Abstract
The standard fuzzy logic and fuzzy set theory are degree-functional and thus susceptible to the problem of penumbral connections. This chapter attempts to radically revise them in order to remove this feature. Whereas the original theory assigns any real number in [0,1] to a proposition or set membership, the revised theory, called Boolean-valued set theory, assigns a value in a Boolean lattice structure B = ⟨D,∧,∨,¬,0,1⟩; consequently, “p or not-p” will receive value 1 and “p and not-p” will receive value 0 for any proposition p in Boolean-valued set theory. The resulting view of vague sets is essentially that depicted in Boolean-valued models of ZFC set theory introduced by Scott and Solovay. On this view, propositions and set membership have many values between 0 and 1, but the values are Boolean-structured and only partially ordered, so we cannot turn them into meaningful numerical degrees in [0,1]. The identity relation between two sets is also given various values, depending on the values of the membership relations involving the two sets; so this theory also endorses vague identity among sets.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
It should be emphasized, however, that semantic supervaluationism already accepts precisified worlds, whether they are called “precisified worlds” or “precisifications.” A version of it that employs the transworld identity theory of individuals also accepts the existence of trans-precisificational objects. The only difference between this version of semantic supervaluationism and ontic supervaluationism I advocate is whether precisifications differ only with respect to semantic facts (i.e., what word refers to which precise object) or also with respect to non-semantic facts (i.e., what trans-precisificational object coincides with which precise object). Thus, the charge often made against my brand of ontic supervaluationism that it assumes a larger ontology than semantic supervaluationism, is totally off the mark.
- 2.
An infinite-valued logic called standard Łukasiewicz logic L 1 (or \( {L}_{{\mathrm{\aleph}}_1}\! \)) is often used as the basis of fuzzy logic. Klir and Yuan (1995, Chap. 8) provides a useful account of fuzzy logic and its connection to fuzzy set theory.
- 3.
See Smith (2008a) for more details about this point.
- 4.
- 5.
- 6.
At the end of the day, I think we should give up this assumption, as I shall argue in the final section. It is a reasonable assumption, however, and whether we should give it up at the end of the day or not really does not affect the current argument. The argument holds up insofar as the value of the disjunction is not constant at the center, right end, and left end.
- 7.
Another possible way to satisfy the desiderata is Edgington’s (1992, 1997). Her idea is to use the standard probability calculus to measure the degree of vagueness; that is,
$$ \begin{array}{c}{\left [\kern-0.15em[ \neg p\right ]\kern-0.15em] }_V=1-\left [\kern-0.15em[ p\right ]\kern-0.15em] {}_V;\\ {}{\left [\kern-0.15em[ p\wedge q\right ]\kern-0.15em] }_V=\left [\kern-0.15em[ p\right ]\kern-0.15em] {}_V\times \left [\kern-0.15em[ q\kern0.5em \mathrm{given}\kern0.5em p\right ]\kern-0.15em] {}_V;\\ {}{\left [\kern-0.15em[ p\vee q\right ]\kern-0.15em] }_V=\left [\kern-0.15em[ p\right ]\kern-0.15em] {}_V+\left [\kern-0.15em[ q\right ]\kern-0.15em] {}_V-\left [\kern-0.15em[ p\wedge q\right ]\kern-0.15em] {}_V.\end{array} $$Edgington contends that just as credence and objective chance may share the same logic even though they are conceptually different (cf. Lewis 1980), the degrees of truth (or what she calls “verity”) pertaining to vagueness may also follow the same logic even though it is different from either credence or objective chance. Discussion of Edgington’s view is beyond the scope of this paper.
- 8.
For the sake of simplicity we shall ignore in this paper the universal and existential quantifiers in the object language. To include those quantifiers, we have to require that the relevant Boolean lattice B be complete. Generally, a complete lattice G is a lattice that has the glb and lub for any subset of D G whether it is finite or infinite. Then we can have
$$ \begin{array}{c}{\left [\kern-0.15em[ \forall x\;\phi (x)\right ]\kern-0.15em] }_B=\underset{a\in U}{\wedge }{\left [\kern-0.15em[ \phi (a)\right ]\kern-0.15em] }_B;\\ {}{\left [\kern-0.15em[ \exists x\;\phi (x)\right ]\kern-0.15em] }_B=\underset{a\in U}{\vee }{\left [\kern-0.15em[ \phi (a)\right ]\kern-0.15em] }_B.\end{array} $$ - 9.
