Abstract
One feature of vague predicates is that, as far as appearances go, they lack sharp application boundaries. I argue that we would not be able to locate boundaries even if vague predicates had sharp boundaries. I do so by developing an idealized cognitive model of a categorization faculty which has mobile and dynamic sortals (‘classes’, ‘concepts’ or ‘categories’) and formally prove that the degree of precision with which boundaries of such sortals can be located is inversely constrained by their flexibility. Given the literature, it is plausible that we are appropriately like the model. Hence, an inability to locate sharp boundaries is not necessarily because there are none; boundaries could be sharp and it is plausible that we would nevertheless be unable to locate them.
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Notes
Evans (1978) has probably the most well known argument against the view that objects can be vague, which is discussed again in Lewis (1988). Briefly, the argument is that if there are vague objects, then there could also be vague identities. A vague object like the Sahara desert, for example, would be indefinitely identical to its sharply bounded counterpart. It turns out however, that if objects a and b are indefinitely identical, that entails that the two objects are not identical. This is an odd result in and of itself, but further strengthening of the assumptions with a definitely operator leads to a flat-out contradiction. Defenses of the view exist (Tye 1990), but has not gained widespread agreement. See Prinz (1998) for an overview.
An epistemicist may insist that I can’t maintain a belief because: (1) I can’t maintain a justified belief, and (2) good epistemic agents refuse to maintain unjustified beliefs. Then the point being made can be recapitulated in terms of (1). That is, we need an explanation for why I cannot maintain a justified belief, where justification is something that can be entangled internally with the psychological inability we are attempting to explain.
‘Synthetic methodology’ is an allusion to the work of Braitenberg (1984).
This is closely related to ideas in contextualism, discussed in “How Plausible is it that We are Like the Model?”
In fact, that would be to miss the purpose of representation altogether; the generality of a representation, i.e., that it can apply to numerous cases, is what makes it useful to an agent. We see this idea already back in Kant in his distinction between concepts (which are general) and intuitions (which are singular) (Kant 1996, B377).
Of course both theories, either broadly construed or in their more detailed developments, differ on how exactly cases are encoded in the system. Prototype theories rely on a process of abstraction to generate a summary representation of statistically significant properties which are then used to determine the concept. In this way, the “prototype” need not correspond to specific cases. Exemplar theories do not rely on an abstract summary, but rather take exemplars to govern a concept. Exemplars are encoded property descriptions of a case or cases. The essential difference between exemplars and prototypes is that the former are encodings of many encountered cases where prototypes are encodings of parameters that characterize those cases. See Smith and Medin (1981) for a general overview and Smith and Medin (1999) for further discussion of the differences between prototypes and exemplar theories. See Malt (1989) who argues for both prototypes and exemplars and that subjects can use either in categorization tasks. See Barsalou (1990) who argues that we cannot empirically distinguish between exemplar and abstracted representations.
Lists don’t literally contain cases like Dorothy, Lassie the dog, or grandma. Talking about adding or taking cases off a list is shorthand for the representational operations in a system of encodings.
One might be inclined to object that such trend-matching could lead to complete expansion or contraction, which is neither advantageous, nor an accurate reflection of our categorization faculty. This objection is addressed in “Objection: Forced Marches”.
I talk as if an projection were a projected entity of a (mental) representation. However, a projection is just the set of cases that an agent is disposed to accept as being equivalent.
It may turn out that some updates are redundant. In this case, the projection at t n is extensionally identical to the projection at t n+1 because they include the same cases. An update is non-redundant when this fails to hold.
One might ask whether adaptive lists are sets. Adaptive lists persist through time, they are not “destroyed” when the encodings are changed. This means their identities are not determined extensionally, and consequently that they are not sets. What ontological commitments we have regarding adaptive lists, especially given that a projection can shift, is a question that deserves more exploration in its own right. The current objective however, is to represent the operational nature of a classification system. For that reason I leave the question about ontology to the side.
Interestingly, peoples’ dispositions, and accordingly their intuitions, vary widely about which examples are and are not instances of knowledge (Goldman 2003), which is evidence that people encode different membership conditions.
