Modelling Metaphysics: The Metaphysics of a Model 9783110326086, 9783110325256

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Uwe Meixner Modelling Metaphysics The Metaphysics of a Model

Philosophische Analyse Philosophical Analysis Herausgegeben von / Edited by Herbert Hochberg • Rafael Hüntelmann • Christian Kanzian Richard Schantz • Erwin Tegtmeier Band 34 / Volume 34

Uwe Meixner

Modelling Metaphysics The Metaphysics of a Model

Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available in the Internet at http://dnb.d-nb.de.

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Alles Vergängliche ist nur ein Gleichnis. Goethe, Faust II

Contents 1

Preface and introduction .................................................................................. 3

2 2.1

The basic description of Model T .................................................................... 9 Mathematical representation1 ..........................................................................15

3 3.1 3.2 3.2.1 3.2.1.1 3.2.2 3.2.2.1 3.3 3.3.1 3.3.1.1

Ontological categories for T ............................................................................17 Basic universals ...............................................................................................17 Individuals which are basic or immediately derivative ..................................20 Spatial configurations ......................................................................................20 Spatial parthood for spatial configurations .....................................................23 Temporal configurations .................................................................................24 Direction and relative position for temporal configurations ...........................27 States of affairs ................................................................................................33 Time-dependent but not time-thematic states of affairs ..................................34 Additional perspectives on not time-dependent, but time-thematic states of affairs ..........................................................................................................41 Time-dependent and time-thematic states of affairs – and the nature of indexical time-thematicness ............................................................................42 Events as sequences of momentary states .......................................................46 Events and states of affairs ..............................................................................49 Basic tropes .....................................................................................................52 Basic spatiotemporal tropes as fundamental or derivative entities .................54 Worlds and modal positions ............................................................................58

3.3.2 3.4 3.4.1 3.5 3.5.1 3.6 4 4.1 4.1.1 4.2

Uniformity and diversiformity of histories and of maximally composite momentary states .............................................................................................61 Regularities of histories ...................................................................................67 Regularities and laws (of nature) ....................................................................71 Uniformity, regularities, laws, and actuality ...................................................75

5 5.1 5.1.1 5.2 5.2.1 5.2.2

Actuality and other modalities for T ...............................................................77 Temporal and historical relativization of statements about T .........................77 The varieties of historical relativization ..........................................................83 The plurality of actuality-predicates, basic and defined .................................87 Actuality – in two ways non-relativized .........................................................92 Actualization** and what is behind it ...........................................................102

1

The designation “Chapter 2” refers to the entire second chapter of this book, whereas the designation “Sect. 2” refers only to the first section of the second chapter. This convention also extends to the other chapters and sections of this book. Thus, “Chapter N” refers to the entire Nth chapter, “Sect. N” refers only to the first section of the Nth chapter.

5.3 5.3.1 5.4

Possibility and necessity for T .......................................................................110 History-relative and/or time-relative necessity and possibility for T ............118 The immanent and transcendent perspective on modality and time .............126

6 6.1 6.2

The physics of H* ..........................................................................................131 The first two of the laws for T/the laws of H* ..............................................133 The Third Law for T and the supervenience of atomic higher continuants relative to H* .................................................................................................136 Democriteanism and four-dimensionalism for H* and the transhistorical identity of atomic higher continuants ............................................................147 Four-dimensionalist counterpartism ..............................................................157 Four-dimensionalism without counterpartism ..............................................159 Final determinations on atomic material objects ...........................................160 The strong essentiality of origin for atomic material objects ........................163 Candidates for further laws for T, and more on the laws for T .....................165 The Candidate Fifth Law for T ......................................................................173 The regulation of collision ............................................................................180 The Candidate Sixth Law for T .....................................................................188 The canon of the laws for T ...........................................................................193 The initial state ..............................................................................................198

6.3 6.3.1 6.3.2 6.4 6.4.1 6.5 6.5.1 6.5.2 6.5.3 6.6 6.7 7 7.1 7.2 7.3 7.4 7.4.1 7.4.2 7.4.3 7.5

Five T-metaphysical issues and the metaphysics of Reality .........................205 The completion of the rules of ACTUHIST ..................................................205 Physicalism and dualism with regard to Model T .........................................213 T-immanent and T-transcendent causation ...................................................223 Composite T-material objects .......................................................................240 Typical material objects in Reality and their T-analogues ............................252 Objections to taking the sequential T-material* objects as the T-analogues of the typical material objects .......................................................................258 The transtemporal and transhistorical identity of T-material* objects .........260 T-metaphysical teleology ..............................................................................262

Index of concepts and principles ................................................................................267

2

1

Preface and introduction

Metaphysics is that part of philosophy which endeavours to elucidate the basic structure of the totality of all there is in a systematic fashion, aiming at completeness, paying special attention to the place and role of human beings in that totality. Exploring the totality of all there is a daunting project, even if it is only the basic – most general – structure of being that one is after. The intention to explore it in a systematic fashion, and with the aim of completeness, makes the project even more daunting. But, through the centuries, metaphysicians have undertaken it nonetheless – fired by the vision of the prize that might be had in the end: the basic, general truth about the world, and human beings in it. Metaphysicians have never returned successfully – at least not in the eyes of their fellow philosophers, their contemporaries or successors in the long history of philosophy – from attempting to sail around the totality of being. The prize is not won. And the prospects of ever winning it are dim. But some clarifications regarding where and what we are may perhaps be nonetheless retrieved from the more or less heroic efforts of the past, present, and future. Recently, some philosophers have advocated availing oneself of the aid of science in the metaphysical project – presumably because science has proven such a great help in medicine and engineering. What needs to be said about this proposal has already been said very well by Arthur Schopenhauer (a metaphysician who was also capable of epistemological reflection) in the 19th century: The height toward which in our times the natural sciences have risen puts, in this respect, all previous centuries into shadow, and is a summit that mankind reaches for the first time. But however great the progress made by physics (understood in the wide sense of the ancients2) may come to be, not yet the smallest step will have come about by this in the direction of metaphysics, just as little as a plane can ever gain cubic content by being extended ever so far.3

2

In this sense, physics coincides with natural science. (Footnote by U.M.) U.M.’s translation. The German original can be found in: Arthur Schopenhauer, Zürcher Ausgabe. Werke in zehn Bänden [Zurich Edition. Works in ten volumes], vol. III, Zurich: Diogenes 1977, p. 207. The quotation is taken from Die Welt als Wille und Vorstellung [The World as Will and Presentation], vol. II, ch. 17. 3

Science is a limiting condition of metaphysics: something that metaphysics should respect and should not contradict without very good reason (noting that it is hard to imagine any reason that would justify metaphysics in contradicting science). But science cannot make metaphysics, it cannot accomplish the metaphysical task, and it cannot even (actively) help to accomplish it. One of the aims of this book is to convince the reader of this (if he or she happens not to believe it already). Even if we were given ideally complete science, the hard metaphysical work would still remain to be done, the adventurous metaphysical journey would still remain to be undertaken. This is just as true if a naturalistic direction is pursued in metaphysics as it is true if the direction followed is a non-naturalistic one. Before undertaking a long journey with uncertain outcome, it seems a good idea to simulate the journey – if that is at all possible. The simulation will not be perfect, of course (otherwise one might as well go on the journey – the journey itself – right away). The simulation will have a small scale, it will drastically simplify (in representation) the terrain to be travelled through, it will radically leave out (in representation) many details and complexities; it will be true to the facts (to the extent they are known) only in those respects which, in the light of the purpose of the journey, are deemed essential to the travellers. And the simulation will be hypothesizing to some extent: it will represent what is likely to happen if vicissitude A comes about, and if, alternatively, vicissitude B comes about, and if vicissitude C comes about … . Regarding representation, the simulation of a journey can be closer to or more remote from the reality of the journey it simulates, and more telling or less telling with regard to that reality. Note that these two pairs of contrasting characteristics do not necessarily parallel each other: a simulation may be close, but not telling, and it may be remote, but telling (this is the case because “tellingness” is a matter of selection, closeness not necessarily so). The contents of this book can be described, in the briefest manner, as a remote but telling simulation of the journey of a metaphysician around the totality of being – or more accurately: as the record of such a simulation. As a simulation-record, the book is not metaphysics fiction: it makes no attempt at make-belief. It is obvious from the start that it is a sandbox game which is being described. But in offering such description, the book has much to tell about many real-life metaphysical matters, often in close connection to each other: matters of metaphysical doctrine, matters of metaphysical method, and matters of the epistemology of metaphysics.

4

The aim of systematic completeness in metaphysics is formidably difficult to realize. But the simulation of metaphysics that is presented in this book is constructed in such a manner that, in it, systematic completeness is accomplishable, namely, with respect to the (comparatively) tiny organized universe of discourse that lies at its centre: Model T, as it will be called. In this book, readers can find a not too dissimilar image (simulation) of what systematic completeness would amount to if it could be had in real metaphysics (the metaphysics of Reality – remotely, but tellingly, modelled by T), an image of its intellectual desirability and of the intellectual satisfaction to be found in the having of it. Even if systematic completeness were not a theoretical value in itself, without it, how could disputes between conflicting metaphysically local positions, with – from the local point of view – equal claims to rational allegiance, be rationally decided? They could not be thus decided; and it seems to me that there are very many such metaphysically local disputes; indeed, that the metaphysical discussions of the present consist almost entirely of them – local disputes which are based on locally dictated, and therefore to some extent accidental, perhaps haphazard, choices of principles. In any case, readers will find in this book a more than usually systematic presentation of the interlocking of metaphysical concepts and ideas, of the making of conceptual choices in metaphysics, of the posing of metaphysical questions, of the discovering of metaphysical depths and heights – all of this, however, in simulation. Almost all metaphysical themes are addressed and brought together with their proper positions relative to each other – but in simulation only. The radical finiteness of the considered metaphysical model (i.e., Model T) draws, one might say, a magic circle around the simulation, giving it the aspect of things happening in an intellectual garden, or on an intellectual playground; the finiteness of the model may even create the illusion – it is still an illusion – that one can hold it all at once before one’s intellectual eyes. But it must be kept in mind that metaphysics in simulation is more than just an intellectually pleasing pastime (though one might extract a moderately exciting computer game from it, calling it “God and the World”). It is a journey around the totality of being that is being simulated. In the book, this intended reality is never lost from sight; indeed, one cannot present a simulation as simulation without persistent reference to what is being simulated. From the contrast and the similarity between the simulation and what is being simulated – which is metaphysical scrutiny itself, and by implication: its objects and results – metaphysical and epistemo-

5

logical insights are to be gained (insights that can be used for judging the metaphysical journeys of the past, recent or remote, or for planning future ones). Simulations are experiments. Like all true experiments, they have the advantage that their conditions can easily be varied by the experimenter. This is also true of the simulation experiment presented in this book. But, due to the limitedness of (philosophy-reserved) lifetime and book-space, only one possibility of simulation – on the whole, though there are some excursions – is fully realized and explored by the author of this book. If this seems few, remember that the simulation is to be that of a journey around the totality of being; even in simulation such a journey is arduous, and does not allow the squandering of one’s strength in lengthy explorations of alternative routes. This is also the reason why the discussion of the views of others is conspicuously absent from this book, certainly not out of disrespect. (I would like to mention that the metaphysical opinions of David Lewis, Alvin Plantinga, Peter van Inwagen, Lynne Rudder Baker, and Jonathan Lowe have been a source of inspiration for me in forming my own metaphysical opinions, which are quite different from theirs.) The single simulation I have chosen to realize – others, I hope will realize other simulations (centring on the same metaphysical model, or on another, more complex one) – is the simulation of a view of the totality of being that is philosophically on the defensive nowadays, as will not remain hidden to anyone who is knowledgeable in recent and contemporary literature on metaphysics (where I am taking metaphysics to also include large parts of the philosophy of mind and of the philosophy of free will). My intention in choosing that particular simulation was to give back to that particular metaphysical view – in showing, by simulation, that it is viable – the freshness of an attractive option. The metaphysical horizon is wider than philosophers of past and present generally have thought it to be. The wideness of the metaphysical horizon – this wideness seems to be revealed, with full impact, by simulationmetaphysics only (and somewhat paradoxically so, considering that simulation-metaphysics is essentially a making smaller): by actually putting the many alternative metaphysical perspectives and directions in a palpable and, as it were, egalitarian form before the metaphysician’s eyes. Even if one chooses – or is forced to choose – to explore only one of the alternatives (in simulation) in full, the others are still there, with a more than usually felt presence (brought about, precisely, by the scaling down of simulation); their strengthened presence is the best medicine against Metaphysi-

6

cian’s Disease: against taking one’s own limited perspective for an absolute absolute. Simulation-metaphysics is enabled to reveal the wideness of the metaphysical horizon by an essential aspect of it which departs from the reality of doing metaphysics in a way that can hardly be overestimated epistemologically. By scaling down the extension and complexity of being, simulation-metaphysics automatically moves to the point where it adopts a God’s-eye view (no matter whether there is in the simulation a place for God-in-simulation, or not); simulation-metaphysics looks from the outside at the-totality-of-being-in-simulation, and it may (and in this book it will) incorporate elements of metaphysical content in its simulating that are directed at, and connect with, the totality-of-being-in-simulation from the outside. Simulation-metaphysics is thus doing what is impossible for real metaphysics to do with regard to the real thing, the totality of being (simpliciter); it adopts – truly adopts, not just imagines itself to adopt – the perspective of complete transcendence. We cannot adopt this perspective with regard to the totality of being, not even if one understands by “the totality of being” – as is reasonable – the totality of being as it is (more or less) known to us. We are immersed in that totality – which means that in doing metaphysics we are in at least one important respect necessarily blind: the respect of finding a well-founded answer to the question whether we have fairly completed our metaphysical work, or are still very far from completing it. It is sometimes forgotten – by amateur and professional metaphysicians alike – that guesswork is all that can be had by way of an answer to this question. Simulation-metaphysics reminds us of this by the very fact that it can give us more in this regard than can be had in reality. The writing of this book was funded for the period of nine months by the German Research Foundation (DFG) as part of a research project (at the University of Regensburg) on the ontology and epistemology of modality. For this financial support, I am grateful. The results of my work that are especially relevant for the philosophy of modality are contained in the Chapters 4, 5, and 6 of this book. Uwe Meixner

7

2

The basic description of Model T

Every metaphysical model has two types of interrelated ingredients: positions and fillings. In every metaphysical model the ontological relationship between positions and fillings is the same: the fillings fill – or do not fill – the positions. Model T, which is a particular metaphysical model, has two types of positions: spatial positions and temporal positions. Moreover, Model T is finite regarding positions: it has 100 spatial positions, and 100 temporal positions. Model T is also finite regarding fillings. In fact, with regard to (possible) fillings of (individual) spatial positions, it is utterly simple: Model T has just one such filling: Full (as we may simply call it). The spatial positions of T are arranged in a certain way, forming the space of T. The simplest way to describe their arrangement is to take the natural numbers from 1 to 10 and to let each of the 100 ordered pairs of these numbers stand for a single spatial position of T, leaving no such position unrepresented, arranging the representation in such a way that the spatial relationships between the spatial positions of T are completely reflected in certain relationships between the pairs of numbers. Consider, for example, and . The spatial position [represented by] is one step to the left of the spatial position [represented by] and three steps down from there (since 3 − 4 = −1, and 6 − 9 = −3), or in other words: the spatial position is one step to the right of and three steps up from there (since 4 − 3 = 1, and 9 − 6 = 3). The general definitions of the notions just employed are the following:

D1 D2 D3 D4

Let a, b, c, and d be elements of {1, …, 10} and x an element of {0, …, 9}: is [in the column which is] x step(s) to the left of =Def (a − c) is negative or zero and x = a − c . is [in the column which is] x step(s) to the right of =Def (a − c) is positive or zero and x = (a − c). is [in the row which is] x step(s) down from =Def (b − d) is negative or zero and x = a − c . is [in the row which is] x step(s) up from =Def (b − d) is positive or zero and x = (b − d).

The four predicates defined above suffice to completely describe the spatial relationships between the spatial positions of T. It should be noted that the monadic predicates “P is [in the column which is] x step(s) to the left/right of ” and “P is [in the row which is] y steps down/up from ” do not yet specify a unique spatial position of T, but several or none. But the conjunctive monadic predicate “P is [in the column which is] x step(s) to the left/right of and P is [in the row which is] y step(s) down/up from ” does specify a unique spatial position of T – or none. Model T has 2100 (possible) fillings of (individual) temporal positions; those fillings are precisely the 2100 fully specific ways (representable by set-theoretical functions) in which T’s unique filling of spatial positions – Full – can be simultaneously distributed over all of the (100) spatial positions of T. Here are two very special fillings of T’s temporal positions: the one in which Full is present at every spatial position of T (that is, the space of T is completely full), and the one in which Full is present at no spatial position of T (that is, the space of T is entirely empty). (No need to say: even these two uniform fillings of temporal positions of T are highly structured in comparison to Full, the unique filling of spatial positions of T.) The temporal positions of T are arranged in a certain way, forming the timeline of T. The simplest way to describe their arrangement is to take the natural numbers from 1 to 100 and to let each of these numbers stand for a single temporal position of T, leaving no such position unrepresented, arranging the representation in such a way that the temporal relationships between the temporal positions of T are completely reflected in certain relationships between the numbers. The two predicates defined by the following two definitions suffice to completely describe the temporal relationships between the temporal positions of T:

D5 D6

Let c and d be an element of {1, …, 100} and x an element of {0, …, 99}: c is x step(s) earlier than d =Def (c − d) is negative or zero and x = c − d . c is x step(s) later than d =Def (c − d) is positive or zero and x = (c − d).

Finally, Model T has (2100)100 (or in other words: 210000) complete histories. Each complete history of T is a way in which T’s 2100 (possible) fillings of (individual) temporal positions can be consecutively distributed

10

over all of the (100) temporal positions of T. (Obviously, in each such way some – indeed, most – fillings of temporal positions will remain unused.) Here are two very special complete histories of T: the one in which the space of T is completely full at every temporal position of T, and the one in which the space of T is entirely empty at every temporal position of T. The basic description of Model T is now already complete. Before we go on, a technical remark: In this book, the explicit (i.e., linguistically expressed) relativization of concepts to T (by “of T”, “for T”, “in T”, “T-”, or whatnot) will frequently be suppressed; this happens for purely stylistic reasons (that is: the relativization is in force wherever it is appropriate, but on occasion it is left tacit). As we go along, the richness of philosophical considerations that Model T gives rise to will become apparent. We now need to consider – first – the character and rationale of T as a metaphysical model. No need to say: T is very far from Reality (as we know it). For example, T is certainly too Newtonian (so to speak) to be a close simulacrum of Reality: with the same total system of spatial positions recurring internally unchanged (though perhaps filled in different ways) at each temporal position. In another respect, T is too medieval (so to speak) to be a close simulacrum of Reality: with the timeline of T having a first temporal position and a last, and with the space of T ending in a final spatial position, no matter which direction is being pursued from no matter which spatial position of T (that is, ending in one of those 36 spatial positions the first or second coordinate of which is either 1 or 10). But on the other hand, T is substantially similar to Reality in the points which are of true metaphysical interest. Model T will serve to clarify our metaphysical theorizing about Reality by making points of metaphysical interest stand out more clearly than in Reality, with all its mind-boggling complexity. In gauging the value of T as a metaphysical model of Reality, one should also keep in mind that, often, at least as much can be learned from contrast as from similarity. It is important to remember what T is not, but can easily be confused with (though certainly not with Reality). If one tries to visualize T, then a square on a white piece of paper (bounded by thin lines) may come to mind, which square is subdivided (by thin lines) into 100 small squares:

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1,10

2,10

3,10

4,10

5,10

6,10

7,10

8,10

9,10

10,10

1,9

2,9

3,9

4,9

5,9

6,9

7,9

8,9

9,9

10,9

1,8

2,8

3,8

4,8

5,8

6,8

7,8

8,8

9,8

10,8

1,7

2,7

3,7

4,7

5,7

6,7

7,7

8,7

9,7

10,7

1,6

2,6

3,6

4,6

5,6

6,6

7,6

8,6

9,6

10,6

1,5

2,5

3,5

4,5

5,5

6,5

7,5

8,5

9,5

10,5

1,4

2,4

3,4

4,4

5,4

6,4

7,4

8,4

9,4

10,4

1,3

2,3

3,3

4,3

5,3

6,3

7,3

8,3

9,3

10,3

1,2

2,2

3,2

4,2

5,2

6,2

7,2

8,2

9,2

10,2

1,1

2,1

3,1

4,1

5,1

6,1

7,1

8,1

9,1

10,1

The piece of paper itself is a page in a book with 100 pages. On each page one finds the same square subdivided into the same 100 small squares. On most pages some of the small squares are blackened, others not, and not always (i.e., not on every page) the same small squares are blackened. On one of the pages, for example, we find: 1,10

2,10

3,10

4,10

5,10

6,10

7,10

8,10

9,10

10,10

1,9

2,9

3,9

4,9

5,9

6,9

7,9

8,9

9,9

10,9

1,8

2,8

3,8

4,8

5,8

6,8

7,8

8,8

9,8

10,8

1,7

2,7

3,7

4,7

5,7

6,7

7,7

8,7

9,7

10,7

1,6

2,6

3,6

4,6

5,6

6,6

7,6

8,6

9,6

10,6

1,5

2,5

3,5

4,5

5,5

6,5

7,5

8,5

9,5

10,5

1,4

2,4

3,4

4,4

5,4

6,4

7,4

8,4

9,4

10,4

1,3

2,3

3,3

4,3

5,3

6,3

7,3

8,3

9,3

10,3

1,2

2,2

3,2

4,2

5,2

6,2

7,2

8,2

9,2

10,2

1,1

2,1

3,1

4,1

5,1

6,1

7,1

8,1

9,1

10,1

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Then one discovers that the book is just one volume in a huge collection of books (counting 210000 volumes). All the books in that collection are, structurally, just like the book already described; if they differ from each other, then only in one respect: for any two books in the collection there will be at least one page (page N, with 1 ≤ N ≤ 100) which is such that it is not the case that the same small squares are blackened on that page in both books. The described collection of books, C, is a concrete picture (a visualization) of T – it is not T itself; for it has many properties that T has not. This being said, it must nevertheless be emphasized that C will often immediately disclose a truth about T (by making it visible) that is otherwise obtainable only by a laborious stepwise procedure. T might also be confused with the abstract structure of which T is, rightly considered, merely a particular instance (along with C, the above-described collection of books), or rather, with the canonical mathematical representation R of that structure (where R, note, is itself a particular instance of the said structure): (1) [the spatial positions =] the ordered pairs that are formable out of {1, …, 10}. (2) [the temporal positions =] the elements of {1, …, 100}. (3) [the fillings of spatial positions =] 1. (4) [the fillings of temporal positions =] the functions (set-theoretically conceived) that assign either 1 or 0 to each ordered pair that is formable out of {1, …, 10} and that are defined for nothing else than the ordered pairs that are formable out of {1, …, 10}. (5) [the complete histories =] the functions (set-theoretically conceived) that assign some function from the functions specified under (4) (not normally always the same function) to each element of {1, …, 100} and that are defined for nothing else than the elements of {1, …, 100}. Plus the following definitions: D1, D2, D3, D4, D5, D6. T is not R – for the same reason that T is not C: both R and C have properties that T has not; though R is abstract and C concrete, this truth remains. T is an ontic system all by itself, the basic description of which is such that both C and R represent T. But T’s complete histories are, in the truth of things, neither certain books nor functions assigning certain numerical functions to certain numbers.

The basic description of T is, in the main, a description of what is intrinsic to T. As will soon become apparent (in Sect. 3.2.2.1), regarding one – objective, non-stipulatory – aspect of T the basic description of T does go beyond what is intrinsic to T. But it remains true that everything that is intrinsic to T can be gathered from its basic description. Therefore: if something cannot be gathered from T’s basic description, then it is not intrinsic to T. (We shall have occasion to make use of this principle – the converse of which, as has just been indicated, is not true! – in Sect. 3.3.2 and in Sect.

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4.1.1.) But this does not mean that everything intrinsic to T has already been explicitly stated in its basic description. Later in the book, new means of description will be introduced, which, being new, are not already employed in T’s basic description and which will be used to further state explicitly what is intrinsic to T. Still, whatever is correctly asserted to be intrinsic to T is already implicit in T’s basic description, that is: can, in principle, be gathered from it – via conceptual construction. One interesting and fundamental aspect of T – but one that is not intrinsic to it, and does not fall within the purview of its basic description, either – should be noted here: there is an outside to T. After all, we are looking at T from outside of it. There is a greater reality – a greater ontic system – in which T is embedded. This suggests – since T is a model of Reality – that there is also an outside to Reality. One might suppose that this suggestion can only be misleading: Reality is the totality of being, and hence there can be no outside to it. But the (stipulatively) synonymous terms “Reality” and “the totality of being” are here understood in the sense of “Reality, or: the totality of being, as we know it” (as was already indicated – in passing – above), and it does make sense to entertain the idea that there is an outside to Reality as we know it, since the totality of being (not as we know it but) in itself may be of greater extent than we know or are even able to know (cf. the end of Chapter 1). The interesting question is this: Is there motivation originating with Reality (as we know it) for assuming that there is indeed an outside to Reality? In answering this question, it will be helpful to further consider Model T, starting with its basic description, asking whether there is motivation originating with T (as a model of Reality) for assuming that there is an outside to T – leaving quite aside the fact that we already know, for independent reasons, that there is indeed such an outside (given our privileged vantage point outside of T). This inquiry will throw light on the parallel issue in the case of Reality, where, in contrast to the situation regarding Model T, we do not seem to have a privileged outside vantage point at our disposal. (For a first step in the envisaged direction, see Sect. 3.2.2.1.) Quite aside from these considerations, analogy suggests that just as there is an outside to Reality as, say, spiders know it, so there is more likely than not an outside to Reality as we know it – notwithstanding the fact that, presumably, we are rather more intelligent than spiders.

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2.1

Mathematical representation

Just like Reality, Model T is amenable to mathematical description (though a much simpler one than is appropriate for Reality). Just as we do with regard to Reality, we can represent spatial and temporal aspects of T mathematically – and we have already done so in the previous section. Given this mathematical representation, it is convenient to call, say, a spatial position of T; but strictly speaking this is not true: strictly speaking, only represents a spatial position of T. Likewise, it is convenient to call, say, 1 a temporal position of T; but strictly speaking this is not true: strictly speaking, 1 only represents a temporal position of T. This being said, I announce that I shall follow the way of convenient parlance, reminding readers that the convenient mathematical manner of speaking is not to be interpreted literally, but merely representationally. Note that mathematical representation does not need to mention numbers: “numberless” set-theoretical representation, too, falls within the purview of mathematical representation. In set-theoretical representation, it is often asserted that a certain entity “is” a certain set – which assertion may seem strange or downright absurd to some. But all that is meant – in settheoretical representation, as opposed to set-theoretical reductionism – is that the set in question represents the entity in question, not that it really is that entity.

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3

Ontological categories for T

I take ontological categories to be highly general kinds of entity (or to be representatives of such kinds: to be the corresponding concepts and predicates). In what follows, there is no ambition to achieve a parsimonious complete system of ontological categories, or to reduce some ontological categories to others, or to show that some ontological category or other is “unfortunately” – contrary to what is believed by the unenlightened – empty. Rather, it is hoped that the following tour through Model T’s rich garden of model-categories may reveal to some metaphysicians of Reality something about Reality itself and its categories (Reality being the original of which T is a metaphysical model) – or at least something about pursuing the general metaphysics (i.e., ontology) of Reality in a way that is more tolerant (but no less rigorous) than the way in which it is usually pursued within the analytic tradition. 3.1

Basic universals

Full – the unique (possible) filling of T’s spatial positions (considered individually) – is a universal, while these spatial positions themselves are individuals. Full can be treated either as a non-predicative universal: type, or as a monadic predicative universal: property, more specifically: as a simple, structureless quality. Talk of Full filling a spatial position, of Full being present at a spatial position, or of Full occurring at a spatial position fits the view that Full is a type – better than the view that it is a property. On the other hand, talk of a spatial position being full, or of Full being identical with Fullness and some spatial position having it fits the view that Full is property – better than the view that it is a type. Not much importance attaches to the question whether Full is a type (and not identical with Fullness) or a property (and identical with Fullness). But it is interesting that the question is difficult to decide. For the time being, we can leave the matter undecided, and I shall employ the idioms fitting the type-view and the idioms fitting the propertyview of Full indiscriminately, noting that it is only slightly inappropriate to say that the property Full (that is, the property of Fullness, or of being full) is present at, occurs at, or fills a spatial position, and that it is only slightly inappropriate to say that a spatial position has the type Full (consider that it is entirely appropriate to say that it is of this type). However, when Full is

being presented again and again as entering into composition with spatial positions to form states of affairs (starting with Sect. 3.3.1), this, indeed, seems to tilt the balance decidedly in favour of regarding Full as a property. No matter whether Full is a type or a property, it is entirely appropriate to say that a spatial position exemplifies or instantiates it. This shows Full to be a universal. Moreover, Full is simultaneously multi-locatable, which also shows it to be a universal: Full can be present at several spatial positions at once, or in other words: several spatial positions can be simultaneously full. Full intrinsically requires that it be exemplified by spatial positions only, and Full intrinsically requires that the T-intrinsic exemplification of it be relative to a temporal position. Accordingly, the canonical way of phrasing an exemplification statement for Full is this: the spatial position P exemplifies Full at the temporal position Z. Aside from these requirements, Full is a free universal; no restrictions are intrinsic to it regarding its Ttemporal and T-spatial distribution (of exemplification), that is: its distribution in the space of T and along the timeline of T. This absence of restrictions makes the number of complete histories of T so enormously large: Full can be distributed in (2100)100 ways in the space of T and along the timeline of T. Full is explicitly recorded – named – in the basic description of Model T; it is a non-relational universal. Certain relational universals are just indirectly referred to in that description, not explicitly recorded, not named. Take, for example, the relation One-step-to-the-right-of (cf. D2), which intrinsically requires that exemplification of it be only by spatial positions, or the relation Two-steps-earlier-than (cf. D5), which intrinsically requires that exemplification of it be only by temporal positions. Aside from the fact that these universals are relational while Full is non-relational, they differ deeply from Full also in other respects. Both relations intrinsically require that their exemplification be not relative to any temporal position: their exemplification is to be atemporal. For example: , exemplify [in the given order] One-step-to-the-right-of; it is not necessary, and indeed misleading, to supply this (true) statement with a relativization to some temporal position or other. Again: 1, 3 [in the given order] exemplify Two-steps-earlier-than; it is not necessary, and indeed misleading, to supply this (true) statement with a relativization to some temporal position or other.

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And in contrast to Full, neither One-step-to-the-right-of nor Twosteps-earlier-than is a free universal. They are totally intrinsically restricted in their (not temporally relative) exemplification. For example: while there are 100 × 100 ordered pairs formable out of the elements of {1, …, 100}, Two-steps-earlier-than intrinsically excludes 9902 of these 10000 ordered pairs (of temporal positions) from being a ordered pair with the following description: N, N´ [in this order] exemplifies Two-steps-earlierthan. But the total intrinsic restrictedness of our two example relations does not only have a negative side: these relations do not only intrinsically exclude certain ordered pairs of temporal, respectively: spatial, positions from exemplifying them, they also intrinsically require that certain pairs of temporal, respectively: spatial, positions exemplify them. Thus Two-stepsearlier-than does not only intrinsically exclude 9902 ordered pairs of temporal positions from exemplifying it, it intrinsically requires that the 98 remaining ordered pairs of temporal positions exemplify it (namely, the 98 elements in the following set {, , … , }). The same is true (mutatis mutandis) of One-step-to-the-right-of: it intrinsically divides the totality of ordered pairs of spatial positions into such that are intrinsically required to [atemporally] exemplify it, and such that are intrinsically required not to exemplify it. Thus, there is zero degree of freedom to the relations Two-steps-earlier-than and One-step-to-the-right-of. The same is true of all relations of any of the following six forms, which relations all are relations between spatial, respectively: temporal, positions of T: (1) N-steps-to-the-right-of, (2) N-steps-to-the-left-of, (3) N-steps-upfrom, (4) N-steps-down-from, (5) N-steps-earlier-than, (6) N-steps-laterthan. The degree of freedom of all these relations (each instantiating one or the other of the six forms) is zero. In this, they differ rather strikingly from Full, of which the degree of freedom is very high: aside from the requirements intrinsic to Full with regard to its field of exemplification, which is to be the set of spatial positions (of T), and its manner of T-intrinsic exemplification, which is to be relativized to temporal positions (of T), there is no further intrinsic restrictedness whatever to the exemplification of this universal, either negatively or positively. I note that one leaves the level of the basic universals of T if one considers properties that are definitionally based (by the operations of filling and disjunction) on the above-considered relations, as, for example, Properly-earlier-than-5 (a property of temporal positions), or Properly-left-of (a property of spatial positions). But what has been observed above

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regarding the atemporality and total restrictedness of exemplification (both positively and negatively) of certain T-basic relations, also applies here (mutatis mutandis): Properly-left-of-, for example, intrinsically requires that all spatial positions of T with a first coordinate that is smaller than 5 [atemporally] exemplify it, and that all spatial positions of T with a first coordinate that is equal to or greater than 5 do not exemplify it. One also leaves the level of the basic universals of T if one considers a property like the following: being full along with being full. This property – for (obvious) reasons intrinsic to it – cannot be exemplified by any spatial position at any temporal position without exemplifying it at that same temporal position (or in other words: without exemplifying Full at that temporal position). Clearly, being full along with being full, though still a universal with a high degree of freedom, does not have the same degree of freedom Full has. 3.2

Individuals which are basic or immediately derivative

The spatial positions of T and the temporal positions of T are the basic individuals of T. As such, they are also atomic individuals of T. Immediately derivative, non-basic, non-atomic individuals of T are, on the one hand, the composite configurations of spatial positions, and, on the other hand, the composite configurations of temporal positions. I shall first consider configurations of spatial positions, simple or composite. 3.2.1

Spatial configurations

Since there are 100 spatial positions, there are (2100 – 1) configurations of spatial positions, or in short: spatial configurations (each non-empty subset of the set of spatial positions – and nothing else – specifies a spatial configuration, and a spatial position belongs to a spatial configuration if, and only if, it is an element of the specifying set for that configuration), of which 100 are simple and identical to the spatial positions themselves, and (2100 – 101) composite, namely, composed by (at least two) spatial positions. Some importance attaches to the distinction between spatial configurations without detached parts, and spatial configurations with detached parts. The distinction can be easily visualized, but its precise description

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requires some preparation. As a first step, the following definition states what is to be understood by the relation of immediate attachment among spatial positions: D7

Let P and P´ be spatial positions of Model T: P´ is immediately attached to P =Def [0] P´ = P, or [1] P´ is 1 step to the left of P and 0 steps up from P, or [2] P´ is 1 step to the left of P and 1 step up from P, or [3] P´ is 0 steps left of P and 1 step up from P, or [4] P´ is 1 step to the right of P and 1 step up from P, or [5] P´ is 1 step to the right of P and 0 steps up from P, or [6] P´ is 1 step to the right of P and 1 step down from P, or [7] P´ is 0 steps to the right of P and 1 step down from P, or [8] P´ is 1 step to the left of P and 1 step down from P.

According to this definition, the following 9 spatial positions – and no more spatial positions – are immediately attached to : first, [0] itself, and then: [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] . In terms of immediate attachment one can define attachment in general: D8

Let P and P´ be spatial positions of Model T: P´ is attached to P =Def P´ is immediately attached to P, or for some natural number N ≥ 1 the following is true: there are spatial positions P1, … [Pi, Pi+1] …, PN of T such that P´ is immediately attached to P1 and … [Pi is immediately attached to Pi+1] … and PN is immediately attached to P.

It is immediately evident on the basis of these definitions (by transferring them to the visualization C of T; see Sect. 2) that every spatial position of T is attached to every spatial position of T. Now, attachment can also be defined relative to spatial configurations (of T):

D9

Let P and P´ be spatial positions of Model T and S a spatial configuration of T: P´ is in S attached to P =Def P´ and P belong to S, and [P´ is immediately attached to P, or for some natural number N ≥ 1 the following is true: there are spatial positions P1, … [Pi, Pi+1] …, PN belonging to S

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such that P´ is immediately attached to P1 and … [Pi is immediately attached to Pi+1] … and PN is immediately attached to P]. Since the space of T is the spatial configuration to which all (and only) the spatial positions of T belong, it is easily seen (according to D8 and D9) that the spatial position P´ is (simpliciter) attached to the spatial position P if, and only if, P´ is in the space of T attached to P. A spatial configuration S is coherent – without detached parts – if, and only if, every spatial position belonging to S is in S attached to every spatial position belonging to S. (An obvious example of a coherent spatial configuration is the space of T itself, and, trivially, every spatial position is ipso facto a coherent spatial configuration.) A spatial configuration is noncoherent – with detached parts (at least one) – if, and only if, some spatial position belonging to S is not attached in S to some spatial position belonging to S. (An obvious example of a non-coherent spatial configuration is the pair of spatial positions and .) Let this much suffice regarding the geometry of Model T, which – though incomparably simpler than the geometry of Reality – is an interesting subject matter all by itself (and a subject matter that could – in principle – be treated in completeness, since T is spatially finite, both with regard to extension and to composition4). I waive the systematic treatment of distance, direction (but see Sect. 3.2.2.1 and, much later, Sect. 6.5.2 for some remarks) and relative position in the space of T, the treatment of the extensions and the forms of the spatial configurations of T, of their touching each other, of their including and intersecting each other, and of their being separate from each other; all of these matters can easily be supplied by the reader who is interested in them, to the extent he or she is interested in them (regarding the three last-mentioned matters – purely mereological matters – some indications will be given below). But even the reader who is not at all interested in the geometry of T (which I take to include T’s mereology of spatial configurations) should have an intuitive grasp of that geometry and should be able to answer the following question correctly and without much effort: Is it always the case that if S is a coherent spatial configuration which is not identical to the space of T, that then the spatial complement of S – that is: the spatial configuration which is specified by 4

An infinite line in a Euclidean space is infinite both with regard to extension and with regard to composition; a finite line in such a space is still infinite with regard to composition.

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the set of all spatial positions that do not belong to S – is also a coherent spatial configuration? Now, from the more narrowly metaphysical – specifically, ontological – point of view, it is to be noted that the spatial configurations of T are, for intrinsic reasons, everlasting continuants; that is, they are non-abstract individuals which are wholly present at every temporal position – whereas continuants simpliciter are non-abstract individuals which are located at more than one temporal position and wholly present at every temporal position at which they are located. Consider that every spatial configuration is a spatial part of the space of T, which space is itself a spatial configuration; hence, with the space of T being wholly present at every temporal position, every spatial configuration is wholly present at every temporal position (for if a spatial configuration is wholly present at a temporal position, then so are all of its spatial parts). As continuants, T’s spatial configurations are not only everlasting, but also (intrinsically) immutable regarding their spatial constitution: At any temporal position, any spatial configuration of T is found to be made up of the same spatial positions of T, in one and the same specific atemporal spatial relatedness. But the immutableness of T’s spatial configurations does not extend further than that. Regarding the being full or not of the spatial positions that belong to them, T’s spatial configurations are of course mutable. 3.2.1.1 Spatial parthood for spatial configurations I hasten to add the definition of spatial parthood for T’s spatial configurations: Let S´ and S be spatial configurations of T: D10 S´ is a spatial part of S =Def Every spatial position that belongs to S´ is a spatial position that belongs to S. On the basis of D10, the static spatial mereology of T is determined – the mereology of T’s spatial configurations –, which turns out to be an extensional, atomistic mereology with unrestricted fusion; this means that the following mereological principles hold true:

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Extensionality For all spatial configurations S and S´ of T: if S´ is a spatial part of S and S a spatial part of S´, then S´ = S. Atomicity 1 For every spatial configuration S of T: there is a spatial configuration S´ which is a spatial part of S, and S´ is spatially atomic [that is: S´ has no spatial part that is different from itself]. Atomicity 2 For all spatial configurations S´ and S of T: if every spatially atomic spatial configuration of T that is a spatial part of S´ is also a spatial part of S, then S´ is itself a spatial part of S. Unrestricted Fusion Let M be any non-empty set of spatial configurations of T: there is a spatial configuration of T of which all the elements of M are spatial parts and which is itself a spatial part of every spatial configuration of T of which all the elements of M are spatial parts. 3.2.2

Temporal configurations

Much of the essential information on temporal configurations of T is parallel to what has been said in the preceding two sections about T’s spatial configurations. Since there are 100 temporal positions, there are (2100 – 1) configurations of temporal positions, or in short: temporal configurations (each nonempty subset of the set of temporal positions – and nothing else – specifies a temporal configuration, and a temporal position belongs to a temporal configuration if, and only if, it is an element of the specifying set for that configuration), of which 100 are simple and identical to the temporal positions themselves, and (2100 – 101) composite, namely, composed by (at least two) temporal positions. Some importance – though perhaps less importance than in the case of spatial configurations – attaches to the distinction between temporal configurations without detached parts, and temporal configurations with detached parts. For obtaining a conceptualization (and not merely a mental pictorial representation) of this distinction, the required steps are similar to

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– but not quite the same as – the steps required in the case of spatial configurations. With respect to temporal positions, the definition of immediate attachment is much simpler than it is with respect to spatial positions (the timeline of T being a much simpler object than the space of T): Let Z and Z´ be temporal positions of Model T: D11 Z´ is immediately attached to Z =Def [0] Z´ = Z, or [1] Z´ is 1 step earlier than Z, or [2] Z´ is 1 step later than Z. The two follow-up definitions are (mutatis mutandis) the same as in the case of spatial configurations: Let Z and Z´ be temporal positions of Model T: D12 Z´ is attached to Z =Def Z´ is immediately attached to Z, or for some natural number N ≥ 1 the following is true: there are temporal positions Z1, … [Zi, Zi+1] …, ZN of T such that Z´ is immediately attached to Z1 and … [Zi is immediately attached to Zi+1] …and ZN is immediately attached to Z. Let Z and Z´ be temporal positions of Model T and V a temporal configuration of T: D13 Z´ is in V attached to Z =Def Z´ and Z belong to V, and [Z´ is immediately attached to Z, or for some natural number N ≥ 1 the following is true: there are temporal positions Z1, … [Zi, Zi+1] …, ZN belonging to V such that Z´ is immediately attached to Z1 and … [Zi is immediately attached to Zi+1] … and ZN is immediately attached to Z]. As in the case of spatial configurations, attachment simpliciter (defined by D12) is easily recognized as a special case of attachment in a configuration (defined by D13) – the timeline of T being the temporal configuration of T to which all the temporal positions of T belong. Now, a temporal configuration V is coherent – without detached parts – if, and only if, every temporal position belonging to V is in V attached to every temporal position belonging to V. Accordingly, the temporal configuration specified by, say, {55, 56, 57, 58} is coherent – without detached parts –, whereas the temporal configuration specified by, say, {55, 58} is non-coherent – with detached parts. It is easily seen that every (gapless) subinterval of the timeline of T and the timeline of T itself and all the

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temporal positions of T are coherent temporal configurations of T; all the remaining temporal configurations of T are non-coherent. Temporal configurations are individuals, and it is not entirely implausible to assume that they are non-abstract individuals. But they are not continuants: it is not true for any temporal configuration of T that it is located at more than one temporal position and is wholly present at every temporal position at which it is located. Simple temporal configurations of T are wholly present at each temporal position at which they are located – because they are each located at only one temporal position and are, in fact, identical with that position; but they are, therefore, not located at more than one temporal position, and hence they are not continuants. Composite temporal configurations, on the other hand, are located at more than one temporal position – they are located at each of the several temporal positions that belong to them; but obviously they are not wholly present at any temporal position at which they are located, and hence they are not continuants. Thus: whereas every spatial configuration of T is a continuant, no temporal configuration of T is a continuant. It also seems – prima facie – to be true that no spatial configuration of T has any temporal parts, whereas every temporal configuration of T has at least one temporal part (though in some cases only itself). The concept of temporal parthood for temporal configurations is defined in the following way (and compare D10): Let V´ and V be temporal configurations of T: D14 V´ is a temporal part of V =Def Every temporal position that belongs to V´ is a temporal position that belongs to V. On the basis of this definition, it is clear enough that every temporal configuration of T has at least one temporal part. On the other hand, D14 leaves us entirely in the dark as to what it would even mean for a spatial configuration of T to have or not to have a temporal part. Note that the definition is conditional to temporal configurations; disregarding this conditionality and applying the definition also to spatial configurations would lead to the strange result that every spatial configuration is a temporal part of every spatial configuration (since no spatial configuration has any temporal positions belonging to it, making the statement “every temporal position that belongs to S´ is a temporal position that belongs to S” trivially true for all spatial configurations S and S´). In view of this, it is perhaps more plausible to hold that one does not even know what the expression

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“temporal part” would mean if applied to spatial configurations than to hold that spatial configurations have no temporal parts. (By the same token, it is perhaps more plausible to hold that one does not even know what the expression “spatial part” would mean if applied to temporal configurations than to hold that temporal configurations have no spatial parts.) In any case, the mereology of the temporal configurations of T is determined by the concept of temporal parthood, presented by D14 above. If one pictures this temporal mereology (the temporal configurations of T being embedded in each other, intersecting each other, being – mereologically – separate from each other), then this mereology may appear to be rather uninteresting as compared to T’s mereology of spatial configurations. But this appearance is illusory. It is due to a factor that is extramereological: the timeline of T is 1-dimensional, whereas the space of T is 2-dimensional. In spite of the reduced complexity that is brought about by the 1-dimensionality of the timeline, T’s purely mereological principles for temporal configurations are just as interesting as T’s purely mereological principles for spatial configurations. Both the mereology of spatial configurations and the mereology of temporal configurations are extensional, atomistic mereologies with unrestricted fusion. The mereological principles found to be true in the previous section for spatial configurations are, mutatis mutandis, also true for temporal configurations. (And note that there are exactly as many temporal configurations of T as there spatial ones: 2100 − 1. Question: Is there a mapping ϕ of the spatial configurations onto the temporal configurations which is such that S´ is a spatial part of S iff ϕ(S´) is a temporal part of ϕ(S), and such that ϕ−1(V´) is a spatial part of ϕ−1(V) iff V´ is a temporal part of V – for all spatial configurations S and S´, and all temporal configurations V and V´, ϕ−1 being the inverse of ϕ?) 3.2.2.1 Direction and relative position for temporal configurations Direction and relative position are extra-mereological matters. Regarding spatial configurations, I shall not much go into them; but regarding temporal configurations they are of sufficient metaphysical importance and, at the same time, of sufficient conceptual simplicity to be accorded a brief, systematically rigorous treatment. That treatment is followed by epistemologico-metaphysical considerations regarding temporal direction (in comparison to spatial direction); structurally similar considerations – with the same basic inside-outside (or immanence-transcendence) structure, but

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with different contents – will recur in several places of this book. – But first, the definitions that befit the present section of the book: Let Z´ and Z be temporal positions of T: D15 Z´ is [properly] earlier than Z =Def Z´ is 1 step earlier than Z, or for some natural number N ≥ 1 the following is true: there are temporal positions Z1, … [Zi, Zi+1] … , ZN of T such that Z´ is 1 step earlier than ZN and … [Zi+1 is 1 step earlier than Zi] … and Z1 is 1 step earlier than Z. D16 Z´ is [properly] later than Z =Def Z´ is 1 step later than Z, or for some natural number N ≥ 1 the following is true: there are temporal positions Z1, … [Zi, Zi+1] …, ZN of T such that Z´ is 1 step later than ZN and … [Zi+1 is 1 step later than Zi] … and Z1 is 1 step later than Z. Let V´ and V be temporal configurations of T: D17 V´ is [at least] partly earlier/later than V =Def Some temporal position belonging to V´ is earlier/later than every temporal position belonging to V. D18 V´ is entirely earlier/later than V =Def Every temporal position belonging to V´ is earlier/later than every temporal position belonging to V. I refrain from offering further definitions, noting that the purely positional concepts presented by D17 and D18 for temporal configurations are intrinsically related to certain purely mereological concepts for such configurations: For example, temporal configurations V´ and V are separate [have no temporal configuration as a common temporal part] if, and only if, V´ is entirely earlier or entirely later than V; and V´ is a temporal part of V if, and only if, V´ is neither partly earlier nor partly later than V. In view of the above-defined positional concepts, which (just like the concepts they are based on5) imply a certain temporal directedness, it is here the proper place to compare the notion of direction as it applies to the timeline of T with the notion of direction as it applies to the space of T. There are two directions intrinsic to the timeline of T: one can call them future and past (instead of saying that Z´ is N steps earlier than Z, one can also say – and mean the same thing – that Z´ is N steps in the past of Z, and instead of saying that Z´ is N steps later than Z, one can also say that 5

For the definitions of these latter concepts, see D5 and D6.

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Z´ is N steps in the future of Z; cf. D5, D6). In contrast, there are four main directions intrinsic to the space of T: we have called them left, right, down, and up (cf. D1, D2, D3, D4). The six linguistic labels used for designating the directions just mentioned show by their very meanings that they refer to temporal, respectively spatial directions (and there is, for T, an objective distinction between these two kinds of direction), and they show by their very meanings that some of these directions are opposed to each other: the temporal direction designated by “future” and the temporal direction designated by “past” are opposed to each other, and so are the spatial direction designated by “left” and the spatial direction designated by “right”, as well as the spatial direction designated by “down” and the spatial direction designated by “up”. But is their any objective reason intrinsic to T why the six labels used for designating the mentioned six directions should be considered fitting designations for these directions to a further extent than has just been indicated? – The answer is “No”. As far as the intrinsic constitution of T is concerned (such is T intrinsically), the temporal direction labelled “past”, for example, might as well be renamed “future”, as long as the temporal direction opposed to it is renamed correspondingly: “past” instead of “future”. There is nothing inside of T that militates against this renaming. In other words: T in itself – T from the inside – gives no indication as to which of the two directions intrinsic to its timeline is past and which is future. At first sight, this may not seem true, since, in view of D5 and D6, one can hold that past is the direction that goes from a natural number (standing for a temporal position) to a smaller natural number (standing for a properly earlier temporal position), while future is the direction that goes from a natural number (standing for a temporal position) to a greater natural number (standing for a properly later temporal position). One can hold that, once temporal positions are represented by natural numbers, pastness and being-earlier-than are intrinsically connected with decreasing numerical size, and futurity and beinglater-than with increasing numerical size. Granted. But this does not help at all with the problem we are concerned with; for the fact remains that T in itself – T from the inside – gives no indication why this temporal position at this particular end of the timeline of T should be [represented by] 1, and not rather [be represented by] 100, and consequently why the next temporal position should be 2, and not rather 99, and so on, till one reaches the temporal position at the other end of the timeline, where T in itself, T from the inside, gives no indication why that temporal position at that other end of the timeline should be 100, and not rather 1. And if the representa-

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tion of temporal positions by natural numbers from 1 to 100 had been the reverse of the actual representation, then T in itself – T from the inside – would have given no indication why it should be that way, and not the other way (i.e., the way it happens to be in fact). In view of this situation there are only two possible reactions: One can hold, (i), that there is no objective justification (at all) for distinguishing past and future in the particular manner they are in fact distinguished in T [which manner is determined by representing the temporal position at this particular end of the timeline of T by the number 1, and not the temporal position at the opposite end], and that there is no objective justification for distinguishing past and future in the way which is the unique alternative to the first-mentioned way (namely, the reverse of it); that, therefore, any manner of distinguishing past and future in T is neither objectively correct nor objectively incorrect, but objectively arbitrary. (Yet, note, there are two temporal directions of T, opposed to each other, and they must be distinguished if T is to be correctly described; there is, according to (i), just no objectively determined unique way to distinguish them as past and future: from the purely objective point of view, one might just as well call them “td1” and “reverse-td1”.) One also can hold, (ii), that there is an objective justification for distinguishing past and future in the particular manner they are in fact distinguished in T; but since that justification is not intrinsic to T (as we have seen), it must come from the outside of T. There is also no objective justification intrinsic to T for distinguishing its four main spatial directions in the manner they happen to be in fact distinguished. Though left and right can be intrinsically tied to the comparison in size between first coordinates, and down and up to the comparison in size between second coordinates [such that, for example, is – properly or vacuously – to the left of if, and only if, X ≤ X´, and – properly or vacuously – up from if, and only if, Y ≥ Y´], the fact remains that there is not only one way (i.e., not only the way in fact employed) to capture the 100 spatial positions of T in the matrix of the ordered pairs of natural numbers from 1 to 10; the four different ways of put-

30

ting the grid of pairs of coordinates over the 100 spatial positions of T are all equally justifiable from the intrinsic point of view, but they lead to quite different assignments of directions within the space of T. Thus, regarding the four main spatial directions, left and right, up and down, the options we are left with are quite similar to (i) and (ii) above: One can hold, (i´), that there is no objective justification (at all) for distinguishing left and right, up and down, in the particular manner they are in fact distinguished in T, and that there is no objective justification for distinguishing these directions in ways which are alternatives to the first-mentioned way; that, therefore, any manner of distinguishing left and right, up and down in T is neither objectively correct nor objectively incorrect, but objectively arbitrary. (Yet, note, there are four main spatial directions of T, in two pairs of opposition, and they must be distinguished if T is to be correctly described; there is, according to (i´), just no objectively determined way to distinguish them as left and right, up and down: from the purely objective point of view, one might just as well call them “sd1” and “reverse-sd1”, “sd2” and “reverse-sd2”, noting that the sd2-directions are orthogonal to the sd1directions.) One also can hold, (ii´), that there is an objective justification for distinguishing left and right, up and down, in the particular manner they are in fact distinguished in T, but since that justification is not intrinsic to T, it must come from the outside of T. The options (i) and (ii), (i´) and (ii´) – taken together – suggest four ways of further developing Model T (building on its basic description), depending on which option one decides to select and adopt in each of the mentioned two pairs of options [(i) and (ii), (i´) and (ii´)]. Three of the four selection-pairs [that is, (i) together with (ii´), (ii) together with (i´), (ii) together with (ii´)] imply that there is an outside to T, one of them [i.e., (i) together (i´)] does not. How should we proceed in further developing T? Since T is to be a metaphysical model of Reality, having a metaphysical affinity to it, one should – always and with regard to the question at hand – see to it that a metaphysical fit between T and Reality is maintained. This fit seems to be served best, as a matter of fact, by further de-

31

veloping T along the lines of the selection-pair (ii) together with (i´). From the point of view of Reality, the justification for this is as follows: Real Space from the inside does not (intrinsically) accord an objective sense and stable reference to left and right, up and down (and, since Real Space is three-dimensional, also not to forward and backward), but merely a subjective sense and changeable reference (depending on the position of the observer), and there is no indication whatsoever as to how this could be otherwise from the outside of Real Space. Since Model T is meant to metaphysically represent Reality, this is motivation enough for adopting (i´). In turn, Real Time from the inside does not (intrinsically) accord an objective sense and stable reference to past and future. Some may be inclined to deny this: Is not the direction of entropic increase intrinsically the objective direction of time, thus making it possible to distinguish intrinsically and objectively between past and future? But though the direction of entropic increase is in fact the objective direction of time, the fact of this is certainly not intrinsic to Real Time (otherwise it would be inconceivable that entropy decreases “as time goes by”, which, however, is not inconceivable at all). Nevertheless, the distinction between past and future, such as we do in fact make it, is, though not intrinsic to Real Time, objective (we certainly cannot arbitrarily reverse it; where, then, does it come from?). Since Model T is meant to metaphysically represent Reality, this is motivation enough for adopting (ii). But (ii) together with (i´) logically implies that there is an outside to T (specifically, this is logically implied by (ii)). Based on the above considerations, the relation of representation between T and Reality – the metaphysical analogy between them – suggests that there is also an outside to Reality (as we know it). How an outside to T can provide objective justification for distinguishing past and future in the specific way (whatever it is exactly) in which they are distinguished in T – thus giving the timeline of T an objectively determined temporal direction – will be fleshed out later on (see Sect. 5.2.1), in expositions which will also throw light on how an outside to Reality can provide objective justification for distinguishing past and future in the specific way they are distinguished in Reality. It should be emphasized that the selection-pair (ii) together with (i´), which represents a fundamental metaphysical stance on time and space (concerning T directly, but indirectly also Reality), is, though reasonably adopted, still freely adopted. We are not rationally forced to adopt that particular selection-pair. The following alternative selection-pair: (i) together with (i´), representing a quite different fundamental metaphysical stance on

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time and space, is defensible, though, in my view, less so than (ii) together with (i´). (The other two selection-pairs are of considerable less interest than the two selection-pairs just mentioned; this marked asymmetry is a reflection of the real-life metaphysical situation.) 3.3

States of affairs

Model T is a rich source of states of affairs, not all of them of the same kind. Time-dependence and time-thematicness seem promising characteristics for categorizing the T-states-of-affairs in an illuminating way. We have for T: (I) time-dependent and time-thematic states of affairs; (II) time-dependent but not time-thematic states of affairs; (III) not time-dependent, but time-thematic states of affairs; (IV) not time-dependent and not time-thematic states of affairs. It seems best to start with category (IV). Examples of T-states-of-affairs that are neither time-dependent nor time-thematic are the following statesof-affairs: that is 1 step to the left of , that is up from , that there are 100 spatial positions, that Full is the filling of spatial positions, etc. These states of affairs do not thematize temporal positions (make them thematic), neither directly nor indirectly, and their obtaining or non-obtaining is not relative to temporal positions. (Note that three of the adduced states of affairs obtain, and that one does not obtain.) Turning now to category (III), I note that the following states of affairs are examples of T-states-of-affairs that are not time-dependent, but timethematic: that 3 is earlier than 7, that is full at 8, that no spatial position is full at any temporal position, etc. These states of affairs thematize temporal positions (directly or indirectly), but their obtaining or nonobtaining is not relative to temporal positions. But an important distinction must be made among the members of category (III): there are members of this category the obtaining or non-obtaining of which is not only not relative to temporal positions, but also not relative to anything else, as the – obtaining – state of affairs that 3 is earlier than 7, or the – non-obtaining – state of affairs that 5 is later than 6; such members of category (III) are like the members of category (IV). And there are members of category (III) the obtaining or non-obtaining of which, though not relative to temporal posi-

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tions, is relative to something else, like the state of affairs that is full at 8, or the state of affairs that no spatial position is full at any temporal position; that the obtaining or non-obtaining of these states of affairs, though not relative to temporal positions, must still be relative to something (depend on something, be conditional to something) is apparent from the fact that merely by considering the states of affairs in themselves it is impossible to determine whether they obtain (time-independently) or not. Leaving category (III) to turn to category (II), I note that this latter category deserves a subsection all by itself. 3.3.1

Time-dependent but not time-thematic states of affairs

Examples of this kind of states of affairs are the states of affairs that is full, that is not full, that is full and empty and full. A whole group of time-dependent but not time-thematic Tstates-of-affairs can be specified as follows: Any state of affairs with a “that”-designation that has one of the following two forms: that P is full

that P is empty,

is a time-dependent but not time-thematic state of affairs. Here “P” is replaceable by any standard name of a spatial position of T: “”, “”, …, “”, “”, and only by such a standard name; “P is empty” is short for, and therefore synonymous to, “P is not full” (this equivalence is justified in view of Full being the only filling of T’s spatial positions). “P is full”, in turn, is short for “P exemplifies Full”; the first statement merely abbreviates what the second one states in full. In consequence, “P is empty” is short for (merely abbreviates) “P does not exemplify Full”. The states of affairs which have a “that”-designation of either of the two above-specified forms are the atomic momentary states of T.6 The atomicity intended in this expression – “atomic momentary states” – must not be confused with atomicity of designation (in that sense, atomic mo6

Here “of T” must not be taken in the sense of “had by T” or “possibly had by T”, but merely in the sense of “belonging to the inventory of T”.

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mentary states would obviously not be atomic, but composite). But even if “atomic” is understood as intended: as designating a certain type of purely ontic atomicity, it must be noted that this intended atomicity does not exclude all sorts of ontic composition. For, indeed, the state of affairs that P is full, and the state of affairs that P is empty can be designated in such a way as to make quite explicit in what sense they are ontically composite, despite of their being – in the sense intended – ontically atomic: [Full, P]

neg([Full, P]).

Thus, for example, the T-state-of affairs [Full, ] – i.e., that is full, that exemplifies Full – and the T-state-of-affairs neg([Full, ]) – i.e., neg(that is full), that is not full, that does not exemplify Full, that is empty – are two atomic momentary states that, notwithstanding their atomicity in the intended sense, are put together in different ways on the basis of the two components Full and ; responsible for this difference in composition is not the elementary composition-operation (indicated by the square brackets),7 but the negation-operation: neg. But, then, in what (intended) sense are those two states of affairs, and the other 198 just like them, ontically atomic? – They are ontically atomic in a purely relative sense, namely, as being the ultimate constituents of a certain manner of composition, which manner is described in the next paragraph. Any consistent conjunction of at least two atomic momentary states of T is a composite momentary state of T, and any composite momentary state of T is a consistent conjunction of at least two atomic momentary states of T. That is, the composite momentary states of T are precisely those states of affairs that have a “that”-designation of the following form: that P1 is full/empty and P2 full/empty and … and PN full/empty, where (i) either “full” or “empty” is used in each conjunct (this is what is indicated by the recurring “full/empty”), and where (ii) N ≥ 2, and where 7

For the elementary composition-operation [X1, …, XN] (N ≥ 2) – in this book, only its dyadic specialization [X, Y] is used – the order of the composed elements matters, since the result of composition may already vary if merely the order of the composed elements is varied; on the other hand, [X1, …, XN] is not simply the ordered sequence , since it is also true that the result of composition may be the same even if the order of the composed elements is varied.

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(iii) “P1”, …, “PN” must each be replaced in such a manner by a standard name of a spatial position of T that no two names occurring in the (entire) replacement designate the same spatial position. (Conditions (i) and (iii) together guarantee the consistency of the conjunction, and conditions (ii) and (iii) together guarantee its genuine compositeness.) Thus, the T-state-of-affairs which was already used as an example above, that is full and empty and full, is a composite momentary state of T. Composite momentary states of T can be more or less comprehensive, depending on the number of atomic momentary states that go into them (as – ultimate – conjuncts). This number – which is equal to the number of spatial positions that a composite momentary state is about – ranges from minimally 2 to maximally 100. Accordingly, those composite momentary states of T that consist of 100 atomic momentary states are called maximally composite momentary states of T. In Sect. 2, the (possible) fillings of Model T’s (individual) temporal positions were characterized as the 2100 fully specific ways in which T’s unique filling of spatial positions – Full – can be simultaneously distributed over all of the 100 spatial positions of T. Those 2100 ways of distributing Full are, as a matter of fact, the maximally composite momentary states of T; for given the conception of identity among momentary states that is adopted in this book (see below in this section), there is no reason not to identify the fillings of T’s temporal positions with the maximally composite momentary states of T. Atomic and composite momentary states – and also Full – may at first sight seem to be continuants. For are they not wholly present at different temporal positions? But, in fact, they fail to be continuants. A continuant simpliciter – according to definition – is a non-abstract individual which is located at more than one temporal position and is wholly present at every temporal position at which it is located (see Sect. 3.2.1). But atomic and composite momentary states (and, in fact, all momentary states) are not individuals – because they are states of affairs (time-dependent but not time-thematic ones); nor is Full an individual – because it is a universal. Their not being individuals effectively blocks their being continuants (whether or not they are non-abstract entities; it seems to me, as a matter of fact, that they are just as non-abstract as spatial configurations are nonabstract). But there is a deeper reason for their not being continuants simpliciter than their failing to be individuals, a reason that touches the core of the concept of continuant: it is impossible to say simpliciter – without (explicit or implicit) relativization – of atomic and composite momentary

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states that they are located at more than one temporal position; the same is true of the universal Full. After all, for each atomic and composite momentary state of T, there are many complete histories of T in which it does not occur at more than one temporal position; the same is true of Full. With appropriate explicit or implicit relativization (there are, in fact, four canonical ways to achieve appropriate relativization, two explicit and two implicit; see Sect. 5.1.1), it is of course quite possible to assert truly that an atomic or composite momentary state, or Full, is located at more than one temporal position. (Note the difference here between Full on the one hand, and atomic and composite momentary states on the other: For the latter entities, temporal locatedness involves no spatial locatedness in a normal sense; for the former entity, temporal locatedness does involve spatial locatedness in a normal, though derivative sense: for intrinsic reasons, Full is located – or occurs – at a given temporal position if, and only if, it is located at that temporal position at some spatial position, or in other words: if, and only if, some spatial position exemplifies Full at that temporal position.) It is quite obvious in what sense atomic and composite momentary states are time-dependent: the obtaining or non-obtaining of these states of affairs is relative to temporal positions (and relative also to something else, in fact, to the very factors of relativization merely gestured at – for the time being – in the previous paragraph). But are those states of affairs really not time-thematic? Why, after all, are they called “momentary states”? Is it not true that is full, for example, is the same state of affairs as the state of affairs that is now full, this latter state of affairs being obviously time-thematic? The answer to this query is this: That is full is identical to [Full, ], and that is now [in another word: presently] full is identical to now([Full, ]). Time is certainly not involved in [Full, ]. It is true that “now” can stand for an operation on states of affairs which is such that now([Full, ]) is the same state of affairs as [Full, ]; this fact explains the applicability of “momentary” to the state of affairs that is full; but if “now” stands for that operation, then now([Full, ]) is just as non-time-thematic as [Full, ] is. It is also true that “now” can stand for another operation on states affairs, which is such that now([Full, ]) is a time-thematic state of affairs; but if “now” stands for this other operation, then now([Full, ]) is not the same state of affairs as [Full, ], since this latter state of affairs is not timethematic.

