Relativistic Linguistics

A study in linguistic relativity.

English—like all human languages—assumes a Newtonian conceptualization of space and time. These languages, and in particular their tense systems, postulate, in addition to three spatial dimensions, a single temporal dimension. These, together, equip space-time with universally valid coordinates; the time dimension, in particular, assigns—or so we imagine—to every event a unique time value, valid for all observers. By comparing these time values, we introduce such notions as simultaneity and precedence; these emerge in language through grammatical tense, as well as through certain prepositions.

In the Newtonian model, all relations of simultaneity and precedence (which might hold between any two space-time events) hold independently of observer. The notion of duration is also well defined and consistent across observers. Finally, geometric notions such as length and angle are independent of observer. The Newtonian framework closely approximates physical reality when all observers travel at speeds well below the speed of light.

Questions such as these could empirically test whether English presupposes Newtonian physics:

  1. Ben and Josh are twins, born on the same day in the same hospital. Ben knows that the supernova of the red supergiant KSN2011d became visible to Earth before his twenty-fourth birthday. Can Josh necessarily say the same? (No.)
  2. Was Ben really born first? (Yes—even under relativity—because our births were close spatially. Sorry Josh.)
  3. Have you ever traveled near the speed of light, in this life, or throughout your evolutionary history? (No. Hence the Newtonian trappings of natural language.)

A Newtonian language ill befits communication between observers moving at speeds approaching the speed of light. This article adapts the temporal logic of Prior [1] to the construction of an alternate system of English verb tenses suited to relativity. (The system should easily adapt to other languages.) Its primary aim is the elimination of ambiguity: this extended language seeks to enable unambiguous communication between observers (e.g., astronauts traveling at high relative velocities). In particular, it permits the discussion of precedence and simultaneity. Furthermore, it aims for flexibility: it seeks to leave remarks as observer-independent as possible, and to facilitate the easy “translation” of phrases between the viewpoints of various observers.

This system raises interesting questions regarding the Sapir-Whorf hypothesis of linguistic relativity, which are discussed below.

This project proceeds largely with an eye towards only special relativity, which accurately models the behavior of weightless observers freely falling through space. I also sketch its extension to general relativity, which in addition models the influence of large gravitational bodies.

A relativistic primer

Measuring space and time

How might an observer, moving through space with a clock, begin to measure distance and time? The notion of simultaneity, for one, raises difficult issues. The light from any given event, after all, takes time to travel; our observer, upon seeing this light, might not know from how far away it originated. If two events were known have occurred at equal distances from the observer, then he could declare them simultaneous just in case their corresponding light waves reach him at the same time.

What if light travelled at irregular, or uneven speeds? This observer could attempt to measure the speed of light’s travel by placing a clock at each of any two positions in space, and recording the clocks’ times as a single beam of light travelled past the two objects, before finally subtracting the two times. For this difference to be meaningful, though, it would need to have been the case that the two clocks were synchronized to begin with, whereas we have yet to define simultaneity (a good discussion can be found here [2]).

The solution is to stipulate the constancy of the speed of light, and to define distance through local measurements of the duration of light’s travel on reflective round trips.

An observer, precisely, should constantly emit pulses of light, or, in practice, radio waves, in all directions. In case a radio pulse, after departing the observer, strikes some space-time event, reflects back, and finally meets the observer once again, the observer must average the radio pulse’s times of departure and arrival (as given by his local clock) and declare this average time the time at which the event occurred. If, for example, a pulse departs from an observer as his clock strikes -1, reflects off of a supernova, and finally returns at time 1, then the observer must declare that the event occurred at his time 0. The spatial distance to the event is, of course, given by the duration of its travel; assuming that the clock counts years, it’s exactly one light-year (this is why it’s so effective to state distances in light-years).

What if a pulse were to travel faster on one leg of the journey than on the other? The observer has no way to detect phenomena of this kind, and must assume that they do not occur. The constancy of light’s speed is not an empirically measurable phenomenon, but a principle that must be stipulated. (Really, it’s a dictum about how distances are understood.) Likewise, what if the observer’s clock were uneven, or unsteady? This again is impossible, or rather, senseless. The observer defines the passage of time by his local clock, and if the round-trip travel of a light beam to and from a repetitively flashing star previously took one year and now takes two, the observer would have no choice but to declare that the star has drifted away.

The observer could choose to keep on hand a “very small box full of cold dilute radium gas” [3, p. 42], and use the gas’s decay to define time. This decay still, though, would only represent the passage of time by fiat, and not by fact. Time, in other words, wouldn’t pass in any sense other than that whereby the radium continued to decay. If the rate of the radium’s decay disagreed with another clock present on board, then there would be no way to settle the dispute between the two. (In practice, there is a preferred rate for any local clock, given by parameterizing the observer’s world line by arc length in the Lorentz metric; see [3, p. 41]. This preferred rate would align the clock with other natural processes, like human aging.)

Therefore a single observer with a clock on hand, by constantly emitting pulses of light and measuring (and averaging) the times of these pulses’ departure and arrival, may assign to every space-time event a set of three spatial coordinates plus a single time coordinate. In this coordinate system, light travels, well, at the speed of light—which is just to say that distances are calculated and drawn in such a way as to make them compatible with the observed elapsed time of light’s travel as well as, of course, the assumed constancy of light’s speed. The time displayed on an observer’s local clock is called his proper time.

For a single observer, the world appears Newtonian. (Each event has a unique place and time.) Things will change when we compare two observers. In short, relations of simultaneity will systematically shift. This is due, roughly, to a Doppler sort of effect concerning the travel of light.

