clause.

126 Modality and Temporality subsumed under the label ‘modal subordination’. Some examples are given in (5) through (7). (5) a. b. (6) a. b. (7) a. b.

A thief might come in. He would take the silver. Mary may come to the party. Sue may come, too. I don’t have a car. It would be parked outside.


In each of these mini-sequences, the (b)-sentence carries certain presuppositions (triggered by he in (5b), too in (6b), it in (7b)). One question that such examples raise is why the second sentence is felicitous in the given context, even though on the face of it, its presuppositions are not satisfied: For instance, the existence of a thief is not asserted in (5a), nor does (6a) assert that Mary (or anyone else) is coming to the party. Furthermore, (5b) and (7b) are intuitively interpreted as the consequents of conditionals whose antecedents are (5a) and (7a), respectively. A variety of proposals for dealing with this phenomenon have been put forth in the literature. Common to all of them is a dynamic perspective on the interaction between sentences and their contexts of interpretation, but regarding the nature of this interaction, we can discern several different approaches. Roberts (1989) appealed to complex inferences and accommodation to explain how sentences like (5b) are interpreted. Others view the dependency as essentially anaphoric (Frank 1996; Geurts 1998; Kibble 1998). Such approaches have been criticized for being too unconstrained and unable to account for the fact that the dependency is, with very few exceptions, limited to the immediately preceding discourse context. Still others draw a crucial distinction between those contexts which result from an update with a licensing expression (such as 5a–7a) and give rise to modal subordination for that reason, and those contexts which do not. Kaufmann (2000) made one such proposal, arguing that this approach is better suited to account for the locality of modal subordination. In the paper ‘A Modal Analysis of Presupposition and Modal Subordination’, Robert van Rooij offers a novel approach in the same conceptual vein, addressing a wider range of data and employing a leaner formal framework. Contexts are represented as modal accessibility relations. An update with a modally subordinating expression results in a context with special properties (rather than a stack of multiple contexts, as Kaufmann would have it). The evaluation


of presupposition-carrying sentences in such contexts is spelled out in a two-dimensional framework which is inspired by Karttunen and Peters’ (1979) account of conversational implicature, but not subject to certain well-documented problems afflicting the latter. The proposal should therefore be of independent interest.

Acknowledgments

CLEO CONDORAVDI Palo Alto Research Center 3333 Coyote Hill Drive Palo Alto CA 94304, USA e-mail: [email protected]

Submitted and accepted: 20.02.05 Final version received: 22.02.05

STEFAN KAUFMANN Department of Linguistics Northwestern University 2016 Sheridan Road Evanston IL 60208, USA e-mail: [email protected]

REFERENCES Condoravdi, C. (2002) ‘Temporal interpretation of modals: Modals for the present and for the past’. In D. I. Beaver, L. Casillas, B. Clark, & S. Kaufmann (eds), The Construction of Meaning. CSLI Publications, Stanford, CA, 59–88. Crouch, R. (1993) The temporal properties of English conditionals and modals. PhD thesis, University of Cambridge. Faller, M., Pauly, M., & Kaufmann, S. (eds), (2000) Formalizing the Dynamics

of Information. CSLI Publications, Stanford, CA. von Fintel, K. (1994) Restrictions on quantifier domains. PhD thesis, University of Massachusetts. van Fraassen, B. C. (1976) ‘Probabilities of conditionals’. In W. L. Harper, R. Stalnaker, & G. Pearce (eds), Foundations of Probability Theory, Statistical Inference, and Statistical Theories of Science, volume 1 of The University of Western Ontario Series


We would like to express our thanks to the eight contributors, twelve anonymous reviewers, and above all to Peter Bosch, the former managing editor of this journal, for his support and patience in seeing the project through. We would also like to thank his successor, Bart Geurts, who oversaw the last stages of this project.

128 Modality and Temporality Lewis, D. (1973a) ‘Causation’. Journal of Philosophy 70:556–567. Lewis, D. (1973b). Counterfactuals. Harvard University Press, Cambridge, MA. Lewis, D. (1975) ‘Adverbs of quantification’. In E. Keenan (ed.), Formal Semantics of Natural Language. Cambridge University Press, Cambridge/New York, NY, 3–15. Lewis, D. (1979) ‘Counterfactual dependence and time’s arrow’. Nouˆs 13:455–476. Ortiz, C. (1999a) ‘A commonsense language for reasoning about causation and rational action’. Artificial Intelligence 111:73–130. Ortiz, C. (1999b) ‘Explanatory update theory: Applications of counterfactual reasoning to causation’. Artificial Intelligence 108:125–178. Pearl, J. (2000) Causality: Models, Reasoning, and Inference. Cambridge University Press, Cambridge/New York, NY. Roberts, C. (1989) ‘Modal subordination and pronominal anaphora in discourse’. Linguistics and Philosophy 12:683–721. Stalnaker, R. (1968) ‘A theory of conditionals’. In J. W. Cornman (ed), Studies in Logical Theory, American Philosophical Quarterly, Monograph: 2. Blackwell, Oxford, 98–112. Tichy´, P. (1976) ‘A counterexample to the Lewis-Stalnaker analysis of counterfactuals’. Philosophical Studies 29:271–273. Veltman, F. (1976) ‘Prejudices, presuppositions and the theory of conditionals’. In J. Groenendijk & M. Stokhof (eds), Proceedings of the First Amsterdam Colloquium [¼Amsterdam Papers in Formal Grammar, Vol. 1], Centrale Interfaculteit, Universiteit van Amsterdam, 248–281.


in Philosophy of Science, D. Reidel. 261–308. Frank, A. (1996) Context dependence in modal constructions. PhD thesis, Institut fu¨r maschinelle Sprachverarbeitung, Stuttgart. Geurts, B. (1998) ‘Presuppositions and anaphora in attitude contexts’. Linguistics and Philosophy 21:545–601. Goodman, N. (1947) ‘The problem of counterfactual conditionals’. The Journal of Philosophy 44:113–128. Karttunen, L. & Peters, S. (1979) ‘Conventional implicature’. In C.-K. Oh & D. Dinneen (eds), Presupposition, volume 11 of Syntax and Semantics. Academic Press, New York, NY, 1–56. Kaufmann, S. (2000) ‘Dynamic discourse management’. In M. Faller, M. Pauly, & Kaufmann (eds), Formalizing the Dynamic of Information. CSLI Publications, Place, 171–188. Kaufmann, S. (2005) ‘Conditional predictions: A probabilistic account’. Linguistics and Philosophy. To appear. Kibble, R. (1998) ‘Modal subordination, focus and complement anaphora’. In J. Ginzburg, Z. Khasidashvili, C. Vogel, J.-J. Le´vi, & E. Vallduvı´ (eds), The Tbilisi Symposium on Language, Logic and Computation: Selected Papers. CSLI Publications, Stanford, CA, 71–84. Kratzer, A. (1979) ‘Conditional necessity and possibility’. In U. Egli, R. Baüerle, & A. von Stechow (eds), Semantics from Different Points of View. Springer, Berlin/New York, NY, 117–147. Kratzer, A. (1981) ‘Partition and revision: The semantics of counterfactuals’. Journal of Philosophical Logic 10:201–216. Kratzer, A. (1989) ‘An investigation of the lumps of thought’. Linguistics and Philosophy 12:607–653. Kratzer, A. (2002) ‘Facts: Particulars of information units?’ Linguistics and Philosophy 25:655–670.

