the logic of causal explanation: an axiomatization

The Logic of Causal Explanation: An AxiomatizationAuthor(s): Robert C. KoonsSource: Studia Logica: An International Journal for Symbolic Logic, Vol. 77, No. 3 (Aug.,2004), pp. 325-354Published by: SpringerStable URL: http://www.jstor.org/stable/20016633 .

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ROBERT C. KoONS The Logic of Causal Explanation An Axiomatization*

Abstract. Three-valued (strong-Kleene) modal logic provides the foundation for a new

approach to formalizing causal explanation as a relation between partial situations. The

approach makes fine-grained distinctions between aspects of events, even between aspects

that are equivalent in classical logic. The framework can accommodate a variety of on

tologies concerning the relata of causal explanation. I argue, however, for a tripartite

ontology of objects corresponding to sentential nominals: facts, tropes (or facta or states

of affairs), and situations (or events). I axiomatize the relations and use canonical models

to demonstrate completeness.

Keywords: causation, causal explanation, strong Kleene, three-valued logic, situation the

ory, Quine, Kim, Davidson, Barwise, Etchemendy, facts, facta, events, causal relata, on

tology, mereology, actuality, modality, modal logic, truth-makers

Introduction

In any investigation of the logic of causal explanation, the inseparability of logic and ontology quickly manifests itself. An explication of the logical form of statements of causation and causal explanation is at the same time the taking of an ontological inventory. Not surprisingly, both the logic and the ontology of causation have been matters of vigorous dispute over the past thirty years, and no consensus has yet emerged. I will propose, not merely a single logic of causal explanation, but also a general framework within which the whole range of proposed accounts of causal explanation could be represented perspicuously.

Causal Relata

Linguistic Data

Since metaphysics is the most general of the sciences, we must bring to bear all of our knowledge of the world, including our most recent scientific

* I would like to thank the University of Texas at Austin, and especially the College

of Liberal Arts, for research support during the spring of 2002. I would also like to thank Mary Ellen Brown, Ivona Hedin of the Institute for Advanced Study at Indiana University - Bloomington for their help during my stay there in January and February, 2002. In addition, let me mention the indispensable contributions to my thinking on this subject

made by Professor Timothy O'Connor at IU during our many conversations there.

Presented by Jacek Malinowski; Received March 1, 2003

Studia Logica 77: 325-354, 2004. ? 2004 Kluwer Academic Publishers. Printed in the Netherlands.



326 R. C. Koons

discoveries. However, we must also not overlook the fact that all of our scientific discoveries depend upon a vast background consisting of ordinary, commonsense knowledge. Consequently, metaphysical inquiry must begin

with the systematization of common sense, although it should not end there.

The great bulk of our commonsense knowledge is tacitly encoded in the structure of natural language. For this reason, philosophy's linguistic turn, beginning with Frege and Russell, proved fruitful. The theory of causation has been enriched by linguistic inquiry into the form of statements of causa tion and causal explanation. This linguistic data has been especially helpful in illuminating the nature of the relata of causation.

In natural language, statements of causation typically employ, in both the subject and object position, noun phrases that have been derived from verb phrases. These nominalization of verb phrases come in two kinds: perfect and imperfect.[5, pp. 2-3] Perfect nominalizations can take the definite or indefinite article, can be pluralized, take adjectives rather than adverbs as

modifiers, and employ prepositions rather than direct-object constructions. Perfect nominals typically include sortal nouns that are derived from verbs, such as 'explosion', 'performance', 'refusal', 'departure', 'death' and 'birth'.

Imperfect nominals often consist of infinite constructions (like 'Caesar to die' or 'the frog to jump') or 'that'-plus-sentence constructions ('that Caesar died' or 'that the frog jumped'). These cannot be pluralized and do not take articles or adjectives but instead take adverbs, direct and indirect objects, and other verb-phrase modifiers: 'that Caesar died quickly', 'Mary to give John the book'.

Gerunds (formed in English by adding 'ing' to the verb's root) can be used either as perfect or as imperfect nominals. We can say either 'Brutus's stabbing Caesar' [imperfect] or 'Brutus's stabbing of Caesar' [perfect], just as we can say either 'Brutus's stabbing quickly' [imperfect] or 'Brutus's quick stabbing' [perfect]. Gerunds can take articles and pluralization, some more

easily than others.

Another significant linguistic datum concerns the role of highlighting in

explanatory contexts, as discussed by Dretske ([10], [11]) and Achinstein

([1]). This highlighting can be accomplished syntactically, transforming 'Mary stole the bicycle' into 'It was Mary who stole the bicycle' or 'It was

the bicycle Mary stole', or by the use of emphasis: 'Mary stole the bicycle', 'Mary stole the bicycle'. Substituting sentences such as these, varying only in the location of such highlighting, can alter the truth-value of statements of causal explanation. Mary's stealing the bicycle can explain John's tardi

ness, in a way that the fact that it was Mary who stole the bicycle may not.



The Logic of Causal Explanation 327

In contrast, Mary's stealing the bicycle may explain Mary's arrest in a way that Mary's stealing the bicycle cannot.

Finally, language use embodies principles of substitutability salve veritate within causal/explanatory contexts. Causal contexts resemble closely con texts of perception reports, including those involving naked-infinitive con structions, of the kind examined by Barwise and Perry in Situations and

Attitudes [3]. The following cases are clearly parallel:

Mary saw John cry. Mary made John cry.

The various principles of substitution that Barwise and Perry observed in the phenomenon of naked-infinitive perception reports can be generalized to certain causal contexts:

(Referential transparency) A made q(t), t = t' = A made 0(t').

For example, from Mary made John cry and John is the class president, it follows that Mary made the class president cry, just as from Mary saw John cry we can obtain Mary saw the class president cry.

(Distribution over Or) A made (B V C) - ((A made B) V (A made C))

(Distribution over And) A made (B&C) ((A made B)&(A made C))

Thus, we can infer John made Mary run or John made Mary walk from John made Mary run or walk, and we can infer John made Mary run and John made Mary walk from John made Mary run and walk.

As is well-known, due to a familiar argument used by Frege, Quine and Davidson, if a context that forms terms from sentential clauses (such as 'the fact that' or 'the event that') satisfies both referential transparency and the substitution of classical logical equivalents, then the context is fully exten sional, allowing any true sentential clause to be substituted for any other true clause. (Davidson takes the proof to demonstrate the existence of just one Big Fact.) Similarly, Barwise and Perry showed that the same result holds for any term-forming context that allows for distribution over disjunction and conjunction and for the substitution of classical logical equivalents. The

moral drawn by Barwise and Perry for naked-infinitive perception contexts (in Situations and Attitudes) and by me for causal contexts (in Realism Re gained) is that the correct principle of substitution is one that allows clauses that are logically equivalent in strong Kleene three-valued logic (or Dunn

Belnap four-valued logic). Equivalence in classical logic is not sufficient. The fact that Mary made John cry and Albert smile is identical to the

fact that Mary made Albert smile and John cry, since the two clauses are strong-Kleene equivalent. The fact that Mary made John cry, is not identical



328 R. C. Koons

to the fact that Mary made John cry and Albert smile or not smile, even though the two clauses are classical equivalent. If Mary made John cry and Albert smile or not smile, then Mary made Albert smile or not smile (by distribution over conjunction), and from this it would follow that either

Mary made Albert smile or Mary made Albert not smile. Clearly, this is invalid, since the causal connection between Mary's action and John's crying

may have nothing to do with an arbitrary fact like Albert's smiling or his not smiling.