An illustration of the relativity of simultaneity

Take a single observer, say A, and suppose—for this basic lesson—that space has just a single dimension. Suppose that, from A’s point of view, at proper time 0 three flashes occur simultaneously, at the physical positions -1, 0, and 1, respectively. This is no more and no less than to say that, apart from a single flash which occurs at A’s immediate location at time 0, two more flashes are detected via pulses sent respectively to A’s left and right, each emitted at time -1 and received at time 1. The situation might look something like this:


To observer A, the flashes at positions -1, 0, and 1 are simultaneous.

Let’s now posit a second observer, B, traveling, according to A’s coordinate system, at half of the speed of light. When do these three flashes—let’s call them -1, 0, and 1, referring to them by their physical positions in A’s coordinates—occur in B’s proper time?

This depends crucially on the speed of B’s clock, which we can, after all, set freely. To enable a meaningful comparison, we will synchronize B’s clock to A’s. The meaning of this is actually somewhat subtle. This means that, according to the global coordinate system provided by A’s clock—which, after all, provides any event with a time (itself computed through averaging)—B’s clock’s ticks occur at the same time that A’s do. (This does not imply the analogous fact with the roles reversed! Also note that this is slightly faster than B’s “preferred” clock speed, as in above.)

Relying on the coordinate system A’s clock provides, we now depict B as moving at half of light speed, crossing A at A’s (and B’s) time 0, say. At what speed should B’s light move? This is the trickiest question. B’s light should move at constant speed, which is to say that it should move one unit of B’s distance per tick of B’s clock. We might as well also scale B’s space axis so as to make his light beams move at the same speed that A’s do when they are charted in A’s coordinates. This question, then, is another question in disguise: how should we spatially plot objects for B, so as to make their distances compatible with B’s clock measurements and with the speed at which we’ve decided B’s light should move? The answer is, conveniently, the same as we drew them for A.

Indeed, light and space being fundamental objects (unlike the measurement thereof), the journey of a beam of light emitted by B at any particular time across any fixed interval of space must take, on A’s clock, just as long as would take that of a beam emitted by A at the same time across the same interval. Realizing any such particular journey within A’s coordinate system, we see that, constrained both by the speed of the beam’s travel (by our above stipulation) and by the total duration of its journey (on A’s clock), we have no choice but to draw its distance for B just as we would have drawn it for A.

Thus we can superimpose B’s world on top of A’s, and consider their experiences together. B will move at half speed, while his light will move at A’s standard light speed. (Spatial distances and rates of change are given meaning only by A’s coordinate system.) When will B establish that the flashes occur? Let’s find out:


B experiences the same three flashes at distinct times.

Determining the proper time of flash 0 on B’s clock is simple. B is right there when the flash occurs, so no averaging need be performed, and the flash occurs at B’s time 0.

The story is different for flashes -1 and 1: in order for B’s pulses to reflect off the space-time events just when they occur, B must send and receive his pulses at different times. Because of his rightward movement, B must emit his rightward pulse early; in order to precipitate its arrival at position 1 at time 0, B must emit the rightward pulse back at proper time -2. Oppositely, his rightward movement causes him to receive this reflected pulse early at time ⅔. Averaging these two, B determines that flash 1 occurs at proper time -4/3. Symmetrically, B’s rightward movement ensures that the reflected flash at -1 does not reach him until later than usual, at time 2. On the other hand, to ensure the arrival of his leftward pulse at the correct time B must emit it late, at time -⅔. Averaging these two, B determines that the flash at -1 occurs at proper time 4/3.

The events -1, 0, and 1, which are simultaneous for A, occur respectively at proper times 4/3, 0, and -4/3 for B. In particular, simultaneity is disrupted. Furthermore, B observes distances differently than does A: B calculates that both flashes occur 4/3 units away from him.

A similar situation could be constructed for a third observer C moving to A’s left at half the speed of light. For C, the situation is reversed: the flashes -1, 0, and 1 occur at proper times -4/3, 0, and 4/3. Here we see the reversal of simultaneity.

More is true. We decided to set B and C’s clocks in such a way that A, “looking into their spaceships”, would see their clocks ticking at the same rate as his. Is the reverse true? B or C, looking into A’s spaceship, will see his clock ticking very, very slowly—and furthermore see him aging very slowly.

In practice, it’s better to align each spaceship’s internal clock speed with its “preferred speed”. In this way, the years passing on each clock will correspond with the years marked on the astronaut’s body. The phenomenon hinted at above—called time dilation—nonetheless persists here, albeit in a more symmetric form. Now, whether A looks into B (or C)’s clock, or vise versa, each astronaut will see the other’s clock, and aging, proceeding at a rate slower than that of his own. (We could always speed up one astronaut’s clock so as to make it appear synchronized to the other. In fact, we did this before. From the perspective of the astronaut with the newly sped clock, though, the other clock will now of course appear to go even slower.)

The effect on the perception of time of switching observers is well understood. Given any space-time event and a particular observer, we can freely assign a time to that event. We can also describe how this proper time changes when we switch observers. These processes are all given by the so-called Lorentz transformations. (The animated GIF on the right-hand side of this web page goes a long way towards explaining the phenomenon.)

The Newtonian special case

We can now be more precise about the world which Newtonian physics presupposes. In Newtonian physics, each event is given a physical location as well as a time coordinate. Critically, now, each such time value is valid for all observers, which is to say that given any two events, the relations less than, equal to, and greater than between their time coordinates correspond, for any observer, respectively to the relations of antecedence, simultaneity, and subsequence in observed occurrence (this is, of course, after factoring in light’s travel time as before; in practice this is not an issue in Newtonian settings).

The Newtonian world can be recovered as a special case of the relativistic situation. Crucial is the notion of synchronization of clocks. The clocks of the members of a collection of observers can be said to be synchronized if the clocks of the members of any pair of observers align (after each observer accounts for light’s travel, as above).

