Evidence is (input or derived) information that has impact on the truth-value of a statement in an inconclusive manner, while proof decides the truth-value a statement in a conclusive manner.
Evidence can be positive or negative, or a mixture of them.
Example: in enumerative induction, a statement summarizes many observations.
Nicod's Criterion: for "Ravens are black" (Is it the same as "All ravens are black" or "The next raven is black"?), black ravens are positive evidence, non-black ravens are negative evidence, and non-ravens are irrelevant.
It is possible to represent evidence qualitatively: all positive (A), all negative (E), some positive (I), and some negative (O). The result is isomorphic to Aristotle's Syllogism.
A quantitative representation is necessary for an adaptive system, since the amount of evidence matters when a selection is made among competitive conclusions.
Example: the multi-extension problem in non-monotonic logic, such as the Nixon Diamond.
The advantage of a numerical measurement of evidence is its generality, not its accuracy. Furthermore, an interpretation of the measurement is required, which usually defines the measurement in an idealized situation.
In ideal situations (ignoring fuzziness, inaccuracy, etc.), amount of evidence can be represented by a pair of non-negative integers. It can be generalized into a pair of non-negative real numbers w+ and w-. We use w for "all available evidence", which is the sum of w+ and w-.
Though in principle, the information is already carried by the amount of evidence, very often a relative and bounded measurement is preferred.
A natural indicator of truth is the frequency (proportion) of positive evidence in all evidence, that is, f = w+ / w.
The limit f, if exists, is the probability for the statement. However, from the value of f alone, whether its limit exists cannot be determined, not to mention where it is.
Under the restriction of insufficient knowledge and resources, the usual assumptions made in mathematical statistics cannot be accepted anymore, and certain interpretations of probability cannot be accepted for degree of believe.
In an open system, all frequency values may be changed by new evidence, and this is a major type of uncertainty — ignorance about the future frequency value.
Related approaches include higher-order probability, probability interval, imprecise probability, Dempster-Shafer theory, etc, though none of them fully satisfies the needs of a non-axiomatic system.
While frequency compares positive and negative evidence, a second measurement, confidence, can compare past and future evidence, in the same manner. Here the key idea is to only consider to a constant horizon in the future, that is, c = w / (w + k).
A high confidence value means the statement is supported by more evidence, so less sensitive to new evidence. It doesn't mean that the statement is "closer to the reality", or the frequency is "closer to the true probability".
The ⟨frequency, confidence⟩ pair can be used as the truth-value of a statement in a non-axiomatic system. It is fully defined on available evidence, without any assumption about future evidence. Also, it captures the uncertainty caused by negative evidence and future evidence.
The three representations: {w+, w}, ⟨f, c⟩, and [l, u] can be transformed into each other. They all represent the system's degree of belief on the statement, or its evidential support.
For the normal statements in an non-axiomatic system, the amount of supporting evidence is finite. There are two limit cases that are discussed in meta-language only: null (zero) evidence and full (infinite, or no future) evidence.
At the input/output interface, the truth-value of a statement can also be represented imprecisely. If there are N verbal labels, the [0, 1] interval can be divided into N equal-width subintervals. Or, default values can be used, so that the users can omit the numbers.
There are reasons to use high-accuracy representations inside the system, while allow low-accuracy representations outside the system.