Events and Facts

As Vendler noted [39], perfect nominals refer to states or events, while im perfect nominals signify facts. Events are located in space and time, while facts are not. Events occur, while facts obtain. At least at first glance, facts are much more finely individuated than are events: Caesar's death is (at least arguably) the same event as Caesar's violent death, while the fact that

Caesar died is distinct from the fact that Caesar died violently. For purposes of simplicity, I will use the term 'event' to designate the

kind of thing picked out by perfect nominals. Such "events" include changes, unchanging states of being, processes and activities. I will not be concerned here with various finer distinctions among such events or eventualities, in

cluding Vendler's well-known distinction of states, activities, accomplish ments, and achievements.[40] I don't intend for the term 'event' to suggest that every event is a kind of change (as Lombard has argued [22]). I take it as reasonably clear that states can be designated by perfect nominals and can serve as both causes and effects (e.g., 'the explosion was caused by the operator's continued sleeping').

I don't intend by making this distinction to beg the question of whether the linguistic distinction between perfect and imperfect nominals

corresponds to a pair of disjoint ontological categories. I want to make room for the possibility that events are simply a special kind of fact (see

Mellor [24]), as well as the possibility that there are only facts, and that

perfect nominals correspond to certain kinds of equivalence classes of facts

(as Bennett argued [5]). I also do not intend to rule out a competing approach, one (like that of

Davidson's) that acknowledges the existence of events but not of facts. [8] A Davidsonian can still accept a event/fact distinction by taking 'facts' to

involve commitment only to true propositions or true eternal sentences.

I will follow a fairly common usage within philosophy (begun, I think, by Davidson) of using the word 'causation' to refer to a relation between




events, and 'causal explanation' as a relation between facts. I am assuming that facts really do explain other facts: that is, I am assuming that 'explain ing' is not limited to what people do in offering explanations. A speech act of correct or felicitous explanation is always anchored in some correspond ing, objective relation between the facts. I don't intend to say much here about the pragmatics of explanation-giving as a speech act. I could instead

make use of Terence Horgan's helpful coinage, quausation, to refer to this explanatory or proto-explanatory relation between facts. Horgan's term has the advantage of lacking the communicative and psychological connotation of explanation.

The linguistic phenomenon of highlighting (discussed in the previous section) reinforces the case for thinking that facts (the relata of explanation) are finely individuated. There is a single event of Mary's stealing the bicycle, not two events, one of Mary's stealing the bicycle and the other of Mary's stealing the bicycle. However, we do not have to go so far as to recognize the existence of intrinsically highlighted facts in order to have enough facts around to ground the distinctions that such highlighting points to. There are a large number of facts grounded in the event of Mary's stealing the bicycle, including the fact that Mary stole something and the fact that someone stole the bicycle. These facts have been called event aspects by Dretske and Sanford [32]. The bolt's giving way is an event; the bolt's giving way suddenly is an event aspect.

The distinction between events and event aspects corresponds to the distinction between token situations and states of affairs (formed by a pairing of a situation token with a type to which it belongs) in Barwise's situation theory. [3] [20] Token situations are concrete things, located in spacetime,

while states of affairs are the exemplification of a type by a token. Situation types can be logically complex (conjunctive, disjunctive, quantificational) and can encompass within themselves a subject/predicate structure - so, the sentence 'a cat is on a mat' corresponds to a particular kind of situation.

Many ontologies include things variously designated tropes, abstract par ticulars, moments (Husserl), individual accidents (medievals), or modes (Locke).1 These are both predicable (like properties) and yet particular rather than universal. Classical examples include such things as the pale ness or wisdom of Socrates or the redness of a particular spot. Such tropes are taken to have spatio-temporal location and to be enumerable (we can count the number of paleness tropes borne by a group of sunbathers, for example). In these respects, tropes resemble events, and many philosophers,

1 See [42], [34], [41], [27], [30], and [31].



330 R. C. Koons

from Leibniz [5, p. 92] to Terence Parsons [29], have suggested the iden tification of events with tropes. Whether this identification is correct will depend upon how finely or coarsely members of each category (viz., tropes and events) are individuated. The identification will fail if, for example, we acknowledge the existence of intrinsically disjunctive or negative tropes but not of intrinsically disjunctive or negative events. However, for most ontologists, intuitions about the essences of events and intuitions about the essences of tropes seem to correspond closely.

If there is a difference between the category of tropes and that of events, it may lie in the fact that the category of events includes certain aggregates or compounds made up of large numbers of interrelated tropes. An aggregate of events is at least in many cases itself an event, even when the events involve distinct sets of participants. Whether an aggregate of tropes involving the same object is itself a trope depends on whether we are willing to accept conjunctive tropes, and whether an aggregate of tropes involving a plurality of objects depends on whether we are willing to accept tropes belonging to aggregate objects.

A tripartite ontology of objects corresponding to nominals may be called for: facts, tropes (or facta or states of affairs), and situations (or events). Facts are in a many-one relation to facta, since one factum can make true several distinct and even logically inequivalent propositions. Events would seem to correspond to certain sets of tropes or facta. The facta or tropes

might be thought of as aspects, modes or even dependent parts of certain events or situations.

The Essences of Events

The task of providing individuation conditions for events and that of spec ifying the essence of events are essentially identical, since we are interested in individuating events in hypothetical as well as in actual cases. Our con cern here is with the metaphysics of events and causation, and not primarily

with epistemology, for which extensionally adequate criteria of identity and distinctness might be sufficient.

Views about the essences of events fall roughly into a spectrum, from

extremely coarse-grained to extremely fine-grained conceptions. On the coarse-grained end, we find the views of Quine and Lemmon [21], accord ing to which events are essentially identified with spatio-temporal regions. Spatio-temporal location clearly provides a lower bound on the essences of events, since everyone agrees that events with different actual spatiotempo ral locations are distinct, and most agree that the spatiotemporal location




of an event is essential to it (although this has been disputed by Chisholm [7] and Lombard [22]).