Journal of Semantics 22: 129–151 doi:10.1093/jos/ffh027 Advance Access publication March 29, 2005

On the Lumping Semantics of Counterfactuals MAKOTO KANAZAWA National Institute of Informatics STEFAN KAUFMANN Northwestern University

Abstract Kratzer (1981) discussed a naı¨ve premise semantics of counterfactual conditionals, pointed to an empirical inadequacy of this interpretation, and presented a modification—partition semantics—which Lewis (1981) proved equivalent to Pollock’s (1976) version of his ordering semantics. Subsequently, Kratzer (1989) proposed lumping semantics, a different modification of premise semantics, and argued it remedies empirical failings of ordering semantics as well as of naı¨ve premise semantics. We show that lumping semantics yields truth conditions for counterfactuals that are not only different from what she claims they are, but also inferior to those of the earlier versions of premise semantics.

1 INTRODUCTION Counterfactuals pose some of the most recalcitrant problems for truth-conditional semantic analysis. The long and rich tradition of writings on this topic, despite substantial advances in many directions, has so far failed to deliver a formally explicit and intuitively accurate account of how their truth conditions depend on those of their constituents and other non-conditional sentences. Among the most influential writings in this area are those of Kratzer (1981, 1989), the latest of which puts forward a theory centred around the novel notion of lumping, which, she argues, solves a number of problems with previous accounts. Given the initial appeal of the use of lumping in Kratzer’s (1989) semantics and its wide influence in linguistics, it is both surprising and worth pointing out that it seems to be in fundamental conflict with other features of her semantics, depriving the theory of much of its predictive power. In this paper, we carefully examine the logical The Author 2005. Published by Oxford University Press. All rights reserved.


STANLEY PETERS Stanford University

130 On the Lumping Semantics of Counterfactuals

n

n

(1) a. h/; )/ p p b. h/; )/ l l c. h/; )/ denote the paired conditionals under the three semantic interpretations. Our main result is that, under certain conditions, the lumping l

semantics of Kratzer (1989) is truth-functional. Specifically, u h/ w is l equivalent to the material conditional u/w; and u )/ w is equivalent to the conjunction u ^ w. It suffices to describe the propositional language since the critical problem with the lumping semantics for conditionals already arises in this case. Models M will be ordered pairs ÆW, Væ of a non-empty set W of possible worlds and a function V mapping propositional variables to subsets of W. For each w 2 W and M propositional variable p, ½½pM w ¼ 1 if w 2 V(p) and ½½pw ¼ 0 if w ; V(p). Below we will refer to propositions by variable names, writing ‘p’ instead of ‘V(p)’. The semantics of truth-functional connectives is as usual: For all formulas u, w and w 2 W, we set M M (2) ½½u ^ wM w ¼ 15½½uw ¼ ½½ww ¼ 1 M M (3) ½½u _ wM w ¼ 05½½uw ¼ ½½ww ¼ 0 M M (4) ½½u/wM w ¼ 05½½uw ¼ 1 and ½½ww ¼ 0 M (5) ½½:uM w ¼ 15½½uw ¼ 0

We suppress the superscript henceforth because no confusion can arise.


consequences of the lumping semantics, and show that the predictions that it makes about counterfactuals are quite different from the ones Kratzer ascribes to it. Although we do not offer a counterproposal of our own, we hope our analysis proves useful for any future attempts to develop a viable theory of counterfactuals that makes crucial use of a notion like lumping. We can best explain Kratzer’s motivations for her 1989 theory as well as present our formal analysis of it by contrasting it with the two earlier theories of counterfactuals discussed in Kratzer 1981. The three theories are all closely related and belong to the class of premise semantics. Each of the three interpretations recognizes dual counterfactual connectives, the would-conditional and the mightconditional, for which we introduce corresponding pairs of binary connectives h/ and )/, respectively. We let the six connectives

M. Kanazawa et al. 131

2 PRELIMINARIES

2.1 Background

(6) If that match had been scratched, it would have lighted. The scratching of the match does not in itself guarantee its lighting: In addition, oxygen has to be present, the match has to be dry, etc. ‘The first problem’ in the interpretation of counterfactuals, Goodman writes, ‘is . . . to specify what sentences are meant to be taken in conjunction with the antecedent as a basis for inferring the consequent’.1 Clearly, for instance, sentences which contradict the antecedent should be excluded, since otherwise many false counterfactuals would come out vacuously true. Less obvious, but far more vexing to Goodman, is the fact that speakers consistently exclude other sentences for non-logical reasons. Why, for instance, is it easy to believe that (6) is true, but unnatural to conclude (7) from the fact that the match did not light? (7) If the match had been scratched, it would have been wet. Goodman was unable to offer an answer to this and related questions that would not make circular reference to counterfactuals: His rule bluntly calls for the selection of those true sentences that would not be false if the antecedent were true. However, his suggestions inspired much subsequent work by authors who continued to grapple with the problem (Rescher 1964; Veltman 1976; Pollock 1981, and others). Kratzer’s writings on premise semantics contribute to this line of research.

1 Goodman’s second problem—that of defining ‘natural or physical or causal laws’—will not concern us in this paper.


Most current theories of conditionals are based on a simple intuition: A conditional asserts that its consequent follows when its antecedent is added to a certain body of premises. This idea was first made explicit in Ramsey’s (1929) influential statement about indicative conditionals, which inspired much subsequent work (cf. Stalnaker, 1968). It is also at the center of Goodman’s (1947) theory of counterfactuals, a close predecessor of Kratzer’s premise semantics. Goodman noted about examples like (6) that while they generally assert that some connection holds between the propositions expressed by their constituent clauses, it is rarely the case that the second follows from the first.

132 On the Lumping Semantics of Counterfactuals

2.2 Basic apparatus Central to Kratzer’s theory is the notion of a premise set. Intuitively, the premise sets associated with a counterfactual at a possible world w represent ways of adding sentences that are true at w to the antecedent, maintaining consistency. We write Premw(u) for the set of premise sets associated with u at world w. Premw(u) determines the truth values at w of both wouldcounterfactuals and might-counterfactuals with antecedent u. Kratzer’s truth conditions can be reproduced as follows.2

‘The would-counterfactual u h/ w is true at w if and only if every set in Premw(u) has a superset in Premw(u) which entails w.’ Definition 2 (might-counterfactual) ½½u)/ww ¼ 1 iff T dX 2 Premw ðuÞ"Y 2 Premw ðuÞ½X4Y/ Y \ w \ W 6¼ ˘ ‘The might-conditional u)/w is true at w if and only if there is a set in Premw(u) all of whose supersets in Premw(u) are consistent with w.’ Remark 1 uh/w iff :ðu)/:wÞ, as intended. All versions of Kratzer’s theory follow this schema. The difference lies in the definition of Premw. In all three versions, Premw depends on a parameter f(w) which identifies the set of propositions relevant to the truth of counterfactuals at w. Kratzer showed that the most naı¨ve implementation of the account is empirically inadequate and sought to improve on it by imposing further conditions on membership in Premw(u). We will discuss three versions of the theory below, distinguishing between them using superscripts: Premn, Premp and n

p

l

Preml give rise to h/; h/ and h/; respectively. 2

The intersection with W is redundant as long as the universe of the model consists only of worlds. We include it here for the sake of generality because the definitions for lumping semantics below will employ a richer ontology.