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It is an interesting question whether the atomic and the composite momentary states are all the momentary states (of T) there are. At first sight, one is inclined to answer “yes”, since “atomic or composite” seems like a complete disjunction. But it is not implausible to admit other momentary states besides atomic and composite momentary states (in the defined senses): The operation of negation, as we have seen, is present on the level of atomic momentary states, since some atomic momentary states are the negations of other atomic momentary states (for example, the atomic momentary state that is empty is the negation of the atomic momentary state that is full, and vice versa). But it would be arbitrary to restrict the operation of negation to atomic momentary states. The operation of negation should also be applicable to composite momentary states – and consequently we have to face momentary states that are neither composite nor atomic, but disjunctive, since the negation of a composite momentary state is a disjunctive momentary state. For example, neg(that is full and empty) – in other words: that not ( is full and empty), that is: that is not full or not empty – is the disjunctive momentary state that is empty or full (where “or” is to be understood in the non-exclusive sense). Waxing even more audacious than that, one could hold that any state of affairs is a momentary state of T that can be built – purely on the basis of atomic momentary states of T – by a finite combination (of uses) of no other operations than, at most, conjunction (conj) and negation (neg). In this vein, for example, neg(conj(that is empty, neg(conj(that is full, that is empty)))) would turn out to be a momentary state, which in the briefest (nonambiguous) way possible can also be designated as follows: that is full or ( full and empty). The above – liberal – position regarding the extension of the set of momentary states of T can be shown to be equivalent to holding that each subset of the set of all (2100) fillings of T’s temporal positions stands for at least one momentary state of T (namely, as the set of all fillings of temporal positions by which that momentary state is intrinsically implied). This,

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obviously, goes very far beyond letting the momentary states be exhausted by the atomic and the composite momentary states. But there is still a decision to be made in this liberal line of thinking: whether to let the number of momentary states go beyond the already required minimal level, or to fix it at that level. For a subset of the set of all fillings of temporal positions may either stand for exactly one momentary state, or for several such states; in the latter case it would indiscriminately represent all of the several momentary states it stands for. Here is an illustration of what is at stake in the decision yet to be made: The set of all fillings of temporal positions stands for the momentary state that is empty or full, since this momentary state is intrinsically implied by all fillings of temporal positions. But the set of all fillings of temporal positions also stands for the momentary state that is empty or full, since this momentary state, too, is intrinsically implied by all fillings of temporal positions. And, of course, one can go on indefinitely with naming momentary states – all of them generated purely on the basis of atomic momentary states by a finite combination of merely the operations of negation and conjunction (and then designated compendiously) – that are each intrinsically implied by all fillings of temporal positions. Now, if the set of all fillings of temporal positions stands for exactly one momentary state, then all those momentary states are really (the grammatical plural is not to be taken seriously) just one momentary state: a single momentary state that can be named in an infinite number of (logically equivalent) ways. If, however, the set of all fillings of temporal positions stands for several momentary states, then Pandora’s Box is open and there seems to be no end to momentary states that correspond to the set of all fillings of temporal positions: setting a limit seems arbitrary – no matter at which point one might decide to set one (which fact, however, does not preclude identifying with each other – by local action, so to speak – at least some of those many momentary states that correspond to the set of all fillings of temporal positions). Just as one is wont to do with respect to Reality, one might here demand also with respect to Model T: entia non sunt multiplicanda praeter necessitatem. – Yes, but what is necessary (i.e., indispensable) or not for what necessary (i.e., unrelinquishable) or not theoretical purposes? That is the question, and there is much room for debate here – even with respect to Model T, which, compared to Reality, is a “tiny little something”. And even if we found a clear-cut answer to the question just posed, an answer we all agreed on, would then Ockham’s Principle really help us (as is

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widely believed) to answer the question of what there is? If, for example, momentary states in a certain conception and number are not indispensable for any unrelinquishable theoretical purposes concerning T, does this really mean, as Ockham’s Principle suggests, that there just aren’t momentary Tstates in that particular conception and number? To my mind, it is not clear what the correct answer to this last question is. How can it be that (our) lack of theoretical need prescribes a limit to what there is? It must, nevertheless, be noted that the willingness to accept momentary T-states in toto (that is, without selecting) certainly decreases in the following sequence of ever more “populated” conceptions of momentary T-states (and this fact, can it be merely a psychological curiosity?): [1] Every momentary state is (qua momentary state) an atomic momentary state. [This leads to the result that there are just 200 momentary states.] [2] Every momentary state is (qua momentary state) an atomic or a composite momentary state. [3] Every momentary state is (qua momentary state) an atomic, a composite, or a disjunctive momentary state (that is, a negation of a composite momentary state). [4a] Every momentary state is (qua momentary state) an atomic momentary state or a construction out of atomic momentary states achieved by applying, at most, conjunction and negation – with the proviso that momentary states that are intrinsically implied by the same fillings of temporal positions are identical. Equivalently: Momentary states can (qua momentary states) be mapped one-to-one onto the sets of fillings of temporal positions. [This leads to the result that there are 2 to the power of 2100 momentary states, because there are 2100 fillings of temporal positions, and therefore 2 to the power of 2100 sets of fillings of temporal positions.] [4b] Every momentary state is (qua momentary state) an atomic momentary state or a construction out of atomic momentary states achieved by applying, at most, conjunction and negation – with the proviso that momentary states that are constructed in different ways out of atomic momentary states are not identical, even if they are intrinsically implied by the same fillings of temporal positions. [This leads to the result that there are infinitely many momentary states.]

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The different provisos of identity that have been added to [4] (implicit above), differentiating it into [4a] and [4b], can also be added to [2] and [3] – and they make a big difference even there. According to the first proviso (the one stated in [4a]), a composite momentary state remains the same composite momentary state no matter how the conjuncts in its “that”designation are permuted; in other words, according to the first proviso, the identity of a composite momentary state does not depend on exactly how that state is constructed by conjunction out of the atomic momentary states that go into it. But according to the second proviso (the one stated in [4b]), there are, on the contrary, just as many composite momentary states as there are non-repetitive ways of construction by conjunction on the basis of the given number of atomic momentary states. (If that number were 2, there would be 2 such ways; if that number were 3, there would already be 12 such ways, hence 12 composite momentary states – whereas according to the first proviso there would be just 4.) Since a decision has to be made, I decide to conceive of momentary states according to conception [4a], accepting therewith also that momentary states that are intrinsically implied by the same fillings of temporal positions are identical. This is a reasonable way to dispose of this matter, though not one that is necessitated by reason. None of the decisions that could be made regarding the conception and number of momentary states is necessitated by reason (let alone by experience) – which is a situation that we shall frequently encounter in ontology and metaphysics, no matter whether we are referring to T or to Reality. 3.3.1.1 Additional perspectives on not time-dependent, but time-thematic states of affairs The position gained at the end of the previous section regarding timedependent but not time-thematic T-states-of-affairs – in other words: regarding the momentary states of T (since there do not seem to be any other time-dependent but not time-thematic T-states-of-affairs than the momentary states of T) – provides additional perspectives on the quite different category of the not time-dependent, but time-thematic T-states-of-affairs. Just as – according to conception [4a] – the momentary T-states correspond one-to-one to (all) the sets of fillings of T’s temporal positions (which fillings – we found in the previous section – are the maximally composite momentary T-states), so, it seems, the not time-dependent, but

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time-thematic T-states-of affairs correspond one-to-one to (all) the sets of complete histories of T (concerning the concept of a complete history, see Sect. 2). If this were correct, then the number of not time-dependent, but time-thematic T-states-of-affairs would turn out to be 2 to the power of 210000, a truly staggering (but finite) number, since the cardinality of the set of all complete histories of T is 210000 (see Sect. 2), which means (according to set-theory) that there are 2 to the power of 210000 subsets of that set, in other words: 2 to the power of 210000 sets of complete histories of T. But, it turns out, the one-to-one correspondence in question – established on the basis of the relational concept being-intrinsically-implied-by – holds only between some not time-dependent, but time-thematic T-states-of-affairs on the one hand, and the sets of complete T-histories on the other. If one ignored this and applied the correspondence in question as a one-to-one correspondence to all not time-dependent, but time-thematic T-states-ofaffairs, then one would be forced to identify with each other the evidently non-identical not time-dependent, but time-thematic states of affairs that 2 is earlier than 3, and that 4 is earlier than 5: Since both these states of affairs are intrinsically implied by every complete history of T, they have the same set of complete histories corresponding to them: the set of all complete histories of T, and therefore, if the correspondence in question were indiscriminately applied as a one-to-one correspondence also to these states of affairs, they would have to be the same state of affairs – which is simply absurd. However, this negative result has a positive follow-up. In Sect. 3.4.1, it will be seen what is the uniform nature and the importance of those not time-dependent, but time-thematic T-states-of-affairs which are such that the correspondence between them and certain sets of complete T-histories is one-to-one. 3.3.2

Time-dependent and time-thematic states of affairs – and the nature of indexical time-thematicness

One example of a time-dependent and time-thematic state of affairs was already mentioned in Sect. 3.3.1 above: that is now full – where “now” functions in such a way as to make the state of affairs that is now full (= now([Full, ])) a state of affairs that is different from the state of affairs that is full (= [Full, ]). This – the preceding, explanatory – clause needed to be added in view of the fact that there is

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also a use of “now” (and of its synonym “presently”) which is no manner indicative of temporal reference, which is, therefore, not constitutive of time-thematicness, in which use the state of affairs that is now full is just the same state of affairs as that is full (see Sect. 3.3.1); in this other use, “now” (and “presently”) can be synonymously replaced by “momentarily”. In what follows, I shall stick to the first-mentioned, more common use, in which “now” (and “presently”) have real modificatory power. Other ways of introducing time-thematicness while maintaining timedependence become apparent by applying the operations of future-tense and of past-tense – which are unary operations, like the operations of present-tense and of negation – to momentary states: that will be full, in other words: fut([Full, ]), is also an example of a time-dependent and time-thematic state of affairs, and so is that was full, in other words: pas([Full, ]). And note: while that is full at 2 is a time-thematic, but not time-dependent state of affairs, that is now, at 2, full is a time-thematic and time-dependent state of affairs (the obtaining or non-obtaining of the latter – but not of the former – state of affairs is relative to temporal positions). Instead of further exploring the huge realm of time-dependent and time-thematic T-states-of-affairs, I shall now look into the nature of indexical time-thematicness regarding Model T. The guiding question here is this: Does the basic description of T (see Sect. 2) already provide the means to account for every kind of time-thematicness regarding T? The item of T which is relevant here is the timeline of T. It consists in the sequence of natural numbers from 1 to 100, which sequence has been provided with a temporal interpretation: (a) those numbers stand for the temporal positions of T; (b) the order of the numbers – the (so-called) natural order: one after the other according to size, always from the smaller one to the greater one – stands for the order of T’s temporal positions in the direction of time: from earlier ones to later ones; (c) the absolute quantities of the differences between the numbers are the quantities of the (temporal) distances between T’s temporal positions.

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It has been determined in Sect. 3.2.2.1, on the one hand, that there is an objective justification for distinguishing past (or earlier) and future (or later) in the particular manner they are in fact distinguished in T (i.e., for putting the sequence of natural numbers from 1 to 100 in that particular manner over the temporal positions of T, and not in the inverse way), and, on the other hand, that this objective justification must come from the outside of T, since it is not intrinsic to T (such is T as a model of Reality). But no matter what may be the foundations for it, already the basic description of T incorporates – for any temporal position – the distinction of present, past and future, and this can be taken to be entirely sufficient for accounting for the time-thematicness in the state of affairs that is now full, the state of affairs that was full, and the state of affairs that will be full. In each of the following three columns, each state of affairs is identical to every other state of affairs in the same column: [a] that is now full at-z*([Full, ]).

[b] that was full pas([Full, ]) at-some-temporal-position-in-the-past-relative-to-z*([Full, ]) at-some-temporal-position-earlier-than-z*([Full, ]).

[c] that will be full fut([Full, ]) at-some-temporal-position-in-the-future-relative-to-z*([Full, ]) at-some-temporal-position-later-than-z*([Full, ]).

In each of these three columns, each state of affairs following the first one in the column is generated by applying one operation, specific to the column, to the atomic momentary state [Full, ]: in the first column, it is the present-tense-operation; in the second, it is the past-tense-operation, and in the third, the future-tense-operation. The last item in each column is named in such a way as to make fully explicit what the time-thematicness of the states of affairs generated in the column consists in. Keeping in mind the definitions D15 and D16 in Sect. 3.2.2.1, it is possible to hold that no new element has been introduced by the columns [a], [b], [c], that is: no element that is not already present (at least implicitly) in the basic descrip-

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tion of T. It does seem to be the case that the indexical designator “z*” – “the present moment” – points precisely to such a new element, though the newness of it would not be constituted by an enlargement of the reservoir of T’s temporal positions, but by some already given temporal position of T possessing a special quality – presentness – singly and fleetingly. But, as a matter of fact, “z*” need not be understood in such a way that it ever stands for a temporal position of T that is qualitatively special. Rather, “z*” may be taken to designate some particular temporal position or other, depending on the occasion (we can stipulate that “z*” never designates anything else than a temporal position), with all the temporal positions of T having the same right to be designated by “z*”, that is: there being an occasion to be designated by “z*” for every temporal position Z (namely, an occasion constituted by focussing on Z), no such occasion having any objective prerogative over any other such occasion. Understood in this way, “z*” never stands for a temporal position that is qualitatively special. And there is no violation of reason involved in understanding it that way. Considered in itself, intrinsically, Model T provides no objective justification for assuming that any of its temporal positions ever deserves the designation “the present moment” (“z*”) more than any of its other temporal positions: regarding this matter, nothing can be gathered from T’s basic description. Thus, we are confronted with a choice that has the very same form as the previous choice (see Sect. 3.2.2.1) regarding the distinction between past and future, or in other words: regarding the determination of the direction of time: We might maintain that there is no objective justification at all for assuming that any of the temporal positions of T ever deserves the designation “the present moment” more than any other temporal position of T, or we might maintain that there is, after all, an objective justification for assuming that some – indeed every – temporal position of T once deserves the designation “the present moment” more than any other temporal position of T, concluding, then, that this justification must come from outside of T, since it does not come from inside of it (such is the intrinsic constitution of T). What should we assume? Our guiding light must be that T is supposed to be a metaphysical model of Reality. Unfortunately, there is far less consensus on Reality having an objective present than there is on Reality having an objective direction of time. Many philosophers argue that there is, in Reality, no objective present, while most of them still believe in the objective directedness of Reality’s time. As we have seen above, tense-operations do not force one to believe in an objective present: there is an understanding of “is now”,

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“was”, and “will” for T (described above) that does entirely without assuming that there is an objective present. Yet, everybody agrees that the present moment always seems qualitatively special, and one is entirely within one’s rights – there is no violation of reason involved – if one takes this seeming seriously, namely, as an indication of an objective present. On this basis, it is here maintained that there is an objective justification for assuming that some – in fact, every – temporal position of T once (but not at the temporal position itself) deserves the designation “the present moment” more than any other temporal position of T – an objective justification that comes from the outside of T. What this objective justification consists in will be seen in Sect. 5.2.1, where a unified T-extrinsic perspective on the direction of time (i.e., past and future), presentness, and actuality will be presented for Model T. This perspective will not fail to be instructive – per analogiam – regarding the parallel (three) issues in the metaphysics of Reality (in relation to which our epistemological status is not as privileged as it is in relation to Model T). 3.4

Events as sequences of momentary states

It is natural to treat events as temporal sequences (not necessarily gapless in time) of momentary states, including single-moment sequences (improper temporal sequences) as limiting cases. Technically speaking, every event is a function the domain of which is some non-empty set of temporal positions and which assigns a momentary state (see Sect. 3.3.1) to every temporal position in its domain. Within this general framework, it is, however, possible to distinguish liberal and less liberal conceptions of events. The most liberal conception of events is constituted by allowing all momentary states (in the sense of conception [4a] in Sect. 3.3.1) – except the momentary state that corresponds to the empty set (of fillings of temporal positions) – to be objects of temporal sequencing (in other words: to be values of functions the domains of which are non-empty sets of temporal positions) in generating events. A less liberal conception of events is constituted by allowing only atomic and composite momentary states (see Sect. 3.3.1) to be objects of temporal sequencing in generating events. A still less liberal conception of events is constituted by requiring not merely that only atomic and composite momentary states be objects of temporal sequencing in generating events, but, in addition, that the domain of an event-generating temporal sequencing must represent a coherent temporal

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configuration (see Sect. 3.2.2). But even at this stage of restrictedness the corresponding concept of event may still seem to be too liberal, too wide for intuitive adequateness. Further restrictions can only be introduced by further requirements of coherence, be it regarding the coherence of the sequenced momentary states amongst each other (cross-time coherence), be it regarding the coherence of each sequenced momentary state in itself (momentary coherence). For example, it is not implausible to require for eventhood that the spatial positions that go into each sequenced (atomic or composite) momentary state form a coherent spatial configuration (in the sense defined in Sect. 3.2.1, below D9). And, in addition, one might require that, from one temporal position in the domain of any given event to the next (not merely relatively to the domain, but relatively to the timeline next), the allotted momentary states (they are just 1 temporal step apart) are not too different from each other. This latter requirement – which is intentionally left vague – takes an event to be a more or less continuous item, not only regarding its time (this is taken care of by requiring that the domain of an event must represent a coherent temporal configuration) but also regarding its content (a content distributed in some manner or other over the time of the event). Talk of continuity must, of course, be understood appropriately here – since the timeline of T is, in the mathematical sense, not continuous but discrete. An example of a temporal sequence of momentary states that satisfies all of the suggested requirements for eventhood is given by the following description: At temporal position 1: all spatial positions of which the second coordinate is 1, 2, or 3 are empty, except . At temporal position 2: all spatial positions of which the second coordinate is 1, 2, or 3 are empty, except . At temporal position 3: all spatial positions of which the second coordinate is 1, 2, or 3 are empty, except . ……… At temporal position 10: all spatial positions of which the second coordinate is 1, 2, or 3 are empty, except .

This sequence of statements describes a temporal sequence of momentary states the domain of which is the coherent temporal configuration specified by the set {1, 2, 3, …, 10} and which assigns to each temporal position in its domain a different composite momentary state. The same spatial positions go into each one of the assigned composite momentary states,

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namely, the 30 spatial positions of which the second coordinate is 1, 2, or 3, and these spatial positions constitute, moreover, a coherent spatial configuration. Finally, from one temporal position in {1, 2, 3, …, 10} to the next, the allotted momentary states are certainly not too different from each other, no matter how this phrase may be precisely understood. As has just been stated, the allotted momentary states all involve the same 30 spatial positions, and in all of the 9 transitions involved only two of those spatial positions change their state: one becomes empty, the other full – two positions that are always next to each other (one always being one step horizontally to the right of the other). The described sequence of momentary states is certainly an event (of Model T), even if the strictest intuitive standards for eventhood are being applied. It is unobjectionable to envisage the event in question as the uniform movement – in the time-interval [1, 10] – of a material atom through space (which is empty in the immediate neighbourhood of the course of the atom), in a straight line that goes from on the left to on the right. But, so far, the conceptual means for a full ontological understanding of this – for the time being – mere picture have not yet been prepared. This will be done in Chapter 6, where Model T’s atomic material objects (its material atoms) will be introduced and scrutinized. Here, it should be noted that if all of the above conditions for eventhood except the last one (concerning coherence of content) are put in effect – each condition being considered necessary for eventhood, and all conditions in conjunction being considered sufficient for it –, that then it is still true that all complete histories of T turn out to be events. (For obtaining this result, it must for example be kept in mind that the fillings of T’s temporal positions are maximally composite momentary states, and that the spatial positions involved in each maximally composite momentary state constitute a coherent spatial configuration of T: the space of T; see Sect. 3.3.1 and Sect. 3.2.1.) And even if all of the above conditions for eventhood without exception are put in effect, it still remains true that very many complete histories are events, namely, all complete histories that have a certain cross-time coherence of content. It is worthwhile to exhibit explicitly and side by side both the logically widest and the logically narrowest definite (entirely non-vague) concept of event that can be gathered from the above considerations: D19 A common event of T (common T-event) is a temporal [proper or improper, but never empty] sequence of momentary states of T [“mo-

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mentary states” in the sense finally decided on in Sect. 3.3.1, i.e., in the sense of conception [4a], there described], where each of those states is intrinsically implied by at least one filling of temporal positions of T [i.e., by at least one maximally composite momentary state]. D20 A rare event of T (rare T-event) is a temporal sequence of momentary states of T that fulfils the following conditions: (i) the temporal positions in the domain of the sequence constitute a coherent temporal configuration of T; (ii) the sequenced momentary states are atomic or composite momentary states, each of which is such that the spatial positions involved in it constitute a coherent spatial configuration. According to these two definitions, it is true that every rare T-event is a common T-event (but not vice versa). But this is a paradox one can live with, I believe. Accordingly, since all T-events, in the global sense, are rare or common, the T-events (in the global sense) just are the common Tevents. 3.4.1

Events and states of affairs

In addition to a certain temporal configuration, certain time-dependent but not time-thematic states of affairs, namely, certain momentary states (sometimes only one such item), are intrinsic constituents of any given event. There is, therefore, an intrinsic relationship between events and time-dependent but not time-thematic states of affairs. There is, however, also a – quite different – intrinsic relationship between events and states of affairs that are not time-dependent, but time-thematic: every event can be translated, according to a general rule of translation, into some such state of affairs, and the translation (according to this general rule) can be reversed in a unique way, returning to the very original it started from. The translation works in the following way: For a given event E, let ST1 be the momentary state assigned to the first (earliest) temporal position Z1 in the domain of E, …, and STN the momentary state assigned to the Nth and last (latest) temporal position ZN in the domain of E (N ≥ 1).

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E is translated into the following not time-dependent, but timethematic state of affairs: Conj(at-Z1(ST1), …, at-ZN(STN)). Here, at-Z(…) is, for each temporal position Z, the unary binding-to-Z operation on momentary states. (Remember in this context that in Sect. 3.3.2 above we have already encountered the operation at-z*(…), where “z*” – such is its stipulated meaning – stands for the present moment.) The operation at-Z(…) yields for each momentary state (hence time-dependent but not time-thematic state of affairs) a non-time-dependent but time-thematic state of affairs. For example, at-2(that is full) is the non-timedependent but time-thematic state of affairs that is full at 2, and at3(that is empty or full) is the non-time-dependent but timethematic state of affairs that is empty at 3 or full at 3.8 Conj, in turn, is the obvious generalization of the already familiar binary conjunction-operation on states of affairs, conj. (Note that Conj(ST) is, for any state of affairs ST, just ST.) Since at-Z1(ST1), …, at-ZN(STN) are – for the momentary states ST1, …, STN – non-time-dependent but timethematic states of affairs, their conjunction – Conj(at-Z1(ST1), …, atZN(STN)) – must also be a non-time-dependent but time-thematic state of affairs. The translation of the event E into Conj(at-Z1(ST1), …, at-ZN(STN)) is uniquely reversible since the (generalized) operation of conjunction preserves – in forming the state of affairs Conj(at-Z1(ST1), …, at-ZN(STN)) – the temporal distinctions between its conjuncts: at-Z1(ST1), …, at-ZN(STN). Thus, Conj(at-Z1(ST1), …, at-ZN(STN)) is comparable to a sum in which, after its formation, the very elements that went into the formation can be reconstructed, without any distinctions lost. (Note that the sum 5 + 4 + 3 – or in other words: 12 – is not simpliciter a reversible sum; but as a sum of immediately successive summands it is indeed a reversible sum: 12 can be decomposed into summands in only one way if it is required that all the summands form a sequence of immediately successive natural numbers.) The states of affairs that are intrinsically correlated one-to-one in the way just described with the common T-events are the common event-like T-states-of-affairs, and the states of affairs that are intrinsically correlated one-to-one in the way described with the rare T-events are the rare eventlike T-states-of-affairs. 8

Note: at-3(that is empty or full) = at-3(disj(neg([Full, ]), [Full, ])).

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Now, each rare T-event E is correlated one-to-one with a certain set of complete T-histories, namely, with the set of complete histories by which it is intrinsically implied – where a rare event E is intrinsically implied by a complete history H if, and only if, for each temporal position Z in the domain of E, the (atomic or composite) momentary state that E allots to Z is intrinsically included – namely, as a coherent field – in the (maximally composite) momentary state that H allots to Z.9 And, derivatively, in view of the above, also each rare event-like T-state-of-affairs – each a non-timedependent but time-thematic state of affairs – is correlated one-to-one with a certain set of complete T-histories. But how is it that the correlation between the rare T-event E and the set M(E) of complete T-histories that intrinsically imply E is one-to-one? – Take M(E) and consider for each temporal position Z the set m(Z, M(E)) of maximally composite momentary states allotted to Z by some complete history in M(E). If m(Z, M(E)) is the set of all maximally composite momentary states of T (i.e., of all fillings of T’s temporal positions), then Z does not belong to the domain of E. Otherwise, Z belongs to the domain of E. In this latter case, because of the one-to-one correlation between the sets of maximally composite momentary states (i.e., fillings of T’s temporal positions) on the one hand and the momentary states on the other (see Sect. 3.3.1), m(Z, M(E)) can be replaced by the unique momentary state that is correlated with it, which will be an atomic or composite momentary state, precisely the one that E allots to Z. Thus: E can be uniquely reconstructed from M(E). It should be noted that not every non-empty set of complete Thistories has a rare T-event corresponding to it one-to-one; quite obviously, there are non-empty sets of complete T-histories that have no rare T-event corresponding to them. It is not even the case that every non-empty set of complete T-histories has a common T-event corresponding to it one-to-one; there are such non-empty sets of complete T-histories that have several common T-events corresponding to them. But if we merely consider the common T-events that do not allot to any temporal position in their domain the momentary state that is intrinsically implied by all fillings of T’s temporal positions, then we have: all such events – and therefore also the not time-dependent, but time-thematic states of affairs into which those events 9

If E were a common and non-rare T-event, the concept of the intrinsic implication of E by a complete history would be the same, except for “intrinsically included in” being replaced, in the defining condition of that concept, by the more general “intrinsically implied by”.

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can be translated (i.e., certain common event-like T-state-of-affairs) – correspond one-to-one to the non-empty and non-full sets of complete Thistories. (A set of Φs is non-full [with respect to Φ] if, and only if, it does not encompass all Φs.) 3.5

Basic tropes

The following three items are one-to-one (intrinsically) correlated: the state of affairs that is full at 2, the event which consists merely in this: the momentary state [Full, ] being allotted to 2 – a certain momentary event –, the particular (or individual) which is the fullness of at 2 – which is a trope. In general: the state of affairs that P is full at Z, the event which consist merely in this: the momentary state [Full, P] being allotted to Z, the particular which is the fullness of P at Z – a trope – are (for inner reasons) one-to-one correlated – for each spatial position P and temporal position Z. This may suggest the identification – nondifferentiation – of the items in the above column (for each spatial position P and temporal position Z). But, in fact, the identification of those items is out of the question, since the relevant categories – state of affairs, event, individual (or particular) – are exclusive of each other. Items that can be designated by expressions of the form “the fullness of P at Z” (or without any remainder of ambiguity: “the particular fullness of P at Z”) – where P must be a spatial position and Z a temporal position of T – are the basic (or atomic) spatiotemporal tropes of T. Obviously, there are 10000 of these items. Items, on the other hand, that can be designated by expressions of the form “the fullness of P” (or without any remainder of ambiguity: “the particular fullness of P”) – where P must be a spatial position of T – are the basic (or atomic) spatial tropes of T. Obviously, there are 100 of these items, each of which corresponds one-to-one to a state of affairs: for any

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spatial position P, the fullness of P corresponds one-to-one to the atomic momentary state that P is full.10 Let P be any spatial position and Z any temporal position. In accordance with the above description of basic spatial and spatiotemporal tropes, we have the following two non-identity principles for basic spatial – respectively, basic spatiotemporal – tropes: The [particular] fullness of P is not identical with the [particular] fullness of P´ if, and only if, P and P´ are different spatial positions.11 The [particular] fullness of P at Z is not identical with the [particular] fullness of P´ at Z´ if, and only if, P and P´ are different spatial positions or Z and Z´ are different temporal positions. The basic spatial tropes of T come very close to being continuants simpliciter (as defined in Sect. 3.2.1). In contrast to T’s momentary states and T’s basic universal Full (cf. Sect. 3.3.1), they have the right category for this: they are individuals, non-abstract ones to boot, and doubtless they are wholly present at every temporal position at which they are located. But it is impossible to say simpliciter of any basic spatial trope that it is located at more than one temporal position, i.e., to assert its temporal multi-locatedness categorically: without (implicit or explicit) relativization to something else (to some complete history or other, for example; or to actuality* – a rather special complete history; see Sect. 5.1.1). Hence basic spatial tropes are not continuants simpliciter. In a relative sense, however, basic spatial tropes can very well be said to be continuants (in this or that complete history, for example). Basic spatiotemporal tropes of T, on the other hand, can only be categorically and truly asserted not to be present at more than one temporal position – and hence they are not continuants (not simpliciter and not relatively).

10

In case one wonders what non-basic tropes are: a non-basic spatial trope is a (proper) collection of basic spatial tropes; a non-basic spatiotemporal trope is a collection of simultaneous basic spatiotemporal tropes. 11 One might object: “Why, the fullness of is identical with the fullness of : the fullness of is Fullness, and the fullness of is also Fullness, which is just the basic universal of T.” This objection, which is based merely on ambiguity, is excluded by inserting (in order to achieve disambiguation) the word “particular” in front of the word “fullness” in the expression “the fullness of”.

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3.5.1

Basic spatiotemporal tropes as fundamental or derivative entities

There are two ways of looking at the basic T-tropes. One can consider them to be entities that are derived from other – fundamental – entities of T: from Fullness and T’s spatial positions in the case of basic spatial tropes, from Fullness, T’s spatial positions, and T’s temporal positions in the case of basic spatiotemporal tropes. Or – the other way around – one can consider the basic spatiotemporal T-tropes to be fundamental entities, from which Fullness, T’s spatial positions, T’s temporal positions, and even T’s basic spatial tropes are derived. Assuming the first perspective, the derivativeness of the basic tropes can be made visible in the language of set-theory, (i), by postulating that the basic spatial tropes are (or properly speaking: are represented by) certain sets with two elements: one of the two elements being Full and the other some spatial position; and, (ii), by postulating that the basic spatiotemporal tropes are certain sets with three elements: one of the three elements being Full, one a spatial position, and one a temporal position. Thus, for example, the fullness of at 2 is (i.e., is set-theoretically represented by) {Full, , 2} (and the three elements can be permuted at will: this has no effect on the adequacy of the representation). Note that the set-theoretic representation of the momentary event correlated with our example-trope – the event which consists merely in this: the momentary state [Full, ] being allotted to 2 – is quite different from that of the trope: {}.12 Assuming the second perspective, the derivativeness of Fullness, T’s spatial positions, T’s temporal positions, and T’s basic spatial tropes can be made visible in the language of set-theory by employing the following equality-predicates: “is qualitatively equal to”, “is spatially equal to”, “is temporally equal to”. Fullness, then, is represented by the single equivalence-set of the predicate “is qualitatively equal to” in the set of T’s basic 12

Temporal sequences of momentary states are functions from temporal positions to momentary states, and such functions are (in set-theoretical representation) sets of ordered pairs – no two of the ordered pairs having the same first component – of which the first component is a temporal position and the second a momentary state. In the limiting case, such a set has merely one element, as in our example.

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spatiotemporal tropes (which equivalence-set is, in fact, precisely this latter set of 10000 elements). The spatial positions of T, in turn, are represented by the equivalence-sets of the predicate “is spatially equal [or isotopic] to” in the set of T’s basic spatiotemporal tropes; there are 100 such sets, each with 100 elements (no two of which are temporally equal), completely dividing up the basic spatiotemporal tropes among them. The temporal positions of T, in turn, are represented by the equivalence-sets of the predicate “is temporally equal [or isochronic] to” in the set of T’s basic spatiotemporal tropes; there are 100 such sets, each with 100 elements (no two of which are spatially equal), again completely dividing up the basic spatiotemporal tropes among them. The basic spatial tropes of T, in turn, are represented by the sets that have as elements just two sets: (1) the equivalence-set of the predicate “is qualitatively equal to” in the set of T’s basic spatiotemporal tropes, and (2) an equivalence-set of the predicate “is spatially equal to” in the set of basic spatiotemporal T-tropes. The approach which takes the basic spatiotemporal T-tropes to be the fundamental T-entities is viable only if order-predicates are added to the equality-predicates, for example, the order-predicate “is N steps earlier than”. Moreover, all order- and equality-predicates need to be characterized – also in relation to each other – by appropriate postulates, all of them general with respect to the basic spatiotemporal T-tropes. Here are seven such postulates (the list is not complete – omitting obvious and less obvious postulates – and is only meant to give an idea of what is involved): 1. For all basic spatiotemporal T-tropes X and Y: if X is spatially and temporally equal to Y, then X is identical to Y. 2. For all basic spatiotemporal T-tropes X and Y: X is at least 1 step earlier than Y or Y is at least 1 step earlier than X or X is temporally equal to Y. 3. For all basic spatiotemporal T-tropes X and Y: if X is temporally equal to Y, then X is N* steps up/down from Y and N*´ steps to the left/right of Y, where N* is a uniquely determined natural number between 0 and 9, and likewise N*´. 4. For all basic spatiotemporal T-tropes X and Y: if X is spatially equal to Y, then X is N* steps earlier/later than Y, where N* is a uniquely determined natural number between 0 and 99. 5. For all basic spatiotemporal T-tropes X, Y, and Z: if X is temporally equal to Y and Y is N steps earlier/later than Z, then X is N steps earlier/later than Z.

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6. For all basic spatiotemporal T-tropes X, Y, and Z: if X is spatially equal to Y and Y is N steps up/down from Z and N´ steps to the left/right of Z, then X is N steps up/down from Z and N´ steps to the left/right of Z. 7. For all basic spatiotemporal T-tropes X and Y: if X is temporally equal to Y, then there is a basic spatiotemporal T-trope X´ that is 1 step earlier or later than X, and a basic spatiotemporal T-trope Y´ that is temporally equal to X´, where X´ and Y´ are such that X´ is spatially equal to X and Y´ spatially equal to Y. Which of the two perspectives described – the first one might be characterized as “compactive”, the second as “abstractive” (but set-theory, this very flexible organon of representation, lends itself equally to both perspectives) – is preferable? Note that we are still dealing with the very same metaphysical model: Model T; all that is at issue is what is ontological fundamental to that model (the issue is not whether to add something to it, or to subtract something from it: in both perspectives the same categories make their appearances, with the same numbers of instances). As a matter of fact, the answer to the question just posed has implicitly – but nonetheless plainly – already been given here: Full, spatial positions, and temporal positions were already introduced in Chapter 2 (called “The basic description of Model T”), but basic spatiotemporal tropes only in Sect. 3.5. Still, the factually preferred way of doing things has been followed quite without reflecting on it, and this is the right time to append reflection. It seems to me that both perspectives and approaches are, in principle, equally justifiable – and therefore, objectively, equally arbitrary. The philosophers with an ontological bias in favour of maximal particularity in indivisible tiny portions might argue that the second perspective is preferable, holding that only basic – atomic – spatiotemporal tropes fundamentally – ultimately – exist. But philosophers who do not share the ontological bias in favour of maximal particularity in indivisible tiny portions – and why, indeed, should they share this bias? – will see things differently. This decision – the one under consideration – concerning what to treat as fundamental in the description of T and what as derivative is entirely in the hands (and in the eyes) of the beholder, who is outside of T. So it is with regard to Model T. Might it not be in similar questions in similar ways – structurally – with regard to Reality (though we are certainly not in the same way outside of Reality as we are outside of T)?

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However, one might doubt that the two perspectives under consideration are in fact objectively equivalent, that they do in fact offer two equally valid – with some effort intertranslatable – descriptions of the same thing: Model T. What about a sentence like “ is empty at 2”? What, according to the second perspective, would make such a sentence true? This seems a hard question, since, under the second perspective, apparently no provision has been made for empty spatial positions. But since the sentence “ is full at 2” is made true by the existence of a certain basic spatiotemporal trope – namely, the trope that (given a high level of theoretical sophistication under the second perspective) can be designated in the following way: “the (particular) fullness of at 2” –, the sentence “ is empty at 2” is made true by the non-existence of that very same trope. Thus, under the second perspective, “ is empty at 2” amounts to this: the fullness of at 2 does not exist.13 Note, however, that the non-existence of that trope – the fullness of at 2 – must not entail the non-existence of the spatial position as (represented by) a certain set of basic spatiotemporal tropes, or the non-existence of the temporal position 2 as (represented by) a certain other set of basic spatiotemporal tropes. Why not? – Because the statement “ is empty at 2” logically entails (implicitly states) both the existence of the spatial position and the existence of the temporal position 2. In fact, both positions must remain existent come what may – even if all the tropes in the sets that are (respectively) representative of the two positions and 2 are non-existent, as may easily happen (it would be the case if is empty at every temporal position and if every spatial position is empty at 2); even then both positions must still be available as existent. Under the first of the above two perspectives, the imperviousness of spatial and temporal positions, as regards their existence, to the massive and even maximal non-existence of basic spatiotemporal tropes represents no problem – since, according to the first perspective, spatial and temporal positions are (much) more basic than basic spatiotemporal tropes. But under the second perspective, according to which basic spatiotemporal tropes are more basic than both spatial and temporal positions, this imperviousness does seem somewhat incongruous.

13

Under the first perspective, in contrast, “ is empty at 2” amounts to this: does not exemplify Full at 2.

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3.6

Worlds and modal positions

As was stated at the beginning of Chapter 2, Model T has two types of positions: spatial positions and temporal positions – and matching these two types of positions, two types of fillings: the (single) universal Full in the case of spatial positions, the (many) ways of distributing Full over all the spatial positions (or in other words: the maximally composite momentary states) in the case of temporal positions. Positions and filling(s) put together – compacted – constitute entities that are distinct from both ingredients that go into the compaction. In the case of a spatial position P and Full, an atomic momentary state, [Full, P], is constituted by their being put together in one manner of compaction, and a spatial trope – representable by the set {Full, P} – is constituted by their being put together in another manner of compaction. In the case of a temporal position Z and a maximally composite momentary state ST, a maximal momentary event – representable by {} – is constituted by their being put together. Unlike maximal momentary events, the complete T-histories (complete histories of T14) are events that are in all respects maximal. The complete T-histories are set-theoretically representable by the sets which have, for every temporal position Z´, exactly one ordered pair as element – where ST´ is a maximally composite momentary state of T – and which have no other elements. Obviously, every complete T-history shares at least one maximal momentary event – a subset {} – with another complete T-history – and is therefore overlapping substantially15 with that other T-history. It may be argued that this fact disqualifies the complete histories of T – though otherwise well-qualified for the job – from being the worlds of T, since worlds are not supposed to overlap. Now, regarding the non-overlap of worlds, one can be of differing rational opinions. But perhaps there is a slight intuitive bias in favour of treating worlds as non-overlapping. Respecting this bias, I abstain from simply identifying the worlds of T with the complete histories of T. On the other hand, worlds and complete histories are certainly not unrelated items, but “almost the same thing”. I herewith propose that the complete histories of T are the fillings of the modal positions of T, and the worlds of T turn out to be what is constituted when these fillings are compacted with those positions. Overlap – modal overlap – disappears in this 14

That is: complete histories that belong to the inventory of T; cf. footnote 6. Regarding the notion of substantial/non-substantial overlap (that is, substantial/nonsubstantial common part), see the following footnote.

15

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approach, since the use of positions always has a separating effect within the positional structure to which the positions belong. (Thus, the spatial trope {, Full} is spatially separate from the spatial trope {, Full}, although the two tropes are qualitatively identical.) However, modal positions (of T) are quite unlike spatial and temporal positions (of T). As far as the intrinsic nature of any spatial position is concerned, any spatial position is open to being filled, and alternatively to being left empty, by Full. Temporal positions do not have the option of emptiness; but still, as far as the intrinsic nature of any temporal position is concerned, any temporal position of T is open to being filled by no matter which maximally composite momentary state. But modal positions neither have the option of emptiness nor are they indifferent as to how they are filled by complete histories: the intrinsic nature of each modal position is such that there is exactly one complete history that fills it, that history being for every modal position a different history. Indeed, modal positions are intrinsically correlated one-to-one with complete histories, no complete history being left without its modal position. (There are, therefore, exactly as many modal positions as there are complete histories.) Prima facie it seems that by introducing modal positions one is stepping beyond T and is moving on to another metaphysical model. This seems so because modal positions do not appear to be intrinsic to T (since they do not seem to be gatherable – via conceptual construction – from its basic description; cf. Sect. 2), but seem to be intrinsic to another metaphysical model (which is, however, in all other intrinsic respect just like T). But, as matter of fact, modal positions can simply be identified with their fillings: each modal position can be identified with the intrinsically unique complete history that it can be filled with. Proceeding along these lines one stays within in T, since complete histories are items intrinsic to T. It is, therefore, assumed that each complete history has itself as its only modal position, there being no other modal positions than those had by complete histories. Thus, in set-theoretical representation, the worlds of T differ from the complete histories of T merely as ordered pairs differ from the Xs. is (i.e., represents) a world of T if, and only if, X is a complete history of T, all worlds of T being such ordered pairs. And if is a world of T, then the complete history X is at once the modal position which is a constituent of that world (say, as the first member of the ordered pair ) and the filling of that modal position, which filling is also a constituent of that world (as the second member of the ordered pair ). This way of treating worlds and complete histories elegantly

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solves the overlap-problem, which (since it was taken seriously) made it necessary to distinguish between worlds and complete histories, but with the stricture of not tearing apart what is “almost the same thing”. The maximal momentary event {} is a common part of many complete histories; but, according to the proposed way of distinguishing between worlds and complete histories, it is not a common part of many worlds – because it is not a part of any world. It is indeed a constituent of many worlds – in virtue of being a common part of many complete histories, each of which is a constituent (in the way described above) of a different world. But commonality of mere constituents – in contrast to commonality of parts – does not constitute overlap. Here, only such constituents are parts that are of the same (the same nearest) ontological category as that which they are constituents of, while constituents that are not of the same category as that which they are constituents of are mere constituents. Thus, for the sake of further illustration, a spatial position P and the universal Full are mere constituents of [Full, P], not parts of it, since [Full, P] is a momentary state, whereas P and Full are not of this category. And P is a common mere constituent of [Full, P] and neg([Full, P]), certainly not a common part of these two momentary states; otherwise these two momentary states would overlap substantially, which is absurd.16 In contrast, the following two momentary states: that and are full, and that and are full, do indeed overlap substantially (in the momentary state that is full). The use of the modifier “modal” in this section is the first indication of matters that will be treated exhaustively in Chapter 5. However, prior to this, something else – also concerning complete histories (and hence worlds) – deserves treatment in a separate chapter.

16

But they do overlap non-substantially, since the momentary state that P is full or empty is a non-substantial common part of them. Here, a non-substantial part of X is a part of X that is also a part of everything with the same (nearest) category as X, whereas a substantial part of X is a part of X that is not also a part of everything with the same (nearest) category as X.

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4

Uniformity and diversiformity of histories and of maximally composite momentary states

Complete histories of T – in short: histories of T, T-histories, or just histories – are, in themselves, more or less uniform, or in other words: more or less diversiform. There are two histories of T that are maximally uniform (i.e., their degree of diversiformity is 0), and therefore they are maximally uninteresting to the metaphysical spectator: the history where at every temporal position every spatial position is empty, and the history where at every temporal position every spatial position is full. On the other hand, the following – laborious – procedure is likely to single out a T-history which is maximally diversiform: Take 100 fair dice, all of them of the same (small) size; three faces of each die are black, three faces white (where black and white are always distributed in the same way over the six faces of each die). Each die is marked, so as to stand for a certain spatial position of T, each die standing for a different position (say, things are arranged in such a way that which spatial position a die stands for can be read off the die directly). Now throw these 100 dice 100 times (out of a bowl containing all of them, with a sweeping motion, onto a plane surface) and record the results in the order they are produced, adhering after each throw to the following instructions regarding each die: If, resulting from the Nth throw, the upward face of the die that is marked with the Pname is black, then record: P is full at N (in the history that is being selected); if, resulting from the Nth throw, the upward face of the die that is marked with the P-name is white, then record: P is empty at N (in the history that is being selected).

Three points regarding this procedure should be kept in mind: (1) It is not a procedure that produces a T-history, or makes a T-history actual; it is merely a procedure that selects a T-history, nominates it, so to speak. (2) Not only on the basis of how things are inside of T, but also on the basis of how things are outside of T, every T-history is as likely to be selected by the procedure as every other T-history. (3) Although each T-history has exactly the same chance of being selected by the procedure (namely, 1 in 210000), T-histories with a high degree of diversiformity are much more likely to be selected by the procedure than T-histories with a low degree of

diversiformity, since there are far more T-histories with a high degree of diversiformity than T-histories with a low degree.17 High uniformity is a, comparatively, rare thing among T-histories, while high diversiformity is not. But how is the degree of uniformity of a T-history determined? Is there a function that assigns to each T-history a determinate number that is the history’s degree of uniformity? The matter is complex, since there are two dimensions of the uniformity of histories: the momentary dimension and the transmomentary. Thus, a history may have at each temporal position – namely, in the maximally composite momentary state it allots to that position – maximal momentary uniformity (that is, all spatial positions are full, or all spatial positions are empty), whereas it is highly diversiform transmomentarily. And vice versa: a history may be maximally uniform transmomentarily (by allotting the very same maximally composite momentary state to each temporal position), whereas it has very low momentary uniformity at each temporal position. The degree of the momentary uniformity of a given history at a given temporal position is just the degree of the uniformity of the maximally composite momentary state which that history assigns to that temporal position. How is the degree of uniformity of a maximally composite momentary state determined? Even this is not an easily answerable question, though it would be an easily answerable question if all that mattered were qualitative uniformity: 17

This statement of comparative likelihoods sounds contradictory, but it is no more contradictory than the following, analogous statement: Although each face of a fair die has the same chance of lying upward after the die is thrown (namely 1 in 6), the diefaces with a number of eyes that is not divisible by 3 are more likely to lie upward than the die-faces with a number of eyes that is divisible by 3, since there are more diefaces of the former kind than of the latter (4 against 2). What one must not do if F is a property with more than one instance (for example, the property of being an upward lying die-face with a number of eyes that is not divisible by 3) is this: to assign to an instance of F (e.g., the upward lying die-face with 2 eyes), as its probability, the probability of an instance of F occurring. If F is a property with more than one instance, then the probability of an instance of F is (normally) not the probability of some F occurring, in other words: it is not the probability of F, or, in other words again, of Fs.

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There are 4950 two-element sets of spatial positions.18 Given a maximally composite momentary state ST, it is determined for each of those pair-sets whether it is qualitatively homogeneous according to ST (which it is just in case both spatial positions in it are full, according to ST, or both empty) or qualitatively heterogeneous according to ST (which it is just in case one spatial position in it is full, according to ST, the other empty). The degree of qualitative uniformity of ST, then, is the number of, according to ST, qualitatively homogeneous two-element sets of spatial positions divided by 4950, and the degree of qualitative diversiformity of ST is the number of, according to ST, qualitatively heterogeneous two-element sets of spatial positions divided by 4950. Note that the degree of qualitative uniformity is 1 for precisely two maximally composite momentary states: the one in which all spatial positions are full, and the one in which all spatial positions are empty; and note that the degree of qualitative uniformity is never lower than 2450/4950 for any maximally composite momentary state, and therefore the degree of qualitative diversiformity – which is 1 minus the degree of qualitative uniformity – never higher than 2500/4950. The lowest point of qualitative uniformity, 0.494949…, is reached when 50 spatial positions are full and 50 empty (then 1225 of the 4950 twoelement sets of spatial positions are homogeneously full, 1225 homogeneously empty, and 2500 heterogeneous); already if 49 are full and 51 empty, or 51 full and 49 empty, qualitative uniformity has become greater than 0.494949…: 0.495151… (in the first case, because of greater uniformity on the side of emptiness – for then 1275 of the 4950 two-element sets of spatial positions are homogeneously empty, 1176 homogeneously full, and 2499 heterogeneous; in the second case, because of greater uniformity on the side of fullness). Unfortunately, it is not possible to identify the (degree of) uniformity of a maximally composite momentary state with its (degree of) qualitative uniformity (although it is true that a maximally composite momentary state has the degree 1 of uniformity precisely if it has the degree 1 of qualitative 18

This number is found out in the following way: There are 10000 ordered pairs of spatial positions (100 × 100). From these, the 100 homo-componential ordered pairs of spatial positions (i.e., the ordered pairs of the form ) must be excluded, and from the remaining 9900 hetero-componential ordered pairs one half must be excluded: for each of those ordered pairs, either the pair itself, , or its inverse, , must be excluded (not both). The remaining 4950 ordered pairs of spatial positions are equinumerous to the sets {P, P´} of spatial positions with P ≠ P´. Hence there are 4950 two-element sets of spatial positions.

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uniformity). The reason for this is clear: uniformity is not just a matter of uniform quality among the spatial positions, but also of uniform spatial relations between qualitatively uniform spatial positions. Thus a maximally composite state with low qualitative uniformity may still have high overall uniformity; an example is the maximally composite momentary state in which 50 spatial positions are full and 50 empty, but in which fullness and emptiness are distributed over the 100 spatial positions in an (horizontally and vertically) alternating pattern (a checkerboard pattern), starting with being full. The qualitative uniformity of a maximally composite momentary state can easily be measured; measuring its overall uniformity is, however, quite a different, forbiddingly difficult matter. Linguistic compressibility is often suggested to be a measure of uniformity. According to this view, the fewer words one has to use in completely describing a maximally composite momentary state, the more uniform it is, its degree of uniformity being taken to be inversely proportional to the number of words in its shortest complete description. But, evidently, this approach makes uniformity a language-dependent matter (for in one language the shortest complete description of a maximally composite momentary state may be very brief, in another very long), and it seems uniformity should not be a languagedependent matter. The difficulties of determining a numerical degree of uniformity for a given history are compounded when one moves from the momentary dimension of the history’s greater or smaller uniformity (which, at each temporal position, is manifested in the greater or smaller uniformity of a maximally composite momentary state) to its transmomentary dimension. – And how, indeed, are the two dimensions to be computationally combined, as they must be for producing the degree of the uniformity of the given history? Measuring the qualitative transmomentary uniformity of a history is, nonetheless, a comparatively easy matter – as was the measuring of its qualitative momentary uniformity (at a certain temporal position, i.e., the qualitative uniformity of the maximally composite momentary state the history allots to that temporal position). There are 4950 two-element sets of temporal positions. The degree of qualitative transmomentary uniformity of a history H is the number of – according to H – qualitatively homogeneous two-element sets of temporal positions divided by 4950, and the degree of qualitative transmomentary diversiformity of H is the number of – according to H – qualitatively heterogeneous two-element sets of temporal

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positions divided by 4950. Here, a two-element set of temporal positions is qualitatively homogeneous – according to H – if, and only if, its two elements have – according to H – the same filling (a maximally composite momentary state), and a two-element set of temporal positions is qualitatively heterogeneous – according to H – if, and only if, its two elements have – according to H – not the same filling (i.e., H does not allot the same maximally composite momentary state to both of them). (In an analogical sense, maximally composite momentary states can be described as qualities of temporal positions, although, properly speaking, they are not qualities but states of affairs. They stand to temporal positions as the quality Full stands to spatial positions. This analogy justifies the use of “qualitative” and “qualitatively” in the present context of scrutinizing the transmomentary uniformity of histories.) In contrast to the qualitative uniformity of maximally composite momentary states, the qualitative transmomentary uniformity of histories sometimes (i.e., for some histories) has the degree 0 (and, note, there are 2100 histories with the degree 1 of qualitative transmomentary uniformity). An example of a history with 0 qualitative transmomentary uniformity but, nonetheless, high overall transmomentary uniformity (and, to boot, high momentary uniformity at each temporal position) is given by the following description: At 1, is full and every other spatial position empty; at 2, is full and every other spatial position empty; ……… at 10, is full and every other spatial position empty; at 11, is full and every other spatial position empty; at 12, is full and every other spatial position empty; ……… at 20, is full and every other spatial position empty; at 21, is full and every other spatial position empty; at 22, is full and every other spatial position empty; ……… at 30, is full and every other spatial position empty; at 31, is full and every other spatial position empty; at 32, is full and every other spatial position empty; ……… at 40, is full and every other spatial position empty; and so on until at 100, is full and every other spatial position empty.

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The image that goes with this complete description of a complete T-history is this: Given that the space of T is visualized as a square (on a white piece of paper) that is subdivided into 100 smaller stationary squares (the spatial positions of T; cf. Sect. 2), our attention is drawn to a lonesome black (i.e., full) movable square. With the exception of the place of this square at any given time of T, all the rest of the space of T is empty (i.e., white) at that time of T. The black square is moving, over the entire time of T (of 100 temporal positions), continuously and in a straight line from the far left to the far right of space, and from the far right to the far left, and back again, and so on, beginning at the lower left-hand corner of the space of T, going one step up at each turnabout, and ending finally at the upper left-hand corner of the space of T. This image (or visualization) of the above-described history – call it “H°” – displays an ontological aspect that has not been treated so far in our exploration of T. H° can be taken to contain (or support, or determine) a higher continuant (not a continuant simpliciter, but a continuant relative to H°19): the lonesome movable black square can be taken to represent this higher continuant – a solid object (hence a non-abstract individual), wholly present in H° at every temporal position at which it is located in H°. This solid object – call it “c°” – is not a basic aspect of H°, since H° can be completely described without even mentioning c° (see above!); but, nevertheless, c° is there (it jumps into our eyes, so to speak, when we visualize H°). We are left with the conclusion that c° is a supervenient aspect of H°. The supervenience of higher continuants on histories (to put the matter summarily) will be treated in Chapter 6 (where the supervenience of the atomic T-material objects will be described). Meanwhile, it is surely correct to note that the rather special regularity of H° is operative in bringing about the supervenience of c° on H°. And that same regularity also gives H° its high degree of (overall) uniformity, whatever may be the exact content of the, intuitively correct, assertion that H° has a high degree of uniformity (this, as should be clear by now, is not easy to explicate in complete exactitude, and I leave it at that) – the high degree of uniformity H° has in spite of the fact that the degree of H°’s qualitative transmomentary uniformity is 0.

19

Regarding the distinction between continuant simpliciter and continuant relative to a history, see Sect. 3.3.1 and Sect. 3.5.

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4.1

Regularities of histories

The end of the previous section leads to a natural question: What is a regularity? In answering this question, I shall restrict my attention to the (diachronic) regularities of T-histories; for a general answer, one would have to consider also the (synchronic) regularities of maximally composite momentary states. Examples of regularities of T-histories are the following four regularities of H°: [R1] that at every temporal position exactly one spatial position is full and all other spatial positions are empty; [R2] that, for each temporal position Z earlier than 100, the spatial position that is full at temporal position Z+1 is not identical to, but still immediately attached to (see D7) the spatial position that is full at Z; [R3] that no spatial position is full at two temporal positions; [R4] that every spatial position is full at some temporal position.20

These four regularities – certain states of affairs that are intrinsically implied by H° – do not characterize H° alone: there are, besides H°, several histories of which R1, R2, R3, and R4 are regularities (for example, the reverse of H°: the history which allots to the temporal position 101−Z the maximally composite momentary state H° allots to Z). Those histories are precisely the T-histories that have a lonesome atom of Fullness (an entity like c°) follow – step by step (that is, continuously21) and without recurrence – a spatial path22 that links all of the 100 spatial positions. (A particularly beautiful path of this kind is the one that starts with , goes 20

This is to be understood just as formulated: the “every” precedes the “some”, so the “some” is to be within the scope of the “every” (and not the other way around). 21 Continuously in the analogical sense Model T allows. 22 Spatial paths are visualized by drawing at least one connecting dash from the middle of a square to the middle of a neighbouring square, the squares representing spatial positions (for the underlying picture of the space of T, see Sect. 2). In drawing dashes to visualize a spatial path, one is following the rule that there can be maximally two dashes originating in the same square, and no more than one dash between one square and another. The following items are examples of spatial paths (designated in a way that is only semi-pictorial): –; ––; –– ; –––. Note that each spatial path is identical to its reverse (for example, –– is identical to ––); for whether a spatial path is considered with this one of its two ends first or with this other one of its two ends first, it does not matter: it is the same spatial path. (–– – is a special spatial path: it is a circular path.)

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to , and then to – and revolves around counterclockwise through the space of T, in ever-widening coils, leaving no gaps and visiting no spatial position twice, ending with .) Each of the four regularities of H° considered is history-specific, that is, it is not shared by all histories of T (whereas non-history-specific regularities – for example, that there are 100 spatial positions, arranged in the very same way, at every temporal position, that Full is, at every temporal position, the only filling of spatial positions,23 and so on – are shared by all histories of T). The conjunction of the four regularities R1, R2, R3, and R4 is itself a – history-specific – regularity of H°, and one may be tempted to call it “the total regularity of H°” (where one can be taken to follow the implicit convention that the regularities of H° that are not history-specific need not be taken into explicit account, since they are the same for every T-history). Intuitively, this seems correct. While it is certain that Conj(R1, R2, R3, R4) does not characterize H° completely, Conj(R1, R2, R3, R4) does seem to be a complete characterization of H° with respect to regularity (though not constituting a simpliciter complete characterization of H°). But for validating the assertion that Conj(R1, R2, R3, R4) is the total regularity of H°, one would have to show that every history-specific regularity of H° is intrinsically implied by Conj(R1, R2, R3, R4) – and one cannot show this, since, as a matter of fact, it is not true that every history-specific regularity of H° is intrinsically implied by Conj(R1, R2, R3, R4): The state of affairs [R5] that at every temporal position Z which is later than 1 the spatial position that is full at Z is either in the same row as the spatial position that is full at temporal position Z–1, or one step upward from that spatial position

is a history-specific regularity of H°, but it is not intrinsically implied by Conj(R1, R2, R3, R4). We have considered several examples of regularities of T-histories. But a general definition of being a regularity of a T-history is not easy to find. One may be tempted to characterize regularities of histories via a certain way of designating them, prominently involving universal quantification over temporal and/or spatial positions. In fact, all of the above examples of regularities are designated in such a way: consider the above thatphrases functioning as singular terms for states of affairs that are regularities. But a state of affairs that is so designated may nevertheless fail to be a 23

It is this regularity which justifies calling a spatial position “empty” if it does not exemplify Full.

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regularity of any history (for example, the state of affairs that every spatial position that is identical with is full at every temporal position that is identical with 1). The following definition, which does not refer in any way to a manner of designation, certainly captures the intuitive idea of what it means to be a regularity of a T-history (and it links this section of the book with the previous section, the notion of regularity with the notion of uniformity): D21 A regularity of a T-history H is a T-state-of-affairs which is intrinsically implied by H and which conveys some, not entirely insignificant degree of uniformity to H. However, this definition is vague, because the concept of the degree of the uniformity of a T-history has remained vague (see the previous section), not to speak of the vagueness of the expression “not entirely insignificant”. – No matter: a vague definition is better than no definition. And it is better to move on than try to remedy a lack of preciseness that may prove irremediable. The very concept defined with a margin of vagueness by D21 is already usefully employed in the following definition: D22 A formal regularity of a T-history H is a regularity of H that, for any temporal position Z and any spatial position P, neither intrinsically implies the state of affairs that P is full at Z, nor the state of affairs that P is empty at Z. All of the above-described regularities of H° are formal regularities of H°, and, in fact, H° does not seem to have any non-formal, material regularities: D23 A material regularity of a T-history H is a regularity of H that, for some temporal position Z and some spatial position P, either intrinsically implies the state of affairs that P is full at Z, or the state of affairs that P is empty at Z. A material regularity of many histories (but obviously not of H°) is the following state of affairs: that is full at every temporal position.

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On the basis of D23, this state of affairs is a material regularity of any history it is a regularity of, because it intrinsically implies the state of affairs that is full at 1. (Note: what a non-time-dependent state of affairs intrinsically implies also includes all intrinsic facts about T, for example, also the fact that 1 is a temporal position. Thus, the state of affairs that is full at every temporal position intrinsically implies the fact that 1 is a temporal position, and therefore – logically – also the state of affairs that is full at 1.) A formal regularity of a history can never contain all of the intrinsic information there is about that history: if it did, it would not be formal regularity of that history (according to D22). For if a regularity of a history H contained all of the intrinsic information there is about H, then it would have to intrinsically imply, for every temporal position Z and every spatial position P, either the state of affairs that P is full at Z or the state of affairs that P is empty at Z, and therefore (because there are temporal positions and spatial positions) for some temporal position Z and some spatial position P, either the state of affairs that P is full at Z or the state of affairs that P is empty at Z – contradicting what it means to be a formal regularity of H (according to D22). It seems generally true that if M is a set of regularities of a history H, that then the conjunction of the regularities in M is also a regularity of H.24 But one need not appeal to intuition to justify this rule. For suppose that all elements of M are regularities of H; hence all elements of M are states of affairs that are intrinsically implied by H, and therefore, doubtless, also their conjunction is intrinsically implied by H. Moreover, since each element of M – qua regularity of H – conveys considerable uniformity to H (see D21), the conjunction of all elements of M can certainly not fail to convey considerable uniformity to H – namely, at least as much uniformity as (and likely enough: more uniformity than) each element of M. The conjunction of all regularities of a history is the total regularity of that history. In some cases, the total regularity of a history determines the history, that is: it contains all intrinsic information about it, or in other words: it allows to specify at each temporal position which maximally composite momentary state the history allots to that temporal position. But the total regularity of H° – the T-history we are presently considering – 24

If M is a set of states of affairs, CONJ(M) – the [state of affairs which is the] conjunction of all elements of M – is the most general concept of conjunction. Conj(ST1, …, STN) is a special case [=Def CONJ({ST1, …, STN})], and so is conj(ST, ST´) [=Def CONJ({ST, ST´})].

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does not seem to determine H°. The reason for this is that the total regularity of H° seems to turn out to be a formal regularity of H°, and therefore it cannot contain all of the intrinsic information about H° (as is explained above). For characterizing H° completely, one must certainly add to the regularities R1, R2, R3, R4, and R5 of H° (which, taken in conjunction, do seem to give the total regularity of H° – as a formal regularity of H°) some other state of affairs that is intrinsically implied by H°. But I forego the labour of finding, and of demonstrating as correct, the most economical way of completely characterizing H°. In the next section, a much more important question than this will be the focus of attention. 4.1.1

Regularities and laws (of nature)

Are the regularities of a history its laws? – In a sense, yes; in another sense, no. There are two conceptions of a law (that is: law of a T-history): the intrinsic conception and the extrinsic. According to the intrinsic conception, given a precise concept of law, a history determines its laws all by itself, intrinsically. Put epistemically: if one is in the possession of a precise concept of law, then, in order to know which states of affairs are the laws of a given history, one merely needs to know that history, in other words: which maximally composite momentary state it assigns to each temporal position. It is entirely appropriate under the intrinsic conception of lawhood to locate the laws of a given history among its regularities. Indeed, it is hard to see under the intrinsic conception of lawhood what else the laws of a history could be if not regularities of that history: states of affairs which are intrinsically implied by it and convey to it some, not entirely insignificant degree of uniformity (see D21). Although it is true, even under the intrinsic conception of lawhood, that not all regularities of a history need be laws of it (perhaps, in fact, R1 – R5 in the previous section are only regularities of H°, not laws25), still, under that conception, every law of a history cannot fail to be regularity of it. But this has the consequence that the concept of a law (of a T-history) cannot be more precise than the concept 25

One might hold that R1 – R5 are already too specific for being laws of H° (or any other history). But the following regularity of H° may well serve as an unequivocal example of a law of H° (a conservation-law): that at every temporal position the same number of spatial positions is full.

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of a regularity (of a T-history), which concept, it turned out, is not precise, but somewhat vague (see the remarks regarding D21 in the previous section). Thus, it must be concluded that the great potential advantage of the intrinsic conception of lawhood – namely, that the laws of a history are already fully determined by the intrinsic constitution of the history if only the concept of law is precisely delineated – is not fully realizable, especially considering that additional vagueness it likely to sit comfortably – in the concept of law – on the ground-vagueness of the underlying concept of regularity. According to the extrinsic conception of lawhood, it is impossible to read the laws of a history in its intrinsic constitution; its intrinsic constitution does not determine its laws (not even if the concept of a law is entirely precise). According to the extrinsic conception of lawhood, laws in the secondary sense are true statements that have the following forms: For all T-histories H: if H is a candidate for being actual, then the state of affairs ST1 is intrinsically implied by H. ……… For all T-histories H: if H is a candidate for being actual, then the state of affairs STN is intrinsically implied by H. But, in the primary sense, laws are just the states of affairs that are named in the consequents of the general conditional statements that are laws in the secondary sense, that is (referring to the above list of statement-forms as a list of laws): just the states of affairs ST1, …, STN. Or in other words: laws, in the primary sense, are precisely the states of affairs that are intrinsically implied by every history which is a candidate for being actual. Three traits of this – now specified – primary concept of a law under the extrinsic conception of lawhood need to be noted, traits which contrast it rather significantly with the – previously considered – concept of a law under the intrinsic conception of lawhood. First, this concept is not yet the concept of a law of a history. This latter concept must be defined (under the extrinsic conception of lawhood) in the following way: A state of affairs ST is a law of a history H if, and only if, ST is a law [that is: ST is intrinsically implied by every history which is a candidate for being actual] and intrinsically implied by H. Accordingly, it may happen that not all laws are laws of a given history H, and even that no law is a law of H (which is equivalent to saying that there are no laws of H).

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Second, the concept of law now under consideration has nothing intrinsically to do with the concept of a regularity (see D21): as far as the present concept of law is concerned, even the state of affairs that is full at 1 – which, surely, is not a regularity (of any history) – could be a law. – Or perhaps it is illuminating to distinguish inner-historical and inter-historical regularities: The present concept of law has nothing intrinsically to do with the concept of an inner-historical regularity – hitherto employed as the concept of a regularity (simpliciter); but to the extent that a law, according to the above definition, is intrinsically implied by every history which is a candidate for being actual, a law could be said to be, according to that definition, an inter-historical regularity (while it may or may not be also an inner-historical regularity). Third, according to the concept of law now under consideration, every law, LW, is a negative selection criterion regarding actuality: Since all histories which are candidates for being actual intrinsically imply LW (LW being any law), only histories which intrinsically imply LW are candidates for being actual, or in other words: if a history does not intrinsically imply LW, it is not a candidate for being actual (and a fortiori that history is not actual). Now, the totality of laws is the conjunction of all laws, and it quite obviously follows that the totality of laws is itself a law (i.e., is intrinsically implied by every history which is a candidate for being actual). This means: A history H is a candidate for being actual only if H intrinsically implies the totality of laws [i.e., every law: every state of affairs that is intrinsically implied by every history which is a candidate for being actual].26 Moreover, it seems appropriate to postulate the following Principle of Sufficiency for Candidacy: A history H is a candidate for being actual if H intrinsically implies the totality of laws [i.e., every law: every state of affairs that is intrin-

26

Consider: If X intrinsically implies the totality of laws, then it intrinsically implies every law, since every law is intrinsically implied by the totality of laws; and if X intrinsically implies every law, then it intrinsically implies the totality of laws, since the totality of laws is the conjunction of all laws (which is itself a law).

