Tópicos Especiais de Ontologia

The Senseless Direction of Time It is often claimed that time is a ‘one-way street’, i.e. that change goes essentially from earlier to later terms. I argue that the debate over the nature of this directedness has suffered from the widespread confusion of this notion with that of the sense of the earlier-than relation. Wannabe realists about time must construe temporal relations as intrinsically and irreducibly dynamic and uniquely directed. Introduction Most of us believe that changes go from earlier to later events. The direction of time as grounded in the passage of time Those who believe in temporal becoming, i.e. those who think that events (objectively) change from being future, to present, to past (the A-theorists), often argue that their view is uniquely capable of accounting for this directionality. It is of the essence of passage, it is claimed, that it occurs in the past-to-future direction. McTaggart, for example, famously claimed that only the passage of time could bestow an undirected series (the C-series) with a direction: “[m]ore is wanted for the genesis of a B-series and time […] than simply the C-series and the fact of change. For change must be in a particular direction. And the C-series, while it determines the order, does not determine the direction.” (1908: 462). Even a staunch denier of passage like Williams concedes that this constitutes a “more serious excuse for the idea of passage” (1951: 464). Despite its intuitive appeal this contention, however, is very problematic. The mere claim that the distinction between past, present and future states of affairs reflect ontological distinctions (the A-theory) does nothing to secure temporal becoming and hence it does nothing to ground its putative directionality. Temporal becoming must consist in more than the current non-existence of past and future states of affairs, since it requires that different times be subsequently present: “[b]ecoming requires that there be a sequence of A-series, i.e., that events change with respect to their A-determinations” (Gale 1968, 86, my emphasis). Kit Fine seems to have had the same point in mind when he wrote that “[t]he passage of time requires that the moments of time be successively present and this appears to require more than the presentness of a single moment of time”. (2005: 287, my emphasis). If the series of events has a direction thanks to the passage of time, therefore, such directedness must be grounded in the relation of subsequence that obtains between different A-series. In short, regardless of one’s preferred ontology of time, directedness must be a feature realized by temporal relations, not by temporary temporal properties alone. The independence of directedness from A-theoretic passage can also be appreciated on purely phenomenological grounds. Our experience (or belief) in temporal becoming is at least partly based on our memories of past states of affairs. We believe that events become past because we have memories of events that we perceived in the past and which we do not presently perceive. But, arguably, we derive our empirical concepts of succession and directedness from direct perceptual experience, in which memory does not play any apparent conscious role: It is important to notice that the relation of succession as given in experience has an intrinsic sense. We perceive directly that there is an intrinsic difference between A succeeding B and B succeeding A. […]. Temporal relations […] have an intrinsic sense as well as an intrinsic order, but […] this intrinsic sense of the fundamental temporal relation of succession is as much an immediately experienced fact as the intrinsic order, and is in no way derived from the distinction between present, past and future. (Braithwaite 1928: 164, my emphasis) . Analogously, our direct awareness of the direction of motion, which is based on our perception of the direction of temporal successions, does not require any conscious recourse to our memories of the past positions of the moving object. Now, since the concept of direction is an empirical concept, from the fact that we can perceive the temporal direction of a process without experiencing some phases of this process as past, it follows that the concept of direction itself is independent from our notion of temporal becoming. The direction of time as merely subjective Some think that, since directedness is absent (or idle) in our best scientific theories we should simply abandon the idea that it is real at all. Thus, for example, Gold (1996: 137) argued that a description of the universe in the opposite temporal direction, though “strange”, is not as description of how it might be but isn’t, but simply a different description of the same universe. Analogously, Mehlberg (1962: 104) argued that since fundamental physical theories are time-reversal invariant, we should conclude that the directedness of temporal relations is unreal or conventional: “it would be either a miracle or an unbelievable coincidence if all the major scientific theories [...] somehow managed to co-operate with each other so as to conceal time’s arrow from us”. More recently, Price (1996: 84) claimed that, until there are good reasons for thinking otherwise, there is no objective matter as to which end of the universe