At the other extreme, we find those (like Mackie [23], Mellor [24], and Zimmerman [45]) who classify events as a kind of fact. Facts are very finely individuated. Corresponding to each fact is an equivalence class of true sentences (and corresponding to each possible fact an equivalence class of possibly true sentences), where one sentence is factually equivalent to an other just in case it can be transformed into the other by the substitution of co-referential terms or the substitution of strong-Kleene (or, equivalently, by Dunn-Belnap [13][4]) equivalent formulas.2 On this events-as-facts view, the distinction I drew in the last section between causation and causal ex planation collapses. The events-as-facts view represents an upper bound on the essences of events, since the relevant equivalence classes of propositions fix the substrate (the participants in the events and their respective roles), the spatiotemporal location, and the core exemplified property that consti tutes the event, and this property is itself very finely individuated, since this approach distinguishes between properties that are classically but not strong-Kleene equivalent

There are good reasons for rejecting both the lower and the upper ex tremes. Contrary to the Quine-Lemmon account, it seems clear that distinct events can occupy the same spatio-temporal region. Two particles or waves could pass through the same region at the same time without any mutual dis turbance or interaction. The two events would have entirely disjoint causal histories and consequences, yet the Quine-Lemmon account would force us to identify them, despite the fact that our best scientific account of the situation would treat them as distinct. Common sense joins science in dis tinguishing, say, the spinning of a globe from its simultaneous alternation in temperature or color, even though the two events coincide in space and time.

At the other extreme, commonly held intuitions resist the postulation of distinct events corresponding to each property that is exemplified in a situation. A spot that exemplifies the property of being red also exemplifies the property of being red or green, but it strains credulity to belief that there is a distinct, intrinsically disjunctive state corresponding to the second property.

Moreover, the facts-as-events account lacks an explanation of why we individuate facts as finely as we do, and not more finely still. If there are distinct facts of the spot's being red and of the spot's being red or green, we

2 See Bas Van Fraassen's account: [36].



332 - R. C. Koons

do we not distinguish the second fact from a third fact of the spot's being green or red.? Treating facts whose corresponding propositions are strong Kleene equivalent as identical makes sense if we think of facts as sets of possible concrete situations (or facta as Mellor calls them [24]), where each situation assigns to each atomic fact one of three truth values (true, false or undetermined). Such situations are saturated: if situation s verifies the disjunctive fact that P V Q, then either s verifies P or it verifies Q, and if it verifies an existentially general fact like the fact that 3x4(x), then there is an x such that s verifies that +(x).

Such an account of situation-tokens or facta provides a new, and more defensible, upper bound on the essential characteristics of events. What is essential to an event is the set of atomic facts that it verifies. This answer raises a further, orthogonal question: is the spatiotemporal location, or the location within the actual causal network, of an event essential to it? Donald

Davidson suggested that an events can be individuated by reference to their causes and effects: if event a and b have the same causes and effects, then a and b are identical (a view which he later recanted - see [9]). It is probable that Davidson did not intend this as an account of the essences of events, because, as Peter van Inwagen has pointed out [37], it is quite implausible to suppose that the effects of an event are essential to it. In an indetermin istic universe, such as ours appears to be, the effects of an event are largely contingent rather than strictly necessary. In contrast, it is plausible to sup pose, as van Inwagen suggested, that the causes of an event are essential to it. In Realism Regained [20, p. 130], 1 provided four independent reasons for thinking this is so, including the intuition that van Inwagen appealed to, namely, that we typically take the origin of a thing to be essential to it. In addition, by assuming that particular events necessitate their particular causes, we can provide a plausible account of causal asymmetry in purely

modal terms: a particular event is causally prior to a second if the actual existence of the second necessitates that of the first, and not vice versa.

Having established a plausible upper bound on the individuation of event

essences, we must now consider the question of the lower bound. Is there a distinct event corresponding to each set of atomic facts? As we have seen, the Quine-Lemmon theorist answers, No. According to that view, only sets of facts that correspond to the set of all facts about some definite space time region correspond to events. This seems unduly restrictive, but other restrictions may be more plausible. Since one of the primary roles of events is to serve as the relata of causation, it would be reasonable to look to causal principles of individuation. Davidson's proposal might be pressed into service her, with one crucial modification. Davidson suggested that two




events be identified just in case they coincided in their causes and effects. Since we are now considering the question of how to individuate events by reference to the corresponding sets of facts, we might consider the following quasi-Davidsonian proposal:

The Explanation (Davidsonian) Account. Two sets of atomic facts A and B correspond to the same event just in case:

1. every fact that explains a member of A also explains a mem ber of B, and vice versa, and

2. every fact that is (potentially) explained by a member of A is also (potentially) explained by a member of B, and vice versa.

We could also consider each of the two sub-conditions separately, leading to the Explanans Account and the Explanandum Account:

The Explanans Account. Two sets of atomic facts A and B correspond to the same event just in case: every fact that explains a member of A also explains a member of B, and vice versa.

The Explanandum Account. Two sets of atomic facts A and B correspond to the same event just in case: every fact that is (potentially) explained by a member of A is also (potentially) explained by a member of B, and vice versa.

In addition to these three possibilities, we must add the upper bound account, which we can call 'the Facta Account'.

The Facta Account. Two distinct sets of atomic facts always correspond to distinct events.

The Facta Account is very close to the proposal made by Lombard, with the difference that Lombard insists on limiting the scope of the term 'event' to changes, excluding unchanging states of being.[22] It is also essentially the proposal made by Menzies. [25]

The Facta account individuates events more finely than the Explanation account, which, in turn, individuates events more finely than either the Explanans or the Explanandum account. Thus, the Facta account provides our upper bound on the fineness of individuation, and the Explanans and Explanandum account a new lower bound.

Whether there is a real difference between the Explanation and the Facta accounts is not obvious. The two accounts will differ if we posit the exis tence of epiphenomenal facta, atomic facts that neither demand nor offer



334 R. C. Koons

Finer

Events as Facts

Facta Account

Explanation Account

Explanans Account Explanandum Account

Quine-Lemmon Account

Coarser

Figure 1. Spectrum of Individuation Coarseness

distinctive causal explanations. If there are such facta, then the Explana tion account will individuate more coarsely than the Facta account, since the epiphenomenal facts will not introduce event-distinctions on the Explanation account, but will do so on the Facta account.

We have not yet considered Jaegwon Kim's account of events as prop erty exemplifications [18][19]. In fact, as Lombard has pointed out, Kim's identification of events with property exemplifications does not, by itself, settle the question of how finely to individuate events. Kim's thesis simply pushes the question back a step: when is the exemplification of property F by a identical to the exemplification of property G by a? The distinctness of F and G as properties is not sufficient to establish the distinctness of their exemplifications in a particular case. As Lombard observes [22, pp. 54-55], Kim actually relies upon the relation of causal explanation to decide the question of event essence and identity. Kim argues that the event of the

bolt's giving way must be distinguished from the event of the bolt's giving way suddenly because the second event, and not the first, is explanatory of the bridge's collapse. Similarly, Kim would distinguish between the event

of Peter's leaving the party and the event of Peter's leaving the party nois

ily if the loudness of the music explained the first but not the second. This

indicates that Kim's account is in fact the Explanation Account given above.

