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.
References
Dummett, M. (1975). Wang’s paradox. Synthese, 30, 301–324; reprinted in Keefe and Smith (1997), 99–118.
Evans, G. (1978). Can there be vague objects? Analysis, 38, 208; reprinted in Keefe and Smith (1997), 317.
Field, H. (2009). Pluralism in logic. The Review of Symbolic Logic, 2, 342–359.
Frege, G. (1884). Die Grundlagen der Arithmetik. Breslau: Koebner; The foundations of arithmetic (J. Austin, Trans., 2nd ed.). New York: Harper, 1960.
Frege, G. (1893). Grundgesetze der Arithmetik 1. Hildescheim: Olms.
Gaifman, H. (2010). Vagueness, tolerance and contextual logic. Synthese, 174, 5–46.
Graff, D. (2000). Shifting sands: An interest-relative theory of vagueness. Philosophical Topics, 28, 45–81.
Graff, D. (2001). Phenomenal continua and the sorites. Mind, 110, 905–935.
Hale, B., & Wright, C. (2001). The reason’s proper study. Oxford: Oxford University Press.
Horgan, T. (1994). Robust vagueness and the forced-march sorites paradox. Philosophical Perspectives, 8: Logic and Language, 159–188.
Kamp, H. (1981). The paradox of the heap. In U. Mönnich (Ed.), Aspects of philosophical logic (pp. 225–277). Dordrecht: Reidel.
Keefe, R., & Smith, P. (1997). Vagueness: A reader. Cambridge, MA: MIT Press.
Lewis, D. (1979). Scorekeeping in a language game. Journal of Philosophical Logic, 8, 339–359.
Lewis, D. (1988). Vague identity: Evans misunderstood. Analysis, 48, 128–130; reprinted in Keefe and Smith (1997), 318–320.
McGee, V., & McLaughlin, B. (1994). Distinctions without a difference. Southern Journal of Philosophy, 33(Supplement), 203–251.
Morreau, M. (2002). What vague objects are like. Journal of Philosophy, 99, 333–361.
Parsons, C. (1990). The structuralist view of mathematical objects. Synthese, 84, 303–346.
Parsons, T., & Woodruff, P. (1995). Worldly indeterminacy of identity. Proceedings of the Aristotelian Society, 95, 171–191; reprinted in Keefe and Smith (1997), 321–337.
Raffman, D. (1994). Vagueness without paradox. Philosophical Review, 103, 41–74.
Raffman, D. (1996). Vagueness and context relativity. Philosophical Studies, 81, 175–192.
Sainsbury, R. M. (1990). Concepts without boundaries. Inaugural lecture, published by the King’s College London, Department of Philosophy; reprinted in Keefe and Smith (1997), 251–264.
Shapiro, S. (2006). Vagueness in context. Oxford: Oxford University Press.
Shapiro, S. (2014). Varieties of logic. Oxford: Oxford University Press.
Soames, S. (1999). Understanding truth. Oxford: Oxford University Press.
Unger, P. (1975). Ignorance. Oxford: Oxford University Press.
Uzquiano, G. (2004). The supreme court and the supreme court justices: A metaphysical puzzle. Noûs, 38, 135–153.
Waismann, F. (1945). Verifiability. Proceedings of the Aristotelian Society, Supplementary Volume, 19, 119–150; reprinted in Logic and language. edited by Antony Flew, Oxford: Basil Blackwell, 1968, 117–144.
Wright, C. (1976). Language mastery and the sorites paradox. In G. Evans & J. McDowell (Eds.), Truth and meaning: Essays in semantics (pp. 223–247). Oxford: Oxford University Press; reprinted in Keefe and Smith (1997), 151–173.
Wright, C. (1987). Further reflections on the sorites paradox. Philosophical Topics, 15, 227–290; reprinted in Keefe and Smith (1997), 204–250.
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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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