On a twist on too-many-thinkers arguments

One of the ways to clinch a too-many-thinkers argument (say, Merricks’ argument against perdurantism, or Olson’s argument for animalism) is to say that the view results in an odd sceptical worry: one doesn’t know which of the many thinkers one is. For instance, if both the animal and the person think, how can you know that you are the animal and not the person: it seems you should have credence 1/2 in each.

I like too-many-thinkers arguments. But I’ve been worried about this response to the sceptical clinching: When the animal and the person think words like “I am a person”, the word “I” refers to the person, even when used by the animal, and hence both think the truth. In other words, “I” means something like: the person colocated with the the thinker/speaker.

But I think I have a good response to this response. It would be a weird limitation on our language if it did not allow speaker or thinker self-reference. Even if in fact “I” means the person colocated with the the thinker/speaker, we should be able to stipulate another pronoun, “I*”, one that refers just to the thinker/speaker. And it would be absurd to think that one not be able to justifiably assert “I* am a person.”

Fun with "now"

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

"Thank goodness it's over!"

Some people think that the sentiment expressed by "Thank goodness it's over!" makes no sense apart from an A-theory of time on which there is an absolute present. But the parallel sentiment expressed by "Thank goodness this isn't happening to me!" surely had better make sense apart from some theory on which there is some "absolute I".

Propositions and the liar

This is an attempt to provide a metaphysical backing to hierarchical theories of truth like those of Tarski, thereby skirting the liar paradox. The details of the construction of propositions can be done in more than one way—the following is more an example of the structure of a theory than a theory.

Step 1: Assume an abundant theory of Armstrong-like first-order states of affairs, namely states of affairs that exist if and only if they obtain. (Intuitively, we will need enough states of affairs so that each first-order true proposition represents a state of affairs.) Now, say that the first-order propositions are ordered pairs (s,v) where s is a first-order state of affairs and v is 0 or 1. We can then define truth for first-order propositions very simply. For any first-order proposition (s,v), the proposition is true if and only if v=1. We can also define truth at a world w as follows: (s,v) is true at w if and only if either s obtains at w and v=1 or s does not obtain at w and v=0.

On the view I am defending, ordered pairs are mere abstractions—they are not first-class members of our ontology. That is a nominalist or perhaps Aristotelian moment in the story. Ordered pairs are not in the domain of ordinary objectual quantification. We need another kind of quantifier, ∃₁x, to handle quantification over first-order propositions. This is like the quantifiers in this post. (Actually, I think states of affairs aren't first-class members of our ontology either. They, too, will be some form of logical construction. This is part of why I am not happy with the details.)

Now, there is one technical issue here. Since what states of affairs exist differs between worlds, likewise what propositions there are differs between worlds. The solution here is to adopt a counterpart theory for first-order propositions. Given worlds w₁ and w₂, the proposition (s,v) has a counterpart in w₂. If s obtains at w₂, then the counterpart is just (s,v). If s does not obtain at w₂, then the counterpart of (s,v) is (~s,1−v), where ~s is the negation of s—a state of affairs that obtains if and only if s does not.

Step 2: We've defined a quantifier ∃₁x over first-order propositions and a truth predicate for them. We do not say that first-order propositions exist simpliciter. Facts about first-order propositions and their truth supervene on first-order facts about the world and are grounded in them. Now, we repeat the process. Define, abstractly, the notion of second-order states of affairs—these are states of affairs that are partly about first-order propositions. In so doing, we're introducing a new quantifier over the second-order states of affairs. And then define a second-order proposition as, again, a part (s,v) where s is a second-order state of affairs and v is 0 or 1. Do everything else as before, defining a quantifier ∃₂x over second-order propositions.

Steps 3 and so on: Repeat.