If van Inwagen and I are right in thinking that the individual essences of events are bound up with their causes and not with their effects, our view provides some support for the kind of asymmetry incorporated in the

Explanans account. Further support is provided by the natural-language phenomenon noted by Zeno Vendler; namely, that the verb 'causes' tends




to take facts as subject and events as object, while no English word has the converse tendency. [38] This suggests that we individuate events by what explains them, and not by what they explain.

As we have seen, the linguistic data (including the phenomenon of high lighting) provides some support for a distinction between events and event aspects. This distinction is often expressed by means of the qua construction, leading Terence Horgan to introduce the helpful term quausation. [17] Mary's stealing of the bicycle lead to her arrest qua act of stealing something, and it led to John's immobility qua event of someone's stealing his bicycle.

This distinction between events and event-aspects can be drawn on any of the four accounts, but the distinction is a weightier, more significant one on the Explanans (Kimian) or Explanandum Accounts than on either the Davidsonian (Explanation) or the Facta Accounts. Even on the Davidsonian and the Facta Accounts, we can distinguish various logically complex facts that are verified by the relevant set of facta: for example, we can distin guish between the fact that Mary stole something and the fact that someone stole John's bicycle. These logically complex facts may interact variously

with different causal laws and thus figure in different explanations. On the Kimian/Explanans and Explanandum Accounts, there is a further variation that can give rise to different event-aspects: distinct sets of facta might correspond to the same effect, and these distinct sets of facta might have different causal properties, either as explananda (on the Kimian/Explanans Account) or as explanantia (on the Explanandum Account).

Suppose, for example, that there are two distinct sets of facta A and B, A verifying the fact that Sebastian's strolls, and B verifying the fact that Sebastian strolls leisurely. (We may suppose that A is a subset of B.)

On the Kimian Account, it may be that these two sets correspond to a single event, since whatever causally explained Sebastian's strolling on this occasion also causally explained his strolling leisurely. The two sets may well differ in what they explain - the leisureliness of his strolling may explain the lifting of his spirits, although his strolling per se does not. In such a case, set B (corresponding to the leisureliness of his stroll) corresponds to a particular aspect of the event of Sebastian's strolling. This kind of aspect cannot be recognized by the Facta account, since distinct sets of facta always correspond to distinct events. It cannot be recognized by the Davidsonian (Explanation) Account, either, since any two sets of facta that differ in any explanatory respect will also correspond on that account to distinct events. Only the Kimian/Explanans and the Explanandum Accounts permit this further distinguishing of event-aspects. Since there seems to be some support for the Kimian/Explanans Account, and none that I know of for



336 R. C. Koons

the Explanandum Account, I will focus in the remainder of the paper on the Explanans (Kimian), Explanation (Davidsonian), and Facta (Lombardian) Accounts.

Although the Davidsonian and Facta Accounts can recognize some dis tinction between events and event-aspects, the distinction remains a prob lematic one on these accounts. The relation between an event and the various properties it exemplifies threatens to be an extrinsic fact about the event (especially if properties are conceived of as transcendent universals), and such extrinsic facts seem ill-suited as the basis or ground of intrinsic causal powers. In addition, this view of the nature of event-aspects seems to get the relationship between causal laws and causal powers backwards. On the

Davidsonian or Facta accounts, different properties are associated with dif ferent causal powers because they bring the event bearing those properties under different causal/explanatory laws. This would pull in the direction of a Humean account of powers, although possibly a kind of "second-order Humeanism", as Brian Ellis has characterized David M. Armstrong's ac count. [14, pp. 348-350] In fact, those causal laws that do hold hold because the corresponding properties have the causal powers they do essentially. The properties do not have the causal powers they do because of the holding of certain causal laws.

In contrast, on the Kimian (Explanans) account, we can distinguish two sets of atomic facts: one sufficient to identify the object of the theft as a bicycle and one omitting such explanatorily superfluous facts. Then we can identify an objective difference between what is explained by the first and the second.

On the Kimian (Explanans) account, the various event-aspects of an event would correspond, not to logically complex propositions made true by the event, but by deletion or subtraction of the some of the atomic facts from the rich totality brought about by a causal history. Event-aspects would be abstract objects in the ancient meaning of the word: proper parts of the totality of atomic facts realized by the event. For example, in the case of

Mary's stealing of John's bicycle, let's apply Davidson's covert-quantification account of action sentences. On Davidson's account, to say that Mary stole John's bicycle is really to say something like this:

There is an x such that (1) x is a stealing, (2) x is by Mary, and (3) x is of John's bicycle.

We then have a single event with three distinct aspects. To explain another event by reference to an event described as 'Mary's stealing of the bicycle' is to identify aspect (2) and not aspect (3) as part of the causal explanans.




The alternative highlighting, 'Mary's stealing the bicycle', asserts that it is aspect (3) rather than (2) that carries the explanatory burden.

On the Kimian (Explanans) account, the existence of real event-aspects corresponds to a real difference in the possibilities of causal explanation. If (and only if) it is possible to explain the instantiation of an event-genus

without explaining the instantiation of a particular species, or to explain the instantiation of a determinable without explaining the instantiation of the corresponding determinate, then there would be event-aspects corresponding to the genus or to the determinable. This provides the Kimian account

with a more rigorous ontological discipline, as opposed to the promiscuous multiplication of event-aspects under the Facta account.

Truth-makers

Events and event-like entities (tropes, Barwise-Perry situations) have played two distinct roles in recent philosophy: as the relate of causation, and as truth-makers. The truth-maker role is in fact the older of the two, dating

back to the semantic theories of Austin and the early Russell ([2], [31]).3 The truth-maker idea is grounded in two intuitions: first, that truth involves some kind of relation between a sentence or proposition and some part of the

world (a correspondence of some kind), and, second, that truth supervenes on being, that the truth-value of a proposition must somehow be determined by what exists.

There has been surprisingly little discussion of what would seem to be a critical issue: why assume that the same class of entities is fitted to play both of these roles (causal relata and truth-makers)? A connection between the two roles might be found in the nature of information, as explicated by Dretske [12] and myself [20, pp. 121-128]. If information depends upon causal relations between the carrier of information and the subject of the in formation, and if the semantic content of sentences and mental states consists in a particular mode of the carrying of information, then the identification of the relata of causation with the ontological foundation of semantic value would follow. What makes a sentence or thought true would be that situ ation that is properly related in the right causal/informational way to the instantiation of the sentence or thought.