Definition 1 (would-counterfactual) ½½uh/ww ¼ 1 iff T "X 2 Premw ðuÞdY 2 Premw ðuÞ½X4Y ^ Y \ W4w

M. Kanazawa et al. 133

3 NAI¨VE PREMISE SEMANTICS In the simplest version of the account, the set Premnw ðuÞ of premise sets associated with antecedent u at w represents all possible ways of adding true sentences to the antecedent, maintaining consistency. Thus the only conditions imposed on each member X of Premnw ðuÞ are that (i) all propositions in X other than u be true at w; (ii) X be consistent; and (iii) u be in X. More concisely: Definition 3 (Naı¨ ve premise set) T Premnw ðuÞ ¼ fX4f ðwÞ [ fug j X 6¼ ˘ and u 2 Xg; where f ðwÞ ¼ fp 2 PðWÞ j w 2 pg:3 n

(8) a. If Paula weren’t buying a pound of apples, the Atlantic Ocean might be drying up n b. (Paula isn’t buying a pound of apples) )/ (the Atlantic Ocean is drying up) This unwelcome consequence is part of a much larger problem which is deeply entrenched in naı¨ve premise semantics: For any false sentence n w that is consistent with the negation of a true sentence u;:u )/w is true. This fact follows from the following equivalences, which were first shown by Veltman (1976): Proposition 1 n w5ðu/wÞ ^ ð:u/hðu/wÞÞ (9) uh/ n (10) u)/w5ðu ^ wÞ _ ð:u ^ )ðu ^ wÞÞ

3 4

PðWÞ denotes the power set of W. n n For )/, see Kratzer (1979). Only h/ is discussed in Kratzer (1981).


n

The truth conditions for the connectives h/ and )/ are as given by Definitions 1 and 2, respectively, where Premnw ðuÞ is substituted for Premw(u).4 Kratzer (1981) discusses at some length the implications of this definition, in particular the predictions it makes about the truth values of would-counterfactuals. It turns out that naı¨ve premise semantics, which she considers the ‘‘most intuitive’’ analysis of counterfactuals, is deeply flawed. Suppose w 2 W is like the actual world in that the Atlantic Ocean is not drying up, and suppose further that Paula is buying a pound of apples. Then the analysis predicts that (8a), interpreted as (8b), is true at w. Intuitively, however, sentence (8a) seems to be false in w.

134 On the Lumping Semantics of Counterfactuals n

n

Thus at a world at which u is true, uh/w and u)/w are both materially equivalent to w, the consequent. This part seems reasonable and is shared with many other logics of conditionals. More problematic n is that at a world at which u is false, u h/w comes down to strict n implication, and u)/w to the statement that u and w are logically consistent. The problem with (8), discussed above, follows from (10). Kratzer (1981) discusses a different but related problem which arises from the equivalence in (9). Naı¨ve premise semantics is, alas, very naı¨ve indeed. 4 PARTITION SEMANTICS

Definition 4 (Partition function) A function f : W/PðPðWÞÞ is a partition function if and only if for T every w 2 W, f(w) ¼ fwg. The set of premise sets for partition semantics is defined in terms of f in the same way as that for naı¨ve semantics. In partition semantics, f is supposed to be indeterminate and allowed to vary from context to context, so it constitutes a new parameter in the definition of the premise sets. Definition 5 (Partition premise set) Let f be a partition function. Then T prempw ðuÞ ¼ fX4f ðwÞ [ fug j X 6¼ ˘ and u 2 Xg: The truth definitions of counterfactuals remain the same as in Definitions 1 and 2. The resulting truth values now depend, via Prempw ; on the partition function. Naı¨ve premise semantics is a special case of partition semantics. As p before, at worlds at which u is true, the conditionals uh/w and


To address the above difficulties, Kratzer (1981) proposed a repair for naı¨ve premise semantics. Rather than treating all true sentences equally for purposes of constructing premise sets, she argued, one has to take into account the fact that speakers, in interpreting counterfactuals, entertain a more coarse-grained conception of the world, analyzing it into agglomerations of facts rather than atomic truths. Formally, Kratzer assumes that only some of the propositions that are true at the world w of evaluation are relevant to the truth of counterfactuals. These relevant propositions are determined by a partition function f. The only condition imposed on f is that the propositions it selects, taken together, uniquely identify w.

M. Kanazawa et al. 135 p

(11) Let a world w be such that a. a zebra escaped; b. it was caged with another zebra; c. a giraffe was also in the same cage. In such a world, the sentence in (12a), interpreted as (12b), is predicted to be false, given the intuitive understanding of similarity between worlds. (12) a. If a different animal had escaped, it might have been a giraffe. p b. (a different animal escaped) )/ (it was a giraffe) The reason behind this prediction is not hard to understand. Given that a zebra escaped in w, among all the possible worlds in which a different animal escaped, the ones where the other zebra escaped are more similar to w than any where the escaped animal was of a different species. Intuitively, however, sentence (12a) seems true in w. The lesson from examples like this is that the relation of ‘similarity’ between worlds that yields the right truth conditions in ordering semantics does not always correspond to the most intuitive notion of similarity. But if the former is simply a theoretical construct, then ordering semantics cannot make concrete predictions about the truth values of particular counterfactuals about which our intuitions are relatively sharp (see Kratzer 1989; 626). T This is due to the requirement that f(w) ¼ fwg There are minor differences with Lewis’ original (1973) formulation, which he argues are immaterial for the resulting semantic theory. 5 6


u)/w are both materially equivalent to w.5 However, where the antecedent is false, the choice of f determines whether they are equivalent to hðu / wÞ and )ðu ^ wÞ ðif f ðwÞ ¼ fp 2 PðWÞ j w 2 pg or f ðwÞ ¼ ffwgg for all wÞ or to some other propositions. Kratzer suggests that, in practice, the range of possible partitions may be further restricted by our ‘modes of cognition’ (p. 211). Lewis (1981) showed that this version of Kratzer’s semantics is equivalent to a version of his own ordering semantics in terms of similarity between possible worlds, as formulated by Pollock (1976).6 While this result attests to the significant expressive power of Kratzer’s theory, it also shows that the latter shares with ordering semantics a number of unwelcome features. Consider the following illustration, discussed in Kratzer (1989):

