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Page 135 of Smith, Quentin, 1998, ?Absolute Simultaneity and the Infinity of Time?, in (ed.) Robin Le Poidevin, Questions of Time and Tense. Oxford: Oxford University Press, pp. 135-183.
ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME Quentin Smith 1. Introduction: Philosophy as a Part of Science Philosophers frequently assume that the nature of things, events, properties, causality, free will and the like is a matter for philosophers to determine, not scientists. But many philosophers assume that the nature of time is the topic of physics, specifically, Einstein's theory of relativity, and that this nature is not something that can be discovered in part or as a whole by philosophical investigation alone. But this is not the case, if only for the reason that philosophy is a part of physics (and of the sciences in general); there is no such division between philosophy and science as is often supposed, or even the "continuum" between philosophy and science that is associated with Quine-type views, which implies that at one end of the continuum there exists some philosophy that is not a part of science. For example, metaphysics and the philosophy of language can be viewed as the part of physics that is used to interpret the mathematical formulae or theorems that are taken to be confirmed by the observational data, and that is used to decide among the many theorems that are taken to be confirmed to some degree by the observational data.
Philosophy is a part of other sciences as well. According to the traditional syntactic theory of science, science consists of a system of theorems, taken as syntactic strings, rules relating these strings to possible observations, and semantic interpretations of these theorems. I would add that philosophy is the "semantic interpretation" part of science. According to the more recent
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semantic theory of science, science consists of models and hypotheses about certain analogies and disanalogies of the models to reality. I would add that philosophy is the hypothesizing about the analogies and disanalogies of the models to reality. With this understanding of philosophy, I will argue that the philosophical portions of the Special Theory of Relativity's concept of time, the General Theory of Relativity's concept of time, and the standard quantum mechanical concept of time, consist in part of unsound verificationist arguments in metaphysics and the philosophy of language and that these theories need to be replaced by a certain non-verificationist theory I shall describe. I shall argue that the correct, non-verificationist, philosophical part of physics implies that abstract objects exist in time, that temporal relations among all physical events are absolute rather than relative to a reference frame, that physical clocks do not measure the metric of time, that there is time before the big bang, that past and future time are infinite, and that time as a whole consists of an infinite number of infinitely long temporal series. The present essay can be viewed as building upon the argument for the tensed theory of time articulated in Smith (1993a), specifically, the argument for the theory of "absolute presentness". This is a tensed theory of time that implies there is a single "absolute tide of becoming" that encompasses all concrete and abstract objects. The theory of absolute presentness, rather than the theory of a tenseless, verificationist and relativist time, is the philosophy of time that arguably belongs to the philosophical part of physics.
2. Real Time and the Orthodox Interpretation of Quantum Mechanics First in the order of things is to show that part of the philosophical theory of time in orthodox quantum mechanics, and Special and General Relativity, is an unsound verificationist argument. The verificationism of these theories has sometimes been noted in a general way in the philosophical literature,[1] but the verificationist arguments need to be pinpointed precisely and their unsoundness made plain to view. Let us begin with the orthodox or Copenhagen interpretation of quantum mechanics. The orthodox interpretation of quantum mechanics has been accepted by virtually all physicists since the late 1920s. Most contemporary philosophers of quantum mechanics, and a small number of contemporary physicists, reject the Copenhagen interpretation; but here I am concerned to discuss the orthodox view that remains prevalent in the scientific community.
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The orthodox theory includes an a prioristic theory of time that is based on an invalid verificationist argument. The inclusion of this a prioristic theory was first evinced by Heisenberg's rejection of his original interpretation of the uncertainty relations between time and energy. Heisenberg originally interpreted these relations epistemically, but Bohr convinced him in private communications to accept a verificationist metaphysics, with its attendant ontological interpretation of the uncertainty relations; hence was born the orthodox interpretation of quantum mechanics. Although this interpretation originated mainly with some papers by Bohr and Heisenberg, the more speculative and somewhat idiosyncratic philosophical theories subsequently developed (in different ways) by Bohr and Heisenberg have not become part of the orthodox interpretation, but only a certain minimal component, a component that is found in textbooks on quantum mechanics (e.g., the typical textbooks used in graduate courses on quantum mechanics). This component involves the idea that time cannot be measured to a precision greater than the so-called Planck time, 10-43second, since the energy utilized by the physical clock process that is making the measurement precludes the possibility of a measurement of a shorter interval. A typical passage reads (Schlegel [1980, p. 77]): "Considering now the measurement of time, we give an indication that one cannot measure a time t for which ct is less than the Planck length [the Planck length is 10-33 centimeter.]. Suppose a clock to have length dimensions of order L. If the clock is to measure a time interval t, the communication time L/c between its parts must be less than t." The author proceeds to characterize the gravitational effect of the clock mass and the energy involved in the workings of the clock. "Now, for a time measurement to be of significance the interval t must be greater than any alterations δt' that are introduced by the gravitational effect of the clock mass and the energy ΔE utilized by the clock process . . . So the Planck length defines a theoretical limit on the smallness of a time interval" (1980, p. 78). Talk of intervals smaller than this length is meaningless. We have the inference: (1) One cannot measure a time t for which ct is less than the Plank length. Therefore, (2) "There is a time for which ct is less than the Planck length" is not a true statement. This inference is based on a verificationist theory of the use of temporal words. Only if the concept expressed by "a time" includes the concept of the means of verifying sentences about time does (2) follow from (1). But the verificationist theory of the meaning of words and sentences has long since been
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shown to be not only false but self-referentially incoherent. The principle of verification is meaningless by its own standards since there are no observations that could verify it. (See Smith [1997, Part One] for an criticism of the argument that the principle of verification is an analytic truth. My criticism also implies that Quine's "holistic verificationism", which retains the thesis that evidence-relations and semantic-relations are coextensive [Quine, 19 p. ], is false). Since (2) does not follow from (1), there is no valid argument of this form that shows that "time breaks down" at the Planck era near the big bang, namely, during what some describe as the first interval of 10-43second after the big bang. Given this, the familiar line that "QM implies that the concept of time ceases to be meaningful near the big bang" does nothing to show that time does not extend infinitely into the past.[2] 3. Real Time and the Special and General Theories of Relativity
What is Einstein's justification for interpreting the variable t in the equations of the Special Theory of Relativity as referring to time? Although the a priori verificationist justification is present in his 1905 article, his most explicit articulation of this justification appears in his 1916 book Relativity. He ask us to consider:
all physical statements in which the conception "simultaneous" plays a part. The concept does not exist for the physicist until he has the possibility of discovering whether or not it is fulfilled in the actual case. We thus require a definition of simultaneity such that this definition supplies us with the method by means of which, in the present case [where two lightning strokes hit near the front and back of a train], he can decide by experiment whether or not both the lightning strokes occurred simultaneously. As long as this requirement is not satisfied, I allow myself to be deceived as a physicist (and of course the same applies if I am not a physicist), when I imagine that I am able to attach a meaning to the statement of simultaneity. See Einstein (1962, p. 22. My italics).
