The S-shaped curve of language change
Figure 1: The logistic function
Rosae: lecture 1
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The equations of the logistic
The logistic function:
ek+st
p=
1 + ek+st
The time derivative of the logistic:
dp
= sp(1 − p)
dt
The logit or logistic transform:
p
ln(
) = k + st
1−p
Rosae: lecture 1
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Figure 2: The logit or logistic transform function
Rosae: lecture 1
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The constant rate effect
Rosae: lecture 1
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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
Rosae: lecture 1
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Figure 7: The loss of V2 word order in French
Rosae: lecture 1
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Figure 8: The rise of Infl-medial word order in Old English
Rosae: lecture 1
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Figure 9: The rise of periphrastic do in English
Rosae: lecture 1
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(1)
(2)
(3)
Rosae: lecture 1
a. How great and greuous tribulations suffered the Holy
Appostels?
b. How great tribulations did the Holy Apostols suffer?
a. ...spoile him of his riches by sondrie fraudes, whiche he
perceiueth not.
b. ...which he does not perceive.
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 by
context
date
1400-25
1426-75
1476-00
1501-25
1526-35
1536-50
1551-75
Rosae: lecture 1
Neg.
Decl.
%
N
0 177
1 903
5 693
8 605
14 651
28 735
38 313
Neg.
Ques.
% N
12 17
8 25
11 27
59 78
61 56
75 84
85 48
Aff.
Trans.
Ques.
% N
0
3
11 56
14 74
24 91
69 26
62 91
74 57
Aff.
Intrans.
Ques.
%
N
0
7
0
86
0
68
21
90
20
76
32 116
42
71
Aff.
wh- obj.
Ques.
% N
0
1
0 27
2 51
11 62
10 63
11 73
36 75
never
% adv-V
–
24
35
69
89
90
89
N
0
154
186
109
170
152
88
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Table 2: Slope (s) and intercept (k) parameter
values for logistic regressions of frequency of non
V-to-I sentences against time by context
Negative
Declaratives
s
k
3.74 -8.33
Rosae: lecture 1
Negative
Questions
s
k
3.45 -5.57
Affirmative
Transitive
Questions
s
k
3.62 -6.58
Affirmative
Intransitive
Questions
s
k
3.77 -8.08
Affirmative
wh- object
Questions
s
k
4.01 -9.26
never
s
3.76
k
-5.37
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The actuation of syntactic change
1. The domain of the logistic function runs from −∞ to +∞, so that
it never reaches 100% and has no beginning (point where the
percentage is 0%). In consequence, the function cannot completely
model any actual linguistic change. We have to add an actuation
mechanism and a completion mechanism.
2. We can suppose that when a new option enters a language, it does
so with some very small positive frequency and that a change goes
to completion when the losing form drops to such a low percentage
that it is no longer learnable. But how do the different linguistic
contexts behave at these points of discontinuity?
Rosae: lecture 1
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Figure 10: Intercept differences among logistic curves
Rosae: lecture 1
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Table 3: Rise in the frequency of do from period 1
(1390–1400) to period 8 (1550–1575) in Ellegård’s
database
per.
1
2
3
4
5
6
7
8
Affirmative
Declarative
do
N
%
6 45000
.01
11
4600
.24
121 45500
.27
1059 59600 1.78
396 28600 1.38
494 18800 2.63
1564 19200 8.15
1360 14600 9.32
Rosae: lecture 1
Negative
Declarative
do
N
%
0
0
–
0 177
0
11 892
1.2
33 660
4.8
47 558
7.8
89 562 13.7
205 530 27.9
119 194 38.0
Affirmative
Question
do
N
%
0
0
–
0
10
0
6 136
4.2
10 132
7.0
41 140 22.7
33
69 32.4
93 114 44.9
72
56 56.2
Negative
Question
do N
%
0
0
–
2 15 11.8
2 23
8.0
3 24 11.1
46 32 59.0
34 22 60.7
63 21 75.0
41
7 85.4
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