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sically implied by every history which is a candidate for being actual].27 The non-relative distinction between actual and non-actual histories – implicit in these two principles – does not occur in the basic description of Model T (see Sect. 2). Nor does occur there the distinction between a history which is a candidate for being (non-relatively) actual and a history which is not such a candidate. In fact, the concept of non-relative actuality and the concept of candidacy for non-relative actuality – the latter concept being needed, under the extrinsic conception of lawhood, for defining the concept of a law – are not even implicit in the basic description of T, that is: they cannot be gathered from it via conceptual construction, and hence they are not intrinsic to T (see Sect. 2 for the form of argumentation just employed). Accordingly, even scrutinizing the complete intrinsic make of all T-histories – that is, which maximally composite momentary state each of them assigns to each temporal position – will not by itself reveal in any manner and degree which T-history is actual, or which one is a candidate for being actual. This must be so because not even the complete intrinsic make of all T-histories taken together determines, by itself, either actuality or non-actuality for any T-history, or even a tendency in this regard, or even a candidacy. A fortiori the complete intrinsic make of all T-histories taken together does not determine the T-laws or any T-history’s laws. (If we set aside the possibility that no T-history is a candidate for being actual, then there still are two extreme eventualities among the countless eventualities that the complete intrinsic make of all T-histories taken together does not suffice to exclude: If all T-histories are candidates for being actual, then the T-laws are precisely the states of affairs that are intrinsically implied by all T-histories; if only one T-history is a candidate for being actual, then the T-laws are precisely the states of affairs that are intrinsically implied by that history.) Both with regard to, (1), the direction of time in T – the T-distinction between past and future – and, (2), the specialness of the fleeting present in T, we found that one must look outside of T for an ontological justification of these aspects, if they are taken to be objective and therefore nonstipulatory aspects of T (see Sect. 3.2.2.1 and Sect. 3.3.2). And if, (3), the non-relative distinction between actual T-histories and non-actual ones and, (4), the distinction between T-histories that are candidates for being 27

The sufficient condition (the clause after the “if”) can also be put in the following way: “all laws are laws of H”.

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(non-relatively) actual and T-histories that are not, and consequently under the extrinsic conception of lawhood, (5), the distinction between states of affairs that are laws and states of affairs that are not, are also taken to be objective, non-stipulatory aspects of T, then these aspects, too, must find their ontological justification outside of T (as we have now seen). In fact, not only the first two but also the last three of the five aspects of Model T mentioned in the previous paragraph are in this book taken to be objective aspects of T.28 This manner of doing T-metaphysics is representationally motivated: it is motivated by the very purpose of T, namely, to be a metaphysical model of Reality; for Reality – according to widespread but by no means common or general philosophical opinion – objectively displays the real counterparts of the five above-mentioned T-aspects. Now, the ontological justification of those aspects of T – taken to be objective – can only come from the outside of Model T. And the analogy between model and original (now employed in the reverse direction: going from the model to the original) suggests that the ontological justification of the corresponding aspects of Reality – taken to be objective – must, analogously, come from the outside of Reality (that is, of Reality as we know it). 4.2

Uniformity, regularities, laws, and actuality

If T is to be a metaphysically faithful model of Reality, then H* – the history which turns out to be the (non-relatively) actual history of T, the one among all the histories of T which is actual – ought to be roughly in the middle between maximal uniformity and maximal diversiformity. Moreover, if T is to be a metaphysically faithful model of Reality, then all laws of H* under the extrinsic conception of lawhood ought to be laws of H* also under the intrinsic conception of lawhood and, therefore, regularities of H*. This – or more accurately speaking: what it implies about Reality – is astonishing, since it could certainly be otherwise. In contrast, it is not astonishing that every law under the extrinsic conception of lawhood is a law of H* (under that same conception); it cannot logically be otherwise, since H*, by being actual, is a candidate for being actual, and therefore every state of affairs that is intrinsically implied by every history which is a candidate for being actual – that is, every law – must certainly be intrinsically implied by H* – that is, be a law of H*. But if T is to be a metaphysi28

Regarding aspects (1) and (2) of T, this was already determined in Sect. 3.2.2.1 and Sect. 3.3.2, respectively.

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cally faithful model of Reality, does this also require that H* be the only history which is such that every law is a law of it?29 If so, then this would mean that the singling out for actuality (the selecting for being uniquely actual) from among all the histories is achieved solely by laws. Or are there – if T is to represent Reality metaphysically faithfully – other histories besides H* which are such that every law is a law of them? According to the Principle of Sufficiency for Candidacy and its companion principle (which is its converse; see Sect. 4.1.1), the answer “yes” to this question is tantamount to the claim that there are other histories besides H* which are candidates for being actual, and this generates the further question of how, if the claim is true, from the several candidates for being actual the one history is selected that is going to be actual (namely, H*). But whether – in faithfully representing the metaphysics of Reality – the laws alone determine the T-history that is going to be actual, or whether also other factors play a role in this (just chance?, just agency other than chance?, or chance and other agency together?), it rather seems that the selection of a single history for being actual is not yet its actualization. It is worth mentioning that there are no miracles in any history which is a candidate for being actual – at least if miracles are defined as counter-nomological occurrences.30 There can, of course, be miracles in histories which are not candidates for being actual; but such histories – and their miracles – are not actual, since they are not even candidates for being actual. (Note that under the intrinsic conception of lawhood there cannot be any miracles – counter-nomological occurrences – in any history.)

29

Since we are from now on sticking to the extrinsic conception of lawhood, we no longer need to add the corresponding explicit qualification. 30 If miracles are defined as occurrences that appear to be counter-nomological – given what we think are the laws – then miracles could, in principle, occur in histories which are candidates for being actual (and, as such, intrinsically imply all the laws). But considering our epistemically privileged vantage point outside of T (which is a rather manageable model), it is difficult to conceive how we (if we put our minds to the matter) could be mistaken about the laws for T and their consequences. Hence we can rest assured that, in the case of T, appearing to be counter-nomological will just coincide – in its true applications – with being counter-nomological. Thus, because there are no counter-nomological occurrences (miracles1) in any history which is a candidate for being actual, there also will be no occurrences that appear to be counter-nomological (miracles2) in any history which is a candidate for being actual.

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5

Actuality and other modalities for T

The previous chapter has finally brought actuality to the foreground, or rather: the non-relativized actuality of histories. It is time for a systematic treatment of the concepts of actuality, and of concepts that revolve, so to speak, around actuality. The following section of the present chapter is a preliminary to that treatment; it does not yet (primarily) concern actuality and non-actuality, but concerns, in effect, the truth and falsity of statements about T. 5.1

Temporal and historical relativization of statements about T

At places in Chapter 3 (in Sect. 3.3.1 and Sect. 3.5) the need was felt for a certain relativization of statements about (aspects of) T, a relativization different from temporal relativization. The following three statements say the same thing, though they vary in their degree of ontological explicitness: (A) is full. exemplifies Full. [Full, ]31 obtains.

The following three other statements say something else than the above; but amongst each other they do again say the same thing, with various degrees of ontological explicitness: (B) is full at 1. exemplifies Full at 1. [Full, ] obtains at 1.

Contrast now the statements listed under (A) and (B) with the following two statements: (C) is one step to the left of . There are 50 temporal positions. 31

That is: the momentary state [Full, ].

The first of the statements under (C) is true, the second false. And for the statements under (C) there is no need of any relativization (over and above the basic semantic relativization that consists in their being about T). But without appropriate relativization it is quite impossible to say truthfully – or even meaningfully – of any of the statements under (A) and (B) that it is true, or that it is false. Like the statements under (C), the statements under (A) are without any explicit relativization – which means that they are without any relativization that is expressed in the sentences themselves. But, in contrast to the (C)-statements, implicit (or tacitly understood) relativization is required to give the (A)-statements a truth-value, namely, (1) to some temporal position (for example, to the temporal position 1), and (2) to some history (for example, to the history H°; for the description of H°, see Sect. 4). The statements under (B), in turn, are with explicit relativization to temporal positions, but not to histories. To give them a truth-value, implicit relativization to some history (for example, H°) is sufficient (and necessary). Contrasting in different ways with the statements under (A), (B), and (C), the following further statements are with complete explicit relativization; they need no implicit relativization to give them a truth-value: (D) is full at 1 in H°. exemplifies Full at 1 in H°. [Full, ] obtains at 1 in H°.

We can conclude that there are among the T-statements, if taken to be true or false, – statements without explicit relativization: not to temporal positions and not to histories, and with implicit relativization to temporal positions and to histories (for example, the (A)-statements, if taken to be true or false); – statements with explicit relativization to temporal positions, but not to histories, and with implicit relativization to histories, but not to temporal positions (for example, the (B)-statements, if taken to be true or false); – statements without explicit relativization: not to temporal positions and not to histories, and without implicit relativization: not to temporal positions and not to histories (for example, the (C)-statements);

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– statements with explicit relativization to temporal positions and to histories, and without implicit relativization: not to temporal positions and not to histories (for example, the (D)-statements); These four instantiated kinds of T-statements correspond to four occurrent patterns of the relativization of statements – with respect to temporal positions and to histories – out of a total of 16 prima facie possible patterns of relativization: Explicit Z H + + + + + + + + + − + − + − + − − + − + − + − + − − − − − − − −

Implicit Z H + + + − − + − − + + + − − + − − + + + − − + − − + + + − − + − −

no. 1 no. 2 no. 3 no. 4 no. 5 no. 6 no. 7 no. 8 no. 9 no. 10 no. 11 no. 12 no. 13 no. 14 no. 15 no. 16

Of these 16 prima facie possible patterns, many are ultima facie impossible – due to logical reasons: If a statement is explicitly relative to X, then – logically – it is not implicitly relative to X; if a statement is implicitly relative to X, then – logically – it is not explicitly relative to X. These two logical restrictions exclude the patterns no. 1, 2, 3, 5, 6, 9, and 11 from occurring. In contrast, the patterns no. 4, 7, 13, and 16 are occurrent: the (D)statements display pattern no. 4, the (B)-statements pattern no. 7, the (A)statements pattern no. 13, and the (C)-statements pattern no. 16. This leaves unaccounted for – so far – the patterns no. 8, 10, 12, 14, and 15. Are they occurrent or not?

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Concerning pattern no. 8: Are there statements (about T) with explicit relativization to temporal positions, but not to histories, and without implicit relativization: not to temporal positions and not to histories? – If the following statement is counted as an example: “[The spatial position] is full or empty at [the temporal position] 1”, then this is reason enough to answer “yes”. But the example is not perfect, since the explicit relativization to temporal positions in the example is merely syntactical: “ is full or empty at Z” remains true whatever name of a temporal position is substituted for “Z”. But here is a better example: “1 is present at 1”. Statements of the form “1 [2, 3, …] is present at Z” are not false whatever name of a temporal position is substituted for “Z”; but neither are they true whatever name of a temporal position is substituted for “Z”. Concerning pattern no. 10: Are there statements with explicit relativization to histories, but not to temporal positions, and with implicit relativization to temporal positions, but not to histories? – Yes, the following statement is an example: “ is presently [in another word: now] full in H°”. Concerning pattern no. 12: Are there statements with explicit relativization to histories, but not to temporal positions, and without implicit relativization: not to temporal positions and not to histories? – Yes, the following statement is an example: “H° is actual in H°”. This statement is true, whereas the statement “H° is actual in H#” is false if “H#” designates a history that is not H°. Concerning pattern no. 14: Are there statements without explicit relativization: not to temporal positions and not to histories, and with implicit relativization to temporal positions, but not to histories? – Yes, the following statement is an example: “[The temporal position] 1 is present”. Concerning pattern no. 15: Are there statements without explicit relativization: not to temporal positions and not to histories, and with implicit relativization to histories, but not to temporal positions? – Yes, the following statement is an example: “H° is actual”. Now, in the statements “H° is actual”, “1 is present” and “ is presently full in H°” – if understood as they must be understood in order to illustrate the relativization patterns no. 15, 14, and 10 – the words “actual”, “present” and “presently” function, in addition to fulfilling their task of explicit qualification (in constituting statemental content), as indicators of the relativization that is required if those statements are to have a truth-value: to histories in the case of “actual”, to temporal positions in the case of “present” and “presently”. None of these words establishes the appropriate

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relativization by itself; it merely indicates which type of relativization must be supplied if the statement in which it occurs is to be true or false – and since that relativization is obviously not supplied explicitly (by the statement itself) it must be supplied implicitly, tacitly. The three just-mentioned statements are therefore, so to speak, explicitly implicitly relativized statements, or in other words: they are overtly indexical statements – in contrast to other statements with implicit relativization: the (A)-statements and the (B)-statements (see above), which are covertly indexical statements. “H° is actual”, “1 is present”, and “ is presently full in H°” – as overtly indexical statements – are closely related in meaning to certain statements without implicit relativization. To which statements without implicit relativization are they closely related in meaning, and what, precisely, is the nature of this close semantic relationship? In order to answer these two questions for all three statements, consider first the following sequence: “H° is actual”, implicity relativized to [history] H°, means the same thing as “H° is actual in H°”; hence it is true. “H° is actual”, implicity relativized to [history] H#, means the same thing as “H° is actual in H#”; hence it is false if H# is not H°; otherwise, it is true. “1 is present”, implicity relativized to [temporal position] 1, means the same thing as “1 is present at 1”; hence it is true. “1 is present”, implicity relativized to [temporal position] Z#, means the same thing as “1 is present at Z#”; hence it is false if Z# is not 1; otherwise, it is true. “ is presently full in H°”, implicitly relativized to 2, means the same thing as “ is full at 2 in H°”; hence it is true (see the description of H° in Sect. 4). “ is presently full in H°”, implicitly relativized to [temporal position] Z#, means the same thing as “ is full at Z# in H°”; hence it is false if Z# is not 2; otherwise, it is true (for checking this, see the description of H° in Sect. 4). Now, the statements “H° is actual”, “1 is present”, and “ is presently full in H°” can also be understood in a way that is significantly different from the way that has just been described (in the above sequence),

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namely, they can be understood in such a way as to be quite without implicit relativization. In order to distinguish in print this other way of understanding those statements from the previously described way of understanding them, I employ “H° is actualI”, “1 is presentI”, and “ is presentlyI full in H°” for indicating that they are meant in the previously described way (hence with implicit relativization), whereas I employ “H° is actual*”, “1 is present*”, and “ is presently* full in H°” for indicating that they are meant in that other way (hence without implicit relativization). Since the statements “H° is actual*” and “1 is present*” are not only without implicit relativization but also without explicit relativization, they fit the pattern no. 16, just like the (C)-statements. But note how different they are in other respects from the (C)-statements. The truth of the (C)statement “ is one step to the left of ” follows from the intrinsic nature of Model T, just like the falsity of the (C)-statement “There are 50 temporal positions”. But the intrinsic nature of Model T is certainly not sufficient for determining a truth-value for “H° is actual*” or “1 is present*” (stably in the first case, mutably in the second). In turn, since the statement “ is presently* full in H°” is without implicit relativization, and with explicit relativization only to a history, not to a temporal position, it fits pattern no. 12, just like the statement “H° is actual in H°”. But note how different it is in other respects from this latter statement. The truth of “H° is actual in H°” follows from the intrinsic nature of Model T; in fact, one might say that this truth is a mere matter of logic. But neither logic alone nor the intrinsic nature of T is sufficient for determining a truth-value for “ is presently* full in H°”. However, the intrinsic nature of T is entirely sufficient for determining a truth-value for each of the statements “H° is actualI”, “1 is presentI”, and “ is presentlyI full in H°”, once the appropriate points of relativization implicitly required by each of these statements have been provided. Those statements are, in being about T, just about the intrinsic nature of T (though they are, in themselves, not fully determinate about it). In contrast, “H° is actual*”, “1 is present*”, and “ is presently* full in H°” are, indeed, also about T, but not just about the intrinsic nature of T. If mere stipulation is out of the question and those statements are to be understood objectively, then something that affects T from the outside is necessary for determining their truth or falsity. With regard to Reality, many philosophers feel unable to make sense of the real-life analogues of “H° is ac-

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tual*”, “1 is present*”, and “ is presently* full in H°”32 as objective statements (which, qua objective statements, can neither be true just by stipulation, nor false just by stipulation). Perhaps the simple paradigm of Model T will prove helpful for opening up new perspectives in this regard. 5.1.1

The varieties of historical relativization

Explicit historical relativization is achieved by inserting the phrase “in H#” into a fitting place in a statement which is amenable to historical relativization (where “H#” schematically represents a name of some complete history of T) – but only if the resulting statement is not logically equivalent to the original statement, but comes closer than the original statement to being true or false just in virtue of how it is as a statement – according to its explicitly stated meaning – about Model T. Implicit historical relativization is achieved, for a statement which is amenable to historical relativization, by relating that statement semantically to one particular history H, not expressis verbis, but still in the manner which is exhibited by the phrase “in H#” when it fulfils its function in the corresponding (achieved) explicit historical relativization (with “H#” standing for a name of H). Is there any other type of historical relativization? – There is none if a type of historical relativization is being looked for which is neither explicit nor implicit. But a certain special type of explicit historical relativization deserves mention. In the previous section, two alternative meanings of “actual” in “H° is actual” were distinguished, which meanings were stipulated to be expressed by, respectively, “actualI” and “actual*”. The two meanings are rather different: whereas “H° is actualI” is an overtly indexical statement – an explicitly implicitly relativized statement – “H° is actual*” is a statement without any relativization at all. As a consequence of the difference in meaning between “actualI” and “actual*”, the definite descriptions “the actualI history” and “the actual* history” must also have a different meaning. If occurring in a statement with implicit historical relativization, the first-mentioned definite description refers to the history the statement is being implicitly relativized to, whichever history that is. In its place of occurrence, it cannot fail to have a uniquely fitting referent, 32

Consider as such analogues: “World w° is [absolutely] actual*”, “Time-point t1 is [absolutely] present*”, and “Space-point is [absolutely] presently* full in w°”.

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though that referent is not stable: if the history of implicit relativization (for the statement) is no longer H, but H´, then also the referent of the designator “the actualI history”, in its place of occurrence, is no longer H, but H´. Note that the inserting of “in the actualI history” into a fitting place in a statement amenable to historical relativization achieves no explicit historical relativization, for it merely produces a statement which is logically equivalent to the original statement. It is quite otherwise with “the actual* history”. There is no guarantee that this definite description has a referent that corresponds to its meaning. Might there not be several histories which are actual*, or none? So far, there is no reason to answer “no” to this question. But if “the actual* history” has a uniquely fitting referent (a referent which corresponds to its meaning, and not some alien referent that is provided by sheer stipulation), then this designator can certainly be used for explicit historical relativization. By inserting the phrase “in the actual* history” – or in short: “in actuality*”33, or putting it even more briefly: “in @*” – into a fitting place in a statement which is amenable to historical relativization, a statement results which is certainly not logically equivalent to the original statement, since the resulting statement comes closer than the original statement to being true or false just in virtue of how it is according to its explicitly stated meaning about T. The explicit historical relativization that is achievable via the phrase “in the actual* history” is, however, rather different from the explicit historical relativization that is achievable via the phrase “in H°” (for example). “H°” – it has already been introduced as the name of the T-history described in Sect. 4 – names that history under all circumstances34 (if only the language in itself stays as it is in itself): “H°” is a rigid designator. In contrast, whatever history is named by “the actual* history” (if that definite description names a history, and not some entity that is alien to its meaning), it does not seem to be named by that designator under all circum33

Here “actuality*” is used as a name of a certain history. The word can also be used – quite differently – as a name of the property that is expressed by “actual*”. Actuality* qua a history is a paradigmatic instance of actuality* qua a property. The typographically unmodified word “actuality” can serve in the very same semantic ways as the word “actuality*”, as long as it is understood that the basic nominal – the (typographically unmodified) word “actual” – is to be taken in its non-indexical and non-relative sense. The word “actual” is meant to be taken in that sense in Sect. 4.1.1 and Sect. 4.2, and also earlier in the book, and hence “actuality” is to be understood there accordingly. 34 This entails: in all linguistic contexts.

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stances (might not the actual* history have been another history?): “the actual* history” does not seem to be a rigid designator. Anticipating, I add: it is indeed not a rigid designator. Consider moreover: On the one hand, the statement “ is full at 1” (one of the (B)-statements in the previous section) is explicitly relativized to a history by inserting “in H°” into it, yielding the statement “ is full at 1 in H°”; this statement is true – hence true or false – just in virtue of how it is according to it explicitly stated meaning about T. On the other hand, the statement “ is full at 1” is explicitly relativized to a history also by inserting “in the actual* history” into it, yielding the statement “ is full at 1 in the actual* history”. Assuming that there is exactly one actual* history, this statement is also true or false just in virtue of how it is according to its explicitly stated meaning about T. But, in contrast to “ is full at 1 in H°”, “ is full at 1 in the actual* history” addresses aspects of T that are not intrinsic to it (which is the reason why, so far, we are quite unable to say whether “ is full at 1 in the actual* history” is true, or whether it is false). Two additional points bring this section to a close: First, both “actualI” and “actual*” must, as non-relational predicates, be distinguished from the relational predicate “actual in”. It is interesting that “X is actualI” is logically equivalent to “X is actual in the actualI history”, whereas “X is actual*” is not logically equivalent to “X is actual in the actual* history”. Assuming, however, that there is exactly one actual* history, it seems indeed to be true (metaphysically, not already logically) that, for all T-entities X, X is actual* if, and only if, X is actual in the actual* history. Second, just as one can distinguish two types of explicit historical relativization, so one can also distinguish two types of implicit historical relativization. It is best to demonstrate this with regard to two examplestatements, both of them requiring (for being true or false) historical relativization only, one of the statements being covertly indexical, the other overtly: Statements available for historical relativization: “ is full at 1”, “H° is actualI”. Explicit historical relativization no. 1: “ is full at 1 in H°”, “H° is actual in H°”. Explicit historical relativization no. 2: “ is full at 1 in @*”, “H° is actual in @*”.

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Implicit historical relativization no. 1: “ is full at 1” [in H°], “H° is actualI” [in H°]. Implicit historical relativization no. 2: “ is full at 1” [in @*], “H° is actualI” [in @*]. Suppose now that the actual* history is H°, in other words: @* = H° (which, as a matter of fact, is not true, but it is here supposed for the sake of the argument). Does this entail that “H° is actualI”, implicitly relativized to H°, means the same thing as “H° is actualI”, implicitly relativized to @*? – It does not, for the following reason: “H° is actualI”, implicitly relativized to H°, means the same thing as “H° is actual in H°”; and “H° is actualI”, implicitly relativized to @*, means the same thing as “H° is actual in @*”; but “H° is actual in H°” and “H° is actual in @*” do not mean the same thing. They do not mean the same thing – unless, of course, “@*” and “H°” mean the same thing. But we have already seen a reason for denying that “@*” and “H°” mean the same thing (though we have not yet seen a reason for denying that “@* = H°” is true): the names “@*” and “H°” are very different in character; for “H°” is a rigid designator, while “@*” is not.35 Thus, “H° is actualI” [in H°] and “H° is actualI” [in @*] do not say the same thing, even if “@* = H°” is supposed to be true (which we have supposed to be true for the sake of the argument). Still, it obviously is rather easy to think that they say the same thing, if “@* = H°” is supposed to be true. At this point, note – with respect to Reality, not to T,36 adapting the meanings of the involved predicates accordingly: transferring them from T to Reality – that it is still necessary to distinguish semantically between “w* is actualI” [in w*] and “w* is actualI” [in the actual* world] even if “w*” is stipulated – defined – to be a rigid designator for the possible (Reality-)world that is in fact designated by the definite description “the actual* world” (the uniqueness condition is taken to be fulfilled for this definite description, which itself is taken to be a non-rigid designator); for that definitional stipulation does not abolish the difference in meaning between “in w*” and “in the actual* world”.

35

The same reason can, of course, be adduced for denying that “@*” and “H##” mean the same thing – no matter which T-history is designated under all circumstances (i.e., rigidly) by no matter which rigid designator represented by “H##”. 36 But keep the matter in mind in order to distinguish semantically between “H*” and “@*” later on.

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5.2

The plurality of actuality-predicates, basic and defined

In Sects. 5.1 and 5.1.1, it has become apparent that there are many actuality-predicates, not just one. According to what is said there, we are facing at least the following actuality-predicates: 1 dyadic actuality-predicate: X is actual in Y with the following meaning postulates formally characterizing it: (MP1) For all X and Y: if X is actual in Y, then Y is a history; (MP2) For all histories X: X is actual in X; (MP3) For all histories X and Y: if X ≠ Y, then X is not actual in Y. 210000 + 6 monadic actuality-predicates: X is actual the meaning of which can be either of the two meanings that are uniformly and unequivocally expressed – respectively – by the following two predicates: X is actualI which is an overtly indexical predicate (if truly or falsely applied, then with implicit relativization to histories only), and X is actual* tivized predicate.

which is neither an indexical nor a rela-

X is actual in H## where, for each T-history, “H##” stands for the standard rigid designator of it (it is indifferent how that designator is formed). Clearly, “X is actual in H##” is a predicateschema which has 210000 instances (because there are 210000 complete T-histories). X is actual in @* which predicate applies to something (i.e., is not empty) only if there is exactly one T-history which is actual*. For, if this condition is not fulfilled, then the term “@*” – “the actual* history” – designates by sheer stipulation something that is alien to T – the moon, say – and then, according to the above meaning postulate, (MP1), “X is actual in @*” can only be false, no matter which X we are looking at.

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which predicate is obviously difX is actual in the actualI history ferent from the predicate “X is actualI”, but nevertheless turns out to be logical equivalent to the latter. X is actual in H* where “H*” is the rigid designator that is stipulated to designate (under all circumstances) the entity that is designated by “@*” (i.e., by “the actual* history”). If “@*” were itself a rigid designator (which, in fact, it is not), then – and only then – “X is actual in H*” would be identical in meaning to “X is actual in @*”. Are there other actuality-predicates? Obviously, the above dyadic actuality-predicate “X is actual in Y” can be used to generate an indefinite number of additional monadic actuality-predicates by putting some name of a history – the name not being a standard rigid designator of it – in the place of the variable “Y”. In contrast, binding “Y” by a quantifier which is, explicitly or implicitly, restricted to T-histories and ranges over at least two of them does not generate a further actuality-predicate. For example, the two obvious ways of binding “Y” in “X is actual in Y” by a quantifier which is restricted to T-histories and ranges over at least two of them do not produce further actuality-predicates: neither “For all histories Y: X is actual in Y” nor “For some history Y: X is actual in Y” is an actualitypredicate. Rather, the first expression (with one free variable, hence a monadic predicate) is a necessity-predicate (or rather – to be precise – it is on the way to such a predicate; see Sect. 5.3); and the second expression (also with one free variable, hence a monadic predicate) is a possibilitypredicate (see also Sect. 5.3). Several prominent modal predicates that are not actuality-predicates (but are logically related to actuality-predicates) will be treated later in this chapter (in Sects. 5.3 and 5.3.1). For the time being, I repeat the question: Are there other actuality-predicates? All of the actuality-predicates listed above can be truly or falsely applied without explicit or implicit temporal relativization; such is their meaning (as is apparent from the previous two sections). The instantiating statements that are formed by using any one of them as predicate are true or false without explicit or implicit appeal to temporal positions (it being understood that the names that are used in forming instantiating statements – statements that assert instantiations of predicates – do not need historical, temporal, or any other relativization for determining their referents). Yet,

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why should there not also be actuality-predicates with temporal relativization (when truly or falsely applied)? In fact, there seems to be a need for such predicates. The above first list of actuality-predicates can, therefore, be complemented by a second list: 1 triadic actuality-predicate: X is actual at Y in U with the following meaning postulate formally characterizing it: (MP4) For all X, Y, and U: if X is actual at Y in U, then Y is a temporal position and U a history. An example of the use of this predicate is the statement “[Full, ] is actual at 1 in H°”. (The relativization pattern of this – true – statement is no. 4 according to the list in Sect. 5.1, that is: + + −−, the same pattern as that of the (D)-statements in Sect. 5.1.) Out of the multitude of temporally relativized dyadic actuality-predicates, I list the following three: with the following meaning postulate X is actualI at Y formally characterizing it: (MP5) For all X and Y: if X is actualI at Y, then Y is a temporal position. An example of the use of this predicate is the statement “[Full, ] is actualI at 1”, which is true relative to H°, but false relative to other histories. (The relativization pattern of that statement is no. 7 according to the list in Sect. 5.1, that is: + − − +, the same pattern as that of the (B)-statements in Sect. 5.1.) with the following meaning postulate X is actualI in Y formally characterizing it: (MP6) For all X and Y: if X is actualI in Y, then Y is a history. If the indicator of indexicality at the end of “actual” is omitted, then this predicate (which forms instantiating statements with the relativization pattern no. 10: − + + −, the same pattern as that of the statement “ is presentlyI full in H°”) becomes indistinguishable in typographical appearance from the dyadic predicate “X is actual in Y” in the first list (which forms instantiating statements with the relativization pattern no. 12: − + − −). Yet, the meanings of the two predicates are very different: “[Full, ] is actualI in H°” is a meaningful statement, it means as much as “[Full, ] is presentlyI actual in H°” (“actual” functioning like “full”); but “[Full, ] is actual in H°” – where the predicate “X is actual

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in Y” is understood as originally introduced at the beginning of this section: without (explicit or implicit) temporal relativization (when truly or falsely applied) – is not even meaningful. which means, by definition, as much as X is actualT in Y “For some temporal position Z: X is actual at Z in Y”. If the indicator of temporal relativization at the end of “actual” is dropped, then also “X is actualT in Y” becomes indistinguishable in typographical appearance from “X is actual in Y”. The semantic difference remains: in contrast to the statement “[Full, ] is actual in H°”, the statement “[Full, ] is actualT in H°” is meaningful – and in fact true. (The relativization pattern of this latter example-statement is neither − + − − nor − + + −; it is, as a matter of fact, + + − −; for that statement does have a temporal relativization, but not an implicit one: an explicit temporal relativization that is merely made invisible by means of definitional abbreviation still remains an explicit temporal relativization.) Out of the enormous number of temporally relativized monadic actualitypredicates, I list the following two: which forms statements the relativizaX is actualII tion pattern of which is no.13: − − + +, the same pattern as that of the (A)-statements in Sect. 5.1. (Note, for contrast, that the relativization pattern of the statements formed by “X is actualI” is this: − − − +.) An example of the use of this predicate is the statement “[Full, ] is actualII” (which is true if implicitly relativized to the history H° and the temporal position 1, but will often – though not always – be false if implicitly relativized in a different way to a history and temporal position). X is actualIT which means, by definition, as much as “For some temporal position Z: X is actualI at Z”. The relativization pattern of the instantiating statements formed by “X is actualIT” is, therefore, the same as that of the instantiating statements formed by “X is actualI at Y”: + − − +. Finally, there is a monadic actuality-predicate that cannot have a place in the above second list – because it is not temporally relativized. It, there-

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fore, fits into the first list, further above, which is meant to list actualitypredicates without temporal relativization. But without the consideration of temporally relativized actuality-predicates in constructing the second list, the predicate in point would not have come into our ken (it can safely be said); for this predicate stands to “X is actualII” as “X is actual*” stands to “X is actualI”. It is, therefore, fitting to write it in the following form: “X is actual**”. The next section will focus on the semantic description of this predicate. In this section, another matter remains to be treated: the matter of definable actuality-predicates which do not carry their being actualitypredicates on their sleeves. The actuality-predicates that I am talking about do not involve the nominal “actual”; instead, they involve the verbs “exemplify” and “obtain”. Consider again the (A)-statements, (B)-statements, and (D)-statements in Sect. 5.1 – not, however, the first statement in each group, only the remaining two: exemplifies Full. ((A)) [Full, ] obtains. exemplifies Full at 1. ((B)) [Full, ] obtains at 1. exemplifies Full at 1 in H°. ((D)) [Full, ] obtains at 1 in H°. In order to make the intended meaning of the predicates that occur in these statements more visible, the predicates can be appropriately indexed, which, if done, leads to the following result: exemplifiesII Full. exemplifiesI Full at 1. exemplifies Full at 1 in H°.

[Full, ] obtainsII. [Full, ] obtainsI at 1. [Full, ] obtains at 1 in H°.

Now, the exemplification of a property Φ by an entity X is just the obtaining of their elementary composition [Φ, X],37 which is a certain state of affairs. Hence, the above statements on the right can be taken to define, line by line, what is meant by the above statements on the left. In turn, to obtain is what it is to be actual for states of affairs. All momentary states belong to the category of state of affairs, and hence also [Full, ]. Therefore: the above-listed statements of obtaining, and mediately also the statements of exemplification they define, can each be

37

Regarding the concept of elementary composition, see Sect. 3.3.1.

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taken to be defined – line by line – by an actuality-statement which uses the appropriate actuality-predicate: [Full, ] obtainsII [Full, ] obtainsI at 1 [Full, ] obtains at 1 in H°

=Def [Full, ] is actualII. =Def [Full, ] is actualI at 1. =Def [Full, ] is actual at 1 in H°.

In virtue of the correspondence between “X is actual**” and “X is actualII” noted above, the first line in this series of definitions suggests that “[Full, ] obtains**” – and hence also “ exemplifies** Full” – should be taken to be merely a different way of saying what is basically said by “[Full, ] is actual**” (just as “[Full, ] obtainsII” – and hence also “ exemplifiesII Full” – is taken to be merely a different way of saying what is basically said by “[Full, ] is actualII”). 5.2.1

Actuality – in two ways non-relativized

The instantiating statements formed by “X is actual**” have the same relativization pattern as the instantiating statements formed by “X is actual*”: – – – – (which is also the relativization pattern of the (C)-statements in Sect. 5.1). Both predicates are without relativization – that is to say: without relativization to temporal positions or to histories, but of course not without any relativization at all. For one thing, they are both about Model T, they serve to describe T, to make distinctions regarding items of T. But in being used relative to T – about T – they apply truly or falsely just in virtue of how they are – according to their explicitly stated meaning – about T: without relativization to temporal positions or to histories. At the same time, the intrinsic nature of T (which can be gathered from the basic description of T) is not sufficient to determine whether an instantiating statement with the one or the other of those two predicates is true or false. Therefore, either both predicates38 apply truly or falsely in virtue of stipulation (or the logical consequences of stipulation) – which would mean that they are not objective predicates, that they, though being about T, capture nothing objective about T – or both predicates apply truly or falsely in virtue of objectively given aspects which are extrinsic to T, come to T from 38

It seems arbitrary to let one of the predicates differ from the other in the two (alternative) features presented in the sentence this footnote is attached to.

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the outside.39 I opt for the second alternative – in view of the fact that T is (is to be) a metaphysical model of Reality, which suggests – no more than suggests – the second alternative. There is, therefore, (to be) an objective justification for the predicates “X is actual*” and “X is actual**” applying truly or falsely. The last paragraph described what the two predicates have in common. In typographical appearance, they are almost the same, and “X is actual” is not infrequently used to express both the meaning of “X is actual*” and the meaning of “X is actual**”. But these meanings are very different nonetheless. For example, “H° is actual*” is a meaningful statement, but “[Full, ] is actual*” is not; in contrast, “[Full, ] is actual**” is a meaningful statement, but “H° is actual**” is not. Moreover, the truthvalue of “H° is actual*” is stable, whereas the truth-value of “[Full, ] is actual**” is mutable – just as mutable as the truth-value of “1 is present*”, which is yet another statement with the relativization pattern no. 16: – – – – (see Sect. 5.1). Obviously, stable and mutable are here employed as temporal categories – involving a reference to time; but the time they refer to cannot be the timeline of T, the sequence of T’s temporal positions. The time they refer to must, therefore, be a time that is “over and above” the timeline of T, though not unrelated to the latter. In order to avoid confusion, it would be best to call that time “time*”, and to call stable and mutable, as they have just now been employed, “temporal* categories”. The correlated truth-value mutability of instantiating statements of the predicates “X is actual**” and “X is present*” suggests that the predicates may be logically equivalent. And, in fact, there is no reason to abstain from assuming their logical equivalence. There certainly is a sense of “actual” in which it connotes “present”, and vice versa: a sense of “present” in which it connotes “actual”. Moreover, it is advantageous to let the predicates “X is actual**” and “X is present*” illuminate each other via logical equivalence. In this spirit, presentness* is taken to entail actuality**, and actuality** to entail presentness*; indeed, actuality* is taken to be presentness**, presentness* to be actuality**. Being identical with presentness*, actuality** cannot apply (truly or falsely) to intrinsically temporally extended items of Model T, as we have already noticed with regard to a special group of temporally extended Titems: T-histories. An item (entity) of T is intrinsically temporally ex39

We do know that Model T has an outside; we do not know whether Reality – Reality as we know it – has an outside (cf. Sect. 2).

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tended if, and only if, a composite temporal configuration of T (see Sect. 3.2.2) is intrinsic to it. In this sense, the following T-entities are not intrinsically temporally extended (i.e., do not intrinsically cover more than one temporal position); they are, therefore, not in principle out of the question for filling the role of a (ontic predication-)subject of actuality**: The temporal positions (1, …, 100; see Sect. 2). The spatial configurations (single spatial positions and configurations of several spatial positions, in particular: the space of T; see Sect. 3.2.1; spatial configurations are everlasting continuants and, therefore, also continuants simpliciter). The fillings of temporal positions (in other words: the maximally composite momentary states; see Sect. 2 and Sect. 3.3.1). The momentary states (mainly, the atomic momentary states – for example, [Full, ], i.e., that is full – and the composite momentary states, in particular: the maximally composite momentary states; see Sect. 3.3.1). The basic spatial tropes (for example, the [particular] fullness of , or the fullness of ; see Sect. 3.5; basic spatial tropes are continuants relative to an appropriate history, not simpliciter). The basic spatiotemporal tropes (for example, the [particular] fullness of at 1, or the fullness of at 2; see Sect. 3.5). The momentary events (for example, the event the event which consists merely in this: the momentary state [Full, ] being allotted to 2; see Sect. 3.5; however, this example is a fairly untypical momentary event: any event – rare or common (see Sect. 3.4) – with a singleton domain is a momentary event). The states of affairs which intrinsically correspond to the basic spatiotemporal tropes (for example, the state of affairs that is full at 2, which intrinsically corresponds to the fullness of at 2; see Sect. 3.5). The states of affairs which intrinsically correspond to the momentary events (if E´ is a momentary event with the domain {Z´} and the value-set {ST´} – Z´ being a temporal position and ST´ a momentary state – then the state of affairs that intrinsically corresponds to E´ is the state of affairs that ST´ obtains at Z´). The time-thematic and time-dependent states of affairs that intrinsically correspond to the two groups of states of affairs just listed in the following way: that P is full at Z ≅ that P is now, at Z, full (see

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the beginning of Sect. 3.3.2); that S´ obtains at Z´ ≅ that S´ obtains now, at Z´. The higher continuants (the object c° described in Sect. 4, in the context of describing the history H°, can serve as an example; c° is a continuant relative to H°, not simpliciter). Some of the listed entities are not only without intrinsic temporal extendedness, but also without a temporal (inner) dimension; those entities are: spatial configurations, momentary states (including the fillings of temporal positions), basic spatial tropes, and higher continuants. The remaining entities in the list do have a temporal dimension, though they are without intrinsic temporal extendedness. All of the listed entities can (in principle) be subjects of actuality**. But which of these entities are the primary potential recipients of actuality**, such that in virtue of their being actual**, if they are actual**, every other entity that is actual** is actual**? – Answer: The primary potential recipients of actuality** are the maximal momentary events, where a maximal momentary event is a momentary event that allots a maximally composite momentary state to the single temporal position in its (singleton) domain (see Sect. 3.6). – And how – in which manner – can these entities be in fact the primary recipients of actuality**? Imagine the (abstract) space of maximal momentary events as a plane which is in the net of a (kind of) Cartesian system of coordinates: the xaxis is formed by the sequence of the 100 temporal positions of T, beginning with 1, ending with 100; the y-axis is formed by the 2100 maximally composite momentary states (the 2100 fillings of temporal positions) of T put in a sequence, beginning (say) with the momentary state in which all (of the 100) spatial positions (of T) are empty, continuing then, firstly, with the 100 momentary states (put in some order) in which all spatial positions except one are empty; continuing, secondly, with the 9900 momentary states (put in some order) in which all spatial positions except two are empty, and so on, ending finally with the momentary state in which all spatial positions of T are full. Clearly, each ordered pair of the described coordinates (as usual, the x-coordinate is in the first position, the ycoordinate in the second) codifies a (unique) maximal momentary event; moreover, each maximal momentary event is (uniquely) codified by an ordered pair of the described coordinates. Imagine each maximal momentary event as a small square (each of the same size) in the finite rectangle (i.e., the two-dimensional rectangular ma-

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trix) that is projected by the two finite axes of the coordinate-system. The coordinates of any maximal momentary event E are determined by going – strictly parallel to the y-axis – from the square that represents E towards the x-axis, and by going – strictly parallel to the x-axis – from that same square towards the y-axis. When doing so, the part of the x-axis that one finally hits upon is (i.e., represents) the x-coordinate of E: a certain temporal position; and the part of the y-axis that one finally hits upon is the ycoordinate of E: a certain maximally composite momentary state. Imagine now the tip of a finger going from one little square to another in the rectangle that comprises, as little squares, all the maximal momentary events (of T), never touching more than one square at a time and never touching a square twice over. This is the fingertip of actuality** (or presentness*) playing on the keyboard of its instrument. The maximal momentary event that is touched by the fingertip becomes actual** (present*), and ceases to be actual** (present*) as soon as it is no longer touched by it. Together with a maximal momentary event E that becomes actual**, the following items become actual**, too – but secondarily, whereas E is the item that becomes actual** primarily: - the temporal position Z that is intrinsic to E (the first coordinate of E); - the maximally composite momentary state ST that is intrinsic to E (the second coordinate of E), and therefore also the space of T and – as spatial parts of T – all spatial configurations of T (considering that the space of T is intrinsic to ST, just as it is intrinsic to every other maximally composite momentary state); - every momentary state that is intrinsically implied by ST, and every basic spatial trope that is intrinsically implied by ST (such a trope intrinsically corresponds to a certain atomic momentary state intrinsically implied by ST); - every basic spatiotemporal trope that is intrinsically implied by E, every momentary event that is intrinsically implied by E, and every state of affairs that corresponds intrinsically (in the ways pointed out above) to an intrinsically E-implied basic spatiotemporal trope or an intrinsically E-implied momentary event. As far as can be seen at present, the fingertip of actuality** is free to go anywhere on its keyboard. Perhaps this is not true. In any case, what it actually does is this (and now we are no longer just imagining):

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Act1 It first touches a (single) square of which the temporal (i.e., first) coordinate is 1, then a square of which the temporal coordinate is 2, then a square of which the temporal coordinate is 3, and so on – the order of the touches precisely following the order of the T-temporal positions – until at last it touches a square the temporal coordinate of which is 100. In what manner this consecutive touching can bring about a process of actualization** has already been spelt out above; in that manner, we now suppose, it actually brings about a process of actualization**. Among many other things that the action of the fingertip of actuality** does for Model T, what it does for it in the first place is this: to bring the timeline of T to life (so to speak). Considered in itself, that timeline is just the sequence of the T-temporal positions in an intrinsic order that is mathematically represented by the natural order of the natural numbers from 1 to 100. Considered in itself, the timeline of T has little temporal meaning; but the action of the fingertip of actuality** gives it that full temporal meaning that it can have for Model T: For one thing, the action of the fingertip of actuality** provides the timeline of T with a uniform non-stipulatory, objective direction, and Model T, therefore, with an objective direction of time: the course of actualization** is always from the numerically smaller temporal position to the next numerically greater temporal position. This objectively justifies calling any temporal position which is (so and so much) numerically smaller than a given temporal position earlier than (and so and so much earlier than) that latter position, and any temporal position which is (so and so much) numerically greater than a given temporal position later than (and so and so much later than) that latter position.40 In consequence, there is an objective justification also for calling the temporal positions that are numerically greater than a given temporal position future with respect to it, and the temporal positions that are numerically smaller than a given temporal position past with respect to it. For another thing, the action of the fingertip of actuality** gives truth and substance to the idea that every temporal position has at some time – which is not the temporal position itself – more of an objective right to be called “present” (or “the present moment”) than any other temporal posi40

To speak in the way described has been our practice from the beginning, but an objective justification for it, though announced earlier, has only been provided just now.

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tion. When the fingertip of actuality** rests on the chosen maximal momentary event, of which a certain temporal position Z is the temporal location, then – at this T-transcending time which is obliquely referred to by “when” and “then” – Z has more of an objective right to be called “present” than any other temporal position and is indeed qualitatively special (namely, present*, actual**). But of course this is only transiently so: because every temporal position Z´ is distinguished in this same manner as soon as the process of actualization** (or, if you will, presentation*) that is brought about by the action of the fingertip of actuality** reaches Z´ (that is, reaches it secondarily, for primarily the process of actualization** reaches the momentarily chosen maximal momentary event of which Z´ is the intrinsic temporal location). The action of the fingertip of actuality**, or rather: the process of actualization** it brings about, corresponds to what one is wont to call, in the context of Reality, the flow of time (the flow of the present would, in fact, be a more appropriate designation), to which flow – which is notoriously difficult to describe consistently and non-trivially – objective existence is denied by many. It seems, however, that without an objective flow of time, time would not have a non-stipulatory, objective direction. Note that the presumable objective fact of (there being a process of) entropic increase does not first determine an objective direction of time, but presupposes an objective direction of time – one just as objective as the fact itself – to be already given; this much is implicit in the very notion of increase (as a process), which per se is increase in the course – and hence: in the direction – of time. Moreover, without an objective flow of time, full reality would have to be denied to the phenomenon of transitoriness – a phenomenon of fundamental importance to human beings in their experience of the world. It is no exaggeration that the deeply felt experience of transitoriness is one of the main motivations for cultural creative activity that is not primarily dedicated to economic purposes: from the construction of the Pyramids to the writing of elegiac poems. What has to be considered next is the relationship between actuality* and actuality**, for they are certainly not unrelated properties. Their relationship can be interpreted in various ways, depending on whether actuality* is considered to be reducible to actuality**, or not. Independently of the question of reducibility, the following statement – the Statement of Linkage – captures the basic fact of the relationship between actuality* and actuality**:

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Act2 For all maximal momentary events E: E is actual* if, and only if, E is once actual**. The word “once” in this statement does, of course, not refer to the line of the 100 temporal positions of T (it does not mean: at a temporal position in that line), but refers to the sequence of the 100 touches of the fingertip of actuality** (it means: on the occasion of a touch in that sequence). Whether actuality* is treated as reducible to actuality** or not, there are no other maximal momentary events that are actual* than, exactly, those which are required to be actual* on the basis of the Statement of Linkage and of the above description of the action of the fingertip of actuality** (Act1, in the main): 100 maximal momentary events, one such event for each temporal position (the temporal position being the single item in the event’s domain). Now, these 100 actual* maximal momentary events – since, taken together, they cover every temporal position – combine to make up one actual* history: the actual* history, i.e., @*, according to the following analytically true principle: Act3 For all (complete) histories H: H is actual* if, and only if, every event that is a momentary phase of H is actual*. The concept of momentary phase, used in Act3, is defined as follows: For all events E´: D24 E is a momentary phase of E´ =Def E is a momentary event, the domain of E is included in the domain of E´, and the momentary state that E assigns to the single temporal position in its domain is identical to the momentary state that E´ assigns to that same temporal position. Though “@*” – as we have now seen – does have a unique referent that fits its meaning, it is entirely undetermined so far which history – to be rigidly designated by “H*” – is referred to by “@*”. Reducing this huge indeterminateness is a matter for the next chapter (Chapter 6) – in preparation of which many other things remain to be considered in the present chapter. One of these is the question whether actuality* is reducible to actuality** or not. If the meaning of “actual*” – in particular, in its application to histories and maximal momentary events – is regarded as being not reducible to

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the meaning of “actual**”, then the actual* history can be considered to be actual*, and to be the only history which is actual*, on grounds that have nothing to do with the action of the fingertip of actuality**. Yes, it remains true then that the fingertip of actuality** follows faithfully the course of the actual* history: the sequence of actual* maximal momentary events. But its doing so can, then, be regarded as doing nothing at all for making precisely that history, precisely those maximal momentary events, actual*; one is within one’s rights then if one supposes that this history being the actual* history and these events being the actual* maximal momentary events are facts determined in ways for which the action of the fingertip of actuality** is entirely otiose. The situation is very different if the relationship that is articulated in Act2 is seen as a relationship of meaning such that, for any maximal momentary event, to be actual* means to be once actual**. In this perspective, the action of the fingertip of actuality** cannot fail to explain (causally in a sense) the singular actuality* of a certain history – together with the actuality* of precisely the maximal momentary events which are that history’s momentary phases (though the explanation provided is far from complete, since the action of the fingertip of actuality** stands itself sorely in need of explanation). Which way is the right way to go in this matter? – Whichever way we like to go. Model T is a model, and how we fashion the relationship of the outside of T to T is entirely up to us. And, this time, the self-imposed guiding principle that Model T should be a metaphysical simulacrum of Reality is of no great help in the decisions to be made. For two options analogous to those that we have just been considering with regard to T are also there with regard to Reality, and the weights are equal on both sides also with regard to Reality: Does time (or better: the present) simply flow down a riverbed that has already been prepared for it, or does it itself make a riverbed for itself as it flows along? – And if the latter question is the one to be answered with “yes”, is time, in finding where to run, guided at least in part by a will of its own (so to speak), or is it exclusively the executant of the will of others (including that strange agent: chance)? Or do all of these questions merely betray a thoroughly misguided conception of time, which really does not “flow” or “run” in any sense at all and is devoid of an objective (non-relative) present and an objective (non-relative) direction? Is there even an objective, non-relative distinction to some possible world which makes it the sole actual* world (of Reality)?

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These questions are genuine metaphysical questions, and questions for which we do not (and cannot) know any answer that enunciates the philosophical analogue of a scientific fact. But fixing on an answer to a metaphysical question by fiat, or because it is “more parsimonious” than another answer, or because we just happen to like it more than another answer (for some metaphysically frivolous reason other than the reason of parsimony), is appropriate only with respect to Model T, which is entirely within our power (and where it does not matter existentially which answer is given); it is not appropriate with respect to Reality, which is certainly not entirely within our power (and where it may matter existentially – deeply so – which answer we give). I shall avail myself of the remarkable advantage that Model T has over Reality: of being entirely within our power, and exercise the privilege that advantage bestows: the privilege of fiat. Therefore, being the (here and now relevant) author of Model T, I do not only make Act1, Act2, and Act3 true (about T), I also make it true that Act2 defines (as an analytic truth) what it means for a maximal momentary event to be actual*, in other words: I consider the actuality* of maximal momentary events as being reducible to actuality**. But this implies that actuality* is unconditionally – and not only with regard to maximal momentary events – reducible to actuality**: The reducibility of the actuality* of histories to actuality** follows on the basis of Act2 (as now understood) and Act3 (already found to be analytically true). And as for any entity X of T that is neither a history nor a maximal momentary event: X’s being actual* means, for many X, that X is intrinsically included in some actual* history or in some actual* maximal momentary event (the latter often already suffices); therefore, X’s actuality*, too, is (ultimately) reducible to actuality**.41 Note that a momentary state ST – being qua momentary state ontologically incapable of being actual* (see what is said about “actual*” and “actual**” at the beginning of the present section) – is not in the relevant sense intrinsically included in any history H or maximal momentary event E. For intrinsic inclusion in H or E in the relevant sense, the temporal configuration that is intrinsic to ST would have to be a temporal part of the temporal configuration of H (which configuration is the timeline of T) or of the temporal configuration of E (which configuration is some temporal 41

What it means for modal positions and worlds of T (see Sect. 3.6) to be actual* is already taken care of by what is means for T-histories to be actual*. (The manner of its being taken care of is obvious in view of the description, in Sect. 3.6, of modal positions and worlds of T.)

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position of T). But ST does not even have a temporal configuration that is intrinsic to it; it does not have a temporal backbone, so to speak. It is true that some momentary states are (at least) once actual**; but, note, this does not make them actual*; for being actual* and being once actual** amount to the same thing only for momentary events, primarily for maximal momentary events (see Act2 above). Nor can ST be actual* qua being intrinsically implied by (though not intrinsically included in) a history or maximal momentary event: the route to actuality* via intrinsic implication is only open for non-time-dependent states of affairs. 5.2.2

Actualization** and what is behind it

The fingertip of actuality**, moving in a certain orderly fashion over the vast plane of the maximal momentary events, is what ultimately (at the final point of action) writes the process of actualization**; it is the instrument of what is behind that process: the instrument of an agent (whatever it is) or of agents (whatever they are). The process of actualization**, though directed at Model T, originates outside of T, in the transcendence of T. This section is dedicated to its further exploration. But before we come to this, it is perhaps advisable to pause for a reflective moment. In the sections of the book prior to the previous section, our view of T has been more or less close-up. But now a great widening of perspective has taken place. Now, Model T is being regarded from a bird’s-eye view (or should we say “God’s-eye view”?), in other words: the outside of T – what surrounds T – is accorded as much attention as T itself. It must be emphasized that we are here assuming an epistemic position with regard to our metaphysical model which is impossible to assume with regard to Reality (even if Reality has an outside). This should throw light on the aspirations of the metaphysics of Reality (in all these many centuries) – and on what it really can achieve. However, not only with respect to Reality, but also with respect to Model T, the possible answers to the question of what explains the world (i.e. – with regard to T – what explains the sole actuality* of H*) are many, and their consideration is purely speculative (“it may be so, it may also be otherwise”) – with the incomparable advantage in the case of Model T that we can decide (unerringly) which one of the many possible answers is true by making it true. As far as we could see in the previous section, the fingertip of actuality** was free to go anywhere on “its keyboard”. Indeed, it still seems it

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could do anything on “its keyboard”: the two-dimensional matrix of maximal momentary events. Why not also touch several “little squares” at once, i.e., why not make several maximal momentary events actual** at once? Why not also touch a square twice over, i.e., why not make a maximal momentary event actual** twice over? But, remarkably, the fingertip of actuality** – ∆**, for short – does not do any such things. We have: L1 L2

∆** never actualizes** more than one maximal momentary event. ∆** never again actualizes** a maximal momentary event that it once actualized**.

And there is more of remarkableness: L3

∆**, in the 1st place, actualizes** a maximal momentary event with the temporal position 1 (in its domain); ∆**, in the 2nd place, actualizes** a maximal momentary event with the temporal position 2; ∆**, in the 3rd place, actualizes** a maximal momentary event with the temporal position 3; ……… ∆**, in the 100th place, actualizes** a maximal momentary event with the temporal position 100.

How very strange! Might not the fingertip of actuality** – even though conforming to L1 and L2 – have jumped wildly about on the plane of the maximal momentary events? Finally: L4

There are no more than 100 consecutive actualizations** (i.e., touchings) of maximal momentary events by ∆**.

Nothing is more strongly suggested by L1, L2, L3, and L442 – and in particular by L3: by the peculiar pre-established harmony that this statement says is holding between the timeline of T and the movement of ∆** – than the following idea: L1 – L4 are the (obeyed) rules of a certain game, ∆** being the executant of the moves of this game. (The executant, note, need not necessarily be a player of the game.) The game is called “Actualiza42

Note that in the presence of L1, L3, and L4, L2 need not be separately asserted, since it follows logically from L1, L3, and L4. Note also that in the presence of L1, L2, and L3, L4 need not be separately asserted, since it follows logically from L1, L2, L3, and what we know about the intrinsic nature of T.

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tion* of a History”. Accordingly, the game is over when a single T-history, in its entirety, has become actual*, which is the case (in view of Act2 and Act3) after 100 consecutive actualizations** – touchings by ∆** – of maximal momentary events have taken place, in accordance with L1, L2, and L3. However, just like any (normal) game, this game, too, can be played again and again: indefinitely many times. Thus, L1 – L4 as rules of the game are the rules for a single round of playing the game – and in this they are, of course, just like the usual rules of games. It should be kept in mind that the terms “actual*” and “actual**” as well as “actualize*” and “actualize**” are, normally, single-full-round-restricted: it is normally (but not always) presupposed that these terms are used within the context of a single full round of the game (though it may not matter which round), in disregard of all other rounds of the game. Are there further rules for the game of Actualization* of a History – for ACTUHIST (in short)? Since L1 – L4 (on the basis of Act2 and Act3) specify what playing ACTUHIST (i.e., a single round of it) is supposed to produce: a single actual* T-history of which each actual* maximal momentary event is a momentary phase, and in what manner: by following the timeline of T strictly consecutively, starting with the temporal position 1, ending with the temporal position 100, further rules for ACTUHIST can only spell out (I) restrictions regarding the eligibility of T-histories for being made the actual* history by playing ACTUHIST, (II) specifications regarding who or what is playing ACTUHIST, and (III) specifications regarding how the action of the executant of ACTUHIST – of ∆**, “the fingertip of actuality**” – is determined by the players of ACTUHIST at each one of the 100 steps of the game. Concerning (I): Restrictions regarding the eligibility of T-histories for being made the actual* history collectively amount to a definition of the set of T-histories which are “candidates for being actual”, as the relevant notion was expressed in Sect. 4.1.1, prior to the typographical differentiation of actuality-predicates according to their various meanings, which has only been introduced in the present chapter. Already in Sect. 4.1.1 the notion of (a history being a) candidate for being actual was introduced as a tool necessary for defining the extrinsic conception of lawhood for T (both in the primary and secondary sense), and the ACTUHIST-rules that specify the totality of the ACTUHIST-restrictions regarding the eligibility of Thistories for being made (the) actual* (one) will, in fact, turn out to be cor-

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relates of the T-laws, of the laws of nature for T. However, the concrete specification of these laws and of their rule-correlates will have to wait till Chapter 6; there the T-laws will be fashioned in such a way as to make higher continuants possible: so as to enable their supervenience – and also in such a way as to fit a purpose that will become apparent only in what is said near the end of the subsection Concerning (II) below. For the time being, our utter freedom in giving laws to T (or what is the same thing: in positing the corresponding ACTUHIST-restrictions) should be properly noted. Might there not be someone (outside of Reality) who is just as free in giving laws to Reality as we (outside of T) are free in giving laws to T – of whom we are images (not totally dissimilar ones) just as T is an image of Reality (not a totally dissimilar one)? Moreover, our utter freedom in giving laws to T should make one ponder the idea that so-called natural laws might not be natural in the sense of being a sort of natural growth, but might be natural only in the sense of holding for whatever it is that is considered nature. Certainly the (natural) laws for T are not a “natural growth”; they are imposed – we impose them (well, I do, and you, gentle reader, please follow me for now) and who else but we also enforces them? Might not also the laws of nature of Reality be laws imposed and enforced by someone? Concerning (II): It is up to us to decide who or what is playing ACTUHIST. We might, for example, decide that there is a sole and entirely unintelligent player: a chance-generator, “Mr Chance”, who selects the maximal momentary event that is to be actualized** at a certain touch of ∆** by a selection procedure that is more or less like (but, in any case, structurally identical to) the procedure described in Sect. 4 (at the beginning of Chapter 4). But will not this produce a totally chaotic – in other words: highly diversiform – actual* T-history? – Leaving aside the possibility that we may be quite content with such a result, it must be noted that Mr Chance need not necessarily be responsible for chaos – though the initial probability of this is indeed high (the reason why is given in Sect. 4). But the initially high probability of chaos can be significantly lowered – indeed, it can become very low – by submitting Mr Chance to significantly restrictive laws for T. After all, every player of ACTUHIST has to abide by the rules – and so must Mr Chance. Such laws for T (corresponding to rules of ACTUHIST) will nullify any selection by Mr Chance for the Nth step of actualization** (i.e., of a maximal momentary event with the temporal position N, according to

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L3) which is not compatible – according to the laws – with the prior nonnullified selections of Mr Chance; and Mr Chance will have to try again to make his selection for the Nth step of actualization** – until he hits, by chance (of course), on a maximal momentary event which the laws allow; this momentary event, then, is the one Mr Chance proceeds to actualize** via ∆**, the fingertip of actuality**. From the point of view inside of T, nothing of this is noticeable – if, indeed, there were anybody there who could take notice. But suppose, for the sake of the argument, the actual* history that is selected by Mr Chance – with, or by utter chance even without, the guidance of laws that restrict the eligibility of T-histories for being (made) actual* – is the T-history H°, with its solitary higher continuant c° (for the description of H° and c°, see Sect. 4). Suppose, just for the sake of the argument, that c° is an intelligent being and prone to metaphysical speculation. Could c° ever know what is responsible for the actuality* of its world – i.e., of H° (or, if you like, of , in view of Sect. 3.6) – and of no other world? It could not, since it cannot adopt “the bird’s-eye perspective” with regard to T. Only in this perspective could it find out the truth about T and H°, namely, that T is the material of a certain game, ACTUHIST, which is played by Mr Chance in accordance with the rules of the game, and that the outcome of it is the sole actuality* of H° among all the T-histories (the actuality* of everything else that is actual* being determined from the actual* modal position,43 which is H°). Since c° is prone to metaphysical speculation, as we have been supposing, it is not excluded that c° may somehow come to believe what we have been supposing is the truth about T and H°; in fact, many of us have come to believe, with regard to Reality, the analogue of what c° has just now not been excluded from coming to believe with regard to T. But that belief – though correct by supposition – would not amount to knowledge. Suppose that c° has in fact come to believe in the Mr-Chancehypothesis. If c° had come to believe instead that solely The Laws of the Universe are responsible for the sole actuality* of H° among the histories of T (the analogue of which, with regard to Reality, many of us came to believe in the 18th and 19th century), or that Mr God alone is responsible for it (indeed, c° had once been a very firm believer in this latter hypothesis), or … …, or that Mr God, The Laws of the Universe, Mr Chance, the soul of c°, and nothing else are in combination, and only in combination, responsible for it – the belief in any of these hypotheses would have had the same status of epistemic rationality for c° as the T-metaphysical hy43

Regarding modal positions, see Sect. 3.6.

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pothesis that we have been supposing to be the correct one (though just for the sake of the argument) and to be the one in fact favoured by c°. But is it not likely – in view of what has been said earlier in this subsection – that H° evolves haltingly (so to speak) if Mr Chance, under the guidance of restricting laws, is behind the actualization** process, and is not this stop-and-go evolution noticed by c°, and does not c° derive respectable epistemic support from this for the Mr-Chance-hypothesis that c° happens to favour? But even if the transcendent production process of the actuality* of H° is stop-and-go with respect to transcendent meta-time, as it may well be (but need not44), this does not mean that the course of ∆**, which, so to speak, writes H° in the colour of actuality*, is also stop-andgo as seen by c° from the inside of H°; c° may well see a smooth flow of actuality** (i.e., presentness*) along the timeline of T (from 1 to 100), vitalizing and devitalizing (we might say) one maximal momentary event of H° after the other – and thereby manifesting (in the ontological and in the epistemological sense) what is, in fact, the physical life of c°. We need not here make a complete decision on the player(s) of ACTUHIST and their abilities. It seems better to wait with this decision until after the laws for T – or in other words, the T-histories that are candidates for being actual* – will have been specified (which will happen in Chapter 6). However, if ACTUHIST is to be a nontrivial game – i.e., a game that needs players or even one player – and, to boot, an interesting game – i.e., a game that has the potential of keeping players mentally occupied for the full extent of 100 moves – then the T-laws must leave many more Thistories than just one T-history as candidates for being actual* (i.e., as candidates for being the one actual* history, in view of L1 – L4). If they left just one T-history, ACTUHIST would become a trivial game and a fortiori a totally uninteresting one; then all the fingertip of actuality** could do would be to run its predetermined course (or to put it rather more metaphorically: all the present* could do would be to flow in the riverbed already prepared for it; cf. what is said near the end of Sect. 5.2.1). – I herewith resolve that ACTUHIST is to be an interesting game. (Would ACTUHIST still be an interesting game if all T-histories were candidates for 44

ACTUHIST could be set up in such a way that Mr Chance is only offered such maximal momentary events to “gamble with” at a given stage of the game as are allowed at this stage by the rules of ACTUHIST (i.e., by L1 – L4 and by the further rules that spell out restrictions regarding the eligibility of T-histories for being made the actual* history). A transcendent stop-and-go of actualization** can be avoided in this way.

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being actual*? – Yes, it would, but it would not be as interesting as it can be.) Concerning (III): At a given stage N of (a round of) ACTUHIST, the act of actualization** has two sides: actualization**-selection and actualization**execution. The object of both these sides of activity is the same: a certain maximal momentary T-event with the temporal position N – no other Tevent than the maximal momentary event that is, fleetingly, actualized** (and in consequence, fleetingly, actual**) at stage N of the game. What has so far been said in this section (Sect. 5.2.2) may have led to the expectation that there is to be a neat distribution of labour behind the actualization**-process: actualization**-selection belongs to the player(s) of the game, actualization**-execution belongs to the executant of the game: to ∆**, the fingertip of actuality**, and that the distinction between player(s) and executant is to be such that the player(s) is (are) powerless with regard to the act of making-actual**, whereas the executant is powerless with regard to the act of selecting-for-being-made-actual**. But, contrary to this possible expectation, I make the following decision: ∆** is purely an instrument – or perhaps even just a useful fiction, the rationale of which is to “impersonate” the (combined) actualizing** choice-act of the player(s). As a mere instrument, ∆** is powerless with regard to the act of selecting-for-being-made-actual** and, considered in itself (abstracting from the user(s) of the instrument), it is also powerless with regard to the act of making-actual**. It would be better to arrange matters in a different way with respect to the power of making-actual** if ACTUHIST were a trivial game, in the sense described in the previous subsection; if ACTUHIST were a trivial game, it would be best to consider ∆** as the (albeit totally constrained: choiceless) actualizer** – in the absence of any player that could have any function for ACTUHIST that is worthy of a player. But it has already been decided above (in the previous subsection) that ACTUHIST is not (to be) a trivial game. Now, the following is true in virtue of the further decision made in the preceding paragraph: if there is just one player of ACTUHIST, then the entire power of making-actual** and the entire power of selecting-for being-made-actual** is concentrated in that one player; Mr Chance, in the role he was put in at the beginning of the previous subsection, is an example of such a – perforce dictatorial – player (but only in that role). But if there are several players, then there are, of course, many possible ways of

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distributing shares of the two powers among them. Again, one should properly appreciate the utter freedom the author of ACTUHIST has in legislating these matters and what this tells us about a – perhaps obtaining – parallel situation with regard to Reality. I herewith adopt as a rule of ACTUHIST one possible manner of initially distributing power-shares among ACTUHIST’s several players – a manner which is modelled (roughly) on the metaphysical position, with regard to Reality, of mainstream theism: L5

The author of all the rules of ACTUHIST (and, in consequence, also of all the laws for T) is also one of the players of ACTUHIST, and the power of makingactual** resides entirely with him. But besides the Author there are also other players of ACTUHIST. Each player has initially (at the beginning of the game) the quantitatively same share of the power45 of selecting-for-being-madeactual** (in the case of players other than the Author, that power-share is conferred to them by the Author). In the case of conflict or indetermination – situations that may arise when the players are employing their powers (i.e., shares of the power) of selecting-for-being-made-actual** (see below, in L6) – the Author is the sole arbitrator.

