Groucho Marxism

Phonology is the branch of linguistics that concerns how languages organize the foundational elements that make their words. These foundational elements are referred to as phonemes. Element Theory is a theory of phonology which posits that phonemes are made up of combinations of a handful of elements or phonological primes which exist across all languages. In its most widely accepted version, there are six such elements which are denoted by A, I, U, ?, L, and H, representing openness, frontness, roundness, stopness, voicing, and frication, respectively. A phoneme may then be defined as a set of elements. Under this definition there are eight possible vowel phonemes: {} (the empty set), {A}, {I}, {U}, {A,I}, {A,U}, {I,U}, and {A,I,U}.

The problem with this is that many languages have more than eight vowel phonemes; for instance, Danish and French both have eleven. The obvious solution is to add another element to the list. For example, we could add an additional element R representing retraction, which results in 16 vowel phonemes: {}, {A}, {I}, {U}, {R}, {A,I}, {A,U}, {A,R}, {I,R}, {I,U}, {U,R}, {A,I,R}, {A,I,U}, {A,U,R}, {I,U,R}, and {A,I,U,R}. There are no languages with more than 16 vowel phonemes. These 16 phonemes can be put in one-to-one correspondence with the cardinal vowels of the International Phonetic Alphabet in the following way: i = {I}, e = {I,R}, ɛ = {A,I,R}, a = {A,I}, ɑ = {A}, ɔ = {A,U,R}, o = {U,R}, u = {U}, y = {I,U}, ø = {I,U,R}, œ = {A,I,U,R}, ɶ = {A,I,U},  ɒ = {A,U}, ʌ = {A,R}, ɤ = {R}, and ɯ = {}.

Alternatively, we can stick with our original three elements A, I, and U, and define a vowel phoneme as a set {X,Y}, where X is an element of {A,I,U} and Y a subset of {A,I,U} which does not contain X. In Element Theory, X is referred to as the head and Y as the operator. Under this definition there are 12 possible vowel phonemes: {A,{}}, {A,{I}}, {A,{U}}, {A,{I,U}}, {I,{}}, {I,{A}}, {I,{U}}, {I,{A,U}}, {U,{}}, {U,{A}}, {U,{I}}, {U,{A,I}}. I am not aware of any language with more than 12 vowel phonemes. These 12 phonemes can be put in one-to-one correspondence with the cardinal vowels of the International Phonetic Alphabet in the following way: i = {I,{}}, e = {I,{A}}, ɛ = {A,{I}}, a = {A,{}}, ɔ = {A,{U}}, o = {U,{A}}, u = {U,{}}, y = {I,{U}}, ø = {I,{A,U}}, œ = {A,{I,U}}, ɤ = {U,{A,I}}, and ɯ = {U,{I}}.

For consonants we must specify voicing, place of articulation, and manner of articulation. Voicing can be specified using the elements ?, L, and H, as follows: plain = {}, voiced = {L}, aspirated = {H}, glottalized = {?}, voiced aspirated = {L,H}, and voiced glottalized = {?,L}. Manner of articulation can also be specified using the elements ?, L, and H, as follows: approximant = {}, liquid = {L}, fricative = {H}, plosive = {?}, trill = {L,H}, nasal = {?,L}, affricate = {?,H}. Place of articulation can be specified using the elements A, I, and U, as follows: labial = {U,{A}}, laminal alveolar = {I,{A}}, apical alveolar = {A,{I}}, laminal postalveolar = {I,{A,U}}, apical postalveolar = {A,{I,U}}, palatal = {I,{}}, labiopalatal = {I,{U}}, velar = {U,{I}}, labiovelar = {U,{}}, uvular = {A,{U}}, pharyngeal = {A,{}}, and glottal = {U,{A,I}},

Thus, a consonant phoneme is specified by a triple (C,V,D), where C and D are subsets of elements {?,L,H}, and V is a set {X,Y} where X is an element of {A,I,U} and Y a subset of {A,I,U} which does not contain X. The vowel phoneme V may be identified with the approximant consonant phoneme ({},V,{}). This leads to the following well-known correspondences: i ~ j, a ~ ʕ, and u ~ w. Vowel phonemes can then be represented using more familiar notation, as follows: i = {I,{}} ~ j, e = {I,{A}} ~ j^ʕ, ɛ = {A,{I}} ~ ʕ^j, a = {A,{}} ~ ʕ, ɔ = {A,{U}} ~ ʕ^w, o = {U,{A}} ~ w^ʕ, u = {U,{}} ~ w, y = {I,{U}} ~ j^w, ø = {I,{A,U}} ~ j^ʕw, œ = {A,{I,U}} ~ ʕ^jw, ɤ = {U,{A,I}} ~ w^ʕj, and ɯ = {U,{I}} ~ w^j. Thus, if V = {X,Y} then the head X determines the primary articulation and the operator Y the secondary articulation.