Ling 726: Mathematical Linguistics, Lecture 12. Phonology. V. Borschev and B. Partee, November 6, 2001 p. 1 Lecture 13. Formal Phonology: Features and their structure. CONTENTS 1. Introduction: the source and the goals. ............................................................................................ 1 1.1. Keenan’s text: mathematical structures in language................................................................ 1 1.2. Goals........................................................................................................................................ 1 2. Features and their structures. ........................................................................................................... 3 2.1. The set of features and the equivalence relation defined by this set. ....................................... 3 3. Features for English. Boolean operations on features...................................................................... 4 3.1. Initial features for English........................................................................................................ 4 3.2. Boolean operations on features................................................................................................ 5 3.3. Redundancy rules on features. ................................................................................................. 6 3.4. Boolean algebra generated by IF.............................................................................................. 6 Homework 13. ......................................................................................................................................... 6 Reading: Keenan. Mathematical structures in language. Ch. 4. Formal Phonology. Pp. 109-128. 1. Introduction: the source and the goals. 1.1. Keenan’s text: mathematical structures in language. This lecture is not about some new mathematical notions and structures. It is about mathematics in linguistics. We would not say that it is about applications mathematics to linguistics, we don’t apply mathematics to language. Really mathematics plays some role in linguistics only when while working with language we manage to find in it some mathematical structures or some structures which may become mathematical when mathematicians work them out. We consider here just one example. This lecture is based on chapter 4 of Keenan’s textbook “Mathematical structures in languages”. Chapter 4 deals with phonology. We are not phonologists. We understand that there exist many approaches to mathematical descriptions of basic phonological structures. But we like this text. It is a wonderful example of methodology and pedagogy: how mathematics can be used in the description of some structures of “real life” and how to write about that. 1.2. Goals. “Thoughts, and expressions of thoughts, are discrete. We can count them. But when you tell me your daughter won a scholarship to the Naval Academy, what impinges in my ears is a continuously varying sound wave. Somehow, perhaps in a successions of steps, I convert your emissions to a discrete (= digital) form, something we represent in written English by a sequence of letters (including space). And letters and sequences of letters are countable. The fundamental cognitive issue in phonology is to account for this conversion of the continuous to the discrete. This is where the magic begins”.1 1 This is a citation from Keenan’s textbook, p. 109. Below we will sometimes cite him just by putting his words in quotation-marks.Ling 726: Mathematical Linguistics, Lecture 12. Phonology. V. Borschev and B. Partee, November 6, 2001 p. 2 Note that such a discretization seems to be a basic cognitive mechanism. When we look on some picture or situation and try to describe it in words the first thing we do (mostly subconsciously) is schematize this continuous picture into some discrete elements: things, their properties, relations, etc. And this is also magic. But let’s return to phonology. We manage to distinguish in speech words and sounds. We will deal with sounds here. But what’s a sound? On one hand different people or even the same person pronounce the same sound in different ways. On the other hand we understand in some way that it is the same sound. But how to formalize this? “Phonologists distinguish sounds according as their properties differ. They refer to such properties as features. Consider for example the initial sounds in pig and big, represented here by the letters p and b. These two sounds are similar in many ways. Both block the air stream from exiting the mouth and the nose. By contrast the initial f and v in fat and vat let sound exit through the mouth (but not the nose)… The m and n sounds in might and night let air exit through the nose but not the mouth. Equally p and b are similar in that both block the air stream by compressing the upper and lower lips together (as does the production of an m)… Where p’s and b’s differ is with regard to voicing. When b is produced the vocal folds in the larynx open just enough to make them vibrate, creating the “voiced” sound in b which is absent in p. Phonologists say that b is voiced, or equivalently, +voice, and p is voiceless, or –voice. The voicing feature similarly distinguishes many other pairs of English sounds, such as t and d, and f and v noted above. Further such pairs are k and g sounds in cot and got and the final sounds in bath and bathe as well as those in bath and badge. Another feature we have mentioned is nasality. The m sound in mig is similar to p and b in that all block air from escaping through the oral cavity. And m is similar to b in that both are +voice. But m is differs from b and p in that producing m allows air to escape through the nasal cavity. So m is +nasal and p and b are –nasal.” So let us formulate, maybe in a fuzzy way, some presuppositions which we have prior to the process of formalization. [This is ours. –VB] (1) The stream of speech can be “sliced” into speech intervals called sounds. (2) Sounds are pronounced with a great diversity but we can say, at any rate in principle, comparing two utterances, ”that is the same sound” or “these are different sounds”. (3) We can formalize differences between sounds (or their sameness) with the help of some set of features. (4) Features are considered as (total) functions defined on sounds (i.e. not as partial ones,