The next question is how the players of ACTUHIST choose, and how on the basis of their individual choices their collective choice is determined. The following rule is the rule of ACTUHIST that answers this question: L6

At each stage N of the game (1 ≤ N ≤ 100), each player selects a non-empty subset of the set MLaw(N), which is the set of all those maximal momentary Tevents that (i) have the temporal position N (as the sole temporal position in their domain) and that (ii) conform to the laws for T, given the maximal momentary events that have been actual** previous to stage N of the game.46 The size (i.e., the number of elements) of the selected subset is initially the same for all players.47 (This expresses the initial quantitative equality – demanded by L5 – of the players’ shares of the power of selecting-for-being-made-actual**.) The provisional collective choice of the players at stage N is the set-theoretical product ΠPlayers(N) of all the subsets of MLaw(N) that are chosen at stage N by

45

The quantitative sameness of the power-share for each of the players at the beginning of the game is one of the reasons why L5 is only roughly representative of mainstream theism. Another reason is that the beginning of the game is the beginning of the game for all of the players: the point where all of them make their first move. 46 For N = 1, there are no such events. In this case, the proviso “given the maximal momentary events that …” is void and has no modificatory effect. 47 The smaller is the subset, the greater is the power of selecting-for-being-madeactual**.

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the players individually. There are three possible outcomes: (1) ΠPlayers(N) has a single element; (2) ΠPlayers(N) has more than one element; (3) ΠPlayers(N) is empty. In the case of (1), the single element in ΠPlayers(N) is the maximal momentary event which is selected for being made actual** (and hence actual*) and which is made actual** (hence actual*) by the Author (via ∆**); in the case of (2) (the case of indetermination), the Author, acting as arbitrator, selects a single element in the set ΠPlayers(N) for being made actual** and makes it actual**; in the case of (3) (the case of conflict), the Author, acting as arbitrator, selects a single element in MLaw(N) for being made actual** and makes it actual**. What results prior to actualization** in the cases (1), (2), and (3) respectively – this is the (final) collective choice of the players at N.

This leaves open the question of how many players of ACTUHIST there are besides the Author, and who or what are those players, and what is the quantity of their initial quantitatively equal shares of the power of selecting-for-being-made-actual**, and how their powers of selecting-for-beingmade-actual** are to be generally characterized in their entirety (initially and also in the further course of the game, when those powers are perhaps no longer equal in quantity for all players). But, as has been remarked near the end of the previous subsection, it is better to keep these questions open until after the laws for T will have been specified. Only then shall we have a concrete idea of what the players of ACTUHIST are dealing with when they are playing ACTUHIST. 5.3

Possibility and necessity for T

Let ST be a non-time-dependent state of affairs of T (perhaps timethematic, perhaps not; see Sect. 3.3 for time-dependence and timethematicness in states of affairs). Consider the following definitions: D#25 ST is possible+ =Def ST is intrinsically implied by48 some T-history. D#26 ST is necessary+ =Def ST is intrinsically implied by every T-history. D#27 ST is possible* =Def ST is intrinsically implied by some T-history that is a candidate for being actual*. # D 28 ST is necessary* =Def ST is intrinsically implied by every T-history that is a candidate for being actual*. 48

“intrinsically implied by” can be logically equivalently replaced by “actual in”, given the restriction of D#25 – D#28 to non-time-dependent states of affairs.

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Not one of these four definitions is a so-called nominal definition, which – for the sake of mere abbreviation – gives meaning to a term that, so far, had no meaning at all. For this reason, there arises the legitimate question whether the four definitions are conceptually adequate. The attempt to answer this question first focuses on D#27: It is evident that D#27 defines a form of possibility. Once the definition is understood, one does not feel the urge to ask, “Why (in the world) is a state of affairs called ‘possible*’ that is intrinsically implied by some Thistory that is a candidate for being actual*? What has the definiens of D#27 to do with being T-possible in some adequate sense?” There are no second thoughts with regard to the conceptual adequacy of D#27. It also seems evident that D#28 defines a form of necessity. Once the definition is understood, one – presumably – does not immediately feel the urge to ask, “Why is a state of affairs called ‘necessary*’ that is intrinsically implied by every T-history that is a candidate for being actual*? What has the definiens of D#28 to do with being T-necessary in some adequate sense?” However, D#28 is – unlike D#27 – a definition about which one can rather easily have second thoughts with regard to its conceptual adequacy. The problem is that necessity (in whatever adequate ontic sense49) entails actuality (in the corresponding adequate ontic sense): whatever is necessary is ipso facto actual; this is a criterion of conceptual adequacy for every definition which purports to define a concept of necessity. But D#28 does not guarantee that whatever is necessary* is ipso facto actual*. Suppose that ACTUHIST is never actually played – although the rules of ACTUHIST have been set down completely. Then the T-histories that are candidates for being actual* are completely specified, and hence it is completely determined on the basis of D#28 which non-time-dependent Tstates-of-affairs are necessary*: a great many of them (we may safely assume). But since no T-history is ever actual*, not even partly, not one of those states of affairs is ever actual* – in striking contrast to what should be the case if D#28 were conceptually adequate (namely, that all of those states of affairs are actual*). The way to amend D#28 is to insert the conjunct “some T-history is actual*” into the definiens of D#28 – given (and keeping in mind) that every actual* T-history is ipso facto a candidate for being actual* and that a non-time-dependent state of affairs which is intrinsically implied by an 49

It is the ontic senses of “necessary” alone that are of interest for the purposes of this book. In an ontic sense, “necessary” lacks any normative or epistemic aspects of meaning.

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actual* T-history is ipso facto itself actual*. Note that the described amendment of D#28 already guarantees not only that every non-timedependent T-state-of-affairs which is necessary* is actual*, but also that every such state of affairs is possible*. However, the amendment of D#28 also produces a result – a result herewith accepted – that runs counter to standard assumptions about the relationship of necessity and possibility: a non-time-dependent T-state-of-affairs ST the negation of which is not possible* is not ipso facto a necessary* T-state-of-affairs – even if we take it to be unproblematic to proceed from “There is no T-history which is a candidate for being actual* and of which it is true that neg(ST) is intrinsically implied by it” to “For every T-history which is a candidate for being actual* it is true that ST is intrinsically implied by it”. All seems well if some T-history is actual*. But for an ipso-factoconnection between not-possible-not, as interpreted by D#27, and necessary, as interpreted by amended D#28, it is not enough that the statement “Some T-history is actual*” is true – that statement must be eo ipso true; it must logically (or analytically) “go without saying” that some T-history is actual*. But it certainly does not go without saying that some T-history is actual*, neither logically nor epistemically; for that proposition is neither logically true nor epistemically trivial. Note that the statement with respect to Reality that corresponds to “Some T-history is actual*” – the statement “Some possible world [non-relatively] actually exists” – is certainly true. But is it true eo ipso, true per se? Any person who believes that Leibniz’s question Why is there something and not nothing? has an interesting answer is rationally bound to deny that it is eo ipso true that some possible world actually exists (for if this were eo ipso true, the answer to Leibniz’s question would be as uninteresting as the answer to the question why dogs are mammals and not reptiles). The analogy between Model T and Reality reveals how such a person could be right in denying that it is eo ipso true that some possible world actually exists. Turning now to the definitions D#25 and D#26, it turns out that problems of conceptual adequacy can also be raised for them – and problems that seem worse than the problem that has just been raised for D#28. Suppose a non-time-dependent T-state-of-affairs ST´ is intrinsically implied by every T-history. This means, in other words, that ST´ is actual in every Thistory (where the actuality-predicate employed is the dyadic one described at the beginning of Sect. 5.2, which predicate has the relativization pattern no. 12 in the list of Sect. 5.1: − + − −). But why (in the world) is ST´ called “necessary+”, as one is licensed by D#26 to call it? Moreover,

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since ST´ is intrinsically implied by every T-history, it is intrinsically implied by some T-history (because there certainly are some T-histories). But why is ST´ even called “possible+”, as one is licensed by D#25 to call it? The designations “necessary+” and “possible+” for ST´ do seem arbitrary. One would not be able to move beyond this appearance of arbitrariness if one limited one’s conceptual horizon to T-intrinsic actualityconcepts. ST´ is actual in some T-history – how does this make ST´ Tpossible in some adequate sense? ST´ is actual in every T-history – how does this make ST´ T-necessary in some adequate sense? No satisfactory answers are forthcoming to these two questions if one merely considers the T-histories in themselves, disregarding all things – among them actuality* – that may come to them from the outside of Model T. The reason for this situation is this: “T-possible” in any adequate sense (as applied to nontime-dependent states of affairs) means as much as “T-possibly actual*”, and “T-necessary” in any adequate sense means as much as “T-necessarily actual*”, in other words: the synonymy of “T-necessary” and “Tnecessarily actual*” and of “T-possible” and “T-possibly actual*” is a requirement for any conceptually adequate interpretation of “T-necessary” and “T-possible”. But as long as one merely considers the T-histories in themselves, that requirement cannot be fulfilled by any interpretation of those two expressions50; for as long as one does so, what is meant by “actual*” (see Sect. 5.2.1) just cannot enter into the picture. Things are otherwise once the outside of T is taken into account, the content of “actual*” having been fully developed in Sects. 5.2.1 and 5.2.2. If one takes into account the outside of T, with special regard to actuality*, then, in the first place, it is immediately evident that “T-possible” (in other words, “T-possibly actual*”) can adequately be taken to mean as much as the predicate “possible*”, as defined by D#27; in this meaning, “Tpossible” leaves nothing to be desired with regard to conceptual adequacy (but there is an afterthought to this: see below); and it is immediately evident that “T-necessary” (in other words, “T-necessarily actual*”) can adequately be taken to mean as much as the predicate “necessary*”, as defined by the amended definition D#28; in this meaning, “T-necessary” leaves nothing to be desired with regard to conceptual adequacy. 50

If the schematic expressions “T-necessary” and “T-possible” are given a specific interpretation, they are replaced by non-schematic expressions, for example, by “necessary*” and “possible*” (corresponding to one specific interpretation of the schematic expressions “T-necessary” and “T-possible”), or by “necessary+” and “possible+” (corresponding to another specific interpretation of those schematic expressions).

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If the outside of T, with special regard to actuality*, is taken into account, then, in the second place, one comes to realizes that “ST´ is Tpossible [T-possibly actual*]” can – alternatively to its above-described meaning, and with complete conceptual adequacy – be also taken to mean as much as this: “in some choice of T-histories as candidates for being actual* there is a T-history which intrinsically implies ST´”. Since, for conceptual reasons, X is a choice of T-histories as candidates for being actual* if, and only if, X is a non-empty set of T-histories, it emerges that the last-mentioned meaning of “ST´ is T-possible” is logically equivalent to the meaning given to “ST´ is possible+” by D#25. Thus, we have after all obtained the – prima facie lacking – justification for the conceptual adequacy of D#25. Correspondingly, “ST´ is T-necessary [T-necessarily actual*]” can – with complete conceptual adequacy – be also taken to mean as much as this: “some T-history is actual*, and in every choice of T-histories as candidates for being actual*, every T-history intrinsically implies ST´”. It emerges that this meaning of “ST´ is T-necessary” is logically equivalent to the one given to “ST´ is necessary+” by D#26 if that definition is amended in the same way as D#28 has already been amended: by inserting the conjunct “some T-history is actual*” into its definiens. Thus, we have after all obtained the – prima facie lacking – justification for the conceptual adequacy of D#26 – but only if D#26 is amended in the same way as D#28. Consider it done. There is an afterthought to all this: Has not the conceptual adequacy of # D 27 been all too easily accepted? Is it really true that being intrinsically implied by a T-history which is a candidate for being actual* is for a nontime-dependent T-state-of-affairs conceptually sufficient for being Tpossible in an adequate sense? Suppose that for some reason it is impossible to play ACTUHIST, although the rules of the game have been set down completely. This has the consequence that there is no actual* T-history, and hence that there is no non-time-dependent T-state-of-affairs which is necessary* or necessary+ (in accordance with amended D#28 and amended D#26). Should the supposed situation not also have the consequence that there is no non-time-dependent T-state-of-affairs which is possible* or possible+ (or T-possible in any adequate sense), in spite of the fact that there is, of course, some non-time-dependent state of affairs ST which is intrinsically implied by a T-history that is a candidate for being actual* – which state of affairs ST would be both possible* and possible+ if D#27

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and D#25 were applied in their current form? –There are several prima facie plausible ways to react to this query: (1) One can accept its rhetorical tendency – and amend the possibilitydefinitions D#27 and D#25 in the same way that the necessity-definitions D#28 and D#26 have already been amended: by inserting the conjunct “some T-history is actual*” into the definiens. This yields the desired result. (2) One might hold that the impossibility of playing ACTUHIST has the consequence that there is no choice of T-histories as candidates for being actual* and that, therefore, there are no T-histories that are candidates for being actual*. – But why should the impossibility of playing ACTUHIST imply the impossibility of determining the T-histories that are candidates for being actual* in some way or other? Is not this determining already done by the rules of ACTUHIST, prior to, and independent of, playing ACTUHIST? Moreover, it has already been asserted as a conceptual truth above that every non-empty set of T-histories is a choice of Thistories as candidates for being actual*. Hence (considering that there is more than one T-history) there are of course various choices of T-histories as candidates for being actual*. There is no reason not to make (or adopt) one of these choices (by formulating the rules of ACTUHIST accordingly) – even though it may turn out to be impossible to play ACTUHIST. (3) One can simply postulate that it is possible to play ACTUHIST. This excludes the situation envisaged above a priori. Of these three options, the first option is, doubtless, the most attractive one. I therefore adopt it. – Hence all of the definitions D#25 – D#28 have now been amended and do no longer have their original form. It seems well to document their (uniform) amendments explicitly, and to give each of the definitions a new name – a name which reflects the now final and official status of it. Let ST be a non-time-dependent T-state-of-affairs: D25 ST is possible+ =Def ST is intrinsically implied by some T-history, and some T-history is actual*. D26 ST is necessary+ =Def ST is intrinsically implied by every T-history, and some T-history is actual*. D27 ST is possible* =Def ST is intrinsically implied by some T-history that is a candidate for being actual*, and some T-history is actual*. D28 ST is necessary* =Def ST is intrinsically implied by every T-history that is a candidate for being actual*, and some T-history is actual*.

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There is another afterthought: That there are 100 spatial positions of T – is not this a non-time-dependent T-state-of-affairs that is T-necessary (and hence T-possible) in an adequate sense come what may, that is: even if there happens to be no actual* T-history (because ACTUHIST just happens to be never played) – and yes, even if there cannot be any actual* Thistory (because ACTUHIST for some reason cannot be played)? This query may be answered in the following way: The state of affairs that there are 100 spatial positions of T is not (in any adequate sense) Tnecessarily actual* if ACTUHIST is never played; and therefore that state of affairs is not T-necessary in any adequate sense if ACTUHIST is never played (this conclusion is reached on the basis of the above determination, which has it that “T-necessary” in any adequate sense is to be taken to mean as much as “T-necessarily actual*”, accordingly understood). But if ACTUHIST is once played, then that state of affairs is of course (in some adequate sense) T-necessarily actual* and hence T-necessary in some adequate sense. It all hinges on whether or not ACTUHIST is once played, in other words: on whether or not some T-history is actual*.51 This conditional view of the matter has at least as much intuitive right as the competing position, which is the position that the T-state-of-affairs that there are 100 spatial positions is unconditionally T-necessary – in some adequate, but hitherto unrecognized, sense. And the conditional view has the significant advantage over the competing position that the concept of actuality that corresponds to that view’s concept(s) of necessity has, by now, become quite clear: it is the concept of actuality*. But this cannot yet be the last word here. If the state of affairs that there are 100 spatial positions is unconditionally T-necessary in some adequate sense, then it must be unconditionally T-actual, though for the time being it is unclear in which sense it must be unconditionally T-actual; we have not encountered the required concept of actuality in our survey of actuality-predicates in Sect. 5.2. This is a problem. It turns out, we cannot leave it a problem (and hold the absence of a solution to it against unconditional T-necessity). For there ought to be a proper concept of actuality which is (not only non-relativized but also) unconditional in application, for the following reason: “There are 100 spatial positions (of T)” is unconditionally true (about T) – and truth unconditional in application requires actuality unconditional in application. 51

It is assumed that a T-history is actual* if, and only if, ACTUHIST (i.e., one full round of it) is once played. But the description of ACTUHIST leaves no doubt that this assumption is true.

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We can sharpen our sense of this so far unrecognized concept of actuality, for which I reserve the predicate “X is actual***” and the name “actuality***”: “ is two steps to the left of ” is unconditionally true, hence the state of affairs that is two steps to the left of is unconditionally actual. For actuality in this sense, it does not matter whether some T-history is actual* or none is (and, indeed, it does not matter whether some possible world of Reality actually exists or none does). If ACTUHIST is never played, then the state of affairs that is two steps to the left of is not actual* (and therefore not necessary*); but nevertheless that state of affairs is actual*** even then, as is the state of affairs that there are 100 spatial positions of T. Clearly, actuality*** is a Timmanent concept of actuality52 – in spite of the fact that actuality*** is neither temporally nor historically relativized; this combination of immanence and non-relativeness should be taken note of. Since we have sufficiently secured the concept of actuality*** (let’s assume), why not now move on to the corresponding necessity-concept: that of necessity***? It is true that in interpreting “T-necessary” by “necessary***” we shall not obtain a synonymy of “T-necessary” and “Tnecessarily actual*” (but only a synonymy of “T-necessary” and “Tnecessarily actual***”). But this might be taken to merely mean that we just have to discard the requirement of the synonymy of “T-necessary” and “T-necessarily actual*” for any conceptually adequate interpretation of “Tnecessary”, notwithstanding the fact that it was previously adopted (see above). There is, however, an effective reason to abstain from introducing necessity*** after all: “T-necessary” in any adequate sense should be a strengthening of “T-actual” in the corresponding sense; this, too, is a requirement for any conceptually adequate interpretation of “T-necessary”. “Necessary*” – as an interpretation of “T-necessary” – fulfils that requirement (as does “necessary+”). On the basis of the fully specified rules of ACTUHIST, necessity* will prove to be a more exclusive (i.e., logically stronger) concept than actuality* (this is entailed by what is said in the subsection Concerning (II) of Sect. 5.2.2); although everything that is necessary* is bound to be actual*, not everything that is actual* is also necessary* (as will be seen in Sect. 6). But “necessary***” – as an interpretation of “T-necessary” – does not fulfil that requirement: “X is necessary***” 52

For its application, it is sufficient to look at Model T in itself, though it may be necessary to look carefully and comprehensively at T.

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just means as much as (and therefore is not a strengthening of) “X is actual***”. What else could “X is necessary***” mean? 5.3.1

History-relative and/or time-relative necessity and possibility for T

The concepts of possibility and necessity presented in the previous section – possibility* and possibility+, necessity* and necessity+ – are concepts that are non-relativized in the same way the concept of actuality* is: they are concepts that are neither explicitly nor implicitly relativized to Thistories or T-temporal positions. At the same time, they are Ttranscendent concepts – just like the concept of actuality*: for determining their true or false application, it is not sufficient merely to look at Model T in itself (however careful and comprehensive the look may be). But we have seen that there are relativized actuality-concepts which are Timmanent. Are there not also relativized necessity- and possibilityconcepts which are T-immanent? It is not difficult to associate some sense with “X is possible in Y” and “X is possible at Y in U”, and with “X is necessary in Y” and “X is necessary at Y in U” – predicates formed in analogy to (and with the same relativization pattern as) the dyadic actuality-predicate “X is actual in Y”, introduced in Sect. 5.2, and the triadic actuality-predicate “X is actual at Y in U”, also introduced in Sect. 5.2. The standard general way (transposed from Reality to Model T) of interpreting these modal predicates is the following: X is possible in Y if, and only if, Y is a T-history and for some Thistory H that stands in the relation R to Y: X is actual in H. X is necessary in Y if, and only if, Y is a T-history and for every Thistory H that stands in the relation R to Y: X is actual in H. X is possible at Y in U if, and only if, U is a T-history and Y a Ttemporal position and for some T-history H that stands in the relation R´(Y) to U: X is actual at Y in H. X is necessary at Y in U if, and only if, U is a T-history and Y a Ttemporal position and for every T-history H that stands in the relation R´(Y) to U: X is actual at Y in H. One will require of the above-invoked relation R and relations R´(Y) – one relation R´(Y) for every T-temporal position Y – [1] that R and R´(Y) be

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intrinsic relations between T-histories, and [2] that, for every T-history U, U stand in R and R´(Y) to itself. These requirements guarantee that everything that is necessary in a history is, for intrinsic reasons, also actual in it; respectively, that everything that is necessary in a history at a temporal position is, for intrinsic reasons, also actual in that history at that temporal position. But these formal requirements leave it rather indeterminate which relations R and R(Y) might be in concreto.53 And the following questions arise: Why should an X that is actual in all – respectively, some – Thistories that stand in a certain relation R to a given T-history Y be called necessary – respectively, possible – in Y? In other words: All T-histories that are related in the way R to Y (including Y itself) “have X in common” – how does this, merely this, make X necessary in Y? Some T-histories that are related in the way R to Y “comprise X” – how does this, merely this, make X possible in Y? – I leave unformulated the entirely analogous questions for the temporally relativized cases. The force of these questions – which are questions of conceptual adequacy, occasioned by an apparent lack of it – will perhaps be felt more strongly if the above-introduced standard way of interpreting the predicates “X is necessary in Y”, “X is possible in Y”, “X is necessary at Y in U”, and “X is possible at Y in U” is contrasted with the interpretation of “X is possible*” and “X is necessary*” in the previous section (see D27 and D28) and with the following also entirely conceptually adequate interpretation of the predicates “X is possibleTP* at Y” and “X is necessaryTP* at Y”. These latter predicates have the relativization pattern no. 8: + – – – (see the list of relativization patterns in Sect. 5.1), and they are defined (interpreted) in the following way:

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One among countless options for the interpretation of R is that, for all histories H and Y, H stands in the relation R to Y if, and only if, H has the same laws [according to the intrinsic conception of lawhood] as Y. One among countless options for the interpretation of R´(Y) is that – for all histories U, H, and temporal positions Y – H stands in the relation R´(Y) to U if, and only if, H has the same laws [according to the intrinsic conception of lawhood] as U and is identical with U up to Y. (Regarding the intrinsic conception of lawhood, see Sect. 4.1.1.) The described interpretations of R and R´(Y) seem non-arbitrary – but only because they are used in specifying Timmanent modalities that are the (fairly remote) analogues of substantial Ttranscendent modalities, namely, possibility* and necessity*, possibilityTP* and necessityTP* (for these latter, see below in the main text). In themselves, they are no better than other interpretations of R and R´(Y) (consonant with the condition that R and R(Y) are to be intrinsic relations).

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D29 X is possibleTP* at Y =Def Y is a T-temporal position; and for some Thistory H which is a candidate for being actual* and which is actual* up to Y: X is actual in H; and some T-history is actual*. D30 X is necessaryTP* at Y =Def Y is a T-temporal position; and for every T-history H which is a candidate for being actual* and which is actual* up to Y: X is actual in H; and some T-history is actual*. Several comments and clarifications are in order: (a) In contrast to D25 – D28 in the previous section, the definitions D29 and D30 are not subject to the condition that ST – or in their case: X – be a non-time-dependent T-state-of-affairs. In the definientia of D25 – D28, one can replace the phrase “ST is intrinsically implied by some/every T-history …” by the phrase “ST is actual in some/every T-history …” (without change in conceptual contents), since, for non-time-dependent Tstates-of-affairs ST, “ST is actual in [T-history] H” just means the same as “ST is intrinsically implied by [T-history] H”. If one replaced “X is actual in H” in the definientia of D29 and D30 by “X is intrinsically implied by H”, then one would do well to submit those definitions to the condition that X be a non-time-dependent T-state-of-affairs (that is: one would do well to limit those definitions to the main area of application for the predicates they define), since for T-entities other than non-time-dependent T-statesof-affairs the synonymy of “X is actual in H” and “X is intrinsically implied by H” is not entirely clear; that is, the following question is not entirely inappropriate: might not a T-entity that is not a non-time-dependent T-state-of-affairs be intrinsically implied by a T-history without being actual in it, or vice versa? (For T-events, however, the said synonymy is clear, noting that between T-events – including T-histories – intrinsic implication is intrinsic inclusion.) But, in any case, “X is actual in H” and “X is intrinsically implied by H” express intrinsic relations of in-being: relations which are solely a matter of X and Y as they are in themselves. (b) A T-history H is actual* up to a temporal position Y if, and only if, (1) the momentary phase of H at Y is actual*54 and (2) for every temporal position Z before Y: the momentary phase of H at Z is actual*. D24 in Sect. 5.2.1 already defines what it means to be a momentary phase of an event, for example: of a complete history. That definition has the following important supplement:

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That is, once actual** – according to Act2 in Sect. 5.2.1.

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For all events E´: D24b E is a momentary phase of E´ at Z =Def E is a momentary phase of E´, and Z is the single temporal position in the domain of E. It is a consequence of D24 and D24b that every event has precisely one momentary phase at each temporal position in its domain. Hence, every Thistory has precisely one momentary phase at each temporal position (since all temporal positions belong to every T-history’s domain). This is the justification for the use of the expression “the momentary phase of H at Y” in the definition (just given) of a history’s being actual* up to a given temporal position. (c) The definientia of D29 and D30 contain the very same adjunct as the definientia of D25 – D28: the statement “some T-history is actual*”, the conjunct that was added as an amendment to the definientia of D#25 – D#28. In the case of D30, that adjunct guarantees that everything which is necessaryTP* at a temporal position is also actual*. For proving this, it is crucial (i) that if a history is actual* it is – ipso facto – a candidate for being actual* and – ipso facto – actual* up to any T-temporal position whatsoever; (ii) that whatever is actual in a history which is actual* is – ipso facto – itself actual*. In the case of D29, the adjunct “some T-history is actual*” guarantees that if for some reason it is impossible – or merely not the case – that there is an actual* history, nothing is possibleTP* at any temporal position. Thus, all three concepts of T-necessity introduced so far (by D26, D28, and D30) and all three concepts of T-possibility introduced so far (by D25, D27, and D29) hinge on ACTUHIST being played once (i.e., on one full round of it being played once; ACTUHIST counts as not having been played at all on an occasion if only part of a round of it was played on that occasion), that is: on there being some actual* T-history. (d) It is a consequence of D30 that everything which is necessaryTP* at a given temporal position Z is also actual* (without relativization to temporal positions). Is it a consequence of D29 that everything which is actual* is possibleTP* at Z? The answer is yes if the following principle is true: (A*) For all X: if X is actual*, then there is an actual* T-history in which X is actual. (A*) is the converse of the principle (ii) mentioned in (c) above; in other words, it is the converse of the following statement:

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(B*) For all X: if there is an actual* T-history in which X is actual, then X is actual*. (B*) is rather more obvious than (A*). (B*) is on the same high level of analytic obviousness as “Every actual* history is a candidate for being actual* and actual* up to any T-temporal position whatsoever” (see (i) in (c)), and on the same high level of analytic obviousness as “For all nontime-dependent states of affairs ST: if there is an actual* T-history by which ST is intrinsically implied, then ST is actual*” – an analogue of (B*) that was invoked in the previous section. In contrast to (B*), (A*), it seems, can turn out to be false. Suppose that in the course of playing the first round of ACTUHIST a maximal momentary event E is actualized** – hence E is actual*; but soon after its actualization** the playing of that round of ACTUHIST is broken off – before having reached the end of the game – and ACTUHIST is never played again. Then there is no actual* Thistory, and a fortiori no actual* T-history in which E is actual – although E is actual*. Hence, according to this scenario, (A*) can turn out to be false – which, of course, does not mean that it is false: if a full round of ACTUHIST is played, then (A*) cannot fail to be true. However, one can make use of an idea introduced at the end of (c) to block the described way of obtaining the result that (A*) can be false: In the envisaged scenario, no full round of ACTUHIST is ever played; but this – see the end of (c) – is tantamount to ACTUHIST being never played at all; hence E is, one can very well say, for conceptual reasons never actualized**, never actual**, and therefore E is – after all – not actual* (in the envisaged scenario). (Note that an analogue of (A*) – namely, “For all non-time-dependent Tstates-of-affairs ST: if ST is actual*, then there is an actual* T-history by which ST is intrinsically implied” – is sufficient, if added to the firstmentioned of the obvious principles pointed out after (B*) above, for making sure that every actual* non-time-dependent T-state-of-affairs is possible* and possible+.) (e) As the previous comments and clarifications show, “X is possibleTP* at Y” and “X is necessaryTP* at Y”, as defined by D29 and D30, express concepts that are rather more complex than the concepts which the predicates “X is possible in Y”, “X is necessary in Y”, “X is possible at Y in U”, and “X is necessary at Y in U” can be made to express by making specific in some manner or other the standard general way of interpreting them (which is described above). Not a negligible amount of care has to be taken even to make sure that everything which is necessaryTP* at a tempo-

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ral position is actual*, and that everything which is actual* is possibleTP* at any temporal position. These disadvantages (if they are disadvantages) are, however, countered by the indisputable conceptual adequacy of “X is possibleTP* at Y” and “X is necessaryTP* at Y”, as interpreted by D29 and D30. If a round of ACTUHIST is being played according to the rules (which is going to be a complete round), then the second predicate serves to describe what is already determined to be actual* by the first step of the game: all that is necessaryTP* at 1, and what is already determined to be actual* by the first and second step of the game: all that is necessaryTP* at 2, and what is already determined to be actual* by the first, second, and third step of the game: all that is necessaryTP* at 3, and so on. If a round of ACTUHIST is being played according to the rules (and is going to be a complete round), then the first predicate – “X is possibleTP* at Y” – serves to describe what is not already excluded from being actual* by the first step of the game: all that is possibleTP* at 1, and what is not already excluded from being actual* by the first and second step of the game: all that is possibleTP* at 2, and what is not already excluded from being actual* by the first, second, and third step of the game: all that is possibleTP* at 3, and so on. Clearly, the amount of what is necessaryTP* increases along the timeline of T, and what is possibleTP* decreases. After the 100th step of the game, the outcome is the same: What is determined by the 100 steps of the game is identical with what is not excluded by the 100 steps of the game: what is necessaryTP* at 100 is what is possibleTP* at 100; for what is necessaryTP*/possibleTP* at 100 is just what is actual in the actual* history: it is what is actual* (on the basis of (A*) and (B*)). Note that if there were just one T-history which is a candidate for being actual*, then the situation that obtains after the 100th step of the (complete round of the) game would already obtain after the first step and for all further steps: at every temporal position that which is necessaryTP* would be that which is possibleTP* – namely, that which is actual*. This is what absolute determinism means for Model T. But the same result is also produced by free-first-step determinism, where there are indeed several T-histories which are candidates for being actual*, but only one which is both a candidate for being actual* and actual* up to 1, in other words: where the first step of the round of ACTUHIST that is being played is not determined, but determines the entire round. (f) In further explanation of what was said in (e), consider: On the basis of Act3 (in Sect. 5.2.1), a T-history which is actual* up to 100 is an actual* T-history (and vice versa), and an actual* T-history is ipso facto a T-

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history which is a candidate for being actual*. Hence, “X is possibleTP* at 100” amounts to “X is actual in some T-history which is actual*” (according to D29, omitting the parts of the definiendum that turn out to be redundant), and “X is necessaryTP* at 100” amounts to “X is actual in every Thistory which is actual*, and some T-history is actual*” (according to D30, omitting the parts of the definiendum that turn out to be redundant). Hence, given that there is exactly one T-history which is actual* (which is the case if one full round of ACTUHIST is played), both “X is possibleTP* at 100” and “X is necessaryTP* at 100” amount to the same: “X is actual in the actual* history”, or (according to (A*) and (B*)): “X is actual*”. And consider: PossibilityTP* and necessityTP* are dynamic modal concepts. The frame for their dynamic nature is set by the following truth, which is a consequence of the nature of the concepts involved: For all temporal positions Z with 1 ≤ Z < 100: {H: H is a T-history which is a candidate for being actual* and actual* up to Z+1} is a subset of {H: H is a T-history which is a candidate for being actual* and actual* up to Z}. But this, by itself, does not yet imply that possibilityTP* and necessityTP* are dynamic concepts. This result follows, however, with the additional assumption – guaranteed to be correct by the constitution of ACTUHIST – that {H: H is a T-history which is a candidate for being actual* and actual* up to 1} has more than one member and that {H: H is a T-history which is a candidate for being actual* and actual* up to 100} has only one member. Clearly, this means that the number of histories which are still candidates for being actual* becomes less in the course of the timeline of T – until finally both possibilityTP* and necessityTP* blend with actuality*. (g) Two pairs of modal concepts are within the close vicinity of possibilityTP* and necessityTP*, and yet significantly different from these latter concepts: D31 X is possible2TP* at Y =Def Y is a T-temporal position; and for some T-history H which is a candidate for being actual* and which is actual* antecedent to Y: X is actual in H; and some T-history is actual*. D32 X is necessary2TP* at Y =Def Y is a T-temporal position; and for every T-history H which is a candidate for being actual* and which is actual* antecedent to Y: X is actual in H; and some T-history is actual*.

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D33 X is possible2TP+ at Y =Def Y is a T-temporal position; and for some T-history H which is actual* antecedent to Y: X is actual in H; and some T-history is actual*. D34 X is necessary2TP+ at Y =Def Y is a T-temporal position; and for every T-history H which is actual* antecedent to Y: X is actual in H; and some T-history is actual*. A T-history H is actual* antecedent to a temporal position Y if, and only if, for every temporal position Z before Y: the momentary phase of H at Z is actual*. Since there is no temporal position before 1, it is trivially true that for every temporal position Z before 1: the momentary phase of H at Z is actual*. It is, therefore, a consequence of the definition just given that every T-history H is actual* antecedent to 1. It is, therefore, a consequence of D27 and D31 that, for every non-time-dependent T-state-of-affairs ST, ST is possible* if, and only if, ST is possible2TP* at 1. And it is a consequence of D28 and D32 that, for every non-time-dependent T-state-ofaffairs ST, ST is necessary* if, and only if, ST is necessary2TP* at 1. (Remember – or see (a) – that for a non-time-dependent T-state-of-affairs to be actual in a T-history is to be intrinsically implied by it.) It is, moreover, a consequence of D25 and D33 that, for every non-time-dependent T-stateof-affairs ST, ST is possible+ if, and only if, ST is possible2TP+ at 1. And it is a consequence of D26 and D34 that, for every non-time-dependent Tstate-of-affairs ST, ST is necessary+ if, and only if, ST is necessary2TP+ at 1. Note also that, for all temporal positions Z with 1 ≤ Z < 100, X is possible2TP*/necessary2TP* at Z+1 if, and only if, X is possibleTP*/necessaryTP* at Z (as is easily verified). (h) “X is possibleTP* at Y” and “X is necessaryTP* at Y” have companion predicates which are relativized implicitly (and are, therefore, indexical) in the same way as the first-mentioned predicates are relativized explicitly: “X is possibleITP*” and “X is necessaryITP*”. These latter predicates have the relativization pattern no. 14: – – + –, whereas the former predicates have the relativization pattern no. 8: + – – –. “X is possibleITP*” is congruent in meaning with “X is possibleTP* at Y” in the following sense: the meanings of these two predicates require that, for all temporal positions Y of T, “possibleITP*” truly applies to X at Y if, and only if, “possibleTP* at Y” truly applies to X (and analogously for “X is necessaryITP*” and “X is necessaryTP* at Y”).

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5.4

The immanent and transcendent perspective on modality and time

Model T is not a metaphysically closed system. There is an outside to Model T, with a time and an actuality of its own, with possibilities and necessities of its own, and Model T is connected to this outside: it is connected to outside time (for example, to the time when a move of ACTUHIST is being made), to outside actuality (for example, to a move of ACTUHIST being actually made), to outside possibility and necessity (for example, to the possibility/the necessity of a certain move of ACTUHIST being actually made). These connections to the outside exert a significant influence on the time-concepts, on the actuality-concepts – which are modal concepts in a broad sense – and on the other, more narrowly modal concepts of Model T. This much should be clear by now. But suppose that there were no outside to Model T, or rather: pretend as best you can that there is no outside to Model T (putting yourself somehow inside of it), and consider the consequences this would have for all temporal and modal matters that concern T. Without going into details, a salient consequence would be that arbitrariness – mere stipulation and convention – would fill all places in the constitution of T-concepts where a connection to the outside of T would come in if there were an outside to T. Another general consequence of the absence of an outside to T – a consequence entailing (and therefore in fact more fundamental than) the consequence described in the previous statement – would be the objective existential homogenization of all temporal positions of T and of all histories of T. If there were no outside to T, the objective existential status of each temporal position of T could only be like that of any other temporal position of T, and the objective existential status of each history of T could only be like that of any other history of T. In other words, if there were no outside to T, all T-temporal positions could only be existentially equal from the objective point of view, and likewise all (complete) T-histories. What has just been counterfactually supposed with regard to Model T is often assumed to be the undeniable truth with regard to Reality: it is often taken to be undeniably true that there is no outside to Reality. The consequences of this assumption can be seen from the consequences of the corresponding assumption regarding Model T: objective existential homogenization – that can only be contravened (ineffectually) by arbitrary stipulation – is the general title for those consequences. But the consequences must be borne, it seems; for whereas it cannot be seriously denied

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that T has an outside (since we are in its outside), it seems that it must be denied that Reality has an outside. How could Reality have an outside? In this book, the word “Reality” is taken to mean as much as “Reality as we know it” (see Sect. 2); in this sense, Reality could, of course, have an outside. And, indeed, there are some indications (though certainly not taken seriously by every metaphysician) that Reality does have an outside.55 But the rhetorical questioner – “How could Reality have an outside?” – may insist on his own way of interpreting the term “Reality”. According to his way of interpreting that term, “Reality” refers to the totality of everything – not as we know it, but in itself (cf. Sect. 2) – and according to this interpretation, indeed, Reality cannot have an outside. Suppose, therefore, for a moment that Reality is the totality of everything in itself, i.e.: of absolutely everything. Then the transcendent perspective on Reality is impossible, only the immanent perspective is possible, and Reality, as the totality of everything, can only be a metaphysically closed system. It follows that the possible worlds of Reality (qua Totality) must all be existentially homogeneous objectively. If one of them is to be distinguished as the (non-relatively) actual world of Reality (qua Totality), then this can only be done by objectively arbitrary stipulation; there is no external point of view that could offer an objective perspective for breaking their existential homogeneity. It is true that the possible worlds are more or less intrinsically (hence objectively) different from each other (just as the T-histories are more or less intrinsically different from each other); but this is not an existential difference: a difference of actuality or, in other words, existence (taking existence in the sense of actual existence). It may so be that we are all intrinsically included in a single possible world and intrinsically absent from all the other possible worlds (as David Lewis would have it); but this is not a difference that is existentially relevant from the objective point of view. Each possible world is actual in itself and nonactual in every other possible world – there is nothing more than this that can objectively and non-arbitrarily be said about the actuality of the possible worlds of Reality qua Totality, and, of course, this does not existentially distinguish any of those possible worlds from any other. Possibility and necessity, then, can only amount to what they are, in some specific

55

The facts of the outside to Reality are unknown to us (where knowledge is taken to require intersubjectively sufficient justification). But Model T – embedded in an outside the facts of which are known to us – affords us the means for reasonable speculation about the facts of the outside to Reality.

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manner taken, to amount to along the lines of the standard general way of giving modal predicates a system-intrinsic meaning; see Sect. 5.3.1. The described consequences cannot, in a truly satisfactory way, be made to go away by simply deciding to use the term “Reality” in a sense that allows one to distinguish between Reality and Totality. Reality qua Reality as we know it may well be a rather insignificant part of Totality (and hence may well have a lot of outside to it); but Totality is still there, is it not? And while one may abstain from calling it “Reality”, one may also – with at least equal justification – not abstain. Indeed, might one not hold that Totality has the greatest purely objective right of being called “Reality”? Therefore: Does not all transcendence end in an ultimate immanence, which, ultimately, does not permit a non-relative actuality of which the delineation is non-arbitrary and objective? What must give us pause here is the question of whether we can be sure that Totality exists (in the sense “exist” has in Free Logic). The words are easily said: “Totality”, “the totality of absolutely everything”; but do these expressions really refer to something? It is well-known that “the set of [absolutely] everything” is a problematic designation. In standard set theory, this designation does not refer to anything (that is, not to anything that matches the literal meaning of “the set of everything”). In standard set theory, there is just the infinite open hierarchy of sets, without a set that is all-embracing. We may take our inspiration and justification from set theory and deny what is doubtful in any case: that Totality exists. If this is correct, then whatever Reality-system Σ we might consider, there is an outside to it, part of which outside is constituted by the smallest Realitysystem that embraces Σ. Therefore: no Reality-system has the greatest purely objective right to be called “the Reality-system”, or simply “Reality”. But there is a curious consequence. The relationship between Model T and its ACTUHIST-surroundings – ACTUHIST being the game that has Model T as its object56 – might summarily be called a relationship of actuality-giving. Analogously, the relationship between a Reality-system Σ and the smallest Reality-system Σ´ that embraces Σ might also be called a relationship of actuality-giving: Σ´ gives (non-relative) actuality to Σ. But since there is no ultimate Reality-system, the chain of Reality-systems that are

56

Note that there is no inkling of the nature of ACTUHIST if one sticks purely to the inside of T.

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connected by the relation of actuality-giving extends to infinity (at least on the side of the actuality-givers). For a (truly) ultimate Reality-system: for Totality – if there were such a Reality-system – non-relativized actuality would (have to) coincide with intrinsic non-relativized actuality, that is: with the actuality-concept of which actuality*** is the specialization with respect to Model T (see Sect. 5.3). But there is no ultimate Reality-system, and non-relativized actuality is for no Reality-system simply identical with intrinsic non-relativized actuality: there is, for each Reality-system, at least some non-relativized actuality which is non-redundantly extrinsic to it, in other words: for which a derivation external to the Reality-system is required. (What this can mean concretely has been modelled; in other words, it has been expounded for a simple model of a Reality-system: for Model T.) Hence: no (truly) ultimate explanation of all the non-relativized actuality that is present in a Realitysystem is possible for any Reality-system – and no ultimate explanation, probably, of all the non-relativized actuality that is present in this particular model of a Reality-system: Model T. This is so because of the fact that there is a smallest Reality-system Σ* in which T is embedded and from which T derives a great part of the non-relativized actuality that is present in it: very likely this T-external derivation of T-actuality has itself such non-relativized actuality to it that it – or in other words: its actuality – does not have an ultimate explanation in Σ* itself, nor, indeed, in any Realitysystem that embraces Σ*.

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6

The physics of H*

The singular term “H*” first occurs (outside the Table of Contents) – with a provisional description of its use – at the beginning of Sect. 4.2 and is finally described in Sect. 5.2 as a rigid designator for the T-entity that is also designated by “@*”, where “@*” is just short for “the actual* (complete T-)history”. Since the rules of ACTUHIST– given that a full round of ACTUHIST is played (which, in effect, was already assumed in Sect. 5.2.1) – make sure that there is exactly one actual* T-history, it is clear that “H*” will rigidly designate that history. Note a rather subtle (but after Kripke fairly familiar) point here: The singular term “H*” can be said to be defined by the singular term “@*” merely in the reduced sense that the latter term fixes the reference of “H*”, not in the full sense that it provides a synonym for “H*”. It is, in fact, impossible for “@*” to provide a synonym for “H*”, since the latter term is stipulated to be a rigid designator whereas the former term is not a rigid designator. In this book, a “rigid designator” is taken to be a designator that designates identically the same referent under all (possible) circumstances. This is not true of “@*” (“the actual* history”), although, given exactly one actual* history, “@*” designates that history in every Thistory, or, for that matter, in every T-world. (Regarding the concept of a T-world and the very close relationship of that concept to the concept of a T-history, see Sect. 3.6.) Thus, the singular term “the actual* history” (“@*”) is after all – though not a rigid designator (simpliciter) – a more rigid designator than, say, the singular term “the actualI history” (which term designates a different T-history in each T-history; regarding the indexical predicate “X is actualI”, see Sect. 5.2). In fact, if what counts as circumstances were restricted to the circumstances that are intrinsic to Model T, “@*” would be a rigid designator. But what counts as circumstances cannot be restricted to the circumstances that are intrinsic to T, since there is an outside to T, which is related to T actuality-wise. It is an indication of the existence of an outside to Reality that the Reality-term corresponding to “@*” – namely, “the (non-relatively) actual [Reality-] world” – seems to be a non-rigid designator (just like its T-analogue “@*”), although it designates the same entity in every possible world of Reality. In Sect. 5.3.1 the principles (A*) and (B*) were introduced:

(A*) For all X: if X is actual*, then there is an actual* T-history in which X is actual. (B*) For all X: if there is an actual* T-history in which X is actual, then X is actual*. If there is exactly one actual* T-history and if the singular term “H*” is used as described, then (A*) and (B*) entail the following: (C*) For all X: X is actual* if, and only if, X is actual in H*. But in contrast to (A*) and (B*), (C*) is far from being analytically true: (C*) states a contingent T-metaphysical fact about H*. So far, not much else is known about H*, except that H* is not identical with H°, “H°” rigidly designating the T-history described in Sect. 4. (See Sect. 5.1.1, where, in one place, “H° = @*” is assumed only for the sake of the argument and declared not to be true to the actual facts.) This chapter is dedicated to enhancing the description of H* by the specifying the laws (of nature) for Model T. The specification of the (natural) laws for T enhances the description of H* because the laws for T are laws of H* in a sense that can be gathered from the following schematic argument (cf. Sect. 4.2): (I) By being actual* H* is ipso facto a candidate for being actual*. (II) The laws for T in the secondary sense (under the extrinsic conception of lawhood) (see Sect. 4.1.1) are the following true statements: Law no. 1: For all T-histories H: if H is a candidate for being actual*, then the state of affairs ST1 is intrinsically implied by H. ………… Law no. N: For all T-histories H: if H is a candidate for being actual*, then the state of affairs STN is intrinsically implied by H. In other words: The laws for T in the secondary sense are the true statements which (a) say for one T-state-of-affairs after another in

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the sequence of T-states-of-affairs ST1, …, STN that it is a law in the primary sense for T, and which (b) spell out just what this means.57 Hence: (III) The laws for T in the primary sense (under the extrinsic conception of lawhood) (see Sect. 4.1.1) are the states of affairs ST1, …, STN, as described in (II). Therefore (as a logical consequence of (I), (II), and (III)): (IV) The laws for T – in the primary sense: the states of affairs ST1, …, STN – are intrinsically implied by H*, i.e., they are laws of H*. Now, the laws (in the primary sense) for T are not only laws of H*, but (all) the laws of H*. The sum of these laws is just the physics of H* – and we have arrived at an explanation of the title of this chapter. 6.1

The first two of the laws for T/the laws of H*

A law for T can be presented in three different ways (and also in a fourth way): 1) It can be straightforwardly named, usually by a that-phrase: this fits a T-law in the primary sense under the extrinsic conception of lawhood. 2) It can be metalinguistically named: this fits a T-law in the secondary sense under the extrinsic conception of lawhood. 3) It can be precisely implied, namely, by a stated rule of ACTUHIST: by precisely the rulecorrelate of the law (the correspondence between T-laws and certain ACTUHIST-rules is first stated in the subsection Concerning (I) of Sect. 5.2.2). Moreover, 4), there is the text-book way of presenting a T-law: it consists in asserting in a salient manner a statement which succinctly expresses a state of affairs that is a T-law in the primary sense; in this fourth way of presenting a T-Law, the best thing to do will often be to simply assert the statement already used in the corresponding first way of presenting a T-Law (that is, used there for forming the that-name of a state of affairs that is a T-law in the primary sense). For the sake of mutual elucidation, all 57

Given that ST is a T-state-of-affairs, “ST is a law in the primary sense for T” means just this: For all T-histories H: if H is a candidate for being actual*, then ST is intrinsically implied by H.

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four ways will be used side by side in each presentation of a law for T that is going to follow. The overarching guideline in decreeing the laws for T is to avoid absolute determinism – the situation that there is just one T-history that is a candidate for being actual* – and also to avoid free-first-two-steps determinism (compare Sect. 5.3.1, comment (e), where free-first-step determinism is explained), indeed to avoid any result which would render the game of ACTUHIST uninteresting, whether from the start or after the first two steps of the game or in a later phase of it that is not close to its end – because such a result would contradict the resolution at the end of the subsection Concerning (II) in Sect. 5.2.2. The First Law for T – presented in the first way – is the state of affairs that at every temporal position exactly four spatial positions are full and all other spatial positions are empty;58 – presented in the second way – is the following true statement: “For all T-histories H: if H is a candidate for being actual*, then the state of affairs that at every temporal position exactly four spatial positions are full and all other spatial positions are empty is intrinsically implied by H”; – presented in the third way – is precisely implied by the rule L759 of ACTUHIST: “At each step of the game the players of ACTUHIST select collectively, for being made actual*, a maximal momentary T-event in which four spatial positions are full and all others empty”; – presented in the fourth way – is given as follows:

At every temporal position exactly four spatial positions are full and all other spatial positions are empty. Comment: The First Law for T is a conservation law. According to it, the quantity of Fullness is constant throughout H*. In addition, the law specifies the numerical value of the constant quantity of Fullness: 4. (Hence the constant quantity of Emptiness is 96, and the constant ratio of Fullness to Emptiness in the space of T is 1/24.) 58

Compare R1 in Sect. 4.1. The numbering is in continuation of the list L1 – L6 (of ACTUHIST-rules) in Sect. 5.2.2. 59

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The Second Law for T – presented in the first way – is the state of affairs that, for each temporal position Z earlier than 100, each spatial position that is full at Z+1 is identical with a spatial position that is full at Z, or is at least immediately attached to such a spatial position;60 – presented in the second way – is the following true statement: “For all T-histories H: if H is a candidate for being actual*, then the following state of affairs is intrinsically implied by H: that, for each temporal position Z earlier than 100, each spatial position that is full at Z+1 is identical with a spatial position that is full at Z, or is at least immediately attached to such a spatial position”; – presented in the third way – is precisely implied by the rule L8 of ACTUHIST: “At each step of the game after the first the players of ACTUHIST select collectively, for being made actual*, a maximal momentary T-event in which each spatial position that is full is identical with a spatial position that is full in the maximal momentary T-event that was selected one step earlier, or is at least immediately attached to such a spatial position”; – presented in the fourth way – is given as follows:

For each temporal position Z earlier than 100, each spatial position that is full at Z+1 is identical with a spatial position that is full at Z, or is at least immediately attached to such a spatial position. Comment: The Second Law for T is a law that regulates change. According to it, change must be continuous – to the extent it still can be continuous in a space and time with discrete positions. The Second Law for T states what natura non facit saltus can still mean for H* despite of the fact that H* is a nature which, given the nature of its time, makes little jumps in time all the time, and which, given the nature of its space, cannot help making at least little jumps in space on each occasion when there is any change in it (already conforming to the First Law) at all.

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Compare R2 in Sect. 4.2. Regarding the concept of immediate attachment, see D7 in Sect. 3.2.1.

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6.2 The Third Law for T and the supervenience of atomic higher continuants relative to H* The presentation of the Third Law for T requires some preparation. First, it is necessary to introduce a new concept: Let H be a T-history: D35 A diachronic path of Fullness in H is a function (i) whose domain is an uninterrupted section of the timeline of T61 and (ii) which assigns to each temporal position Z in its domain a spatial position which is full at Z in H – in such a manner that at each temporal position Z´ before the last temporal position in that domain the spatial position that is assigned to Z´+1 is identical with or immediately attached to62 the spatial position assigned to Z´. D36 A complete diachronic path of Fullness in H is a diachronic path of Fullness in H whose domain is the timeline of T. Besides the predicate “X is a diachronic path of Fullness in H”, which is explicitly relativized to histories, there is the accompanying indexical predicate “X is a diachronic path of Fullness”, which is implicitly relativized to histories – such that “X is a diachronic path of Fullness” is true of X in a history H if, and only if, “X is a diachronic path of Fullness in H” is true of X. Neither of these two predicates is relativized, explicitly or implicitly, to temporal positions. The predicate “X is a diachronic path of Fullness” will be used below in the formulation of the Third Law. The use of a predicate which is implicitly relativized to histories, but is without implicit relativization to temporal positions is not without precedent in the formulation of the laws for T: both in the formulation of the First Law and of the Second Law, the predicate “X is full at Z” has been employed, the relativization pattern of this predicate being no. 7 of the list in Sect. 5.1, 61

Note that the set of all temporal positions and also all sets with a temporal position as sole element trivially count as uninterrupted sections of the timeline of T; it is useful to allow degenerate diachronic paths of Fullness – paths the domains of which comprise but a single temporal position. 62 Actually, “is immediately attached to” is already entirely adequate for saying what needs to be said here, since the identity of spatial positions is a special case of their immediate attachment (see D7 in Sect. 3.2.1). But I choose “is identical with or immediately attached to” for the sake of explicitness.

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that is: + – – + (whereas the relativization pattern of “X is a diachronic path of Fullness” is no. 15 in that list: – – – +). The First and the Second Law for T do not by themselves guarantee that H* is a history of the spatially continuous movements of 4 atomic higher continuants (relative to H*), that is: of 4 atomic T-material objects (compare Sect. 4). Those two laws do not guarantee that each of the intended four atomic objects is at each temporal position unequivocally located in the space of T, at a location that is distinct from the locations of the other three. They do not guarantee that it is unequivocally determined at each temporal position Z after 1 whether the atomic object that is taken to be located at one of the four spatial positions that are full at Z is identical or not with the atomic object that is taken to be located at one or another of the four spatial positions that are full at an arbitrary temporal position previous to Z. It is true that the First and the Second Law for T guarantee that there is a diachronic path of Fullness that can be traced backwards from any spatial position that is full (in H*) at a temporal position after 1 to a spatial position that is full (in H*) at 1. But so far nothing forbids that two or more diachronic paths of Fullness (in H*) lead back to different origins although (not only the first, but also) the last temporal position in the domains of these diachronic paths is the same and although they assign the same non-empty (in H*) spatial position to that latter temporal position. One can also put it in the following way … But first, another definition needs to be stated: D37 An atomic momentary Fullness-event is a momentary event which assigns an atomic momentary state ST to the sole temporal position in its domain, where ST is such that, for some spatial position P, ST is identical with the state of affairs that P is full.63 Now, so far, nothing forbids that one and the same atomic momentary Fullness-event in H* (such an event is in H* if, and only if, the atomic momentary state it assigns to the temporal position in its domain is a conjunct of the maximally composite momentary state that H* assigns to that same temporal position) can with equal justification be claimed by two or more of the intended higher continuants – the intended 4 atomic material 63

For more regarding atomic momentary states, see Sect. 3.3.1. An example of a momentary event – and in fact of an atomic momentary Fullness-event – can be found at the beginning of Sect. 3.5. (Compare also maximal momentary events, which were introduced in Sect. 3.6.) On events in general, see Sect. 3.4.

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objects – as an “assertion of their presence”. Let me make this plain by presenting a very simple example-situation: Consider the lower left-hand corner of the space of T. Suppose that the spatial positions and are full at 1 in H*. The other two spatial positions that are full at 1 in H* (in accordance with the First Law) are supposed to be fairly far away, say, in the fifth row of the space of T. Suppose further that the spatial positions and are full at 2 in H* ( being empty at 2 in H*). We leave out of consideration (for the time being) what has happened in going from the first to the second temporal position in the fifth row of the space of T; but the entire change that has occurred in H* will of course conform to the First and the Second Law. There is, however, no uniquely correct way of interpreting this change as a totality of movement of the intended four atomic objects. Each way of interpreting a change (including, as a limiting case, constancy) in H* – for example, from one momentary phase of H* to the next – as the totality of movement (including, as a limiting case, rest) of the four intended atomic objects in H* from one time to another is regulated by the following six postulates – regulated in the sense that every such way of interpretation must make true (relative to H*) the conjunction of them; in the postulates, “T-CH” stands for the uninterrupted timeline-section of T in the course of which the relevant change occurs: Postulate 1 At all times [i.e., temporal positions] in T-CH there exist the same 4 atomic material objects. Postulate 2 An atomic material object exists at a time Z in T-CH if, and only if, it is at Z in some place [i.e., spatial position]. Postulate 3 No atomic material object is in more than one place at any time in T-CH. Postulate 4 No two atomic material objects are in the same place at any time in T-CH. Postulate 5 For all times Z in T-CH that are earlier than the last time in TCH: every place an atomic material object is in at Z+1 is identical with or immediately attached to the place the object is in at Z. Postulate 6 There is an atomic material object in place P at time Z in T-CH if, and only if, P is full at Z. Now (referring to what was assumed about H* above), is it the case that the atomic material object that is located at at 1 is still in that

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place at 2 whereas the atomic material object that is located at at 1 has moved diagonally down to be at at 2? Or is it the case that the atomic material object that is located at at 1 has moved straight to the right to be at at 2 whereas the atomic material object that is located at at 1 has moved straight down to be at at 2? There is no way of finding out which one of these two questions is to be answered affirmatively and which one negatively – because there exists no unique fact of the matter that could uniquely determine an affirmative or negative answer to either one of the two questions. The reason for this is the following: there is a way of interpreting the relevant entire change – making true all of the above six postulates – according to which the first question is answered affirmatively, the second negatively; and there is a way of interpreting that same entire change – also making true all of the above six postulates – according to which the second question is answered affirmatively, the first one negatively.64 The analysis of the situation in the lower left-hand corner in terms of diachronic paths of Fullness is the following: As matters have been presented above, there are four short diachronic paths of Fullness in H* in the lower left-hand corner of the space of T, each having for domain the section {1, 2} of the timeline of T: χaA(1) = χaB(1) = χbA(1) = χbB(1) =

χaA(2) = χaB(2) = χbA(2) = χbB(2) =

[a-path of object A]; [a-path of object B]; [b-path of object A]; [b-path of object B].

Clearly, the diachronic paths χaA and χaB are those that suggest the interpretation in terms of atomic-material-object-movements which is stated in the first of the above two questions, and the diachronic paths χbA and χbB are those that suggest the interpretation in terms of atomic-material-objectmovements which is stated in the second of the above two questions. But note that χaB(2) = χbA(2) but χaB(1) ≠ χbA(1), and note that χaA(2) = χbB(2) but χaA(1) ≠ χbB(1). The identity of termination and diversity of origin for χaB and χbA and for χaA and χbB show that the described situation does not of itself determine a uniquely correct interpretation of it in terms of 64

For demonstrating these assertions, one must, of course, also take into account what has happened, in going from 1 to 2, in the above-mentioned fifth row (of the space of T), and not only what has happened in the lower left-hand corner. A fitting specification of what has happened in the fifth row is given further below.

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atomic-material-object-movements: it cannot be read off the described situation whether the object that is at at 2 is atom B or atom A, just as it cannot be read off the described situation whether the object that is at at 2 is atom A or atom B. Note, moreover, that χaA(1) = χbA(1) but χaA(2) ≠ χbA(2), and χaB(1) = χbB(1) but χaB(2) ≠ χbB(2) – that is, we have identity of origin and diversity of termination for χaA and χbA and for χaB and χbB. Thus, it also cannot be read off the described situation whether atom A is at 2 still at or has gone to , and whether atom B is at 2 at or, on the contrary, at . Once again, the described situation does not of itself determine a uniquely correct interpretation of it in terms of atomic-material-object-movements. What can be done about this, given that H* is to be a history of the spatially continuous movements of 4 atomic T-material objects? One thing one might do is to forbid certain ways of movement for the intended atomic objects. If diagonal movements were forbidden, then there would be only one interpretation in terms of atomic-material-object-movements for the above-described situation (namely, the interpretation suggested by the diachronic paths χbA and χbB in H*); a fortiori it would be the uniquely correct interpretation. But this forbidding of diagonal movements seems utterly ad hoc – and, in any case, it would not suffice for making go away all situations which are ambiguous regarding the atomic-material-objectmovements that occur in them. For consider now the fifth row of the space of T. Suppose that the spatial positions and are full at 1 in H* (H* being ruled by the First and the Second Law). Suppose further that the spatial positions and are full at 2 in H*. Suppose finally that these same spatial positions are also full at 3 in H*. Thus, the analysis of the supposed situation in terms of diachronic paths of Fullness is this: χaC(1) = χaD(1) = χbC(1) = χbD(1) =

χaC(2) = χaD(2) = χbC(2) = χbD(2) =

χaC(3) = [a-path of object C]; χaD(3) = [a-path of object D]; χbC(3) = [b-path of object C]; χbD(3) = [b-path of object D].

The interpretation in terms of atomic-material-object-movements suggested by χaC and χaD is that atom C and atom D move towards each other – touching – and then stop. The interpretation in terms of atomic-materialobject-movements suggested by χbC and χbD, however, is that atom C and atom D move towards each other – touching – and then move through each

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other. Clearly, the situation in the fifth row of the space of T is just as ambiguous regarding atomic-material-object-movement as is the previously considered situation in the lower left-hand corner of the space of T, notwithstanding the fact that diagonal atomic-material-object-movements cannot be extracted from the situation in the fifth row. There shall be no tinkering. In order to make H* a history of the spatially continuous movements of 4 atomic T-material objects, the following law is decreed in addition to the First and the Second Law: The Third Law for T – presented in the first way – is the state of affairs that all diachronic paths of Fullness65 that differ from each other but have the same domain do not assign the same spatial position to any temporal position; – presented in the second way – is the following true statement: “For all T-histories H: if H is a candidate for being actual*, then the following state of affairs is intrinsically implied by H: that all diachronic paths of Fullness that differ from each other but have the same domain do not assign the same spatial position to any temporal position”; – presented in the third way – is precisely implied by the rule L9 of ACTUHIST: “At each step of the game after the first the players of ACTUHIST select collectively, for being made actual*, a maximal momentary T-event which is related to the maximal momentary T-event selected one step earlier in the following way: Let M1 be the set of spatial positions that are full in the maximal momentary event selected one step earlier, let M2 be the set of spatial positions that are full in the currently selected maximal momentary event. Consider the ordered pairs in the Cartesian product M1×M2. Restrict your attention to those ordered pairs in M1×M2 which are such that their second member is identical to their first member or is (at least) immediately attached to it. No two of these ordered pairs have their first or second member in common”; – presented in the fourth way – is given as follows:

All diachronic paths of Fullness that differ from each other but have the same domain do not assign the same spatial position to any temporal position.

65

Here the indexical predicate “X is a diachronic path of Fullness”, described further above (see below D36), makes its appearance.

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Comments: (1) It is a consequence of the Third Law together with the First and the Second Law that there are exactly four complete diachronic paths of Fullness in each T-history which is a candidate for being actual* – and therefore also in H*; indeed, there are exactly four diachronic paths of Fullness in H* for each uninterrupted section of the timeline of T. (2) Each spatial position that is full in H* at a temporal position Z belongs at Z to (i.e., is assigned to Z by) exactly one of the four complete diachronic paths of Fullness in H*. (3) The First, Second, and Third Law make it possible to found the transtemporal identity of the intended four atomic higher continuants relative to H* on the gen-identity of atomic momentary Fullness-events in H*.66 According to these laws, each atomic momentary Fullness-event in H* is connected via a unique (uninterrupted) sequence of atomic momentary Fullness-events in H* – namely, via the event-sequence which is oneto-one correlated (in the obvious way) with a certain diachronic path of Fullness in H* – to exactly one of the four absolutely original atomic momentary Fullness-events in H* (absolutely original in the sense that they have the temporal position 1 as the sole element in their domains); furthermore, different but simultaneous atomic momentary Fullness-events in H* are connected in the described manner to different absolute origins in H* (i.e., to different absolutely original atomic momentary Fullness-events in H*). Then we have: For all atomic momentary Fullness-events E and E´ in H*: D38 E is gen-identical with (or: to) E´ in H* =Def E and E´ have (i.e., are connected to) the same absolute origin in H* (in the manner described above). For all atomic material objects c in H* and temporal positions Z that are such that c exists at Z (in H*): D39 ε(c, Z) is the atomic momentary Fullness-event in H* that represents c at Z (that “asserts the presence of c at Z”). For all atomic material objects c and c´ in H* and all temporal positions Z and Z´ that are such that c exists at Z in H* and c´ at Z´ in H*: 66

For the definition of atomic momentary Fullness-events, see D37 above. Following that definition, it is also explained what it means for such events to be in H*.