Suppose that the members of a collection move so that the relative velocity between the members of any given pair of observers is zero. (This is possible even when the observers are far apart; here we assume special relativity). In this case, though the observers’ clocks might not be synchronized, they can differ from each other by at most constant terms; that is, they “tick in unison”. When this is true, it’s possible for all the members of the collection to synchronize their clocks. This takes an act of coordination. One leader must take charge. This leader should, to each other observer, send a pulse and later receive it again, only to later insist to this other observer that the event of reflection occurred at the proper time (on the other observer’s clock) that the leader computes to be his average [3, p. 137] (this process is called Einstein-Poincarè Synchronization). Provided that the relative motion is indeed zero, clocks thus synchronized will stay synchronized. For such a group of observers—equipped with synchronized clocks—the world might as well be Newtonian.

Remarks on general relativity

This section contains a few remarks aimed towards readers with background in general relativity. The objects of interest in general relativity are spacetimes, connected, 4-dimensional, oriented, and time-oriented Lorentzian manifolds [3, p. 27]. An observer, in general relativity, is a future-pointing timelike curve with its arc-length parameterization under the Lorentz metric. Its proper time is given by the parameter variable. The image of the curve is called the observer’s world line.

Simultaneity is difficult or impossible to define in general relativity. One possible definition could take a space-time event to occur simultaneously with an observer’s proper time t if and only if some space-like geodesic, passing between that event and the image of t in the observer’s world line, has a tangent vector which lies in the observer’s local rest space at t [3, p. 49]. One could also define, perhaps somewhat arbitrarily, some time function, and take as hypersurfaces of simulaneity its level sets [3, p. 53]. These definitions coincide in the special-relativistic case, where the spacetime is nothing other than 4-dimensional Minkowski space. In general relativity, they may separate or fail to exist altogether.

In any case, any working definition of simultaneity—or rather, more generally, the ability to associate, given any observer, a proper time to any space-time event (in some possibly proper subset of space-time)—is sufficient to proceed in what follows.

Absent a notion of simultaneity, one is quite nearly out of luck. The only option would be to refer only intrinsically to space-time events, and to avoid the use of tense altogether.

Interestingly, the condition permitting the synchronization of clocks admits a natural expression even in the language of general relativity. Indeed, the possibility of so-called proper time synchronization corresponds precisely to the condition that the differential 1-form physically equivalent to the vector field whose integral curves describe our observers is exact (see [3, p. 53]).

Tense logic and relativity

Preliminaries on tense logic

We begin the linguistics. We introduce a modified version of a system of “temporal logic” originally developed by Prior [1] and described, for instance, in [4]. Our system takes, as starting point, a model M, which consists of an ordered sequence of times TM together with the ability to assign, to any given instantaneous world event (in some possibly proper subset of space-time), a corresponding time t in TM.

We introduce a “morphological” system of affixes; each particular such, when attached to a verb, shall enrich it with additional information concerning time. For a model M and any verb V, we define:

    1. IM,t[V]: In model M at time t, V.
    2. PM,t[V]: In model M at some time t’ < t, IM,t’[V].
    3. FM,t[V]: In model M at some time t’ > t, IM,t’[V].

We always insert finite verbs V in the present tense, though we conjugate them as usual according to person and number. Aspect and mood will be discussed below.

The time t corresponds to the time at which a sentence is spoken, and I, P, and F correspond respectively to the tenses of present, past, and future. These affixes may be “composed” (right-to-left) in a natural way, and many standard tenses in English can be recaptured using I, P, F, and their various combinations. In the following examples, we describe temporal logic expressions together with their roughly equivalent English tenses. We assume that 0 is the time of speech.

    • The star IM,0[flashes]: ?The star flashes. (This English version is infelicitous because the verb flash, a semelfactive, can’t be expressed in the present. See below remarks on aspect.)
    • The star PM,0[flashes]: The star flashed (or has flashed).
    • The star FM,0[flashes]: The star will flash.
    • The star PPM,0[flashes]: The star had flashed.
    • The star FPM,0[flashes]: The star will have flashed.
    • The star PFM,0[flashes]: The star would flash (or was to flash).
    • The star FFM,0[flashes]. ?The star will be to flash. (English lacks a standard apparatus for dealing with the FF “tense”.)

Thus we see the compact representations afforded by this system. Absent the assumption that 0 is the present time, these “tenses” would cease to have direct English analogues.

Inspired by Prior [4], we mention additional operators, which we will not use:

    1. HM,t[V]: In model M at all times t’ < t, IM,t’[V].
    2. GM,t[V]: In model M at all times t’ > t, IM,t’[V].

These correspond respectively to the notions it has always been the case that and it will always be the case that [4]. The flexibility of this system invites the introduction of two further operators, inspired by mathematical analysis:

    1. EM,t[V]: In model M there exists some t’ > t such that at all times t’’ > t’, IM,t’[V].
    2. OM,t[V]: In model M for infinitely many values of t’ > t, IM,t’[V].

These correspond to the mathematical notions often referred to respectively as eventually and often.

The above system differs from Prior’s—or rather its characterization in [4]—in important ways. In Prior’s system, the operators P, F, etc. attach to entire propositions (effectively sentences), rather than verbs alone; both the original propositions and the modified ones are then truth-valued. The function that takes a proposition to its truth value in a model M at time t is denoted IM,t. Prior’s approach has the advantage that the various operators can be defined somewhat more formally, and their composition defined more naturally. In our system, only the verb is modified. This evokes the more linguistic fact that tenses are verbal categories, in the sense that when they’re morphologically realized, the relevant morphemes attach to the verb [4]. Finally, this approach affords a more compact notational system.

Models, sentences, and syntax trees

In practice, we’ll deal with just a few particularly important models. The Newtonian global time coordinate will serve as one canonical example (more relativistically, one could take the common time represented by a collection of observers with mutually synchronized clocks). In the Newtonian situation, observers will be able to use the Newtonian model to communicate unambiguously about time.

