Another weird discrete theory of time

Suppose time is discrete. The usual story then is that we are always at some point of time. But what if, instead, we are always between times? I.e., the times themselves are something like imaginary points—we don’t occupy them on their own. It is only the interval between two successive times that we occupy, and we occupy the interval as a whole. Such an interval is a “now”.

If t_n and t_n + 1 are successive times, then we say that at (t_n,t_n + 1) (think of this as an ordered pair or an interval—your choice of mathematical representation!):

• x is F iff x is F at t_n and at t_n + 1

• x is non-F iff x is non-F at t_n and at t_n + 1

• x exists iff x exists at t_n and at t_n + 1

• x non-exists iff x is does not exist at t_n or at t_n + 1

• x is changing from F to non-F iff x is F at t_n but not at t_n + 1

• x is changing from non-F to F iff x is non-F at t_n and at t_n + 1

• x is coming into existence iff x exists at t_n + 1 but not at t_n

• x is ceasing to exist iff x exists at t_n but not at t_n + 1

• x is coming to be F iff x is F at t_n + 1 and either does not exist at t_n or exists at t_n but is not F then

• x is ceasing to be F iff x is F at t_n and either does not exist at t_n + 1 or exists at t_n + 1 but is not F then.

Here is a plausible thesis:

1. x fails to exist or x is F or x is non-F.

On the theory we are exploring, this is false in a now. Instead:

2. x non-exists or is coming into existence or is ceasing to exist or is F or is non-F or is changing from F to non-F or is changing from non-F to F.

This theory is a variant of one I tried out in an earlier post, minus the possibility of the now being a point.

On two knowledge arguments

There is a structural similarity between the main reasons for adopting the A-theory of time and the knowledge argument against physicalism. In both cases, it is claimed that there is some information that we know but which is left out of the reductive theory:

1. I know that Alice is sitting now

and

2. I know what it feels like to be sitting.

The first piece of knowledge cannot be derived from data about tenseless reality and the second cannot be derived from data about physical reality, or so it is claimed.

The similarity between the two arguments suggests that there should be a correlation between dualism and adherence to the A-theory of time: for if one is convinced by one argument, one is more likely to be convinced by the other, and if one is unconvinced by one, one is less likely to be convinced by the other. Speaking for myself, I am a B-theorist dualist, and while I am unconvinced by the time argument, I go back and forth on the mind one.

It is interesting, though, to see if we can go beyond superficial similarity. One way to do that is to see if the best responses to one of the arguments can generate plausible responses to the other.

The best response to the time argument seems to be the Kaplan story that “now” is a mere indexical, and that the content of “Alice is sitting now” is the proposition that Alice sitting is at t₁ (if t₁ is now), though the character or linguistic meaning of “Alice is sitting now” is something different from the character of “Alice is sitting at t₁” (specifically, a character is a function from world-utterance pairs to propositions, and this character assigns to an utterance of “Alice is sitting now” at t in w the proposition that Alice is sitting at t).

Is there a similar story about mind argument? It’s not so clear to me. Perhaps a start would be to say that what makes me it true that I know what it feels like to be sitting is that:

3. I know that sitting feels like this.

The physicalist analogue to the Kaplan story would then be that “Sitting feels like this” expresses the proposition that sitting feels like ϕ where ϕ is some physical state of affairs, but the character or linguistic meaning of “Sitting feels like this” and “Sitting feels like ϕ” are different. I don’t think this works, however. There are two ways of taking this approach:

1. ϕ is a specific neural state that I have when I feel like I’m sitting (say, S-fibers firing)

or

2. ϕ is a complex functional state that anything has when it feels like it’s sitting, a state implemented by different neural or other physical states in different beings.