136 On the Lumping Semantics of Counterfactuals 5 LUMPING SEMANTICS

Definition 6 (Situation Model) A situation model is a triple M ¼ ÆS; PðEjBÞ

204 Toward a Useful Concept of Causality for Lexical Semantics 7 GENERAL PROPERTIES OF CAUSALITY Domain knowledge about what kinds of eventualities cause what other kinds of eventualities is encoded in axioms of form (3). These are usually very specific to domains—e.g. flipping switches causes lights to go on. These are the most common sufficient conditions for causality. The predicate cause appears in the consequent. A candidate for a general sufficient condition is the idea that every eventuality has a cause. The axiom would be stated as follows:

It is not uncontroversial that we would want this axiom. Certainly, very often we have no idea of what the cause of something is. For most of human history, people did not know what caused the wind, although they may have had theories about it. There is in commonsense reasoning, one can argue, the scientifically erroneous notion of an ‘agent’, an entity capable of initiating causal chains. Either agents could appear as the first argument of the predicate cause, or some primitive action on the part of agents, such as will(a), would initiate the causal chain and these actions would be exempt from the axiom. There is not very much that can be concluded from mere causality, without any further details. That is, there seem to be very few axioms stating general necessary conditions for causality, in which cause is in the antecedent. I will mention two. The first relates causality and existence in the real world. If we were to state this in its strongest, monotonic form, we would use the predicate causal-complex: ð"s; eÞ½causal-complexðs; eÞ ^ RexistsðsÞ RexistsðeÞ If s is a causal complex for an effect e and s really exists, then e really exists. When we state this using the predicate cause, ð4Þ ð"e1 ; eÞ½causeðe1 ; eÞ ^ Rexistsðe1 Þ RexistsðeÞ the axiom is only defeasible, because it requires the rest of e ’s causal complex, the presumably true part, to be actually true. Axiom (4) can be used with axiom (3) to show that specific causes occurring will cause their effects to occur.


ð"e2 Þ½Rexistsðe2 Þ ^ eventualityðe2 Þ ðde1 Þ½Rexistsðe1 Þ ^ causeðe1 ; e2 Þ

Jerry R. Hobbs 205

Another general necessary condition for causality is its relationship to time. Effects can’t happen before their causes: ð"e1 ; eÞ½causeðe1 ; eÞ :beforeðe; e1 Þ Similarly, ð"e1 ; eÞ½causeðe1 ; eÞ :causally-priorðe; e1 Þ Now we come to the question of whether cause should be transitive:

Let us analyze the question in terms of causal complexes. Suppose we know that an eventuality e1 is a member of a set s1, which is a causal complex for eventuality e2, which is in a causal complex s2 for eventuality e3. It is possible that s1 [ s2 is inconsistent. Shoham’s example is that taking the engine out of a car (e1) makes it lighter (e2) and making a car lighter makes it go faster (e3), so taking the engine out of the car makes it go faster. The problem with this example is that the union of the two causal complexes is inconsistent. A presumable eventuality in s2 is that the car has a working engine. When it is consistent, then we can say that s1 [ s2 is a causal complex for e3. ð5Þ ð"s1 ; s2 ; e2 ; e3 Þ½causal-complexðs1 ; e2 Þ ^ e2 2 s2 ^ causal-complexðs2 ; e3 Þ ^ consistentðs1 [ s2 Þ causal-complexðs1 [ s2 ; e3 Þ Since ð"e1 ; e2 Þ½causeðe1 ; e2 Þ ðds1 Þ½causal-complexðs1 ; e2 Þ ^ e1 2 s1 we can define the function ccf ðe1 ; e2 Þ ¼ s1 That is, ccf(e1, e2) is a causal complex by virtue of which e1 causes e2.


ð"e1 ; e2 ; e3 Þ½causeðe1 ; e2 Þ ^ causeðe2 ; e3 Þ causeðe1 ; e3 Þ

206 Toward a Useful Concept of Causality for Lexical Semantics Now suppose cause(e1, e2) and cause(e2, e3) are true. We can conclude that causal-complex(s1, e2), e2 2 s2, and causal-complex(s2, e3) are all true for some s1 and s2. If s1 [ s2 is consistent, then by (5) we can conclude causal-complexðs1 [ s2 ; e3 Þ By the definition of cause, all the eventualities in s1 fe1g and s2 fe2g are presumably true, and e2 is triggered by e1 in the causal complex s1 [ s2. Thus, we can identify e1 as the cause in s1 [ s2 for e3. This means that cause(e1, e3) holds. We have established the rule

^ consistentðccf ðe1 ; e2 Þ [ ccf ðe2 ; e3 ÞÞ causeðe1 ; e3 Þ If we take :consistent(ccf(e1, e2) [ ccf(e2, e3)) to be the abnormality condition for the axiom, then we can state the defeasible rule ð"e1 ; e2 ; e3 Þ½causeðe1 ; e2 Þ ^ causeðe2 ; e3 Þ ^ :ab1 ðe1 ; e2 ; e3 Þ causeðe1 ; e3 Þ That is, causality is defeasibly transitive. This rule is used heavily in commonsense reasoning for deducing causal chains between an effect and its ultimate cause. Several writers have argued against the transitivity of causality on the basis of examples like The cold caused the road to ice over. The icy road caused the accident. The cold caused the accident. (Hart and Honore´ 1985; Ortiz 1999b) and John’s leaving caused Sue to cry. Sue’s crying caused her mother to be upset. John’s leaving caused Sue’s mother to be upset. (Moens and Steedman 1988; Ortiz 1999b)


ð"e1 ; e2 ; e3 Þ½causeðe1 ; e2 Þ ^ causeðe2 ; e3 Þ

Jerry R. Hobbs 207

ð"e1 ; e2 Þ½causeðe1 ; e2 Þ possibleðe1 Þ ^ possibleð:e1 Þ ^ possibleðe2 Þ ^ possibleð:e2 Þ

8 SUMMARY The key move in the present development has been to distinguish between the notion of ‘causal complex’ and ‘cause’. Causal complexes can be reasoned about monotonically but can rarely be completely explicated. Causes constitute the bulk of our causal knowledge but must be reasoned about defeasibly. A precise picture of how cause and causal-complex are related was described. This has led to more precise