is the “bang” and which is the “crunch”. Although the aim of this paper is not to argue against fully anti-realist conceptions of temporal reality, it is worth noting that denying objectivity to temporal characteristics while admitting that they are represented in conscious experience should always be handled with care. Unlike other aspects of reality, where the vehicles of experience do not share the character of what they represent (a mental representation of red apple need not and could not be red), the vehicles of temporal experience can, and arguably do share the characters of the reality that they represent. As noted by Helmotz (1925: 445): Simultaneity, succession, and the regular return of simultaneity or succession, can obtain as well in sensations as in outer events. Events, like our perceptions of them, take place in time, so that the time-relations of the latter can furnish a true copy of those of the former (Helmotz, 1925, 445, my emphasis). Some further think (correctly, I argue) that the perceptions of temporal objects actually do share the temporal character of what they represent. Broad (1921: 151), for example, claimed that: It is a matter of direct inspection that the immediate objects of some of our states of mind have temporal characteristics. It is as certain that one note in a heard melody is after another in the same specious present and that each has some duration as that some objects in my field of view are red or square […]. It is then quite certain that some objects in the world have temporal characteristics, viz. the immediate objects of some states of mind. (Broad 1921, p. 151) This point can be expressed borrowing from a famous metaphor by Hermann Weyl. The idea that the gaze of my consciousness “[crawls] upward along the life-line of my body” (1949: 116) contains the idea of something crawling upward (note: not downward!), i.e. towards the future. In a nutshell, it is absurd to contend that we have no justifiable notion of objective directedness, since our own stream of consciousness is so objectively directed, and that this is so is a matter of direct inspection. Criticism of Farr and Deng here Reductionism about temporal directedness. Some philosophers suggested that the experienced directedness of processes should be reduced to features of experience that merely match some asymmetry of events in time, without entailing any objective directedness of time. The observation that the direction of increasing memory aligns with (or that it itself involves) that of increasing entropy, for example, has sometimes been offered as an explanation for this putative illusion (Grümbaum 1971: 196; Baker 1979: 343). But our question is why change and time should appear to be directed forward rather than backward and, as Shuster 1986 has rightly remarked, “Grünbaum and Baker, rather than answer the question, merely indicate certain other conditions that are also correlated with earlier and later” (712, my emphasis). Analogously, the contention that the direction of processes simply is the direction of increasing entropy, rather than merely correlating with it, has the unwanted consequence o making the second law of thermodynamics true a priori, by definition. We could not, even in principle, suppose any violation of the law, since any such violation would be accompanied by a corresponding inversion of the direction of time. Maudlin and the stock market. The standard B-theory of time: time without dynamicity There has been a dominant tendency among scientifically minded philosophers to conceive reality as objectively temporal, but devoid of dynamicity and transiency. Philosophers of this persuasion (the B-theorists) agree with McTaggart that temporal becoming is subjective or illusory. But they disagree with him that this entails that time is altogether unreal. Temporal reality is conceived as nothing more than the subsequent occurrence of sets of simultaneous events, ordered by genuinely temporal relations. Directedness, arguably, is an objective (and directly perceived) feature of these relations, one that realists of all persuasions must account for. But what is it? What is its ontological underpinning? According to the widely held view of temporal relations (the standard B-theory), their directedness is completely captured by their formal features. Temporal relations are antisymmetric, irreflexive and transitive. These features, according to this view, bestow the temporal series with an order and thereby with their direction: It might be replied that science does not supply an arrow for time because it has no need of it. But I think it plain that time does have a sense, from early to late. I only think that it can be taken care of on much less draconian principles than absolute passage. There is nothing in the dimensional view of time to preclude its being generated by a uniquely asymmetrical relation, and experience suggests powerfully that it is so generated. But the fact is that every real series has a "sense" anyhow. (Williams 1951: 465). I shall argue, to the contrary, that asymmetry is insufficient to provide a relation with a direction. The formal features of the series do not bestow a direction on it, any more than the less-than relation bestows a direction on the series of real numbers. Wannabe realists about temporal