On (a), we have an analogy to the time case, for we can take the character of “Sitting feels like this” to be a function that assigns to world-utterance pair the proposition that sitting feels like ϕ where ϕ is the physical state that is the feeling for the utterer in that world. But there is also a serious disanalogy: for in the time case, the B-theorist knows (or can claim to know) the character, since the B-theorist knows a priori the specific rule by which a referent is assigned to “now” at a world-utterance pair. But the physicalist does not know a priori the specific functional story which assigns a referent to “like this” at a world-utterance pair.

On (b), we have a disanalogy, since the character is constant: at every world-utterance pair, the same proposition is assigned as the content of “Sitting feels like this.”

Still, maybe there is still a fundamental analogy, in that the time case teaches us (if we accept the Kaplan story) that one proposition can be expressed by two sentences s₁ and s₂ such that it is correct to say “I know s₁” but not correct to say “I know s₂”. Thus, I know that I am sitting now but I don’t know that I am sitting at t₁. And similarly, maybe, I know that sitting feels like this but I don’t know that sitting feels like ϕ.

What if we go the other way around, and see if the best answer to the mind argument helps with the time argument?

I guess what is generally thought to be the best answer to the mind argument is something like this: there is a conceptual difference between the “like this” of the feeling and the physical or functional state ϕ, but ontologically they are the same. And this seems very close to Michelle Beer’s defense of the B-theory.

Another prominent answer to the mind argument is to deny that the knowledge claim expresses factual knowledge, as opposed to something like know-how or imaginative mirroring. It seems to me that a know-how story could be told about the time argument: to know that Alice is sitting now is to have certain kinds of know-how concerning dealing with Alice’s sitting. The “imaginative mirroring” case might be harder.

Fun with "now"

I am now writing this and you are now reading this.

Antipresentism

Presentists think that the past and future are unreal but the present is real. I was going to do a tongue-in-cheek post about an opposed view where we have the past and future but no present. But as I thought about it, the position grew a little on me philosophically, at some expense of the tongueincheekness. Still, please take all I say below in good fun. If you get a plausible philosophical view out of it, that's great, but it's really just an exercise in philosophical imagination.

One way to think about antipresentism is to imagine the eternalist's four-dimensional universe, but then to remove one slice from it. Thus, we might have 1:59 pm and 2:01 pm, but no 2:00 pm. Put that way, the view isn't particularly attractive. Still, I do wonder why it would be more unattractive to remove just one time slice than to remove everything but that one time slice as the presentist does. It would, of course, be weird for the antipresentist to say that events first exist in the future, then pop out of existence just as one would have thought that they would come to be present, and then pop back into existence in the past. But perhaps no weirder than events coming out of nothing and going back into nothing, as on presentism. This way to think about antipresentism makes it a species of the A-theory.

But the antipresentisms I want to think about are ones that might be compatible with the B-theory. Start with the famous puzzles of Zeno and Augustine about the now. Augustine worried about the infinite thinness of the now. Zeno on the other hand worried about the fact that there are no processes in the now; there is no change in the now since within a single moment all is still.

One way of taking these ideas seriously is to see the present as an imaginary dividing line between the past and the future. There is in fact no dividing line: there is just the past and the future. (I think Joseph Diekemper's work inspired this thought.)

We might, for instance, instead of thinking of times as instants think of the basic entities as temporally extended events or time intervals, not made out of instantaneous events or moments. An event or interval might be past, or it might be future, or—like the writing of this post—it might be both past and future. (Thus, "past" and "future" is taken weakly: "at least partly past" and "at least partly future".) Some events or time intervals have the special property of being both past and future. We can stipulate that those events or time intervals are present. But they aren't real because they are present. They're just lucky enough to have two holds on reality: they are past and they are present. (In this framework, the presentist's claim that only present events are real sounds very strange. For why should reality require both pastness and futurity—why wouldn't one be enough?) There are no events or time intervals that are solely present.