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Here we see Einstein defining "simultaneity" in terms of a method of verifying statements in which this word occurs. Indeed, he says that apart from such verificationist definitions, "simultaneity" and relevantly similar words are meaningless, both for the physicist and for everyone else. Clearly, this is an a priori philosophical theory and belongs to the philosophy of language and metaphysics. Einstein adopts the theory in the philosophy of language: (3) The meaning of temporal words is a method of verifying the sentences in which these words occur. Note that Einstein provides no argument for this philosophy of language; he simply assumes it (as if it is obvious). These and other theories of Einstein show that the philosophy of language is indeed a part of physics. Thus, it is entirely appropriate to consider a theory of time that is based on a certain philosophy of language, such as the philosophy of language defended in Smith (1993a), to be a theory that is a part of physics (rather than some separate discipline that falls outside the realm of the sciences). One difference is that Einstein assumed, without argument, his philosophy of language, and drew inferences about time from these philosophical assumptions, whereas in Smith (1993a) I argued for a certain philosophy of language and drew inferences about time from this philosophy. There is no difference in kind between Smith (1993a) and Einstein's book Relativity; one difference in degree is that there is a greater number of mathematical equations in Einstein's book. Einstein's formulation of the General Theory of Relativity is also based on a verificationist metaphysics. For example, he writes in his seminal 1916 paper "The Foundation of the General Theory of Relativity":
That this requirement of general covariance [viz., that laws of motion are preserved by arbitrary transformations between any two coordinate systems], which takes away from space and time the last vestige of physical objectivity, is a natural one, will be seen from the following reflection. All our space-time verifications invariably amount to a determination of space-time coincidences. If, for example, events consisted merely in the motion of material points, then ultimately nothing would be observable but the meeting of the material points of our measuring instruments with other material points, coincidences between the hands of a clock and points on the clock dial, and observed point-events happening at the same place at the same time. The introduction of a system of references serves no other purpose than to facilitate the description of the totality of such coincidences. [See Einstein et. al. (1952, p. 117)]
This passage suggests the inference: (4) Verifications of statements about time are observations of coincidences between material particulars (e.g. marks on a clock dial and the hands of a clock). Therefore,
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(5) Statements about time are statements about observable coincidences between material particulars. "Time" is reductively definable in terms of an observable spatial relationship between one physical thing (the hands of a clock) and another physical thing (points on the clock dial). With the advent of big bang cosmology, many cosmologists reductively defined General Relativistic time in terms of the expanding radius of the universe. For example, Christopher Isham states that the expanding universe solutions of Einstein's equations allow a slicing of "spacetime" in which each three-space can be viewed as the three-dimensional spherical boundary of a four-dimensional ball. "The time variable associated with this decomposition is the radius of the sphere. . . An absolutely crucial idea here is that 'time' can be defined internally in terms of a particular property (i.e., the radius) of the curvature of the three-dimensional space." See Isham (1988, p. 391). Einstein came to accept a substantival theory of spacetime by 1920 and most philosophers of physics now accept a substantival interpretation of GTR and STR. Verificationism, however, also lies at the basis of these substantival interpretations, as I shall now argue. 4. Verificationism and Substantival Interpretations of Relativity Theory It may be objected that my above criticisms pertain only to interpretations of STR, GTR and QM that assume a reductionist and relational (Leibnizian) theory of time. If we adopt a substantival theory of time or spacetime, as many philosophers of physics have done since the 1970s, then (it may be alleged) my above criticisms of these theories do not hold. I believe that the familiar substantival interpretations of STR, GTR and the orthodox version of quantum mechanics are also based on a false verificationist metaphysics. I shall show this only for STR; it will be immediately apparent how my argument about STR applies also to GTR and the orthodox interpretation of QM. Recently some philosophers who argue for a substantival theory of space-time have argued that a non-verificationist formulation of STR is possible if we take it to be about (not luminal or causal relations but) space-time, where space-time is characterized in terms of Lorentz invariance. For example, Nerlich (1994, p. 251) says that "Einstein's early infatuation with positivism, and the ensuing darkness which this shed on the deep riches of his discoveries is an unhappy chapter in the brilliant tale of science in this century ". Philosophers
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such as Nerlich argue that Minkowski spacetime is a substantival entity and that the constant speed of light is not a matter of convention. However, this substantival and nonconventionalist interpretation of STR still rests on a verificationist metaphysics. This theory requires that the variable t in the Lorentz transformations and s in the Minkowski formula for spacetime separation refers, under certain conditions, to time.[3] But what justifies the assumption that, given a substantival and nonconventionalist theory of STR or Minkowski spacetime, t or s refers in suitable cases to a time or temporal interval? A perspective on this question can be obtained by comparing STR with the theory it "replaced", Lorentz's theory. Note that Lorentz himself did not interpret the t and t' in the Lorentz transformation equations as referring to times, since he took time to be the referent of a variable in the Galilean transformation equation. In Lorentz's theory, the Lorentz transformations have two steps, the first step giving us the Galilean coordinates (of events in absolute space and time) and the second step giving us "apparent" spatial and temporal coordinates, which is what measuring instruments record in frames that are either moving or at rest in absolute space. (Lorentz called these apparent times "local times".) The Galilean transformations give us the values that time and space have relative to an inertial frame at rest in absolute space. The Galilean coordinates are the ontologically real coordinates of events in time and space. The electromagnetic laws hold only in the inertial frame at rest in absolute space, even though they appear to hold in all inertial frames (they appear to be covariant, i.e., to have the same form in all inertial frames). But it is not physically possible to verify which inertial frame is at rest in absolute space, so we cannot know the relevant values in the Galilean transformation. All that can be observed are the values in the second step of the Lorentz transformation, which give us the measurable or apparent spatial and temporal values in a given inertial frame. (In one frame, the one at rest in absolute space, the apparent coordinates are also the real coordinates, but we cannot know which frame this is.) Now the Special Theory of Relativity, be it formulated in terms of a relationalist and conventionalist metaphysics, or in terms of a substantivalist and nonconventionalist metaphysics, entails that nothing real corresponds to the Galilean transformation, that there are no Galilean coordinates, and that the apparent coordinates are the only ontologically real coordinates. This