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D40 c, at Z, is the same object as c´, at Z´ =Def ε(c, Z) is gen-identical with ε(c´, Z´) in H*. Now, let P3a, P3b, P3c and P3d be the four spatial positions that are full at 3 in H* (to these spatial positions, the letters “a”, “b”, “c”, and “d” are arbitrarily assigned) and let P20a, P20b, P20c and P20d be the four spatial positions that are full at 20 in H* (to these spatial positions, too, the letters “a”, “b”, “c”, and “d” are arbitrarily assigned; hence P20a, for example, need not be the same spatial position as P3a, nor be in any other interesting spatiotemporal way related to P3a). The four atomic momentary Fullness-events in H* that correspond to P3a, P3b, P3c, and P3d are {}, {}, {}, and {}; and the four atomic momentary Fullness-events that correspond to P20a, P20b, P20c, and P20d are {}, {}, {}, and {}. Let, finally, P1a, P1b, P1c, and P1d be the four spatial positions that are full at 1 in H* (again the letters “a”, “b”, “c”, and “d” are assigned arbitrarily); the four atomic momentary Fullness-events in H* that correspond to these latter spatial positions – that is, the absolutely original atomic momentary Fullness-events in H*, or the absolute origins in H* – are {}, {}, {}, and {}. Preliminary to the main act of theory in this section, I state the following indispensable meaning-postulate for the predicate “E represents c at Z” (which predicate is first used in D39 above): Postulate for “E represents c at Z” For every atomic momentary Fullness-event E in H*, temporal position Z, and atomic material object c in H*: E represents c at Z only if the domain of E is {Z}. There will be occasion to use this postulate, but, in view of its fairly trivial nature (which is apparent, once the postulate is explicitly acknowledged), I shall do so tacitly. Now, the pivotal principle that establishes the supervenience of the atomic higher continuants relative to H* – that is, of the atomic material objects in H* – on the atomic momentary Fullness-events in H* is the following postulate, consisting of three sub-postulates:

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Supervenience Postulate (i) For every atomic material object c in H*: there is exactly one atomic momentary Fullness-event E with the domain {1} in H* that represents c at 1. (ii) For every atomic momentary Fullness-event E with the domain {1} in H*: there is exactly one atomic material object c in H* that is represented by E at 1. (iii) For all atomic momentary Fullness-events E and E´ in H* and all temporal positions Z and Z´ such that the domain of E is {Z} and the domain of E´ is {Z´}: E´ represents at Z´ the same atomic material object in H* that E represents at Z if, and only if, E´ is gen-identical with E in H*. On the basis of the First, Second, and Third Law for T, the Supervenience Postulate, one might say, guarantees that the basic H*-facts about the atomic material objects in H* – that is, where each of these objects is at each time – are reducible to the atomic momentary Fullness-events in H*. The Supervenience Postulate, one might say, makes those facts (and the objects they are about) projections of these events (only thus justifying the use of the term “supervenience”), rendering the former nothing over and above the latter. According to the Supervenience Postulate, one might say, the basic H*-facts about the atomic material objects in H* are no more than an interpretation, a reading of the atomic momentary Fullness-events in H*. Note, however, that what is suggested is itself according a reading to what is stated by (i), (ii), and (iii) – a reading which is not the only possible one. There is another interpretation of (i), (ii), and (iii), according to which these statements simply describe the intrinsic relationship between two sides of the whole of H*-being, two sides that are very much equals with regard to their H*-ontological status. This is the interpretation (and implied view of supervenience) that I shall adopt here. Still, the atomic momentary Fullness-events in H* have a certain T-ontological (though not necessarily also an H*-ontological) priority over the basic H*-facts about the atomic material objects in H*. The reason for this is that in every Thistory (except one: the completely empty one) there occur some atomic momentary Fullness-events, whereas very many T-histories do not qualify for hosting (lasting) atomic material objects (let alone more complex higher continuants). In this sense, the atomic momentary Fullness-events in H* are more basic in their T-ontological status than the basic H*-facts about the atomic material objects in H*, and therefore those events are T-

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ontologically prior to these facts. Note also – in the matter of comparative T-ontological basicality – that every atomic momentary Fullness-event occurs, in an obvious and elementary sense, in many more than one history; the transhistorical identity of atomic material objects, however, is neither an obvious nor an elementary matter; the rather tricky subject is treated in Sect. 6.3. According to the Supervenience Postulate, each of the absolutely original atomic momentary Fullness-events in H* represents – at 1 – exactly one atomic material object in H* (according to (ii)), and no two of those events represent – at 1 – the same atomic material object in H* (otherwise (i) would be contradicted), and there is no atomic material object in H* that is not represented – at 1 – by an absolutely original atomic momentary Fullness-event in H* (according to (i)). Thus, there is a perfect match between the absolutely original momentary Fullness-events in H* and the atomic material objects in H*. Representing “full” suggestively by “„” (cf. p. 12), we have: {}, {}, {}, {} ↑1↓ ↑1↓ ↑1↓ ↑1↓ c+ c# c> c~.

Suppose now that one of the four complete diachronic paths of Fullness in H* goes through P1a, P2d, P3c, … P20d …, ending in P100b, another through P1b, P2a, P3d, … P20a …, ending in P100c, another through P1c, P2b, P3b, … P20c …, ending in P100d, and yet another – the fourth one – through P1d, P2c, P3a, … P20b …, ending in P100a: χ1: χ2: χ3: χ4:

P1a P1b P1c P1d

P2d P2a P2b P2c

P3c P3d P3b P3a

… … … …

P20d P20a P20c P20b

…… …… …… ……

P100b. P100c. P100d. P100a.

The sequences of atomic momentary Fullness-events in H* that are one-toone correlated with these diachronic paths of Fullness in H* are the following: σ1: σ2:

{}, {}, {}, …, {}, … …, {}. {}, {}, {}, …, {}, … …, {}.

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σ3: σ4 :

{}, {}, {}, …, {}, … …, {}. {}, {}, {}, …, {}, … …, {}.

All atomic momentary Fullness-events in H* that belong to σ1 are genidentical to each other in H* (employing D38). It follows on the basis of (iii) of the Supervenience Postulate that all atomic momentary Fullnessevents in H* that belong to σ1 represent, at their respective temporal positions, the same atomic material object in H*, namely c+ (since c+ is the object that is represented by {} at 1; see above); this object is in place P1a at 1, in place P2d at 2, in place P3c at 3, …, in place P20d at 20, … …, in place P100b at 100. Entirely analogous results can be obtained for each of the three other sequences: σ2, σ3, and σ4. For each of these sequences, those results concern the atomic material object in H* that is correlated with the sequence’s origin (i.e., with its first member): for σ2 it is c#, for σ3 it is c>, and for σ4 it is c~ (see above). By employing the definition that has just (in the previous paragraph) been made use of already (but without explicit mention), For all atomic material objects c in H*, spatial positions P, and temporal positions Z: D41 c is in place P at Z =Def {} is an atomic momentary Fullness-event in H* and {} represents c at Z, the above six postulates for atomic material objects are derivable on the basis of the already given three laws for T and the Supervenience Postulate – no matter with which uninterrupted section of the T-timeline T-CH is being identified (T-CH may even comprise just a single temporal position). In order to derive those postulates, Postulate 2 must be treated as a mere definition, and the predicates “X is an atomic material object”, “X is in place P at time Z”, “X exists at time Z”, “Y is full at time Z”, which occur in the postulates, must of course be explicitly or implicitly supplemented by the appropriate reference to H*. It may be helpful for readers to keep in mind the following remarks on matters of formulation. The implicit way of H*-relativization can be explicitly represented as follows: “X is an atomic material object (in H*)”, “X is in place P at time Z (in H*)”, “X exists at time Z (in H*)”, “Y is full at time Z (in H*)” (and square brackets may replace the round ones). The explicit way of H*-relativization results from the explicit

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representation of the implicit way by simply omitting the brackets. One may go the implicit way of H*-relativization with a predicate at one time, and the explicit way with it at another time, and one may go the implicit way with one predicate, and the explicit way with another predicate. The explicit representation of implicit H*relativization is merely an occasional reminder for the reader; normally, implicit H*relativization is, of course, tacit (since it is implicit).

Suppose the question arises whether a certain atomic material object c in H*, which exists at time Z (in H*), in other words (according to the definitional Postulate 2): which is located at Z in some place, is identical or not with a certain atomic material object c´ in H*, which exists – is located in some place – at time Z´ (in H*). On the basis of the Supervenience Postulate and the already given three laws for T, there is always an unequivocal answer to this question: Since c is located in some place at Z, there is, according to D41, some spatial position P which is such that {} is an atomic momentary Fullness-event in H* and such that {} represents c at Z; and since c´ is located at some place at Z´, there is, according to D41, some spatial position P´ which is such that {} is an atomic momentary Fullness-event in H* and such that {} represents c´ at Z´. Now (employing D39), ε(c, Z) = {}, and ε(c´, Z´) = {}.67 Therefore, if {} and {} are gen-identical in H*, then c, at Z, is identical with c´, at Z´ (according to D40); if, however, {} and {} are not gen-identical in H*, then c, at Z, is not identical with c´, at Z´ (according to D40). 6.3

Democriteanism and four-dimensionalism for H* and the transhistorical identity of atomic higher continuants

The definitions D39 and D40 in the previous section are both restricted to atomic material objects c in H* and to temporal positions Z which are such that c exists at Z (in H*). As a matter of fact, it is unnecessary to include “c 67

For suppose that there were another atomic momentary Fullness-event in H* besides – for example – {} that also represented c at Z: {}, P´´ being be a spatial position that is different from P. Hence, according to (iii) of the Supervenience Postulate, {} and {} would have to be gen-identical (in the sense of D38). But since P´´ is different from P, the already given laws for T do not allow this gen-identity.

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exists at Z” (and the same is of course true of “c´ exists at Z´”) in the formulation of the respective restricting conditions to those definitions – because it is easily seen that every atomic material object in H* exists at every temporal position. First, according to (i) of the Supervenience Postulate, every atomic material object in H* is represented at 1 by an atomic momentary Fullnessevent in H* with the domain {1}. Hence (making use of D37), for every atomic material object c in H*, there is some spatial position P which is such that {} is an atomic momentary Fullness-event in H* and such that {} represents c at 1. Hence (according to D41) every atomic material object in H* is in some place at 1, and therefore (according to Postulate 2, T-CH being {1}): Every atomic material object in H* exists at 1. Second, suppose some atomic material object c in H* that exists at 1 does not exist at a later temporal position Z. Hence c is in no place at Z (according to Postulate 2, T-CH being {Z}), and therefore: there is no atomic momentary Fullness-event in H* that represents c at Z (according to D41, D37). But, on the other hand, there is an atomic momentary Fullness-event in H* with the domain {Z}, ε(Z), that belongs to a sequence of momentary Fullness-events in H* (correlated with a diachronic path of Fullness in H*), of which the absolute origin in H* is this: ε(c, 1), in other words (according to D39): the atomic momentary Fullness-event in H* that represents c at 1. This much follows on the basis of the already given three laws for T. Hence ε(Z) is gen-identical with ε(c, 1) in H* (according to D38), and therefore (according to (iii) of the Supervenience Postulate): ε(Z) represents at Z the same atomic material object in H* that ε(c, 1) represents at 1 – which is c. Hence there is an atomic momentary Fullnessevent in H* that represents c at Z – contradicting what has already been concluded on the basis of the initial supposition. That supposition is therefore shown to be false, and therefore: Every atomic material object in H* that exists at 1 exists at every later temporal position. Taken together, the (italicized) statements demonstrated in the two preceding paragraphs establish what was to be shown: Every atomic material object in H* exists at every temporal position. Thus, the atomic material objects in H* – c+, c#, c>, and c~ – are everlasting continuants relative to H*. Their everlastingness as continuants they have in common with the spatial configurations of T (see Sect. 3.2.1). However, the spatial configurations of T are, as everlasting continuants tout court, also continuants simpliciter (for this concept, see Sect. 3.2.1), “simpliciter” indicating not

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merely their fulfilment of the minimal criteria for being continuants, but also the absoluteness of their being continuants; they are continuants simpliciter, but the atomic material objects in H* are not. Even if it is not always made explicit (in order not to burden sentences overmuch with the relativizing phrase “in H*”), existence at a time, being located at a time in a place, etc are – for c+, c#, c>, and c~ – relative to H*. (There is also a reason other than the mere avoidance of clumsiness why the phrase “in H*” does not occur in the definienda of D40 and D41 and in the other predicates that need to be understood in a way that is relative to H*: H* is the actual* T-history, and “H*” rigidly designates that history. This kind of absoluteness – though non-intrinsic – gives H* a certain privilege: the relativizing reference to H* can be left tacit, implicit.) As one will remember, Democritus’ atoms were everlasting continuants (in a world-relative sense), too. In all other respects, the atomic material objects in H* are, of course, utterly different from the atoms of ancient atomism68 – and, for that matter, utterly different also from today’s elementary particles (though, in contrast to today’s “atoms”, elementary particles are regarded as being really atomic). Obviously, the physics of H* is not physics – and yet it may teach us many things about metaphysics. Here is an example. The laws for T – already the first three, the ones already given – together with the Supervenience Postulate guarantee that there is a one-to-one correlation between the atomic continuants in H* and the complete diachronic paths of Fullness in H*. Now, each complete diachronic path of Fullness in H* is one-to-one translatable into a temporally complete sequence of atomic momentary Fullness-events in H*, and each temporally complete sequence σ of atomic momentary Fullness-events in H* is, in turn, one-to-one translatable into an event which is a part-event of H*, namely, into the event the domain of which is the entire timeline of T and which assigns to each temporal position Z the momentary state that the atomic momentary Fullness-event with the domain {Z} in the sequence σ assigns to Z. The obviousness of the translations is best demonstrated by an illustrative example (using material from the previous section): χ1: σ1:

P1a P2d P3c … P20d … … P100b. {}, {}, {}, …, {}, … …, {}.

68

Suffice it to mention that the ancient atomists saw their atoms in a limitless space and a limitless time.

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ε1:

{, , , …, , … …, }.

χ1 is a complete diachronic path of Fullness in H*, σ1 is the temporally complete sequence of atomic momentary Fullness-events in H* which is (T-ontologically) intertranslatable with χ1, and ε1 is the temporally complete worm of Fullness in H* which is (T-ontologically) intertranslatable both with σ1 and χ1. Just like each of the momentary events in σ1, ε1 is a part-event of H* (and therefore in H*), according to the following definition: For all T-events E and E´: D42 E is a part-event of E´ =Def The domain of E is a subset of the domain of E´, and for every temporal position Z in the domain of E: the momentary state that E allots to Z is intrinsically implied by the momentary state that E´ allots to Z. Since the atomic material object c+ is one-to-one correlated with the first atomic momentary Fullness-event in σ1 (see the previous section), it is also one-to-one correlated with σ1 itself (the laws for T and the Supervenience Postulate guarantee this) – and therefore also with χ1 and ε1 (because χ1 and ε1 are, in their turn, one-to-one correlated with σ1, as we have just seen). But the atomic material object c+ does not seem to be positively identifiable with either χ1, or σ1, or ε1. Rather, χ1, σ1, and especially ε1 merely seem to impersonate – in different but intertranslatable ways – the individual history of c+ in H*. Not every one-to-one correlation is a case for identification. Nevertheless, if a position in the metaphysics of material objects, which many believe to be true of Reality and which is known as four-dimensionalism, were – mutatis mutandis – also true of Model T, then c+ would be identical with ε1; and c#, c>, and c~ would be identical with ε2, ε3, and ε4 respectively, where ε2, ε3, and ε4 are related in the same manner to σ2, σ3, and σ4 as ε1 is related to σ1, with c# corresponding to σ2, c> to σ3, and c~ to σ4 just as c+ corresponds to σ1 (the basis for this is found in the previous section). For, according to four-dimensionalism as applied to Model T and in particular to H*, the atomic material objects in H* just are their individual histories in H*, and histories, as regards ontological category, are best considered to be events – which makes ε1, ε2, ε3, ε4 (which are events) better

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candidates for identification with c+, c#, c>, c~ than χ1, χ2, χ3, χ4 and σ1, σ2, σ3, σ4 (which are not events). Four-dimensionalism for H*, therefore, has a different conception of what it is to be a material object than the conception that has here been adopted. According to four-dimensionalism for H*, c+, c#, c>, and c~ – being events – have a fourth, a temporal dimension, in addition to their two proper spatial dimensions and their one improper – stunted – spatial dimension (I wish to use the word “four-dimensionalism” also in speaking about T); and they have temporal parts in H* (and in every T-history of which they are part-events) in the following sense: For all T-events E and E´: D43 E is a temporal part of E´ =Def The temporal configuration of E [i.e., the domain of E] is a temporal part of the temporal configuration of E´69, and to every temporal position in the domain of E, E assigns the same momentary state that E´ assigns to it. However, according to the conception that has here been adopted, c+, c#, c>, and c~ are higher continuants (relative to H*), and therefore they do not have a temporal dimension70 besides their three spatial dimensions (two of these being proper dimensions, and one an improper, stunted one), with the consequence that temporal parts are inconceivable for them, whereas they do have – in H* – a spatial part at each time at which they exist. (But note that c+, c#, c>, and c~ – since they are the atomic material objects in H* – have, at all times, only a single, improper spatial part: the atomic material object itself. Thus: What are the spatial parts of c+ at 85 in H*? – It is c+, and only c+.) It is a point in favour of four-dimensionalism that, according to it, the consideration of atomic T-material objects relative to H* does not require the introduction of a separate T-ontological category (i.e., higher continuant relative to H*), since, according to four-dimensionalism, atomic Tmaterial objects in H* belong to a category of entities that is already in use: event. Thus, regarding the atomic material objects in H*, the maxim known 69

Compare D14 in Sect. 3.2.2. Often, in the literature, the word “continuant” is used in such a manner that fourdimensional entities, too, are called “continuants” (provided they are not momentary entities). This is not in accordance with what seems to be the original use of the expression “continuous in time” as a properly ontological term – by Eddy Zemach in his 1970 paper “Four Ontologies” (The Journal of Philosophy 67, pp. 231-247).

70

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as “Ockham’s Razor” – entia non sunt multiplicanda praeter necessitatem – seems to be better served by four-dimensionalism than by the supervenience theory adopted in the preceding section, according to which theory the basic H*-facts about the atomic material objects in H* and these objects themselves are indeed strictly based on the atomic momentary Fullness-events in H*, but are not reducible to those events; nor are they reducible, according to that theory, to the larger part-events of H* that can be formed out of the atomic momentary Fullness-events in H*. Given the apparent Ockham-advantage of four-dimensionalism, why should one resist the identification of the atomic material objects in H* with their individual histories in H*? – One should resist this identification because that apparent advantage is merely apparent. Ockham’s Razor does not enjoin us to be ontologically parsimonious at all costs; it merely enjoins us not to go in our ontological assumptions and distinctions beyond what is needed – not praeter necessitatem. And there is considerable need to distinguish the atomic material objects in H* from their individual histories in H* – because statements about the former may easily be true that do not easily translate into true statements about the latter. As matters have been presented in Sect. 6.2, the following statement is true: “c+ is in place P2d at 2 [in H*]”. Suppose the following statement is also true: “The state of affairs that c+ is in place Pg at 2 is possible*”, where Pg is a spatial position that is different from P2d. According to D27 in Sect. 5.3, this implies: the state of affairs that c+ is in place Pg at 2 is intrinsically implied by some T-history that is a candidate for being actual*. Let this history be called “Halt”. Halt is not identical with H*. In H*, c+ is in place P2d at 2, whereas, in Halt, c+ is in a different place at 2: in Pg; Halt cannot be identical with H*, for otherwise c+ would be in two places at the same time (in H*) – contradicting Postulate 3 (which can be shown to be true for H*, with {2} as T-CH; see the remarks in Sect. 6.2). It is clear by now how it is that c+ is in place P2d at 2 in H*; it is not clear, so far, how it is that c+ is at the same time in place Pg (which is different from P2d) in Halt (which is different from H*). However, since Halt is a T-history which is a candidate for being actual*, the First, Second, and Third Law for T describe Halt just as truly as they describe H*, and it is therefore entirey natural to generalize the Postulate for “E represents c at Z” and the Supervenience Postulate (see the preceding section) in the following way:

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Postulate for “E represents c at Z in H” 71 For every history H, atomic momentary Fullness-event E in H, temporal position Z, and atomic material object c in H: E represents c at Z in H only if the domain of E is {Z}. Generalized Supervenience Postulate72 Let H be any history which is a candidate for being actual*: (i) For every atomic material object c in H: there is exactly one atomic momentary Fullness-event E with the domain {1} in H that represents c at 1 in H. (ii) For every atomic momentary Fullness-event E with the domain {1} in H: there is exactly one atomic material object c in H that is represented by E at 1 in H. (iii) For all atomic momentary Fullness-events E and E´ in H and all temporal positions Z and Z´ such that the domain of E is {Z} and the domain of E´ is {Z´}: E´ represents at Z´ in H the same atomic material object in H that E represents at Z in H if, and only if, E´ is genidentical with E in H. It is, in principle, easy to transform the definitions D38 – D41 in accordance with these generalized postulates, which transformation involves four measures (following the prefixing of the condition “For all histories H that are candidates for being actual* ...” to D38 – D41): (1) replacing “H*” by “H”; (2) explicitly relativizing the definienda with the phrase “in H” (if not already thus relativized after (1)) and, in the case of D39, replacing “ε(c, Z)” by “ε(c, Z, H)”; (3) replacing “ε(c, Z)” and “ε(c´, Z´)” in the definiens of D40 by “ε(c, Z, H) and “ε(c´, Z´, H)”, (4) adding “in H” to “represents c at Z” in the definiens of D41. It is also easy to understand the Postulates 1 – 6 (turning Postulate 2 into a definition) relative to H – where H can be any T-history which is a candidate for being actual* (and need not be H*). Thus, we have established the parity of theoretical environment (so to speak) for H* and Halt, which parity is necessary for answering the ques71

This postulate entails the earlier Postulate for “E represents c at Z” because H* is a history and the predicate “E represents c at Z” is logically equivalent to the predicate “E represents c at Z in H*”. 72 This postulate entails the earlier Supervenience Postulate because H* is a history that is a candidate for being actual* and the predicate “E represents c at Z” is logically equivalent to “E represents c at Z in H*”.

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tion of how it is that c+ is in place P2d at 2 in H* and in place Pg at 2 in Halt. Is it that c+ is in person (so to speak), and not via a counterpart, in one place in one history and, at the same time, in another place in another history? This, initially, may say seem implausible, even inconceivable. Nevertheless, I propose to pursue precisely this seemingly implausible line of thinking. But note, for going ahead in the chosen direction, it is essential not to identify c+ with its individual history in H*, which is a certain temporally complete worm of Fullness in H*: ε1. For if c+ were ε1, it simply could not be in person in Halt (and hence not in person anywhere anytime in Halt): ε1 is obviously not a part-event of Halt, and there is no other way in which ε1 could be in person in Halt. In contrast, by being a part-event of H*, ε1 is of course in person in H*, and, derivatively, ε1 is in person in a place P at a time Z in H* if, and only if, the atomic momentary Fullnessevent {} is a temporal part of ε1. The central question is this: What makes it be the case that the atomic material object in Halt which is in place Pg at 2 in Halt is identical with the atomic material object in H* which is in place P2d at 2 in H* (which latter object – it was determined – is no other object than c+)? – There is a fairly straightforward answer to this question: the (maximally composite) momentary state that Halt assigns to 1 is identical with the (maximally composite) momentary state that H* assigns to 1; or equivalently: both Thistories have the same origin, that is, the same maximal momentary event with domain {1} is a part-event of both. For seeing how this helps, consider the following: What are the atomic material objects in H* and where they are in H* initially is determined by the momentary state that H* assigns to 1. In turn, what are the atomic material objects in Halt and where they are in Halt initially is determined by the momentary state that Halt assigns to 1. But now: the momentary state that Halt assigns to 1 is identical with the momentary state that H* assigns to 1. Therefore, the atomic material objects in Halt are identical with the atomic material objects in H*, and the former are initially in Halt in the very same places that the latter are in initially in H*. Therefore, c+ is in place P1a at 1 in H*, and c+ is also in place P1a at 1 in Halt. That c+ is in place Pg at 2 in Halt, not in P2d, and that c+ is in place P2d at 2 in H*, not in Pg, is explained by the fact that {}, but not {}, is genidentical with {} in H*, whereas {}, but not {}, is gen-identical with {} in Halt.

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There is no other way than the identity of origins of T-histories that are candidates for being actual* for explaining – for making fully transparent – the transhistorical identity of atomic higher continuants. For suppose the origins of Halt and H* were different: in the (maximally composite) momentary state that Halt assigns to 1, the four spatial positions that are full – it is supposed – are not the same four spatial positions that are full in the momentary state that H* assigns to 1. This immediately introduces a problem of identification. Even if the origins of Halt and H*, though different, are partly identical regarding Fullness: even if a spatial position P is full at 1 in H* and full at 1 in Halt, it now seems somewhat arbitrary to say that {} represents at 1 in Halt the same atomic material object that it represents at 1 in H*. (One might stipulate this identity of represented object, but stipulation always has an element of – objective – arbitrariness.) But there arises a disturbing question: Even though the origins of Halt and H* are identical, might not those origins (or rather: the atomic momentary Fullness-events in them) still represent differently in Halt and H* with regard to atomic higher continuants? For H*, we have noted the following correspondence between the (represented) atomic material objects in H* and the (representing) absolutely original atomic momentary Fullnessevents in H* (see Sect. 6.2): {}, {}, {}, {} ↑1↓ ↑1↓ ↑1↓ ↑1↓ c+ c# c> c~.

If the origins of Halt and H* are identical, then the absolutely original atomic momentary Fullness-events in Halt are exactly the same as the absolutely original atomic momentary Fullness-events in H*: {}, {}, {}, {}. But, still, might we not have a different correspondence for Halt between the atomic material objects in that history and the absolutely original atomic momentary Fullness-events in it than we have for H*? Might not the correspondence look like this (for example): {}, {}, {}, {} ↑1↓ ↑1↓ ↑1↓ ↑1↓ c# c~ c+ c>,

or – worse – like this:

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{}, {}, {}, {} ↑1↓ ↑1↓ ↑1↓ ↑1↓ c§ c& c? c!,

where each of the atomic material objects in Halt – c§, c&, c?, and c! – is distinct from each of the atomic material objects in H*: c+, c#, c>, and c~? However, if we allow that such things might be, we are, in effect, ignoring what was said above: [I], What are the atomic material objects in H* and where they are in H* initially is determined by the momentary state that H* assigns to 1, and, [II], what are the atomic material objects in Halt and where they are in Halt initially is determined by the momentary state that Halt assigns to 1. Hence, [III], the same situation regarding atomic material objects and their initial positions is determined for Halt and H* if the momentary state that Halt assigns to 1 is identical with the momentary state that H* assigns to 1. Note, however, that the determination-statements [I] and [II] and their stated consequence [III] – though belonging to the supervenience-of-higher-continuants project – are not covered by the Generalized Supervenience Postulate and the rest of the already given supervenience apparatus. Those statements need additional grounding: Postulate of Original Supervenience For all histories H and H´ which are candidates for being actual*: if the momentary state that H´ assigns to 1 is identical with the momentary state that H assigns to 1, then the atomic material objects in H´ are the atomic material objects in H and each atomic material object in H´ is located at 1 in H´ where it is located at 1 in H [and adjoin the same consequent again, but with “H” and “H´” interchanged – which consequent, however, is already logically implied]. We thus arrive at the conclusion that the (according to supposition) true statements “c+ is in place P2d at 2 in H*” and “c+ is in place Pg at 2 in Halt” – with Pg ≠ P2d – are true for three reasons: (a) Halt and H* have the same origin (since they assign the same maximally composite momentary state to the temporal position 1), and (b) the atomic momentary Fullness-event {} is gen-identical in Halt with the atomic momentary Fullness-event that represents c+ at the temporal position 1 both in Halt and in H*, and (c) the atomic momentary Fullness-event {} is gen-identical in H* with that same event.

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Note that (a) entails what is presupposed in (b): that there is precisely one atomic momentary Fullness-event that represents c+ at the temporal position 1 both in Halt and in H*; the deduction that demonstrates the italicized statement employs the Generalized Supervenience Postulate, the Postulate of Original Supervenience, the transformation of D41,73 and, of course, the fact that there is a momentary Fullness-event that represents c+ at 1 in H*: {}. 6.3.1

Four-dimensionalist counterpartism

What does four-dimensionalism, from its point of view, have to offer in order to make both the statement “c+ is in place P2d at 2 in H*” and the statement “c+ is in place Pg at 2 in Halt” true? – It could turn to a version of counterpartism: According to four-dimensionalism, c+ is the event ε1: a certain temporally complete worm of Fullness in H*. Now, as things are (that is: as things are assumed to be, according to the suppositions made), there is precisely one complete worm of Fullness f in Halt which is such that f(1) = ε1(1); let this worm of Fullness be called “εalt”. One can take the fact that εalt is the only worm of Fullness in Halt which has the same origin as ε1 as the basis for asserting that εalt is the counterpart in Halt of ε1, that is: the counterpart in Halt of c+. The next step is to interpret the statement “c+ is in place Pg at 2 in Halt” as meaning – properly understood – the same as the statement “the counterpart in Halt of c+ is in place Pg at 2 in Halt”. Moreover, both to ε1 and to εalt there applies the same way of explicating what it means for them to be in a certain place at a certain time: “ε1 – that is, c+ – is in place P2d at 2 in H*” is explicated by “ε1(2) = that P2d is full, and ε1 is a part-event of H*”; and “εalt – that is, the counterpart in Halt of c+ – is in place Pg in Halt” is explicated by “εalt(2) = that Pg is full, and εalt is a part-event of Halt”. Since the following statement is true (according to the suppositions made): “ε1(2) = that P2d is full, and ε1 is a part-event of H*”, and since this other statement is also true: “εalt(2) = that Pg is full, and εalt is a part-event of Halt”, we finally arrive – now in the framework of four73

The transformation of D41 – the definition D41 transformed to fit the Generalized Supervenience Postulate – is this: For all histories H that are candidates for being actual*, all atomic material objects c in H, spatial positions P, and temporal positions Z: c is in place P at Z in H =Def {} is an atomic momentary Fullnessevent in H and {} represents c at Z in H.

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dimensionalism – at the desired conclusion: “c+ is in place P2d at 2 in H*” is true, and “c+ is in place Pg at 2 in Halt” is also true. For evaluating this four-dimensionalist counterpartism, one should note that, according to this doctrine (and on the basis of the rest of Tmetaphysics), c+ has at least as many counterparts in other histories than H* as there are histories which, though different from H*, have the same origin as H* and are candidates for being actual*.74 If there are many such histories, the attractiveness of four-dimensionalism, based on Ockham’s Razor, is somewhat compromised – but only somewhat, for it is still true that four-dimensionalist counterpartism does not stand in need of new entities: it makes use merely of entities that are already there in any case – namely, of certain events – for identifying the atomic material objects with them. On the other hand, according to four-dimensionalist counterpartism, the counterparts of c+ in histories other than H* are events that are not part-events of H*; these counterparts of c+ will, therefore, function as merely possible* atomic material objects. It is certainly a point in favour of the higher-continuant (or supervenience) approach to atomic material objects that it is not forced to assume merely possible*, hence non-actual*, atomic material objects merely to account for the truth of certain modal statements about c+, for example, to account for the (assumed) truth of “The state of affairs that c+ is in place Pg at 2 is possible*”. But the weightiest reason against four-dimensionalist counterpartism seems to be this: it is unable to accord to the statement “c+ is in place Pg at 2 in Halt” (which is true according to supposition) the straightforward sense that is suggested by the wording of the statement, but must construe it as being, properly understood, not about where c+ is in Halt at 2, but about where in Halt at 2 is the counterpart in Halt of c+. This manoeuvre of reinterpretation is necessary for four-dimensionalist counterpartism; it is not necessary for all approaches to atomic material objects and their transhistorical identity. The supervenience approach, as we have seen in Sect. 6.3, does entirely without counterpartist reinterpretation; that approach, we have seen, does indeed allow what initially seemed implausible: to interpret “c+ is in place Pg at 2 in Halt” as being directly about the atomic material object c+.

74

I leave it here an open question whether c+ could also have counterparts in histories that have the same origin as H*, but are not candidates for being actual*. However, the Limiting Postulate in Sect. 6.4 requires a negative answer: No.

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6.3.2

Four-dimensionalism without counterpartism

There is a way for four-dimensionalism to avoid counterpartism. In going that way, one must construe atomic material objects in H* in a way that is more complex than identifying them simply with the temporally complete worms of Fullness in H*. The general idea is this: Principle of Sophisticated Four-Dimensionalism If P is a spatial position which is such that the momentary state that P is full is intrinsically included in the maximally composite momentary state ST and if there is a history H, with H(1) = ST, which is a candidate for being actual*, then we have: c(P, ST) =Def , and c(P, ST) is an atomic material object in those histories, and only in those, which are elements of {H: H is a history, candidate for being actual*, and H(1) = ST}. There are no other atomic material objects in any history than those that can be defined in the way just indicated. By applying this principle, we obtain the following result (since, as we have supposed, P1a is a spatial position which is such that the momentary state that P1a is full is intrinsically included in the maximally composite momentary state H*(1), H* being – trivially – a history which is a candidate for being actual*): c(P1a, H*(1)) =Def , and c(P1a, H*(1)) is an atomic material object in every history which is an element of {H: H is a history, candidate for being actual*, and H(1) = H*(1)}. Obviously, if c+ is to be identified with an atomic material object according to sophisticated four-dimensionalism, then c(P1a, H*(1)) is precisely that object [and for c# it is c(P1b, H*(1)), for c> it is c(P1c, H*(1)), for c~ it is c(P1d, H*(1))]. According to sophisticated four-dimensionalism, an atomic material object c is a (set-theoretical) construct out of three components: (1) a cer-

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tain spatial position P; (2) a maximally composite momentary state ST in which P is full and which is the initial state of some history that is a candidate for being actual*; (3) the set of histories which are candidates for being actual* and of which ST is the initial state. Being this construct, c is an atomic material object in every history that belongs to component (3) of its construction, and it is not an atomic material object in any other history. In each history that belongs to component (3) of its construction, c is initially located at P. Moreover, c – always c itself, not a counterpart of it – is located at a spatial position P´ at a temporal position Z´ in a history H that belongs to component (3) of c’s construction if, and only if, P´ is full at Z´ in H and there is a diachronic path of Fullness in H that connects P´ with P. If one considers the conceptual material that sophisticated fourdimensionalism uses for the construction of atomic material objects, then it is quite apparent – in view of component (3) – that the spirit of fourdimensionalism has been preserved (and that the designation “fourdimensionalism” is still deserved). Yet, it is undeniable that fourdimensionalism in its sophisticated form loses much of its intuitive appeal. Whatever they are, atomic material objects should not appear to be rather complex constructs. Thus, the supervenience theory of atomic material objects, which treats them as supervenient higher continuants, seems to be, after all, the winner over four-dimensionalism, simple or sophisticated; this has emerged in the course of the analysis of an assumedly true, non-trivial possibility*-statement – an entirely unproblematic one – about an atomic material object in H*. Note, however, how the situation would change if there were just one history that is a candidate for being actual* – the actual* history, which is H*: then there would be no true and non-trivial possibility*-statement about any atomic material object in H* (since possibility* would collapse into actuality*). But Model T is not to be developed in this way (as has already been decided in Sect. 5.2.2, Concerning (II)). 6.4

Final determinations on atomic material objects

The first three laws for T are object-constituting laws. No matter which (complete) T-history that is a candidate for being actual* we are looking at, those laws guarantee that we shall always see exactly four atomic material objects in it, the careers of which objects are unequivocally traceable through T-space from the beginning of that history to the end. Certain four atomic material objects (for example, c+, c#, c>, and c~) will be seen to be

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numerically the same (four) objects in several histories that are candidates for being actual* – given that those histories have the same initial state. But if histories that are candidates for being actual* do not have the same initial state, then identifying an atomic material object in one of them with an atomic material object in the other and, likewise, differentiating an atomic material object in one of them from an atomic material object in the other become somewhat arbitrary affairs. For the sake of T-metaphysical definiteness, we must decide this matter in a general way. How to decide it in a general way – this is already implicit in the Principle of Sophisticated Four-Dimensionalism, considered in the previous section, in which the matter in question has, in fact, already been decided implicitly. Here we explicitly posit the Complement to the Postulate of Original Supervenience For all histories H and H´ which are candidates for being actual*: if the momentary state that H´ assigns to 1 is not identical with the momentary state that H assigns to 1, then no atomic material object in H´ is identical with any atomic material objects in H [and adjoin the same consequent again, but with “H” and “H´” interchanged – which consequent, however, is already logically implied]. If P is a spatial position that is full in a maximally composite momentary state ST which is assigned to 1 by some history H that is a candidate for being actual*, and if P´ is a spatial position that is full in a maximally composite momentary state ST´ which is assigned to 1 by some history H´ that is a candidate for being actual*, then the ordered pair determines an atomic material object (which is in H): c, and the ordered pair also determines an atomic material object (which is in H´): c. This much follows already on the basis of the supervenience postulates stated before the above Complement to the Postulate of Original Supervenience. Hence: (1-IdentityAMO)

If = [i.e., P = P´ and ST = ST´], then c = c.

And on the basis of the supervenience postulates stated before the Complement, we also have: If P ≠ P´ and ST = ST´, then c ≠ c. Now, taking into account the Complement to the Postulate of Original Su-

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pervenience, we have in addition: If ST ≠ ST´, then c

≠ c.

If ≠ [i.e., P ≠ P´ or ST ≠ ST´], then c ≠ c.

Statements that are entirely analogous to the principles (1-IdentityAMO) and (2-IdentityAMO) follow on the basis of the Principle of Sophisticated Four-Dimensionalism, specifically on the basis of the definition that is stated therein; just replace “c” and “c” in (1-IdentityAMO) and (2-IdentityAMO) by “c(P, ST)” and “c(P´, ST´)”. And an analogue of the following Limiting Postulate can also be gotten from the Principle of Sophisticated Four-Dimensionalism (just replace “c” in the Limiting Postulate by “c(P, ST)”, and attend also to the italicized part of the Principle): Limiting Postulate For any (T-entity) X: if X is an atomic material object, then X = c, where P is a spatial position, and ST a maximally composite momentary state in which P is full and which is the initial state of some history that is a candidate for being actual*. (In other words: Each atomic material object is an atomic material object that is uniquely determined by a spatial position P and a maximally composite momentary state ST, ST being the initial state of some history that is a candidate for being actual*, and P being full in ST.) It is an easily obtainable consequence of the Limiting Postulate together with (1-IdentityAMO) and (2-IdentityAMO) that the atomic material objects of Model T are one-to-one correlated with the ordered pairs of the following kind: P is a spatial position of T, and ST a maximally composite momentary T-state in which P is full and which is the initial state of some (complete) T-history that is a candidate for being actual*. Not every one-to-one correlation is a case for identification (or in another word: reduction), and the present one-to-one correlation is a perfect example of a one-to-one correlation that is not a case for identification: atomic material objects are not identical with the (just-described) ordered pairs that they are correlated with. They are supervenient T-objects, and the attempts (including four-dimensionalism, with or without counterpartism) to reduce them to more basic T-entities merely specify various objectifications, so to speak, of their supervenience – objectifications which are

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clearly perceived (by me and, I trust, also by you) to be distinct from the supervening T-objects themselves. The supervenience of the atomic T-material objects in combination with their non-reducibility gives rise to certain questions: Could it be that the non-reductively supervening atomic T-material objects, and the other higher continuants of T that are composed of them, exist merely in the eye of the beholder, who is looking at T and its various histories from the outside of T? Could it be that the beholder is, so to speak, seeing atomic Tmaterial objects (qua higher continuants) into T, from the outside of T? This would not make those objects free-floating illusions, since the seeing which constituted them would be based on – indeed, determined by – hard facts (which in no way already involve or refer to the objects that are to be constituted out of them). Nevertheless, if matters were as described, then, in a sense, atomic T-material objects (qua higher continuants) would not really be there. – Now, what if it is with Reality just the way it has just been tentatively considered to be with Model T? 6.4.1

The strong essentiality of origin for atomic material objects

Consider c+ again: c+ is in place P1a at 1 (in H*), as we have supposed. Could c+ have not been in P1a at 1? We can make the “could” in this question precise in the following way: Is it possible+ that c+ is not in P1a at 1? According to D25 in Sect. 5.3, the answer to this question is yes if there is a T-history which intrinsically implies the state of affairs that c+ is not in P1a at 1. And we can be sure that there is a history of this kind, for example, a history in which c+ does not exist (is in no place75) at 1. The next question is more interesting: Is it possible* that c+ exists at 1 and is not in P1a at 1? According to D27 in Sect. 5.3, the answer to this question would be yes if there were a T-history H that is a candidate for being actual* in which c+ exists at 1, but is not in P1a at 1 (or putting it more explicitly: which intrinsically implies the state of affair that c+ exists at 1, but is not in P1a at 1). But there is no such T-history H. For suppose, for reductio, that there is a T-history H that is a candidate for being actual* in which c+ is in some place P at 1 (i.e., exists at 1), but is not in P1a at 1. The spatial position P [which – according to supposition and the supervenience of atomic material objects – is full in the maximally 75

Compare Postulate 2 in Sect. 6.2.

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composite momentary state H(1): the initial state of history H that is a candidate for being actual*] and H(1) together determine the following atomic material object: c (which is in P at 1 in H). Since c+ is (supposed to be) in P at 1 in H, c+ must be identical with c. But c+ is also – has been presupposed to be – in place P1a at 1 in H*. Hence c+ must also be identical with c. Therefore: c = c. Consequently we obtain in accordance with (2-IdentityAMO): = , that is: P = P1a and H(1) = H*(1). But this implies that c+ is in P1a at 1 in H – contradicting the above supposition for reductio. It has now been shown to be impossible* that c+ exists at 1 and is not in P1a at 1. Therewith it has also been shown to be impossible* that c+ exists [in the non-time-dependent but time-thematic sense in which “exists” means as much as “exists at some temporal position”, with implicit relativization to histories] and is not in P1a at 1, since c+ exists in an arbitrary history H´ which is a candidate for being actual* if, and only if, c+ exists at 1 in H´.76 Therefore, its initial location – its origin, its place at 1 in H* – is in a certain (explicated and readily understandable) sense essential to c+.77 And the result can be generalized (since c+ is, in fact, just an arbitrary example of an atomic material object): its initial location is essential – essential in the above-exhibited sense – to every atomic material object (of T) which is located in P at 1 in H (P being a spatial position and H a history that is a candidate for being actual*). But there is even more of “essentiality of origin” for c+: not only its initial location is essential to c+, its initial setting is also essential to c+. For suppose, for reductio, that there is a T-history H that is a candidate for being actual* in which c+ is in some place P at 1 (i.e., exists at 1), but H(1) ≠ H*(1). Along the lines made explicit above (in the second to last paragraph), one obtains [with the assumption for reductio and using the (presupposed) fact that c+ is in place P1a at 1 in H*]: P = P1a and H(1) = H*(1), contradicting the assumption for reductio. Thus, the metaphysics of T has here been constructed in such a way that atomic material objects are not context-free entities; rather, their initial context (a certain maximally composite momentary state, with their posi-

76

This follows on the basis of the supervenience apparatus for atomic material objects, including the already established three laws for T. 77 The initial location of c+ is essential to c+ qua initial location, not already qua location: it is essential to c+ that it be in that location at 1, not, of course, that it be in that location at all times.

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tion in that state specified) is part of their essence (as their initial context, not simpliciter; cf. footnote 77). 6.5

Candidates for further laws for T, and more on the laws for T

Are there (that is: should there be) other laws for T – laws of H* – in addition to the three laws that have already been established? – It seems indeed: if T is to be a metaphysical model of Reality, then there should be more laws. In this section and its subsections (6.5.1, 6.5.2, and 6.5.3), I shall present candidates for such laws, and also provide further clarifications regarding the laws already accepted. In Sect. 6.6, a final determination concerning the canon of the laws for T will be made. The initial situation of H* with H*’s (supervening) atomic material objects – which is also the initial situation of every history that is a candidate for being actual* and has the same initial state as H*, hence the same (supervening) atomic material objects – is completely described as follows (regarding the – schematic – content of the description, compare the correspondence diagram in Sect. 6.2): 1. P1a, P1b, P1c, and P1d are full at 1. All other spatial positions are empty at 1. P1b P1c P1d P1a ↑1↓ ↑1↓ ↑1↓ ↑1↓ c+ c# c> c~.

Note that this description is schematic (i.e., it is a schema of a description properly speaking: of a specific description) as long as P1a, P1b, P1c, and P1d are not concretely specified. (Note also that the description is schematic only with regard to P1a, P1b, P1c, and P1d.) And one can continue in this schematic style (regarding the schematic content conveyed, compare the schemata of complete diachronic paths of Fullness in H* presented in Sect. 6.2): 2. P2a, P2b, P2c, and P2d are full at 2. All other spatial positions are empty at 2. P2a P2b P2c P2d ↑2↓ ↑2↓ ↑2↓ ↑2↓ c# c> c~ c+.

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3. P3a, P3b, P3c, and P3d are full at 3. All other spatial positions are empty at 3. P3b P3c P3d P3a ↑3↓ ↑3↓ ↑3↓ ↑3↓ c~ c> c+ c#.

And so on, up to 100 – which gives us a complete (schematic) description of H*. In it, the letters “a”, “b”, “c”, and “d” are arbitrarily assigned to the four spatial positions that are full at each temporal position Z (with 1 ≤ Z ≤ 100). But, in it, the atomic material objects c+, c#, c>, and c~ are not – not for any temporal position Z´ with 1 ≤ Z´ < 100 – arbitrarily assigned to the four spatial positions that are full at Z´+1. For example, the description – given the supervenience apparatus, which includes the First, Second, and Third Law, but no other law for T – implies the following part of a significant assignment restriction: Since c+ (for example) is assigned to P3c at 3, P3c must be immediately attached to P2d, the spatial position assigned to c+ at 2, which spatial position in turn must be immediately attached to P1a, the spatial position assigned to c+ at 1. In fact, by following the assignments of spatial positions to the four atomic material objects c+, c#, c>, and c~ throughout the entire description of H*, one is bound to retrieve from it four complete diachronic paths of Fullness in H*. Furthermore, the Third Law entails the following: If a spatial position P´ that is full at Z´+1 (for any temporal position Z´ with 1 ≤ Z´ < 100) is immediately attached78 to a spatial position P that is full at Z´, then P is the only spatial position that is full at Z´ to which P´ is immediately attached. Thus (for example): since P3c is full at 3 and immediately attached to P2d, which spatial position is full at 2, P2d is the only spatial position that is full at 2 to which P3c is immediately attached. And (also according to the Third Law): If a spatial position P´ that is full at Z´+1 is immediately attached to a spatial position P that is full at Z´, then P´ is the only spatial position that is full at Z´+1 which is immediately attached to P. Thus: since P3c is full at 3 and immediately attached to P2d, which spatial position is full at 2, P3c is the only spatial position that is full at 3 which is immediately attached to P2d.

78

Remember that immediate attachment can also be achieved by identity (which is, so to speak, the most immediate kind of immediate attachment).

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Thus, one is bound to find that the four complete diachronic paths of Fullness in H* which one can retrieve from the description of H* by following the assignments of spatial positions to the atomic material objects c+, c#, c>, and c~ throughout that description are, in fact, all the diachronic paths of Fullness there are in H*. Should there be additional regulations for the movements of c+, c#, c>, and c~? Note that no additional regulations – laws – are required for constituting c+, c#, c>, and c~ as higher continuants relative to H* (and relative to every other history that is a candidate for being actual* and that has the same initial state as H*). The basis of their being is already complete, and c+, c#, c>, and c~ already stand before us as (supervenient) higher continuants. Further laws will, therefore, be regulations that are not necessary for the constituting of c+, c#, c>, and c~; they will move the apparatus of laws closer to determinism. And yet there is motivation for further laws. This motivation is acquired by considering Reality, and in particular by considering the actual physical world. The law of the conservation of momentum and the law of the conservation of energy seem good originals for having a simulacrum in Model T. A joint simulacrum of both of these Reality-laws would be the T-law of the conservation of (the total quantity of) motion – if the T-state-ofaffairs of the conservation of motion were made a T-law. In order to present that state of affairs as a candidate for being a T-law, some conceptual preparations are necessary: Let H be a history that is not excluded by the first three laws for T from being a candidate for being actual*, c any of the four atomic material objects in that history, Z a temporal position: D44 l(c, Z, H) =Def the location of c at Z in H [this being a spatial position]. Let H be a history that is not excluded by the first three laws for T from being a candidate for being actual*, c any of the four atomic material objects in that history, Z a temporal position before 100: D45 =Def the state of motion for c in H from Z to Z+1. If l(c, Z, H) = l(c, Z+1, H), then the state of motion for c in H from Z to Z+1 is a state of rest simpliciter, and also a (in fact, the) state of rest for c in H from Z to Z+1. If, however, l(c, Z, H) ≠ l(c, Z+1, H), then the state of

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motion for c in H from Z to Z+1 is a state of movement simpliciter, and also a (and, in fact, the) state of movement for c in H from Z to Z+1. D46 X is a transition =Def For some temporal position Z (of T) before 100: X = . Instead of the formulation “the state of motion for c in H from Z to Z+1”, we can also use the formulation “the state of motion for c in H in the transition ”; and instead of formulating “U is a/the state of movement/rest for c in H from Z to Z+1”, we can also formulate “U is a/the state of movement/rest for c in H in the transition ”. As a last step in the preparations for presenting the T-state-of-affairs of the conservation of motion as a candidate for being a T-law, we have: Let H be a history that is not excluded by the first three laws for T from being a candidate for being actual*, Z a temporal position before 100: D47 U is a state of movement in H in the transition =Def For some atomic material object c in H: U = and l(c, Z, H) ≠ l(c, Z+1, H). For defining “U is a state of rest in H in the transition ”, simply replace “l(c, Z, H) ≠ l(c, Z+1, H)” in D47 by “l(c, Z, H) = l(c, Z+1, H)”. Like other predicates, “U is a state of movement in H in the transition ” has a history-indexical version. Its history-indexical version is “U is a state of movement in the transition ”, which predicate, when truly or falsely applied, is implicitly relative to T-histories that are not excluded by the first three laws for T from being candidates for being actual*. This indexical predicate will be used immediately below, in the presentation of the Candidate Fourth Law for T. (Previously, in Sect. 6.2, the history-indexical version of the predicate “X is a diachronic path of Fullness in H” – “X is a diachronic path of Fullness” – was used in the presentation of the Third Law for T, which predicate, when truly or falsely applied, is implicitly relative to T-histories.) We are now ready for the following legislative act, which is only one step short (see below Comment (2)) of a completed T-legislation:

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The Candidate Fourth Law for T – presented in the first way – is the state of affairs that, for all transitions X and Y, the number of the states of movement in X = the number of the states of movement in Y. – presented in the second way – is the following true statement: “For all T-histories H: if H is a candidate for being actual*, then the following state of affairs is intrinsically implied by H: that, for all transitions X and Y, the number of the states of movement in X = the number of the states of movement in Y”; – presented in the third way – is precisely implied by the following candidate rule of ACTUHIST: “At each step of the game after the first two steps the players of ACTUHIST select collectively, for being made actual*, a maximal momentary T-event which is such that the number of accomplished movement-changes for it relative to the maximal momentary T-event E that was selected one step earlier is equal to the number of accomplished movement-changes for E relative to the maximal momentary T-event selected immediately before E”; – presented in the fourth way – is given as follows:

The number of the states of movement is in each transition the same as in every other transition. Comments: (1) The first three laws for T establish that there are 0, 1, 2, 3, or 4 states of movement (hence 4, 3, 2, 1, or 0 complementary states of rest) in a given transition within a T-history H that is a candidate for being actual*. The Candidate Fourth Law for T adds to this that the total quantity of movement in H in the first transition (that is, in , 1 and 2 being temporal positions) neither diminishes nor increases (or more accurately: is not replaced by another quantity) in the course of H; hence the total quantity of motion (whether conceived of positively: as total quantity of movement, or negatively: as total quantity of rest) always remains the same in H. (2) The Candidate Fourth Law for T is not yet a law for T. But all it lacks for being one is the fiat by the Author; if that fiat is provided, the Candidate Fourth Law for T becomes a law for T; if it is not provided, it stays a non-law for T. The final decision is made in Sect. 6.6. (3) It may seem mandatory (and not only more to the point) to use the expression “movement-changes” instead of the simple word “changes” in the third way of presenting the Candidate Fourth Law. For the phrase “accomplished changes for [the maximal momentary event] E” can be under-

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stood in such a way that there do seem to be less accomplished changes than movement-changes for E relative to the maximal momentary event selected immediately before E. Suppose – for the sake of having a concrete example – that the initial situation of H* (with H*’s atomic material objects) is this: 1. , , , and are full at 1. All other spatial positions are empty at 1. ↑1↓ ↑1↓ ↑1↓ ↑1↓ c+ c# c> c~.

Suppose further that the second situation of H* is this: 2. , , , and are full at 2. All other spatial positions are empty at 2. ↑2↓ ↑2↓ ↑2↓ ↑2↓ c+ c# c> c~.

For the maximal momentary event that is given by 2., there are, apparently, 4 accomplished movement-changes relative to the preceding maximal momentary event that is given by 1.: c+ has moved from to , c# has moved from to , c> has moved from to , and c~ has moved from to . (There seem to be, therefore, 4 states of movement – in H* – in the first transition,79 and hence, according to the Candidate Fourth Law, there would have to be 4 states of movement in all further transitions.) But on the other hand, for the maximal momentary event that is given by 2., there are just 2 accomplished changes relative to the preceding maximal momentary event that is given by 1.: the spatial position that was full one moment ago is empty now, and the spatial position that was empty one moment ago is full now; all other spatial positions that were full are still full, and all other spatial positions that were empty are still empty. However, 1. and 2. – the conjunction of 1. and 2. – is, as a matter of fact, not a description that is in accordance with the laws for T already ac79

, for example, seems to be one of them, since it seems to be a (and, in fact, the) state of movement for c+ (in H*) in the transition .

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cepted; for 1. and 2. is not in accordance with the Third Law. Consider the diachronic paths of Fullness in H* that can be gathered from 1. in conjunction with 2.; they all have the same domain: {1, 2}. At first, it may seem that there are just 4 of those paths, as is required by the Third Law in order to enable the supervenience of the 4 atomic material objects in H* (their moving or resting) on the spatiotemporal distribution of Fullness in the course of H*. But this first impression is false: the function that allots to 1 and to 2, and the function that allots to 1 and to 2, and the function that allots to 1 and to 2 are also diachronic paths of Fullness in H* that can be gathered from 1. in conjunction with 2., and there are still further such diachronic paths of Fullness in H*. It is obvious, then, that the Third Law is violated; for example, the diachronic path of Fullness in H* with the domain {1, 2} which allots to 1 and to 2 (and which is naively taken to be identical with the initial part of the path of c+ in H*) allots the same spatial position to 2 as the different diachronic path of Fullness in H* with the domain {1, 2} that allots to 1 and to 2. It has thus become plain that in the third way of presenting the Candidate Fourth Law – i.e., when it is presented as exactly corresponding to a stated candidate rule of ACTUHIST (as being precisely implied by it) – the expression “movement-changes” can after all be equivalently replaced by the simple word “changes”, this word being understood precisely in the sense in which it was used above (namely, in order to describe the merely putative fact of there being 4 accomplished movement-changes but only 2 accomplished changes when going in H* from the temporal position 1 to the temporal position 2). The Third Law does not allow that the number of changes – i.e., atomic-state-changes – should ever differ from the number of movement-changes. The above scenario is excluded by that law; for the Third Law requires, that, no matter which immediately consecutive maximal momentary events in H* we are looking at, the number of spatial positions that have switched their status regarding Fullness between those two events (which number is the relevant number of atomic-state-changes) is equal to the number of atomic material objects in H* that have changed their spatial position between those two events (which is the relevant number of movement-changes). (4) Although the Candidate Fourth Law is the present focus of interest, something about the Third Law has now become apparent that may not have been as clearly visible before. The Third Law does allow contact between atomic material objects, that is, it allows that such objects are in

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touch at a time, diagonally or laterally. What it does not allow, however, is that those objects stay in diagonal contact, or stay in lateral contact, and, therefore, it does not allow that they move in (diagonal or lateral) stable contact. The Third Law does not allow stable contact because stable contact ineluctably involves, when an event-basis for it occurs in a history H, the occurrence in that history of differing diachronic paths of Fullness with the same domain that assign the same spatial position to some temporal position. The occurrence in H of such diachronic paths of Fullness not only disqualifies H from being a candidate for being actual*, according to the Third Law, but also has the consequence that a uniquely determined correct interpretation of H as a history of the moving and resting of certain atomic material objects becomes impossible and that the supervenience of atomic material objects in H on H fails. For illustration, consider the following spectacular case: The four spatial positions in the middle of the space of T – that is: , , , and – are full at 1 in a history H, all other spatial positions are empty at 1 in H. And at 2, those four spatial positions are still full in H, and all other spatial positions are empty. Nothing has moved in H, so it seems: 4 atomic material objects are just lying still, in (diagonal and lateral) contact with each other. But the 4 diachronic paths of Fullness with the domain {1, 2} that suggest this interpretation – the path that assigns to 1 and to 2, the path that assigns to 1 and to 2, the path that assigns to 1 and to 2, and the path that assigns to 1 and to 2 – are far from being all the diachronic paths of Fullness in H with the domain {1, 2}. There also are, for example, the following 4 other diachronic paths of Fullness in H with the domain {1, 2}: the path that assigns to 1, but to 2; the path that assigns to 1, but to 2; the path that assigns to 1, but to 2; and the path that assigns to 1, but to 2. These latter diachronic paths of Fullness suggest, on the contrary, that 4 atomic material objects have begun a counter-clockwise circular movement in H, in contact with each other. But, of course, there are also 4 diachronic paths of Fullness in H with the domain {1, 2} which suggest that 4 atomic material objects have begun a clockwise circular movement in H, in contact with each other. (For obtaining these latter diachronic paths of Fullness, take the last four descriptions of diachronic paths of Fullness in H with the domain {1, 2} and replace in each description “1” by “2” and “2” by “1”.) And there also are 4 diachronic paths of Fullness in H with the domain {1, 2} which suggest somewhat weirder goings-on, as for example, 4 atomic material objects interchanging their places crosswise. One can continue, but there is no need to do so: the massive ambiguity of history H – already in the first transition – with regard to the movement (and rest) in it of atomic material objects has become obvious, making the supervenience of those objects on H impossible. This massive ambiguity – which, interestingly, coexists with the non-ambiguity of H regarding the movement (in the transition ) of the square, all-encompassing composite material object which is, in H, at the centre of space both

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at temporal position 1 and temporal position 2 – is the result of a massive violation of the Third Law.

The Third Law forbids the situation presented in our example. Given that this example can be seen to involve a composite material object (at rest at the centre of space), one might come to believe that the Third Law also forbids the existence of composite material objects. It does not. The Third Law only forbids composite material objects in which the atomic material objects that constitute them are in stable contact (which means not only that they stay in contact, but also that the manner of contact between them does not change80). Stable contact between atomic material objects, therefore, is not a way in which any composite material object – a composite higher continuant – is ever constituted in histories that are candidates for being actual*. But stable distances that are greater than 0 between atomic material objects can also be used in order to make atomic material objects constitute a composite material object. (Moreover, one can very well question whether even the stability of distances between atomic material objects is necessary for their constituting a composite material object. For more on composite material objects, see Sect. 7.4.) 6.5.1

The Candidate Fifth Law for T

Both the Candidate Fourth Law and the First Law are conservation principles. The First Law demands that the total quantity of matter be preserved (throughout any T-history that is a candidate for being actual*); the Candidate Fourth Law demands that the total quantity of motion be preserved 80

Note that the Third Law can seem to allow staying in contact between atomic material objects if the manner of contact changes. Consider two atomic material objects: one is located at at 1, and one at at 1, and the surroundings (say, in the square with the sides to and to ) are empty. These two atomic material objects are in lateral contact at 1. At 2, however, both and are empty, whereas and are full, the surroundings being still empty. Nothing, it seems, contradicts the Third Law here: One of the two material objects has moved from to , and the other from to ; and now, at 2, they are in diagonal contact; they have, therefore, stayed in contact, but have changed their manner of contact – without violation of the Third Law, it appears. But appearances are wrong in this case: the Third Law is violated, because, in the described situation, the diachronic path of Fullness with the domain {1, 2} that assigns to 1 and to 2, overlaps with the different diachronic path of Fullness with the same domain that assigns to 1 and to 2.

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(throughout any T-history that is a candidate for being actual*). The First Law, however, provides a specific number which gives the preserved total quantity of matter: 4, whereas the Candidate Fourth Law does not provide a specific number which gives the preserved total quantity of motion. In other words, the First Law specifies a nomological constant of nature (for T, or: of H*): the matter-constant, while the Candidate Fourth Law would not by itself specify a nomological motion-constant even if it left its status of candidacy. There would be a motion-constant in H*, of course, if the Candidate Fourth Law became a law (and if ACTUHIST is played, which, however, is taken for granted); it would amount to 0, 1, 2, 3, or 4; but in contrast to the matter-constant – which is identically the same number in H* and in every other history which is a candidate for being actual* – that motion-constant in H* might not be nomological and, in any case, it would not be nomological already in virtue of the matching candidate law having become a law (although that motion-constant might after all be nomological: after all the laws for T have been specified). All of this could have been arranged differently: The Candidate Fourth Law could have been presented in such a way that it would by itself provide a nomological motion-constant (simply called “the motion-constant”) if it became a law; the First Law could have been presented in such a way that the matter-constant is, say, 10 – and so on and so forth. Therefore, on the strength of the analogy in metaphysics between Model T and Reality, one can very well ask: Can we be sure that Reality’s laws of nature could not have been different? And if they could have been different, why, then, are they the way they in fact are? But here is a question that is less big: If the Candidate Fourth Law for T were a law for T, should yet another law for T be established, or even further laws? A need for further regulation may be felt with regard to the touch and the near-touch of atomic material objects. Two such objects are in touch at a temporal position Z if, and only if, they touch diagonally or laterally at Z; and two such objects are in near-touch at Z if, and only if, they are at Z just one (i.e., exactly one) spatial position apart. (What, exactly, does this mean: “they are at Z just one spatial position apart”? This is clear enough if the positions of the atoms are connectable at Z by a vertical, horizontal, or diagonal line: the line crosses exactly one spatial position. But if, for example, one atom is at and the other at , then the atoms are also just one spatial position apart: for, from , just one step is needed in order to reach a position which is laterally or diagonally in touch with – and vice versa, interchanging and .)

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Moreover, a need for further regulation may be felt with regard to the situation that an atomic material object has moved into a border-position. An atomic material object, c, has moved into a border-position at a temporal position Z if, and only if, (i) the first coordinate of c’s spatial position at Z is either 1 or 10, or (this “or” is non-exclusive) the second coordinate of that spatial position is either 1 or 10, and (ii) there is a temporal position Z´ immediately before Z such that neither the first nor the second coordinate of the spatial position of c at Z´ is either 1 or 10. If a need for regulation is felt with regard to having moved into a border-position, then it is felt because this situation requires (except if it occurs at the end of a history), immediately subsequent to it, a modification in the movement of the atomic material object concerned – and one desires to see that modification regulated, in addition to the regulation of it that is provided by the laws already established (those laws hypothetically including the Candidate Fourth Law). Touch and near-touch, too, can require, immediately subsequent to them, a modification in the movement of at least one of the atomic material objects concerned. Consider, for example, two atomic material objects, all by themselves in a certain region of Tspace, heading in a straight line towards each other; they are on collision course – and now, at temporal position Z, they touch or, alternatively, they are just one empty spatial position apart. They cannot go on as before: a modification in the movement of at least one of them is required next – and one desires to see that modification regulated, in addition to the regulation of it that is provided by the laws already established. Now, the desire for additional regulation in the considered situations is felt particularly strongly if the following candidate for being a law for T – a candidate which is obviously reminiscent of Newton’s First Law – is hypothetically considered (as is already the Candidate Fourth Law) to be a law: The Candidate Fifth Law for T – presented in the first way – is the state of affairs that, for any atomic material object c and any temporal position Z after 1 and before 100, if c has not moved into a border-position at Z and is not at Z in confrontation with any other atomic material object, then the state of motion for c in the transition is a state of rest if the state of motion for c in the transition is a state of rest, and if this latter state of motion is not a state of rest, then the state of motion for c in the transition is a state of movement with the same direction as the state of movement for c in the transition .

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– presented in the second way – is the following true statement: “For all T-histories H: if H is a candidate for being actual*, then the following state of affairs is intrinsically implied by H: that, for any atomic material object c and any temporal position Z after 1 and before 100, if c has not moved into a border-position at Z and is not at Z in confrontation with any other atomic material object, then the state of motion for c in the transition is a state of rest if the state of motion for c in the transition is a state of rest, and if this latter state of motion is not a state of rest, then the state of motion for c in the transition is a state of movement with the same direction as the state of movement for c in the transition ”; – presented in the third way – is precisely implied by the following candidate rule of ACTUHIST: “At each step of the game after the first two steps the players of ACTUHIST select collectively, for being made actual*, a maximal momentary T-event which is such that the accomplished changes for this maximal momentary T-event relative to the maximal momentary T-event selected one step earlier conform – if interpreted as changes in the location of atomic material objects c1, c2, c3, c4 – to the following description: If ci [i ∈ {1, 2, 3, 4}] has not moved into a border-position one step earlier and is not, at that step, in confrontation with any other (of the four) atomic material objects, then the state of motion for ci in going from one step earlier to the present step is a state of rest if the state of motion for ci in going from two steps earlier to one step earlier is a state of rest, and if this latter state of motion is not a state of rest, then the state of motion for ci in going from one step earlier to the present step is a state of movement with the same direction as the state of movement for ci in going from two steps earlier to one step earlier”; – presented in the fourth way – is given as follows:

For any atomic material object c and any temporal position Z after 1 and before 100: If c has not moved into a border-position at Z and is not at Z in confrontation with any other atomic material object, then the state of motion for c in the transition is a state of rest if the state of motion for c in the transition is a state of rest, and if this latter state of motion is not a state of rest, then the state of motion for c in the transition is a state of movement with the same direction as the state of movement for c in the transition .

