The S-shaped curve of language change

Figure 1: The logistic function

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The equations of the logistic

The logistic function:

p =ek+st

1 + ek+st

The time derivative of the logistic:

dp

dt= sp(1− p)

The logit or logistic transform:

ln(p

1− p) = k + st

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Figure 2: The logit or logistic transform function

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The constant rate effect

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Figure 3: Different slopes different intercepts (Bailey)

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Figure 4: Different slopes same intercept (Common origin)

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Figure 5: Same slope different intercepts (CRE)

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Figure 6: the replacement of have by have got in British English

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Figure 7: The loss of V2 word order in French

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Figure 8: The rise of Infl-medial word order in Old English

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Figure 9: The rise of periphrastic do in English

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(1) a. How great and greuous tribulations suffered the HolyAppostels?

b. How great tribulations did the Holy Apostols suffer?

(2) a. ...spoile him of his riches by sondrie fraudes, whiche heperceiueth not.

b. ...which he does not perceive.

(3) a. Quene Ester looked never with swich an eye.

b. Queen Esther never looked with such an eye.

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Table 1: Frequency of non V-to-I sentences bycontext

Aff. Aff. Aff.Neg. Neg. Trans. Intrans. wh- obj.Decl. Ques. Ques. Ques. Ques. never

date % N % N % N % N % N % adv-V N1400-25 0 177 12 17 0 3 0 7 0 1 – 01426-75 1 903 8 25 11 56 0 86 0 27 24 1541476-00 5 693 11 27 14 74 0 68 2 51 35 1861501-25 8 605 59 78 24 91 21 90 11 62 69 1091526-35 14 651 61 56 69 26 20 76 10 63 89 1701536-50 28 735 75 84 62 91 32 116 11 73 90 1521551-75 38 313 85 48 74 57 42 71 36 75 89 88

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Table 2: Slope (s) and intercept (k) parametervalues for logistic regressions of frequency of non

V-to-I sentences against time by context

Affirmative Affirmative AffirmativeNegative Negative Transitive Intransitive wh- object

Declaratives Questions Questions Questions Questions nevers k s k s k s k s k s k

3.74 -8.33 3.45 -5.57 3.62 -6.58 3.77 -8.08 4.01 -9.26 3.76 -5.37

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The actuation of syntactic change

1. The domain of the logistic function runs from −∞ to +∞, so thatit never reaches 100% and has no beginning (point where thepercentage is 0%). In consequence, the function cannot completelymodel any actual linguistic change. We have to add an actuationmechanism and a completion mechanism.

2. We can suppose that when a new option enters a language, it doesso with some very small positive frequency and that a change goesto completion when the losing form drops to such a low percentagethat it is no longer learnable. But how do the different linguisticcontexts behave at these points of discontinuity?

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Figure 10: Intercept differences among logistic curves

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Table 3: Rise in the frequency of do from period 1(1390–1400) to period 8 (1550–1575) in Elleg̊ard’s

database

Affirmative Negative Affirmative NegativeDeclarative Declarative Question Question

per. do N % do N % do N % do N %1 6 45000 .01 0 0 – 0 0 – 0 0 –2 11 4600 .24 0 177 0 0 10 0 2 15 11.83 121 45500 .27 11 892 1.2 6 136 4.2 2 23 8.04 1059 59600 1.78 33 660 4.8 10 132 7.0 3 24 11.15 396 28600 1.38 47 558 7.8 41 140 22.7 46 32 59.06 494 18800 2.63 89 562 13.7 33 69 32.4 34 22 60.77 1564 19200 8.15 205 530 27.9 93 114 44.9 63 21 75.08 1360 14600 9.32 119 194 38.0 72 56 56.2 41 7 85.4

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