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is entailed by the Principle of Relativity and the Light Principle, for if the electromagnetic laws really hold in all inertial frames and light really has the constant velocity c in all inertial frames, it follows that the apparent coordinates are the only real coordinates. What justifies these two principles of STR? The Lorentz theory and STR are observationally equivalent (cf. Zahar [1989]); furthermore, Lorentz's theory is not ad hoc, as has recently come to be recognized (Zahar [1989]; Craig, [forthcoming]). In light of such facts as these, what justifies omitting the Galilean transformation? The standard or "official" justification offered by contemporary defenders of substantival interpretations of STR is the appeal to simplicity. It is said that it is simpler to assume that the only temporal relations, time series, etc., that exist are the ones postulated by the substantivalist formulation of STR, than it is to assume that there also are other, absolute temporal relations, an absolute time series, for which there is no observational evidence. But this standard justification is undermined by the fact that substantival STR is in fact a much less simple theory than the theory of absolute time. Substantival STR postulates infinitely or indefinitely many real time series in addition to the observable physical clock processes. But the theory of absolute time postulates only one real time series in addition these clock processes. The theory of absolute time implies the Lorentzian "local times" are merely apparent times; that is, they are mere appearances, and are not temporal series that belong to the furniture of reality. There exist real clocks, but the "local times" these clocks may be said to "measure" do not in fact exist. All that exists is one real time series, absolute time. The STR substantivalist and the defender of absolute time both posit (i) indefinitely many observable physical clocks, but the STR substantivalist in addition postulate (ii) indefinitely or infinitely many real time series, whereas the defender of absolute time instead postulates (iii) only one real time series. This is reason to believe that the theory of absolute time is simpler and more ontologically economical than substantival STR. Furthermore, STR posits infinitely many "contents" of the laws of nature, specifically, of the mechanical and electromagnetic laws, one for each co-moving inertial frame, even though these contents have the same "form" in each frame. The Lorentz theory, however, postulates only one content and one form, the content and form the laws possess in the inertial frame at rest in absolute space. In Maxwell's electromagnetic laws, E = electric field and H = magnetic field. In a Lorentzian theory, these laws have one content; Maxwell's equations have the content [E, H; x, y, z, t] in
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absolute space and time, where x, y and z are the spatial coordinates and t the temporal coordinate. But in STR, these laws have a different content in each co-moving inertial frame; in one frame there obtains the laws [E, H; x, y, t, z], in another frame there obtains [E', H'; x', y', z', t'], and so on ad infinitum. In STR, the laws have the same form in each frame (are covariant), but the infinite multiplicity of different contents makes its postulation of laws infinitely less simple than the Lorentzian postulate of one content. A widely accepted view today is Zahar's "conspiracy of silence" argument against the Lorentzian theory, developed at greatest length in his (1989). In an earlier essay (1983) he gives a clear encapsulation of the argument:
This argument is manifestly unsound, for a series of coincidences or accidents having a single global effect are compounds of nomologically independent events that have a single global effect. A coincidence is inexplicable and the compound of events constituting the coincidence have necessary and/or sufficient conditions (or probabalistic conditions) that are independent of one another. However, in a Lorentzian theory, what Zahar calls the "compensatory factors which exactly nullify these asymmetries" are nomologically explained and have the same sufficient and necessary conditions (or have conditions that are not independent of one another). There is an ether in absolute space that transmits both the electromagnetic forces and molecular forces. This transmission causally affects (is a causally sufficient condition of) various properties of the forces. Lorentz deduces the Lorentz-Fitzgerald contraction hypothesis and the measured constancy of the velocity of light from this transmission causal hypothesis. The Galilean transformation gives real temporal and spatial coordintes which, due to the time-dilations and length-contractions that are derived from the transmission causal hypothesis, appear as the measured "local" or "apparent" coordinates that are governed by the Lorentz transformations. This is exactly the opposite of "a series of coincidences and accidents" that have a single global affect. In fact, the real crux of Zahar's objection to the Lorentzian theory is a verificationist assumption, despite Zahar's extensive denials that verificationism is a premise of his critique of Lorentzian theories. He writes: "But there is something paradoxical in that, through postulating a universal medium, we are driven to conclude that it must be undetectable. Was it not dissatisfaction
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with this paradox so closely connected with the crucial experiment which caused Einstein to look for another explanation?" (Zahar, 1989, p. 92, my emphasis). This shows Zahar's fundamental objection is that the postulation of an undetectable existent is by itself "paradoxical". It is only a verificationist assumption that could justify the thesis that the undetectability of the universal medium is a sufficient reason for concluding that its postulation is paradoxical. The hypothesis "x exists and is undetectable" is paradoxical by itself if and only if it is true a priori that "whatever exists is detectable". I suggest that, faced with this rebutal of the "argument against absolute time based on simplicity or the 'conspiracy of silence' objection'", the STR substantivalist would nonetheless feel theoretically reluctant to abandon STR in favor of the theory of absolute time. I suggest, further, that the reason for their reluctance is a deep but implicit committment to certain a priori theses in metaphysics and the philosophy of language. (I do not mean they would say or admit this, but that this is the ultimate justification that is behind--whether they recognize it or not--their allegiance to STR.) They share with the relational interpreters of STR the fundamental ontological decision to eliminate "physical unobservables and immeasurables", i.e., to take "the real" and "the physically measurable or verifiable" as expressing logically equivalent concepts. One main difference between the STR relationalist and the STR substantivalist is that the substantivalist does not identify time with observable physical clock processes, but rather takes observable physical clocks as giving us the accurate measurements of time. The substantivalist assume that statements about time, even though they are not about observable clocks, are nonetheless about something that is accurately measurable by observable clocks. STR substantivalists and relationalists are implicitly committed to the a priori theses: (6) Necessarily, sentences about the topological and metrical properties of time are verified or falsified by, and only by, the observable properties and relations of physical clocks; and (7) Necessarily, the topology and metric of time are accurately measurable by observable physical clocks.