The truth-maker approach gives rise to two related questions: what are the truth-makers for negated atomic propositions, and what are the truth

3See the related work of John Cook Wilson [42], G. F. Stout [34], Donald C. Williams [41], Bas van Fraassen [36], Mulligan, Simons and Smith [27], and Michael Pendlebury

[30].



338 R. C. Koons

makers for universal generalizations? Some, like David Lewis, have sug gested that negated atomic propositions are made true by the non-existence of a truth-maker for the atomic proposition itself. Similarly, one might sup pose that a universal generalization is made true by the non-existence of a falsity-maker for any of its instances. However, these hypotheses involve abandoning one of the motivating intuitions for the truth-maker idea: the intuition that truth always involves a relation between the proposition and some part of the world.

Moreover, if the non-existence of a falsity-maker account is accepted, then the thesis that there is a relation of causation that takes events as relata is in jeopardy. Mellor has used the supposedly obvious non-existence of negative events or states as conclusive reason for rejecting events as the relata of causation. [24, p. 135]

A number of alternative accounts have been proposed. It is possible to collapse the two problems into one. If there is such a thing as a totality situation that acts as a truth-maker for the corresponding universal general ization, and if there is a primitive kind of distinctness fact that grounds the distinctness of two propositions (or abstract situation-types), then a negated atomic proposition -'Fa could be made true by the totality situation that

makes true the universal generalization that every atomic proposition made true by an existing situation is distinct from Fa.4

Alternatively, if there are simple situations that make true negated atomic propositions, including propositions involving any possible individ ual, whether actually existing or not, then a universal generalization could be made true by the sum of all the situations making true each its instances.

Thus, neither account provides an elimination of negativity from the do main of truth-makers, but each posits a limited basis of primitive negative facta (in the one case, primitive facts of distinctness, and in the other, prim itive facts of non-actuality), upon which all other negative facts supervene.

A Modal Logic of Truth-makers

Truth-maker theory includes claims of the form: situation (or event) s makes true proposition q$. Similarly, the meta-theory of modal logic includes such things as: proposition b is true at world w. This parallel suggests a simpli fication in our metaphysical theory: that of identifying the relation of being

made true by a situation and the relation of being true at a world. This iden tification would require identifying possible worlds with with certain possible situations (events or states).

4 See Hochberg [16] for a recent version of such an account.




Modal metaphysicians have thought of possible worlds in a variety of ways: as actual, concrete, causally isolated universes (Lewis), as maximal 'states of affairs', as maximal properties of a certain kind, and as maximal

propositions or sets of propositions. The possible-world-as-possible-situation account could be a variant of either the second or third of these. A possi ble situation could be identified with a certain kind of situation-type or situation-property: one that includes everything that would have been essen tial to a situation that would have arisen from particular causal antecedents. Such a type would include all of the intrinsic character of the possible sit uation, the identities and essences of all of its causal antecedents, and the appropriate situation-haecceity (if such exist).

A possible world would be a possible situation s that could not have been co-instantiated with any possible situation s' not part of s. This is the sense in which possible worlds are maximal situations.

Once we have made the shift from worlds to situations, however, a radical new possibility opens up: namely, an interpretation of modal logic in which all of the situations are assumed to be actual, parts of the one and only actual world. In fact, in formalizing the relations of causation and causal explanation, non-actual situations are irrelevant. We can, in place of the usual modal operators of possibility or necessity, use modal operators of in clusion (expressing the fact that a fact is realized by some part of the present situation) and of explanation (expressing a causal relationship between two actual situations).

A semantic theory for situational modal logic would introduce a domain of situations and a partial (three-valued) interpretation function defined over these situations. Here are the three-valued strong Kleene truth tables for negation, disjunction, and conjunction.

P T F U_

-p F T U

(pVq) p

T TIT T q F T F U

U T U U



340 R. C. Koons

(p&q) p T F U

T T F U

qF F F F U U F U

It will be convenient to add a formula, T, that is made true by every situa tion, and its negation, I, that is made false by every situation.

In partial logic, there are no logically true propositions: no proposition is true in every three- or four-valued interpretation. We do, however, have non trivial logical implication. In fact, there are a variety of relations that are species of logical implication or consequence in partial logic. In three-valued logic, there are three notions of logical consequence that seem most natural: verification validity, falsification validity, and double-barreled validity. A set F verifiably entails a set A just in case every interpretation that verifies every member of F verifies some member of A. A set F falsifiably entails a set A just in case every interpretation that falsifies every member of F

falsifies some member of A. The relation of double-barreled implication (first suggested by [6]) holds between a set F and A just in case r both verifiably and falsifiably implies A. Whenever I talk about implication in three-valued logic, I will mean double-barreled implication, since this comes closest in many ways to the classical case.

Muskens [28] has proved that the following system of rules (rL+*) is complete for double-barreled implication in three-valued propositional logic:

* (Ri) -qH

* (R2) -(&b) H XV-' * (R3) -(q V b) H _q&-b

* (R4) q&H $ 0 and O&VH F A

* (R5) $Hq0V and ' Fb-q$VyV)

* (R6) If Up F X, and b,pH X, then (VO),pF X.

* (R7) If X F p, X, and X ' ,p, then X p(X & ),p

* (R8*)& -- V

* (R9) If q $HO and4F -, X) then F X.

* (R10) r H A if and only if there are nonempty sets {j1, .. ., am} C r and { B C)..4 An} Asuch that al & ... &am F3i V ...V/3n.




Models will also include a part/whole relation, E, familiar from formal mereology. I will introduce two distinct pairs of operators:

El J

* 1, 1: mereological operators, representing true in all sub-situations and true in some sub-situation, respectively.

* O. Z- 0: causal explanation operators, indicating that some for

mula is true in all (some) situation causally explanatory of the present situation-index.

The Mereological Operators

In situation logic, formulas are evaluated at partial situations, not at worlds. Since such situations can be more or less partial, there is a significant part whole relationship between situations. In particular, a given situation will contain a number of still-more-partial sub-situations. To evaluate the truth of a claim that some fact, such as q, is explainable in situation s, we need to look at sub-situations of s that verify q, to see if any of those sub-situations stand have been caused, and if so, by what sort of cause. This will require

the addition of two special operators, t (true in some subsituation), and its

dual, {.

The semantic clauses for the part-of operators are as follows:

FI * M, s VX (x E s - M, x ).

* M,s x 3(sMxS&Mx X .

? M, S XX 3(x E s&M, x t0

? M, s V=( + X F~ s -- M, x )

These are defined in such a way that every l-closed formula takes a classical truth value at every situation.