relations must add an extra, non-formal ingredient if they want their series to be genuinely temporal. This will provide new grounds for a view of temporal reality advocated over the past few decades by N. Oaklander: the R-theory of time. Quote Nathan The reason why so many B-theorists failed to take seriously the contention that their view ‘spatializes’ time, I argue, lies in the confusion between two concepts: that of the direction of processes and that of the putative “direction” of asymmetric relations in general. This made standard B-theorists less sensitive than they should have been to criticism coming from the friends of passage. The aim of this paper is to expose this confusion and indicate how it should be corrected. The ontology of asymmetric relations and the ontology of directedness To expose this confusion, it is worth making a digression into the ontology of asymmetric relations. Noting that all asymmetric relations yield different states of affairs depending on the order in which their terms feature in them (aka ‘differential applicability’), many philosophers have supposed that all asymmetric relations are essentially directed: It might perhaps be supposed that every relation has one proper sense, i.e. that it goes essentially from one term to another. In the case of time relations, it might be thought of that it is more proper to go from the earlier to the later term then from the later to the earlier. (Russell 1913: 86-7) Russell, who held this view himself in his 1903 (95-6) called this the “from-and-to character” of asymmetric relations. If this view is correct then every asymmetric relation R has a converse R^(-1), such that if a stands in relation R to b then b stands in relation R^(-1) to a. Accordingly, the earlier-than relation between events is said to go from earlier to later terms, while its converse (the later-than relation) is said to go in the opposite sense. As I said, many treat this notion of sense as synonymous with the notion of directedness. This is apparent in the passage from Williams quoted above and is explicitly or implicitly assumed by most B-theoretic accounts of time. Tegtmeier 1997, for example, claimed that “[t]ime has a direction insofar as it is a linear order, a series or quasi-series. Every series has a direction.” (p. 186, my emphasis). This view is also assumed implicitly by all attempts to reduce directedness to some asymmetry of events in time: “a sufficient difference of sense”, claimed Williams 1951 (465), “[…] would appear to be constituted, if nothing else offered, by the inevitably asymmetrical distribution of properties along the temporal line (or any other).” Prominent and common manifestations of this view are all the attempts to reduce directedness to the entropic gradient, or to think that it could “emerge” at a macroscopic level out of it. Quote Hawking. This common conception is also explicit in Reichembach’s seminal book The Direction of Time (1956). Taking a closer look at his arguments will help to clarify further the view that I wish to criticize. Like many before him (McTaggart 1908, Broad 1923, Braithwaite 1928) Reichembach stresses that order should not be confused with directedness: “a line, though serially ordered, does not have a direction” (26). As it turns out, however, the notion of “directedness” that they have in mind is radically different from that which should be at issue in the discussion of change and time. Reichembach thinks that the points in a line lack a “direction” not because the asymmetric relations between them are not temporal, but because their asymmetry is not intrinsic to the line: When we say that a line, though serially ordered, does not have a direction, we mean that there is no way of distinguishing structurally between left and right, between the relation and its converse. […] That is, the relation ‘to the left of’ has the same structural properties as the relation ‘to the right of’ (ibid.). What series do have a direction according to him, then? Endorsing the received view of asymmetric relations, he thinks that all series generated by an intrinsically asymmetric relation have one: The numbers are governed by the relation smaller than, which is asymmetrical, connected, and transitive, like the relation to the left of; therefore the numbers have an order. But in addition, the relation smaller than has a direction; That is, it is structurally different from its converse, the relation larger than. (ibid.) He concludes that the relation earlier-than must be “of the same kind” as the relation smaller-than, meaning that it “differs structurally from its converse, the relation later-than” (ibid.). In short, he thinks that the notion of change and time having a direction is essentially the same as that of Russell’s from-and-to character of all asymmetric relations, embellished by the perfection of intrinsicality. This common conflation of the notion of sense with that of direction appears to permeate, implicitly or explicitly, most debates on the nature of time, from physics, to ontology, to phenomenology. I argue that it is untenable, and that it has significantly hampered progress in the debate. According to the received view, in the words of McTaggart (1927: ¶ 695): “[the B-series] has two senses, according as we take the generating relation