Semantic Values and Rabbit Pictures
Robbie Williams pointed out that in my recent musings on worms and stages, I ignore the following straightforward characterizations:
Worm Theory: the semantic value of predicates like "rabbit" is a set of 4D worms.
Stage Theory: the semantic value of predicates like "rabbit" is a set of 3D stages.
He's right. I believe that these theories both cannot work, so I don't want to define stage and worm theory that way.
Why can't they work? Consider Ralph the rabbit. On the simple Stage semantics (on which I'll focus here), the semantic value of "Ralph" is a rabbit stage, and the value of "rabbit" a set of such stages:
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The main problem with this is that different things can share the same stage. In other words, there is no unique correct way to find the past and future counterparts for a given stage. Given unrestricted mereological composition, there exist things like the fusion of today's Ralph stages, and the fusion of today's Ralph stages with yesterday's stages of the Eiffel tower. For any fusion, we could have a corresponding predicate, say, "rabbit-segment" and "tower-rabbit". And we could have names for rabbit-segments and tower-rabbits, say, "Segralph" and "Towralph". Then here are some facts about Ralph, Segralph and Towralph:
| existed yesterday | was once 324m high | is a rabbit | is a rabbit- segment | |
| Ralph | true | false | true | false |
|---|---|---|---|---|
| Segralph | false | false | false? | true |
| Towralph | true | true | false | false |
However, Ralph, Segralph and Towralph all share the same current stage. Hence if the contribution "Ralph", "Segralph" and "Towralph" made to the truth-value of sentences was just that stage, there could never be different truth values in these columns. But there are. So the semantic value of "Ralph" (and "Segralph" and "Towralph") is more than just a stage.
In response, one could maintain that there really is a unique correct way to find the past and future counterparts for a given stage. Perhaps, one might suggest, Ralph, Segralph and Towralph do not really share a stage. Rather, there are many different stages colocated at the same place and time, differing in their hidden haecceity. This haecceity uniquely determines their past and future counterparts. On this view, there are probably infinitely many objects with different haecceities wholly located at any place and time,
Alternatively, one could say that Segralph and Towralph simply do not exist, or that it is for some other reason impossible for a language to have expressions that work like "Segralph", "rabbit-segment", etc. That's also bad, for several reasons. First, such expressions exist in English: I have introduces these new words simply as abbreviations for slightly longer English expressions. Second, it is very implausible that our way of carving reality should be the only possible way. Why couldn't people in other cultures be interested in rabbit-segments or tower-rabbits? (The examples were of course particularly gruesome; there are many less gruesome ways to draw temporal boundaries around Ralph's present stage; and much more so for artefact stages.) Thirdly, the very same problems arise with our familiar "statue" and "lump of clay" (and "person"/"body", etc.). A statue and a lump of clay can share the same present stage, even though "the statue existed yesterday" is true and "the lump of clay existed yesterday" is false. It will hardly do to say that either of these expressions cannot exist.
So many predicate values operate not simply on stages, but on stages together with their counterparts at other times. That is, we need to assign to names not only stages, but also a function from times to further stages (an intension), so that we can let the predicate values operate on them:
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I've used a mapping from intensions to truth values instead of a set of intensions as the semantic value of "rabbit" just to prevent people from interpreting me as saying that rabbits are strange functions from times to stages. Of course they are not. We have to reject the idea that rabbits are whatever is in the semantic value of "rabbit". We also have to reject the idea that Ralph is the semantic value of "Ralph" (that would also make him a function from times to stages). These are very important lessons, I believe.
The present semantics is still too simple because it can't distinguish fission from time travel. When things have double stages at a certain time, that can be either because they are time travelers or because they have fissioned. That makes a difference for predications: By time traveling, Ralph can kick his younger self, but not by undergoing fission. Whatever makes a difference for the truth-value of predications needs to be put into the semantic values:
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The '=' sign indicates that at t3, Ralph has time traveled, whereas at t2 he has merely fissioned.
Then of course there are index-shifting operators like "tomorrow", for which we have to add a time argument to our predicate values:
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Finally, all these semantic values systematically vary with the context of utterance. This holds in particular for the counterpart relation that provides the intensions (and also explains the difference between time travel and fission multiplicity). So we need two-dimensional intensions:
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World and place coordinates are still missing, but this is roughly how I think a stage theorist's semantics should look like. It is admittedly somewhat more complicated than it was on the initial characterization. But I don't see any way to get by with less complexity.
Once you've got all this complexity, you can see how the mapping into worm theory goes. At the bottom level, instead of sets of stages, we put fusions of stages. (There might be a few more bells and whistles to account for fissioning and time travel, but that's the basic idea, I take it.)
I find it a bit confusing that the functions from times to rabbit stages are called "intensions", since it looks like what plays the familiar intension role, e.g. interacting with time-shifting operators, is the function from times to [functions from times to rabbit stages]. But that may just be me...
I still think we should stick with the simple (naive?) characterization. That means that on stage semantics, segralph and towralph will be rabbits. That's a bullet, but I think it's biteable. (It's closely related to Achille Varzi's puzzle about tenors and turnips in the AJP 2003.). Or at least, I think it's better to bite the bullet than complicate stage theory in the way that you go on to suggest (which I think you're right in thinking is, in effect, a kind of worm theory).
We'll have to do something about unbiteable cases. e.g. ``was once 324 ft high'' (for stage theorist) or ``is essentially human'' (for the modal counterpart theorist). In the latter case, one familiar trick is to paraphrase the 'predicate' as a combination of operator and predicate "Nec (statue(x))", and then rely on the inconstancy of the salient counterpart relation to do the rest. We could do the same thing for the stage case: "Was: 324ft high(x)": the different temporal counterpart relations invoked by "TowRalph" and "Ralph" then give the difference in truth value.
Of course, if you insist (to pick another example) that "is 1 day old" is to be treated as a predicate, then paraphrase in terms of operator won't work (the difference between paraphrasing and giving a semantic interpretation becomes important here).
Even so, I think that it's better for the stage theorist to disambiguate e.g. "is one day old qua rabbit" from "is one day old qua rabbit-segment". Both Ralph and SegRalph satisfy the second but not the first. Maybe these are different predicates, or perhaps different properties (so that "is one day old" becomes indexical, depending on the salient counterpart relation).