The situations in the canonical model consist of consistent, saturated theories of the language, that is, sets of formulas that are closed under (double-barreled) logical consequence, and that contain at least one disjunct

whenever they contain a disjunction. In the canonical model, the part/whole relation can be defined in a familiar way:



342 R. C. Koons

Ir F-C A X o(I I 1 XvE -X r

Vq(q E IF qEl' CA)

As a normal modal logic, the system will have the following axioms and rules:

* (Ml) 1 ( bV-'{q$)

* (M2) H (- q < -?,)

* (M3) H (. q$& 1 ) -4 (X$&4X)

* (M4) If s b, thenH(1qX-41b)

These axioms, especially M3 and M4, are critical to satisfying the intu itions about causal explanation discussed above in section 1.1. The general form of statements of causal explanation (for the Explanans account) is as follows:

K> (Type of explanandum & (Type of explanans, whether relevant

or not +-ORelevant type of explanans))

Axiom M3 expresses the principle of the Distribution over Or with re spect to the explanandum, as discussed in section 1.1, since M3 is equivalent to the following: K K K

(1 (O V X) ,+ (l 1V l x))

Axiom M4 guarantees that a situation explains all of the strong-Kleene con sequences of the features of any situation that it explains. This encompasses the Principle of the Distribution over And in section 1.1. It is the use

of non-classical, strong-Kleene logic that prevents this logic of explanation from collapsing into triviality. Explanation is not closed under classical log ical consequence: If the fire explains the water's boiling, it need not explain the logically equivalent situation of the water's boiling and the moon's be

ing eclipsed or the water's boiling and the moon's not being eclipsed. If it

were, then M3 (Distribution over Or) would entail that the fire must explain either the presence of a lunar eclipse (if the moon is indeed being eclipsed).

* (M5) (tb&-'q) I-(HIV--b)

* (M6) H (I 4 -1+ I )




It is not possible to ensure that the part/whole relation is antisymmet ric in the canonical model of the propositional version of situational logic.

However, in the Quantified Situation Logic developed in Realism Regained, antisymmetry can be ensured. [20, pp. 305-309]

The soundness of these axioms and rules is fairly obvious. Rule M5 must

be stated as it is, rather than as the tradition T axiom (i.e., (1 Q -?

since we are in a three-valued setting. I q$ can be true in a situation s where q itself is neither true nor false. In such a situation, the instance of the T axiom will also be truth valueless. The lefthand side of M5 cannot be verified in any situation, and its righthand side can never be falsified. This ensures that M5 satisfies the requirement for double-barreled logical implication: it is vacuously true that any model that verifies the left side also verifies the right, and any model that falsifies the right side also falsifies the left. See the Appendix for a sketch of the completeness proof.

The Causal Explanatory Operators

The causal explanatory operators will be either a one-place, two-place, or three-place operator, depending upon which account of event-individuation

we adopt. If we adopt something like the Quine-Lemmon account, then the causal explanation relation between two events (situations) will involve two qua- constructions: situation s qua q explains situation s' qua 4. In this case, the operator will have three places: the formula (X H-LM, 4') will be true at situation s just in case the formula X is true in every situation s' of such a kind that s' qua X explains s qua 4.

In contrast, if we adopt the Facta account, we will need only a one-place operator: the formula X +- F will be true at situation s just in case X is true in every situation s' such that s' explains (simpliciter) s. On the Facta account, there will be no explanatorily significant distinction between event aspects. What might naively be thought of as distinct event-aspects of a single event will, on this account, constitute distinct events.

The Explanans account gives rise to an intermediate situation, a two place operator. The formula (X -LDq) will be true at a situation s just in case X is true in every situation s' such that s' qua X explains s. In this case, the

causal explanatory operator will resemble the operators of dynamic logic. [15] The formula (X +- Dq$) corresponds to the formula [a]X of dynamic logic, where a designates some action or program. There are two key differences: in the logic of causal explanation, a formula (representing some event-aspect of the cause) takes the place of an action-designating term, and the causal



344 R. C. Koons

explanatory operator will be backward-looking, in contrast to the forward looking operator of dynamic logic. (X +-Lq) means that X is true in every causally prior situation that brings about the current situation by virtue of its s-aspect, not that X is true in every causally posterior situation that would be brought about by the current situation by virtue of its +-aspect.

In this paper, I will sketch the semantics for the Explanans account, with its two-place explanation or quausation operator. For each formula q of the language, there will be a binary relation -<?, on the set of possible situations.

We will have s -Fl s' just in case the situation s is a cause-qua-q of situation s', that is, situation s does make 0b true, and ? represents an aspect of s under which s is causally explanatory of s'. I will assume that whenever 4 and 0 are logically equivalent, -<,=-<,0.

The semantic clauses for the two-place causal explanatory operators will be as follows:

* M , s

(,0

<

0X) 8s V

(s' -<, s -*M , s' F

* M,s 3= ( MAX) Vs'(s' -<, s-&M,s' t P)

* M,s t '( -X) V ]s'(s' -< s&M,s' ?i)

As for the other modal operators, this gives rise to a bivalent interpretation for modally closed formulas.

Here is the appropriate axiomatization of the basic logic:

* (El) F (t +-D2) V --Q(b +-Dq)

* (E2) H -b( +-1bo) <-* (--l O-Xb)

* (E3) H ((s V x) +-00) -* (b ?-00) V (X V-00) * (E4) If b F- X, then F (b -h) - (X '-E0).

* (E5) If q H H, then - (X < 0X) (X

* (E6) H (4 <-L $0)

These axioms fits our pre-theoretic intuitions about causal explanation. El simply relates the two dual operators, and E2 reflects the fact that we are suppressing non-classicality at the level of the explanation relation itself, for the sake of simplicity.

Axioms E3 and E4 support principles of Distribution of Or and And with respect to non-relevant information about the explanans. These prin ciples reflect an ontological commitment about explanantia: viz., that they




are situations, concrete wholes consisting of atomic facta, and not fact- or proposition-like. If the explanans verifies the information V V X, then either it verifies 4' or it verifies X. There are no irreducibly disjunctive or abstract explanantia. Explanantia are parts of the concrete world, not abstractions from it. Here is an example of the kind of inference supported by E3:

Mary's stealing the bicycle or the scooter, qua her stealing something, ex plained her arrest. Therefore, either Mary's stealing the bicycle, qua her stealing something, explained her arrest, or Mary's stealing the scooter, qua her stealing some thing, explained her arrest.

Axiom E5 encapsulates the principle that strong-Kleene equivalent event-aspects are intersubstitutable in the explanans-context. This seems right: we don't want to distinguish event-aspects so finely that we can't carry through the following inference:

The container's pressure and volume explained the gas's temperature. Therefore, the container's volume and pressure explained the gas's temper ature.

There seems to be no need, for example, to move to sub-structural logic here.

Finally, axiom E6 corresponds to the quite reasonable assumption that if one event is a cause, qua A, of a second event, then the first event must in fact be a q-event. The qua construction is factive.

