A statement S is evidence for (against) a
theory T iff S is known to be true and there is a theory T' that is not known
to be false and T' implies that T is more (less) probable given S.
Put otherwise, and with less appeal to probability theory: Any statement S is evidence for or agains a theory T if and only if theory T becomes more or less probable, plausible, credible or supported once it is known S is t
example is with T = a is honest, S = a is nouveau riche T' = if a is nouveau
riche, a is not honest. The last may be (and fairly should be) a probabilistic
claim, to the effect that the nouveaux riches tend to grow rich with dishonest
means, without insisting this is invariably so.
Note this is a probabilistic characterization of what counts as evidence for
or against a theory T and that it depends on there being another theory T'.
That is formally: S is evidence for T iff (ET')(p(T|T'&S)>p(T|T'), while S is
evidence against T iff (ET')(p(T|T'&S)<p(T|T')
This also covers the cases when T'&S implies T is true or T is false, i.e.
the cases of deductive proof and deductive refutation. (Consider T = s is a
straight line, T' = s is inspected and S = s contains a bend.)
If S is not (yet) known to be true, then S is at best potential evidence
for or against T. And if T' implies that T given S is just a bit more or less
probable than T when not given S then S is weak evidence for or against T.
If S makes T much more or much less probable than when not given S then S is
strong evidence for or against T. Finally, the strength of the evidence
of S for or against T depends on the probability of T': The more probable T'
is the better the support S gives to T, and the less probable T' is the
worse the support S gives to T.
Hence the better sort of evidence one can provide for a theory T is by
means of strong evidence from a theory T' with good support, and the best sort
of evidence one can provide is from a true theory T' that entails a statement
that directly proves or refutes T deductively.
A classic on the importance and proper use of evidence is W.K.
Ethics of Belief". This expounds a simple but
adequate theory that is summed up by
is wrong always, everywhere, and for anyone, to believe anything upon