Our most important models will be those given by the proper time of a single space-time observer. The notion of simultaneity, discussed above, permits the assignment to each space-time event a proper time on such an observer’s clock. This assignment will depend, of course, on the observer, and these discrepancies will be central to our project.

The tenses P and F should be regarded, then, as lying “somewhere between” past and future: while a choice must be made in any remark, this choice depends on the observer, and could change. This evokes the way that a quantum computer bit can be either 0, 1, or “somewhere in between”.

Tenses in our new linguistic system, then, will operate by specifying an observer—or, in actuality, a model—and indicating an instant in that observer’s proper time. More precisely, each finite verb will be enriched with this information, using the above system. Before we may proceed, though, more must be said. How are references to events outside of the verb—say, in the subject or object positions—to be interpreted? Consider the sentence

  1. ?The supernova of KSN2011d is large.

Suppose that we’ve selected an observer A and a proper time in TA with reference to which to interpret the tense of the sentence’s verb “is”. Even having done so, in order to determine this tense, we must situate the event described in the sentence’s subject. Depending on whose proper time with respect to which we first interpret the time of the supernova, its eventual translation into A’s proper time may or may not occur before the time of the sentence’s utterance. Of course, the natural choice is to place the event immediately into A’s timeline.

More strikingly, though, take the example:

  1. ??Ben’s twenty-fourth birthday happens and Josh’s twenty-fourth birthday happens.

Soon we’ll enrich the verbs happens with observer information. For now, though, suppose that we’ve fixed a single observer A and a time t in TA to use for both verbs. Even in this case, though, in order to assign a tense to each one we need to know in whose timelines the two birthdays are to be interpreted. One natural choice could be to interpret each birthday in its celebrator’s timeline. Of course, both will need to be translated into A’s time in order to interpret the verbs, and this is equivalent to interpreting both events in A’s timeline. Stranger things happen if we desire, for instance, to interpret both events in a single twin’s timeline before passing both events to A. In this case, one birthday might occur far before the other (in the timeline we’ve chosen, and so also in A’s). In particular, not only precedence and simultaneity, but also tense, will in general differ.

We could force each of a clause’s events into the same timeline as that of its finite verb. For flexibility, though, it’s better to permit the independent interpretation of each of a sentence’s constituents.

This leads to a general solution. We purposefully remain somewhat agnostic on its details. These details are difficult, for one, though a few subtleties are discussed below. More importantly, though, this proposal is rather a schematic which must eventually be refined.

The solution, then, is this. The information communicated will be a syntax tree, with each node decorated with an observer. Passing from the bottom of the tree upwards, each reference to an event’s time will be given meaning in light of the timeline of its parent node. Statements concerning geometric quantities such as length and angle—which also undergo relativistic effects—will also be given meaning in this way. An event proceeding up the tree will, in effect, “bounce between observers” under the effects of repeated Lorentz transformations. At the level of the verb, of course, each event will be translated into the verb’s timeline and given a tense. If we decline to explicitly assign observers to a syntax tree’s nodes, we will assume that each event is interpreted in the timeline of its clause’s verb. This will be the most common situation in practice.

Expression and translation

Let’s see how this modified tense system will operate. Inspired by the discussion above, we have the following examples, spoken at time 0:

    1. Star -1 IA,0[flashes] and star 1 IA,0[flashes].
    2. Star -1 FB,0[flashes] and star 1 PB,0[flashes].
    3. Star -1 PC,0[flashes] and star 1 FC,0[flashes].

These are all true and grammatical. The following could be true of two twins Ben and Josh traveling far apart and at high relative velocities:

    1. Ben’s twenty-fourth birthday PBen,25[happens] and Josh’s twenty-fourth birthday FBen,25[happens].
    2. Ben’s twenty-fourth birthday FJosh,25[happens] and Josh’s twenty-fourth birthday PJosh,25[happens].

Adding a suitably situated observer A, we get:

    1. Ben’s twenty-fourth birthday IA,0[happens] and Josh’s twenty-fourth birthday IA,0[happens].

Many natural expressions involve iteration of tense operators. For example, take the following expression, preceded by its “Newtonian translation”:

    1. By the time the supernova Chomsky3843 occurred, I had turned 24.
    2. By the time the supernova Chomsky3843 PBen,25[occurs], I PPBen,25[turn] 24.

We also have Josh’s point of view:

    1. At the time the supernova Chomsky3843 occurred, I was to turn 24.
    2. At the time the supernova Chomsky3843 PJosh,25[occurs], I PFJosh,25[turn] 24.

The supernova Whorf9548, on the other hand, has yet to occur:

    1. By the time the supernova Whorf9548 occurs, I will have turned 25.
    2. By the time the supernova Whorf9548 FBen,24[occurs], I FPBen,24[turn] 25.

From Josh’s point of view:

    1. At the time the supernova Whorf9548 occurs, I will be yet to turn 25.
    2. At the time the supernova Whorf9548 FJosh,24[occurs], I FFJosh,24[turn] 25.

An important aspect of this system is that any statement can be translated from the perspective of one observer to that of another. We’ve already seen this occur a number of times. In short, a fully decorated syntax tree provides enough information to situate each event in someone‘s timeline. When we change observers, all such events are rearranged through the Lorentz transformation. As reference frame changes, some Ps will change to Fs and vise versa.

The idiosyncracies of translation between observers

Though we’ve seen how sentences can be translated between observers, strange things can happen with prepositions. Notice, for example, that in sentence (7) the English distinction between “by the time” and “at the time” fails to remain meaningful after the passage to relativistic language. More accurately, though I’ve preserved this distinction to assist in generating intuitions, the translation of a sentence from one reference frame to another leaves prepositions untouched (only interchanging Ps and Fs), and consequently the sentence’s prepositions, which are preserved, might become somewhat strange. In short, coherence of prepositions is not invariant upon change of observer.

