323 Notes on Phonemic Theory in Terms of Set Theory 1. Notes on Phonemic Theory Here I will discuss phonemic theory in terms of set theory. A phoneme is a set of allophones which are not contrastive and share a common phonological feature or features. For example [t] and [t h ] are not contrastive and they form a set containing apical voiceless stops. We may write the name of this with anything we want, but it is strongly advisable to use a symbol that can be associated with the set. Let us select T for this set: E.g. T = {[t], [t h ]} or /T/ = {[t], [t h ]} It helps if the symbol comes close to the default. T is not complete. The two allophones are members of /T/, but there are at least two more allophones in the set: E. g./T/ = {[t], [t h ], [ ], [ ʔ ]} (the empty slot is for the tap.) When completing exercises, I will expect phonemic notation, not phonetics ones. I.e. = [t h ] is incorrect for /T/. Not that [t h ] in English is constrained to intervocalic positions after a stressed vowel (relatively standard) and in V _ V _ … Where the first vowel is stressed and the second one unstressed. This position is not common in Canadian dialects; it is an American pronunciation in some dialects. This allophone is much more restricted than [t]. Having the widest distribution, [t] is the default, which may write as [#t]. # is not a phonetic symbol but it simply means that [t] or whatever allophone is the default: E.g. T = {[t h ], [ ], [ ʔ ], [#t]} 2. Morphophonemes Depending on ones point of view, /s/ and /z/, for example, could be seen as two distinctive phonemes in all positions. The is the Bloch axiom: Once a phoneme,

2. Morphophonemic Theory always a phoneme. Some now consider this axiom outdated. If no contrast exists between them in a certain position, then they may be members of the same phoneme set. The plays a role in two English paradigms involving [s] and [z]. Traditionally, the phonemic representation for dogs and cats is /dagz/ and /kæts/. The plural marker is /z/ and /s/, respectively. In this position, there can be no contrast between /z/ and /s/. Their phonetic representations are [z] and [s], which can be predicted. Suppose we consider the plural marker /s/. If /s/ occurs after a voiceless obstruent, then it is realized as [z]. This means that there is a set {[s], [z]}, whose distribution is determined by the voiceless obstruent. What should we name this set? Let us call it /Z/: E.g. /KÆTZ / [k h æts]. Here the double ended arrow refers to mapping: the phonemic form is mapped to its phonetic form and vice versa. Where [s] and [z] do contrast, then /S/ is distinctive from /Z/: E.g. [hæwz] vs. [hæwz] = /HÆWS/ vs. /HÆWZ/. Note that this contrast is not possible after voiceless obstruents. The set for /Z/ is given E.g. /Z/ = {[s], [#z]} and S = {[#s]}. [z] is the default since [z] everywhere else (not after a voiceless obstruent): E.g. sofas is /sofəz/ [só w fəz] Now can state that the plural morpheme in English is the single /Z/, and not as two phonemes /s/ and /z/ as in traditional analyses. This leads to a state of greater elegance, since at least one awkward stop is removed. It is now possible to state that in terms of this theoretical framework, the sign for the English nominal plural is /Z/. This one of two allomorphs. I will return to this shortly.

2. Morphophonemic Theory Before we leave /S/, there is another case we must look at. It is well known now that a barred I [ ɨ ] appears phonetically between fricatives and a word-final [z]: [p ʊ š ɨ z]. This is predictable. Suppose we add [ ɨ z] to /Z/. And now let us put this in a mapping relation: E.g. /Z/ [ ɨ z]/ ___##, where ## is a word boundary. OrE.g. /Z##/ [ ɨ z##]. The former is the more traditional way of writing the rule. [ ɨ ] is not a distinctive phoneme in English. It only occurs in unstressed syllables in certain contexts, different from [ə] in many dialects of English. These two unstressed phones are often viewed as contrastive: E.g. Rosas = [ro w zəz] and roses = [ro w z ɨ z]. However, this differentiation is predictable. The first contains a stem final schwa [ə], whereas in the latter, [ ɨ ] is epenthetic: E.g. /ROZə/-/Z/ [ró w zəz]. Here, /z/ is adjoined to the stem and is spelled out as [z], as expected. In the latter case, rose = /ROZ/. When it is plural, the affix with the sign /Z/ is adjoined to the stem: E.g. /ROZ-Z/. The first /Z/ [z], whereas the second /z/ [ ɨ z]. Note that this schema will cause traditionalists either to turn over in their graves or cause them to lose more hair and risk having a stroke. The set /Z/ is now modified to include [ ɨ z]: E.g. /Z/ = {[ ɨ z], [s], [#z}]. The contexts for mapping the phoneme /Z/ to its allophones is well known. In a given context, on of the three allophones occurs. I will adopt the filtering or constraint method employed by Optimality Theory. The first constraint is that a fricative plus another fricative is unacceptable at the end of a word: E.g. *[+Fricative][+Fricative]##. Note that the word boundary ## does not have be written in the classical way.

2. Morphophonemic Theory The second constraint is that a voiced obstruent cannot follow a voiceless one within the same syllable: frets = [FRET-Z] [fr ɛ t-s]. E.g. *$ … [+voiced][-voiced] … $], where $ represents a syllable boundary. In the remaining cases, the default is selected. The past tense sign for regular English verbs is /D/. The classical English phoneme /d/ contains two allophones: [d] and [flap]. Note that [t] and [d] do not contrast when they follow voiceless alveolars: seized = [sijzd], ceased = [sijst]. Again we should add [t] as a member of the phoneme set /D/: E.G. /D/ = {[flap], [t], [#d]}. If an alveolar stop follows another alveolar stop, an epenthetic vowel is inserted (in traditional phonology) between the two alveolar stops. E.g. raided = [re: flap ɨ d], rated = [re flap ɨ d]. An extra complication occurs here as both /T/ and /D/ a flap. The preceding vowel is phonetically long before a voiced obstruent, whereas the preceding vowel is phonetically short before before a voiceless obstruent. In open syllables and before sonorants the vowel is long. The concept is similar to the above. A morphophoneme is a set of phonemes that alternate in a given paradigm