Any time we speak of an update, we mean a non-redundant one.
With the way we have designed the system, updating occurs when a test yields a positive result. To handle cases of updating where the first test yields a negative result, we consider the complementary concept for which it is a positive result and then proceed likewise. This is not an ad hoc amendment, for it is quite plausible that a concept and its complement are connected by a rule so that updating one automatically updates the other. We simply consider tests with positive results first for ease of proof.
The reader may nevertheless want to know which semantic views are compatible with Sortal Mobility. Obviously, it is compatible with contextualist views of the kind mentioned. One might even extend our model, for example, to something like what appears in the work of Barker (2002). Roughly speaking, we could let the sortal be a gradable adjective in a shared discourse and let the updating rules of our model be determined by how uses of the term update the shared knowledge in discourse.
Generally speaking, however, Sortal Mobility is compatible with a view that vague predicates express properties whose extensions have exact boundaries, where the given property a predicate expresses varies across contexts. It is also compatible with views where the properties expressed do not vary across contexts, though one needs to be more subtle about what the constituents of a context are. If mental states are constituents of context, then clearly a change in mental state is a change in context, which is the preceding contextualist view. If, however, the properties expressed by vague predicates do not vary by context, then there seem to be at least two options. One is to give up that properties have crisp boundaries. Though compatible, this would unnecessarily double up explanatory work for our inability to locate sharp boundaries. The second option is to maintain that properties are crisp and then explain why appearances are to the contrary. This would need to allow for some separation between the property and its representation, which is perfectly compatible with Sortal Mobility.
Alternatively, one may hold that predicates do not express properties at all. Semantic content may be, for example, entirely ‘in the head’. The details such a view would take us too far adrift. Suffice it to say that Sortal Mobility is relatively non-committal to views on semantic content, as it is a claim about our classification dispositions and not the meanings of terms.
See RoboCup (http://www.robocup.org). It is an international research initiative targeted to combine technology from artificial intelligence and robotics. The ultimate goal is to create soccer-playing humanoid robots that can eventually play at a competitive level with human players.
Natural lighting in particular changes much more frequently than controlled indoor lighting.
The way ACT does this is by calculating the mean value.
One obvious reason why it received all the attention it did was because it overthrew the classical view that concepts or categories encode the necessary and sufficient conditions for their application. See Margolis and Laurence (1999) for further reasons why prototypical theories received widespread attention. See Rosch and Mervis (1975) for the influence of Wittgenstein in this regard.
One difficulty of the view is that it is not compositional (Fodor and Lepore 1996), which has been disputed (Hampton and Jönsson 2009; Hampton 2007). We can safely ignore this debate, since we are not concerned with the thesis that concepts, the constituents of compositional thoughts, are prototypes. Even if that thesis turns out to be false, prototype theory stands as a thesis of categorization behaviour.
One might not find this response entirely compelling. If concepts compete for members, the result might be that we end up with (briefly) immobile boundaries, and a forced march would reveal them. For example, let us consider a scenario as above, where we march our model’s middle age class from 45 up to 68, at which point its old age class kicks in and forces it to say that 68 is not middle aged (although it has just said ‘yes’ that 67 is middle aged). Has the forced march thereby exposed a boundary? If it has, it is very short lived. By having considered 68 as old aged, the old age class may expand its boundary as well, requiring the middle age class to retract. So if we were to test the model’s middle age class against 67 again, the model would respond in the negative.
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Acknowledgments
Thanks goes first and foremost to Bernard Molyneux for his extensive feedback on this work, from its conception to the final version. Thanks also to Aldo Antonelli, Adam Sennet, Paul Teller, and an anonymous referee for invaluable comments on drafts. For helpful discussions at various points in the development of this paper I am indebted to Lawrence Barsalou, Jeff Schank, Miguel Sebastian, and Matthew Stone.
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Baumgaertner, B. Vagueness Intuitions and the Mobility of Cognitive Sortals. Minds & Machines 22, 213–234 (2012). https://doi.org/10.1007/s11023-012-9273-3
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DOI: https://doi.org/10.1007/s11023-012-9273-3