Neither of these examples is very compelling. Certainly the starred sentences are not about direct causes, but they are about indirect causes. Very frequently, newspapers attribute some number of deaths to a heat wave, even though the direct causes might be a variety of medical and other conditions. And we can imagine Sue’s mother complaining about the wide repercussions of John’s actions—‘Look what he did to me!’ Direct causality is, of course, not transitive. Shoham (1990) believes that cause should be antisymmetric and antireflexive. Two eventualities cannot cause each other, and an eventuality cannot cause itself. I am not sure of this. If two books are leaning against each other and keeping each other in an upright position, it seems quite reasonable to say that the one book’s condition of leaning toward the other is causing the other’s condition of leaning toward the first. If this instance of symmetry is allowed, then reflexivity follows. Each book’s position is causing its own position, though not directly. It is possible to view this as a reasonable statement, in spite of its initial implausibility. There is a strong temptation in writing about causality to confine oneself to events, that is, changes of state. This is surely not adequate, since we would like to be able to say, for example, that the slipperiness of the floor caused John to fall, and that someone spilling vegetable oil on the floor caused the floor to be slippery. A state like slipperiness can be both a cause and an effect. Nevertheless, there is something to the temptation. Whether a state is a cause or effect will not normally become an issue unless there is the possibility of a change into or out of that state. That requires that both the state and its negation be possible. Thus, the focus on events could be seen to result from this requirement. With the proper notion of ‘possible’, we could state the following axiom:

208 Toward a Useful Concept of Causality for Lexical Semantics

Acknowledgements I have profited from discussions with Lauren Aaronson, Cleo Condoravdi, Cynthia Hagstron, Pat Hayes, David Israel, Srini Narayanan, and Charlie Ortiz about this work, and from the comments of the anonymous reviewers. The research was funded in part by the Defense Advanced Research Projects Agency under Air Force Research Laboratory contract F30602-00-C-0168 and under the Department of the Interior, NBC, Acquisition Services Division, under Contract No. NBCHD030010, and in part by the National Science Foundation under Grant Number IRI-9619126 (Multimodal Access to Spatial Data). The U.S. Government is authorized to reproduce and distribute reports for Governmental purposes notwithstanding any copyright annotation thereon. The views and conclusions contained herein are those of the author and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of any of the above organizations or any person connected with them.

JERRY R. HOBBS Information Sciences Institute 4676 Admiralty Way Marina del Rey, California 90292 USA e-mail: [email protected]

Received: 30.06.03 Final version received: 07.09.04 Advance Access publication: 11.04.05


characterizations of some of the properties of causality, such as transitivity. These concepts cannot be defined in terms of necessary and sufficient conditions. But it has been possible to specify a number of necessary conditions and a number of sufficient conditions, thereby constraining what a causal complex and a cause can be. All of this puts us in a good position to study a number of linguistic phenomena in terms of their underlying causal content. Section 2 gives the example of modals such as ‘would’. Other such phenomena are the meanings of explicitly causal words like ‘because’ and ‘so’, and causal uses of prepositions like ‘from’, ‘before’, ‘after’, ‘for’, and ‘at’; causative lexical decompositions, e.g., ‘kill’ as ‘cause to become not alive’ and ‘teach’ as ‘cause to learn’ or ‘cause to come to know’; coherence relations in discourse such as explanation; and many others. One often senses an uneasiness in accounts of linguistic phenomena in terms of causality, as though one were building on very shaky foundations. What I have tried to do in this paper is put the concept of causality on a firm enough basis that we no longer have to be embarassed by an appeal to causality in linguistic analyses.

Jerry R. Hobbs 209

REFERENCES printed in M. Ginsberg (ed.) Readings in Nonmonotonic Reasoning, 145–152, Morgan Kaufmann Publishers, Inc. Los Altos, California.) Moens, M. & Steedman, M. (1988). ‘Temporal ontology and temporal reference’. Computational Linguistics 14(2):15–28. Ortiz, C. L. (1999a). ‘A commonsense language for reasoning about causation and rational action’. Artificial Intelligence 111(2):73–130. Ortiz, C. L. (1999b). ‘Explanatory update theory: Applications of counterfactual reasoning to causation’. Artificial Intelligence 108(1–2): 125–178. Pearl, J. (2000). Causality. Cambridge University Press. Shoham, Y. (1990). ‘Nonmonotonic reasoning and causation’. Cognitive Science 14:213–252. Shoham, Y. M. (1991). ‘Remarks on Simon’s comments’. Cognitive Science 15:301–303. Simon, H. A. (1952). ‘On the definition of the causal relation’. The Journal of Philosophy. 49:517–528. Simon, H. A. (1991). ‘Nonmonotonic reasoning and causation: Comment’. Cognitive Science 15:293–300. Suppes, P. (1970). A Probabilistic Theory of Causation. North Holland Press.


Frank, A. & Kamp, H. (1997). ‘On context dependence in modal constructions’. Proceedings, SALT 7, Stanford University. March. Hart, H. L. A., & Honore´, T. (1985). Causation in the Law. Clarendon Press. Hobbs, J. R. (1985a). ‘Ontological promiscuity.’ Proceedings, 23rd Annual Meeting of the Association for Computational Linguistics, Chicago, Illinois, July 61–69. Hobbs, J. R. (1985b). ‘On the coherence and structure of discourse’. Report No. CSLI-85-37, Center for the Study of Language and Information. Stanford University. Hobbs, J. R., Stickel, M., Appelt, D. & Martin, p.(1993). ‘Interpretation as abduction’. Artificial Intelligence 63(1–2):169–142. Hobbs, J. R. (2003). ‘The logical notation: Ontological promiscuity’. Available at http://www.isi.edu/ hobbs/disinf-tc.html Lewis, D. K. (1973). Counterfactuals. Harvard University Press. Cambridge, MA. Mackie, J. L. (1993). ‘Causes and conditions’. In E. Sosa and M. Tooley (eds), Causation. Oxford University Press, 33–55. McCarthy, J. (1980). ‘Circumscription: A form of nonmonotonic reasoning’. Artificial Intelligence, 13:27–39. (Re-

Journal of Semantics 22: 211–229 doi:10.1093/jos/ffh023

Schedules in a Temporal Interpretation of Modals TIM FERNANDO Trinity College Dublin

Abstract

1 INTRODUCTION Just as semantic accounts of modality commonly invoke possible worlds, theories of temporality (concerning, for instance, aspect) often appeal to eventualities. But what are eventualities? And what are worlds? The present work analyses eventualities and worlds uniformly as certain relations s 4 TI 3 ED between a set TI of times t and a set ED of eventuality-descriptions u, with ‘s schedules u at t:

‘

sðt; uÞ pronounced

Insofar as eventuality-descriptions apply to eventuality-types, we may call s a schedule of eventuality-types. Exactly what eventualitydescriptions are and how they pick out eventuality-types depend on the application at hand: the fragment of English to be analysed, and the bit of reality that is conceptualised (to serve that end). In particular, we may derive ED from certain words and phrases under consideration, while basing eventuality-types on additional conceptualisations of, for instance, time. A concrete and illuminating illustration is provided by the temporal interpretation of modals in Condoravdi 2002, henceforth CON2. Some sentences with which CON2 is concerned are listed in (1). (1) a. b. c. d.

He He He He

might might might might

be be be be

here right now. here any day now. here next week. here yesterday.

The Author 2005. Published by Oxford University Press. All rights reserved.


Eventualities and worlds are analysed uniformly as schedules of certain descriptions of eventuality-types (reversing the reduction of eventuality-types to eventualities). The temporal interpretation of modals in Condoravdi 2002 is reformulated to bring out what it is about eventualities and worlds that is essential to the account. What is essential, it is claimed, can be recovered from schedules that may or may not include worlds.