Since the causal relation is transitive, we shall assume that if s -<0 s', and s' -<, s", then s -A s". That is, if the b aspect empowers s to cause s',

and some aspect of s' empowers it to cause s", then the 4 aspect empowers s to cause s'. This gives rise to the following axiom:

* (E7) V ((X *-om) +-{4) -( (X +0-)

I think it is also plausible to assume that if q logically entails 4, (in partial logic), and if a situation s makes 4 true, then any situation that the +-aspect of s can explain can also be explained by the stronger, more inclusive q$-aspect. As a model condition, this means that when 0 entails 4, then the restriction of -A, to the set of situations making q$ true is a subset of -<O, which, in turn corresponds to the following axioms:

* (E8) If q H 4, then b H (X *-OF) -* (X +-00).

* (E9) H (X +-(q V 4)) -* ((X -C0X) V (X +-0C))



346 R. C. Koons

Finally, there is the question of the introduction of disjunctive explanans. In a case of overdetermination, in which a situation has a certain power qua

q, and has the very same power qua 4', does it follow that the situation has that power qua q V 4'? Here the ontological questions that divide the Explanans account from the Facta and Explanation account begin to have real bite. If event-aspects correspond to real, abstracted parts of the whole

event, then there would be no reason to recognize any aspect corresponding to (q V 4).5 So, the Explanans account would entail rejecting the inference to a disjunctive explanans in such cases. For the Facta and Explanation account, the crucial question is: by what properties is the event in ques

tion brought, together with its effect, under an explanatory law? Whether

the event's making true the proposition (q V 4) brings the event under a

subsuming explanatory law will depend on one's view of the extension of lawhood. Presumably, if there is a law that q explains X, and a second law

that 4 explains X, then we can infer from (4 V 4) that X can be explained, but whether there is a single law of the form '(Q V 4) explains X' is a further matter. Still, even a staunchly anti-Humean theorist might well think that lawhood should be preserved by such a logical implication, so the inference to the disjunctive explanans seems plausible in that case.

If we wish to add such a principle, we will require that, for every 0 and

4', <(CV)c (-t 0 -<4 The corresponding axiom will be:

* (EO0) F (X +-O q)&( -X +-O') -* (X <- (Q V 4))

In the canonical model, the causal-explanatory relation -<, can be defined

in a familiar way:

V4'(4 E F - '(4' -00)'A)

The completeness proof for the axiomatization above are sketched in the

appendix.

Causal Explanation Formalized

With this formal machinery in place, we are now in a position to give the log ical form of statements of causal explanation. First, on the Quine-Lemmon account, it is essential that event-aspects of both the cause and the effect be

specified in a complete statement of causal explanation.

5 This could be disputed, but only at the price of a wildly inflated ontology. Yablo [43]

[44] and Shoemaker [33] accept the existence of real, distinct events corresponding to the

exemplification of arbitrary disjunctions and generalizations of properties.




(T 0 )A

Here there is an implicit reference to some part s' of the indexed situation that has been caused by some situation s, but now we have explicitly desig nated that it was qua 0 that s caused s', and that it was qua Vb that s' was caused by s. Explicit reference to the cause is no longer required.

By way of illustration, let us suppose that Mary's stealing the bicycle (qua act of stealing) caused her to be convicted by the judge and jury (qua conviction of the crime of theft). This causal explanation could be stated as:

(By judge and jury & (Of a bicycle *-O((Stealing & By Mary), (Conviction of Mary of theft))

In a situational version of modal logic, instead of explicitly referring to a situation s and saying that s is of the type Stealing, we instead treat the situation as the index (Kaplan's context of evaluation) at which a proposition like Stealing is evaluated as true (if the index is of the corresponding type), false (if the index belongs to the negation of the type), or neither. The above formula states that the current situation (let's call it s) is an action of which the judge and jury are the agents and an action of conviction of Mary of theft, and this situation s' is explained by a situation (let's call it s') which is an act of stealing of which Mary is the agent and a bicycle the direct object.

Moreover, it states that it was qua act of stealing by Mary that s' achieves the status of explanans, and that it is qua act of conviction of Mary of theft that s is explained by s'. The formula thus singles out the causally relevant aspects of both the explanans (stealing by Mary) and the explanandum (conviction of Mary of theft), while permitting the introduction of causally irrelevant features of both situations, namely, that it was a bicycle that was stolen, and that it was a judge and jury that issued the conviction.

In most cases, the explanandum (Mary's conviction) will not be the context of evaluation; instead, the context of evaluation may include the explanandum as a proper part. In such cases, we need to make use of the

mereological operators. We simply must append an inclusion operator 1 to the beginning of our formula of explanation:

{(By judge and jury & (Of a bicycle +-O((Stealing & By Mary), (Conviction of Mary of theft))

In a similar way, we can distinguish between the effects of Sebastian's strolling and Sebastian's strolling in a leisurely way, or between a bolt's



348 R. C. Koons

snapping and its snapping suddenly. More importantly, perhaps, we can distinguish between the effects a mental state has qua mental state from those which it has qua brain state.

On the Explanans account, we must specify the relevant event-aspect of the cause (s), but not that of the effect, s', since s' is, in all its aspects, an effect of the q-grounded power of s.

1 (,0&(T 5)

Again, by way of illustration, we can express the causal explanation of Mary's conviction of theft as follows:

l(By judge and jury & Conviction of Mary of theft & (Of a bicycle

*-O (Stealing & By Mary)))

In this case, we cannot distinguish between causally relevant and irrel evant features of the explanandum. The implication of the formula is that everything that is stated about the conviction situation (the sub-situation of the present index) is causally explained by Mary's stealing of the bicycle, qua an act of stealing by Mary.

Finally, when we turn to the Facta account, we immediately encounter a serious problem. Since, on this account, the qua expression is superfluous, we have at hand only a one-place operator, 0. We cannot, however, use this operator to express the fact that the fact that 0 explains the fact that

V, nor even that the fact that b is causally explainable. The best we can do is something like the following:

1 (b & q0 *-{)

This states that there is a subsituation s' at which Vb is true, and there is

a cause s of s' at which q$ is true. However, we cannot infer from this that s

is the fact that A, or that s' is the fact that b, merely that the constitutive content of facts s and s' entail, in strong Kleene logic, the facts that 0 and that 4, respectively. This is probably a harmless defect in the case of the explanandum (s' or 4), since a Facta theorist might well accept the

principle that whenever a set of facta explain a second set of facta, the first also explains whatever is a strong-Kleene consequence of the second set. It is, however, a fatal defect in the case of the explanans (s' or s), since no one

would want to commit himself to the general principle that whenever one fact explains another, any consequence of the first also explains the second.