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Thesis (6) belongs to the philosophy of language and (7) belongs to metaphysics. (Clocks allegedly measure the topology of time at least inasmuch as they allegedly measure the order in which events or times occur; the metric involves the temporal distance between events or times.) If the reader supposes that "clock" analytically expresses a concept of an instrument that accurately measures the metric of time, we should substitute a more neutral expression for "clock", such as an "apparently periodic physical process" or "a type of motion of a physical particular between spatial points that is apparently regularly repeated". The question then would be whether such apparently regular motions are clock processes. But I shall assume "clocks" refers only to such moving physical systems and does not analytically entail that they accurately measure the metric of time. I take "x is accurately measurable by observable physical clocks" to entail "x is knowable by observing the clocks". Let us consider the metaphysical thesis (7), which is weaker than (6) since it asserts a sufficient condition ("by"), not both a sufficient and necessary condition ("by and only by"). The necessity operator is not that of physical or nomological necessity, since (7) is used as a criterion to determine what could be physically necessary. There are many metaphysically possible worlds (or what Plantinga [1974] and Forbes [1985] call "broadly logically possible worlds") in which material bodies can be accelerated to arbitrarily high velocities (not limited by a finite velocity of light) and in which the operand of (7) is true. (The operand of [7] is expressed by "the topology and metric of time are accurately measurable by observable physical clocks".) These worlds are physically impossible, according to STR. This is a consequence of the fact that the necessity operator in (7) does not have a merely physical modality (its truth does not depend on the truth of STR), but a metaphysical modality. (7) is supposed to be a true proposition that belongs to several different physical theories (the true physical theory and several false ones) and that determines which physical theories are possibly true. This metaphysical assumption is at the basis of both the relational and substantival interpretations of STR and is required to rule out the Galilean transformation. But this metaphysical assumption, like the principle of verification, is not true. There is a metaphysically possible world in which the Lorentz theory is true; in this world, the structure of time is given in the Galilean transformations, but the values of these transformations cannot be known by observing any clock. But we need not rely on Lorentz's theory to show this; any world in which time has a topology and metric that cannot be known by observations of physical clocks (but in which Lorentz's laws also do not obtain) is a world that shows (7) is false. (6) is also false, for observations of physical clocks would neither verify nor falsify statements about the metrical and topological properties of time in these worlds.
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(6) and (7) are false a priori metaphysical assumptions that are a part of STR and that are used to interpret the observational data provided by physical clocks. Without these verificationist metaphysical assumptions, STR collapses; there is no longer any justification for accepting the Principle of Relativity and the Light Principle. The observational data require us to accept the hypotheses that the laws of electrodynamics (and mechanics) appear to hold (with same form) relative to every inertial frame, and light appears to have a constant velocity c, independent of the state of motion of the emitting body, relative to every inertial frame. But the passage from "seeming" to "being" is made through an a priori verificationist theory of "being". In order to develop an adequate theory of time, we need to reject the a priori propositions (6) and (7) and consider a relevant a posteriori proposition. The proposition that is relevant to determining the nature of time a posteriori is: (8) The topology and metric of time are accurately measurable by observable physical clocks. The truth value of (8) is knowable via the correct metaphysical part of physics in conjunction with the observational data relevant to the mathematical equations in physics. In the following several sections, I shall present some arguments that imply that (8) is false. First, I need to show that abstract objects exist in time (sections 5 and 6). Second, I will show (in section 7) that it follows by a series of steps that all simultaneity relations between physical events are absolute (are two-termed relations) and are not relative to a reference frame (three-termed relations). This indicates that proposition (8) is false, for observable physical clocks do not record absolute temporal relations between distant physical events. These conclusions enable us to argue that time probably preceded the big bang, that the past and future are probably infinite, and that time probably consists of an infinite sequence of infinitely long temporal intervals (see section 8).
5. Necessary and Sufficient Conditions for an Object to Exist in Time My argument for absolute simultaneity and the infinity of past and future time hinges on the admittedly controversial assumption that there are some abstract objects whose existence is not dependent on the existence of any concrete object. There exist universals that are not exemplified, there exist propositions even if no one is thinking of them, and there are numbers even if there are no minds or bodies. In other words, I assume platonic realism.
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For a large number of philosophers, platonic realism is a preposterous theory they cannot imagine believing. But it is also true that a large number of philosophers find anti-platonism, especially in its nominalist version, a preposterous theory they cannot imagine believing. (Anti-platonism includes Aristotelian realism, conceptualism, physicalism, trope theory and the many varieties of nominalism.) The debate between platonists and anti-platonists has been going on for over two thousand years without any sign of a "knock-down argument" or a consensus of opinion among philosophers in sight. However, I do not want to make my argument of interest only to platonists. The largest group of philosophers who find platonism distasteful are physicalists. To make platonism seem more palatable to physicalists, I would point out that most physicalists, such as Quine, have argued that physicalism requires some abstract objects, sets (to which numbers arguably may be reduced), since the postulation of sets is required by the physical sciences. If Quine is right, then even physicalists are committed to the existence of abstract objects. Quine plausibly argued that the physical sciences require sets to be postulated, but Tooley (1987) plausibly argued in a similar spirit that the physical sciences require uninstantiated properties to be postulated. Given this, I do not see that platonism can be so unpalatable to physicalists. I do not see much difference between believing in the existence of the unit set of Socrates or the null set or the infinite hierarchy of power sets of sets, and holding the platonist belief that there is an uninstantiated property of having 867 sides. In any case, my entire argument below, given in terms of platonic realism, is logically equivalent to an analogous argument that does not assume platonic realism but is about merely the null set, or about unit sets, or ordered n-tuples of physical particulars, or sets of sets. Thus, physicalists such as Quine would be able to accept the basic premise of my argument without abandoning their philosophical scruples. I shall briefly present one argument for absolute simultaneity that is based solely on physicalism, with its sets, after I have presented the detailed argument in terms of platonic realism. First, we need to give a real or true (as opposed to stipulative) definition of an object's existence in time. A stipulative definition is neither true nor false and merely records one's decision to use a word in a certain way; for example, one may stipulate that one will use "God" to refer to the universe, so that x is God if and only if x is the universe. We may object to this stipulative definition that it does not facilitate communication but makes it more difficult, but we cannot object that it is false. By contrast, a real definition purports to describe the nature or essence of something. If we want to give a real definition of being human, we will specify the logically necessary and sufficient conditions for something to have the nature of a human. If we offer the real definition, something is human if and only if it is a rational animal,
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then this definition is either true or false. This definition is false if some humans are not rational (e.g., are born and live without a cerebral cortex) or if some humans are not animals (e.g., have their animal parts gradually replaced by bionic parts) or if some rational animals are not humans (e.g., whales). In the following real definitions of an object's existence in time, the variable x ranges over concrete and abstract objects, but not over instantaneous events or enduring processes. By "n-adic property" I mean a monadic property or a relation of any sort. The crucial aspect of these definitions is that the second order variable F ranges over real and Cambridge n-adic properties.
