An 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. Clifford's "The Ethics of Belief". This expounds a simple but adequate theory that is summed up by Clifford's dictum:

"It is wrong always, everywhere, and for anyone, to believe anything upon insufficient evidence".