Indeed, consider the following transmission from observer A to observer B. I’ve replicated its Newtonian translation together with its translations into A’s and B’s frames of reference:

    1. After the supernova Chomsky3843 occurs, I will begin repairs on the ship.
    2. After the supernova Chomsky3843 FA,0[occurs], I FA,0[begin] repairs on the ship.
    3. After the supernova Chomsky3843 PB,0[occurs], I FB,0[begin] repairs on the ship.

All three of these sentences are both grammatical and true, given a suitable space-time situation—say, that in which B is traveling rapidly to A’s right and the supernova occurs far to A’s right. Yet the translation of sentence 8. C. into Newtonian language reads:

    1. *After the supernova Chomsky3843 occurred, I will begin repairs on the ship.

This is an ungrammatical—or at least bizarre—use of the preposition after. Many examples of such strange behavior can be constructed.

Interestingly, however, I’m forced consider this phenomenon a feature rather than a bug. After all, while it’s surely useful for B to indicate that the supernova has occurred in his past and that the repairs will begin in his future, it may also be useful for B to know—and this information is furnished by the sentence’s prepositions—that A “is waiting” for the latter to occur (in his own time, of course) “before” he begins repairs. Thus though the supernova may have happened (for B), the repairs may be yet to begin.

Of course, if B desires the functioning of A’s ship, he might be interested—rather than in the time on his own clock simultaneous with the supernova—instead in the time on his own clock simultaneous with that time on A’s clock which itself is simultaneous with the supernova. (Here we see the iterated or “bouncing” use of the Lorentz transform.) In this case, B should decorate the syntax tree by casting the introductory prepositional phrase in A’s reference frame, while casting the verbs, as before, in his own. (Here we see our first example of a “non-trivial” decoration of the tree.) If B did this, then both verbs would recover their future tense (although the distance of these events into the future would likely change).

Aspect and mood

The aspect of a verb refers to its internal time structure. Some verbs—all of the verbs we’ve used so far, in fact—occur instantaneously. Other verbs, like know, run, cross the street, and recognize [4], occur over time.

The literature on aspect is vast and fascinating, and many surprising phenomena occur. I have no hope of mastering the topic, and I can only plead that the reader consult section 4 of [4]. My only point, here, is that the system we’ve created adapts fairly easily to the use of verbs with more complicated aspects.

While the difference between, say, flash and know resides within the words themselves—these verbs feature different lexical aspects—many languages permit the modification of a given verb’s aspect using free or inflectional morphemes. For example, in English, the difference between I walked and I was walking is one of aspect; though both occur in the past, one is presumably instant while the other occurs over time.

Confusingly, many languages regularly permit the modification of aspect morphologically, and some bundle aspectual and temporal modifications into a single inflectional system. For example, in the Romance languages, the imperfect “tense”—which is inflected morphologically—is actually a combination of the past tense and the imperfective aspect; these cannot be naturally separated. Perhaps better is the treatment given Slavic languages such as Russian, which provides, for each concept, a parallel pair of (infinitive) verbs—each of which can be independently inflected and conjugated—which handle the imperfective and perfective aspects, respectively. Interestingly, it appears that the languages most apt to refrain from conflating tense, aspect, and mood (see below) are analytic and creole languages (see [5]).

In any case, let’s explore how aspect might function under our system. As usual, we modify the verb (in this case, to be washing) appropriately, but keep it in the present tense. Implicit is that the action indicated by a verb in imperfective aspect spans a block of time, and that this block too can be given relative meaning via the Lorentz transform. I’ll provide a Newtonian example and then two translations. (Here B is moving quickly to A’s right, and the supernova occurs far to A’s left.)

    1. When the supernova Chomsky3843 occurred, I was washing the dishes.
    2. When the supernova Chomsky3843 PA,0[occurs], I PA,0[am washing] the dishes.
    3. When the supernova Chomsky3843 FB,0[occurs], I PB,0[am washing] the dishes.

Of course, translating back to English gives the ungrammatical

    1. *When the supernova Chomsky3843 occurs, I was washing the dishes.

By now, this is something to be expected.

English, quite unusually, distinguishes between past and present perfect, via the have + past participle construction, as in I have done. If we believe that this indeed represents a separate aspect, we could implement this construction in relativistic language through the present-tense verb “have done”. In particular, our relativistic language distinguishes between (I) PM,t[have done] and PPM,t[do], though both appear in English as had done. The details here are subtle, and I leave them to the reader.

Mood, in language, is another fascinating phenomenon in the spirit of aspect. The mood of a verb indicates additional information concerning its “degree of necessity, obligation, probability, ability” [5]. Familiar examples include the imperative, conditional, and subjunctive moods. Pirahã, famously, features a mood describing “evidentiality (whether evidence exists for the statement, and if so what kind)” [5].

Like aspect, mood is inflected morphologically in many languages, though it is distinct from both aspect and tense. Also like aspect, it can be handled naturally by this system. I include just one example, using the subjunctive and conditional, in order:

    1. If my telescope had been working, I would have seen the supernova Chomsky3843.
    2. If my telescope PA,0[has been working], I PA,0[would see] the supernova Chomsky3843.

In particular, this system allows for the expression of the future conditional, which fails to exist as a distinct form in all of the languages I’ve studied.

Things become slightly tricky for the imperative tense in English, which is largely indistinguishable from indicative at the written level. On the other hand, the mood is conveyed—even in written language—using pragmatic clues. I don’t see why the same couldn’t continue in the relativistic situation.

Though tense, aspect, and mood are to at least some degree conflated in almost all languages, the language we propose here completely separates, at least, tense from the other two. This facilitates a degree of both flexibility and conceptual clarity not present in standard English or in other languages. In particular, this relativistic language permits many heretofore-unavailable combinations of tense, mood, and aspect.

