Sets, rules and natural classes: [ ] vs. { }




phonology, features, set theory, logic, unification, phonological rules


We discuss a set-theoretic treatment of segments as sets of valued features and of natural classes as intensionally defined sets of sets of valued features. In this system, the empty set { } corresponds to a completely underspecified segment, and the natural class [ ] corresponds to the set of all segments, making a feature ± Segment unnecessary. We use unification, a partial operation on sets, to implement feature-filling processes, and we combine unification with set subtraction to implement feature-changing processes. We show how unification creates the illusion of targeting only underspecified segments, and we explore the possibility that only unification rules whose structural changes involve a single feature are UG-compatible. We show that no such Singleton Set Restriction can work with rules based on set subtraction. The system is illustrated using toy vowel harmony systems and a treatment of compensatory lengthening as total assimilation.


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Bale, A., Papillon, A., & Reiss, Ch. (2014). Targeting underspecified segments: A formal analysis of feature changing and feature filling rules. Lingua, 148, 240–253.

Bale, A., & Reiss, Ch. (2018). Phonology: A Formal Introduction. Cambridge, MA: MIT Press.

Bendjaballah, S., & Haiden, M. (2008). A typology of emptiness in templates. In J. Hartmann, V. Hegedus, & H. van Riemsdijk (Eds.) Sounds of Silence: Empty Elements in Syntax and Phonology (pp. 23–59). Amsterdam: Elsevier.

Borrelli, D. A. (2013). Raddoppiamento Sintattico in Italian. A Synchronic and Diachronic Cross-dialectical Study. New York: Routledge.

Chomsky, N. (2000). Language as a natural object. In New Horizons in the Study of Language and Mind (pp. 106–133). Cambridge: Cambridge University Press.

Chomsky, N., & Halle, M. (1968). The Sound Pattern of English. New York: Harper & Row.

Côté, M.-H. (2008). Empty elements in schwa, liaison and h-aspiré: The French Holy Trinity revisited. In J. Hartmann, V. Hegedus, & H. van Riemsdijk (Eds.) Empty Elements in Syntax and Phonology (pp. 61- 103). Amsterdam: Elsevier.

Dell, F., & Elmedlaoui, M. (1985). Syllabic consonants and syllabification in imdlawn tashlhiyt berber. Journal of African Languages and Linguistics, 7, 105–130.

Flemming, E. (2005). Deriving natural classes in phonology. Lingua, 115, 287–309. DOI:

Halle, M. (1964). On the bases of phonology. In J. J. Katz, & J. Fodor (Eds.) The Structure of Language (ch. 9, pp. 324–333). Englewood Cliffs, NJ: Prentice-Hall.

Hayes, B., & White, J. (2015). Saltation and the p-map. Phonology, 32, 267–302.

Hyman, L. (1985). A Theory of Phonological Weight. Dordrecht: Foris.

Inkelas, Sh. (1995). The consequences of optimization for underspecification. In J. Beckman (Ed.), Proceedings of the North East Linguistic Society, 25 (pp. 287–302). Philadelphia, PA: University of Pennsylvania Graduate Linguistic Student Association.

McCarthy, J. (1988). Feature geometry and dependency: a review. Phonetica, 45, 84–108.

Mielke, J. (2008). The Emergence of Distinctive Features. Oxford: Oxford University Press.

Peterson, G. E., & Harary, F. (1961). Foundations of phonemic theory. In R. Jakobson (Ed.) Structure of Language and Its Mathematical Aspects. Proceedings of Symposia in Applied Mathematics, 12 (pp. 139–165). Rhode Island: American Mathematical Society.

Reiss, Ch. (1995). A Theory of Assimilation, with Special Reference to Old Icelandic Phonology. Doctoral Dissertation, Harvard University, Department of Linguistics.

Reiss, Ch. (1996). Deriving an implicational universal in a constrained OT grammar. In K. Kusumoto (Ed.), Proceedings of North East Linguistic Society, 27 (pp. 303–318). Cambridge, MA: Harvard University & MIT Graduate Linguistic Student Association.

Reiss, Ch. (2003). Quantification in structural descriptions: Attested and unattested patterns. The Linguistic Review, 20, 305–338.

Reiss, Ch. (2019). Are Saltations Real? Ms. Concordia University.

Sagey, E. C. (1986). The Representation of Features and Relations in Nonlinear Phonology. Doctoral Dissertation, Massachusetts Institute of Technology, Cambridge, MA.

Shen, D. T.-Ch. (2016). Precedence and Search: Primitive Concepts in Morpho-phonology. Doctoral Dissertation, National Taiwan Normal University, Taipei.

Trommer, J. (2015). Moraic affixes and morphological colors in Dinka. Linguistic Inquiry, 46, 77–112.



How to Cite

Bale, A. ., Reiss, C. ., & Ta-Chun Shen, D. . (2020). Sets, rules and natural classes: [ ] vs. { }. Loquens, 6(2), e065.




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