We must interpret the formula (b& q ) as claiming that there is a causal explanation of the fact that / that logically entails that q. To do bet ter than this, we would have to supplement our formal language in two ways: first, by introducing explicit quantification over situations, and, second, by introducing term-forming operators that pick out situations (facts) in terms of their maximal internal content. The first addition consists in the Quanti fied Situation Theory that I developed in Appendix A of Realism Regained [20, pp. 303-9]. Quantified Situation Logic introduces a special predicate A, expressing that a situation is actual (an actual part of the current index). The second addition would require either something like the level-dropping operators of Montague's Intensional Logic [26] or first-order theory of facts that mirrors Richmond Thomason's first-order theory of propositions.[35]

On a Montague-like approach, we could introduce new terms of the form q$, referring to the situation whose maximal content corresponds to the

propositional-type A. On the second, we would construct a term by means of function symbols that mirror the logical structure of /. I'll use 4* to abbreviate this term.

Armed with this additional machinery, we can now say that the fact that q causally explains the fact that b in one of the following two ways:

0 1 (0, & A-?,-O

t (,O& AOd* +

0)

This states that there is some sub-situation s' of the indexed situation which verifies both 4 and the fact that that situation is caused by the fact that Q. However, once we have this additional expressive power, there would seem to be little reason to use anything like a modal operator (such as + 0) to express the quausation/explanation relation. To say that the fact that q explains the fact that 0 we could instead introduce an ordinary, first order binary predicate, such as Exp, expressing the explanation relation as Exp(q*, 0*). The usefulness of modal logic depends on there being a real, ontological difference between events and facts, of the sort incorporated in either the Quine/Lemmon or the Explanans account.

Abductive Laws

Given a logic of causal explanation, we are now in a position to formulate general laws of abduction. Such laws could hold in a radically indeterministic world, a world in which causes not only do not necessitate their effects but do



350 R. C. Koons

not even raise the objective probability of those effects above some negligible, even infinitesimal level. In such a world, projection from a hypothesized situation to its probable effects would be impossible, but abduction from observed effects to their causes might still be tractable.

To express the generality of an abductive law, we must make use of the Quantified Situation Logic developed in Appendix A of Realism Regained [20, pp. 303-309]. An abductive law licensing the inference from 4' to b could take the following form:

Vx( ( (Ax&4A) - (Ax & (T -q)))

A still more useful form of abductive law is one that would enable one to locate the inferred cause in relation to the observed effect. Let f represent some particular functional relation between situations:

Vx(I (Ax&+O) -4 (Ax&(Af(x) +-q 0)))

This abductive law permits the inference to the actuality of an explanatory b situation f-related to the observed 4-situation. For example, suppose

there is an abductive law to the effect that a particular band in the spectral analysis of a star is always caused by the presence of iron in that star. In this case, we can let 4' be the situation-type that is verified by spectral analyses containing the relevant band, q be the situation-type that is verified by stars that contain iron, and the f relation will be a relation mapping spectral analyses to the stars from which they originate. The abductive law above then states that any sub-situation of the current index that contains the appropriate kind of spectral analysis is a situation causally explained by another situation, and the second situation explains the first qua being the situation of an iron-containing star.

Canonical Completeness Proofs

The completeness proof for the usual connectives of propositional logic is given in Muskens[28]. I'll give here the clauses for the mereological and

explanatory operators. I show by induction the usual lemma for canonical

models: C, IF[ = b E F.

To 1. 1X

(->) Assume l r. By axiom Ml, we have l r F. By Ml,

1 E F. By the definition of Cc, for any A such that A Ec F,




Q A. By induction, it follows that, for any such A, A L 0. Hence, K>

U-) Assume1 q F. Let A := 4 --' C F}. We need to show

that none of the consequences of b belong to A. To prove this, it K>

is sufficient to show that if b is a consequence of 0, then I b C IF. This follows from axioms M2 and M4. Thus, we can expand the set of consequences of X to a saturated set 0 such that o n A = 0 and e rc r, by a generalization of Lindenbaum's Lemma. If 0 entails a

K> disjunction (qVx), then r contains 1 (4O V X) (by axioms M2 and M4). By axiom M3, and the fact that r itself is saturated, it follows that

oK K> r contains either I 4 or I X. Thus, we can successfully extend the

consequences of q$ to a set 0 that is both saturated and that satisfies the condition 03 -c r. Since 0 contains 0b, we have by induction that

0 = q. Hence, r 1=1 nb.

2. (, OX

( -+) Assume that (4 r-0q) f F. By axiom El, we have that '(4 -

Oq) C r), and, by E2, that (--b< +-Dq) E F). By the definition of <?, for any A such that A x-< r, 4 V A. By induction, it follows that for any such A, A a 4'. Hence, rF ( (4' +-{q).

( +-) Assume that (4 -q) C r. Let A -

{X: (--X +-L E rF}. We

need to show that none of the consequences of 4 belong to A. To show this, it suffices to show that if X is a consequence of 4, then

(x *-oq) C r. This follows from axiom E4. Given axiom E3, we can

extend the consequences of 4 to a saturated set 0 such that 0 -< r (by another application of Lindenbaum's Lemma). Since e contains 4', we know by induction that 0 I= 4. Hence, r t= (4 *-(>).

In addition, we need to prove that in the canonical model the part-whole relation is reflexive and transitive. Transitivity is an immediate consequence of M6, given the definition of the part-whole relation for the canonical model. Suppose that r cc A and A EC 0. We need to show that r E:c 0. First,

O1 nn El suppose '4 q' E 0. By axiom M6, '14 I' E 0. So, '-' I O' V A. By axiom

LI Ml, it follows that '4 q' C A. So, by definition of Cc, ' IF5' ' F. Next,

0 00 0 assume that '4,' e F. Then 'l b' C A and ' el q' C. By axiom M6, 4 q$'



352 R. C. Koons

E e. So, IF c E. (A very similar proof can be used to show that the causal-explanatory relation is transitive in the canonical model, given axiom E8.)

In the case of reflexivity, we need to show that, for an arbitrary F, we a

have F Cc F. First, suppose that '{ q$' C r. We need to demonstrate that I F. Suppose for contradiction that ' Eq' E F. By an application of

rule M5, we would have '(1 -b V q0)' E F (substituting '--lo' for the variable

'+'). Given the consistency of F and axiom M2, we know that '1 --ib' V F. Given the saturatedness of F, this means that '4' E F. But this contradicts the consistency of r, since we assumed that ' -'' c F.

Next, suppose that 'q' E F. We need to show that '. qb'E F. Suppose

for contradiction that '1 Ib' F F. Given MI, this means ' E j q' I F, and,

given M2, it follows that 'f -q' E F. By applying M5, we get '(1 q V -q)'

E F. Since F is consistent, we cannot have '1 sPE IF. Since F is saturated, we must have '--b' E F, but this again contradicts the consistency of F.

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ROBERT C. KooNs Department of Philosophy University of Texas at Austin 1 University Station

Austin, Texas, U. S. A. rkoonsfmail . utexas . edu



the logic of causal explanation: an axiomatization

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