to be earlier-than or later-than”. This is of no concern to McTaggart, since he thinks that the only real series is the atemporal and (hence) undirected C-series. But this is of concern to anyone who thinks that there are genuinely temporal relations since these surely don’t ‘go’ in both directions! A concern about the standard view of asymmetric relations, one that is directly relevant for the question of temporal directedness, stems from the (too) intimate connection that exists between converse relations. Peter Geach, for example, argued that the two converses of asymmetric relations are never observable independently from one another, and that therefore these notions (and their distinction) cannot have an empirical origin: A relation neither exists nor can be observed apart from its converse relation; what is more, the concept of a relation and of its converse is one and the same indivisible mental capacity, and we cannot exercise this capacity without actually thinking of both relations together. (1957, p. 33, my emphasis). Indeed, it is tempting to think that both converse relations necessarily enter in all the same facts. Isn’t the fact that the cat is on the mat the same fact as that the mat is under the cat, after all? And aren’t these two circumstances experientially indistinguishable? This is of special importance to the B-theorists, for they typically think that we derive our concept of temporal succession (and hence of direction) by direct perceptual acquaintance. When I perceive a rapid sequence of events, say the notes of a melody, I cannot fathom whether I perceive the earlier-than or the later-than relation. It seems to me that I perceive both or, rather, that I only perceive one directed series of events going from earlier to later ones. If converse relations are empirically indistinguishable, then their distinctness cannot ground the empirical concept of directedness. The intimate relation between converse relations strongly suggests that they should not be treated as distinct. Some relations make it seem intuitive to think that they come imbued with a fundamental direction. As an example (other than time) think of the logical relation between the premises and the conclusion of an argument. But what about asymmetric relations that are clearly undirected? What about the number series, for example? Is the series in any sense ‘increasing’, from smaller to greater numbers? Is a monotonic function “increasing” or “decreasing”? It seems like the answer to these questions should be a resounding: “no!”. Not so much because these claims are false, but because they seem to commit a category mistake. As Weyl (1949: 4) said: “Two propositions such as ‘5 follows upon 4’ and ‘4 precedes 5’ are expressions of one and the same relations between 4 and 5. It is unwarranted to speak here of two relations inverse to each other.” Since in these cases it seems utterly arbitrary to pick one converse at the expense of the other, realists about senses should conclude that (in these cases at least) both relations exist and are realized. Russell himself radically changed his mind about the received view: In the cases where there seems to be a natural direction, this will be found to have no logical foundation. In a dual complex, there is no essential order as between the terms. The order is introduced by the words or symbols used in naming the complex, and does not exist in the complex itself. (Russell 1913/1992: 86-7) Many expressed similar doubts about the tenability of the standard ontology of asymmetric relations. These arguments stem from the observation that the (putative) senses of asymmetric relations are logically equivalent (Armstrong 1978: 42), inseparable (Geach 1957: 33), indistinguishable (Fine 2010, Reichembach 1956, Weyl 1949) or semantically indeterminate or defective (Williamson 1985, VanInwaagen 2006). These authors then proceed to provide alternative accounts of the differential applicability of asymmetric relations that construe them as ‘neutral’ with respect to their direction. To illustrate how these considerations bear on the issue of temporal directionality, consider the following argument, adapted from Williamson 1985. Assume that the relation earlier-than (the universal denoted by E) is distinct from its converse, later-than (denoted by E*), and consider the following three possible languages. In language L, expression E stands for the relation earlier-than (E), and the convention as to the order of the terms is, like in English, that the earlier term is to be written to the left of the later one): [L] E(a,b) means that a is earlier than b. In language L1 too, E stands for the relation earlier-than, except for a different convention: the earlier term is to be written to the right of the later one. E(a,b) means in L1 what the marks E(b,a) mean in L: [L1] E(a,b) means that b is earlier than a. In language L2, finally, E stands for the converse relation E* (later-than), but the convention as to the order of the marks is like L. The marks E(a,b) mean the same in L2 as in L: [L2] E(a,b) means that a is earlier than b. Now, are L and L2 different languages? Under the assumption that asymmetric relations are distinct from their converses, they must be! E stands for the universal E in L1, and for E* in L2. However, L and L2 are utterly