Speaking this language

Though this discussion has proceeded largely with an eye only toward written language, it’s only natural to provide for this language’s spoken communication. (Presumably, all communication—whether written or verbal—will be transmitted by radio wave at light speed.) In any case, we’ll propose an approach. We suppose throughout that the reference frames assigned to the nodes in any given clause are identical to that assigned to the clause’s finite verb.

We might speak a verb by stating its model (observer), followed by its proper time, followed by its operators, followed by the present form of the verb. Thus for example, we have:

    1. Star -1 will flash and star 1 flashed.
    2. Star -1 FB,0[flashes] and star 1 PB,0[flashes].
    3. “Star -1 B-0-F-flashes and star 1 B-0-P-flashes.”

Sentence (c) is clunky, but it sure beats the ambiguity of (a).

A remark on pathologies in quantification

Though I have yet thus far to forbid any of the constructions of standard English—even prepositions of time, which are ill-behaved upon change of observer, are permitted—the pathologies we will soon encounter might give us pause.

In short, consider what could go wrong in sentences featuring quantification, such as

    1. Every star in that galaxy flashed before my twenty-fourth birthday.

As has been observed, for instance in [6, p. 3], the subject noun phrase “every star in that galaxy” is actually not a noun phrase in the simple sense in which is, say, “the brightest star in that galaxy”. Indeed, the noun phrase “every star” is represented, at the level of logical form, as a quantification over many objects, and the verb “flashed” applies to each of them in turn. Roughly, we have:

    1. x[(star in that galaxy)(x) → (flashed before my twenty-fourth birthday)(x)]

Of course, there’s no single time—in the speaker’s proper time, or in anyone else’s—at which the “event” every star flashed could be said to occur, and the sentence cannot be translated using our previous techniques. If we placed the times of each of the individual stars’ flashes into another observer’s timeline, it could well happen that some occur before the event of the birthday while some occur after. This would be very difficult to translate—into any language (even one which introduced relativity at the level of logical form).

Thus, sentences with quantification should perhaps be permitted—they could be valuable in certain situations—with, however, the qualification that they cannot be translated into the viewpoints of other observers.

Related pathologies could arise in the presence of scope ambiguities. I will not explore this matter here. Perhaps this suggests, though, that the syntax trees observers use to communicate should be enriched also with information resolving the possible semantic ambiguities associated with, say, scope and binding.

Remarks on the Sapir-Whorf hypothesis

The well-known Sapir-Whorf hypothesis, named after Edward Sapir and Benjamin Lee Whorf, proposes, in short, that differences across languages lead to differences in the workings of the minds of the respective speakers of those languages. The superficial similarity between relativistic linguistics and linguistic relativity, though a pun, is an “incestuous” one: Whorf coined the term linguistic relativity intending direct analogy with Einstein’s corresponding “physical relativity”, whose implications he believed were similar [7].

Famously, Sapir and Whorf studied the American Indian language Hopi, and argued that native speakers of Hopi conceptualize time differently than do speakers of “Standard Average European” languages. The debate over whether Hopi actually represents an example of linguistic relativity has raged on [8]. Many linguistics have successfully demonstrated language-based differences in color perception [7].

In the face of these discussions, my (highly uneducated) impression has been one of skepticism of linguistic relativity. I cannot speak to the issue of Hopi. As for color perception, my feeling is that even if the experiments seeking to demonstrate language-based differences in color perception were successful, the linguistic-relativistic effect so validated would not be sufficiently significant to justify the claims of Sapir and Whorf. Furthermore, the universality subsequently introduced by Chomsky is appealing from a formal standpoint, and from that of one who prefers to imagine that thoughts can occur without being couched in language.

In other words, before “believing in” the Sapir-Whorf hypothesis, I would like to be shown evidence of what I took to be a language’s role in producing significant, material differences in the way that its speakers thought. It is this that I have yet to see.

What could such a material difference look like? It’s hard to imagine a way of thinking highly different from our own. A plausible case study, though, appears in the form of the physical relativity we saw above. This, one is ready to believe, represents a truly different way of thinking about the world. This project, indeed, was undertaken with the following guiding idea: If ever there was an example of linguistic relativity, this would be one. Show me, in other words, a natural language whose handling of, say, time is as different from English’s as is that of the language we propose above. Then I’ll accept the Sapir-Whorf hypothesis.

For now, though, the operation of the human mind seems—differing languages notwithstanding—largely universal. Indeed, the Sapir-Whorf hypothesis is appealing partly because it conjures the intriguing prospect of encountering a way of thinking different from that which we’re used to. The study of foreign languages is made all the more enthralling when one faces, aside from their interesting formal properties, the prospect of encountering an entirely novel way of thought. The speakers of foreign languages are similarly made more enthralling by the same prospect.

Alas, until we encounter relativistic linguistics, the hypothesis of linguistic relativity might be just too much to ask for.


  1. Arthur Prior. Past, Present, and Future.
  2. Einstein Online: The Definition of “Now”.
  3. Sachs, Wu. General Relativity for Mathematicians.
  4. Stanford Encyclopedia of Philosophy. Tense and Aspect.
  5. Wikipedia. Tense-aspect-mood.
  6. Barbara Partee. Opacity and Scope.
  7. Wikipedia, Benjamin Lee Whorf.
  8. Wikipedia. Hopi Time Controversy.