Imagine the process completed. What have we done? It is tempting to say:

1. We have defined nth order propositions for all n, and a truth predicate for each level.

And then it is tempting to ask:

2. But what, then, about truths about propositions of any order whatsoever, and a truth predicate for them?

But the temptation should be resisted. We should not say (1), since (1) quantifies in a way not allowed by the theory. We have the quantifiers ∃₁x, ∃₂x, .... We do not have a quantifier over these quantifiers. (Can't we do ∃n∃_nx? No: the subscript in "∃₁x" is just a marker that cannot be replaced by a variable.)

Likewise, there isn't a single truth predicate over all the levels. Rather, what we defined are analogical truth predicates. That's another Aristotelian moment in the theory.

For convenience, we can define cumulative quantifiers and cumulative truth predicates. Thus, we can define ∃₃*xFx as the disjunction: ∃₁xFx or ∃₂xFx or ∃₃xFx.

The liar disappears. Never in the process do we get to define a paradoxical sentence. The sentence of the form "The proposition expressed by this sentence is not true" needs a level of quantification. But when we fix the level of quantification, we get something like:

3. ∀_n*x (if this sentence expresses x, then x is not true)

But there is no nth or lower order proposition expressed by (3), and so (3) is trivially true.

We then need a thesis about the level of quantifiers in ordinary language. Perhaps the thesis is that we charitably raise the level of quantifiers in ordinary language sentences until we get something meaningful and relevant. Or perhaps we have quantifier-indexicality: which quantifier level is used in a sentence depends on the length of the chain of truth-nesting that goes on in the grounding of this sentence.

But what about the usual sorts of tricks for breaking out hierarchies, like taking an infinite set of sentences with different levels of quantifiers and then asserting that all the sentences in the set are true? We don't get to do that. For corresponding to the levels of quantifiers and propositions, there are levels of expression relations between sentences a propositions. When we try to say that all the sentences in a set are true, we mean something like: "the proposition expressed by each sentence in the set is true." But there is no univocal sense of "the proposition expressed" that holds for all the sentences.

But what if a clever person then goes on and defines infinite order propositions, with a new quantifier over them, just as I did it for all the finite orders. That game can, indeed, go on. But it does not generate a paradox. The paradox always appears to loom one step away, but when we get there, it's further off--it's like "tomorrow is always a day away".

"You"

My wife just got a wrong-number call from some woman: My wife asks whom her caller wants to speak with. The caller responds:

1. "You!"

The caller than says:

2. "The boss says the next time you come in, you are fired."

Whom did the caller's "you" refer to? If the extension-determining rule for "you" is "the speaker's interlocutor", then it referred to my wife. In that case, (2) was a false statement, and probably so was (1).
A more charitable reading would be that here we have an "attributive use" not of a definite description but of an indexical, and hence "you" refers to the person who is to be fired. This would have the paradoxical hypothetical consequence that had the caller added at the end of the conversation: "You didn't hear anything I just said", that would have been true—for "you" would seem to still be referring to the person to be fired, not to my wife. Maybe, though, the "you" would have shifted in reference by this point?
I think the story about attributive use of definite descriptions transfers to attributive use of indexicals. Suppose I am watching a soccer game on the tube. I falsely believe the game is being broadcast live, but there is in fact a ten minute delay. Suddenly upon seeing a pass I exclaim: "Now is the best moment in a soccer game that I have ever seen!" The "Now" does not, I think, refer to the time at which I am watching, but to the time at which the pass actually happened. Nothing surprising about attributive use of indexicals, because after all an indexical is essentially a degenerate case of a definition description. "The next shortest man after me" is a definite description even though it uses "me" and likewise "The entity identical with me" is a definite description. And we can just say "I", which is briefer.

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.

Communication boards and indexicals (Language, Part I)

This is the first in what may be a series of posts developing a view of language that erases the distinction between language and context. The view may self-destruct before it's fully developed. I'm having fun here. No originality is claimed.

Some disabled people communicate with a communication board. A communication board is a board with printed pictures, some representing objects like shoes and chairs, some representing verbs like sitting, and some representing emotions like happy or sad. One communicates with a communication board by pointing to a sequence of pictures. High-tech communication boards will say the word, but it's important to my argument that I be talking of a low-tech one, which is just a pre-printed board. If a communication board is sufficiently large and extensive, and there is sufficient syntactic structure in the order in which one points to the pictures, this will be a language.