indistinguishable: whatever E(a,b) means in L1, it means the same as it does in L2. Ditto for all expressions purported to mean that a term is later than another. Even if this does not strictly contradict the hypothesis, it shows that the supposed distinction between the senses is a distinction without a difference. Worse still, it shows that we ourselves do not know what we mean by this distinction. No causal link between facts and language would allow us to tell which is which. Whatever perception is caused by a fact that would be truthfully described by “a is earlier than b”, in fact, will be also caused by a fact truthfully described by “b is later than a” (cf Geach 1957: ?). So, even if we share the same phenomena, we could not merely “point” at the relevant facts in the hope to disambiguate the semantics. If we were to ‘point’ at two rapidly succeeding notes, for example, we would still be unable to decide whether your language differs from mine in this respect or not. Even pointing at the earlier note saying “this is the earlier term” won’t do the trick, since we would agree about that regardless of whether we speak language L or L2. Whatever the semantics cum ordering conventions for English I and you use, we would agree about everything. We would agree about which is the ‘earlier term’ and which the ‘later term’ whenever we were presented with the same temporal facts. We would insist that we know what we mean by ‘earlier-than’ and ‘later-than’. We would agree that whenever it is true that a is earlier than b, it cannot also be true that b is earlier than a, and viceversa. Finally, whenever we were presented with a fact that we would truthfully describe by “a is earlier than b” we would agree that the intrinsic direction of the succession goes from the earlier to the later term. The same would hold if we were to describe the same fact by saying that “b is later than a”. In short, the supposed distinction between my use of E and yours would be a distinction without a difference. Crucially, however, the directedness of the succession relation would be invariant across the supposed distinct idiolects that we might have learned: whether your English is like L or L2, you would agree that E(a,b) “goes” from a to b. From these considerations (meant to apply to any asymmetric relation), Williamson draws the conclusion that the hypothesis with which we started, that E and E* are distinct relations, is false. While I have some sympathy for these views, I do not wish to take issue with this question here (although I’ll say something about it towards the end of this paper). What matters for our purposes is that the direction of change, which is the direction of temporal successions, is invariant with respect to %%%%% The argument from time reversal A fourth and last line of argument comes from the philosophy of physics. Change is customarily represented mathematically by a curve φ:R⟼S on a space of instantaneous states S (with a given initial condition φ(0) 〖=s〗_0). Time reversal is represented by the operator T:S⟼S, which maps the evolution curve φ(t) onto the curve φ(-t), and which is thought to reverse the order of events. This familiar way of representing time reversal is perfectly apt to expose the difficulties discussed in this paper. Russell (1903: 95-6) showed how to construct mathematical signs out of order, so that the difference of signs corresponds to the difference between the senses. The procedure is trivial for natural numbers, and can be easily extended to all numbers, including the reals. Let R be the relation in virtue of which a number is next after the first: mRn is equivalent to the proposition m+1=n. Now, put mR^2 p whenever mRn and nRp, and so on for higher orders. For every number a, given the standard view of relations, the asymmetric relation R^(+a) has an inverse R^(-a), such that 〖mR〗^(+a) q corresponds to the proposition m+a=q, and 〖mR〗^(-a) q to m-a=q. Hence the distinction between mathematical signs corresponds to (and can be used to represent) the distinction between the two senses of the magnitude relation between the numbers. Therefore, anything that can (or cannot) be represented by inverting the sign of the time variable, can (or cannot) be represented by replacing a time relation by its converse. If the arguments put forward in this paper are correct, then, all the difficulties related to conceiving directionality as corresponding to the sense of the generating relation will resurface as difficulties in representing time inversion by inverting the sign of the time variable. And so it is! For every process (represented by φ(t)) we can construct another representation, called the “converse description”, by using ‘negative time’, i.e. by replacing the time parameter t by its inverse: t^'=-t. Thus, for example, if φ(t)=vt-1/2 gt^2 is a representation of the parabolic trajectory (in two dimensions) of an object subject to Newtonian gravity in positive time, φ^' (t)=vt'-1/2 g〖t^'〗^2 represents the same trajectory in negative time. However, we can use this converse description in positive time, in which case it is taken to represent the reverse process: “this is easily done by simply cancelling the prime mark” (Reichembach 1956: 20). Since t and t’ range over exactly the same values (the times), the converse description