212 Schedules in a Temporal Interpretation of Modals The oddness of (1d), marked by , is broadly compatible with the idea of historical necessity (e.g. Thomason 1984), according to which the past is settled, and only the present and future are open to branching (whence the acceptability of (1a–c)). But if there are no might’s about the past, then how do we explain (2)? (2) She might have won. What (2) has, which each sentence in (1) lacks, is the perfect (have -en), an analysis of which leads CON2 to two readings of (2), given in (3). (3) a. For all we know now, she might have won. b. She might have at an earlier point won.

(4) a. MIGHT (PERF (she-win)) b. PERF (MIGHT (she-win)) Flying against the surface form (4a) of (2), the scoping of the perfect over might in (4b) is not uncontroversial. It is, however, crucial in CON2 for imposing historical necessity on (3b)/(4b) relative to a notion of history shifted towards an earlier point in the past. But is (4b) based on a flawed interpretation of the perfect? To understand this question, let us examine some of the assumptions underlying CON2. CON2 draws on a generous inventory of worlds, states, events and times to form, on the one hand, eventive and stative properties, and, on the other hand, temporal properties. As made precise in section 3 below, eventive and stative properties serve as interpretations of eventuality-descriptions. To interpret the modals and the perfect, CON2 steps up to temporal properties, replacing the specific states and events in stative and eventive properties by times, alongside worlds that figure in all properties. Now, it is easy enough to convert a stative or eventive property to a temporal property by mapping states or events to their temporal trace. Going back from a temporal property to a stative or eventive one, however, runs into the problem that too many states and events may have the same temporal trace. For instance, does an interpretation of (5) as a temporal property allow us to extract the consequent state of Pat being away (left out from CON2, which focuses on the so-called existential perfect)? (5) Pat has left. And if we were to sharpen the temporal property interpreting PERF(A) to a stative property, could PERF still scope over MIGHT


CON2 defines operators for the perfect and for might, deriving (3a) from the scoping (4a), and (3b) from (4b).

Tim Fernando 213

Table 1.

From temporal properties to schedules in 3 steps.

Section Section Section Section

Given: temporal property (from CON2) Step 1: turn world w into schedule sw (satisfies ~) Step 2: generalize sw to smaller schedules s (forces y) Step 3: reconstruct sw from (generic) set G of schedules

2 3 4 5

u(w)(t) sw , t ~ u s y t,Su sw ¼ G

(6) She might have at an earlier point won (had she followed my advice . . .), but she didn’t. By contrast, accepting she did not win makes (3a) untenable and, in that sense, unstable. (7) a. She didn’t win. But for all we know now, she might have. b. She didn’t win. But she might have (had she . . .).


as in (4b)? Not if PERF were to require states or events (in its inputs), whereas MIGHT returns times (in its outputs). Under the reformulation below, the opposition between temporal properties and stative/ eventive properties evaporates. The world-time pairs in temporal properties become schedules, encompassing worlds and eventualities alike. (4b) is kept viable, and so the reader wishing to rule out (4b) must seek other grounds for doing so. The reformulation of CON2 below is intended as a first step at pinning down the semantic entities CON2’s modals and perfect characterise—a first step, that is, to isolating what Schubert (2000) calls characterised situations. The main thrust is to strip world-time pairs down to the essentials—or, at least, to schedules, from which, it is claimed, the essentials can be extracted. This is carried out below in three steps, outlined in Table 1. We proceed in the next section, section 2, from the semantic set-up in CON2, converting worlds into schedules in section 3. We introduce schedules other than those induced by worlds in section 4, before making do without world-induced schedules in section 5. Precisely what the symbols in Table 1 mean will be explained in due course. That said, let us note at the outset that appeals to forcing y (as in sections 4 and 5) are not new in philosophical semantics, stretching at least as far back as van Fraassen (1969). Forcing lurks at the background of the data semantics of Veltman (1984), where its impact is diminished by the failure of (what is termed there) ‘stability.’ Stability relates to (3) above roughly as follows. (3b) is stable insofar as it is tenable even if we accept that she did not win.

214 Schedules in a Temporal Interpretation of Modals More formally, stability coincides in section 4 below with the persistence of y relative to the subset relation 4 on schedules s, s# s y t; u and

s 4 s#

implies

s# y t; u:

s y Æeæu

iff

ðds# 2 RÞ s# y u

for some set R of schedules specifying the epistemic possibilities of y (without regard to s). As s appears only in the left side (not the right) of the biconditional, we can restore t to get (8),1 making persistence with respect to Æeæu unproblematic. (8) For all schedules s, s# in the domain of y, s y t; Æeæu

iff

s# y t; Æeæu:

What then becomes of the instability in (7a)? Rather than analysing (7a) in terms of a single non-persistent forcing relation y, we appeal to context change of the kind advocated in Veltman (1996). The first sentence of (7a), she didn’t win, changes the epistemic base R to R#, effectively inducing a new forcing relation y#, relative to which (3a) fails (whether or not it holds for the initial relation y).2 Notice that if we are to make sense of discourses such as (7b), the first sentence in which rules out possibilities entertained in the second, we must keep the epistemic base for (3a)/(4a) separate from the modal base for (3b)/(4b), called metaphysical in CON2.3 Accordingly, we shall 1 This would suggest that the situation characterised by Æeæu is not so much a schedule s that forces Æeæu but rather the set fs# 2 R j s# y ug of schedules in R that force u. 2 My apologies for the notational clash with Veltman 1996, where might / is described as nonpersistent relative to a predicate y that takes on its left side not s, but rather a state r corresponding (here) to a set such as R of s ’s. My y is just a slice of Veltman’s y, fixed by a choice of r/R in the background. (In saying this, I am putting aside times t that appear to the right of my y, but not Veltman’s. Variations in t should, of course, not be confused with Veltman’s updates of r/R.) 3 We must, so to speak, immunise metaphysical might from updates that infect epistemic might.