Let's say that the pointing is done with a finger. Now, what are the words or other linguistic units in this language?

Here is a really bad suggestion: The words are constituted by pointings with a finger and two token pointings count as of the same type if and only if the finger directions in the two pointings have the same relationship to the natural axes of the speaker's body (within some measure of precision; I will use the word "speaker" regardless of whether a language is spoken or not). The pictures on the board, on this suggestion, are simply context.

What's wrong with this suggestion? Well, for one, it means that if the speaker points to parent, fruit and happy, expressing (let's suppose) the proposition that the parent is happy with the fruit, and I shift the board over by an inch, and the speaker again points to parent, fruit and happy, then the speaker has used different word types, because her pointings are now in different directions. That is absurd--surely the speaker has said the same sentence.

Another way to see the absurdity of this view is that it will be impossible to give a story about the syntax of the language in terms of the arrangement of word types, since whether a given sequence of finger pointings, identified by direction relative to the speaker's body, is syntactically correct depends crucially on what the pictures pointed to are, and not just on the angles (again, think of a case where the board gets shifted over). But we don't want context to be the primary determiner of syntax!

One might think that the mistake in this story is that it is not the angles relative to the speaker body that matter for identifying the word type, but rather the direction of the finger as measured in some natural coordinate system based on the configuration of the board. (E.g., run the x-axis along the horizontal side of the board, the y-axis along the vertical side, the z-axis upward from the board, and then specify the cartesian coordinates of the tip of the finger and the finger's big joint.) But that's silly, too. Suppose that the speaker's board gets upgraded by getting a few new pictures, and with existing pictures moved a bit to accommodate the new ones. The speaker's language, thus, becomes extended. But now if we identified word types in terms of the coordinates of the finger relative to the board, the same sequence of finger positions as before would now be expressing something completely different. More seriously, previously syntactically correct sequences of word types would no longer be syntactically correct. In other words, we have a completely new language. But that is surely a hamfisted way of describing what happened in the board upgrade. There is something that is obviously wrong with the previous two accounts. The crucial thing to note is that the pictures that are pointed out are not mere context. They are crucial for the syntax: whether a sequence of three pointings is syntactically correct depends precisely on what parts of speech the pictures represent. Clearly, the thing to do is to either identify word-types with the pictures that are pointed out (more precisely: picture-types, in order to allow for upgrades of the board), or with pointings-at-x, where x ranges over the pictures (or picture-types) on the board.

Hypothesis: What happens with the communication board is also what happens with demonstratives. The thing pointed to is not context: it either is a part of the sentence (much as some folks think that items referred to de re are parts of the proposition) or a pointing at (de re) it is a part of the sentence. And something like this happens with all indexicals. At this point I am offering no argument, except the suggestive analogy of the communication board language.

Apparent disanalogy: In the communication board language, it is not the the picture tokens that function as word types (or, equivalently, it is not the pointings-at-picture-tokens), but the types of pictures (or the pointings at a type of picture). But in true demonstratives, there is no similar type/token distinction on the side of the things pointed out. One simply points at Alexander, not at something of the sort of Alexander.

This disanalogy is due to the fact that in a typical communication board, none of the pictures refer to the picture-token there, and that in typical demonstratives, we are trying to refer to particulars. But I submit that these are mere accidents. We could imagine that some of the pictures indicate particulars, like George, Socrates, etc. And there would be nothing absurd about a picture that indicates the picture-token that it is. (Maybe it's a very a beautiful and emotionally significant picture, so it's worth talking about as an individual. When a board is upgraded, it gets scraped off the old board and pasted on the new one.)

Moreover, we do in fact have cases where demonstratives point to a type, it's just that we don't use them quite as much as ones where we point to particulars. We've learned this from Kripke. Point to water and say: "We will call this 'water'." The "this" refers to the natural kind, not the particular bunch of water.