It’s not as if there is absolutely no way of assigning numbers between 0 and 1 that conforms to the ordering relation ≤. For instance, Zhang (1982) came up with one way, in which the paths in the lattice structure and the locations in the paths are given binary interpretations, which are then translated into real numbers in [0,1]. However, on Zhang’s assignment, if, for instance, 〚p〛 B = 5/8 and 〚q〛 B = 3/8, then even though 〚p〛 B ∧ ¬〚p〛 B = 0 and 〚p〛 B ∨ ¬〚p〛 B = 1, 〚p〛 B ∧ 〚q〛 B = 1/8 and 〚p〛 B ∨ 〚q〛 B = 7/8. These are hardly the values one would expect or can make sense of. As this example illustrates, while it is possible for us to assign numbers between 0 and 1, such an assignment lacks intuitive meaning.
- 10.
Or, in a multiple-conclusion formulation such as that of the classical sequent calculus,
$$ \Gamma \vdash \Delta \iff \underset{p\in \Gamma}{\wedge }{\left [\kern-0.15em[ p\right ]\kern-0.15em] }_B\le \underset{q\in \Delta}{\vee }{\left [\kern-0.15em[ q\right ]\kern-0.15em] }_B\kern0.5em \mathrm{for}\ \mathrm{any}\kern0.5em B. $$ - 11.
Needless to say, classical logic can be defined semantically, as the bivalent logic. But, then, classical logic, thus defined, is deductively no different from the logic defined by the general Boolean-valued semantics.
- 12.
Unless you embrace Edgington’s aforementioned view.
- 13.
Williamson (1994) and Keefe (1998, 2000, Chap. 5) also criticize fuzzy set theory for its linear value assignments. Goguen (1969), even though he is generally sympathetic to fuzzy set theory, also considers this feature a shortcoming of the theory and develops a system in which values are only partially ordered. Goguen’s theory is rather different from the present proposal; most importantly, his general framework is not Boolean, though he accepts Boolean structures as special cases. We will not discuss Goguen’s theory in this paper. See Williamson (1994, pp. 131–135) for a brief critical examination of Goguen’s theory and other similar theories.
- 14.
Weatherson’s general framework is similar to this chapter’s, although his main concern is with introducing “≤” (or “<,” “truer than”) in the object language.
- 15.
I would like to thank Richard Zach for pressing this point.
- 16.
As far as I know, Zhang is the first researcher to point out the significant connections between fuzzy set theory and Boolean-valued set theory.
- 17.
One might still contend that = in Boolean-valued set theory is not really identity but something else. (I myself argued in Akiba (2000a, b, 2004) that what is often considered vague identity between vague individuals is not really identity but mere coincidence.) The issue of vague (or indeterminate) identity is without doubt an extremely delicate one. The root problem is that we, collectively, do not seem to have a clear idea of what vague identity ought to be. Still, I am inclined to think that vagueness of = here is as good a candidate of vague identity as anything can be. But a full defense of this claim is beyond the scope of this paper. See Williamson (1996, 2002) and Smith (2008b) for arguments for the unintelligibility and nonexistence of vague identity, and see Akiba (MS) for my defense of vague identity. Thanks to the anonymous reviewer of this chapter for raising this issue.
- 18.
Of course, it may not be, because of quantum indeterminacy, for instance. For quantum indeterminacy, see Darby (2014) and the papers cited therein.
- 19.
I owe much to the anonymous reviewer for seeing this point.
- 20.
I would like to thank the anonymous reviewer of this chapter for valuable comments. His/her comments on the relation between the possible worlds semantics and Boolean-valued semantics were particularly helpful. I also would like to thank for comments Nick Smith, Richard Zach, and the audience at the Society for Exact Philosophy 41st annual meeting, May 24, 2013, Université de Montréal, where a part of this chapter was presented.
References
Akiba, K. (2000a). Vagueness as a modality. Philosophical Quarterly, 50, 359–370.
Akiba, K. (2000b). Identity is simple. American Philosophical Quarterly, 37, 389–404.
Akiba, K. (2004). Vagueness in the world. Noûs, 38, 407–429.
Akiba, K. (MS). A defense of indeterminate distinctness. In preparation.
Alxatib, S., & Pelletier, F. J. (2011). The psychology of vagueness: Borderline cases and contradictions. Mind and Language, 26, 287–326.
Barnes, E. (2010). Ontic vagueness: A guide for the perplexed. Noûs, 44, 601–627.
Barnes, E., & Williams, J. R. G. (2011). A theory of metaphysical indeterminacy. In K. Bennett & D. W. Zimmerman (Eds.), Oxford studies in metaphysics (Vol. 6, pp. 103–148). Oxford: Oxford University Press.
Bell, J. L. (2005). Set theory: Boolean-valued models and independence proofs (3rd ed.). Oxford: Clarendon Press. (1st ed., 1977.)
Bonini, N., Osherson, D., Viale, R., & Williamson, T. (1999). On the psychology of vague predicates. Mind and Language, 14, 377–393.
Chow, T. Y. (2008). A beginner’s guide to forcing, version 2. http://arxiv.org/pdf/0712.1320v2.pdf
Cohen, P. J. (1966). Set theory and the continuum hypothesis. New York: Benjamin.