3 comments on “Relativistic Linguistics

  1. Ben says:

    We could call this language “Relativistic English”, along the vein of Standard English, Southern English, etc…

  2. Richard says:

    I must say, I enjoyed this one quite a bit. The central issue, generally speaking, is something like this: If simultaneity, pastness, futurity, and other temporal notions are relativised to observers, so that different observers may have different pasts, presents, futures etc, then we face a certain problem of communication among different observers. This problem of communication is not that certain meanings cannot be communicated, or even that they cannot be communicated in English, but rather that certain meanings given in your ‘Relativistic English’, if communicated using an intuitive translation into normal English, result in ungrammatical sentences. 8. C. brings this out nicely. The decision to use Prior’s operators on verbs only was clever. It allows us to focus tense-logical analysis on the parts of the sentence that actually are tensed.

    Your skepticism about Sapir-Whorf seems justified for all but the most anodyne versions of that hypothesis. Though it certainly was worth noting that grammaticality isn’t preserved by this intuitive translation that maps meanings from one observer to another via a relativistic language which modifies the tense of verbs relative to observers, we shouldn’t confuse this with the idea that relativistic meanings are not expressible in ordinary English, or thinkable by native English speakers.

    I had been toying with the idea that your sentence 8. C. doesn’t really express anything especially relativistic though. The problem with its translation is that it yields a sentence roughly of the ungrammatical form:

    (1) *After S occurred, R will occur

    We can grasp the meaning of this, though, in some sense. First, ‘after’ seems a bit odd. Like ‘until’ and ‘since’, it’s a binary tense operator (so we’ll have constructions of the form After(S,R)), but other than this there seems some room for conflict with our intuitions. In the sentence “After x occurred, y occurred”, the meaning appears to be:

    (i) P(x occurs & F(y occurs)) & P(y occurs) .

    But, contrasting this with the sentence: “After x occurs y occurs” appears to be ambiguous between:

    (ii) [If x occurs, then y occurs]

    And something like

    (iii) [Always(If x occurs, y occurs)]

    Where ‘always P’ is defined as: [H(P) & G(P) & P]

    The ungrammaticality of 8. C. could be connected to the fact that the sentence *“After x occurred, y will occur” expresses these meanings:

    (iv) P(x occurs) & F(y occurs)

    (v) P(x occurs & F(y occurs)) & F(y occurs)

    Both of these are in fact logically equivalent. But the most ‘natural’ reading of (iv) in English is ‘x occurred and y will occur’, which of course implies (tacitly) that y occurs after x. But the ‘after’ here is redundant, whereas it is not redundant in ‘after x occurred, y occurred’, formulated as in (i). For notice that (i) is not equivalent with

    (vi) P(x occurs) & F(y occurs)

    Whereas, again, (v) is equivalent with (iv). It could be that the ‘after’ in the sentence (1), above, is logically redundant in a way that it is not in the translation of the grammatical sentence

    (2) “After x occurred, y occurred”

    Mutatis mutandis, similar points apply to show the non-redundancy of ‘after’ in the sentence

    (3) “After x will occur, y will occur”

    So supposing this feature of ‘after’ were somehow relevant to the ungrammaticality of sentences like your 8. C. Can we find a sentence with the same content that doesn’t use ‘after’ but is also ungrammatical? How about:

    (4) * “x occurred, and then y will occur”

    (possible instance: *“the supernova occurred, and then I will repair the ship”)
    Once again, there is no semantic problem here, but again the attempt to use certain tense-logical expressions which relate events past and future seems problematic. It seems that when these expressions (‘After’ ‘and then’ etc.) are used to relate events past to subsequent events past, there is no problem, ditto for events future to subsequent events future. But for events past and subsequent events future the problem arises. When the path from the past event to the subsequent one passes through the present moment (of the observer), there is a problem. Compare also relational tense-logical operators which relate events conversely, e.g. ‘before’. Of the following sentences only (7) is ungrammatical:

    (5) x occurred before y occurred

    (6) x will occur before y will occur

    (7) * x occurred before y will occur (possible instance: *“the supernova occurred before I will begin repairs”)

    This problematic feature is exactly what the relativity of simultaneity imposes upon us: the ‘movement’ of the present moment so that it occupies a temporal position between the events related by these relational tense-logical expressions. But the coherence of these meanings (these event orderings) is hardly dependent upon relativity. It’s just that relativity clearly prompts their consideration.

    For example, if we were looking at a timeline of the events surrounding the birth of Isaac Newton, I might point to two points – one event, x, before the birth of Sir Isaac and one event, y, after it – and say *‘After x occurred, y will occur’. What’s happening here, perhaps, is that I am infelicitously speaking from two contexts: one in which I am the omniscient, top-down observer viewing the timeline as something finished (‘x occurred’) and one in which I am the situated observer, adopting the perspective of Newton’s time (‘y will occur’). The infelicitous remark (*‘After x occurred, y will occur’) is the result of trying to relate in a single context (what we might call a ‘relativistic context’) events which are tensed with respect to two different contexts (Newton’s, and mine the omniscient observer of the timeline).

    No commitment to relativistic physics is required to prompt this analysis, though the physics certainly implies that relativistic contexts may exist in a physical (as opposed to merely logical) way. Though, again, I think you’re right that it is significant that this shifting of present moments (this merging of contexts/perspectives) allows constructions which, though perfectly coherent semantically, are ungrammatical when translated in the most natural way. This could lend some support to the view that grammaticality is not only insufficient for semantic coherence, or sense (as Chomsky argued), but that it is also unnecessary for it. Grammar and meaning are pried apart even more. It may still be that a coherent sense must be expressible in some grammatical sentence, sure, but the construction of such a sentence might not follow in the most uniform/natural/expected/systematic way given how such sentences are constructed for distinct meanings which are structurally similar.

    This is all very interesting. Good choice of topic!