There is a natural weakly-earlier-than relation e on events. If we had instants of time, we would say that EeF if and only if some time at which E happens is earlier than some time at which F happens. But that's just to aid intuition. Because there are no instantaneous events, every event is weakly earlier than itself: e is reflexive. It is not transitive, however. The antipresentist theory I am sketching takes e to be primitive. There is also a symmetric temporal overlap relation o that can be defined in terms of e: EoF if and only if EeF and FeE.

If we like, we can now introduce abstract times. Maybe we can say that an abstract time is a maximal pairwise overlapping set of time intervals (or of events, if we prefer). We can say that t₁ is earlier than t₂ provided that some element of t₁ is strictly earlier than some element of t₂ (where E is strictly earlier than F provided EeF but not FeE). I haven't checked what formal properties this satisfies—I need to get ready for class now (!).

An affixal theory of some indexicals (Language, Part V)

This post continues from earlier reflections on indexicals, but in a slightly different way. I want to offer a very strange theory of indexicals like "I", "now" and "this". They are not words at all. They are affixes, like the "-ing" in "walking" and the "in-" in "indoor". An affix is added to a root, and thereby yields a word. The affix indicates how the sentence makes use of the concept indicated by the root. In highly inflected languages, affixes may play a significant role in indicating whether a given noun is, say, the subject, direct object, indirect object, etc. of the sentence.

We might more accurately think of affixes as functions from partial words to words, remembering that adding an affix may force changes in the "root" part. Moreover, in my sense, an affix might not actually be contiguous with a word. Thus, in Polish the personal endings of past-tense verbs are sometimes allowed to float free of the verb and attach to something else. Thus, one can say: "My w domu bylismy" (="We at home were"), where "bylismy" is "byli" (=past tense of "to be" with a plural "-i" suffix) plus "-smy", the first personal plural ending for past-tense verbs, but one can also move the "-smy" to be after the "My" (="We"): "Mysmy w domu byli." One way—maybe not the way most Slavic grammarians will do it—to read this is as a sentence whose words are "My" (="We"), "-smy ... byli" (="were"), "w" (="at") and "domu" (="home"). The "-smy ... byli" is a scattered word there, written non-contiguously.

"I", "now" and "this" are such non-contiguous affixes. But where is the partial word to which the affix is attached? In ordinary-speech cases of "I" and "now", the answer is easy. With "I", the root is the speaker, and with "now", the root is the time of speech. In other words, when I say "I am now at home", the first word in the sentence is a scattered word consisting of two parts: me (the six-foot-tall guy having a body and a soul) and "I". The third word is a scattered word consisting of two parts: the actual time of speech (whatever the ontology there is—I do not think we need to insist on words or parts of words being something we take ontologically seriously, so we probably don't need to worry whether there are times) and "now". "This" is harder. My inclination is to take it as an affix attached to an activity of pointing at something, perhaps including just the pointing gesture (mental or physical, perhaps contextual), but perhaps including that which is pointed at. I don't have a very good analysis of that.

Fictional speech is puzzling for this view. When the narrator uses "I", if that is the affix, where is the root? I don't know what to say. But fictional speech is anyway problematic. Perhaps I can say that a "sentence" said by a fictional narrator is not really a sentence but a fictitious sentence, just as a "murder" committed by a fictional narrator is not really a murder but a fictitious murder.

And what about cases of "I" and "now" that do not really refer? For instance, you tell me: "The other day, George told me: 'I am not at home.'" But George is your imaginary friend. Whom does the "I" refer to? Well, there is no "I" in your sentence. There is only an "'I'". What is between quotation marks is not a sentence, and not even a candidate for a sentence, because it is incomplete, in the way "George Bush was -ing down the street" is.

This is a really revisionary view of language. Does it have any advantages? I think it does. For one, it makes indexicals not stand out as some disparate category. They are just affixes to a root that is often Lagadonian. Moreover, I think this is the sort of view at which one may end up if one has a general enough view of the possibilities for language—dropping any insistence on a linear structure, for instance, allowing Lagadonian languages, etc.

