Abstract
An abstraction principle is any proposition in the form
where a and b are variables of a given type (typically individual objects or properties/sets of objects); Σ is a higher-order operator, denoting a function from items of the given type to objects; and E is a relation over items of the given type. In any standard, non-free logic, it follows from (ABS) that the embedded relation E is an equivalence relation: it is reflexive, symmetric, and transitive. In ordinary, nonmathematical discourse, concerning ordinary physical objects, we sometimes invoke what look like abstraction principles (in the form (ABS)). For example, we speak of the weights of individuals, indicating whether those are the same or different. We might say that Harry has the same weight as Sarah, but not as Joe. This seems to involve the following, which we may call the Weight Principle:
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(W) The weight a is identical to the weight of b if and only if a and b are equi-weighted,
where a and b are variables ranging over physicals objects or people. However, on a plausible reading of this, in line with ordinary usage, the right-hand side is not an equivalence relation, since being equi-weighted is not an equivalence relation due to vagueness. The purpose of this essay is to develop an account of such “quasi-abstract” objects as weights.
This chapter is based on part of Chapter 6 of my Vagueness in context (Shapiro 2006).
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Notes
- 1.
Charles Parsons (1990) coined the term “quasi-concrete” for abstract objects (or properties) that have concrete instances. A typical example is the type shared by all tokens of a 12-point Times Roman letter “e.” In a sense, the present quasi-abstract objects are also quasi-concrete.
- 2.
I suppose we are assuming that like molecules have exactly the same weight, whether or not this makes sense in light of what is known at the quantum level.
- 3.
Delia Graff (2001) challenges the nontransitivity in the case of colors. If she is right, then we can drop this example. There are plenty of others. Shapiro (2014) floats the possibility that, in effect, synonymy is not transitive. See also Field (2009). This would bring meanings into the present focus.
- 4.
Shapiro (2006) employs the term “determinately” for what McGee and McLaughlin call “definitely.”
- 5.
My use of the phrase “open texture” is not quite Waismann’s (1945). He writes, “Vagueness should be distinguished from open texture. A word which is actually used in a fluctuating way (such as ‘heap’ or ‘pink’) is said to be vague; a term like ‘gold’, though its actual use may not be vague, is non-exhaustive or of an open texture in that we can never fill up all the possible gaps though which a doubt may seep in. Open texture, then, is something like possibility of vagueness.”
- 6.
Shapiro (2006) also contains a model-theoretic development, but that is omitted here in the interest of brevity. The system is similar to the Kripke semantics for intuitionistic logic (a variation on the modal logic S4), except that, at each node of each frame, vague terms have both an extension and an anti-extension. A node represents a way that vague terms can competently be deployed in a conversation, including one in which a tolerance principle is in force.
- 7.
Recall that we are holding external contextual factors, like comparison class and the like, fixed.
- 8.
See Uzquiano (2004) for an illuminating account of dynamic sets.
- 9.
An extreme version, dubbed the “Copenhagen view of vagueness” in Chapter 5 of Shapiro (2006), is that this holds for every object in the field of the predicate.
- 10.
Terence Parsons and Peter Woodruff (1995) defend the coherence of indeterminate identities. They point out that the Evans argument invokes the contrapositive of the Leibniz principle of the indiscernibility of identicals and that in a three-valued system, the contrapositive of a valid inference need not be valid. Moreover, they suggest that predicates that invoke indeterminacy need not express properties. Parsons and Woodruff also provide a nice model, in a crisp, bivalent metalanguage, to illustrate the coherence of vague identity. Shapiro (2006, Chapter 6) provides another.
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Shapiro, S. (2014). Vagueness and Abstraction. In: Akiba, K., Abasnezhad, A. (eds) Vague Objects and Vague Identity. Logic, Epistemology, and the Unity of Science, vol 33. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-7978-5_10
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