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Comments: (1) The first comment must address the two concepts employed in the Candidate Fifth Law that have not been previously introduced: (a) the direction of a state of movement, and (b) the confrontation of atomic material objects. Regarding (a): According to D47 and D44, states of movement (in whatever history which is not excluded by the first three laws from being a candidate for being actual*, and in whatever transition) are ordered pairs of non-identical spatial positions – which spatial positions are, moreover, immediately attached to each other (as is demanded by the supervenience apparatus). The direction of a state of movement, then, is the direction which is represented by the (imaginary) arrow that goes from the first spatial position in the state of movement to the second, other spatial position in it. That direction will be one of the, in all, eight directions in the space of T: either it is one of the four spatial main directions – left, right, down, and up (these directions first appear in Sects. 2 and 3.2.2.1) – or it is one of the four in-between directions: diagonal-left-down, diagonal-right-up, diagonal-left-up, diagonal-right-down. Moreover, a state of movement is directed at every spatial position that lies on the prolongation of the arrow that goes from the first to the second spatial position in it; this concept of being directed at a spatial position is going to be used immediately, in the next paragraph. Regarding (b): An atomic material object c is at a temporal position Z (Z > 1) in confrontation with another atomic material object c´ if, and only if, either c and c´ are in touch at Z, or c and c´ are in near-touch at Z and the respective states of motion of c and c´ in the transition are states of movement which are such that there is at Z a spatial position between c and c´ at which they (the states of movement) are both directed. (A spatial position is at Z between c and c´ – which atomic material objects are in near-touch at Z – if, and only if, it is immediately attached both to the spatial position of c at Z and the spatial position of c´ at Z. Thus, supposing that c is in at Z and c´ in at Z, not only is between c and c´ at Z, but also and .) (2) It may seem that there are cases where an atomic material object is at a temporal position Z (with Z > 1) in touch with another atomic material object – but where the consequent of the Candidate Fifth Law should also be postulated. Consider the atomic material objects c and c´ in a region of space all by themselves. Suppose the atomic material object c has moved in the transition from to , whereas the atomic mate-

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rial object c´ has moved in that same transition from to . Hence c and c´ are in lateral touch at Z. But, it might be asked, why not apply the Candidate Fifth Law also in this case and accordingly conclude what is required by this candidate law (if applied): in the transition c has moved from to , whereas c´ has moved in that same transition from to ? Indeed, why not? – Because this would violate the Third Law. In the envisaged situation, there is a diachronic path of Fullness with the domain {Z, Z+1} which assigns to Z, and to Z+1, and there is a diachronic path of Fullness with the domain {Z, Z+1} which assigns to Z and to Z+1. But, unfortunately, there is in the envisaged situation also the diachronic path of Fullness with the domain {Z, Z+1} which assigns to Z and to Z+1, and the diachronic path of Fullness with the domain {Z, Z+1} which assigns to Z and to Z+1. We therefore have diachronic paths of Fullness with identical domain that differ from each other but assign the same spatial position to some temporal position – contradicting the Third Law. As a consequence of this violation of the Third Law, the movements of atomic material objects cannot be unambiguously read off the temporally evolving pattern of fullness and emptiness (and the supervenience of such objects on that pattern fails). Is it that an atomic material object has moved from to , and another such object from to – or is it that an atomic material object has moved from to , and another such object from to ? But there are other cases of fleeting touch between atomic material objects where an extension of the validity of the Candidate Fifth Law would not get into conflict with the Third Law. Consider the atomic material objects c´´ and c´´´ in a region of space all by themselves. Suppose the c´´ has moved in the transition from to , whereas c´´´ has moved in that same transition from to . Hence c´´ and c´´´ are in diagonal touch at Z. And now, indeed, one could apply the Candidate Fifth Law also in this case (which is a case in addition to the cases covered by the given antecedent of that candidate for being a law for T: by the ifclause that is incorporated in its above-presented formulation) and accordingly conclude: in the transition c´´ has moved from to , whereas c´´´ has moved in that same transition from to . The resulting temporally evolving pattern of fullness and emptiness would not contradict the Third Law. (3) Since (the validity of) the Candidate Fifth Law can be extended, one can consider extending it. This can be done in two ways: by supple-

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menting the Candidate Fifth Law, as it now is, by a candidate law with the same consequent, which covers in its antecedent precisely the cases in extension of the presently given Candidate Fifth Law (among them the case considered last in the previous comment); or by replacing the Candidate Fifth Law by a stronger version of it with the same consequent, which covers in its antecedent, besides the cases that are taken care of by the Candidate Fifth Law in its presently given form, also the cases in extension of the presently given Candidate Fifth Law. And one can let the cases in extension of the presently given Candidate Fifth Law be as many as possible – that is, as many as are not excluded by the laws for T already established. By extending the Candidate Fifth Law in this way one would obviously be making a large step towards determinism. But the confrontation of an atomic material object c with another such object (diagonal touch being a special form of confrontation) should, I decide, always be (at least) the opportunity for a modification in the state of motion of c. I therefore abstain from extending the Candidate Fifth Law in such a way as to have it (or rather: its supplement or stronger version) cover also certain cases of confrontation. (4) In the formulation of the Candidate Fifth Law heavy use is made of indexical versions of non-indexical predicates (some previously introduced, others not; for the latter, see Comment (1) above); that is, precisely speaking, heavy use is made of predicates which – when truly or falsely applied – are implicitly relative to histories that are not excluded by the first three laws for T from being candidates for being actual*. In the fourth way of presenting the Candidate Fifth Law the places for implicit relativization can be made explicit in the following way: “For any atomic material object c [in the history H that is not excluded by the first three laws for T from being a candidate for being actual*] and any temporal position Z after 1 and before 100: If c has not moved [in H] into a border-position at Z and is not [in H] at Z in confrontation with any other atomic material object [in H], then the state of motion for c [in H] in the transition is a state of rest if the state of motion for c [in H] in the transition is a state of rest, and if this latter state of motion is not a state of rest, then the state of motion for c [in H] in the transition is a state of movement with the same direction as the state of movement for c [in H] in the transition .” (5) Like the First Law and the Candidate Fourth Law, the Candidate Fifth Law is a conservation principle. But, in contrast to these other conservation principles, the Candidate Fifth Law is conditional. It demands, if

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certain conditions obtain, that the preceding state of motion of an atomic material object be preserved by this object’s subsequent state of motion with regard to quality (namely, rest or movement) and direction. This demand may seem entirely harmless (putting aside the worry that a large step towards determinism would be made by adopting the Candidate Fifth Law as a law), but, as a matter of fact, it is not harmless. We shall see – in Sect. 6.6 – a compelling reason to abstain from making the Candidate Fifth Law for T a law for T: it turns out that it can rather easily get into conflict with the first three laws for T. 6.5.2

The regulation of collision

For any temporal position Z with 1 < Z, for any history H that is not excluded by the first three laws for T from being a candidate for being actual*, for any atomic material object c in H: if c has moved at Z in H into a border-position or is at Z in H in confrontation with some other atomic material object in H, then, and only then, we have: c is in a collision at Z in H (in a technical, broad sense of the word “collision”). This is a simple definition. But the variety of collision-situations is huge, and thus a regulation that provides a single determinate outcome to every collision-situation will not be easy to find – and may, in fact, not be desirable: for, if the Candidate Fourth Law and the Candidate Fifth Law were adopted in addition to the first three laws for T, an all-comprehensive single-outcome collisionregulation would lead to the result that everything regarding the selection of H* is determined after the first two steps of ACTUHIST. Much concerning the outcome of collision-situations has already been regulated. This can be seen by contemplating simple but typical collisionsituations. Let the other two atomic material objects be stowed away (constantly without movement) in an upper corner of the space of T (far from the scene of collision), but let the atomic material objects c and c´ confront each other – in the lower half of the space of T, the time being Z (with 1 < Z < 100) – in the following paradigmatic way: [1] c is in at Z–1, c´ is in at Z–1, and c is in at Z, c´ is in at Z [lateral head-on pair-collision with touch, both collisionpartners moving].

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Or – similarly, and yet with a considerable difference – let c and c´ confront each other in the following way: [2] c is in at Z–1, c´ is in at Z–1, and c is in at Z, c´ is in at Z [lateral head-on pair-collision without touch, both collisionpartners moving]. Consider situation [2]. What is going to be the case at Z+1? On the basis of the established laws for T and the Candidate Fourth Law for T, there are many things that cannot be the case at Z+1, for example: – c and c´, emptying their former places, cannot merge in the place at Z+1 (everything else staying as it was); for this would contradict the First Law; – c and c´ cannot be in the same places still at Z+1 that they were in at Z (and also everything else staying as it was); for this would contradict the Candidate Fourth Law; – it cannot be that and (or and , or …) are full at Z+1 (all the other spatial positions being empty – with the exception of those two in an upper corner of the space of T); for this would contradict the Third Law; – it cannot be that and (or and , or …) are full at Z+1 (all the other spatial positions being empty – with the exception of those two in an upper corner of the space of T); for this would contradict the Second Law. And, so far, there still are many things that can be the case at Z+1. Reality, of course, suggests that there is just one kind of thing that can be the case at Z+1 – both with regard to situation [2] and situation [1]: – As regards situation [1], c is in at Z+1, and c´ in ; as regards situation [2], c is in at Z+1, and c´ in . In other words: in the transition , c and c´ reverse the movements they make in the transition (whether situation [1] is being considered, or situation [2]). Regarding the further regulation of collision, strengthening the Candidate Fourth Law seems the best thing to do. The Candidate Fourth Law concerns the total quantity of motion; one could make it concern the total

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character of motion (inclusive of its total quantity). Or one could supplement the Candidate Fourth Law with another candidate law that concerns the total character of motion, leaving the Candidate Fourth Law as it is. This latter option is obviously preferable, since it involves no meddling with what has already been given a determinate form and has been discussed in that form. But the important question, of course, is this: what is the total character of motion? The 8 directions in the space of T (the 4 main directions: left, right, down, and up, and the 4 in-between directions: diagonal-left-down, diagonal-right-up, diagonal-left-up, diagonal-right-down) can be suggestively symbolized in the following way: , , , , , , , . These directions are also the possible directions of the state of motion for an atomic material object c [in a history H not excluded by the first three laws from being a candidate for being actual*] in a given transition [c being in H, Z being a temporal position earlier than 100]. The possible qualities of the state of motion for c in – rest or movement – can be suggestively symbolized in the following way: 0, 1. The possible characters, then, of the state of motion for c in are the pair-wise products of that state’s possible directions with that state’s possible qualities. Treating directions like numbers, this yields the following 9 possible characters of the state of motion for c in : 0, , , , , , , , . Now, the character of the state of motion for c [in H] in is one of these 9 possibilities. Furthermore, the total character of motion [in H] in the transition is canonically given by a designation of the following form: (csm1 ⊕ csm2 ⊕ csm3 ⊕ csm4). In this, “csm1”, “csm2”, “csm3”, and “csm4” each stand for the character of a state of motion [in H] in , and if (the Arabic numeral) k is different from (the Arabic numeral) j, then “csmk” stands for the character of a state of motion – smk – that is different from the state of motion – smj – whose character “csmj” stands for. [Note: though k ≠ j indicates smk ≠ smj, this does not mean csmk ≠ csmj! For “(csm1 ⊕ csm2 ⊕ csm3 ⊕ csm4)” is just an abbreviation of “(c(sm1) ⊕ c(sm2) ⊕ c(sm3) ⊕ c(sm4))”.]

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Canonical fillings of the schema (csm1 ⊕ csm2 ⊕ csm3 ⊕ csm4) look like this (to give four examples): (0 ⊕ 0 ⊕ 0 ⊕ 0), ( ⊕  ⊕  ⊕ 0), ( ⊕  ⊕  ⊕ ), ( ⊕  ⊕  ⊕ ). Note that the concatenation operation ⊕ is commutative, which makes ( ⊕  ⊕  ⊕ ) be identical with ( ⊕  ⊕  ⊕ ), and associative, which means: an expression of the form (csm1 ⊕ csm2 ⊕ csm3 ⊕ csm4) needs no internal brackets. The total quantity of motion in the transition can be easily extracted from the total character of motion in : just count the number of occurrences of arrows in the canonical designation of the total character. In most cases, the total character of motion in the transition contains more information than is given by the total quantity of motion in – the exception being the case where the total character of motion in is (0 ⊕ 0 ⊕ 0 ⊕ 0). Consider now again the above-described collision-situations [1] and [2], and the outcomes for them that are suggested by Reality. According to the descriptions provided, the total character of motion in the transition is in both situations this: (0 ⊕ 0 ⊕  ⊕ ) (using a canonical and suggestive designation). And according to the descriptions provided, the total character of motion in the transition – the total character which is suggested by Reality – is in both situations this: (0 ⊕ 0 ⊕  ⊕ ) (using a canonical and suggestive designation). But because of the commutativity of the ⊕-operation, (0 ⊕ 0 ⊕  ⊕ ) is identical with (0 ⊕ 0 ⊕  ⊕ ) [and identical with ( ⊕ 0 ⊕  ⊕ 0), and identical with ( ⊕ 0 ⊕ 0 ⊕ ), and identical with (0 ⊕  ⊕  ⊕ 0), and so on]. Hence it is revealed that not only the total quantity of motion, but also the total character of motion (inclusive of the total quantity of motion) is preserved in the collision according to [1] and in the collision according to [2] – if the outcomes of these collisions are both taken to be of the one kind that is suggested by Reality. In the transition , the atomic material objects c and c´ have merely exchanged their positions under the same total character of motion which they were already under in the transition . This can be put in the following way, using a selfexplanatory diagram:

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::(0 ⊕ 0 ⊕ c ⊕ c´), in other words: ::(0 ⊕ 0 ⊕ c´ ⊕ c). ⇓ ⇓ ::(0 ⊕ 0 ⊕ c´ ⊕ c), in other words: ::(0 ⊕ 0 ⊕ c ⊕ c´).81

Now, on the basis of the particular (though paradigmatic) examples considered, one might be tempted to postulate a sweeping generalization: The total character of motion is in each transition the same as in every other transition. This entails (but is not entailed by): The total quantity of motion is in each transition the same as in every other transition, which latter statement is just another way of giving a text-book presentation of the Candidate Fourth Law (compare, in Sect. 6.5, the fourth way of presenting the Candidate Fourth Law). But there are two worries with regard to this generalization: Worry (I): Suppose the atomic material object c has moved into a border-position P at the temporal position Z (with 1 < Z < 100), with no other atomic material objects around. Evidently, in this case, the total character of motion in the transition will not, under any circumstances, be the same as the total character of motion in the transition , barring extraordinary measures of adaptation that aim at preserving the latter character (i.e., “motion-compensation at a distance”). Worry (II): Suppose the total character of motion in the transition is identical to the total character of motion in the transition . Does not this mean – on the basis of the laws given,82 no matter which constellation is considered – that the spatial positions of all (four) atomic material objects at Z+1 are determined by their spatial positions at Z? 81

If all of the four atomic material objects [in H] under (0 ⊕ 0 ⊕  ⊕ ) in are indicated, then the total character of motion may be designated like this: (c10 ⊕ c20 ⊕ c ⊕ c´) [or like this: (c20 ⊕ c10 ⊕ c´ ⊕ c)]. And if all of the four atomic material objects under (0 ⊕ 0 ⊕  ⊕ ) in are indicated, then the total character of motion may be designated like this: (c10 ⊕ c20 ⊕ c´ ⊕ c) [or like this: (c20 ⊕ c10 ⊕ c ⊕ c´)]. 82 The laws given (or the given laws) are the First, Second, and Third Law for T.

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For the moment, I set aside Worry (I) – I don’t worry about it now – and concentrate on Worry (II). Consider, then, for further illustration, the following case: At Z−1, c is in place and c´ in place , and at Z, c is in place and c´ in place . Supposing that the other two atomic material objects are, for the time being, at rest in some corner of the space of T, the total character of motion in the transition < Z−1, Z> is this: (0 ⊕ 0 ⊕  ⊕ ); and by putting those atomic material objects into the picture that are in an interesting manner under this total character of motion, this same total character of motion comes to be designated in the following way: (0 ⊕ 0 ⊕ c ⊕ c´). Now, where are c and c´ at Z+1 if the total character of motion in the transition < Z, Z+1> is also (0 ⊕ 0 ⊕  ⊕ )? It may seem prima facie that the total character of motion in the transition < Z, Z+1> simply cannot be (0 ⊕ 0 ⊕  ⊕ ) if the given laws are observed. But that prima facie impression is false. If one designates the total character of motion in the transition < Z−1, Z> less suggestively, that is, if one designates it like this: (0 ⊕ 0 ⊕  ⊕ ) (which can be done because of the commutativity of ⊕), then one is bound to see that putting c in place at Z+1 and c´ in place preserves (0 ⊕ 0 ⊕  ⊕ ), such that in the transition c and c´ merely have exchanged their positions under the same total character of motion which they already were under in the transition : ::(0 ⊕ 0 ⊕ c ⊕ c´) ⇒ ::(0 ⊕ 0 ⊕ c´ ⊕ c),

or in other words: ::(0 ⊕ 0 ⊕ c´ ⊕ c) ⇒ ::(0 ⊕ 0 ⊕ c ⊕ c´),

or putting it in the most suggestive (but, though entirely correct, slightly confusing) way: ::(0 ⊕ 0 ⊕ c ⊕ c´) ⇒ ::(0 ⊕ 0 ⊕ c ⊕ c´).

And, note, putting c in place at Z+1 and c´ in place is the only way to preserve (0 ⊕ 0 ⊕  ⊕ ) which is consistent with the given laws – as can easily be checked. This result, so to speak, inductively confirms Worry (II).

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But is there not a general argument that shows that the suspicion which is voiced in Worry (II) is entirely justified? – No, there cannot be such an argument; for in some cases the identity of the total character of motion in the transition with the total character of motion in the transition does not entail (on the basis of the given laws) that the spatial positions of all (four) atomic material objects at Z+1 are determined by their spatial positions at Z. This can be illustrated by an example. Consider again a case that we have already contemplated in the context of discussing the Candidate Fifth Law (see the second comment regarding that law). That is, consider the atomic material objects c and c´ in a region of space all by themselves and suppose the atomic material object c has moved in the transition from to , whereas the atomic material object c´ has moved in that same transition from to (c and c´ are, therefore, in diagonal touch at Z). Suppose the other two atomic material objects are, for the time being, at rest in some corner of the space of T. Hence the total character of motion in the transition is this: (0 ⊕ 0 ⊕  ⊕ ). Suppose the total character of motion in the transition is also (0 ⊕ 0 ⊕  ⊕ ). It may seem that the only location for c at Z+1 and for c´ at Z+1 which is lawfully compatible with this is for c and for c´. But in fact the location for c at Z+1 and for c´ at Z+1 is also lawfully compatible with the assumed fact that (0 ⊕ 0 ⊕  ⊕ ) is the total character of motion both in the transition and in the transition . According to the first(-mentioned) outcome of the encounter between c and c´ at Z (according to which c is in at Z+1, and c´ in ) we have: ::(0 ⊕ 0 ⊕ c ⊕ c´) ⇒ ::(0 ⊕ 0 ⊕ c ⊕ c´).

But according to the second(-mentioned) outcome of the encounter between c and c´ at Z (according to which c is in at Z+1, and c´ in ) we have: ::(0 ⊕ 0 ⊕ c ⊕ c´) ⇒ ::(0 ⊕ 0 ⊕ c ⊕ c´),

or in other words (making use of the commutativity of ⊕): ::(0 ⊕ 0 ⊕ c ⊕ c´) ⇒ ::(0 ⊕ 0 ⊕ c´ ⊕ c).

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Thus, according to the first outcome, it is not only the case that the total character of motion is in both transitions the same, it is also the case that c and c´ have the same positions under it; whereas according to the second outcome, it is only the case that the total character of motion is in both transitions the same: c and c´ – exchanging their positions – do not have the same positions under it. The laws given do not determine which outcome will come about; they allow both outcomes, with the result that c and c´ can be found at Z+1 either in , respectively , or back again in , respectively . There are, therefore, cases where it is not the case that the spatial positions of all (four) atomic material objects at Z+1 are (lawfully) determined by their spatial positions at Z, even if the total character of motion in the transition is the same as in the transition . Worry (II) has been dispelled: the envisaged Candidate Sixth Law (presented above – right before the Worries – in text-book form), if assumed in addition to the laws given, would not invariably lead to determinism (that is: once the first two maximal momentary events of H* have come about). It is another question whether the addition of the envisaged Candidate Sixth Law to the laws given (by making it a law) would still allow a sufficient amount of indeterminism to keep the game of ACTUHIST interesting. But be that as it may, the idea of adding The total character of motion is in each transition the same as in every other transition to the given laws is, in fact, not a good idea. The reason for this is that Worry (I) remains. There is no way to dispel it. Moreover, one can well worry whether there are not many further cases of collision – in addition to those that Worry (I) refers to – in which the total character of motion just cannot be preserved without extraordinary measures of adaptation, and whether, if all such cases of collision were nomologically excluded (one might think of this), it would not have a crippling effect on ACTUHIST. In fact, already the nomological exclusion of the cases Worry (I) refers to would impose a severe restriction on the moves of the players of ACTUHIST. The most reasonable thing to do in reaction to Worry (I) (and the further worries it gives rise to) seems to be this: to modify the envisaged Candidate Sixth Law (which – in its text-book form of presentation – is the

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statement in italics in the previous paragraph) while preserving of that merely envisaged law as much as can be preserved of it. 6.5.3

The Candidate Sixth Law for T

Modifying the envisaged Candidate Sixth Law along the lines indicated at the end of the previous section yields the following result: The Candidate Sixth Law for T – presented in the first way – is the state of affairs that, for all temporal positions Z, with 1 < Z < 100, the total character of motion in differs as little from the total character of motion in as is possible without contradicting the first three laws for T. – presented in the second way – is the following true statement: “For all T-histories H: if H is a candidate for being actual*, then the following state of affairs is intrinsically implied by H: that, for all temporal positions Z, with 1 < Z < 100, the total character of motion in differs as little from the total character of motion in as is possible without contradicting the first three laws for T”; – presented in the third way – is precisely implied by the following candidate rule of ACTUHIST: “At each step of the game after the first two steps the players of ACTUHIST select collectively, for being made actual*, a maximal momentary T-event E3 which compares to the two just antecedently selected maximal momentary T-events – E1 and (after E1) E2 – in the following way: the total character of motion [i.e., the collective motion-character of the four atomic material objects already established in the relevant round of the game] which can be gathered from the difference between E2 and E3 differs as little from the total character of motion which can be gathered from the difference between E1 and E2 as is possible without contradicting the first three laws for T”; – presented in the fourth way – is given as follows:

From any transition to the next (as long as there is a next transition): the amount of modification in the total character of motion is the least amount allowed by the first three laws for T.

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Comments: (1) The Candidate Sixth Law involves a triple comparison. Suppose we are looking at a history not excluded from being a candidate for being actual* by the first three laws for T. Suppose we have a transition before us, and the transition next to it, (Z being a temporal position with 1 < Z < 100). In the first comparison, a maximal momentary event E1, with Z−1 as its time, and a maximal momentary event E2, with Z as its time, are juxtaposed, and the difference between them is interpreted on the basis of the first three laws for T as the movements or rests of four atomic material objects. In the second comparison, the same maximal momentary event with Z as its time, E2, is juxtaposed with a maximal momentary event E3, with Z+1 as its time, and the difference between them is interpreted on the basis of the first three laws for T as the movements or rests of (the same afore-mentioned) four atomic material objects. The first comparison yields (among other things) the total character of motion in the transition , the second comparison yields the total character of motion in the transition . In the third comparison involved by the Candidate Sixth Law, these two total characters of motion are juxtaposed. Is their difference as small as it can be on the basis of the first three laws for T? If yes, then the Candidate Sixth Law has been respected; if no, then the Candidate Sixth Law is violated (and therefore, if the Candidate Sixth Law for T became a law for T, any history H with E1 and E2 and E3 as momentary phases83 would not be a candidate for being actual*, even if H is not excluded by the first three laws for T from being a candidate for being actual*). (2) Suppose we have before us two non-identical total characters of motion, following each other from one transition, X, to the next, Y: (csm11 ⊕ csm12 ⊕ csm13 ⊕ csm14) and (csm21 ⊕ csm22 ⊕ csm23 ⊕ csm24). Since they are non-identical, the amount of modification in the total character of motion between X and Y is not 0 (if they were identical, it would be 0). Now, their being non-identical cannot be a matter of the order in which the constituent motion-characters that go into them – csm11, csm12, csm13, csm14 on the one hand, and csm21, csm22, csm23, csm24 on the other – are presented (because of the commutativity of ⊕); there must be more substantial differences. If the absolute difference between the number of occurrences of arrows in (csm11 ⊕ csm12 ⊕ csm13 ⊕ csm14) and the number of occurrences of arrows in (csm21 ⊕ csm22 ⊕ csm23 ⊕ csm24) is the number 83

See D24 in Sect. 5.2.1.

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N, then the amount of modification in the total character of motion between X and Y is N plus some fraction, and the converse of this is also true. Clearly, the amount of modification in the total character of motion between X and Y is 0 plus some fraction if, and only if, the total quantity of motion that can be gathered from (csm11 ⊕ csm12 ⊕ csm13 ⊕ csm14) [by counting the occurrences of arrows in this character] is identical to the total quantity of motion that can be gathered from (csm21 ⊕ csm22 ⊕ csm23 ⊕ csm24) [by counting the occurrences of arrows in this latter character], or in other words: if, and only if, there is no modification in the total quantity of motion between X and Y. The fraction to be added to the number N [which number can be determined in the way just described and which can only be 0, 1, 2, 3, or 4] is fixed (except in the case of N being 484) by finding out how many constituent motion-characters in (csm11 ⊕ csm12 ⊕ csm13 ⊕ csm14) are identically matched with constituent motion-characters in (csm21 ⊕ csm22 ⊕ csm23 ⊕ csm24). Consider csm11 first. Does it have an identical match among csm21, csm22, csm23, csm24? If yes, then the first constituent motion-character in the sequence csm21, csm22, csm23, csm24 that matches csm11 identically is deleted from the sequence (the sequence remaining the same in all other respects), we call “one” and move one to considering csm12; if no, then we move on to considering csm12 right away. After csm11, csm12, csm13, and csm14 have all been considered (in the way described for csm11), we count the “one”-calls made in the course of processing (as described) csm11, csm12, csm13, and csm14. The number of those “one”-calls must be 0, 1, 2, 3, or 4; it is also the number of identical matches between (csm11 ⊕ csm12 ⊕ csm13 ⊕ csm14) and (csm21 ⊕ csm22 ⊕ csm23 ⊕ csm24). Then: (a) If the number of “one”-calls is 4, then N (that is, the absolute difference in the total quantity of motion between X and Y) can only be 0 and the fraction 0 is added to N.85 (b) If N is 4, then the fraction 0 is added to N. (c) If N is smaller than 4, then: if the number of “one”-calls is 3, the fraction 1/5 is added to N; if it is 2 the fraction 2/5 is added to N; if it is 1, the fraction 3/5 is added to N; if it is 0, the fraction 4/5 is added to N. 84

For this case, see the stipulation (b) below. In the schematic example we have been considering, this result is excluded, since we have been assuming that the total character of motion in Y is not identical with the total character of motion in X. 85

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Here are examples illustrating all possible amounts of modification in the total character of motion between transitions X and Y (those amounts being determined from the examples, and denoted in the form N.F, as described above): X::(0 ⊕ 0 ⊕  ⊕ ) X::( ⊕  ⊕  ⊕ ) X::(0 ⊕ 0 ⊕  ⊕ ) X::( ⊕  ⊕  ⊕ 0) X::( ⊕  ⊕  ⊕ ) [impossibility] X::( ⊕  ⊕  ⊕ 0) X::( ⊕  ⊕  ⊕ ) X::( ⊕  ⊕  ⊕ 0) X::( ⊕  ⊕  ⊕ 0) [impossibility] [impossibility] X::( ⊕ 0 ⊕  ⊕ 0) X::( ⊕ 0 ⊕  ⊕ 0) X::( ⊕ 0 ⊕  ⊕ 0) [impossibility] [impossibility] [impossibility] X::(0 ⊕ 0 ⊕ 0 ⊕ 0) X::(0 ⊕ 0 ⊕ 0 ⊕ ) X::(0 ⊕ 0 ⊕ 0 ⊕ 0)

Y::( ⊕ 0 ⊕  ⊕ 0) Y:: ( ⊕  ⊕  ⊕ ) Y::(0 ⊕ 0 ⊕  ⊕ ) Y:: ( ⊕  ⊕  ⊕ 0) Y::( ⊕  ⊕  ⊕ ) Y::( ⊕ 0 ⊕  ⊕ 0) Y::( ⊕  ⊕  ⊕ 0) Y::( ⊕  ⊕  ⊕ ) Y::( ⊕  ⊕  ⊕ ) Y::( ⊕  ⊕  ⊕ ) Y::( ⊕  ⊕  ⊕ ) Y::( ⊕  ⊕  ⊕ )

Y::( ⊕  ⊕  ⊕ 0) Y::( ⊕  ⊕  ⊕ ) Y::( ⊕  ⊕  ⊕ )

amount of modification: 0.0 amount of modification: 0.2 amount of modification: 0.4 amount of modification: 0.6 amount of modification: 0.8 amount of modification: 1.0 amount of modification: 1.2 amount of modification: 1.4 amount of modification: 1.6 amount of modification: 1.8 amount of modification: 2.0 amount of modification: 2.2 amount of modification: 2.4 amount of modification: 2.6 amount of modification: 2.8 amount of modification: 3.0 amount of modification: 3.2 amount of modification: 3.4 amount of modification: 3.6 amount of modification: 3.8 amount of modification: 4.0

(3) The Candidate Sixth Law is special in two respects: (i), it has a teleological character (it is a principle of minimization for a certain quantity: the amount of modification in the total character of motion), and (ii), in specifying that teleological character it explicitly refers to the given laws: the first three laws for T. These laws leave open a wide space of indetermination, which the Candidate Sixth Law is meant to fill to a certain extent, though not to completion. (Consider in this regard the next comment.) (4) If the Candidate Sixth Law is lifted to the status of a law for T and added to the first three laws for T (and no other law-candidates are made laws), then there still is a margin of indetermination (as desired). As we have seen in the previous section: even if the total character of motion in the transition is – without violation of the first three laws for T – identical to the total character of motion in the transition (that is,

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even if we have a case of compliance with the Candidate Sixth Law where the amount of modification in the total character of motion is the absolute minimum: 0.0), this does not invariably mean that the positions of the four atomic material objects at Z+1 are lawfully determined by their positions at Z. (5) Under a certain assumption, the Candidate Fourth Law can be deduced from the Candidate Sixth Law – given the first three laws for T. This is the Assumption: Given the first three laws for T to regulate the selection of a history for being made actual*, the following is true for all temporal positions Z with 1 < Z < 100: no matter which maximal momentary events have already been selected up to (and including) Z, the first three laws for T allow that the number of the states of movement in the transition be the same as the number of the states of movement in the transition . With this assumption in place, the Candidate Sixth Law requires that the number of the states of movement is in each transition the same as in every other transition. For, taking as given the first three laws for T to regulate the selection of a history for being made actual*, suppose for reductio that in some transition the number of the states of movement is not the same as in every other transition. Hence there is some temporal position Z with 1 < Z < 100 such that in the transition the number of the states of movement is not the same as in the (immediately preceding) transition . But according to the Assumption, the first three laws for T allow – no matter which maximal momentary events have already been selected up to Z – that the number of the states of movement in be identical with the number of the states of movement in . Therefore, from transition to (the next) transition , the amount of modification in the total character of motion is not the least amount of modification in the total character of motion that is allowed by the first three laws for T; rather, it is greater than the least amount allowed (for if the number of the states of movement did not change from to – and it lawfully could do so –, then this would necessarily mean a lesser amount of modification in the total character of motion than if the number of the states of movement changed from to – as it in fact does). We, therefore, have a consequence which contradicts the Candidate Sixth Law – and the deduction of the Candidate Fourth Law from the Candidate Sixth Law and the Assumption (given the first three laws for T) is complete. But is the Assumption true? I shall come back to this question in the next section.

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6.6

The canon of the laws for T

In the previous section we have considered three candidates for being Tlaws in addition to the first three laws for T. Which of these candidates should be made a law for T? As has already been announced (see the end of Sect. 6.5.1), there is a compelling reason to abstain from making the Candidate Fifth Law for T a law for T. Consider the following initial segment of a history that evolves, in that segment, in accordance with the first three laws: At 1: , , , and are full and all other spatial positions are empty. At 2: , , , and are full and all other spatial positions are empty.

Now, if the Candidate Fifth Law for T is taken to be a law for T, the next step can only be this (if the evolving history is to be a candidate for being made actual*): At 3: , , , and are full and all other spatial positions are empty.

But what is the next step to this? It turns out, there is no next step which is in accordance with the first three laws, in particular, with the Third Law – although, according to the Second Law, there has to be a next step. No matter how one resolves the above-described situation in accordance with the First Law and the Second Law, there are always differing diachronic paths of Fullness with the same domain (namely, {3, 4}) which assign the same spatial position to some temporal position. A sufficient reason for this is that there is no spatial position immediately attached to which is not also immediately attached to one or another of the other three spatial positions that are full at 3 (that is, to , , or ). Thus, the lawful evolution of any history with the above-described initial segment runs into antinomy just one moment after that initial segment: At 3, it reaches a point where it cannot be continued in accordance with the first three laws, though these laws require it to be continued; the abovedescribed maximal momentary event with the temporal location 3 just contradicts the first three laws and must be excluded from any lawful Thistory; but, on the other hand, that event is precisely the maximal momen-

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tary event which – given the two preceding maximal momentary events – is determined to come next by the Candidate Fifth Law if taken to be a law for T. The situation considered is a situation of conflict between the Candidate Fifth Law, if taken to be a law for T, and the first three laws for T. Without assuming the Candidate Fifth Law, without taking it to be a law for T, this antinomic situation can, of course, be avoided, and since the first three laws are firmly entrenched, the hypothetical T-nomological status of the Candidate Fifth Law is what must be made responsible for that antinomic situation. The Candidate Fifth Law and the first three laws are, however, not (simpliciter) incompatible as laws; this would only be the case if there were no T-history that respects all four of these principles – which is certainly not the case. But the situation considered gives one pause nonetheless, for it is not exceptional. Others like it can easily be constructed. There is, therefore, reason to fear that adding the Candidate Fifth Law as a law to the first three laws for T will have a crippling effect on the game of ACTUHIST. One, therefore, had better not make that principle an additional law for T. As regards the Candidate Fourth Law, there is no good reason to make it a separately postulated further law for T if the Candidate Sixth Law is made a law for T in addition to the first three laws. For either the Assumption (that is, the crucial assumption that was used in Comment (5) at the end of the previous section) is true, or it is not true. If the Assumption is true, then the Candidate Fourth Law follows from the Candidate Sixth Law, given the first three laws (see Comment (5)), and therefore, if the Candidate Sixth Law is made a law for T in addition to the first three laws, the Candidate Fourth Law is automatically turned into a law for T, too; hence there is no need, and therefore no good reason, to postulate it separately as a further law for T. If, however, the Assumption is not true, then the relationship between the first three laws and the Candidate Fourth Law is comparable to the relationship between the first three laws and the Candidate Fifth Law, with the consequence that the Candidate Fourth Law cannot without qualms be made a law for T in addition to the first three laws (though the Candidate Sixth Law may still be made such a law without qualms), and again – this time even independently of making the Candidate Sixth Law a law for T in addition to the first three laws – there is no good reason to postulate the Candidate Fourth Law separately as a further law for T.

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The point made in the last sentence can be more closely considered in the following way: The Candidate Fourth Law is presented in the third way as being precisely implied by the following candidate rule of ACTUHIST (see the presentation of the Candidate Fourth Law in Sect. 6.5): “At each step of the game after the first two steps the players of ACTUHIST select collectively, for being made actual*, a maximal momentary T-event which is such that the number of accomplished movement-changes for it relative to the maximal momentary T-event E that was selected one step earlier is equal to the number of accomplished movement-changes for E relative to the maximal momentary T-event selected immediately before E”. But if the Assumption is not true, then the first three laws for T are given to regulate the selection of a history for being made actual* and some temporal position Z with 1 < Z < 100 is such that, with certain maximal momentary events having been selected up to (and including) Z, the first three laws for T do not allow that the number of the states of movement in the transition be the same as the number of the states of movement in the transition . Therefore: if the Candidate Fourth Law were made a law for T in addition to the first three laws, then the candidate rule of ACTUHIST by which it is precisely implied, would be made a rule of ACTUHIST, although – under the supposition that the Assumption is not true – it can get into conflict with the rules of ACTUHIST already in place: under certain circumstances – perhaps there are many such circumstances –, the new rule cannot be obeyed together with the old rules. This is a reason to abstain – under the supposition that the Assumption is not true – from making the Candidate Fourth Law a law for T in addition to the first three laws. (Compare the above discussion that led to abstaining from making the Candidate Fifth Law a law for T in addition to the first three laws.) Thus, whether the Assumption is true or not true, there is no good reason to make the Candidate Fourth Law a separately postulated further law for T if the Candidate Sixth Law is made a law for T in addition to the first three laws. I herewith declare that the Candidate Sixth Law for T is a law for T, thereby making it a law for T. This automatically means (among other things) that the candidate rule which is referred to in the third way of presenting the Candidate Sixth Law is made a rule of ACTUHIST: it is the rule L10 of ACTUHIST (the rule L9 of ACTUHIST is presented in the context of the presentation of the Third Law for T in Sect. 6.2). It is somewhat misleading to call the Candidate Sixth Law for T – now that it is a law for T and has left the status of candidacy – “the Sixth Law for T”, since there is, as a matter of fact, no separately postulated Fourth

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and Fifth Law for T. It is also somewhat misleading to call the Candidate Sixth Law for T – now that it is a law for T – “the Fourth Law for T” (although it is indeed the fourth law for T which is separately postulated), since, understandably, one might take that designation to be referring to the Candidate Fourth Law, supposing it to have been made a law. In this situation, I choose to call the Candidate Sixth Law, in its new status as a law for T, “the Law of Minimization (of Change in the Total Character of Motion)”. It should be noted that the Law of Minimization preserves everything which is – given the firm basis of the first three laws for T – safe and sound in the Candidate Fourth Law and the Candidate Fifth Law. I surmise, moreover, that the Assumption is in fact true, but I have no proof for this. A proof of the Assumption would also prove – in view of what was said in Comment (5) in Sect. 6.5.3 – that the Candidate Fourth Law is contained as a law in the Law of Minimization. The Law of Minimization is all that should be explicitly assumed (in addition to the first three laws) for regulating collision, not setting aside those cases of collision where an atomic material object has moved into a border-position. In view of the huge variety of collision cases, the suggested policy of regulation certainly provides the safest manner of regulating them in a completely general way: it will not lead to surprising Tnomological conflicts (on the basis of seemingly harmless assumptions).86 It is tempting to view the space of T as enclosed on all sides by a barrier from which an atomic material object that has moved into a borderposition is taken to be lawfully reflected – if nothing interferes – at an angle that is equal to the angle of impact: in analogy to what a billiard ball lawfully does when it hits the bounds of the billiard table. But, on closer inspection, there is not sufficient motivation – expectation of elucidative payoff – for pursuing this line of nomological development. Borderpositions in the space of T are special in the following way: they intrinsically require a change in the movement of any atomic material object that has moved into (one of) them – a change that may or may not be deter86

A T-nomological conflict is a conflict between T-principles P1, …, PN, taken to be T-laws. Such a conflict can be absolute – then there is no (i.e., cannot be any complete) T-history which is in accordance with P1, …, PN. Or it can be conditional: given a certain (usually innocent-looking) assumption, some principles (or principle) out of P1, …, PN require(s) something which is forbidden by the rest of the principles in P1, …, PN, and hence there is no (i.e., cannot be any complete) T-history which satisfies that assumption and is in accordance with P1, …, PN.

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mined by the laws for T now given. There is nothing interesting for the elucidation of metaphysics to be gained from not leaving things at that. Thus we have: the First Law for T (corresponding to the rule L7 of ACTUHIST), the Second Law for T (corresponding to the rule L8 of ACTUHIST), the Third Law for T (corresponding to the rule L9 of ACTUHIST), and the Law of Minimization (corresponding to the rule L10 of ACTUHIST). There is some motivation for adding another law (for an exploration of this option, see the next section). But I decide against it: The laws just named are all the (explicitly presented) laws for T, and therefore, by taking each law for T in its first way of presentation, they are all the (explicitly presented) laws of H* (for this replacement of “for T” by “of H*”, see the end of Sect. 6). And what about the T-states-of-affairs that were not explicitly presented as laws, but are nevertheless intrinsically implied by every T-history that is a candidate for being actual*? – Those are also laws for T/of H*, in the sense employed in Sect. 4.1.1; but they are all logically implicit in the (explicitly presented, axiomatic) four laws for T/of H*. The physics of H* is, therefore, complete. Metaphysical and epistemological insight is to be gained from comparing this physics with physics. It is not so much the comparison of content that is interesting (it goes without saying that the physics of H* – despite some remote similarities – is very different from physics). What is interesting is this: Since I am the author of the physics of H*, I can be sure that that physics is complete: I declared it to be complete after having specified four laws – of which the physics of H* is simply the sum (taking each law in its first way of presentation). But how could we ever be sure that physics is complete? For one thing, we do not have the overview necessary for this, because we are not outside of Reality – at least not in the clean and, fairly, all(of T)-seeing manner in which we are outside of Model T. For another thing, no one of us can be – nor can we be collectively – author of any law of nature, let alone of the laws of nature. Since the physics of H* – one might also call it “the physics for T” – is complete, the set of the complete T-histories which are candidates for being actual* is completely specified (for the basis of this result, see Sect. 4.1.1, and there, specifically, the Principle of Sufficiency for Candidacy and the principle which is the converse of that principle, stated immediately in front of it). This provides a firm basis on which claims of possibility* and necessity* can be securely judged (using the definitions D27 and D28 in Sect. 5.3). There is no comparably firm basis for judging analogous

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– therefore: intended to be purely objective – claims of nomological possibility and necessity with regard to Reality – a fact which is frequently disregarded in metaphysical discussions. 6.7

The initial state

In the previous section, I presented (without explicitly mentioning it) a maximally complete momentary state which cannot occur before the temporal position 100 in any T-history that evolves in accordance with the first three laws; a fortiori, it cannot occur before 100 in any T-history that is a candidate for being actual*. The description of that maximally complete momentary state (provided in Sect. 6.6 within the description of a maximal momentary event) is this: , , , and are full and all other spatial positions are empty.

Though this state is barred from occurring before 100 in any T-history that is a candidate for being actual* (the reason for this is given in Sect. 6.6), there is no apparent reason for believing that its occurrence is also excluded at the end (at the temporal position 100) of a T-history that is a candidate for being actual*. But it is inscrutable – at least to me – whether there is indeed some T-history that is a candidate for being actual* which ends with that state. (The attempt to construct such a history backwards seems to me an overly arduous – and metaphysically pointless – exercise.) There is some temptation to postulate for some maximally composite momentary state or other that it is the ultimate state of every T-history which is a candidate for being actual*. Postulating such a state would amount to having another law (of nature) for T (a fifth one); it would also amount to having it as a fixed truth that, in the end, such and such is necessarily* bound to be the case. But considerable worries regarding the new law’s compatibility with the already established laws and also a sense of pointlessness are rather strong deterrents from giving in to that temptation. However, there is the much stronger temptation to postulate for some maximally composite momentary state or other that it is the initial state of every T-history which is a candidate for being actual*. After all, the rules of many games (for example, the rules of chess) specify an initial state. In the case of a law for the beginning worries regarding its compatibility with the already established laws arise to a lesser extent than in the case of a law 198

for the end (or, for that matter, a law for the middle). A law for the beginning – a law specifying a maximally composite momentary state as the initial state of every T-history which is a candidate for being actual* – would have interesting consequences. Such a law would imply that all T-histories that are candidates for being actual* emerge from the same origin, and therefore: that they form a single treelike structure, a single “tree of possibility*” – which is rather pleasing aesthetically (though branches reuniting after dividing seem not to be excluded). It also would imply (in view of what was said in Sects. 6.3 and 6.4) that there are just four atomic Tmaterial objects – not just in every T-history that is a candidate for being actual*, but overall (overall in the following sense: there would be just four atomic T-material objects after having collected the atomic T-material objects from every T-history that is a candidate for being actual*; such objects are to be found nowhere else but in such T-histories). Those four atomic T-material objects would be c+, c#, c>, and c~ – that is, they would be the four atomic material objects in H*. Hence all atomic T-material objects would be actual* (since they all would have an individual history which is actual* via being a part-event of H*87), none non-actual* – which is attractive to everyone with actualist inclinations. Moreover, it is easily seen that c+ and its three colleagues would each be necessarily* actualI88 (on the basis of D28, given what is anyway assumed: that some T-history is actual*) – which is attractive to traditional materialists (who confer on matter – or more recently: matter-energy – the divine attribute of necessary existence). Nevertheless, in spite of these attractions, I resist the temptation to have a T-law for the beginning. The beginning is a place of freedom for the players of ACTUHIST, a place of freedom also for that special ACTUHIST-player: the Author of the game – who is not willing to forgo that freedom by binding himself (though freely) once and for all to a T-law that (completely) specifies the initial state of every T-history that is a candidate for being actual*. And the aspect of similarity with Reality – always to be respected in the construction of Model T (if the similarity in question is metaphysically relevant) – renders support to the chosen policy of worldmaking: the laws of physics have nothing to say about the beginning of the physical world (except, of course, that they allow it). 87

For the concept of the individual history of an atomic material object, see Sect. 6.3; the definition of part-event – D42 – can also be found there. 88 Regarding the overtly indexical predicate “X is actualI”, see the list of predicates at the beginning of Sect. 5.2.

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Even given merely the four laws decided on in the previous section, postulated to be all the laws, the freedom of the beginning is far from unlimited. We have seen that there is no T-history that is both a candidate for being actual* and that has, in initial position, the maximally composite momentary state which is given by the following statement: , , , and are full and all other spatial positions are empty.

A (round of the) game of ACUHIST has not even started if the players hit on this maximally composite momentary state as initial state. And there are many other maximally composite momentary states that are out of the question as objects of initial choice in the process of selecting a certain Thistory for actualization*. But, note, the following maximally composite momentary state – or speaking of the primary object of choice: the maximal momentary event which has that state as (sole) functional value, and the set {1} as domain – can be lawfully selected initially: that , , , and are full and all other spatial positions are empty.

This state – according to which the four positions in the middle of the space of T, and only they, are full – already caught our attention in Sect. 6.5; it is a state that cannot be lawfully upheld for more than a moment. Therefore, if one is interested in obtaining the closest representation in Model T of Big Bang Cosmology that is possible in Model T, then one merely needs to select first the following maximal momentary event: {}.

For what will happen next – what must happen next in accordance with the laws of nature for T – is this: the four atomic material objects in the middle of the space of T fly apart as in an explosion. And even the manner of the explosion is determined: the only maximal momentary event that can be the next one in accordance with the laws for T is this: {}.

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Given the two maximal momentary events we have just been looking at, a total character of motion is determined for the first transition, that is, for . Indicating the T-spatial directions from to , from to , from to , and from to – in this very order – yields ( ⊕  ⊕  ⊕ ).89 And now the Law of Minimization can go into action. From transition to the next transition, , the least amount of modification in the total character of motion that is allowed by the first three laws is 0, or in other words: the first three laws allow that the total character of motion be preserved (entirely) from transition to transition . What must be the case in accordance with the Law of Minimization is, therefore, this: the total character of motion in the transition is identical with the total character of motion in the transition . It follows that the total character of motion in the transition is no other than ( ⊕  ⊕  ⊕ ), since ( ⊕  ⊕  ⊕ ) is the total character of motion in the transition . Prima facie this may seem to entail that the maximal momentary event which follows the two previously presented maximal momentary events is the following: option (a): {}.

But, as a matter of fact, this is just one possibility out of several possibilities of continuation, a possibility which one can easily misbelieve to be the only one – perhaps because one is still under the spell of the Candidate Fifth Law, or perhaps because one underestimates just in how many different ways, on occasion, one and the same total character of motion can be maintained in going from one transition to the next. The total character of motion in remains also the same as in if the maximal mo89

Remember that the arrows – due to the commutativity of ⊕ – can freely switch their places in this expression without the denoted total character of motion thereby becoming a different one. ( ⊕  ⊕  ⊕ ) is perhaps the most suggestive designation of the total character of motion under consideration.

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mentary event after the first two is not the maximal momentary event just mentioned, but is, for example, the following one: option (b): {}.

If this is the third maximal momentary event, then the total character of motion in the transition is likely to be designated in the following way (given the way in which the total character of motion in the preceding transition was designated): ( ⊕  ⊕  ⊕ ). This looks rather different from ( ⊕  ⊕  ⊕ ). But it is easily seen that ( ⊕  ⊕  ⊕ ) is just a permutation of ( ⊕  ⊕  ⊕ ); the motion-characters designated by these two expressions are, therefore, identical (because of the commutativity of the ⊕-operation). If “c1” denotes the atomic material object that is in place at 1, “c2” the atomic material object which is in place at 1, “c3” the atomic material object which is in place at 1, and “c4” the atomic material object which is in place at 1, then we have the following two descriptions (of a type already used in Sect. 6.5.2): According to option (a): ::(c1 ⊕ c2 ⊕ c3 ⊕ c4) ⇒ ::(c1 ⊕ c2 ⊕ c3 ⊕ c4).

Or in other words (making use of the commutativity of ⊕):

::(c1 ⊕ c2 ⊕ c3 ⊕ c4) ⇒ ::(c3 ⊕ c4 ⊕ c1 ⊕ c2).

According to option (b): ::(c1 ⊕ c2 ⊕ c3 ⊕ c4) ⇒ ::(c1 ⊕ c2 ⊕ c3 ⊕ c4).

Or in other words (making use of the commutativity of ⊕):

::(c1 ⊕ c2 ⊕ c3 ⊕ c4) ⇒ ::(c3 ⊕ c4 ⊕ c1 ⊕ c2).

These descriptions show that not only according to option (a) but also according to option (b) the same total character of motion – namely, ( ⊕  ⊕  ⊕ ) – is given both in transition and in transition . But whereas according to option (a) the four atomic material objects also maintain their positions under that same total character of motion, they ex-

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change them according to option (b): c1 exchanges positions with c3, c2 exchanges positions with c4 (as the descriptions show). The laws for T do not determine whether option (a) or option (b) is the one to be followed. Which option is to be followed is up to the players of ACTUHIST to decide (and there are still other lawful options than the two options just described for what is to be the case at temporal position 3). But note that the situation would have been very different if the Candidate Fifth Law had been included among the principles chosen to be the laws for T in the previous section. Then option (a) would have been the only option, and, in fact, all further developments would have been determined, too; then every step of ACTUHIST after the freely made first step – that is: after letting the four central spatial positions, and only they, be full at 1 – would have been determined, and the players of ACTUHIST would have had very little to decide (nothing after the first step). This is seen as follows. Under the Candidate Fifth Law, if hypothetically assumed to be another law for T (besides the First, the Second, and the Third Law, and the Law of Minimization), each of the above-named four atomic material objects has to continue on the diagonal path that it was put on – already without having a choice in this – in the transition , until each has moved at 5 into a corner-position (a special border-position); that corner-position is for c1, for c2, for c3, and for c4. What next? The laws for T – and in particular the Law of Minimization – leave just one possibility: at 6, c1 is in place , c2 in place , c3 in place , and c4 in place . For only in this way (the first three laws are obeyed and) the amount of modification in the total character of motion from transition to transition is as little as it can be in accordance with the first three laws: zero. And then the – hypothetical – Fifth Law takes over again (but of course the other laws must also be obeyed): c1, c2, c3, and c4 can only retrace their steps – until, at 9, they are exactly where they were at 1. At 10, then, the very same things have to happen as at 2 (the laws leave no choice) – and the whole cycle is repeated until, at 17, the four atomic material objects are again exactly where they were at 1 (and at 9), and so on. (It is easy to predict where c1, c2, c3, and c4 will be at 100.90) It has, therefore, become manifest: if the Candidate Fifth Law had been included among the principles canonized in Sect. 6.6, the number of 90

There are eight transitions in each of the identical cycles of their movements. Thus they have returned to their initial places at 9, 17, 25, 33, …, 81, 89, and 97.

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T-histories which have a central Big Bang in their beginning and are not excluded by the laws for T would have been very small: that number would have been 1, whereas in fact – on the basis of merely the principles canonized in fact – it is a larger number than 1. Such is the power of determination of the Candidate Fifth Law (and if more motivation were needed not to make it a law for T, the consequences of making it a law for T that have just been pointed out would certainly help to provide that motivation).

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7

Five T-metaphysical issues and the metaphysics of Reality

In this chapter, the presentation of Model T will be completed, and some of the philosophical fruits of Model T will be garnered. These fruits are not the incontrovertibly correct answers to the metaphysical questions that we have regarding Reality; rather, they consist in a clarification of the nature of these questions, in a renewed sense of the variety of reasonable answers to them, and in a renewed sense of the large uncertainty which must be connected with adopting – from the inside of Reality (as we know it) – any one of those answers as correct. 7.1

The completion of the rules of ACTUHIST

In Sect. 5.2.2, an answer to the question “of how many players of ACTUHIST there are besides the Author, and who or what are those players, and what is the quantity of their initial quantitatively equal shares of the power of selecting-for-being-made-actual**, and how their powers of selectingfor-being-made-actual** are to be generally characterized in their entirety” was postponed until after the laws for T would have been specified. The specification of the laws for T (as well as of the corresponding rules of ACTUHIST: L7, L8, L9, and L10) has been accomplished in the previous chapter. It is time to answer the question(s). For what follows, it will be helpful to reread the rules L5 and L6 of ACTUHIST, presented in Sect. 5.2.2; they constitute the background to the following rule: L11

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Besides the Author (in short: AU*), there are four players of ACTUHIST, each of them associated (for the entire time of the game) with exactly one of the four atomic T-material objects in H*: A+ with c+, A# with c#, A> with c>, and A~ with c~. At each step (stage) N of the game, each player that is not AU* individually selects a place in the space of T to be in at (temporal position) N – not for itself, of course, but for the atomic T-material object that is associated with it.91 The individual selection made by AU* at N, in turn, is to be identified with the selection made by A+ at 1, 5, 9, …, 97, with the selection made by A# at 2, 6, 10, …, 98, with the selection made by A> at 3, 7, 11, …, 99, with the selec-

I use the form “it” as a gender-neutral grammatical form. This use is not meant to suggest that the players of ACTUHIST are non-persons.

tion made by A~ at 4, 8, 12, …, 100. The individual selection of each player is restricted to what the laws for T allow at stage N, given the course of the game previous to N, in other words: given the process of actualization**92 previous to N, in other words again: given at each stage N´ previous to N, first, the provisional collective choice – the collection of the individual selections – of the players, second, its correction93 by AU*, and, third, the actualization** by AU* of the correction-result, that is: of the (final) collective choice of the players.

Comments: (1) For N = 1, there is no course of the game previous to N. Hence, in this case (that is, for N = 1) the condition “given the course of the game previous to N” is void (it could be omitted, in this case, without changing the import of what is being said). At stage 1, the individual selection of each player is restricted to what the laws for T simpliciter allow at stage 1. (2) For connecting L11 with L6, the following needs to be said: The law-abiding selection of a place P (in the space of T) for an atomic Tmaterial object c to be in at N amounts to selecting a certain non-empty subset S(c, P, N) in the set MLaw(N), which is the set of all those maximal momentary T-events that (i) have the temporal position N (as the sole temporal position in their domain) and that (ii) conform to the laws for T, given the maximal momentary events that have been actual** previous to stage N of the game. S(c, P, N) is simply the set of maximal momentary Tevents in MLaw(N) according to which c is in P at N, there being some such maximal momentary T-event in MLaw(N).94 (3) At the first step A+ (and AU* in unison – so to speak – with A+, according to L11) and A# and A> and A~ make their choices: they select a place for c+, respectively c#, respectively c>, respectively c~, to be in at 1. Two things can go wrong: (I) the players select less than four different places; (II) they select four different places all right – each player doing so law-abidingly (which is easy at step 1) – but the corresponding maximal momentary event (that is, the maximal momentary event with the domain {1}, allotting to 1 the maximally composite momentary state in which those four places are full, all others empty) is not allowed by the laws for T (and we have seen that there are such events in Sect. 6.7). Problem (II) is, as a matter of fact, already taken care of to a large extent by L6. Suppose A+ selected place P+ for c+ to be in at 1 and A# se92

Regarding the concept of actualization**, see Sect. 5.2.2. For the cases which make a modificatory correction necessary, see L6. 94 Note that S(c, P, N) will turn out to be identical to the set of maximal momentary Tevents out of MLaw(N) according to which P is full at N. The laws for T guarantee this. 93

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lected P# for c# to be in at 1, A> selecting P> for c>, and A~ P~ for c~, to be in at 1. The players made their selections law-abidingly. The corresponding subsets of MLaw(1) – S(c+, P+, 1), S(c#, P#, 1), S(c>, P>, 1), and S(c~, P~, 1) are, therefore, all non-empty. But, under Problem (II), their set-theoretical product S(c+, P+, 1) ∩ S(c#, P#, 1) ∩ S(c>, P>, 1) ∩ S(c~, P~, 1) – in other words, the set ΠPlayers(1)95 – is empty. – In this case, according to L6, the Author (i.e., AU*), acting as arbitrator, selects a single element in MLaw(1) and makes it actual**. And we can add: In this case, the Author also determines which of the four spatial positions that are full at 1 – according to the sole element selected by him in MLaw(1) – corresponds to which player and associated atomic T-material object (the latter being, at 1, in the spatial position which the Author-determines to correspond to it). But by adding this, we go beyond L6. Problem (I) occurs if two or more players lawfully select one and the same place for their associated atomic T-material objects. This can also happen later in the game, at stage/step N > 1, but then it needs no special treatment and is already taken care of by L6. The reason for this is the following: Given, for N > 1, the already collectively selected (and actualized**) maximal momentary events with temporal positions before N, the laws for T are such that a nomologically possible maximal momentary event E with temporal position N, according to which the atomic Tmaterial object c is in place P at N, is in no case identical with a likewise nomologically possible maximal momentary event E´ with temporal position N, according to which an atomic T-material object c´, different from c, is in the same place P at N; the events E and E´ have to differ. This implies – given that player A lawfully selects place P for c to be in at N and that player A´ lawfully selects that same place P for c´ to be in at N – that the set-theoretical intersection of S(c, N, P) and S(c´, N, P), S(c, N, P) ∩ S(c´, N, P), is empty and hence that ΠPlayers(N) is also empty and that we, therefore, have a case of conflict before us, for which cases, however, L6 is already prepared. But it is a different matter for N = 1. Additional regulation is needed (which will also take care of what has not already been taken care of by L6 regarding Problem (II)): 95

ΠPlayers(1) is identical with S(c+, P+, 1) ∩ S(c#, P#, 1) ∩ S(c>, P>, 1) ∩ S(c~, P~, 1) because the selection-set for AU* at 1, SAU*(1), coincides with S(c+, P+, 1) according to L11. Therefore: ΠPlayers(1) = SAU*(1) ∩ S(c+, P+, 1) ∩ S(c#, P#, 1) ∩ S(c>, P>, 1) ∩ S(c~, P~, 1) = S(c+, P+, 1) ∩ S(c#, P#, 1) ∩ S(c>, P>, 1) ∩ S(c~, P~, 1).

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L12

If the players do not manage at stage 1 to place the four atomic T-material objects associated with them in such a manner that each of these objects is connected with a different location in a single maximal momentary event (with domain {1}) which is allowed by the laws for T, then AU* does this placing for them.

(It remains true – also under the condition formulated in L12 – that the players of ACTUHIST each initially have the quantitatively same share of the power of selecting-for-being-made-actual**, as is required by L5; but it is AU* that, in the end, exercises their powers for them.) (4) L11 and L12 complete the list of the rules of ACTUHIST. As is appropriate, the rules of ACTUHIST remain schematic with regard to the players of the game. It is not being said who or what are A+, A#, A>, and A~. The schema will be filled out concretely when ACTUHIST is being played in a concrete situation, say, by five people. And only when these five people have made their initial choice in the (round of the) game of ACTUHIST they play will it become palpable and identifiable which objects are c+, c#, c>, and c~. Their initial choice (if made in accordance with the rules of ACTUHIST) already decides which four objects are the four atomic T-material objects in H* (in view of what was said in Sect. 6.3), though the making of H* has only just begun. And if one takes into account which of the players chooses which spatial position in that initial choice (no matter whether the individual choices that go into it be sovereign choices or choices made – in effect – by AU*), then that initial choice also decides which of those four objects is c+, which c#, which c>, and which c~. Note here the very interesting contrast between the T-immanent and the T-transcendent perspective on a (round of the) game of ACTUHIST. In the T-immanent perspective the game is its outcome, that is: it is a complete chronological sequence of actual* (at the respective step actualized**) maximal momentary T-events, a sequence which is organized by the laws for T. The actuality* of those maximal momentary T-events and the actuality* of the complete T-history that is “made out of them” (as a history is made out of its momentary phases) – which history is no other Thistory than H* – suggest even in the T-immanent perspective that there is a realm which is transcendent to Model T. But this is no more than a suggestion. One can read H* in the T-immanent perspective as the history of four atomic T-material objects (of their movements and rests), but which one of those objects is c+ (associated with A+), which c# (associated with

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A#), which c> (associated with A>), and which c~ (associated with A~) is inscrutable in the T-immanent perspective. The T-immanent perspective cannot provide even an inkling of A+, A#, A>, and A~ – let alone of AU* – in their respective roles with regard to the coming about of H*. In the Timmanent perspective (which, by the way, is a manner of speaking that cannot be taken literally: there is, literally, no one in T who has that perspective) one could call the supposition that there are such entities “utterly speculative”, “superfluous” (constituting “a flagrant violation of Ockham’s Razor”), and even “meaningless”. But, in the truth of things, that supposition is neither superfluous nor meaningless: it is true and truly explanatory. (5) In H*, what is substantial – c+, c#, c>, c~, and what they do (i.e., their moving and resting) – supervenes on what is non-substantial: a temporal sequence of certain positions in the space of T being full, respectively empty. And therefore this sequence – the mere H* – might be said to be what is, in a sense, really substantial in H*. But this fits only the Timmanent perspective. For if the T-transcendent perspective is assumed, one sees the acting players behind the atomic material objects of H*, and then these atomic objects, the riders of the vehicle, appear to be what is really substantial in H*, and no longer the mere H*, the vehicle – the supervenience of the riders of the vehicle on the vehicle notwithstanding. (6) In one instance the rules of ACTUHIST are not schematic with regard to the players of the game; one of the players, a special one, is AU*, the Author, who is a concrete person: I, the author of this book. It greatly decreases the chance of ACTUHIST’s ever being played – validating the assumption made at various points in this book: that it is played – if I have to be one of the players in person. I herewith authorize anyone who would like to do so to act as my plenipotentiary deputy in a game of ACTUHIST. (7) The role of AU* in setting up ACTUHIST – including the ontological realm which that game refers to: Model T – is large (AU* does so single-handedly). But in a game (i.e., a round of the game) of ACTUHIST, the role of AU* seems comparatively small, at least as regards the selection (not regarding the actualization**) of the consecutive maximal momentary events. At each step of the game, as far as selection is concerned, AU* seems to hide, so to speak, behind one of the other four players, seems to efface himself in that player – the hider of AU* being always a different player and AU* not preferring any player as his hider (see the respective clause of L11). This seems a weak role for AU* to play, but, note, it need not be a weak role – not if AU* cooperates in the selection made by his hider at a given stage of the game, co-determines that selection. Aside

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from that, there are indispensable roles to play for AU* in the resolution of conflict and initial dysfunction (see L6 and L12). (8) L6 also gives AU* (the Author) a role in the resolution of cases of indetermination, which comes about at a stage N of the game if, and only if, the (provisional) selection product ΠPlayers(N) has more than one element. But it is easily seen that L11 does not allow the occurrence of indetermination at any stage of the game which is later than 1. Suppose that A+ (obeying L11) law-abidingly selects at a stage N later than 1 – that is, at a stage N > 1 – the place P´ for c+ to be in at N, A# law-abidingly selects at stage N the place P´´ for c# to be in at N, and A> and A~ follow suit. The corresponding subsets of MLaw(N) – S(c+, P´, N), S(c#, P´´, N), S(c>, P´´´, N), and S(c~, P´´´´, N) – are, therefore, all non-empty. Suppose now that ΠPlayers(N) has more than one element; this means that SAU*(N) ∩ S(c+, P´, N) ∩ S(c#, P´´, N) ∩ S(c>, P´´´, N) ∩ S(c~, P´´´´, N) has more than one element, which, in turn, means that S(c+, P´, N) ∩ S(c#, P´´, N) ∩ S(c>, P´´´, N) ∩ S(c~, P´´´´, N) has more than one element [because SAU*(N) is, according to L11, identical with S(c+, P´, N), or with S(c#, P´´, N), or with S(c>, P´´´, N), or with S(c~, P´´´´, N)]. It follows that P´, P´´, P´´´, and P´´´´ are four different spatial positions; for otherwise S(c+, P´, N) ∩ S(c#, P´´, N) ∩ S(c>, P´´´, N) ∩ S(c~, P´´´´, N) would be empty instead of having more than one element (this is true in view of N > 1; cf. Comment (3) above). But it also follows that it cannot be the case that S(c+, P´, N) ∩ S(c#, P´´, N) ∩ S(c>, P´´´, N) ∩ S(c~, P´´´´, N) has more than one element (so that we have a reductio). For every element of S(c+, P´, N) ∩ S(c#, P´´, N) ∩ S(c>, P´´´, N) ∩ S(c~, P´´´´, N) must be a maximal momentary Tevent, with the domain {N}, according to which c+ is in P´ at N, c# is in P´´ at N, c> is in P´´´ at N, and c~ is in P´´´´ at N. Hence we have (a): each of those elements must be a maximal momentary T-event, with the domain {N}, according to which P´ is full at N, P´´ is full at N, P´´´ is full at N, and P´´´´ is full at N. But each of those (afore-mentioned) elements must also be an element of MLaw(N) [because S(c+, P´, N) ∩ S(c#, P´´, N) ∩ S(c>, P´´´, N) ∩ S(c~, P´´´´, N) ⊆ MLaw(N)]. Hence (as required by the First Law for T) we have (b): each of those elements must be such that according to it four spatial positions are full at N and all other spatial positions empty. Therefore (according to (a) and (b) and the above-derived consequence that P´, P´´, P´´´, and P´´´´ are four different spatial positions): every element of S(c+, P´, N) ∩ S(c#, P´´, N) ∩ S(c>, P´´´, N) ∩ S(c~, P´´´´, N) must be a maximal momentary T-event, with the domain {N}, according to

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which the four spatial positions P´, P´´, P´´´, and P´´´´ are full at N and all other spatial positions are empty. But, evidently, there is at most one such maximal momentary T-event. (9) Problem (I) which was treated above (in Comment (3)) – that is, the problem that the players of ACTUHIST select at stage 1 less than four different places for the atomic T-material objects to be in at 1 – can amount to a situation of indetermination, namely, if there are in MLaw(1) several maximal momentary events (with exactly four full spatial positions) according to which the places de facto selected by the players at 1 (which are less than four) are full at 1. In this case, L6 applies (but evidently without making L12 superfluous). (10) Consider MLaw(N) – that is, the set of the maximal momentary Tevents that (i) have the temporal position N and that (ii) conform to the laws for T, given the maximal momentary events that have been actual** previous to stage N of the game. Instead of referring here to the laws for T, one can also refer to the rules L7 – L10 of ACTUHIST. That a maximal momentary event E with the temporal position N conforms to the rules L7 – L10 of ACTUHIST, given the maximal momentary events that have been actual** previous to stage N of the game, means that E can be, as far as L7 – L10 are concerned, the (final) collective choice of the players of ACTUHIST at N, given the maximal momentary events that have been actual** (hence: that have been actualized** by AU*, hence: that have been collectively selected by the players of ACTUHIST) previous to stage N of the game. Thus the individual choices of the players at N are, according to L6 and L11, already subject to what collective choices are possible for them (collectively) at N. (11) Is MLaw(N) guaranteed to be non-empty at a each stage N of the game? Or putting it in other words: Is MLaw(N) non-empty at each stage N of any round of ACTUHIST that might ever be played? – Yes, trivially so: there is, by definition, no (possible) round of ACTUHIST in which MLaw(N) is empty at some stage N (since such a round is by definition a complete round). If one wants to have a real question, one must ask something else: Is MLaw(N) non-empty at each stage N of every (entirely ruleabiding) attempt at playing a round of ACTUHIST that might ever be made? – This question must be answered by “yes” if the following is true: The Proposition No matter how far the five players have played and no matter how, if they have done so according to the rules without lapse, and therefore

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entirely law-abidingly, then there is always a complete T-history which is a candidate for being actual* and which integrates the maximal momentary events already chosen to be actual* (i.e., which has those events as momentary phases). The rules of ACTUHIST seem to allow the truth of this proposition. However, it may seem instead that the Proposition can be straightforwardly refuted by using an example already previously employed (see the beginning of Sect. 6.6). Suppose the players have played in the following way: At 1: , , , and are full and all other spatial positions are empty. At 2: , , , and are full and all other spatial positions are empty. At 3: , , , and are full and all other spatial positions are empty.