(D1) x exists in time if and only if there is some time t at which x possesses some n-adic property F and some different time t' at which x does not possess F.
The tensed version is as follows:
(D2) x exists in time if and only if x now possesses F and either did not or will not possess F; or x will possess F and either does not now possess F or did not possess F; or x used to possess F, and either now does not possess F or will not possess F. (The "or" expresses an inclusive disjunction, meaning and/or.)
My subsequent arguments that all abstract objects exist in time, that simultaneity and presentness are absolute, and that the past and future consist of infinite sequences of infinitely long temporal intervals, all hinge upon the fact that D1 and D2 are true and, specifically, upon the fact that Cambridge properties that are included in the domain over the which the second-order variable F ranges. If there are no Cambridge properties and no Cambridge changes, then abstract objects do not exist in time and my arguments for absolute simultaneity, for infinite time, and for the falsity of the "clock principle" (8) presupposed by STR, GTR and QM, are unsound arguments. Thus, I shall first refute a series of arguments purporting to show that Cambridge changes and properties do not exist. After this, I will give some positive arguments that Cambridge changes and properties exist. I will then show that it is logically necessary that something exists in time if it undergoes a Cambridge change. First, I would note that the "real/Cambridge change" terminology is unfortunate, since it conversationally implicates (in Grice's sense) that Cambridge changes are not "real" in the sense of not existing. Moreover, the real/Cambridge distinction is both vague and equivocal and is often used to mark several different distinctions. For example, one distinction may be called the "internal change/external change" distinction, which is the difference between a change
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in the parts or monadic properties of an object x and a change in y's relation to the object x that is due only to a change in the parts or monadic properties of x. However, I shall use the "real/Cambridge" terminology since (even though an adequate definition is hard to come by) most philosophers can intuitively recognize what is a Cambridge change and what is a real change. Do there exist Cambridge changes and Cambridge properties, such as the null set's relational property of being apprehended by John or a proposition's relational property of being believed by Jane? Consider one argument for the nonexistence of these properties: There are some propositions about related items that change their truth-value entirely due to changes in one of the objects, without implying the other relational term has any monadic property in virtue of which the truth-value changes. For example, if a proposition p is believed by Jane at 12:00 a.m. and is not believed by Jane at 12:01, then a different proposition, p is occurrently believed by Jane, changes its truth value entirely due to changes in Jane's mental states. There is no monadic property of p by virtue of which the proposition, p is occurrently believed by Jane, undergoes a change in truth value. One can (allegedly) infer from this that p does not undergo any existent change or acquire or lose any property and thus that Jane's occurrent belief does not give any reason to think that p exists in time. However, this argument (and each logically related argument for the nonexistence of Cambridge changes) is flawed for a number of reasons. (a) The inference that the proposition p does not acquire or lose any property, and thus that there is no reason to think p exists in time, is invalid. All that follows is that p does not acquire or lose a monadic property. The premises of the argument are consistent with the thesis that p acquires and loses the relational property of being believed by Jane, that this is an existent Cambridge change that p undergoes, and that p thereby exists in time. (b) An additional problem with this argument is that the proposition, p is occurrently believed by Jane, does not change its truth value by virtue of a monadic property of either Jane or p. The proposition changes its truth value by virtue of a dyadic property being transiently exemplified by Jane, namely, the dyadic property of occurrently believing p. However, this is logically equivalent to the fact that the proposition, p is occurrently believed by Jane, changes its truth value by virtue of p transiently exemplifying the dyadic property of being occurrently believed by Jane. Consequently, this account gives no logical distinction between real and Cambridge changes. Further, this doctrine is self-contradictory, since it says that the proposition, p is occurrently believed by Jane, changes its truth value; but a proposition changing its truth value is a paradigmic instance of a Cambridge change, which supposedly are not existent changes at all.
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(c) Perhaps the anti-Cambridge theorist can formulate an argument that does not include a premise that implies a certain proposition undergoes a Cambridge change. The anti-Cambridge theorist may say that acquiring and losing a relational property is not a change at all (a change that exists), and therefore that there are no Cambridge changes. But this statement is provably false, since if a change is defined as the acquisition or lose of a monadic property, the absurd result follows that the people at Hiroshima being blown up by an atomic bomb is a not a change they underwent, since being blown up by a bomb is a relational property. In fact, the paradigmic instances of changes are acquisitions or loses of relational properties, e.g., being killed by, being injured by, being broken by, being reshaped by, moving from x to y, etc. These changes cannot all be reductively analyzed into changes of monadic properties, if only for the reason that movement or change of place consists primitively in relations of the moving object to other objects or places. (d) There is another strategy for denying that Cambridge changes exist. One can define changes (i.e., all changes that exist) in terms of acquiring or losing an n-adic property by virtue of a causal event or process. But this strategy for eliminating Cambridge changes is logically untenable for one of two reasons. First, it implies that it is logically possible that changes in mental states are not changes. If agent causality and the libertarian theory of free will are true, such that mental events are freely caused by the agent, then the mental event E of believing the proposition p will be caused by the agent A, but the more complex event of A causing E will not itself have a cause. If the next mental event E' that A experiences is also caused by A, then the event, A causing E, is neither an effect of anything nor a cause of anything. (The brain events would be parallel or subvenient to the mental events but not causally related to them.) It would follow that the agent's causing her mental event E and then causing her mental event E' is not a mental change. But this is absurd, since such changes of mental state are paradigmic instances of changes. Second, we may argue that Cambridge changes satisfy this causal definition of a change. (This argument does not depend on libertarionism or the theory of agent causality.) A mental belief-event E does bring about something, the instantiation of an n-adic property; it brings about the instantiation of such a property as being believed by Jane. Since the proposition p acquires this property as a result of the event E occurring, p's acquisition of this property is due to a causal event occurring, the event E. It may be objected that p's acquisition of this property is not an effect of the event E since p is an abstract object and thus cannot be affected by anything.