Persistence is indispensable to the application we shall make of forcing (e.g. Proposition 3, section 5). Now, while (3b)/(6), (7b) may pose no problem for persistence, (3a)/(7a) is a different matter. Suppose s forced (3a) and s# encoded she lost, whereas s did not. Then surely s# could not force (3a)? In fact, it could, provided we analyze epistemic might not as in Veltman 1984’s data semantics, but more along the lines of Veltman’s (1996) update semantics. Dropping t for the sake of clarity (at the cost of correctness) and writing Æeæu for ‘might epistemically u,’ let

Tim Fernando 215

assume y comes with two sets Re and Rm of schedules specifying the epistemic and metaphysical possibilities, respectively. To avoid cluttering the notation, we will refrain from hanging the sets Re, Rm as subscripts on y. Such a practice would be useful were we to encode a dynamic interpretation of conjunction involving changes to Re (and possibly also to Rm, s and t). But the present paper stops short of that, keeping Re and Rm frozen.4 Holding Re, Rm constant, we will have enough to do sorting out complications involving time, the perfect and metaphysical might (omitted in Veltman 1984, 1996). 2 TEMPORAL PROPERTIES IN CON2

(i) a set PT of temporal points/moments/instants linearly ordered by a, and a set TI 4 Pow(PT) - f;g of times consisting of non-empty subsets t of PT such that for every z 2 PT, z2t

whenever x a z a y

for some x; y 2 t

(that is, time is a non-empty a-interval) (ii) sets WO, EV and ST of worlds, of events and of states, respectively, along with a function s: (EV [ ST) 3 WO / (TI [ f;g) that specifies the temporal trace s(e, w) 2 TI [ f;g of an event or state e in world w, where sðe; wÞ ¼ ;

iff

e is not realized in w;

the intuition behind s(e, w) 2 TI being that e is a single token/ occurrence in w (as opposed to a type that recurs in w). CON2 calls a function P from worlds (i) eventive if for every world w, P(w) is a unary predicate on events (so P(w)(e) is either true or false for every event e) (ii) stative if for every world w, P(w) is a unary predicate on states (so P(w)(e) is either true or false for every state e) (iii) temporal if for every world w, P(w) is a unary predicate on times (so P(w)(t) is either true or false for every time t) (iv) a property if P is eventive or stative or temporal. 4 Thus, other instances of instability noted in Veltman (1984) need not concern y, which in its static form, is not designed to cover all formulas that arise in natural language interpretation.


The semantic set-up in CON2 takes the following ingredients for granted:

216 Schedules in a Temporal Interpretation of Modals To turn any property to a temporal property, a world-time pair w, t is assigned sets EV(w, t) and ST(w, t) of events and states as follows. An event is located at w, t if its temporal trace in w is contained in t Evðw; tÞ ¼ fe 2 Ev j ; 6¼ sðe; wÞ 4 tg whereas a state is located in w, t if its temporal trace in w overlaps with t Stðw; tÞ ¼ fe 2 St j sðe; wÞ \ t 6¼ ;g:

AT is used to formalize both the perfect and the modals. A function PERF mapping properties P to temporal properties is defined by ðPERF PÞðwÞðtÞ ¼ ðdt# a tÞ ATðt#; w; PÞ where the linear order a on PT is extended to a relation on TI by quantifying universally over the points t a t# iff

ð"x 2 tÞð"x# 2 t#Þ x a x#:

To analyse modals, a modal base function MB is assumed that maps a world-time pair (w, t) to a set of worlds, relative to which a function MIGHTMB maps a property P to the temporal property satisfying ðMIGHTMB PÞðwÞðtÞ ¼ ðdw# 2 MBðw; tÞÞ ATðtN ; w#; PÞ where (expanding time forward, as in Abusch 1998)5 tN is the indefinite extension of t to the future fx 2 Ti j ðdy 2 tÞy d xg 5

Gennari (2003) makes a claim related to the idea that modals expand time forward.


(Viewed from outside t, events give the impression of being bounded while states do not. Events occur, states hold.) A property P is then mapped to the temporal property kwktAT(t, w, P) by existentially quantifying over the events and states located in w, t 8 0 si formed from chains s0 s1 s2 . . . in R. Such objects [i>0 si are captured in the usual completeness theorems in logic as maximal consistent extensions. For reasons to be explained shortly, forcing arguments refine the notion of a maximal consistent extension to that of a generic set (e.g. Keisler 1973)—generic not in the sense of ‘lions have manes’ but rather in connection with y. To spell out this connection, it is useful to extend y to formulas u 2 U with negation : s y t; :u

iff

ð"r #sÞ not r y t; u

(i) for all s 2 G and s# 2 Rm [ Re, s# 4 s implies s# 2 G (ii) every pair s, s# 2 G has a common extension s$ 2 G: s$ s [ s# (iii) for all u 2 U and t 2 TI, there is an s 2 G such that either s y t, u or s y t, :u. Conditions (i) and (ii) essentially make G an ideal. What differentiates generic sets from plain maximal consistent extensions is condition (iii), as illustrated by the following example. Over the time intervals Ti ¼ fð0; 1Þ; ð1; 2Þ; ð2; 3Þ; . . .g and eventuality descriptions ED ¼ fdieðSocratesÞ; aliveðSocratesÞg; let us define the schedules s0 ¼ ; and for i > 0, si+1 ¼ si [ fðði; i+1Þ; aliveðSocratesÞÞg s#i ¼ si [ fðði; i+1Þ; dieðSocratesÞÞg: Suppose these schedules constituted Rm and Re Rm ¼ fsi j i>0g [ fs#i j i>0g ¼ Re : Clearly, Socrates is immortal in the maximal consistent extension ð"t 2 TiÞðdi>0Þ si ðt; aliveðSocratesÞÞ:

S

i>0 si


(recalling that # is the restriction of to Rm [ Re). Now, G is generic if

Tim Fernando 225

On the other hand, applying condition (iii) to u ¼ [m]die(Socrates) and t ¼ (0, 1), it follows from ðzÞ ð"s 2 Rm Þðdt# 2 TiÞ s [ fðt#; dieðSocratesÞÞg 2 Rm that Socrates must die in every generic set G ðdt# 2 TiÞðds 2 GÞ sðt#; dieðSocratesÞÞ:

[R ¼ f[G j G is generic; and G 4 Rg; the intuition being that for a 2 fm, eg, a generic set G contained in Ra induces the a-world [G. As for (Q), we take it to say [ðRm [ Re Þ ¼ ð[Rm Þ [ ð[Re Þ and ED and Ti are finite or countable: Under these assumptions, Proposition 3 can be proved along standard lines in forcing, using the persistence of y (Proposition 2(a) above) and the fact that every s 2 Rm[Re belongs to a generic set, provided TI 3 ED is finite or countable (see e.g. Lemma 1.4 in Keisler 1973, page 101). Two other facts, (15) and (16), are worth recording. (15) The forcing of doubly negated formulas s y t; ::u

iff

ð"r #sÞðdp #rÞ p y t; u

reduces (as usual) to generic sets s y t; ::u

iff

ð" generic G such that s 2 GÞ [ G ~ t; u:

(16) If Wm [ We is an anti-chain, then [YW ¼ W

for

W 2 fWm ; We ; Wm [ We g


This is not to say that generic sets commit us to the mortality of Socrates. Only that if we want to entertain the metaphysical possibility that Socrates is immortal in a generic set, then we had better choose Rm so that (z) fails. (This is easy enough to arrange say, by including infinite schedules in Rm, or by introducing an eventuality description immortal(Socrates) with a suitable constraint on its schedules in Rm.) Armed with the notion of a generic set, we can devise more useful choices of [ and (Q) for Proposition 3 than (14). For all R 4 Rm [ Re, let [R form the unions of generic sets contained in R

