Peirce proposed his ''thermometer for the proper intensity of belief'' [6] for several reasons.

- First because of
considerations that when the odds
go to zero or to infinity, then the intensity of belief on either
hypothesis goes to infinity;
^{46}when ``an even chance is reached [the feeling of believing] should completely vanish and not incline either toward or away from the proposition.'' [6] The logarithmic function is the simplest one to achieve the desired feature. (Another interesting feature of the odds is described in footnote 16.) - Then because
(expressing the question in our terms),
if we started from a state of indifference (initial odds equal to 1),
each piece of evidence should produce odds equal
to its Bayes factor [our
]. The combined odds will be
the product of the individual odds
[Eq. 19].
But, mixing now Pierce's and our terminology,
when we combine several arguments (pieces of evidence),
they ``ought to produce a belief equal to the sum
of the intensities of belief which either would produce
separately''. [6] Then ``because we have seen that the chances
of independent concurrent arguments are to be multiplied together
to get the chance of their combination, and therefore
the quantities which best express the intensities
of belief should be
such that they are to be
*added*when the*chances*are multiplied...Now, the logarithm of the chance is the only quantity which fulfills this condition''. [6] - Finally, Peirce justifies his choice by the fact that human perceptions go often as the logarithm of the stimulus (think at subjective feeling of sound and light - even `utility', meant as the `value of money' is supposed to grow logarithmically with the amount of money): ``There is a general law of sensibility, called Fechner's psychophysical law. It is that the intensity of any sensation is proportional to the logarithm of the external force which produces it.''[6] (Table 1 provides a comparisons between the different quantities involved, to show that the human sensitivity on probabilistic judgement is indeed logarithmic, with a resolution about the first decimal digit of the base 10 logarithms.)

As far as the logarithms in question, I have
done a short research on their use,
which, actually, lead me to discover Peirce's
*Probability of Intuition* [6]
and Good's *Probability and the weighing of
Evidence* [7].^{47}As far as I understand, without pretension
of completeness or historical exactness:

- Peirce' `chances' are introduced as if they were our odds,
but are used if they were Bayes factors
(``the chances of
independent concurrent arguments are to be multiplied together
to get the chance of their combination'' [6]).
Then he takes the
*natural*logarithm of these `chances', to which he also associates an idea of*weight of evidence*(``our belief ought to be proportional to the weight of evidence, in the sense, that two arguments which are entirely independent, neither weakening nor strengthening each other, ought, when they concur, to produce a belief equal to the sum of the intensities of belief which either would produce separately'' [6]). - According to Ref. [8]
the modern use of the logarithms of the
odds seem to go back to I.J. Good, who used
to call
*log-odds*the*natural*logarithm of the odds.^{48} - However, reading later Ref. [8] it is clear that
Good, following a suggestion of A.M. Turing,
proposes a decibel-like (db)
notation
^{49}, giving proper names both to the logarithm of the odds and to the logarithm of the Bayes factor:- ``
db ...
may be also described as the
*weight of evidence*or amount of information for given '' [7]; - ``
db may be called the
*plausibility*corresponding to odds '' [7].

- ``
db ...
may be also described as the
- Decibel-like logarithms of the odds are used
since at least forty years with under the name
*evidence*. [23].

- `plausibility' is difficult to defend, because it is too similar to probability in everyday use, and, as far as I understand, has decayed;
- `weight of evidence' seems to be a good choice, for the reasons already well clear to Peirce.
- `evidence' in the sense of Ref. [23]
seems, instead, quite bad for a couple of reasons:
- First, because `evidence' has already too many meanings, including, in the Bayesian literature, the denominator of the r.h.s. of Eq. (3).
- Second, because this name is given to the logs of the odds (including the initial ones), but not to those of the Bayes factors to which no name is given. Therefore, the name `evidence', as used in Ref. [23] in this context, is not related to the evidence.

So, in summary, these are the symbols and names used here:

**JL**- is the
*judgement leaning*in favor of hypothesis and against , with the conditions in parenthesis. If we only consider an hypothesis () and its opposite , that could be possibly related to the occurrence of the event or its opposite , also the notation JL, or JL, will be used (as in table 2 of Appendix I).

(Sometimes I have also tempted to call a JL `intensity of belief' if it is clear from the contest that the expression does not refer to a probability.) **JL**- , with the
same meaning of the subscript and of the argument, is the
variation of judgement leaning produced by
a piece of evidence and it is called here
*weight of evidence*, although it differs by a factor from the analogous names used by Peirce and Good^{51}.

Giulio D'Agostini 2010-09-30