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However, it is false that abstract objects cannot be affected by anything. Even though Quine is a concrete object, any singular or de re proposition that includes him as a part is an abstract object. The de re proposition consisting of Quine and the property of being alive, {Quine, being alive}, is affected if one of its parts is affected. If Quine is blown up by a bomb, then this de re proposition is affected; indeed, the explosion causes this proposition to cease to exist, since the impact destroys one of the parts of the proposition (Quine). This singular proposition ceases to exist in the sense that it exists at t but does not exist at the later time t'. There is a difference between two types of effects, abstract effects (which are types of Cambridge changes) and concrete effects. This does not imply Neoplatonism is true, since both abstract effects and concrete effects can be brought about by concrete causes, such that there are no abstract causes of concrete events. But some abstract effects have abstract causes; a de re proposition's ceasing to exist is an abstract event that causes my de dicto belief that this proposition exists to undergo the abstract event of becoming false. (Of course, this account of the causal definition of change implies that my first argument against the causal definition of change [the argument about the complex event, agent A freely causing E] does not tell against the causal definition of a change; it tells against this definition only if all causes and effects are concrete occurrences.) (e) A fallacy of equivocation is one reason many philosophers believe that Cambridge changes do not exist, a fallacy that is tempting due to the conversational implicature of "real change". They believe "x undergoes a merely Cambridge change" does not imply "there is some change that x undergoes". But the relevant argument (x underwent a change c that is not real; therefore, c does not exist) is based on a fallacy of equivocation on "real"; in the premise it means something such as "a change in a concrete thing's parts or monadic properties" and in deriving the conclusion it is taken to mean instead "a change that exists". Let us now pass to a positive argument that Cambridge changes exist. It can be argued that "c is a Cambridge change and c does not exist" is implicitly self-contradictory. A familiar example of a Cambridge change is being remembered and then not being remembered. If I remember my dead grandmother at noon, August 24, 1996 and cease to remember her by 12:01, August 24, 1996, my grandmother has undergone a Cambridge change. The familiar line is that the only change that exists is a change in my mental states or myself; I have changed from the mental state of remembering her to the mental state of not remembering her. I acquired and lost a property, but by grandmother did not, or so the familiar line goes.
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But the familiar line is implicitly self-contradictory. If my grandmother did not acquire and later lose any property, then she did not acquire and lose the n-adic property of being remembered by me. She did not stand to me in the relation of being remembered, and then cease to stand to me in this relation. One of two possibilities occurred instead; (i) my memory image stands to me in the relation of being remembered, and then no longer stands to me in this relation, or (ii) my act of remembering is a monadic property, such that nothing stands to me in the relation of being remembered. But the first option is self-contradictory, since I did not remember my memory image, but my grandmother. My memory image exists simultaneously with my occurrent mental act of remembering, but the relation of remembering (by definition) is a relation to some thing or event that exists earlier than the act of remembering. At 12:05 I can remember the memory image I formed at 12:00, but this is different then remembering my grandmother walking across her lawn in 1970. The second option, that my act of remembering is a monadic property, is also self-contradictory. If it is a monadic property, then it is an intentional act that has no intentional object. It is a conscious act but there is nothing of which it conscious. It is not "about" anything. However, remembering, by definition, has an intentional object, viz. something remembered. The proposition, there occurred an act of remembering in which nothing was remembered, is analytically false. Of course, I can seem to remember something and, due to a defect in my epistemic faculty, not remember anything. But I am here talking about rememberings (i.e., seeming rememberings that are true seemings and thereby are rememberings or acts of remembering.) Each intentional act, by virtue of having an intentional object, does not require the extra-mental existence of the intentional object. If I fantasize a unicorn walking on a tightrope, the intentional object is a fantasized unicorn, and "x is a fantasized unicorn" implies "x exists dependently on the act of fantasizing". The only option left is the third one, that my grandmother stands to me in the relation of being remembered, and later does not stand to me in this relation. Since "standing in a relation" implies "exemplifies an n-adic property", it logically follows that my grandmother exemplified an n-adic property and at a later time did not exemplify it. My grandmother underwent this Cambridge change and therefore there is a change my grandmother underwent. A logically analogous argument can be used to show that there is some n-adic property acquired and lost in each case of Cambridge change, and that in each such case there exists a Cambridge change. So far, I have argued that Cambridge properties exist and that my definitions D1 and D2 of existing in time cannot be refuted by showing that there are no Cambridge properties. However, the rebutal of these arguments
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does not imply that D1 and D2 are true. For example, one could concede that Cambridge changes exist but still hold that temporal existence should be defined in terms of real changes. Thus, I need some positive arguments that show D1 and D2, with their intended implication that undergoing Cambridge changes is logically sufficient for existence in time, are true definitions of temporal existence. According to D1 and D2, it is irrelevant what sort of n-adic property is acquired or lost (be it real or Cambridge), since it is logically impossible for an object to acquire at one time some n-adic property and lose at a later time that property if that object exists timelessly. "x exists timelessly and yet possesses different n-adic properties at different times" is an implicit logical contradiction. I shall demonstrate this by first considering some arguments that this is not a logical contradiction. In William F. Vallicella's interesting article, "No Time for Propositions" (1995), he argues that there is no reason to accept a certain premise of one of my arguments in Smith (1990), namely, that "Something is in time if it stands in a relation at a time". Vallicella writes: "suppose proposition p stands to John at time t in the relation of being believed. If propositions are timeless, then [Smith's premise that something is in time if it stands in a relation at a time] is false" (Vallicella, [1995: p. 474]). Vallicella concludes that my argument is question-begging. I did not offer a justification for this premise since it seemed obvious to me, but a justification can be offered. If p stands in R to x at time t, it follows that what is expressed by the adverbial modifier "at time t", modifies what is expressed by "stands in". In other words, being at t is a property of p's standing in the relation R, and if something exemplifies an n-adic property at a certain time, then that object exists in time. Vallicella says that p's standing in relation R exists in time, but it does not follow that p exists in time. Vallicella suggests that I commit a fallacy of division. Consider the whole event, John's occurrently believing that p. This event occurs in time. But the fact that this whole belief-event occurs in time does not imply that each part of this belief-event, such as the proposition p, occurs in time. Vallicella notes that the Cambridge belief-event of p's standing in a belief-relation to John is an event that is simultaneous with other events; this is "an event and therefore the kind of thing that can be simultaneous with other events, and thus unproblematically in time. But p, although it stands in the belief-relation, is not an event. . ." (Vallicella, [1995: p. 474]). Vallicella concludes that my argument that p exists in time is unsound. If Vallicella's argument is that only events exist in time, that p is not an event, and therefore that p does not exist in time, we can dismiss his argument