[Edited to fix typo and remove an embarrassing slip.]

"Now" and other indexicals (Language, Part IV)

In English, the pitch and rate at which one speaks typically do not affect the types of the tokens that one is using. Whether you say "home" quickly or slowly, with a low or a high pitch, your utterance is a token of one and the same type. But this need not be a universal truth. One can easily imagine a language where an utterance of "home" means one thing when spoken quickly and another when spoken slowly. In that case, there are two word types here, and which type one's token falls under will be partly determined by the sequence of phonemes and partly by the rate at which one speaks. (We might represent the two word types in writing as "home" and "h-o-m-e" if we wish.)

A language provides a mechanism for classifying linguistic tokens under linguistic types. There are very few restrictions on how the classification scheme can work, except contingent ones derived from our recognitional abilities. Imperceptible differences between tokens will not do the job.

It is quite possible, then, to have a language where the classification system is time-dependent. Thus, "home" at odd-numbered hours of the day means a tank, and at even-numbered hours it means an F-16. There are, thus, two word-types with the same phonemes, and to distinguish between them you have to check what time it is. Such a language might well be useful for confusing an evesdropping enemy.[note 1]

Imagine now a language L1 where the sound "chow" when uttered at a time t denotes the time equal to t+7minutes, when t is during an even-numbered hour, and denotes the time equal to t-7minutes, when t is during an odd-numbered hour, and where the type of the word is identified by both the sound "chow" and the time of utterance. This language can be understood as containing continuum many word-types, identified partly by the time at which the word-type is tokened and partly by the phonemes. This is a very odd language, but a possible one.

Observe that in L1 no utterance of "chow" is the utterance of an indexical. What an indexical refers to depends on both the token's type and on the context of utterance. But what "chow" refers to does not depend on the context of utterance, but only on the token's type. The token's type depends on the time of utterance, but that is a different matter.

Consider now a language L2 where the sound "fow" when uttered at a time t denotes the time t, and where the type of the word is identified by both the sound "fow" and the time of utterance. Just as no utterance of "chow" in L1 was an indexical, so no utterance of "fow" in L2 is an indexical. Rather, it is the utterance of a fine, upstanding, context-free referring term.

But now a question: How do we know that utterances of "now" in English are utterances of an indexical? Why not analyze utterances of "now" in English precisely the way utterances of "fow" are to be analyzed in L2? There are, I submit, no facts of linguistic practice (normative or not) that allow us to distinguish between English's "now" and L2's "fow". If linguistic facts supervene on facts of linguistic practice, there is no fact of the matter whether an utterance of "now" should be read as an indexical whose type is identified by the phonemes or as a non-indexical whose type is partly identified by the phonemes and partly by the time of utterance.

If we understand "now" along the lines of "fow", then any argument for the A-Theory of time based on our use of "now" is likely going to fail. For "fow" is perfectly at home in the eternalist world of the B-Theory. And what I said about "now" goes for tenses as well.

This strategy is closely parallel to the old failed B-theoretic attempt to translate "now" into the time of utterance. That attempt failed because when one translated "It is now 11:56 am", it translated into "It is 11:56 am at 11:55 am", and hence a sentence that one could reasonably be wrong about got translated into one that no one could be reasonably wrong about, which is absurd. On the present strategy, an utterance of "now" at 11:56 am does refer to the 11:56 am, indeed is rather like a proper name for it. In a sense "It is now 11:56 am" may be a tautology, but it is not a trivial tautology. Rather, it is like "Cicero is Tully" or "London is Londres."

If all this is right, then no deep facts about language hang on the distinction between indexicals and non-indexicals. There may be more than one way of classifying bits of utterances into types, and for any way of classifying that makes a bit of utterance into a token of an indexical, there is a way of classifying that makes that bit of utterance into a token of a non-indexical identified in some non-phonemic way. Each classification should give rise to the same proposition as expressed by the utterance as a whole.