Proof of Well-Formedness Condition clauses

In addition to the given machinery of linguistic time, which is assumed for any linear representation of elements, if we additionally assume that there are two places, a vowel ``place'' (call it V), and a tone ``place''(call it T), and we refer to all and only the elements occurring at V as vowels and we refer to all and only the elements occurring at T as tones, then all three parts of the WFC follow. Multilinearity (condition C above) says that at each time there exists an element at each place. Therefore

D: At each time there is an element at V (a vowel).
E: At each time there is an element at T (a tone).

Proof of WFC #1: Choose an arbitrary vowel, call it v. By universality and immediacy, v occurs at some time, 77#77. But E states that there exists a tone at every time so there must a tone at 77#77. Therefore there is a tone at the same time as that vowel. Since the vowel was chosen arbitrarily, this is true for every vowel.

Proof of WFC #2: A symmetrical argument applies, with ``tone'' and ``vowel'' interchanged, and ``D'' replacing ``E''.

Proof of WFC #3: Association lines in non-linear phonology correspond to simultaneous occurrence in an LT system. The corresponding claim made by clause 3 is that if an element 78#78 precedes another element 79#79, then elements cotemporal with 78#78 and 79#79, say 80#80 and 81#81, may not occur in the opposite order as their cotemporal elements. Considerthe following propositions, i-iv, where 19#19 encodes ``precedes'' and @ encodes association lines or simultaneity.

i: 78#78@80#80.
ii: 79#79@81#81.
iii: 82#82.
iv: 83#83.

Clause 3 of the Well-Formedness Condition essentially states that given i-iii, iv is ruled out. However, this follows trivially from fundamental assumptions about Linguistic time-and-place. By universality and immediacy, 78#78 occurs at some time; let's call it 1. By proposition i, 80#80 is also at time 1. Similarly, 79#79 and 81#81 occur at some time, which we may call 2. Since 78#78 precedes 79#79, the time, 1, at which 78#78 occurs, precedes the time, 2, at which 79#79 occurs. But since 80#80 follows 81#81, the time at which 80#80 occurs, 1, follows the time at which 81#81 occurs, 2. But by linearity, one time may not both precede and follow another. Therefore 1 cannot both precede and follow 2, and one of our assumptions must be incorrect. To maintain propositions i-iii, we must abandon iv. QED.

The Well-Formedness Condition is at the very heart of non-linear phonology. The derivation of Clauses 1, 2, and 3 of the WFC from independent principles may therefore be considered an important achievement of an explicitly spatio-temporal perspective on linguistic representations.^A.6