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out of hand, for his theory would imply that persisting objects, continuants, do not exist in time. But surely electrons, monkeys and galaxies exist in time. Perhaps his argument may be more charitably construed as the argument that some objects exist in time, but that "an object O's exemplification of an n-adic property F is an event simultaneous with other events" does not imply "the object O exists in time". The reason there is no implication, the argument may go, is that there are Cambridge properties and exemplifying Cambridge properties simultaneously with other events is not a logically sufficient condition for an object to exist in time. An object must transiently exemplify a real n-adic property in order to exist in time. This suggests that Vallicella's argument in effect reduces to the second main argument against my definitions D1 and D2. The first main argument is that Cambridge properties do not exist. The second main argument is that Cambridge properties exist but that objects exist in time only if they acquire or lose some real property. However, this is no argument at all but merely a question-begging assertion. There are no discernable premises to "the argument" that an object must undergo a real change in order to exist in time. This assertion seems instead to be a stipulative definition of "exists in time" or a linguistic convention that most philosophers and scientists have tacitly agreed to adopt. But stipulative definitions and linguistic conventions are non-alethic items and thus cannot threaten the thesis that D1 and D2 are true definitions of time's nature. Suppose, however, that some objectors assert that it is true that objects have to undergo real changes to exist in time, and claim that my contrary assertion (viz., that D1 and D2 are true) is no more or less a question-begging assertion than their own assertion. The objectors may be content with a "stand off" with me, which would allow them to maintain their "real change" definition without fear of refutation. However, I shall show the thesis that Cambridge properties exist, and yet that temporal existence is defined in terms of real properties, is inconsistent with relevance logic. Our objector believes that the sentence:
(9) p stands in a belief-relation at time t1 and does not stand in this relation at the later time t2, does not entail
(10) p exists in time.
But they claim that:
(11) x stands in a causal relation at time t1 and does not stand in this relation at the later time t2, entails
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(12) x exists in time.
However, if (11) entails (12), in the sense of relevance logic, then (9) entails (10). The parts of the sentence that are relevant to the entailment are exhibited in the following argument:
(13) x stands in an R-relation at time t1 and does not stand in this relation at the later time t2;
therefore,
(14) x exists in time.
The word "causal" is not the word that makes the premise (11) relevant to the conclusion (12), viz., "x exists in time". Rather, it is the temporal words in the premise, "at time t1" and "at the later time t2". Note, for example, that there is no relevant entailment in this argument:
(15) God stands in a causal relation to abstract objects;
therefore,
(16) God exists in time.
There is no part of (15) that enables it to relevantly entail (16). To provide such an entailment, an extra premise is needed, such as "whatever stands in a causal relation exists in time". (Even if causes and effects are only events, and never objects, an object can stand in a causal relation at one time and not at another time in the sense that it exemplifies a causal property, or undergoes a change that is a causal event, at one time but not at a later time.) Some philosophers argue that the open sentence "x causes y" strictly implies "x is earlier than y"; other philosophers argue that "x causes y" strictly implies "x is simultaneous with y"; still others argue that it implies "x is earlier than or simultaneouw with y"; and still others that it implies "x is either earlier than, simultaneous with or later than (backwards causation) y". And other philosophers argue "x causes y" strictly implies "x is earlier than, simultaneous with or later than y, or x exists timelessly and y exists in time, or x and y exist timelessly". Perhaps one of these views is correct. But the correctness of any such view would do nothing to impugn my impugn my argument that "x causes y" does not entail in the sense of relevance logic that "x exists in time." If p strictly implies q, then it is logically necessary that p materially implies q. The proposition q is true in every possible world in which q is true. But this falls short of entailment in the sense of relevance logic. For there to be
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such entailment, a n-adic predicate in the conclusion must appear in at least one of the premises and in the present case "( ) exists in time" or "is temporal" appears in the conclusion but not in the premise, which features instead the n-adic predicate "causes". Because there is no relevant entailmement of conclusions about time from premises about causes, there can be an open and debatable question about which temporal or atemporal conclusions (if any) are strictly implied by premises about causal relations. Similar arguments hold for other real n-adic properties in terms of which temporal existence may (allegedly) be defined, such as light-connectability, movement and the like. In short, the thesis that Cambridge properties exist but that temporal existence is defined in terms of real properties is not consistent with relevance logic. It is interesting that Vallicella (1995) concedes that some propositions exist in "wide time", but not the "narrow time" of the concrete causal order. This concession is all I need for my definitions of existence in time to be true, for "x exists in wide time" entails "x exists in time". The adjectives "wide" and "narrow" refer to nontemporal properties of the items that exist in time. "x exists in narrow time" means "x exists in time and is a concrete particular that bears causal relations to other concrete particulars". "x exists in wide time" means "x exists in time and need not be a concrete particular that bears causal relations to other concrete particulars". Note that "x exists in time" in my definitions and in the conclusions of my arguments does not entail "x endures from the first time (at which it possesses a Cambridge property F) to the later time (at which it does not possess F)". If we are talking about a permanently existing de dicto proposition or null set, we are entitled to infer that it remains present from the first time (when I come to be aware of it) to the second time (when I cease to be aware of it). In terms of the tenseless theory of time, we are entitled to infer that it is temporally located at the same time, t1, that my first act of awareness is located, is temporally located at the same time, t2, at which the event of my ceasing to be aware of the abstract object is located, and that the object is located at all the times in between t1 and t2. But if a past object, such as my dead grandmother, is first remembered and later not remembered, it does not follow that my grandmother is present from the first time to the second time. It does follow that my grandmother existed at some times throughout the time period from the first time (the time of my remembering) to the second time (the time of my not remembering). My grandmother existed at times with different degrees of pastness; that is, she became more and more past from the first time to the second time. In terms of the tenseless theory of time, my dead grandmother has a greater temporal distance from the second time, 12:01 a. m., then she does from the first time, 12:00 a.m.; specifically, she is earlier than 12:01 a.m. by one more second than she is from 12:00 a.m.