Here, there is no complete T-history which is a candidate for being actual* and which has the just described three maximal momentary events as momentary phases, and MLaw(4) is empty. This attempt at playing a round of ACTUHIST has stalled, and there is nothing the players can do than to start over again and do things differently. They could do so in the following way: their steps 1 and 2 yield the same maximal momentary events as in their previous attempt; but step 3 yields the maximal momentary event that is described by At 3: , , , are full and all other spatial positions are empty.

The Law of Minimization requires that from transition to transition the amount of modification in the total character of motion is the least amount allowed by the first three laws for T. Now, the first three laws for T do allow that this amount of modification is 0. That is why the maximal momentary event E32 – corresponding to the above second description of what is the case at 3 – is in MLaw(3);96 that is also why the maximal mo96

In the transition , the total character of motion is this: ( ⊕  ⊕  ⊕ 0); and if E32 is chosen, the total character of motion in the transition is this: ( ⊕  ⊕  ⊕ 0), in other words (due to the commutativity of ⊕): the same total character of motion as in the preceding transition.

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mentary event E31 – corresponding to the above first description of what is the case at 3 – is in MLaw(3). Or is E31 in MLaw(3)? Note that we have a refutation of the Proposition only if E31 is in MLaw(3): for only if it is in MLaw(3) have the players in the example played entirely law-abidingly up to (and including) 3. Connecting with what has been said at the beginning of Sect. 6.7, the answer to the question at hand can only be the following: E31 does not conform to the laws for T; it is, therefore, not included in MLaw(3). And therefore: no refutation of the Proposition has in fact been given. 7.2

Physicalism and dualism with regard to Model T

The correct immanent Weltanschauung for Model T is physicalism, or to be precise: T-physicalism. Each concrete T-entity97 is T-physical in a primary sense, namely, in the sense which is appropriate for its T-ontological category. Thus, spatial configurations and, among them, the spatial positions P are T-physical in a different sense than the sense in which the universal Full and the spatial relations in T are T-physical. And the state of affairs that P is full (or: [Full, P]), the negation of that state of affairs (neg([Full, P])), and the entire group of momentary states are T-physical in a different sense again. The event {} and its many fellows (including maximal momentary events and complete histories), the state of affairs that P is full at 1 and its many fellows (each of them corresponding one-to-one to an event) – they are T-physical in, respectively, yet another sense of the term, and the same is true also (and especially) of the atomic material objects of T. In spite of these many primary senses of the word “T-physical”, there is a recognizable family-resemblance – an analogy – among the T-entities which are called “T-physical” in this or that sense specific to their category. And this analogy justifies applying the word “Tphysical” with a global meaning to them all, which meaning, though not entirely clear (but one might define it by a laborious disjunctive procedure that makes use of every category-specific predicate of T-physicalness), is clear enough. Thus, each concrete T-entity is T-physical also in a secondary sense, namely, in the global and category-transcending (but T-relative) sense just described. 97

Are there abstract T-entities? – Perhaps. T-spatial and T-temporal directions and total (T-)characters of motion may well be abstract enough for being abstract Tentities.

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Temporal positions (and configurations) are the one kind of concrete T-entity about the T-physicalness of whose members one might be hesitant – not without reason. But the somewhat subservient position of temporal positions does not give them a strong enough status to break the uniformity of T-physicalness among the concrete entities of T: they are, so to speak, carried along physically by the other concrete T-entities. That temporal positions are natural numbers (and that spatial positions are ordered pairs of natural numbers) is, by the way, no reason to deny them T-physicalness. For the “are” in the preceding statement is merely the “are” of the beingrepresented-by relation. (Note at this point that the “are” of the representation relation – which is the inverse of the being-represented-by relation – is used when a proud father puts his finger-tip on a photograph and says: “These are my children”.) Mathematical representations (meaning: mathematical representatives) are used as abstract stand-ins for the concrete things themselves, the very descriptors for those things being predicated of them (see Sect. 2.1), because they carry the all-important structural information regarding those concrete things on their sleeves (so to speak), which is a great convenience. This is why mathematical (and, in extension, set-theoretical) representation is widely used. No ontological confusion should result from this (though, unfortunately, it often does). T-physicalism is true for Model T (as one can say even without having scrutinized every concrete T-entity for T-physicalness), and consequently it is also true for the T-history H*. From the T-immanent point of view, H* is a history – if one looks at the highest T-ontological level – of T-matter in motion through T-space, and of nothing else. Immanently regarded, H* is a miniature model of implemented “materialism” – by which name many people, who are more interested in historically motivated terminological continuity than in descriptive correctness, still prefer to call the ontological position which, from the point of view of descriptive correctness, is better called “physicalism”. For materialism, if one takes it by its word (i.e., understands its name as literally descriptive of the position that is designated by that name), is just false, and already Democritus and Epicurus would have concurred: it will not do to call spatial configurations, among them space itself, “material entities” (though they certainly are physical entities). Analogously, T-materialism, if the designation is taken literally (with its Trelative sense) and not “nostalgically”, is just false, too: T-spatial positions are not T-material entities. Now, if the T-transcendent point of view is adopted, T-physicalism is still true for H* and for T. But, from the T-transcendent point of view,

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there are things to be said about, or relating to, T that are quite beyond the pale of T-physicalism. For one thing, actuality*, which is a concrete property and in a sense even a T-property (since it is instantiated only by Tentities – if “actual*” is taken in the literal sense and not in the analogical sense the word has if predicated also of items in Reality), is certainly not a T-physical property, and it does not seem to be a physical property in any sense. For another thing, the stages or steps of a game of ACTUHIST – taken as mere positions – are not obviously physical in any sense and are obviously not T-physical (though isomorphic to the temporal positions of T and mathematically represented in the same way as the latter, even named in the same way: “1”, “2”, “3”, …). But the most striking supplementation of T-physicalism is the relationship between the four atomic T-material objects c+, c#, c>, and c~, and the players A+, A#, A>, and A~. In view of that relationship (as ruled by the rules of ACTUHIST, and especially by L11), c+ might be said to be the Tbody of A+, and A+ the soul of c+, and c# might be said to be the T-body of A#, and A# the soul of c#, etc. Regarding the appropriateness or inappropriateness of this manner of speaking, several comments come to mind: (a) In Reality, souls seem to be attached exclusively to composite and highly complex material objects. This speaks against the suggested terminology, in view of the fact that c+, c#, c>, and c~ are simple objects. Answer: Agreed. (b) There is no third party X such that, for example, c+ is the body of X and A+ the soul of X; instead, c+ and A+ (for example) “go straight for each other”, with no higher unit uniting them. This is not the usual way of relating body and soul.98 Answer: Agreed, though an artificial higher unit that unites c+ and A+ can easily be provided: {A+, c+}. (c) Normally, souls are not regarded as transcendent entities; they are regarded as entities that are just as immanent as bodies. Here, however, “the souls” – A+, A#, A>, and A~ – are transcendent, while “the bodies” – c+, c#, c>, and c~ – are immanent, because the players A+, A#, A>, and

98

According to that usual way, my body is not in the primary manner of speaking the body of my soul; it is primarily my body. And my soul is not in the primary manner of speaking the soul of my body; it is primarily my soul. According to the usual story, my soul and my body are integrated in a higher unit that unites them: in me, the human being; only because of this higher uniting unit (so the usual story goes) can it be said – in a secondary manner of speaking – that this body (mine) is the body of that soul (mine), and that this soul (mine) is the soul of that body (mine).

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A~ (whoever or whatever they may turn out to be) are transcendent to Model T, whereas c+, c#, c>, and c~ are not. Answer: Agreed. (d) In view of the transcendence just commented on, the relationship between A+ and c+ (for example) curiously reminds one of a mainstay of orthodox Christian dogma: the doctrine that there are two natures in Christ, a divine nature and a human nature, unmixed and unseparated (as the Ecumenical Council of Chalcedon determined in AD 451). Answer: That relationship reminds one of this doctrine, but in the presently considered, supposedly analogous T-related case there is nothing to which one could ascribe the two natures – unless one ascribed it to that artificial unit: {A+, c+}, in which A+ and c+, and therefore their natures, are indeed united “unmixed and unseparated” (taking into account how intimately the rules of ACTUHIST connect A+ with c+ nomologically). In spite of these dissuasive comments and in spite of the fact that I agree (though with qualifications) with what they say, I shall speak of A+ as the soul of c+, and of c+ as the T-body of A+, and likewise for the other three pairs consisting of a player of ACTUHIST on the one hand and the correlated atomic T-material object in H* on the other. My reason for this is the following. The relationship between A+ and c+ (for convenience, I stick to this example) is, as a matter of fact, analogous to the relationship between me and my body, if viewed in a dualistic perspective (which is the perspective I happen to consider correct99). The former relationship is an image of the latter – not in all metaphysically relevant respects (my body is a complex composite material object, whereas c+ is an atomic T-material object), but in the most important ones. Thus, the former relationship illustrates, and therefore helps to elucidate, the latter. And the illustration works better if, in giving expression to it, the same terminology is used as for describing what it illustrates. The Physical World, and my body along with it, has a certain amount of self-sufficiency (which self-sufficiency many people consider to be complete) – just like H*, and c+ along with it, has a certain amount of selfsufficiency. This self-sufficiency is large enough to make the Physical World a well-rounded realm of being (so to speak), a sort of garden: a garden of physical existence; the same holds true of H*. But since I exist in consciousness and action in a manner that cannot be resolved into physicalness, I transcend the garden of physical existence; I transcend, along with it, my body; I also transcend the realm of physical possibility that includes in its vast extent the Physical World. Analogously, A+ transcends 99

I have comprehensively defended dualism in my 2004 book The Two Sides of Being.

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H*; along with H*, A+ transcends c+; A+ also transcends the large realm of physical possibility (in a modified, T-related sense) which surrounds H*; in fact, A+ transcends the entire Model T. This transcendence of A+ is an analogue of my transcendence. And the analogy can be brought out even more strongly. I transcend my body, the Physical World, and the surrounding realm of (purely) physical possibility; but I am nomologically – by psycho-physical laws – related to them in my consciousness and action, that is: in my activities. Similarly, A+ transcends c+, H*, and Model T; but A+ is nomologically (by the rules of ACTUHIST) related to them in its activities: in its action and, likely enough, in its consciousness (for being conscious is very likely an aspect of A+ as a player of ACTUHIST). Due to our nomological relatedness, I and my body are one, though not one in the sense of identity; analogously, A+ and c+ are one, but not one in the sense of identity. The unit that corresponds to me and my body is called “a human being”. However, this unit, this human being, does not constitute the unity of me and my body; rather, I, my body, and our psycho-physical nomological relatedness – that is, our nomological unity – constitute this human being, this unit. If the nomological unity between me and my body is kept in mind, it is not amiss to represent the human being U.M. by the pair-set {I, my body}, and not amiss even to say that the human being U.M. is the pair-set {I, my body}; it is not amiss to say this if (and only if) one steadfastly remembers that this “is” is merely the “is” of the being-representedby relation and not the “is” of identity (compare further above in this section). I and my body, and their respective natures, are united – truly united – “unmixed and unseparated” in U.M., and in fact so closely united that it is not amiss to say: “I am U.M.” – if it is steadfastly remembered that this “am” (notwithstanding the closeness of the relationship that it stands for) is merely the “am” of the being-represented-by relation, not the “am” of identity (compare further above in this section). That the being-represented-by relation – and not identity – is the relation which underlies “is” and “am” in asserting “U.M. is {I, my body}” and “I am U.M.” can easily be seen in the following way: If “is” and “am” are interpreted as expressing identity, then from “U.M. is {I, my body}” and “I am U.M.” one can logically deduce “I am [identical with] {I, my body}”, which is false, and therefore at least one of the two premises from which that falsehood has been deduced must be false, too – which does not seem right since both premises are straightforwardly asserted (by dualists

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like me) with the firm belief of speaking the truth. If, however, “is” and “am” are interpreted as expressing the being-represented-by relation, then “I am [represented by] {I, my body}” is true, just as true as “U.M. is {I, my body}” and “I am U.M.” (and if the being-represented-by relation is transitive, then the former statement is a logical consequence of the latter two statements). I am certainly represented by {I, my body}, because I am already represented by {my body}, because I am already represented by my body. Since I am represented by my body (which is due to our psychophysical nomological relatedness: our nomological unity), it is true, in a sense, that I am my body. But the sense in which this is true (the sense of being-represented-by) is not the sense which the physicalists have in mind (the sense of identity); in their sense, it is false that I am my body (contrary, of course, to what they themselves believe). As I am represented by my body, so A+ is represented by c+. In other words, my body is my incarnation, and c+ is the T-incarnation of A+. The way incarnation works is, to a certain extent, illustrated by the manner in which T-incarnation works. This is just another aspect of the cognitive value of Model T and of the T-surrounding game of ACTUHIST for (real-life) metaphysics. By incarnation, a certain body becomes my body, and I become its soul; by T-incarnation, a certain atomic material object of T becomes the T-body of A+, and A+ becomes its soul. How T-incarnation works has been described in detail, though without mention of the word, in the preceding section; that is, all the information on exactly how c+ (for example) represents A+ – on exactly how c+ is A+ representationally – can be found in that section (given, of course, the basis provided in the two chapters preceding the present chapter). In this section, I shall point out some general aspects of T-incarnation which are of special interest in relation to the real thing: incarnation. First, it is important to note that the atomic material objects of T have been T-ontologically characterized without reference to the players of ACTUHIST (see Sects. 6.2, 6.3, and 6.4); the players of ACTUHIST are strictly external to the atomic T-material objects, even far remote from them (so to speak); the former certainly do not “mix” with the latter. Nevertheless, there is, even prior to T-incarnation, a deep connection between these two sides of T-related being. What atomic T-material objects there are is determined by the four laws for T that were passed in the preceding chapter. However, these same four laws also determine which complete Thistories are candidates for being actual*, in other words: which are candi-

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dates for being made actual* (for being actualized*) via the stepwise progressive actualization** (strictly following the timeline of T) of each of their momentary phases100 – an actualization** which is at each step effected by AU* on the basis of the collective choice of the players of ACTUHIST of an object for actualization** at that step: a certain maximal momentary event. In each T-history that is a candidate for being actual*, each of the four atomic T-material objects in that history has its determinate, unambiguously traceable individual history – and the atomic T-material objects101 are capable of representing players of ACTUHIST that make choices between alternatives. The laws for T and the rules of ACTUHIST have been made for this. Precisely which manner of representation of players of ACTUHIST by atomic T-material objects is meant to be – this has been explained in the preceding section. From the exposition there given we can gather that, according to the rules of ACTUHIST, the representation (the incarnation) of four of the five players of ACTUHIST, by one atomic Tmaterial object each, is contingent: it could have been otherwise. (But no player of ACTUHIST can change the representation it has once that representation is established at the very first step of the game.) Moreover, if all goes well, the representation of each represented player is the result of each represented player’s own choice. (If it is not the case that all goes well, AU* will lend a helping hand; see L12.) The simplicity of the incarnator (i.e., the entity which incarnates, in contrast to the incarnatee: the entity which is incarnated) is one radical difference between T-incarnation and real incarnation. And that T-incarnation is the result of “each represented player’s own choice” is another aspect in which T-incarnation differs radically from real incarnation. For this body is not my body by my own choice; whereas, given that a round of ACTUHIST is played, a certain atomic T-material object is indeed by A+’s own choice the T-body of A+ (that is: c+) – at least if all goes well (if not, then it is AU* who determines which atomic T-material object is the T-body of A+). 100

The foundation for this is laid by Act1, Act2, and Act3 in Sect. 5.2.1. Regarding actualization**, see Sect. 5.2.2. 101 The set of the atomic T-material objects is the union of all sets X which satisfy the following description: there is a T-history H that is a candidate for being actual* and X is the set of the atomic T-material objects in H. (As things have been set up, the sets that satisfy this description have no element in common if they differ at all, and they always have four elements.)

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Though the incarnated players of ACTUHIST normally come by their T-bodies as a result of their own choosing and have at least some control over their T-bodies (though no further than to the extent that is specified by rule L11), their T-bodies also have a considerable movement-independence from them. This fact is a true image of the considerable movementindependence of my body from me, in spite of its also being controlled by me to some extent. (Matters are not essentially different for you and your body, I trust.) The described T-fact is rooted in the laws for T: they are the foundation of the very constitution of the atomic T-material objects, to which the T-bodies belong, and they prescribe the possible movements of those objects. Similarly, the movement-independence of my body is rooted in the laws of physics, of which I am not the author (whereas I am indeed the author of the laws of the physics for T [= the physics of H*], which laws are simply the laws for T; see the end of Sect. 6.6). The connection between an ACTUHIST-player and the atomic Tmaterial object that is its T-body is entirely T-transcendent; the two do not mix in any sense. And yet, as long as the relevant round of ACTUHIST is being played, the two cannot be separated. But note (and what follows stands in curious contrast to what is asserted in the preceding sentence): An atomic T-material object which is a player’s T-body is, so to speak, transparent; we can, so to speak, look right through it to what is at its basis: a certain chronologically evolving (lengthening) sequence of consecutively actualized** atomic momentary Fullness-events (see D37 in Sect. 6.2). The spatial positions of these events102 go into another chronologically evolving sequence (implicit in the first-mentioned sequence) which in the end will be a complete diachronic path of Fullness (see D35 and D36 in Sect. 6.2) in the actualized* complete history. The laws for T make is possible to interpret this latter sequence as the evolving 1-to-100 movementand/or-rest in T-space of a higher continuant of T (one relative to H*), namely, of an actual* atomic T-material object, of an object, to boot, which is not immune to spontaneous modifications in its state of motion (uncaused modifications regarded from the inside of T, but player-caused modifications regarded from the outside of T). But all of this is the result of a certain interpretation of that sequence, albeit a result that is strictly determined (in all of its details) as soon as the interpretation is tried out.103 As 102

From each atomic momentary Fullness-event exactly one spatial position can be extracted. 103 We ask: “Can we not read the behaviour of a continuant object in space out of this sequence of momentary events?” By the very act of asking this question we are al-

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was suggested at the end of Sect. 6.4,104 what is seen in the sequence – a higher T-reality that strictly supervenes on a more basic, lower T-reality – might merely be in the eye of the beholder, and not something that is ontically given independently of the beholder. This would mean that atomic Tmaterial objects are not really (T-)there, that they are the T-relative equivalents of fictions, though entirely non-arbitrary fictions (necessary ones, so to speak). But be that as it may (the issue seems to be one that can only be decided by sheer decision), if they were T-fictions, it would yet remain true that four atomic T-material objects are in the transcendent perspective – given a round of ACTUHIST – more than fictions, since they are the Tbodies, the T-incarnations105 of four of the five players of (the round of) ACTUHIST. Yet, the above-described transparency of the T-bodies remains. Might not also all of the true reality of this body, taken to be a continuant, be the result of its being my body? For this body, too, appears to be a supervenient object – namely, supervenient on the fluctuating microphysical event-reality underlying it – and this body, too, can seem to be a mere fiction in comparison to that reality. The only difference might be that the supervenience of this body on microphysical eventreality is incomparably more difficult to describe – to make transparent – than the supervenience of a T-body on its basis of T-events, the reason being that this body is an utterly complex composite material object.

There is a third radical difference between T-incarnation and real incarnation (the other two radical differences are mentioned further above in this section) – the most radical difference (if it really exists). ACTUHIST can be played an indefinite number of times, yielding an indefinite number of rounds of ACTUHIST. One and the same player can participate in several of those rounds, and nothing forbids that a multiply participating player be differently T-incarnated (but there is no necessity for this, either). This, note, does not mean that there can be such a thing as T-reincarnation. The occurrence of T-reincarnation would require that a player changes its ready beginning to try out a certain interpretation of the sequence. And as soon as we do so, we find that our question has a unique and entirely determinate answer: “Yes, we can read this behaviour, and no other behaviour, of a continuant object in space out of this sequence of momentary events.” 104 Compare regarding what is said in this paragraph also Comment (5) in Sect. 7.1. 105 T-incarnation – in effect, the first collective choice of the players of (a round of) ACTUHIST – turns certain atomic T-material objects into T-incarnations of four of those players.

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T-body – under certain circumstances that are appropriate for this – in the course of one and the same round of the game; but ACTUHIST does not allow this. In contrast, multiple T-incarnation of one and the same player can easily be produced without hurting the rules of ACTUHIST – so easily that one can consider it a fact (perhaps it will some day be a fact in fact). However, [1)] it does not seem to be the case that I am, or anyone else is, a player in more than one round of that immense game that stands to the physical part of Reality as ACTUHIST stands to Model T – simply because there seems to be only one round of that game, call it “the Great Game”.106 Moreover, [2)] it does not seem to be the case that I am, or anyone of us is, a player in the – apparently – sole round of the Great Game for more than a very short time, a time so short that it dwindles to utter insignificance in comparison to the entire time of the sole round of the Great Game – whereas each player of a round of ACTUHIST participates in it throughout its entire duration. But [3)] it does seem to be the case that I am, along with others, a player in the sole round of the Great Game and that my playing within that sole round is bound up with one, unvarying incarnation of me and is coeval with my being incarnated by it – just as the playing of, say, A+ within a round of ACTUHIST is bound up with one, unvarying Tincarnation of A+ and coeval with A+’s being incarnated by it. We can therefore conclude (from 1), 2), and 3)): [4)] It appears to be the case that incarnation regarding me (as incarnatee) does not numerically vary from one round of the Great Game to another – trivially so, since there is (apparently) just one round of the Great Game –, that it does not numerically vary within one and the same round of the Great Game, either, and that it takes place only for an exceedingly short time compared to the entire duration of the sole round of the Great Game. From this last statement (i.e., 4)), large difference as well as large agreement in metaphysics between the Great Game, myself, and incarnation on the one hand and ACTUHIST, A+ (for example), and Tincarnation on the other hand can be inferred. However, “it does not seem to be the case that”, “it does seem to be the case that”, “it appears to be the case that” – modifiers that were used in the preceding paragraph (see 1), 2), 3), and 4)) – indicate that there is at least some uncertainty attached to 106

As ACTUHIST – if played – results in the selection and actualization of a certain complete T-history, the Great Game – which is being played right now, and has been played for thousands of millions of years – results in the selection and actualization of a certain temporally complete physical universe.

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the statements they modify (an uncertainty that is due to the metaphysical nature of those statements). If five people decide to play the game of ACTUHIST, they pre-exist their playing, and if they play the game and get tired of it, they can leave it and, say, go outside to look at the sunlit garden, and resume playing the game later (when it starts to rain). The reality outside of Model T – the reality beyond T in which the playing of the game is done that has T as its play-material: ACTUHIST – is large enough to allow such time-outs. And of course there are more important things for people to do than to play ACTUHIST; it is no serious matter, it is “just a game”. However, there does not seem to be a more important thing for people to do than to play the Great Game; its seriousness is supreme, it is not “just a game” at all. And, fittingly, Reality beyond the physical part of Reality does not seem to be large enough to allow doing anything else to its “inmates” than playing the game that has the entire physical part of Reality as its play-material: the Great Game. Hence it does not seem to be the case that there is room for time-outs from that game, or, indeed, for either pre-existing or postexisting the playing of it. Yet, particularly in view of the sad fact that playing the Great Game is painful (more or less so, but very considerably for the majority of the players),107 the human metaphysical desire of all times seeks deliverance from the Great Game – not extinction, but rather what would be the much enlarged equivalent of getting up from a game of ACTUHIST to go outside and look at the sunlit garden, never to return to the game. 7.3

T-immanent and T-transcendent causation

There are two kinds of causation relative to Model T: T-immanent [or “horizontal in T”] causation and T-transcendent [or “vertical to T”] causation. When talking about T-immanent causation, the word “causation” in the expression “T-immanent causation” requires no index in order to indicate the essential relatedness of that causal relation to T. It is otherwise with the second kind of causation relative to T; for we are not interested in some T-transcendent relation of causation or other, but in T-transcendent 107

In contrast, playing ACTUHIST cannot be painful in any considerable way (leaving aside the remote possibility that someone might get so emotionally involved in the game as to suffer seriously from its not running the course he/she wants it to run).

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causation relative to T. When the time comes to discuss this causal relation, I shall designate it by “T-transcendent causation→T”. The two kinds of causation just indicated do not only coexist, they are also intertwined. Important and somewhat surprising lessons about causation in Reality can be drawn from this. In order to avoid getting too far afield in the exploring and judging of alternatives of theory,108 causation will here be treated, without further ado, as being tantamount to sufficient causation (hence as being not tantamount to conditio-sine-qua-non causation – counterfactual or otherwise – or to probabilistic causation). This is legitimate even if one doubts (as I do not) that, all things considered, it is best to identify causation with sufficient causation (if a conceptual decision must absolutely be made, and one is not content to have several concepts – “theories” – of causation side by side). Treating causation as sufficient causation is legitimate because for many central purposes (though of course not for all non-marginal purposes) causation is sufficient causation. In sufficient causation, every cause is (qua cause) sufficient for the coming about of its effect: each cause makes its effect come about (which is not the case in conditio-sine-qua-non causation or in probabilistic causation). Making another long story short, it is best to take it that effects – the second terms of causation-relations – are events. Hence – since we are here dealing with Model T and with causation that concerns T, aims at T, so to speak – the second terms of the intended relations of causation will be Tevents. Now, in T-immanent causation, also the causes – the first terms of causation-relations – are T-events. Moreover, it is an obvious consequence of the physicalism which is true of Model T (T-physicalism) that all Tevents are T-physical,109 whether in the global and category-transcending sense of “T-physical”, or in the appropriate category-specific – that is, Tevent-specific – sense (see the beginning of Sect. 7.2). Therefore: if we introduce the predicate “X is an IT-cause of Y” for expressing T-immanent causation, we have as characteristic for that type of causation: ITC1 For all X and Y: if X is an IT-cause of Y, then X and Y are Tphysical T-events.

108

I have tried to master these alternatives comprehensively in my 2001 book Theorie der Kausalität. 109 For deducing “All T-events are T-physical” from T-physicalism, one needs to make use of the fact that all T-events are concrete T-entities.

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The following statement is easily seen to be a logical consequence of ITC1: ITC2 For all X: if X is an IT-cause of something T-physical, then X is itself T-physical. And the following statement, in turn, is easily seen to be a logical consequence of ITC2: ITC3 For all Y: if Y is T-physical and has an IT-cause, then Y also has a T-physical IT-cause. Do ITC2 and ITC3 look somewhat familiar? They should. Just drop the prefixes “IT-” and “T-”, and from ITC2 there emerges the Strong Principle of the Causal Closure of the Physical, or briefly: the Strong Closure Principle, which is C2

For all X: if X is a cause of something physical, then X is itself physical.

By the same procedure (of prefix-dropping), there emerges from ITC3 the Weak Principle of the Causal Closure of the Physical, in short: the Weak Closure Principle, which is C3

For all Y: if Y is physical and has a cause, then Y also has a physical cause.

ITC2 is the T-analogue of C2, and ITC3 the T-analogue of C3. ITC2 and ITC3 look familiar because the originals of ITC2 and ITC3 – C2 and C3: the Closure Principles – play a considerable role in modern philosophy of mind as premises of arguments for mental-event-physicalism (in Reality), stated by “Every mental event is physical”. Unlike C2 and C3, ITC2 and ITC3 are obviously, even trivially, true: they are obvious logical consequences of ITC1, which is incontrovertibly true because T-physicalism is incontrovertibly true (and because Timmanent causation relates T-events). No complete analysis of Timmanent causation is necessary for obtaining these results. (But a complete analysis of T-immanent causation will nevertheless be provided in this section; see below.)

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C2 and C3 would be just as obviously true as are ITC2 and ITC3 if C1

For all X and Y: if X is a cause of Y, then X and Y are physical events

were as incontrovertibly true as is its T-analogue, ITC1; for C2 and C3 follow logically from C1. But C1 is certainly not incontrovertibly true. As is easily understandable psychologically, physicalists, in arguing on the basis of C2 or C3 for mental-event-physicalism, already subconsciously presuppose the truth of event-physicalism (“Every event is physical”). Hence – given their unquestioning adoption of an exclusively eventrelating conception of causality: a conception according to which causation is, and can only be, event-causation – physicalists subconsciously presuppose also the truth of C1. And therefore – now on the conscious level – C1’s straightforward logical consequences C2 and C3 seem obviously correct to physicalists, and to be very plausible premises indeed in arguing for mental-event-physicalism. But what we are looking at here is just a somewhat indirect, somewhat subtle way of begging the question. For people who are not already convinced of event-physicalism (though they do not deny it either, let’s say) – because they are not already convinced of its special case: mental-event-physicalism (though not denying it) – there is no good reason to accept C1, or C2, or C3. Why, in the first place, should such people accept causal event-physicalism (that is, C1)? And if they do not accept causal event-physicalism, why should they, then, accept the Strong or the Weak Closure Principle (C2 or C3)? There are no cogent answers to these questions. It is a logical consequence of C2 that nothing non-physical is a cause of anything physical; and it is a logical consequence of C3 (and C2) that nothing non-physical is a physically unparalleled cause of anything physical. In other words, C2 entails the causal powerlessness of the nonphysical regarding the physical, and C3 entails (what is usually taken to be) the causal superfluousness of the non-physical regarding the physical.110 Causal powerlessness and/or causal superfluousness regarding the 110

Regarding the parenthesis in the statement that this footnote refers to, consider that parallel causation – two (sufficient) causes of very different kinds (for example, one physical, the other non-physical) for one and the same effect E – does not ipso facto imply that one of the two causes in it is superfluous (or “over-determinative”), since it could be the case that neither of the two causes of E can be a cause of E without the other also being a cause of E.

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physical are considered to be telling against the existence of non-physical mental events: this is the basis on which C2 or C3 are used in arguing for mental-event-physicalism In contrast, there is no point in using ITC2 or ITC3 to argue for Tphysicalism (or any specialization of it), since T-physicalism is evidently true (and therefore also every specialization of it); no need to argue for it. Moreover, ITC2 does not even entail the causal powerlessness of the nonT-physical regarding the T-physical, and ITC3 does not even entail the causal superfluousness of the non-T-physical regarding the T-physical (nor does ITC2). What ITC2 does entail is that nothing non-T-physical is an ITcause of anything T-physical, which is merely the IT-causal powerlessness of the non-T-physical regarding the T-physical; and what ITC3 (and ITC2) does entail is that nothing non-T-physical is a T-physically unparalleled IT-cause of anything T-physical, which is merely the IT-causal superfluousness of the non-T-physical regarding the T-physical. And all of this is entirely trivial, given the incontrovertible truth of ITC1, and does not in itself exclude that the concrete non-T-physical is, regarding the T-physical, neither causally powerless nor causally superfluous – in a specifiable sense of “causally”, “causal”, and “cause” which is quite separate from ITcausation (i.e., T-immanent causation), but which nonetheless is an entirely legitimate, even indispensable sense. Analogously, the logical consequences of C2 and C3 do not, taken by themselves, exclude that the non-physical is neither causally powerless nor causally superfluous regarding the physical – in a specifiable sense of “causally”, “causal”, and “cause” which is separate from C1-fulfilling causation and from event-relating causation in general, but which nonetheless is an entirely legitimate, even indispensable sense. This fact is overlooked by physicalists, and it is easy to overlook it because the word “cause” is used in C2 and C3 (in contrast to ITC2 and ITC3) without a modifying prefix or other indicator of relativization. This linguistic nakedness almost irresistibly insinuates that the word “cause”, and its cognates “causal” and “causally”, have just one legitimate sense. 111 Looking at a model – that is, at the fruitful (not just peaceful) coexistence of T-transcendent causation→T

111

Note that there is already more than one sense of “cause” in which C2 and C3 are true. For example (among several other such senses), there is a sense of “cause” in which C1, and therefore C2 and C3, are analytically true. Simply define causation as (being nothing else than) physical event-causation. Many contemporaries would find nothing fishy in this at all (“What else, in the world, could causation be?”).

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with T-immanent causation – may, perhaps, be an eye-opener in this particularly unfortunate case of conceptual blindness regarding Reality. In addition to the predicate “X is an IT-cause of Y” for expressing Timmanent causation, I introduce the predicate “X is a TT-cause of Y” for expressing T-transcendent causation→T. The two predicates are defined as follows: D48 X is an IT-cause of Y =Def [1] X and Y are T-events [and hence Tphysical], [2] every temporal position in the domain of X is earlier than every temporal position in the domain of Y, [3] X is actual*, [4] every complete T-history which is a candidate for being actual* and has X as a part-event also has Y as a part-event, and [5] there is no proper part-event Z of X such that every complete T-history which is a candidate for being actual* and has Z as a part-event also has Y as a part-event. D49 X is a TT-cause of Y =Def [a] X is {A+, A#, A>, A~, AU*}, [b] there is a T-event Z such that (i) Y is a part-event of Z and (ii) each momentary phase E of Z is the (final) collective choice of {A+, A#, A>, A~, AU*} at the step of ACTUHIST which corresponds to the sole T-temporal position in the domain of E. Comments: (1) There is an assumption which underlies much of what follows below. It is the assumption of Normalcy. Normalcy (in the present context) is taken to be what is described by the following statement: There is a certain (complete) round of ACTUHIST, with indeterminism even after the first two steps, and “A+”, “A#”, “A>”, “A~”, and “AU*” are taken to refer to the human players of that round, who at no stage of it violate the rules of ACTUHIST, and the predicates “actual*”, “actual**”, “actualize*” [“make actual*”], and “actualize**” [“make actual**”] – with everything that has been said in this book in characterization of their use – and “H*”, too, are understood in the way that is relative to that round of ACTUHIST. In what follows it is assumed that this statement is true; hence I shall speak of the assumption of Normalcy. So far, the above statement has not been made true – because ACTUHIST has not been played by anybody so far; it

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has not, so far, reached the normality-status for games such as ACTUHIST: the status of having been correctly played by human players at least once and with some step in the game, in addition to the first two steps, not already rule-determined, regarding its content, on the basis of the preceding steps. But I trust that ACTUHIST will reach that status. If not, then the assumption of Normalcy will nevertheless have rendered its service. (2) It follows from D48 that if X is an IT-cause of Y, then Y is an actual* T-event; and it follows from D49 that if X is a TT-cause of Y, then Y is an actual* T-event. This is seen as follows: Suppose (for the first deduction) that X is an IT-cause of Y (the statement after “that” is the supposition). Hence X is an actual* event (according to D48, [1] and [3]), and therefore X is a part-event of H*. The latter consequence is obtained on the basis of (C*) in Sect. 6. (C*) is bound to be true, given the assumption of Normalcy, and implies that X, by being actual*, is actual in H*, and the way for a T-event (here: X) to be actual in a Thistory (here: H*) is to be a part-event of it. It follows, moreover, from the supposition (according to D48, [4]) that Y is a part-event of every complete T-history which is a candidate for being actual* and has X as a part-event. But H*, which has just been shown to have X as a part-event, is a complete T-history that is a candidate for being actual* (after all, H* is the one candidate actualized* of all T-histories that are candidates for being actual*). Hence Y is a part-event of H*. And therefore Y is an actual* event. This consequence is obtained on the basis of (C*) in Sect. 6. (C*) implies that Y is actual* if it is actual in H*. But, doubtless, Y is actual in H*, because it is a part-event of H* (as we have seen). We have now established (by the first deduction):

ITC0

For all X and Y: if X is an IT-cause of Y, then Y is an actual* T-event.

Now suppose (for the second deduction) that X is a TT-cause of Y. Hence (according to D49, [b]) Y is a part-event of a T-event Z which is such that every momentary phase E of Z is the collective choice of {A+, A#, A>, A~, AU*} at the step of ACTUHIST that corresponds to the sole T-temporal position in the domain of E. Hence every momentary phase of Z is, according to L6, a maximal momentary event which is once [that is: at a certain step in that round of ACTUHIST which – following the assumption of Normalcy – is being referred to] actualized** by AU*, the Author. Hence every momentary phase of Z is a maximal momentary event that is once actual**, and hence every momentary phase of Z is actual* (according to the principle Act2 in Sect. 5.2.1). But if all momentary phases of Z are actual*, then, doubtless, Z is itself actual* (cf. the principle Act3 in Sect. 5.2.1).

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Therefore: Y is actual* – since Y has already been concluded to be a part-event of the event Z, now established to be actual*, and since an event (here: Z) certainly cannot be actual* without all of its part-events being actual*, too. [Alternatively, one can reason in the following way: Since event Z is actual*, it follows according to (C*) that Z is actual in H*, which means that Z is a part-event of H*. Therefore, since Y has already been concluded to be a part-event of Z, Y is also a part-event of H* (by the transitivity of parthood), which means that Y is actual in H*. Hence we have according to (C*): Y is actual*.] We have now established (by the second deduction):

TTC0

For all X and Y: if X is a TT-cause of Y, then Y is an actual* T-event.

(3) What has just been shown to be true for IT-causation (i.e., Timmanent causation) and TT-causation (i.e., T-transcendent causation→T) amounts merely to this: the above definitions of these concepts of causation fulfil a minimal criterion of conceptual adequacy (for definitions of concepts of causation). As far as that criterion goes, those definitions pass. But the conceptual adequacy of D49 needs, in fact, no further defence: there can be no reasonable doubt that it is a relation of (sufficient) causation (regarding T, of course) which is defined by D49 – and evidently it is a relation of agent-causation (its first term being a composite agent). In contrast, a few words need to be said in justification of the conceptual adequacy of D48, which, if adequate, presents a relation of eventcausation (its first terms being events). The definitional clauses [2] and [5] – which played no role in the deductions in Comment (2) – are meant to support the conceptual adequacy of D48 (and, evidently, in this role [2] is much more important than [5]). The central idea behind D48 is that ITcausation is T-nomological determination between actual* T-events which fulfils certain additional requirements. The major part of that idea – namely, that IT-causation is a relation of nomological determination between actual* T-events – is spelled out by the definitional clauses [1], [3], and [4]. The use of the word “nomological” is justified by the inner correspondence between the laws for T on the one hand and the T-histories that are candidates for being actual* – referred to in [4] – on the other. The minor part of the idea behind D48 – namely, the sum of the additional requirements that delimit IT-causation from T-nomological determination between actual* events – is spelled out by the definitional clauses [2] and [5].

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[5] is meant to exclude the trivial cases of event-causal overdetermination according to [1], [3], and [4] from being cases of ITcausation – by allowing only such (sufficient) IT-causes as are in themselves minimally sufficient for the effect; this agrees with widespread intuitions regarding event-causation. It is an interesting question to what extent there can be non-trivial cases of event-causal over-determination according to [1], [2], [3], [4], and [5] – but it is a question I leave open. It seems to me that such cases can occur under certain circumstances – and if they occur, then there is nothing problematic in counting these cases as cases of IT-causation. (Certainly not all possible cases of event-causal overdetermination should be excluded from being actual cases of eventcausation already on conceptual grounds, that is, by simply defining concepts of event-causation in such a way as to exclude all possible cases of event-causal over-determination from being actual cases of eventcausation.) [2], too, agrees with widespread intuitions regarding event-causation; but one may well wonder why it is so widely believed that the cause must temporally precede the effect. I offer a speculation – not entirely baseless – that tries to answer this question: The temporal precedence of the cause over the effect is, in event-causation, the sole foundation of the ontological asymmetry between cause and effect, which asymmetry is required by the idea – central to (sufficient) causation – that the cause makes the effect come about. Note how differently, how effortlessly, in agent-causation – in particular, in TT-causation – the required ontological asymmetry of cause and effect is taken care of: an agent (the cause) is (onto-)categorially distinct from an event (the effect). In TT-causation, in particular, the cause is a Ttranscendent (composite) agent – {A+, A#, A>, A~, AU*} – while the effect is a T-immanent event. Thus, TT-causation relates very different entities indeed; but notwithstanding this fact there is nothing particularly incomprehensible about TT-causation. (4) Nothing in D48 forbids that there are many T-events that are ITcauses. According to D49, however, there can be at most one TT-cause: {A+, A#, A>, A~, AU*}. Why not also have the non-empty subsets of {A+, A#, A>, A~, AU*} as potential TT-causes? – For one thing, any subset of {A+, A#, A>, A~, AU*} that does not include AU* cannot make a T-event come about; for according to L5 the power of making-actual** – and hence also the power of making-actual* – resides entirely with AU* (the Author). For another thing, the act of actualization** by AU* is never

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primarily directed at a T-event chosen by an AU*-including proper subset of {A+, A#, A>, A~, AU*},112 but always – at any stage of the game of ACTUHIST played – at the maximal momentary T-event which is the (final) collective choice of all the agents in {A+, A#, A>, A~, AU*}. This does not preclude that sometimes, and even frequently, clearly delineable causal shares can be attributed to the individual agents after all (i.e., after all of them have chosen and done). But such attribution is secondary: TTcausation lies primarily – and therefore simpliciter – with {A+, A#, A>, A~, AU*}, and with {A+, A#, A>, A~, AU*} alone (but with AU* playing a special role in this). I suggest that the implications by analogy of these Tcausal matters for our practice of allotting causal (and, in consequence, moral) responsibility to individuals in Reality can hardly be overestimated. We – and all other created souls – are, in our causal roles, just parts of the plural Great Will, of the Maker of History, whose single, unobstructable willing is no mere (i.e., untransformed) sum of countless obstructable willings; the Author of the Laws of Nature Himself is just a part of the Great Will (but plays a special role – of which the role of AU* in ACTUHIST is, I submit, to some extent a structural analogue – in determining what it is that the Great Will wills, and in making it come about). (5) There is a curious question: “When was X a [sufficient] cause of Y?”, or putting the same question more briefly: “When did X cause Y?” Some philosophers think it very important that this question have an informative answer in any given case of genuine causation, and hold it against the very idea of agent-causation that – allegedly – the timingquestion cannot be answered (in an informative way) for cases that are supposed to be cases of agent-causation. Well, how is the timing-question to be answered for IT-causation – and for TT-causation? For IT-causation, the time of causation is the last temporal position in the domain of the cause-event; for this is the time at which the cause is complete. For TT-causation, the time of causation is the entire time-stretch from the first temporal position in the domain of the effect-event Y, Zfirst(Y), to the last temporal position in that domain, Zlast(Y); for the temporal positions between (and including) these two temporal positions correspond one-to-one to the steps of the round of ACTUHIST in 112

It does sound strange to say that a set chooses, or causes, anything. But it is merely a harmless façon de parler (which can be very useful for keeping formulations from getting over-complicated): An AU*-including subset M of {A+, A#, A>, A~, AU*} chooses (or causes) X if, and only if, all the members of M collectively choose (or cause) X.

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the course of which Y is made actual* (as a part-event of the section of H* that begins with Zfirst(Y) and ends with Zlast(Y)). (6) The Principle of Sufficient Cause – “Everything that happens has a (sufficient) cause” – is the causal form of the Principle of Sufficient Reason. Corresponding to the two kinds of T-directed causation we are considering, the Principle of Sufficient Cause has two T-related analogues: ITC4

For all actual* T-events Y: there is an X which is an IT-cause of Y.

TTC1

For all actual* T-events Y: there is an X which is a TT-cause of Y.

Note that ITC4 is the logical converse of ITC0 in Comment (2), and that TTC1 is the logical converse of TTC0, also in Comment (2). For ∀y[P(y) ⊃ ∃xR(x, y)], which is the logical form of both ITC4 and TTC1, is the logical converse of ∀y[∃xR(x, y) ⊃ P(y)]; and therefore (due to elementary logical equivalence) it is also the logical converse of ∀x∀y[R(x, y) ⊃ P(y)], which latter formula is the logical form of both ITC0 and TTC0. Now, there is a trivial reason for the falsity of ITC4, and there is a non-trivial reason for the falsity of ITC4. The trivial reason for the falsity of ITC4 is this: For many actual* T-events, there are no actual* T-events that temporally precede them in the manner demanded by [2] of D48. The least-lasting such event is the first maximal momentary event that goes into the making of H*, and the longest-lasting is H* itself. (Under the assumption of Normalcy, the designator “H*” refers as intended.) The trivial reason for the falsity of ITC4 cannot fail to falsify ITC4 as long as one does not deny the status of T-event to what I consider to be T-events which begin at the beginning: T-events whose first temporal position is the temporal position 1. And why should one deny the status of T-event to those entities, considered to be T-events by me? To deny them that status seems utterly ad hoc. The non-trivial reason for the falsity of ITC4 is this: For some actual* T-event E* there are many actual* T-events that precede it (in the manner required by [2]) – but none of these events (T-lawfully) determines E*, that is: it is not true for any actual* T-event X prior to E* that E* is a part-event of every complete T-history which is a candidate for being actual* and of which X is a part-event. Therefore: condition [4] of D48 is not fulfilled by any actual* T-event X prior to E*, and therefore there is no IT-cause of E*.

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This non-trivial reason for the falsity of ITC4 is not a merely possible reason for it. There is a non-initial step N (1 < N ≤ 100) of the relevant round of ACTUHIST – see the assumption of Normalcy – which is such that what happens at that step is not determined by the rules of ACTUHIST on the basis of what has happened before. Therefore, the initial section of H* that the players have selected and actualized* up to (and including) step (N−1) – in short: H*//(N−1) – has at step N more than one possible T-lawful momentary continuation. Of course, some maximal momentary event E* with temporal position N is actualized* and added to H*//(N−1), yielding H*//N; but E* is not the only such event that can be Tlawfully actualized* given the (already accomplished) actualization* of H*//(N−1). This means: H*//(N−1) does not (T-lawfully) determine E*, and consequently no actual* T-event prior to E* determines E* (for if some such event X did determine E*, then H*//(N−1) would, of course, also determine E*113]). But if no actual* T-event prior to E* determines E*, then there is no IT-cause of the actual* T-event E* – and ITC4 is falsified. The remarkable fact is this: Whereas ITC4 – the Principle of Sufficient Cause for T-immanent causation – is false, even trivially false, TTC1 – the Principle of Sufficient Cause for T-transcendent causation→T – is demonstrably true. Suppose Y is an actual* T-event. Hence, (i´), Y is a part-event of H* – because, according to principle (C*), everything actual* is actual in H* and because the manner for a T-event (here: Y) to be actual in a Thistory (here: H*) is to be a part-event of that history (see Comment (2) above). But, (ii´), each momentary phase E of (the T-event) H* is the (final) collective choice of {A+, A#, A>, A~, AU*} at the step of ACTUHIST that corresponds to the sole T-temporal position in the domain of E; for this was the whole point of playing ACTUHIST: H* is the T-history selected and actualized* by {A+, A#, A>, A~, AU*} in the course of the 100 steps of the – according to the assumption of Normalcy – relevant round of ACTUHIST. It follows from (i´) and (ii´), using D49, that {A+, A#, A>, A~, AU*} is a TT-cause of Y. 113

X [by being actual* and prior to E*] is a part-event of H*//(N−1), and therefore every complete T-history that is a candidate for being actual* and of which H*//(N−1) is a part-event is also a complete T-history that is a candidate for being actual* and of which X is a part-event (by the transitivity of parthood). Therefore, if E* is a partevent of every complete T-history that is a candidate for being actual* and of which X is a part-event, then E* is a part-event of every complete T-history that is a candidate for being actual* and of which H*//(N−1) is a part-event.

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For the metaphysics of Reality, an important lesson can be drawn from these T-related results. Although there is a natural conception of eventhood – its T-analogue is presented in Sect. 3.4 – and a natural conception of event-causation – its T-analogue is given by D48 above – which make the Principle of Sufficient Cause (regarding Reality) trivially false if it is interpreted in accordance with them,114 and although there are reasons derived from modern physics for holding that the Principle of Sufficient Cause is very probably false under all physico-event-causal interpretations of causation (for such interpretations, C1 is analytically true – cf. footnote 111 –, with causes being sufficient causes) – notwithstanding all this, the Principle of Sufficient Cause may be true in an agent-causal interpretation of it, roughly (very roughly) along the lines pointed out above. The only reason why one is adverse to exploring this option that saves the Principle of Sufficient Cause – which has been considered a mainstay of rationality for, roughly, the last two and a half thousand years – is the circumstance that one would be forced to assume causes which are beyond the physical part of Reality. Modern physics strongly suggests that some occurrent physical events – in fact, physical occurrences with a time that precedes them – do not have any physical cause. This, in itself, falsifies the Principle of Sufficient Cause under all physico-event-causal interpretations of causation, according to which interpretations causes are, by definition, physical events (the occurrence of which is sufficient for bringing about the – likewise physical – effect). For under any such interpretation the following is true: if an event has no physical cause, then it has no cause at all; if occurrent, the event is, therefore, absolutely accidental. Thus, if one wishes to maintain the Principle of Sufficient Cause in the face of occurrent physical events that have no physical cause, then adopting a physico-event-causal interpretation of causation is out of the question. If one wishes to save the Principle of Sufficient Cause in the face of modern physics, without denying what is very likely the case: that some occurrent physical events have no physical cause, then accepting non-physical causes of occurrent physical events that are without physical cause cannot be avoided (which implies discarding both C2 and C3, beloved by physicalists, in the corresponding general interpretation of “cause”). Then, in addition to physical event-causation of the physical – of which IT-causation is an analogue – non-physical agent-causation of the physical – of which TTcausation is an analogue – deserves special attention and respect. 114

In Reality, too, there is no lack of occurrent events that are not temporally preceded by any occurrent events.

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Note the – prima facie – curious fact that I have not provided a Trelated analogue of the non-physical event-causation of the physical (which would have amounted to the introduction of an event-causal kind of T-transcendent, but T-directed causation). It is not implausible that there is a causal link between the conscious states of the human players in the relevant round of ACTUHIST (compare the assumption of Normalcy) – which “states” are non-physical occurrent events – and the T-physical events that turn out to be actual*. But that causal link, if it exists, is highly indirect. In the causal foreground are the collective actions (the selections-cumactualization*) of the players, which actions are overt and regulated by known rules, therefore easily describable (in striking contrast to the psychological processes that lead up to these actions, perhaps cause them). (7) Since all actual* T-events have a TT-cause (the composite agent {A+, A#, A>, A~, AU*}) and since some actual* T-events have an ITcause* (this being some actual* T-event preceding them), some actual* Tevents are T-causally “over-determined” in a non-trivial way – in fact, they are T-causally determined in a twofold way already at the level of the ontological categories of their causes. In fact, only the IT-chance-events, the IT-accidents – that is, the actual* T-events that do not have an IT-cause: the events that appear to be accidental in the T-immanent perspective – are not T-causally “over-determined”. I put the last word of the preceding sentence in scare quotes because that word (like the word “over-reacted”) suggests that something untoward is designated by it; but there is nothing untoward in the union of IT- and TT-causation. Both types of causation have a common backbone (so to speak): the rules of ACTUHIST, and in particular, the rules L7 – L10 of ACTUHIST, each of which rules precisely implies a law for T: L7 precisely implies (i.e., there is an exact rule-to-law correspondence between L7 and) the First Law, L8 precisely implies the Second Law, L9 the Third Law, and L10 the Law of Minimization, there being no other (axiomatic) laws for T than the mentioned four (as emerges in Chapter 6). The rules of ACTUHIST regulate the T-directed actions of {A+, A#, A>, A~, AU*} – and thereby TT-causation – and they determine, via the laws for T, which T-histories are candidates for being actual*, thereby determining what might be called the hub of IT-causation. TT- and IT-causation are not only joined by their common nomological backbone, there are also bound together by a relation of existencedependence: the existence of IT-causation requires the existence of TTcausation. There would not be any IT-causation if there were no TTcausation; for if there were no TT-causation, then there would not be any

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actual* T-event, hence there would not be any T-event that fulfils condition [3] of D48, and hence there would not be any IT-causation. ITcausation – since it is an event-causal relation – does not initiate actuality*, it only transmits it, so to speak; initiating actuality* is the privilege of TTcausation. This means that TT-causation is fundamental to IT-causation, that IT-causation is secondary to TT-causation. It is also true that TTcausation, given a whole round of ACTUHIST, cannot fail to bring along with it IT-causation, but it would be very misleading to say, in view of this fact, that the existence of TT-causation requires the existence of ITcausation. (8) But is there not – in the near vicinity of IT-causation – an eventcausal concept of T-causation that does not involve actuality* and that has no relations with TT-causation? – One might consider a concept of Tcausation that is not absolute, but relative to this or that T-history H; such a concept would not involve the actuality* of events – it would only involve their actuality-in-H, that is, their being part-events of H. But in order to make it a completely T-immanent concept, it must not only be dissociated from the concept of actuality* itself, but also from the notion of T-histories being candidates for being actual*. Here follows a specific proposal of a concept of T-causation which is as envisaged (the proposal is produced by modifying D48 in the required ways): D50 X is a CIT-cause115 of Y in H =Def [1´] X and Y are T-events and H is a complete T-history, [2´] every temporal position in the domain of X is earlier than every temporal position in the domain of Y, [3´] X is a part-event of H [that is, X is actual in H], [4´] every complete Thistory which is nomologically identical to H and has X as a partevent also has Y as a part-event, and [5´] there is no proper partevent Z of X such that every complete T-history which is nomologically identical to H and has Z as a part-event also has Y as a partevent. The above-invoked notion of nomological identity between T-histories simply means that they have the same laws – the same laws under the intrinsic conception of lawhood (see Sect. 4.1.1), while the extrinsic conception of lawhood, otherwise adopted here, is left out of play (since it essentially refers to actuality*). 115

“CIT-cause” stands for “completely immanent T-cause”.

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Well, what is wrong with the completely T-immanent concept of history-relative causation that is presented by D50? – Nothing much is wrong with it, but it is not the concept of T-immanent causation that we are (or in any case: ought to be) mainly interested in; doubtless, it is definition D48, not D50, that presents the concept of T-immanent causation that we are (or ought to be) mainly interested in. From this fact a paradigm-setting thought regarding the analysis of causation in Reality can be derived: Analyzing causation equirelatively (if I may be allowed this neologism) to all possible worlds, without reference to what is absolutely actually the case [in such analyzing every possible world is considered, with regard to its objective status of actualness, to be just like any other possible world], is not enough (to say the least). (9) If we trace to its roots the causality that relates to Model T in a given round of the game of ACTUHIST, then we discover that this causality has three roots. Each of these roots gives life and being to both branches of T-causality: to T-immanent causation and to T-transcendent causation→T: first, the rule/law-making by AU* (the Author); second, the rule/law-conform (final) choices of the players A+, A#, A>, A~, and AU*; third, the choice-conform actualizations* by AU*. The central role of AU* for T-causation is conspicuous, yet AU* is certainly not what might be called a causal dictator. All things considered, it seems appropriate to introduce the following distinction: If we consider any actual* T-event E, the IT-causes of which are collected in the set ITC(E), then the events in that set are the second (and secondary) causes of E, whereas {A+, A#, A>, A~, AU*} is the first (and primary) cause of E. This is not the place for a detailed comparison of the present theory of first cause and second causes with the traditional theory – found, for example, in the writings of Thomas Aquinas. But it seems to me that the present theory has three advantages over the traditional one. Some of these advantages are due to the fact that the present theory of first cause and second causes refers to Model T, where metaphysical matters are far less complex than in Reality and must therefore be far clearer to us; yet not all of these advantages have this merely T-related character. The said three advantages of the present theory of first cause and second causes over the traditional theory are the following: (1) The present theory is entirely clear and specific; in particular, it is entirely clear and specific about how the first cause and the second

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causes are related to each other: it spells out “the mechanism” of their relatedness. (2) According to the present theory, the first cause is not a singularity but a plurality (but with AU* having a special position in it; why the inner plurality of the first cause116 is an advantage will become plain in Sect. 7.5). (3) Although second causes are secondary causes, it can, according to the present theory, be clearly and plausibly explained why they are nevertheless not superfluous: For one thing, causation by an event is an entirely different affair from causation by an agent. Even if both kinds of causation aim at the very same event (in relation to T, all event-causal effects are agent-causal ones!), both kinds of causation do so in very different ways. Therefore, saying that we have no use for second causes of actual* T-events, no use for their ITcauses (themselves actual* T-events) – because we already have the first cause of all actual* T-events, the TT-cause of them all (a single Ttranscendent composite agent) – is like saying that we have no use for apples because we already have one pear. For another thing, IT-causation is so far from being superfluous that it indicates the extent to which TTcausation has an automatic character; this is so because the instances of IT-causation with causes (entirely) prior to a given temporal position N and with effects beginning not earlier than N collectively indicate the extent to which what has been done by {A+, A#, A>, A~, AU*} before stage N in the selection and actualization* of H* already fixes what is going to be done in this respect by {A+, A#, A>, A~, AU*} at N and later in the game. (10) Given the present intellectual climate (in extreme contrast to the intellectual climate of the Middle Ages), an apparent superfluousness of second causes is not an issue of present-day metaphysics (of Reality). The apparent superfluousness, indeed, the apparent straight non-existence of a first cause – this is an issue of present-day metaphysics. Perhaps the intimate union of T-immanent causation and T-transcendent causation→T, of IT-causation and TT-causation – in other words: the intimate union of the many second (event-)causes relating to T and of the one first (agent-)cause relating to T – can serve as an orientation for rethinking some all-too-quick 116

There is no outer plurality of first cause, since there is just one first cause for everything that happens (see TTC1 and combine it with D49). But there is an inner plurality of the first cause, since it is a composite agent.

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theoretical decisions underlying the widespread belief that there is no first cause. 7.4

Composite T-material objects

In considering composite T-material objects, I shall concentrate on the Thistories which are candidates for being actual* and which have the same initial state as H*; in other words (in view of the results of Chapter 6), I shall concentrate on the lawful T-histories in which c+, c#, c>, and c~ – the actual* atomic T-material objects – are the atomic T-material objects. In order to be able to express myself briefly, I shall call T-histories which are candidates for being actual* and have the same initial state as H* “H*relatives” (and I shall call only such entities “H*-relatives” that are Thistories with the same initial state as H* and candidates for being actual*). Given the declared restriction of attention, only a certain specialization of the general relation of material composition (constitutive of composite T-material objects) will here be considered. For expressing that special relation, I shall use the predicate “X is composed* at Z in H by Y”. This predicate is analytically characterized by the following principle: MO1

For all X, Z, H, and Y: if X is composed* at Z in H by Y, then Z is a T-temporal position, H an H*-relative, and Y a subset of the set of atomic T-material objects in H.

Since in all H*-relatives the atomic T-material objects are the same objects, namely: c+, c#, c>, and c~, what is said by MO1 can also be put in the following way: MO1´ For all X, Z, H, and Y: if X is composed* at Z in H by Y, then Z is a T-temporal position, H an H*-relative, and Y a subset of {c+, c#, c>, c~}. Now, the predicate that is characterized by MO1´ (or MO1) cannot be perspicuously defined, because there is not just one way of material* composition and because the various ways of material* composition differ greatly. But several of these ways will be considered below, which will give considerable content to the concept of material* composition. And of course also without defining the predicate “X is composed* at Z in H by

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Y”, treating it as a primitive, it can be used in defining other predicates, for example: D51 X is a T-material* object [broadly conceived] =Def There is an H which is such that (i) for every T-temporal position Z: X is composed* at Z in H by a subset of {c+, c#, c>, c~},117 and such that (ii) for some T-temporal position Z: X is composed* at Z in H by a nonempty subset of {c+, c#, c>, c~}. The following principle of existence holds true for all T-material* objects: MO2

For all T-material* objects Y, T-histories H, T-temporal positions Z: Y exists at Z in H if, and only if, Y is composed* at Z in H by a non-empty subset of {c+, c#, c>, c~}.

It is easily verified that every element of {c+, c#, c>, c~} is a Tmaterial* object in the sense that is specified by D51. Take c+, for example. It is obviously true, for every T-temporal position Z, that c+ is composed* at Z in H* [which is an H*-relative of itself] by the subset {c+} of {c+, c#, c>, c~}, and consequently [since 1 is a T-temporal position] it is true that c+ is composed* at 1 in H* by that same subset {c+} of {c+, c#, c>, c~}. Since the definiens of D51 is fulfilled by c+, we therefore have: c+ is a T-material* object, as defined by D51. But it is also clear that c+ and its colleagues are very special T-material* objects. We can fix their common nature by positing: MO3

The atomic T-material* objects [which are no other entities than the – long-familiar – actual* atomic T-material objects: c+, c#, c>, and c~] are precisely those T-material* objects that are representable by the sets {c+}, {c#}, {c>}, and {c~}.

Before continuing about T-material* objects, a brief note on representability (a very important notion that does not only occur in MO3 but also in the principles MO4, MO5, and MO7 below): The G-entities are 117

Note that the quantificational phrase “a subset of {c+, c#, c>, c~}” is meant to be inside the scope of the quantificational phrase “for every T-temporal position Z”, and not vice versa. Note also that “X is composed* by a subset of {c+, c#, c>, c~}” is just a useful façon de parler for what is properly, but awkwardly, expressed in the following way: “X is composed* by precisely the objects in a subset of {c+, c#, c>, c~}”.

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representable by the K-entities if, and only if, no G-entity is a K-entity, but the G-entities and the K-entities are (nevertheless) one-to-one correlated with each other in a particularly telling uniform manner. Note that instead of “is representable by” one may as well say “is represented by”; the former formulation merely emphasizes that there may be several, perhaps many, ways of representing the G-entities. In any case, if the G-entities are representable/represented by the K-entities, then also each G-entity is representable/represented by a (unique) K-entity – and each K-entity represents a (unique) G-entity. Returning to T-material* objects: in the near vicinity of c+, c#, c>, and c~, there are T-material* objects that are in all respects just like c+, c#, c>, and c~ – except that they are not atomic, but composite: MO4

The simply composite T-material* objects are precisely those Tmaterial* objects that are representable by the subsets of {c+, c#, c>, c~} that have at least two elements.