and the description of the reverse process are representationally indistinguishable. This should come as no surprise, by now. Again, these observations cause no trouble for antirealists about the direction of time. Reichembach, for example, thinks that “positive and negative time supply equivalent descriptions, and it would be meaningless to ask which of the two descriptions is true” (1956: 127); analogously, Gold claims that “the description of our universe in the opposite sense of time [...] is not describing another universe, or how [our universe] might be but isn’t, but it is describing the very same thing” (1966, p. 327). Indeed, far from causing troubles for their view, these observations can and have been used precisely to argue for the unreality of directedness (Gold 1966, Price 1996, Farr 2020). I have already expressed my cartesian reservations about these positions. Yet again, however, these results are surely unacceptable to those who wish to be fully realists about temporal relations. Conclusions. I have argued that the standard treatment of temporal relations according to which their orientation is realized by the formality of intrinsic asymmetry is implicitly committed to an antirealist (C-theoretic) conception of time. This, I think, casts new light on Bergson’s contention that mathematics ‘spatializes’ time: standard mathematical representations of temporal reality are blind to the distinction between the B- and the C-theory. It also suggests that the tentative thesis that Earman (1974) called ‘the heresy’ is on the right track: [The heresy] states first of all that if it exists, a temporal orientation is an intrinsic feature of space-time which does not need to be and cannot be reduced to nontemporal features, and secondly that the existence of a temporal orientation does not hinge as crucially on irreversibility as the reductionist would have us believe. (p. 20) B-theorists, I argue, are faced with a choice. Either they concede that what we perceive as a temporal series is really an undirected (albeit intrinsically ordered) series, in which case their view relapses into a C-theory of time; or they construe temporal relations as intrinsically and irreducibly dynamic and uniquely directed (the R-theory). References Armstrong, D.M. (1978). A Theory of Universals (Universals & Scientific Realism: Volume II), Cambridge: Cambridge University Press. Baker., L. (1979). On the Mind-Dependence of Temporal Becoming. Philosophy and, Phenomenological Research 39. Braithwaite, R. B. (1928). Symposium: time and change. Aristotelian Society Supplementary Volume, 8 (1), 162–174. Boltzmann, L. (1964). Lectures on Gas Theory 1896-1898. Berkeley: University of California Press. Broad, C. D. (1923). Scientific Thought. New York: Harcourt, Brace, and Company. Earman, J. (1974). An attempt to add a little direction to “the problem of the direction of time”. Philosophy of Science 41(1), 15–47. Eddington, A. S. (1928). The Nature of the Physical World. Cambridge: Cambridge University Press. Farr, M. (2020). C-Theories of Time: On the Adirectionality of Time. Philosophy Compass 15(12): 1–17. Fine, K. (2005). Tense and Reality, in Kit Fine (ed.), Modality and Tense: Philosophical Papers (Oxford: Oxford University Press): 261–320. Fine, K. (2000). ‘Neutral Relations’, Philosophical Review 199, 1–33. Gale, R. M. (1968). The Language of Time. London: Routledge and Kegan Paul. Geach, P. T. (1957). Mental Acts: Their Content and their Objects, London: Routledge & Kegan Paul. Gold, T. (1966). Cosmic processes and the nature of time. In R. G. Colodny (Ed.), Mind and Cosmos, pp. 311–329. Pittsburgh: University of Pittsburgh Press. Grünbaum, A. "The Meaning of Time," Basic Issues in the Philosophy of Time, , eds. Eugene Freeman and Wilfred Sellars (LaSalle, Illinois: Open Court, 1971) Husserl, E. (1991). On the phenomenology of consciousness of internal time (1893-1917), translated by J.B. Brough. Dordrecht: Kluwer Academic Publishers. Mehlberg, H. (1962). Review of The Direction of Time. The Philosophical Review 71(1), 99–104. McTaggart, J. M. E. (1908). The Unreality of Time, Mind 18: 457-84. (1927). The Nature of Existence (Volume II). Cambridge: Cambridge University Press. Oaklander, N., (2002). Mctaggart’s Paradox Defended, Metaphysica 3 (1). Price, H. (1996). Time’s Arrow and Archimedes’ Point: New directions for the physics of time. Oxford: Oxford University Press. Reichenbach, H. (1956). The Direction of Time. Dover Publications. Russell, B. (1903). The Principles of Mathematics, London: George Allen & Unwin Ltd. (1913). The Theory of Knowledge: The 1913 Manuscript, edited by E. R. Eames in collaboration with K. Blackwell, London: Routledge, 1992. Schuster, M. M. (1986). Is the Flow of Time Subjective? The Review of Metaphysics, 39(4), 695–714. Tegtmeier, E. (2008). Direction of time. In L. Nathan Oaklander (ed.), The Philosophy of Time. Routledge. pp. 4--246. van Inwagen, P. (2006). Names for Relations. Philosophical Perspectives. 20: 453–77. Weyl, E. (1949). Philosophy of Mathematics and Natural Science. Princeton University Press, Princeton. Williams, D.C. (1951). The Myth of Passage, Journal of Philosophy 48: 457-72. Williamson, T. (1985). Converse Relations. Philosophical Review. 94: 249–62.

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