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There may be a different sort of objection to D1 and D2 than the sorts I considered in the preceding paragraphs. It may be objected that my definition of temporal existence, if true, would imply, by parallel reasoning, that the definition of spatial existence would imply the absurd conclusion that abstract objects exist in space. An objection of this sort was suggested to me by Graham Nerlich, in connection with my argument in Smith (1993a, pp. 209-10) that abstract objects exist in time. I argued that if some abstract object possesses at one time t1 the property of being referred to by John and does not possess this property at the later time t2, then the abstract object acquires and loses the property of being referred to by John and thus exists in time. A parallel argument of the sort Nerlich suggested concerns "exists in space". If some abstract object O possesses the property of being referred to by John when John is at place P, and loses this property when John moves to another place P', then O acquires and loses the property of being referred to by John when John is at different places and thus O exists in space. I would respond that this argument is not parallel, since there is no relevant analogy. The thesis that the abstract object O exists in time is not inferred from the premise that John exists at different times when he refers and does not refer to O. Rather, it is inferred from the premise that John's referring to O exists at one time and not at another time. But John's referring to O does not exist at any place (even though John exists at different places). John's mental act of referring to O lasts for two seconds, but is not extended through any space (it does not have a certain height, width and depth) and is not a mass point that is located at any place.
6. All Abstract Objects Exist in Time
(a) First Argument. So far I have argued that if a proposition stands in a belief-relation to somebody, that proposition exists in time. This is not sufficient to show that all propositions exist in time, for many propositions are not believed by anybody. However, if I have a belief about all propositions, e.g., that all propositions are nonspatial, then all propositions exist in time, for in this case all propositions would acquire and lose some property. Vallicella believes this argument is invalid; He writes: "Now surely the following argument is invalid: I believe that every proposition is nonspatial q is a proposition Therefore I believe that q is nonspatial" (1995: p. 477).
Page 158 of Smith, Quentin, 1998, ?Absolute Simultaneity and the Infinity of Time?, in (ed.) Robin Le Poidevin, Questions of Time and Tense. Oxford: Oxford University Press, pp. 135-183. Vallicella notes that "my thinking doesn't 'reach' q due to the opacity of belief contexts". (1995, p. 477). Vallicella believes that my argument only shows that the general proposition, every proposition is nonspatial, acquires and loses a belief-relation to me. This is true; however, each proposition, such as q, acquires and loses a different relation, not to me, but to my occurrent act of believing the general proposition; each proposition acquires and then loses a truth-making relation to my occurrent believing. The proposition q, and each other proposition, transiently stands to my occurrent believing in the relation of making it true. After I no longer hold this belief, q and each other proposition no longer stand in this truth-making relation to my belief. Thus, my argument about believing a general proposition about all propositions does, in fact, show that all propositions exist in time. Of course I am not saying that q is the truth-maker of my belief in the proposition, every proposition is nonspatial. Rather, the sort of truth-making relation in which it transiently stands to my occurrent believing is being a necessary part of the truth-maker of the believing. The truth-maker of my occurrent belief is the conjunctive state of affairs whose conjuncts are q is nonspatial, r is nonspatial, . . ., and so on for each proposition. This state of affairs is also the truth-maker of the proposition, every proposition is nonspatial. This conjunction permanently stands in a truth-making relation to the proposition and transiently stands in a truth-making relation to my occurrent believing.
(b) Second Argument. Vallicella notes that my argument regarding beliefs shows only that propositions exist in time contingently; they exist in time if there happen to exist intelligent organisms that believe propositions. However, I think there are more general arguments that show that propositions, and all other abstract objects, exist in time if time exists (regardless of whether or not intelligent organisms exist.) For each event e, when e is present, there is a true tensed proposition, e is present, that corresponds to the state of affairs composed of e's being present. When e becomes past, then this proposition no longer has the property of corresponding to any state of affairs, since there no longer is the state of affairs of e's being present. The proposition acquires and then loses the n-adic property of being true and thus the proposition exists in time. The same holds for propositions of the form, e is past by two seconds, or e is future by two seconds, or any other transiently true proposition. The same holds for each transiently exemplified universal; e.g., the universal, three-sidedness, is
Page 159 of Smith, Quentin, 1998, ?Absolute Simultaneity and the Infinity of Time?, in (ed.) Robin Le Poidevin, Questions of Time and Tense. Oxford: Oxford University Press, pp. 135-183. first exemplified by a sand formation on the beach and is later not exemplified by it. A related argument shows that each nontransiently truth-valued proposition and each unexemplified universal exists in time. Each nontransiently truth-valued proposition p is a conjunct of a more complex proposition p and q, such that q transiently corresponds to a state of affairs. This implies that p acquires and loses the property of being a part of a conjunctive proposition one of whose conjuncts is transiently corresponding to a state of affairs. Further, each unexemplified universal is a part of some such proposition p and thus acquires and loses the property of being part of a conjunctive proposition one of whose conjuncts is transiently corresponding to a state of affairs. (c) Third Argument. There is a more general argument that shows that any object, concrete or abstract, exists in time if time exists. If time exists, then there is some temporal state of affairs, the presentness of the instant x, that is presently different from each concrete and abstract object. When x is present, each concrete and abstract object has the corresponding relational property of being presently different from the obtaining state of affairs, the presentness of x. When x becomes past, each concrete and abstract object loses the relational property of being presently different from (what was) the obtaining state of affairs, the presentness of x; they do not now stand in the relation of being different from x's being present, since there now is no state of affairs consisting of x's being present. (d) Fourth Argument. L. Nathan Oaklander believes (private communication) that my arguments show only that if the tensed theory of time is true, all abstract objects exist in time. Oaklander argues that my tenseless definition D1 does not show that three-sidedness exists in time. The tenseless definition is: (D1) x exists in time if and only if there is some time t at which x possesses some n-adic property F and some different time t' at which x does not possess F. Oaklander argues that universals do not exist in time. He argues that if a sand formation on the beach is three-sided at t1 and is not three-sided at t2, it does not follow that three-sidedness acquires and loses the property of being exemplified by the sand formation. He states that the sentence,
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