For example, the pair-group /c+, c#/ is a simply composite T-material* object. The argument for showing that /c+, c#/ is a T-material* object (in the sense of D51) already exhibits much of the way in which /c+, c#/ is just like c+ (and just like c#): For every T-temporal position Z it is obviously true that /c+, c#/ is composed* at Z in H* [which is an H*-relative of itself] by the subset {c+, c#} of {c+, c#, c>, c~}, and consequently [since 1 is a T-temporal position] it is true that /c+, c#/ is composed* at 1 in H* by that same subset {c+, c#} of {c+, c#, c>, c~}. Since the definiens of D51 is fulfilled by /c+, c#/, we therefore have: /c+, c#/ is a T-material* object, as defined by D51. Just like the atomic T-material* object c+, the simply composite Tmaterial* object /c+, c#/ is composed* at all times by the same subset of {c+, c#, c>, c~}: in the case of c+, that subset was {c+}; in the case of /c+, c#/, it is {c+, c#}. Moreover, c+ and /c+, c#/ are composed* by {c+}, respectively {c+, c#} not only in H* at all times, but in all H*-relatives at all times; and nothing else than {c+}, respectively {c+, c#} goes into their composition*. This is what makes c+ representable by {c+}, and /c+, c#/ representable by {c+, c#}. A simply composite T-material* object is impervious to any changes in the spatial relationships of the atomic T-material objects that compose* it: it survives any degree of scattering. In the metaphysics of Reality, many philosophers are disinclined to call something scattered (for example, the

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group consisting of a certain hair on my head and the moon) a “material object” – for a scattered something does not seem to have that certain amount of unity that is required for being a material object. However, few have qualms about accepting the totality of existing material objects: the Material World – which is a maximally scattered something – as a material object. Why not then also dignify the less comprehensive scattered somethings (all of which are less scattered than the Material World) with the title of material object? I presume the principal reason why so many philosophers balk at this idea is this: they believe that there is just one, homogeneous ontological kind of material object. Such a belief is bound to make one somewhat miserly in bestowing the designation “material object”. But taking a look at Model T may help to revise that (false) belief. For there is not just one kind of T-material* object. Besides atomic and simply composite T-material* objects, there are, as will become evident below, other composite Tmaterial* objects, with greater degrees of unity than the minimal degree that simply composite T-material* objects have. That the last-mentioned objects do not fulfil higher standards of unity is no reason to deny them the status of being T-material* objects. They are T-material* objects all right, but with a differentia: they are simply composite, that is, they have a minimal degree of unity in composition*. Note that this minimal degree of unity is not a defect, not a privation. It is precisely the low degree of unity simply composite T-material* objects have which is responsible for their maximal stability over time. The simply composite T-material* object which is the T-analogue of the Material World is /c+, c#, c>, c~/ – an object that is representable simply by {c+, c#, c>, c~}: by the set that it is composed* by at all T-temporal positions in all H*-relatives (including H* itself). Now, there is a composite T-material* object, call it “Cc”, which is different from /c+, c#, c>, c~/ and which is, in fact, not a simply composite T-material* object, of which nevertheless the following is true: Part of the nature of Cc For all H, all Z, and all Y: if Cc is at Z in H composed* by Y and exists at Z in H, then Y = {c+, c#, c>, c~}. This has the following consequence (given what has been said before):

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For all T-histories H and all T-temporal positions: if Cc exists at Z in H, then Cc is composed* at Z in H by the same set by which /c+, c#, c>, c~/ is composed* at Z in H [or in other words: Cc is composed* at Z in H by the same atomic T-material objects by which /c+, c#, c>, c~/ is composed at Z in H] – which implies that the spatial location of Cc at Z in H is precisely the spatial location of /c+, c#, c>, c~/ at Z in H [this location being simply the sum of the spatial locations at Z in H of c+, c#, c>, and c~]. But notwithstanding these facts, Cc and /c+, c#, c>, c~/ are different Tmaterial* objects, I have said. How can this be? – Though Cc is composed*, whenever it exists in any T-history, by {c+, c#, c> c~} and is in this respect just like /c+, c#, c>, c~/, Cc does not exist at every T-temporal position in every H*-relative – quite unlike /c+, c#, c>, c~/. And of course Cc is not representable by {c+, c#, c> c~} – quite unlike /c+, c#, c>, c~/. However, Cc is representable by {c+, c#, c> c~} plus a certain Tspatial* form. There is a huge reservoir of such forms – all of them spatial universals that potentially form (in other words: potentially organize) collections of atomic T-material* objects (including even the 1-collections of atomic T-material* objects, which comprise just one such object) and only such collections. Fittingly, the identity-principle for T-spatial* forms is this: For all T-spatial* forms Φ and Φ´: if it is true for every H*-relative H and every T-temporal position Z that Φ and Φ´ form (organize) at Z in H the same collections of atomic T-material* objects118 [if, in other words, each collection of atomic T-material* objects that collectively has Φ at Z in H also collectively has Φ´ at Z in H, and vice versa], then Φ and Φ´ are identical. And there is a plausible constraint on all T-spatial* forms, whether they are of variable quantitative forming capacity, or are potentially forming only four atomic T-material* objects, only three, only two, or (in the limiting case) only one T-material* object; one could call – and I shall call – that constraint “the principle of temporal locality for T-spatial* forms” (as will be noticed, it is far from imposing spatial locality): 118

If both forms happen to form no such collections at Z in H, then they still form the same such collections at Z in H.

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The having, or not having, of a T-spatial* form Φ by certain atomic T-material* objects – their being, or not being, arranged according to Φ – at a temporal position Z in an H*-relative H is (due to the nature of Φ as a T-spatial* form) merely a local matter of the atomic Tmaterial* objects and of the maximally composite momentary state that H allots to Z. Other maximally composite momentary states, other temporal positions, other histories have nothing to do with the fact that certain atomic T-material* objects are arranged – or not arranged – according to Φ at Z in H. The principle of temporal locality for T-spatial* forms will guide our theorizing about such forms. However, if it comes to mirroring typical material objects of Reality in T-material* objects, then that principle does appear to be too restrictive (see Objection 5 in Sect. 7.4.2). Note that a T-spatial* form Φ, which is had by an X at a temporal position Z in an H*-relative H, is had in the primary sense by X at Z in H if, and only if, X is a collection of atomic T-material* objects – the collections of atomic T-material* objects being representable precisely by the 15 non-empty sets of atomic T-material* objects (and there is no reason not to add: those collections are the simply composite T-material* objects in union with the atomic ones). And that same form Φ (had by X at Z in H) is had in a secondary sense by X at Z in H if, and only if, X is a T-material* object which is such that the collection of atomic T-material* objects that goes into the constitution of X (one may need to add: in H at Z) has Φ in the primary sense at Z in H. As the spatial form that goes into the constitution of our present example, the T-material* object Cc, I choose a T-spatial* form that is fairly “geometrical”, but still easy to describe. The constitutive spatial form of Cc, or simply: the spatial form of Cc, is Four atomic T-material* objects in a diagonal line, each object separated from its next neighbour(s) in the line by a single empty spatial position.

This form is designated by “Φ0”, and Cc is representable by {{c+, c#, c>, c~}, Φ0}. According to MO2: (α) Cc exists at a T-temporal position Z in a Thistory H if, and only if, Cc is composed* at Z in H by a non-empty subset of {c+, c#, c>, c~}. And according to part of the nature of Cc (see above):

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(β) Cc is composed* at Z in H by a non-empty subset of {c+, c#, c>, c~} if, and only if, C is composed* at Z in H by {c+, c#, c>, c~}. Taking into account the entire nature of Cc, we have moreover: (γ) C is composed* at Z in H by {c+, c#, c>, c~} if, and only if, H is an H*-relative and {c+, c#, c>, c~} is arranged at Z in H according to Φ0 [in other words: … and the objects in {c+, c#, c>, c~} are arranged at Z in H according to Φ0]. Therefore (via chaining together logically (α), (β), and (γ)): (δ)

Cc exists at a T-temporal position Z in a T-history H if, and, only if, H is an H*-relative and {c+, c#, c>, c~} is arranged at Z in H according to Φ0.

Cc is an example of a hylomorphic-composite T-material* object. Such objects are defined by the following statement: MO5

The hylomorphic-composite T-material* objects are precisely those T-material* objects that are representable by the pair-sets {M, Φ}, where M is a subset of {c+, c#, c>, c~} with at least two elements and where Φ is a T-spatial* form which is such that the objects in M are arranged according to Φ at some temporal position in some H*-relative.

According to MO5, Cc is a hylomorphic-composite T-material* object if, and only if, {c+, c#, c>, c~} is arranged according to Φ0 at some temporal position in some H*-relative. In assuming Cc as an example of a hylomorphic-composite T-material* object, I have, therefore, also assumed that {c+, c#, c>, c~} is arranged according to Φ0 at some temporal position in some H*-relative. I have not shown that this is true. Whether it is true or not rather seems to depend on what the initial state of H* is like; but, so far, ACTUHIST has never been played (the assumption of Normalcy in the preceding section cannot help here, because that assumption is too unspecific). Suffice it that ACTUHIST can certainly be played in such a way that the above condition in italics is true and that Cc is, accordingly, a hylomorphic-composite T-material* object. To assume that ACTUHIST is played in that way in fact is unproblematic as long as we merely want to have an example for illustrative purposes. If Y is a hylomorphic-composite T-material* object and is represented by the pair-set {M, Φ} in the way described in MO5, then M is called “the

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matter (or hyle) of Y” and Φ is called “the form (or morphé) of Y”. Thus, in the case of our example, Cc, we have: the matter of Cc is {c+, c#, c>, c~}, and the form of Cc is Φ0. This object-relative terminology of matter and form is very useful for the formulation of a general principle of which (δ) above is just a particular consequence: MO6

For all hylomorphic-composite T-material* objects C, T-histories H, T-temporal positions Z: C exists at Z in H if, and only if, H is an H*-relative and the matter of C is arranged at Z in H according to the form of C.

Since the subsets of {c+, c#, c>, c~} with at least two elements represent the simply composite T-material* objects, it is always possible to say that the matter of a hylomorphic-composite T-material* object is a certain simply composite T-material* object. The relationship between a hylomorphic-composite T-material* object and the simply composite T-material* object that is its matter (for example, the relationship between Cc and /c+, c#, c>, c~/) is easily seen to be an analogue of the relationship between a body (broadly conceived, in the sense in which one also speaks of “heavenly bodies”, the sense in which a body is not necessarily a body of anything119) and the particular collection of atoms it consists of – but only if one considers that particular collection of atoms to be essential for the body’s identity, which, indeed, is not the typical attitude towards the matter of a body. (If it were the typical attitude, then few people would be in doubts regarding the question whether Theseus’s Ship is or is not numerically the same ship when the first original plank of it has been replaced by a new one: it would be utterly evident to them that it is not numerically the same ship.) T-material* objects that are in line with the typical – and entirely legitimate – attitude towards the matter of a body will be introduced in Sect. 7.4.1. Staying for a while yet with hylomorphic-composite T-material* objects, consider that the form Φ0 has specializations as well as generalizations. Selected specializations and generalizations of Φ0 can be arranged in a two-sided sequence with Φ0 in the middle, for example: Φ−3 Φ−2 Φ−1 Φ0 Φ1 Φ2 Φ3

119

But for the explicitly or implicitly relative sense of “body”, compare Sect. 7.2.

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In this sequence, each form is a specialization of every form that precedes it in the sequence (precedes it according to the usual, conventional way of reading the sequence). Therefore: the further to the left in the sequence a form is, the more unspecific it is; and the further to the right in the sequence a form is, the more specific it is. The following selection of generalizations and specializations of Φ0 has precisely the structure just described (if transformed from a horizontal to a vertical sequence); but it is just one of many possibilities of filling out that structure: Φ−3: Four atomic T-material* objects. Φ−2: Four atomic T-material* objects, each object separated diagonally or laterally from its next neighbour(s) by a single empty spatial position. Φ−1: Four atomic T-material* objects in a line, each object separated from its next neighbour(s) in the line by a single empty spatial position. Φ0: Four atomic T-material* objects in a diagonal line, each object separated from its next neighbour(s) in the line by a single empty spatial position. Φ1: Four atomic T-material* objects in a diagonal line, each object separated from its next neighbour(s) in the line by a single empty spatial position, with c+ followed by c#, c# followed by c>, and c> followed by c~. Φ2: Four atomic T-material* objects in a diagonal line, each object separated from its next neighbour(s) in the line by a single empty spatial position, with c+ followed – from left to right – by c#, c# followed by c>, and c> followed by c~. Φ3: Four atomic T-material* objects in a diagonal line, each object separated from its next neighbour(s) in the line by a single empty spatial position, with c+ located in , c# located in , c> located in , and c~ located in .120

Each of these seven forms, when combined with the required matter, uniquely determines a single hylomorphic-composite T-material* object – a single one, because there is just one way to provide the required matter (the four atomic T-material* objects referred to in each of the seven forms can only be c+, c#, c>, and c~), and uniquely, because none of the other six forms determines that same object. The seven hylomorphic-composite Tmaterial* objects determined (for the sake of having examples, we assume that the H*-relatives are such that the seven determined objects are hylomorphic-composite T-material* objects) are the following:

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The first part of the description of Φ3 is superfluous given the second part (beginning with “with”); it is, nevertheless, not omitted: in order to make the comparison with the preceding forms easier.

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C−3, which is represented by {{c+, c#, c>, c~}, Φ−3}; C−2, which is represented by {{c+, c#, c>, c~}, Φ−2}; C−1, which is represented by {{c+, c#, c>, c~}, Φ−1}; C0, or Cc, which is represented by {{c+, c#, c> c~}, Φ0}; C1, which is represented by {{c+, c#, c>, c~}, Φ1}; C2, which is represented by {{c+, c#, c> c~}, Φ2}; C3, which is represented by {{c+, c#, c> c~}, Φ3}.

Each of these seven T-material* objects exists at a T-temporal position Z in a T-history H only if all the objects preceding it (i.e., higher than it) in the above sequence also exist at Z in H. This result follows on the basis of MO6 and the (chosen) contents of Φ−3, Φ−2, Φ−1, Φ0, Φ1, Φ2, and Φ3: the matter of Cn (for n > −3) in the above sequence – that is, {c+, c#, c>, c~} – is arranged at Z in the H*-relative H according to the form of Cn [i.e., Φn] only if it is also arranged at Z in H according to the form of Cn−1 [i.e., Φn−1]. Thus, for C3, there are six T-material* objects in the above sequence – namely, C−3, C−2, C−1, C0, C1, and C2 – that coexist with C3 whenever it exists (in whatever H*-relative); and whenever they coexist with it (and hence with each other), they are located in the very same place as C3 (due to the fact that all the objects in the above sequence are made up of the very same matter: {c+, c#, c>, c~}). The principle that there can be no two T-material* objects at a given T-temporal position in one and the same Tplace is, therefore, seen to be false as soon as hylomorphic-composite Tmaterial* objects are drawn into consideration (it is, however, true if restricted to atomic and simply composite T-material* objects). The conclusion for Reality that is analogous to the T-conclusion just reached is unavoidable and should be properly noted. If we consider these atoms of this statue (of Marcus Aurelius) to be essential for it (as we may) and to be essential also for this coherent lump of metal (as we may), then we are confronted with two existing hylomorphic-composite material objects – notwithstanding the fact that, necessarily, the lump of metal exists in the same place as the statue whenever the statue exists,121 that they, whenever the statue exists (for example, now), must occupy the very same place at the very same time. The reason is this: the two hylomorphiccomposite material objects, though they have the same constitutive matter, have different constitutive forms (the constitutive form of the statue being a specialization of the constitutive form of the lump of metal). 121

But note that this is not true vice versa: the lump of metal can exist at some time in some place without the statue existing there (because it does not exist at the time).

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Each object in the above sequence of seven T-material* objects can exist at a temporal position Z in a T-history H even if all the objects following it in the sequence do not exist at Z in H. Thus C−3 can exist at Z in H without C−2, C−1, C0, C1, C2, and C3 existing at Z in H. In fact, the Tspatial* forms in the sequence Φ−3, Φ−2, Φ−1, Φ0, Φ1, Φ2, Φ3 have been chosen in such a manner that the extremes of the sequence – Φ−3 and Φ3 – are at the same time extreme forms with regard to the four atomic Tmaterial* objects to be arranged according to them: Φ−3 is, so to speak, the genus supremum regarding c+, c#, c>, c~, taken in collection, and Φ3 is, so to speak, an infima species regarding them (one of many such infimae species). Thus, the hylomorphic-composite T-material* object C−3 is seen to be a limiting case of such objects: it is a duplicate of the simply composite T-material* object /c+, c#, c>, c~/, and as such it is as changeable without detriment to its identity or existence as a four-composite T-material* object (that is, one composed of four atomic T-material* objects) can be: maximally changeable regarding the ways in which four-composite T-material* objects can change. On the opposite end, there is quite a different limiting case of hylomorphic-composite T-material* objects: C3, which is an absolutely static hylomorphic-composite T-material* object; if it changes, its identity and existence are destroyed. Note, in this context, that the form of a hylomorphic-composite Tmaterial* object C – the form (simpliciter) of C– is matched by the complete form which the object has when existing at this or that temporal position Z in this or that H*-relative H: the complete form of C at Z in H; this latter form of C (had by C at Z in H) is usually not the form of C but an ultimate specialization of the form of C, and usually it is not the same form at each instance Z at which C exists in an H*-relative H. The form (simpliciter) of C is the substantial form of C, whereas the complete form of C at Z in H (had at Z in H by the at-Z-in-H-existing C) is an accidental form of C – if (and only if) it is not identical with the form of C. Thus, the substantial form of C−3 is matched by a huge number of accidental complete forms of C−3 (had by C−3 at some temporal position in some H*-relative), whereas the substantial form of C3 is matched by no accidental complete form of C3 at all. The substantial form and the accidental complete forms of a hylomorphic-composite T-material* object C always have in common that the entire matter of C is arranged according to them (whenever the matter of C is arranged according to them): they are total forms of C, not merely partial forms. The completeness regarding which they differ (note, however, that

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also substantial forms are sometimes complete: see the substantial form of C3) consists precisely in this: to each atomic T-material* object that belongs to the matter of C a spatial position is assigned by naming both the atom and the position. The form Φ3 (see above) is, therefore, complete; but neither the forms Φ−3, Φ−2, Φ1, Φ0, and Φ1, nor the form Φ2, nor the form Φ´, which is described by Four atomic T-material* objects in a diagonal line, each object separated from its next neighbour(s) by a single empty spatial position, with one atom located in , another located in , another in , and another in ,

are complete. Hylomorphic-composite T-material* objects have been considered above in their four-composite variety. But of course there is also the threecomposite and the two-composite variety of them. An example of a hylomorphic-two-composite T-material* object (the H*-relatives are assumed to be accordingly) follows below; the example-object is given via its representative (according to MO5): {{c#, c>}, Two atomic T-material* objects in a horizontal line, separated by two empty spatial positions}.

In representing hylomorphic-four-composite T-material* objects it is redundant to specify their matter, since that matter can only be {c+, c#, c>, c~}. Such objects could also be represented merely by their form, it being understood that they all have the same matter. Things are otherwise with hylomorphic-two-composite T-material* objects: the object represented by {{c#, c>}, Two atomic T-material* objects in a horizontal line, separated by two empty spatial positions} is an object different from the object represented by {{c+, c>}, Two atomic T-material* objects in a horizontal line, separated by two empty spatial positions}, although the substantial forms of the represented objects are (evidently) identical. In addition to the one form just used in introducing two different hylomorphic-two-composite T-material* objects, consider two other salient forms: one which both of these objects invariably have whenever they exist in whatever T-history (it is an essential, though not substantial form of the objects); and one which only one of those objects may occasionally – accidentally – have: (1) the least specific form for hylomorphic-two-composite T-material* objects: Two atomic T-material* objects, and (2) a most spe-

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cific form for certain hylomorphic-two-composite T-material* objects, a (for such objects) complete form: Two atomic T-material* objects in a horizontal line, separated by two empty spatial positions, with c# located in and c> located in . Each of these two forms is also fit to serve as the substantial form of certain hylomorphic-two-composite Tmaterial* objects: the first as the substantial form of the six hylomorphictwo-composite T-material* objects that are duplicates of, respectively, /c#, c>/, /c#, c~/, /c#, c+/, /c>, c~/, /c>, c+/, and /c~, c+/ (each of these latter objects being a simply two-composite T-material* object); the second as the substantial form of a single absolutely static hylomorphic-twocomposite T-material* object (the matter of which is {c#, c>}). Not all hylomorphic-composite T-material* objects are sempiternal in the H*-relatives – whereas all atomic and all simply composite T-material* objects are sempiternal in them. However, for a hylomorphic-composite Tmaterial* objects that is not sempiternal in the H*-relatives it is no inprinciple problem to resume existing after a spell of non-existence and to exist for just a single moment. Some may hold that also for this reason hylomorphic-composite T-material* objects cannot serve as T-analogues of typical material objects in Reality. But the main and – in contrast to the just-mentioned purported reason – quite uncontroversial reason for the inability of hylomorphic-composite T-material* objects to serve as Tanalogues of typical material object in Reality will be treated in the next section. (Later – in Sect. 7.4.2 – I shall consider more closely the issues of momentary existence and “resurrection” for typical material objects in Reality.) 7.4.1

Typical material objects in Reality and their T-analogues

For the typical – that is, from the point of view of our everyday experience most familiar – material objects O in Reality (typical bodies, broadly conceived), it makes no sense to speak of the matter (simpliciter) of O. One can only speak of the matter of O at this or that time of its existence; for the matter of O will not be the same at different times of O’s existence. In fact, the matter of O at a later time of O’s existence may have no part in common with the matter of O at an earlier time of O’s existence. Moreover, typically, it will not even be entirely determined what belongs and what does not belong to the matter of O at a given time.

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All of this – including the indeterminacy of matter – is true of my body (for example) and true also of the Ship of Theseus. Regarding the Ship, remember that its original matter was gradually replaced, piece by piece, and after some time came to the point of having been completely replaced. And if there be doubts whether the indeterminacy of matter is true of my body and the Ship, consider that, at the microscopic level, one will certainly find a great number of atoms of which it is not determined whether they belong at a given time to the matter of my body – respectively, the Ship –or not. But though the indeterminacy of matter is a trait of the typical material objects in Reality, I shall disregard it in the T-modelling of such objects. The indeterminacy of matter characteristic of them is, it seems to me, less a matter of objective ontological fact than of human indecision when faced with the absence of sufficient reason (in other words: with arbitrariness) in determining precisely which precise object is to be designated by a given designator (with a given meaning: “my body”, “this ship”). A typical material object in Reality is constituted in each world [of Reality] in which it exists by its (substantial) form, and by a succession of matter, which is at least in some world of the typical material object’s existence varied regarding its diachronic composition (in fact, the actual world [of Reality] will serve as such a world). For modelling this in T, I treat the successive matters (or matters in succession) which constitute a succession of matter as being always precise matters (cf. the previous paragraph). In preparation to T-modelling, I introduce the concept of a matter*sequence: D52 A matter*-sequence is a function the domain of which is the timeline of T and which assigns to each item in its domain a subset of {c+, c#, c>, c~}. (Only such functions are matter*-sequences.) In each H*-relative (that is: in each T-history that is a candidate for being actual* and has the same initial state as H*) a more or less trivial matter*sequence can be assigned to each of the T-material* objects that have already been considered, that is: to each atomic, simply composite, and hylomorphic-composite T-material* object. For the atomic T-material* object c>, the matter*-sequence that is assigned to it in every H*-relative is the one with the constant value {c>}. For the simply composite T-material* object /c#, c+/, the matter*-sequence that is assigned to it in every H*relative is the one with the constant value {c#, c+}. And for the hylomor-

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phic-composite T-material* object C that is represented by {{c#, c>}, Two atomic T-material* objects in a horizontal line, separated by two empty spatial positions} we have: the matter*-sequence Σ*(C, H) that is assigned to C in the H*-relative H allots the value {c#, c>} to each temporal position at which C exists in H, to all other temporal positions it allots ∅ (the empty set). There are T-material* objects that have matter*-sequences in H*relatives that are quite unlike the more or less trivial ones for the already considered T-material* objects. Consider a very simple case (the H*relatives are to be accordingly): the T-material* object that is representable by the form One atomic T-material* object located in , in short: by Φ°. This T-material* object, C°, exists at a temporal position Z in a Thistory H if, and only if, H is an H*-relative and some atomic T-material* object is arranged at Z in H according to Φ°, in other words: if, and only if, H is an H*-relative and some atomic T-material* object is located in at Z in H. The matter*-sequence Σ*(C°, H) that is assigned to C° in the H*-relative H functions in the following way: if some atomic Tmaterial* object is located in at the temporal position Z in H, then there is exactly one such object and Σ*(C°, H) allots to Z the set which contains only that object; if no atomic T-material* is located in at Z in H, then Σ*(C°, H) allots ∅ to Z. Clearly, if Σ*(C°, H) allots to the temporal position Z a value M which is not ∅ and to another temporal position Z´ a value M´ which is also not ∅, then M and M´ need not be identical – and let’s assume that they are not identical. Thus, the matter*-sequence that C° has in the H*-relative H – that is, Σ*(C°, H) – is indeed quite unlike the ones for the T-material* objects that have been considered in Sect. 7.4. Before I specify the nature of the T-material* objects of which C° is an example, the following entirely general definition of the main concept employed in the preceding two paragraphs will further illuminate what has been said there: Let C be a T-material* object (as defined by D51), H an H*-relative: D53 Σ*(C, H) [“the matter*-sequence assigned to C in H”] =Def the matter*-sequence which allots to each temporal position Z the subset of {c+, c#, c>, c~} by which C is composed* at Z in H. Now, I give the name “sequential T-material* objects” to the T-material* objects of which C° is an (untypical) example:

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MO7

The sequential T-material* objects are precisely those T-material* objects that are representable by a T-spatial* form Φ which is (a) such that, for any H*-relative H and any temporal position Z, at most one non-empty subset of {c+, c#, c>, c~} is arranged at Z in H according to Φ, and (b) such that in some H*-relative a nonempty subset of {c+, c#, c>, c~} is arranged according to Φ at some temporal position and another non-empty subset of {c+, c#, c>, c~} at another temporal position.

The form that represents a sequential T-material* object C is the (substantial) form of C. Regarding the existence of sequential T-material* objects, we have: MO8

For all sequential T-material* objects C, T-histories H, T-temporal positions Z: C exists at Z in H if, and only if, H is an H*-relative and some non-empty subset of {c+, c#, c>, c~} is arranged at Z in H according to the form of C.

Note the similarity as well as the difference of MO8 and MO6. In fact, what is said by MO6 can also be formulated in a way that is obtainable from MO8: by simply replacing in MO8 the word “sequential” by the expression “hylomorphic-composite”: MO6´ For all hylomorphic-composite T-material* objects C, T-histories H, T-temporal positions Z: C exists at Z in H if, and only if, H is an H*-relative and some non-empty subset of {c+, c#, c>, c~} is arranged at Z in H according to the form of C. MO6´ is just as true as MO6 because, for hylomorphic-composite Tmaterial* objects C, some non-empty subset of {c+, c#, c>, c~} is arranged according to the form of C at a given time in a given H*-relative if, and only if, the matter (simpliciter) of C (i.e., the non-formal element of C’s representation according to MO5) is arranged according to the form of C at that time in that H*-relative. A sequential T-material* object need not be synchronically composable. Indeed, the above-considered example of a sequential T-material*, C°, is not synchronically composable: there is no H*-relative H which is such that a subset of {c+, c#, c>, c~} with at least two elements is arranged according to Φ° (the form of C°) at any temporal position in H. But all se-

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quential T-material* objects are diachronically composable: for each sequential T-material* object C it is true that in some H*-relative a nonempty subset of {c+, c#, c>, c~} is arranged according to the form of C at some temporal position and another non-empty subset of {c+, c#, c>, c~} at another temporal position. This is simply a consequence of the definitional statement for sequential T-material* objects, MO7, and it distinguishes sequential T-material* objects from the T-material* objects considered in Sect. 7.4. Note, in particular, the difference between a synchronically incomposable sequential T-material* objects (like C°) and an atomic T-material* object. For the sake of completeness and contrast I add: There is yet another kind of T-material* object; to this kind belong T-material* objects which are neither atomic, nor simply composite, nor hylomorphic-composite, nor sequential T-material* objects: MO9

The hylomorphic-atomic T-material* objects are precisely those Tmaterial* objects that are representable by the pair-sets {M, Φ}, where M is a subset of {c+, c#, c>, c~} with a single element and where Φ is a T-spatial* form which does not refer to an element of {c+, c#, c>, c~} and which, moreover, is such that the object in M is arranged according to Φ at some temporal position in some H*relative.

MO10 For all hylomorphic-atomic T-material* objects C, T-histories H, T-temporal positions Z: C exists at Z in H if, and only if, H is an H*-relative and the matter of C is arranged at Z in H according to the form of C. The T-material* object that is represented by {{c+}, Φ°} is an example of a hylomorphic-atomic T-material* object (or speaking properly: ACTUHIST can be played in such a way as to verify the assumption that {{c+}, Φ°} is an example of a hylomorphic-atomic T-material* object). The clause “which [i.e., Φ] does not refer to an element of {c+, c#, c>, c~}” was included in MO9 in order to exclude over-representation: in the absence of that clause, the object that is represented by {{c+}, Φ°} would also be represented by {{c+}, Φ°°}, where Φ°° is the form The atomic Tmaterial* object c+ located in (whereas Φ° was introduced above as the form One atomic T-material* object located in ). In the special case of hylomorphic-atomic T-material* objects, already those T-

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spatial* forms that specify a precise location for a single atomic Tmaterial* object without naming such an object can serve as substantial and complete forms (for example, Φ°), which fact, however, makes it necessary to exclude, in order to avoid over-representation, from the ability to serve as substantial forms of hylomorphic-atomic T-material* objects those [appropriate and properly speaking] complete T-spatial* forms122 (for example, Φ°°) that one would prima facie expect to be able to serve in this role (in view of the forms that can serve as substantial and complete forms of hylomorphic-composite T-material* objects, whether four-, three-, or two-composite). The important question, to which I now turn, is this: Are the Tmetaphysical analogues of the typical material objects in Reality to be seen in the sequential T-material* objects (leaving out of consideration the indeterminacy of matter of the former objects: treating them as if there were no indeterminacy of matter for them)? Is there an analogy between the sequential T-material* objects, as described by MO7 and MO8, and my body, surely a typical material object in Reality? If there were, my body would be representable by a certain spatial form: the substantial form of my body, Ψ. And the following would have to be true: For each world W [of Reality] which has the same laws and the same initial state as the actual world [of Reality] and for each time Z [of Reality] it is either the case that there is exactly one non-empty set of elementary particles that is arranged at Z in W according to Ψ, or the case that there is no such set (in which case the empty set is taken to be arranged at Z in W according to Ψ). Indeed, this statement seems to be true. Moreover, also the following statement seems to be true: In the actual world, a non-empty set of elementary particles is arranged according to Ψ at some time, and another non-empty set of elementary particles is arranged according to Ψ at another time. In fact, there even appear to be two sets of elementary particles, arranged according to Ψ (the substantial form of my body) at their respective times, of which the following is true: they do not have the same size (cardinality), 122

For the characterization of completeness for T-spatial* forms, see Sect. 7.4.

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and they do not have any element in common. Thus, it is plausible to conclude that the T-metaphysical analogues of my body are not to be found among the hylomorphic-composite T-material* objects; rather, the closest T-metaphysical analogues of my body and of the other typical material objects in Reality are sequential T-material* objects, more specifically: lifelong synchronically composite sequential T-material* objects that are sequential in H* (hence in actuality*).123 ACTUHIST can be played in such a way that an example of such an object – namely, the maximally extended symmetrical three-in-a-line thing, represented by the following T-spatial* form: Three atomic T-material* objects in a line (horizontal, vertical, or diagonal), each separated from its neighbour(s) in the line by three empty spatial positions – is actually* given. 7.4.2

Objections to taking the sequential T-material* objects as the Tanalogues of the typical material objects

Several objections can be raised against the view that the sequential Tmaterial* objects are the T-analogues of the typical material objects in Reality. As an example of a typical material object in Reality, I shall again be using my body. Objection 1: If my body were a sequential material object – in analogy to the sequential T-material* objects, as described by MO7 and MO8 –, then my body could exist for just a single moment. This seems implausible. Response: It seems undeniable that there is a first moment of my body’s existence; surely that moment could easily have been also the last moment of its existence. Objection 2: If my body were a sequential material object, it could exist again after not having existed for a while. This seems implausible in principle. Response: That my body comes into existence again is, in principle, not harder to conceive than that it comes into existence once (and that it certainly did).

123

A sequential T-material* object C is lifelong synchronically composite if, and only if, at every temporal position Z at which C exists in H* a subset of {c+, c#, c>, c~} with at least two elements is arranged at Z in H* according to the form of C; and C is sequential in H* if, and only if, in H* a non-empty subset of {c+, c#, c>, c~} is arranged according to the form of C at some temporal position, and another non-empty subset of {c+, c#, c>, c~} at another temporal position.

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Objection 3: Suppose my body were a sequential material object. Then former matter of it that was arranged according to my body’s substantial form once – say, 40 years ago – could (in principle) be arranged according to that same form again now, right beside my existing body, creating a problem. For then there would be two coexisting sequential material objects with the same substantial form – which cannot be, since sequential material objects are represented by their substantial forms (cf. MO7), which requires that two sequential material objects must have two different substantial forms. Response: This attempted reductio ad absurdum commits a non sequitur: from the assumption that my body is a sequential material object it does not follow that the matter that was in it 40 years ago could (in principle) be again arranged according to my body’s substantial form at this very moment. All that follows is this: the matter that was in my body 40 years ago could be again arranged according to my body’s substantial form at this very moment if (and only if) no other non-empty set of elementary particles is at that moment arranged according to that form (cf. MO7). Clearly, this condition is not fulfilled, and hence once cannot conclude that the matter that was in my body 40 years ago could at this very moment be again arranged according to my body’s substantial form (on the contrary, one must conclude that it cannot be thus arranged, not even in principle). Objection 4: If my body were a sequential material object, my body would be represented by its substantial form (see the preceding section). Hence it would be identifiable with its substantial form: reducible to it. But this is absurd. Response: As a sequential material object, my body is constituted in each world [of Reality] in which it exists (such a world must be a world that has the same laws as the actual world [of Reality] and the same initial state) by its substantial form and by a succession of matter, which is at least in some world of my body’s existence varied regarding its diachronic composition (the actual world [of Reality] will serve as such a world). My body is certainly not constituted by its substantial form alone. In view of this fact, my body cannot be identified with its substantial form, Ψ. It is, however, represented by Ψ, and there is no need to mention anything else but Ψ in representing my body. This is so because in each world W in which my body exists Ψ has a unique succession of matter (the matter-sequence of my body in that world; cf. D53) as its complement in constituting my body in W, and this complementing succession of matter is not an independent variable, but a functional value of Ψ and W.

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Objection 5: Do two perfectly identical twins (suppose there were such animals) have they same substantial form or not? If they have the same substantial form (as they seem to), then, obviously, they are not sequential material objects (since such objects, if they are different, must have different substantial forms). If, however, they do have different substantial forms, it is not to be seen what forms those substantial forms could be if they are to be spatial forms (in analogy to the substantial forms of sequential T-material* objects, which are T-spatial* forms). Response: The two substantial forms of two “identical” sequential material objects are not temporally local forms and therefore not necessarily distinguishable on the (temporal) spot: on the spot, they may seem to be one and the same form, their difference becoming apparent only in an extended interval of time. It must, however, be admitted: if substantial forms for sequential material objects are such that the having or not having of them is not a temporally local matter, then one cannot regard them as having their T-analogues among the T-spatial* forms (for which “the principle of temporal locality” has been assumed; see Sect. 7.4). 7.4.3

The transtemporal and transhistorical identity of T-material* objects

Each T-material* object does not exist at any temporal position in any Thistory that is not an H*-relative (this is a consequence of MO2 and MO1). Each T-material* object exists at some temporal position in some H*relative (this is a consequence of D51, MO1, and MO2). Thus, in treating the transtemporal and transhistorical identity of T-material* objects, we can leave out of consideration all T-histories that are not H*-relatives and need only consider H*-relatives. The problem of how atomic T-material124 objects stay, in a recognizable way, numerically the same object – from one T-temporal position to another, and from one T-history to another – has already been solved (in Sects. 6.2 and 6.3). Therefore, the staying-traceably-the-same-object problem, as I shall call it, is also already solved for the atomic T-material*125 objects (since all atomic T-material* objects – that is, c+, c#, c>, and c~ – are atomic T-material objects). But then that problem is also solved for the 124

Note the absence of the asterisk, which indicates (here) the greater generality of the unmarked predicate in comparison to the marked one. 125 Note the presence of the asterisk, which indicates the lesser generality of the marked predicate in comparison to the unmarked one.

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simply composite T-material* objects. All information concerning the individuation, and hence the identity, of T-material* objects is contained in the representatives of such objects. Now, the representatives of simply composite T-material* objects are the sets of atomic T-material* objects with at least two members (see MO4). The transtemporal and transhistorical identity of a simply composite T-material* object C is, therefore, determined and traceable on the basis of the – already known to be traceable – transtemporal and transhistorical identity of the atomic T-material* objects that are the elements of the representative of C. But then the staying-traceably-the-same-object problem is also solved for the hylomorphic T-material* objects (whether hylomorphic-composite or hylomorphic-atomic). The transtemporal and transhistorical identity of the matter of a hylomorphic T-material* object C´ is determined and traceable on the basis of the – already known to be traceable – transtemporal and transhistorical identity of the atomic T-material* objects that are the elements of the matter of C´: a non-empty set of atomic T-material* objects, and one of the two elements of the representative of C´, the following pair-set: {the matter of C´, the [substantial] form of C´} (see MO5, MO9). For finding, for any T-temporal position Z and any H*-relative H, that we are dealing with C´ at Z in H, that is: with the very same object C´ we are dealing with, say, at temporal position 5 in history H*, we merely need to find that the matter of C´ (always the selfsame matter) is arranged at Z in H according to the form of C´ (and the existence of C´ at Z in H, the location of C´ at Z in H – they are found out together with this). For sequential T-material* objects, the staying-traceably-the-sameobject problem has obviously a different nature than for (simply) atomic and hylomorphic-atomic, simply composite and hylomorphic-composite Tmaterial* objects, the reason for this being that sequential T-material* objects are represented by their forms alone. But the solution of that problem for sequential T-material* objects is nevertheless clear: the question whether the sequential T-material* object C´´, which is found to exist at the T-temporal position Z in the H*-relative H, recurs as numerically the same object at the T-temporal position Z´ in the H*-relative H´ and how it can do so has a clear answer: C´´ recurs at Z´ in H´ if, and only if, a nonempty set of atomic T-material* objects is arranged at Z´ in H´ according to the substantial form of C´´. If there is such a set, then there is exactly one such set (in accordance with MO7), which set is the matter of C´´ at Z´ in H´: the matter in which C´´ – C´´ itself, not a counterpart of it – makes it

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appearance at Z´ in H´ and in which it obtains existence and location at Z´ in H´. But perhaps C´´ is not the only sequential T-material* object that makes it appearance in this matter in H´ at Z´. Note that sequential Tmaterial* objects C´´ and C´´´ that are two are two in virtue of the fact that their substantial forms – the forms that represent them – are different Tspatial* forms, which means according to the identity-principle for Tspatial* forms (see Sect. 7.4) and according to the specific nature of the Tspatial* forms that are the substantial forms of sequential T-material* objects (see MO7) that in some H*-relative at some T-temporal position the substantial forms of C´´ and C´´´ do not form the same collection of atomic T-material* objects: there, either each of the two forms a different single collection of such objects, or one of them forms one such collection, the other no such collection at all. Notwithstanding this fact, the two sequential T-material* objects C´´ and C´´´ may also have the same matter in some H*-relative at some T-temporal position, and hence may be located in the very same place at that time in that H*-relative. This is nothing to balk at: it can also happen in Reality. The (famous) cat Tibbles and its (defined to be) tailless remainder Tib are two sequential material objects in Reality (not merely such objects, of course, but also such objects). If Tibbles loses its tail at time Z, then Tib and Tibbles have, afterwards, the same matter and are located in exactly the same place. Nevertheless, they are two sequential material objects, not just one. 7.5

T-metaphysical teleology

There is no purpose which is intrinsic to Model T, and there is even no purpose which is intrinsic to the game of ACTUHIST (which game deals with Model T and, in a sense, encompasses it). A round of the game of ACTUHIST begins with a first step, and ends when 100 steps have been made. The rules of ACTUHIST state how the game is to be played in those 100 steps; the rules define the game, they are intrinsic to it: if those rules are ever violated during the 100 steps, no round of ACTUHIST has in fact been played. But there is nothing in the rules of ACTUHIST which tells the players of ACTUHIST for what, when playing the game for 100 steps according to the rules, they must, or at least ought to, use the freedom that the rules leave them: for what to achieve. Hence there is no purpose which is intrinsic to the game of ACTUHIST. And there is, of course, no purpose

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which is intrinsic to Model T – just as there is no intrinsic purpose to the world of arithmetic or geometry. On the other hand, ACTUHIST and Model T have been constructed – in the way they have here been constructed – precisely with the purpose of their having purposes that are extrinsic to them. There is, for example, the philosophical purpose had by ACTUHIST and Model T, highly extrinsic to them, of presenting in a nutshell – but fairly completely and systematically – metaphysical issues of Reality, in order to illuminate them. (But even the – far from complete – description of the contents of the nutshell takes up hundreds of pages.) Governed by this general philosophical purpose (extrinsic to it), ACTUHIST has, among many other specific philosophical purposes (extrinsic to it), also the philosophical purpose to show that what has no intrinsic purpose may yet have the general extrinsic purpose of being an opportunity for the coming into play of many specific purposes extrinsic to it. Fulfilling this purpose, ACTUHIST has no intrinsic purpose, but it does have the general extrinsic purpose of being an opportunity for the coming into play of many specific purposes extrinsic to it: the gamerelated purposes of the ACTUHIST-players when playing ACTUHIST. In this respect, the game of life (which certainly seems to be without intrinsic purpose) and the game of human history (which also seems to be without intrinsic purpose) may well be just like ACTUHIST. What are possible game-related purposes of ACTUHIST-players when playing ACTUHIST? – There is no end to such purposes. For example, also the purpose of conveying a secret love-message to another player – by making a certain move in a round of the game of ACTUHIST – is counted among the possible game-related purposes of ACTUHIST-players when playing ACTUHIST. Whatever the actual game-related purposes of ACTUHIST-players when playing ACTUHIST will turn out to be, each of these purposes – though always extrinsic to ACTUHIST – will have a higher or lower degree of extrinsicness (i.e., of distance from the matter of ACTUHIST); each of them will be either short-range (tactical) or longrange (strategic); and each of them – if juxtaposed with simultaneous game-related purposes of other players in the same round of ACTUHIST – is likely to be oriented in one of the following three social directions (so to speak): isolation, confrontation, cooperation. ACTUHIST can be played in such a way that a considerable amount of T-geometrical and T-kinematic beauty is actualized; it can also be played in such a way that no such beauty is actualized. It goes without saying that persistent cooperation among all of the five players will tend to

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favour the actualization of T-geometrical and T-kinematic beauty in a round of the game of ACTUHIST. But, note, cooperation among the ACTUHIST-players is neither a sufficient nor a necessary condition for the actualization of Model-T-beauty. I note in passing that a certain amount of instant cooperation – it can be highly non-local cooperation, say, between player A+, incarnated by c+ in one corner of T-space, and player A>, incarnated by c> in the opposite corner – is already enforced by the rules of ACTUHIST, and especially by L10, corresponding to the Law of Minimization (of Change in the Total Character of Motion).126 As there are no purposes that are intrinsic to Model T or to ACTUHIST, so there are no values that are intrinsic to Model T or to ACTUHIST. All purposes and values that relate to ACTUHIST or Model T are extrinsic to both. All such purposes and values come from the players of ACTUHIST, and are brought to bear on ACTUHIST and Model T through the actions of the players of ACTUHIST. Taken collectively, those actions produce, in the course of playing a round of ACTUHIST, the actual* Thistory (which T-history is rigidly designated by “H*”– in the context of the round of ACTUHIST taken to be relevant). In this, as it were, petrified track one can read to some extent the T-transcendent actions that produced it, and to a much lesser extent the T-transcendent values and purposes that guided those actions. But one cannot read these things to any extent in the actual* T-history if one does not know whether the players of ACTUHIST are persons (and, of course, also not if one does not know the game itself); for then one is not in a position to exclude the possibility that a generator of genuine chance or a generator of inscrutable necessity – that is: a machine that generates deterministic chaos which is beyond the observer’s capacity of computation127 – produced the actual* T-history without any regard to values or purposes. 126

The parenthesis – which is meant to show the full name of the law simultaneously with its abbreviated name – indicates in which respect the Law of Minimization (and therefore L10) requires cooperation between the players of ACTUHIST. (That law is given this name in Sect. 6.6, but is stated as “The Candidate Sixth Law for T” already in Sect. 6.5.3; there, in the context of stating the law, also the rule L10 is stated, though without the name “L10”, which is given to it only in Sect. 6.6.) 127 Inscrutable necessity can seem to be genuine chance, though it is, of course, not genuine chance: it is just a simulacrum of genuine chance – a simulacrum which is indistinguishable from genuine chance for any observer who must remain without insight into the manner of the simulacrum’s production. Inscrutable necessity can be a perfect substitute for genuine chance.

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In view of the absence – in fact, the unattainability – of knowledge analogous (mutatis mutandis) to the knowledge just referred to, many philosophers choose to believe that the actual world of Reality is, ultimately, just this: a mere product of genuine chance and/or inscrutable necessity. If they were right, this same view would, in the end, also have to be correct with respect to the actual* T-world,128 despite appearances to the contrary: despite it, presumably, appearing to be the case that the making of the actual* T-world is in the hands of five value-and-purpose-guided persons abiding by the rules of ACTUHIST. This must be so since the production of the actual* T-history (if it comes about) is of course completely embedded in the production of the actual world of Reality. However, I do not believe it likely that those philosophers are right. It is as unlikely as that genuine chance and/or inscrutable necessity have, by themselves, brought forth these very words that I am now writing, and indeed this book, reducing me, who wrote them, to a mere transmission wheel, or to the ball of a ballpoint pen (which is, perhaps, the more fitting image in this context). But suppose I am indeed the Author of ACTUHIST and Model T – in the full sense of the word, not just in the transmission-wheel-sense. In what manner and to what extent am I responsible for the outcome of, say, the first round of the game of ACTUHIST to be played? – This question cannot be answered in a specific way without describing in detail how the first round of ACTUHIST is played, that is: without making further assumptions (though only for the sake of the argument). I abstain from making such assumptions. But in a general way the question has already been answered: I am the Author of ACTUHIST, of the rules that define this game, and these rules allot to me (or my deputy – his or her responsibility is my responsibility) a certain role and influence in the playing of any round of ACTUHIST. I need not here repeat the description of that role and influence. I merely note that, in ACTUHIST, I am one in a plurality of five (I myself decreed it so), who in selecting the moves of the game has a rather limited – arbitrative – say (I myself decreed it so); my responsibility for the content of the actual* T-history that is the outcome of a round of ACTUHIST is, therefore, rather limited. Don’t blame me for this content, at least not me alone. Indeed, the rules of ACTUHIST do not exclude the situation (rather, favour it somewhat) that the affairs in a round of ACTUHIST run in such a way – whether in an undesirable way or not – that uninitiated observers (observers who are kept ignorant of the relevant facts) are led to conclude – perhaps rationally in their situation (given certain preconcep128

Regarding the very close relationship of T-worlds and T-histories, see Sect. 3.6.

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tions), but still falsely – that, for the round in question, there is no fifth player, no player who has the role that the rules of ACTUHIST ascribe to the Author (or his deputy). “What is he needed for?”, they ask. – Well, they can be told what he is needed for, and in such a way that what they are told cannot fail to rationally convince them. In contrast to the attitude of Author-denial just described, consider now another such attitude for uninitiated observers: first to assign to the fifth player – because “he is the author of the game, it is supposed” – the crucial responsibility for the undesirable way in which the affairs in a given round of ACTUHIST are running (for their ugly, boring, chaotic, or destructive character) and then to conclude that, because the fifth player is thus responsible, there is in fact no fifth player: he has, so to speak, no moral right to exist (“I cannot believe in such an author”). This latter attitude of Author-denial is not only wrong like the one previously described, it verges on the absurd. Fortunately, nobody is likely to adopt it – for the simple reason that the number of uninitiated observers of a round of ACTUHIST is bound to be small, in monumental contrast to the number of uninitiated observers of what is happening in Reality.

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Index of concepts and principles The index gives the page-numbers of the first substantial thematization of the listed concepts and principles, and in many cases the page-numbers of important follow-ups to that first thematization. @* see the actual* history absolute staticness (said of hylomorphic-composite material* objects) 250, 252 accidental form of C 250 actual* (actuality*) 84, 92-93, 98-103, 113, 199, 215 actual* antecedent to (temporal position) Y 125 actual* up to (temporal position) Y 120 actual** (actuality**) 92-94, 96, 98103 actual*** (actuality***) 117, 129 actual existence see existence in the sense of actuality actualism 199 actuality, non-relative 129 actuality, non-relative, for histories 74 actuality, non-relative and extrinsic 129 actuality, non-relative and intrinsic 129 actuality, unconditional 116-117 actuality-giving 128 actuality-predicates 87, 89-92 actualize* (actualization*) 104 actualize** (actualization**) 103, 107 ACTUHIST 104 agent-causation 230-231, 235 amounts of modification in the total character of motion, possible 191 atomicity for spatial configurations 24 atomic momentary Fullness-events [D37] 137 atomic momentary Fullness-events in H* 137, 142, 144-146 atomic momentary Fullness-events in H*, absolutely original 142, 145 atomic-state-changes 171 attached to P [D8] 21

attached to Z [D12] 25 AU* see the Author Author-denial 266 Big Bang 200, 204 bodies of the players of ACTUHIST 215-216, 218-219 bodies, broadly conceived 247, 252 c and c´ is in near-touch at Z 174 c and c´ is in touch at Z 174 candidacy for being actual 73-74, 104, 197, 219 candidacy for being actual* see the preceding entry Candidate Fifth Law for T 175-176, 193-194, 203-204 Candidate Fourth Law for T 168, 173174, 184, 192, 194-196 Candidate Sixth Law as first envisaged 187 Candidate Sixth Law for T 188, 194196, 264 causal over-determination 231, 236 causal powerlessness 226-227 causal superfluousness 226-227 Cc (a composite material* object) 243247 characters of the state of motion for c in , possible 182 c has moved into a border-position at Z 175, 196 c is at Z in confrontation with c´ 177 c is in a collision at Z in H 180 CIT-cause of Y in H [D50] 237 coherence of a spatial configuration 22 coherence of a temporal configuration 25

Complement to the Postulate of Original Supervenience 161 completeness of forms of hylomorphiccomposite material* objects 250-251 composability (of sequential material* objects), diachronic 256 composability (of sequential material* objects), synchronic 255 composition-operation, elementary 35 conjunction 35, 38, 50, 70 conservation laws 134, 167 conservation principles 173-174, 179180 contact between atomic material objects 171-173 continuants, everlasting 23, 94, 148 continuants, atomic higher see material objects, atomic continuants, higher 66, 95, 137, 151, 163 continuants in H*, atomic see material objects in H*, atomic continuants relative to a history 66, 94 continuants relative to H* 137 continuants relative to H*, atomic higher see material objects in H*, atomic continuants relative to H*, everlasting 148 continuants simpliciter 23, 94, 148 counterpartism, four-dimensionalist 157158 ∆** see the fingertip of actuality** determinism 123, 134, 169, 179, 187, 192 determinism, absolute 123, 134 determinism, free-first-step 123, 203 determinism, free-first-two-steps 134, 180 diachronic path of Fullness in H [D35] 136 diachronic path of Fullness in H, complete [D36] 136, 149 directed at (for states of movement) 177

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direction of time, objective 97-98 directions, spatial 29, 177, 182 directions, temporal 28 directions of the state of motion for c in , possible 182 diversiformity see uniformity earlier than Z [D15] 28, 97 elementary composition 35 entirely earlier/later than V [D18] 28 essentiality of origin for atomic material objects 164 event-causal over-determination 231 event-causation 226, 230-231, 235-236 event-causation, physical 227 event-physicalism 226 event-physicalism, causal [C1] 226 events [E, E´, …] 46, 49, 94, 150-151, 224 events, common [D19] 49 events, maximal momentary 58, 95, 99 events, momentary 52, 94, 137 events, rare [D20] 49 exemplification and actuality 91-92 existence in the sense of actuality 127 existence in the sense of Free Logic 128 existence of material* objects [MO2, MO6, MO8, MO6´, and MO10] 241, 247, 255-256, 260 exists at Z (for atomic material objects) 138, 146 extensionality for spatial configurations 24 extrinsic conception of lawhood 72, 104, 237 fillings 9, 58-59 fillings of modal positions 59 fillings of spatial positions 9 fillings of temporal positions 10, 36, 94-95 first cause and second causes 238-239 first cause of E 238 four-dimensionalism 150 four-dimensionalism, sophisticated 159

four-dimensionalism for H* 151 Full 9, 18 Fullness see Full future-tense operation 43-44 gen-identity in H* of atomic momentary Fullness-events in H* [D38] 142 genuine chance 264 identity among momentary states 41 identity-principle for spatial* forms 244 identity-principles for atomic material objects 161-162 immediately attached to P [D7] 21 immediately attached to Z [D11] 25 incarnation of players of ACTUHIST 218-219, 221 incarnation of players of ACTUHIST, multiple 222 indeterminacy of matter 253 indetermination see determinism indexicality, covert 81 indexicality, overt 81 individual histories of atomic material objects in H* 150, 152, 199 individuals 20 initial state (of history) 198 in place P at Z (for atomic material objects) [D41] 138, 146 in S attached to P [D9] 21 inscrutable necessity 264 intrinsically temporally extended 93-95 intrinsic conception of lawhood 71, 237 intrinsic implication between events 120 intrinsic implication of a rare event by a history 51 intrinsic implication of a state of affairs by a history 67 intrinsic inclusion between events 120 intrinsic inclusion in a history or in a momentary event 101 in V attached to Z [D13] 25 IT-causation see the next entry IT-cause of Y [D48] 228

IT-chance-event 236 H* (a history) 75, 88, 99, 131, 137, 209, 216-217 H*-relatives 240 histories (complete) [H, H´, …] 10, 48, 58, 99, 150 later than Z [D16] 28, 97 Law for T, the First 134, 173-174, 197 Law for T, the Second 135, 197 Law for T, the Third 141, 171-173, 197 law for the beginning (of history) 198199 law for the end (of history) 198-199 Law of Minimization 196-197, 264 laws (of nature) for T 74, 104-105, 132, 174, 197-198 laws in the primary sense under the extrinsic conception of lawhood 72, 133 laws in the secondary sense under the extrinsic conception of lawhood 72, 132-133 laws of H 71-73 laws of H* 132, 197 laws under the extrinsic conception of lawhood 72, 132-133 laws under the intrinsic conception of lawhood 71 Limiting Postulate 162 materialism 214 material objects, actual* atomic 199, 240 material objects, atomic [c, c´, …] 155, 159, 162-165, 199, 218-219 material objects, composite 173 material objects, merely possible* atomic 158 material objects in H, atomic 219 material objects in H*, atomic [c+, c#, c>, c~] 137-138, 142, 144-146, 149152, 199, 205, 209 material* composition 240

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material* objects [D51; C, C´, …] 241, 243, 260 material* objects, atomic [MO3] 241, 249, 252-253, 256, 260 material* objects, four-composite 250 material* objects, hylomorphic 261 material* objects, hylomorphic-atomic [MO9] 256, 261 material* objects, hylomorphic-composite [MO5] 246-255, 261 material* objects, hylomorphic-fourcomposite 251 material* objects, hylomorphic-twocomposite 251-252 material* objects, sequential [MO7] 255-258, 261-262 material* objects, sequential synchronically incomposable 256 material* objects, simply composite [MO4] 242-243, 247, 249, 252-253, 261 material* objects, simply two-composite 252 mathematical representation 15, 214 matter*-sequences [D52] 253-254 mental-event-physicalism 225-227 miracles 76 modal positions 59 momentary phase of E [D24] 99 momentary phase of E at Z [D24b] 121 momentary states (all) [ST, ST´, …] 41, 94-95 momentary states, atomic 34, 94, 137 momentary states, composite 35, 94 momentary states, disjunctive 38 momentary states, maximally composite 36, 65, 94 motion-constant in H* 174 movement-changes and atomic-statechanges 169-171 movement-changes and changes see the preceding entry movement-independence of bodies 220 Mr Chance 105

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necessary* [D#28, D28] 110, 115, 197, 198 necessary+ [D#26, D26] 110, 115 necessary*** (necessity***) 117 necessaryTP* at (temporal position) Y [D30] 120, 123-124 necessaryITP* 125 necessary2TP* at (temporal position) Y [D32] 124 necessary2TP+ at (temporal position) Y [D34] 125 necessity, relativized 118-119 necessity, unconditional 116 negation 35, 38 nomological conflict 196 nomological constant of nature (for T/of H*) 174 nomological identity between histories 237 nomological matter-constant 174 nomological motion-constant 174 non-identity principle for basic spatial tropes 53 non-identity principle for basic spatiotemporal tropes 53 Normalcy 228 N steps down from P [D3] 9 N steps earlier than Z [D5] 10 N steps later than Z [D6] 10 N steps to the left of P [D1] 9 N steps to the right of P[D2] 9 N steps up from P [D4] 9 obtaining and actuality 91-92 parallel causation 226 part-event of E [D42] 150 partly earlier/later than V [D17] 28 past-tense operation 43-44 path of c+ in H* 171 physical in the primary, categoryspecific senses 213, 224 physical in the secondary, categorytranscending sense 213-214, 224 physicalism 213-215, 224-225

physico-event-causal interpretation of causation 235 positions 9, 58-59 possibility, relativized 118-119 possible* [D#27, D27] 110, 115, 197, 199 possible+ [D#25, D25] 110, 115 possibleTP* at (temporal position) Y [D29] 120, 123-124 possibleITP* 125 possible2TP* at (temporal position) Y [D31] 124 possible2TP+ at (temporal position) Y [D33] 125 Postulate for “E represents c at Z” 143, 153 Postulate for “E represents c at Z in H” 153 postulate for “X is actual at Y in U” [MP4] 89 postulate for “X is actualI at Y” [MP5] postulate for “X is actualI in Y” [MP6] 89 Postulate of Original Supervenience 156 postulates for atomic material objects [Postulate 1 – Postulate 6] 138, 146 postulates for the basic spatiotemporal tropes as fundamental entities 55-56 postulates for “X is actual in Y” [MP1 – MP3] 87 precise implication of a T-law by an ACTUHIST-rule 133, 236 present* (presentness*) 93, 96, 107 present-tense operation 37, 42-44 principle (A*) for “actual*” and “actual in” (and an analogue) 121-122, 132 principle (B*) for “actual*” and “actual in” (and an analogue) 122, 132 principle (C*) for “actual*” and “actual in” 132 principle for “X is composed* at Z in H by Y” [MO1, MO1´] 240 Principle of Sophisticated FourDimensionalism 159, 162

Principle of Sufficiency for Candidacy (for being actual) and its converse 73-74, 197 Principle of Sufficient Cause 233, 235 Principle of Sufficient Cause for Timmanent causation 234 Principle of Sufficient Cause for Ttranscendent causation→T 234 principle of temporal locality for Tspatial* forms 245, 260 Principle of the Causal Closure of the Physical, Strong [C2] 225 Principle of the Causal Closure of the Physical, Weak [C3] 225 principles for material* objects [MO2 – MO10] 241-242, 246-247, 255-256 principles for the action of the fingertip of actuality** [Act1, Act2, Act3] 97, 99, 101 principles for T-immanent causation [ITC0 – ITC4] 224-225, 229, 233 principles for T-transcendent causation→T [TTC0 – TTC1] 230, 233 process of actualization** 97-98, 102, 206 purposes of ACTUHIST 262-264 qualities of the state of motion for c in , possible 182 Reality 14, 127-128 Reality-systems 128-129 regularities, history-specific 68 regularities, inner-historical 73 regularities, inter-historical 73 regularities of histories 67, 69, 73 regularity of H [D21] 69 regularity of H, formal [D22] 69 regularity of H, material [D23] 69 regularity of H, total 71 reincarnation of players of ACTUHIST 222 relativization of T-statements, temporal and historical 78-79

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representability of the G-entities by the K-entities 241-242 representation and identity 217-218 rigid designator 84, 131 rule for translating events into states of affairs 49-50 rule L5 of ACTUHIST 109 rule L6 of ACTUHIST 109-110, 206 rule L7 of ACTUHIST 134, 197 rule L8 of ACTUHIST 135, 197 rule L9 of ACTUHIST 141, 197 rule L10 of ACTUHIST 195, 197, 264 rule L11 of ACTUHIST 205-206 rule L12 of ACTUHIST 208 rules of ACTUHIST, the first four [L1, L2, L3, L4] 104 second causes of E 238 sempiternal existence of atomic material objects in H* 148 sempiternal existence in the H*relatives 252 sequential material objects (in Reality) 258-260, 262 spatial configurations [S, S´, …] 20, 94-95 spatial part of S [D10] 23 spatial part of atomic material object 151 spatial path 67 spatial positions [P, P´, …] 9, 59 spatial* forms 244-245, 247-248, 250252, 260 souls of the atomic material objects in H* 215-216, 218 Statement of Linkage 98 state of movement 168 state of movement for c in H from Z to Z+1 168 state of movement for c in H in the transition see the preceding entry state of movement in H in the transition [D47] 168 state of rest 167

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state of rest for c in H from Z to Z+1 167 state of rest for c in H in the transition see the preceding entry state of rest in H in the transition 168 states of affairs (all) [ST, ST´, …] 33 states of affairs, event-like and common 50 states of affairs, event-like and rare 51 states of affairs, not time-dependent and not time-thematic 33 states of affairs, not time-dependent, but time-thematic 33, 42, 49 states of affairs, time-dependent and time-thematic 43, 94 states of affairs, time-dependent but not time-thematic 34, 41, 94 subjects of actuality** 94-95 subjects of actuality**, primary 95 sufficient causation 224 supervenience of atomic material objects in H on H 172 supervenience of higher continuants 66, 105, 158, 160, 167 supervenience of the atomic material objects 66, 162-163 supervenience of the atomic material objects in H* 143-144, 152, 167, 209 supervenience on the atomic momentary Fullness-events in H* 143-144, 152 supervenience on the spatiotemporal distribution of Fullness in the course of H* 171, 209 Supervenience Postulate 144-145, 153 Supervenience Postulate, Generalized 153 temporal configurations [V, V´, …] 24, 214 temporally complete worms of Fullness in H* 150, 154, 159 temporal part of V [D14] 26 temporal part of E [D43] 151

temporal positions [Z, Z´, …] 9, 45, 59, 94, 214 T-entities, abstract and concrete 213214 the (non-relatively) actual history 75, 84 the actual* history 84, 99, 131 the amount of modification in the total character of motion 190 the Assumption 192, 194-196 the assumption of Normalcy 228 the atomic momentary Fullness-event in H* that represents c at Z [D39] 142 the Author 109 the character of the state of motion for c [in H] in 182 the Closure Principles 225 the complete form of C at Z in H 250 the direction of a state of movement 177 the fingertip of actuality** 96-97, 100, 103-104, 108 the first three laws for T see the given laws the flow of time 98 the form (simpliciter) of C 247, 249250, 255 the (axiomatic) four laws for T 197, 218-219 the given laws 184 the Great Game 222-223 the Great Will 232 the individual history of c+ in H* 150, 154 the laws given 184 the location of c at Z in H [D44] 167 the Material World 243 the matter-constant 174 the matter (simpliciter) of C 247, 249, 255 the matter of C at Z in H 261-262 the matter of O (at a time) 252-253 the matter*-sequence assigned to C in H [D53] 254

the motion-constant 174 the Physical World 216-217 the physics of H* 133, 197 the players of ACTUHIST [AU*, A+, A#, A>, A~] 205-206, 218 the Proposition 212-213 the Ship of Theseus 247, 253 the space of T 9, 22 the state of motion for c in H from Z to Z+1 [D45] 167 the state of motion for c in H in the transition see the preceding entry the state of movement for c in H from Z to Z+1 168 the state of movement for c in H in the transition see the preceding entry the state of rest for c in H from Z to Z+1 167 the state of rest for c in H in the transition see the preceding entry the Strong Closure Principle 225 the substantial form of C 250, 255 the timeline of T 10, 43, 93, 97 the totality of being 7, 14 the totality of everything 127-128 the totality of laws 73, 197 the total character of motion [in H] in the transition 182-183 the total quantity of matter 173-174 the total quantity of motion 169, 173174 the total quantity of motion [in H] in the transition 183, 190 the total quantity of movement in H in the first transition 169 the Weak Closure Principle 225 time* 93 time of causation 232 T-laws see laws for T Totality 127-128 touch of atomic material objects at Z 174

273

transhistorical identity of atomic material objects 145, 155-156, 260 transitions [D46] 168 transtemporal and transhistorical identity of material* objects 260-262 transtemporal identity of atomic material objects in H* [D40] 142-143 tropes 52 tropes, basic 54 tropes, basic spatial 53, 94-95 tropes, basic spatiotemporal 52, 94 TT-causation see the next entry TT-cause of Y [D49] 228 typical material objects in Reality [O, O´, …] 245, 252-253, 257-260

uniformity of histories 61, 64 uniformity of histories, momentary 62 uniformity of histories, momentary and qualitative 64 uniformity of histories, transmomentary 62 uniformity of histories, transmomentary and qualitative 64 uniformity (overall) of maximally composite momentary states 62, 64 uniformity of maximally composite momentary states, qualitative 63 universals 17 unrestricted fusion for spatial configurations 24

ultimate state (of history) 198

worlds 59

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22 Fred Wilson Body, Mind and Self in Hume’s Critical Realism ISBN 978-3-938793-79-4 512pp., Hardcover, EUR 139,00

23 Paul Weingartner Omniscience From a Logical Point of View ISBN 978-3-938793-81-7 188pp., Hardcover, EUR 79,00

24 Simone Gozzano, Francesco Orilia Tropes, Universals and the Philosophy of Mind Essays at the Boundary of Ontology and Philosophical Psychology

ISBN 978-3-938793-86-2 289pp., Hardcover, EUR 79,00

ISBN 978-3-938793-96-1 306pp., Hardcover, EUR 89,00

27 Holger Gutschmidt , Antonella Lang-Balestra, Gianluigi Segalerba (Hrsg.) Substantia - Sic et Non Eine Geschichte des Substanzbegriffs von der Antike bis zur Gegenwart in Einzelbeiträgen ISBN: 978-3-938793-84-8 565pp., Hardcover, EUR 149,00

28 Rosaria Egidi, Guido Bonino (Eds.) Fostering the Ontological Turn Gustav Bergmann (1906-1987) ISBN 978-3-86838-008-8 274pp., Hardcover, EUR 89,00 29 Bruno Langlet, Jean-Maurice Monnoyer (Eds.) Gustav Bergmann Phenomenological Realism and Dialectical Ontology ISBN 978-3-86838-035-4 235pp., Hardcover, EUR 89,00 30 Maria Elisabeth Reicher (Ed.) States of Affairs ISBN 978-3-86838-040-8 219pp., Hardcover, EUR 79,00 31 Richard Schantz (Hrsg.) Wahrnehmung und Wirklichkeit ISBN 978-3-86838-042-2 252 Seiten, Hardcover, 89,00 EUR

ISBN 978-3-938793-83-1 196pp., Hardcover, EUR 69,00

EditedBy • HerbertHochberg • RafaelHüntelmann ChristianKanzian • RichardSchantz • ErwinTegtmeier

PhilosophischeAnalyse PhilosophicalAnalysis 32 Javier Cumpa & Erwin Tegtmeier (Eds.) Phenomenological Realism Versus Scientific Realism Reinhardt Grossmann – David M. Armstrong. Metaphysical Correspondence ISBN 978-3-86838-051-4 139pp., Hardcover, EUR 69,00

33 Christan Kanzian Ding – Substanz – Person. Eine Alltagsontologie

ISBN 978-3-86838-057-6 353 Seiten, Hardcover, EUR 39,00

34 Uwe Meixner Modelling Metaphysics The Metaphysics of a Model ISBN 978-3-86838-060-6 274pp., Hardcover, EUR 79,00

EditedBy • HerbertHochberg • RafaelHüntelmann ChristianKanzian • RichardSchantz • ErwinTegtmeier