THE QUIETIST'S GAMBIT.
| Autor | Mena, Ricardo |
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Introduction
It has been said that Chrysippus offered the following advice: if your chariot is heading towards a precipice, stop it well before it hits the edge. The next piece of advice has been also attributed to him: if you are confronted with a forced march sorites series, stop answering questions while you still know the answers. Both pieces of advice are worth considering seriously. (1)
The second piece of advice requires some explanation. Suppose you are faced with a series of people. The first member is very tall and the last one very short. The difference in height between adjacent members of the series is only 1 mm. This is a sorites series for the predicate "x is tall". If you say that one of them is tall, you may want to say that the next one in the series is also tall: you do not want to imply that a millimeter can make a difference between someone who is tall and someone who is not tall. Suppose someone asks you: "Is the first one tall?". If you feel like playing the game, you should say "Yes". Next you are asked whether the second member is tall. Since a millimeter does not make a difference, you should answer "Yes". Then, you get asked whether the next member is tall, and whether the one after that is tall, and so on. You'd better keep answering "Yes", following a forced march. Also, your chariot is heading towards a precipice. If you keep answering, you should answer "Yes". Keep doing that and you will speak falsely--you will say, of people who are not tall, that they are. You do not want to do that, so Chrysippus recommends to stop answering questions altogether. But when to stop? It is hard to tell, but if you want to make sure you are not answering falsely, you had better shut up while you still know the right answer to the question before you. That is to say, you had better stop answering at some point where you still know that, say, the [n.sub.th] member is tall. (2)
There is a crucial assumption in play. (3) When marching down the series, there is no way to know exactly when one should offer an answer other than "Yes"--whether it be "No", "Neither", "No fact of the matter", or what have you. This, of course, is due to the fact that "Tall" is vague. Knowing such a thing would be to know that, say, Julia is tall, whereas Hector, who is only a millimeter shorter, has some other status. It is plausible that this is something we do not get to know. Given this, Chrysippus recommends to stop playing the forced march sorites game at some point. There is nothing you can do to assure that you answer all questions correctly--perhaps it is not even possible to do it, not even by luck. (4) As such, either you lose, or you stop playing. The game is broken, so it is alright to stop playing.
Let us think about a different game. For this one I want to make a substantial assumption: throughout this paper I shall assume that vagueness is a semantic phenomenon, as opposed to an epistemic or metaphysical one. (5) Here is the game: we are faced with the task of offering an adequate semantic model for a language containing vague predicates. For simplicity I will take vague predicates to be those that can be used to set up a forced march game--i.e., "tall", "old", "rich", "near downtown". (6) To complete our task we are given standard mathematical resources--i.e., sets, fuzzy sets, functions, objects--and a classical metalanguage. It is controversial whether or to what extent this task can be completed. Put another way, it is controversial whether this game is broken, just like the forced march game seems to be. The reasons to think that it is are quite compelling, although not decisive. If one is persuaded by them, I would recommend a Chrysippian attitude towards this game--one can play it for a while, but at some point one has to fall silent. Explaining what it means to adopt a Chrysippian attitude when offering a semantic model is one of the main objectives of this paper. First let us take a look a the reasons one may have to think that this game is broken, then we can see what the Chrysippian attitude is and why it is appealing.
Whether or not one can offer an adequate semantic model for vague languages depends on what the phenomenon of vagueness is. Let us take a look at one of the central and most puzzling features of the phenomenon. We shall proceed by way of an example.
1.1. Good Runners
You are observing today's 5k. First come the leading runners; they are very fast and in excellent shape. The speed and athletic excellence of the runners gradually decreases as time goes by. The runners towards the middle are not quite as fast and athletic. After some time you observe the last participants. They are slow and out of shape. A friend approaches you and asks: "Did you have a chance to see good runners?" To which you reply: "Yes, all and only the fast ones were good runners." Here we can find a sorites series. The first members of the series are clearly good runners, the last members are clearly not good runners, and the running ability of the members of this series gradually decreases as they go by. Crucially, the running abilities of adjacent members of the series are indistinguishable for any practical purpose.
Now, based on your assertion, we can certainly classify some members of the series as good runners. The leading runner is clearly a good runner, given that she is very fast, and others close to her count as good runners as well. It is also clear that based on your assertion you did not classify some members as good runners; the last ones were not classified in that way. Thus, you have used "good runner"--and "fast"--to classify some members of the series in a certain way and not others. This much is uncontroversial, or at least it should be.
It is also clear that based on your assertion there is no piece of information available to us that could help us point at the last member of the series that has been classified as a good runner. The dominant position is that this is so because vague predicates--like "x is a good runner" and "x is fast"--do not draw sharp boundaries between cases where the predicate applies and all the rest. (7) Relative to the sorites series in the previous example, "x is fast" is not the kind of predicate that applies to, say, Julia but not Eli, who has been a centimeter behind Julia during the whole race. (8) This is why this phenomenon is so puzzling. If there is no sharp boundary between the positive cases of application and all the rest, how can it be that there are both positive and negative cases of application of the predicate? How can it be that some of the 5k runners are fast whereas others are not? This is the kind of consideration that fuels the sorites paradox.
So here we are with our standard mathematical tool-box and a classical metalanguage trying to model this phenomenon. Let me present you with an old worry directed at this kind of model-theoretic approach to the phenomenon of vagueness. (9) The objective of the presentation is not to prove that the worry is correct--although I suspect that it is. For my purposes it is enough to show that it is reasonable and that we should take it seriously. In the rest of the paper I will argue that in the event that this worry is correct, we should adopt something like a Chrysippian attitude when modeling vague languages.
Here is the worry. Among the things we want to model is the range of application of vague predicates relative to suitable soritical domains. Much of what we say about the phenomenon of vagueness depends on this. (10) Now, as we have seen, what is distinctive about vague predicates is that they do not draw sharp boundaries. What is distinctive about sets--our core mathematical tools--is that they draw sharp boundaries. A set clearly divides the objects that are members of it from all the others--a fuzzy set clearly divides the objects that are members of that set to a certain degree from all the rest. That is precisely what the range of application of a vague predicate does not do. As such, it seems odd to try to model the range of application of vague predicates by using sets--it may seem that sets are simply not the right kind of tool for this task.
There are some complications--the first one has to do with the notion of borderline case, and the second one with higher-order vagueness. A serious attempt to model vagueness as a semantic phenomenon has to be more ingenious. One may think that in between the positive and negative cases of application of the predicate there are borderline cases. (11) If one thinks this can be modeled with three disjoint and mutually exhaustive sets--or with two disjoint but not mutually exhaustive sets--one would face a well known objection. Vague predicates do not draw sharp boundaries between positive and borderline cases just as they do not draw those boundaries between positive and negative cases. However, sets used in this way do draw those boundaries. Thus, the reasoning goes, if we use sets in this way we can only misrepresent the semantic values of vague predicates. It is worth pointing out that most theorists of vagueness would agree with this. However, most of them will argue that if we take into account the phenomenon of higher-order vagueness, the worry can be dismissed.
It is tempting to think that the transition from the positive to the borderline cases is vague, just as the transition from the positive to the negative cases is vague. Perhaps we can make sense of this while representing the range of application of vague predicates using sets. This line of defense typically uses a determinacy operator (D) to argue that there is no sharp division between the clear cases and the borderline cases, and the negative cases and the borderline cases. The thought is that just as there are borderline cases between the positive and the negative cases, there are second-order borderline cases between the determinate cases and the borderline cases. (12) The...
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