226 Schedules in a Temporal Interpretation of Modals where [ is defined relative to Rm ¼ YWm and Re ¼ YWe. Concerning (15), note that s y t; u implies

s y t; ::u

6 CONCLUSION We have carried out Steps 1–3 of Table 1, reformulating the temporal properties of CON2 in terms of a forcing relation s y t; u involving schedules s that may or may not be worlds. But why bother with a schedule s# forcing t, u that is -smaller than a world w? Because s# picks out more schedules that force t, u than w. By the persistence of y, any schedule bigger than s# must force t, u, including w, all schedules bigger than w (of which there are none, if w is a world), and other worlds bigger than s# (of which there may be any number). In other words, small is beautiful. We should try to make a schedule s that forces t, u smaller (and not only bigger, as we do when forming worlds via generic sets). With this in mind, let us close by considering the prospects of truncating schedules. More precisely, given a schedule s and time t, let st be the restriction of s to times a or 4 t st ¼ fðt#; PÞ 2 s j t# a t or t# 4 tg: The question is: for which u 2 U can we count on (17)? (17) s y t; u iff

st y t; u

(17) goes through without a hitch for nearly all the clauses of y. The only problematic cases are the metaphysical modalities Æmæu and [m]u,


but that in general, the converse fails (and hence so would the last equivalence in (15), if we were to drop :: from its left hand side). As for (16), what it says essentially is that if we feed our machinery a set of worlds (that is, an anti-chain Wm [ We), then we get it back. No more, no less. But of course, what makes y and genericity interesting is that worlds are not, at the outset, required. Notice that in the Socrates example of this section, no schedule in Rm ¼ Re is a world (i.e. 4-maximal).

Tim Fernando 227

which can be saved if we strengthen the prefix "r # s to "r # st, yielding s y t; Æmæu iff ð"r #st Þðds# 2 mbm ðr; tÞÞðdt#do tÞ s# y t#; u and s y t; ½mu iff ð"r #st Þð"s# 2 mbm ðr; tÞÞðdt#do tÞ s#y t t#; u:

s½t ¼ st [ fðt; RÞg for some schedule s and time t. Then we can encode y in statements s 8 u (pronounced ‘‘s pins u’’) between R-skeds s and formulas u satisfying (18). (18) s½t8u iff

s y t; u

To construe (18) as a definition of 8, we need to show that the choice there of s and t for the sked s ¼ s[t] is immaterial — that is, whenever s[t] ¼ s#[t#], s y t; u

s# y t#; u:

iff

But that is immediate from (17) and R ; ED. Alternatively (instead of deriving 8 from y), we might define 8 from scratch12 and read (18) from right to left as a definition of y from 8. This route to y,

11

As before, the prefix "r# st has no effect on the epistemic modalities, so we can replace m above by a 2 fm, eg for the sake of uniformity. 12 This is easy, albeit tedious. Given an R-sked s, let us write last(s) for the unique time t such that s(t, R), and write sRfor the schedule s – f(last(s), R)g obtained from s by removing R. Then s8P s8u 4 s8PerfðuÞ s8Æaæu s8½au

iff iff iff iff iff

sðlastðsÞ; PÞ for P 2 ED ðdt 2 domainðsÞÞ t 4 lastðsÞ and sR ½t8u ðdt 2 domainðsÞÞ t a lastðsÞ and sR ½t8u ð"r # sÞðds# 2 mboa ðr; lastðsÞÞÞs#8u ð"r # sÞð"s# 2 mboa ðr; lastðsÞÞÞðds$ lastðsÞ s#Þ lastðs$Þdo lastðsÞ and s$8u

where mboa ðs; tÞ ¼ fs#½t# j s# 2 mba ðsR; tÞ and t#do tg:


With this modification,11 we secure (17) for all u 2 U, without (remarkably) losing any of our previous results (including Propositions 2 and 3, and (15),(16)). Moreover, suppose we introduce a fresh symbol R ; ED (for Reichenbach’s reference time) and define an R-sked to be a relation of the form

228 Schedules in a Temporal Interpretation of Modals however, depends on (17), which may well fail for u incorporating the progressive.13

Acknowledgments I am indebted to Cleo Condoravdi, Stefan Kaufmann and Frank Veltman for helpful comments, and to two anonymous referees for thoughtful criticisms. My thanks also to the organizers and participants of the 7th Symposium on Logic and Language in Pecs, Hungary (August 2002), where an early version of this paper (here superseded) was presented under the shameless title ‘Between events and worlds under historical necessity.’

Received: 02.03.03 Final version received: 28.08.04

REFERENCES Abusch, D. (1998) ‘Generalizing tense semantics for future contexts’. In S. Rothstein, (ed.), Events and Grammar, Kluwer. Dordrecht; 13–33. Condoravdi, C. (2002) ‘Temporal interpretation of modals: Modals for the present and for the past’. In D. Beaver, S. Kaufmann, B. Clark, & L. Casillas, (eds), The Construction of Meaning. CSLI. Stanford, 59–88. Dowty, D. R. (1979) Word Meaning and Montague Grammar. Reidel. Dordrecht. Fernando, T. (2003) ‘Reichenbach’s E, R and S in a finite-state setting. Sinn und Bedeutung 8 (Frankfurt), (www.cs.tcd. ie/Tim.Fernando/wien.pdf) van Fraassen, B. C. (1969) ‘Facts and tautological entailments’. Journal of Philosophy, 66(15):477–487. 13

Gennari, S. P. (2003). ‘Tense meanings and temporal interpretation’. Journal of Semantics, 20(1):37–71. Keisler, H. J. (1973) ‘Forcing and the omitting types theorem’. In M. Morley, (ed.), Studies in Model Theory. The Mathematical Association of America, 96–133. Plotkin, G. (1983) Domains (‘Pisa notes’). Department of Computer Science, University of Edinburgh. Schubert, L. (2000) ‘The situations we talk about’. In J. Minker, (ed.), LogicBased Artificial Intelligence. Kluwer. Dordrecht, 407–439. Steedman, M. The Productions of Time. Draft, ftp://ftp.cogsci.ed.ac.uk/pub/ steedman/temporality/temporality.ps. gz, July 2000. (Subsumes ‘Temporality,’ in

Details in Fernando (2003), where schedules are reduced further to regular languages over the alphabet Pow(ED [ fR, Sg), with ED assumed finite, towards a finite-state formulation of ideas described in Steedman 2000.


TIM FERNANDO Computer Science Department Trinity College, Dublin 2 Ireland e-mail: [email protected]

Tim Fernando 229 J. van Benthem & A. ter Meulen, (eds), Handbook of Logic and Language. Elsevier North Holland, 895–935,1997.) Thomason, R. (1984) ‘Combinations of tense and modality’. In D. Gabbay & F. Guenthner, (eds), Handbook of Philosophical Logic. Reidel, 135–165.

Veltman, F. (1984) ‘Data semantics’. In J. Groenendijk, T.M.V. Janssen & M. Stokhof, (eds), Truth, Interpretation and Information, Foris, 43–63. Veltman, F. (1996) ‘Defaults in update semantics’. Journal of Philosophical Logic 25:221–261.


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