Evidence and Defeat

Here is a plausible principle that I think ought to be denied:

Necessarily, if one has some evidence for p and no evidence against p, then one's total body of evidence supports p.

I think that this principle should be denied since there is both 'direct' and 'indirect' evidence:

p is direct evidence for q (for S) iff p evidentially supports q all by itself.

p is indirect evidence for q (for S) iff (i)p is not direct evidence for q,(ii) p is direct evidence for not-r, (iii) r is direct evidence for q.

Undercutting defeaters are examples of indirect evidence. I take there to be two kinds of defeaters:

Necessarily, (ignoring the individual) for all propositions x, y and z, x is an undercutting defeater of y regarding z iff y is evidence for z, x is not evidence for or against z, and (y&x) is not evidence for z.

Necessarily, (ignoring the individual) for all propositions x, y and z, x is a rebutting defeater of y regarding z iff y is evidence for z, x is evidence for not-z, and (x&y) is evidence for not-z.

So long as there are undercutting defeaters, then the above principle is false. This is because one can have evidence against what supports p without having evidence against p. As such, one's total body of evidence can fail to support p (though not support not-p) while there being some evidence for p and no evidence against p itself.

This all comes from a discussion I'm involved in here.