    • Ben says:

      Richard, good comment. I’ll get to the content of the thing in a second, but first, a few details in the interest of nitpicking. Before we speak of things like “a timeline of the events surrounding the birth of Isaac Newton” or an “omniscient, top-down observer viewing the timeline as something finished”, we should make clear that only a preliminary (and arbitrary) specification of observer can make such talk possible/meaningful/unambiguous in the first place. We may of course choose, as our observer, Newton himself; in any case, a choice must be made. There is no “preferred” or “objective” choice, and that of Newton himself is then not “objective” so much as it is, say, “distinguished” for our purposes. From a physical point of view, again, this choice is no more objective than any other, and our talk of precedence and simultaneity should keep this fact in mind.

      Once we’ve fixed Newton as our observer, a proper (dare I say Newtonian?) timeline does emerge, and the points you’ve made carry through without change. What you point out, after all (and I’ll say more about this later) is that the sorts of linguistic problems we run into here are grounded not so much in relativity as they are in the infelicitous treatment of the sorts of timelines we work with on a daily basis in Earth’s inertial reference frame. Again, more about this later.

      More importantly perhaps, I’d take issue with your statement “This problem of communication is not that certain meanings cannot be communicated, or even that they cannot be communicated in English, but rather that certain meanings given in your ‘Relativistic English’, if communicated using an intuitive translation into normal English, result in ungrammatical sentences.” We’ve talked about this, and the matter might seem inconsequential, but I think it’s worth discussing. For whatever it’s worth, I feel that these meanings can not be communicated in English — at least not, that is, without supplying the relevant details “artificially” and using many more words than we otherwise would have liked to. For example, the sentence “The supernova occurred before Ben’s 24th birthday” is, I contend, somewhere between unambiguous and just plain meaningless in standard English. The proposition that this sentence’s speaker intends to express fails to come through. We could have supplemented the sentence by saying something like “Interpreting all events in Ben’s timeline, the supernova occurred before Ben’s 24th birthday.” Here the sentence attains meaning, but given the bulk of the unwieldy additional language we have to use, I don’t know if it’s fair to argue that these meanings can “be communicated in English”. In fact, it seems that you might have been arguing that even without the artificial supplement the meanings can be conveyed in English. With this I’d certainly disagree. I think the plain English fails to convey the meanings we desire. (You’ve hinted in conversation that this might lead into some thorny philosophical consequences. We can try to deal with these as they come. Maybe things won’t turn out so badly.)

      Furthermore, I don’t know if I’d agree with you that the essence of the problem lies in the ungrammaticality of the sentences’ “re-translations” into standard English. The essence of the problem as I see it is, as you may have gathered above, that the standard English sentences fail to convey the desired meanings to begin with, and that a relativistic enrichment is necessary to make communication possible at all. Again, I disagree that the sentences convey meaning before the relativistic step. As I understood things, the ungrammaticality of the re-translated sentences was little more than an amusing artifact of the comparative non-robustness of standard English in the face of relativistic change-of-observer.

      You’re right to point out, on the other hand, that the ungrammaticality of the re-translated sentences does shed important light on the (as it turns out) somewhat bizarre grammatical requirements of the English words “after” and “before”. I found interesting your observation that sentences using these prepositions are grammatical when both arguments occur before or after the time of speech, but not when one occurs before and the other occurs after. It’s also worth recalling your observation that in the latter case, because the events’ antecedence and subsequence to the speaker’s time are conveyed anyway through tense, the precedence which would have been indicated by an “after” or “before” can be deduced a fortiori through a sort of transitive property of precedence, and thus that the preposition is actually redundant. This may go some way towards explaining its ungrammaticality — English perhaps has, speaking anthropomorphically, taken steps to forbid the use of the prepositions “after” and “before” when the precedence relation that they’d have indicated can be deduced from tense alone.

      I’m not sure how to make sense of your treatment of this phenomenon from the viewpoint of infelicitously trying to merge two “temporal viewpoints” into a single sentence. Though that could have been one way to arrive at the incorrect sentence we landed on, it seems that that’s not actually how we got there (we got there through relativistic change of observer). Perhaps what you’ve done is give an explanation for why it sounds strange to us, or why we instinctively reject it. Again, though, I don’t think that this alone can explain its ungrammaticality. Such a thing could perfectly well have been spoken by Newton himself — that is, without any improper merging of temporal viewpoints — were it not for the incidental prohibition of such constructions in English, which is of course what we set out to explain in the first place.

      I should add one final thing. While in the case of the sentence *”After S occurred, R will occur” we’re led to wonder why a sentence like this, which seems in fact so relatively harmless upon deeper inspection, should be forbidden in English, there indeed are other sentences which transform in less benign ways upon change of observer. For example, take the sentence lower down: *”When the supernova Chomsky3843 occurs, I was washing the dishes.” Here, the semantics of “when” — which, presumably, require its two arguments to coincide or overlap in some way — are totally violated in the ungrammatical re-translated sentence, and furthermore in a way which cannot be explained merely by the existence of actually-somewhat-arbitrary rules of English use. To be more explicit, we could have come up with a sentence using “after” that, after change of observer and re-translation, results in an English sentence in which precedence is completely disrupted altogether. For example, given an appropriate space-time setup, the sentence “After the supernova occurred, I celebrated my twenty-fourth birthday” could become “After the supernova occurs, I celebrated my twenty-fourth birthday”. Here, the preposition “after” is not made redundant by tense and the transitive property, but actually contradicted by them. Furthermore, it was a matter of luck that the sentence above resulted in the former and not the latter. Similar outrages could be constructed using words like “since” and “until”.

      The character of re-translated sentences’ ungrammaticality cannot be captured exhaustively by the template you’ve identified and explored, in which the speaker’s time is placed between two events when previously it occurred before both or after both. While this special case certainly leads to interesting observations about how English works, the trappings of relativity are more varied.

      All of this serves to emphasize a broader point I’d like to make, though, which is that it’s not these flimsy ungrammatical English sentences I’m most interested in, but rather their relativistic counterparts. Those are the primary objects of this study — and they, by construction, resist the ravages of change of observer.

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