Darby, G. (2014). Vague objects in quantum mechanics? In K. Akiba & A. Abasnezhad (Eds.), Vague objects and vague identity (pp. 69–108). Dordrecht: Springer.
Edgington, D. (1992). Validity, uncertainty and vagueness. Analysis, 52, 193–204.
Edgington, D. (1997). Vagueness by degrees. In R. Keefe & P. Smith (Eds.), Vagueness: A reader (pp. 294–316). Cambridge, MA: MIT Press.
Evans, G. (1978). Can there be vague objects? Analysis, 38, 208.
Fine, K. (1975). Vagueness, truth and logic. Synthese, 30, 265–300.
Goguen, J. A. (1969). The logic of inexact concepts. Synthese, 19, 325–373.
Gottwald, S. (1979). Set theory for fuzzy sets of higher level. Fuzzy Sets and Systems, 2, 125–151.
Gottwald, S. (2006). Universes of fuzzy sets and axiomatizations of fuzzy set theory. Part I: Model-based and axiomatic approaches. Studia Logica, 82, 211–244.
Keefe, R. (1998). Vagueness by numbers. Mind, 107, 565–579.
Keefe, R. (2000). Theories of vagueness. Cambridge: Cambridge University Press.
Klir, G. J., & Yuan, B. (1995). Fuzzy sets and fuzzy logic: Theory and applications. Upper Saddle River: Prentice Hall.
Lakoff, G. (1973). Hedges: A study in meaning criteria and the logic of fuzzy concepts. Journal of Philosophical Logic, 2, 458–508.
Lewis, D. (1980). A subjectivist’s guide to objective chance. In R. C. Jeffrey (Ed.), Studies in inductive logic and probability (Vol. 2, pp. 263–293). Berkeley: University of California Press.
Lewis, D. (1988). Vague identity: Evans misunderstood. Analysis, 48, 128–130.
Machina, K. (1976). Truth, belief and vagueness. Journal of Philosophical Logic, 5, 47–78.
Morreau, M. (2002). What vague objects are like. Journal of Philosophy, 99, 333–361.
Parsons, T. (2000). Indeterminate identity. Oxford: Oxford University Press.
Rasiowa, H., & Sikorski, R. (1963). The mathematics of metamathematics. Warsaw: PWN.
Sattig, T. (2014). Mereological indeterminacy: Metaphysical but not fundamental. In K. Akiba & A. Abasnezhad (Eds.), Vague objects and vague identity (pp. 25–42). Dordrecht: Springer.
Serchuk, P., Hargraves, I., & Zach, R. (2011). Vagueness, logic and use: Four experimental studies on vagueness. Mind and Language, 26, 540–573.
Smith, N. J. J. (2008a). Vagueness and degrees of truth. Oxford: Oxford University Press.
Smith, N. J. J. (2008b). Why sense cannot be made of vague identity. Noûs, 42, 1–16.
Weatherson, B. (2005). True, truer, truest. Philosophical Studies, 123, 43–70.
Williams, J. R. G. (2008). Multiple actualities and ontically vague identity. Philosophical Quarterly, 58, 134–154.
Williamson, T. (1994). Vagueness. London: Routledge.
Williamson, T. (1996). The necessity and determinacy of distinctness. In S. Lovibond & S. G. Williams (Eds.), Essays for David Wiggins: Identity, truth and value (pp. 1–17). Oxford: Blackwell.
Williamson, T. (2002). Vagueness, identity and Leibniz’s law. In A. Bottani, M. Carrara, & P. Giaretta (Eds.), Individual essence and identity (pp. 273–304). Dordrecht: Kluwer.
Zadeh, L. A. (1965). Fuzzy sets. Information and Control, 8, 338–353.
Zhang, J. (1980). A unified treatment of fuzzy set theory and Boolean-valued set theory – Fuzzy set structures and normal fuzzy set structures. Journal of Mathematical Analysis and Applications, 76, 297–301.
Zhang, J. (1982). Between fuzzy set theory and Boolean valued set theory. In M. M. Gupta & E. Sanchez (Eds.), Fuzzy information and decision procedures (pp. 143–147). Amsterdam: North-Holland.
Zhang, J. (1983). Fuzzy set structure with strong implication. In P. P. Wang (Ed.), Advances in fuzzy sets, possibility theory, and applications (pp. 107–136). New York: Plenum.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
Akiba, K. (2014). Boolean-Valued Sets as Vague Sets. 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_8
Download citation
DOI: https://doi.org/10.1007/978-94-007-7978-5_8
Published:
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-007-7977-8
Online ISBN: 978-94-007-7978-5
eBook Packages: Humanities, Social Sciences and LawPhilosophy and Religion (R0)