How do I reduce a negative hint with multiple possibilities?

The more conclusive way to solve these puzzles is using a grid with more possible interactions - not only the interactions with, say: name and hat; and name and drink; but also hat and drink. Try a grid like this:

What you call a "positive clue" say "The coffee drinker likes dogs" would be incorporated by ticking the box that intersects "Coffee" and "Dogs", and crossing out all the other possibilities like "Coffee" and "Cats", "Horses", ...

A negative clue like "The coffee drinker does not like dogs" would be incorporated by crossing this box.

Then things can start to interact - if there are all but one crosses in a row/column, the remaining one must be the answer. - if the British has White ticked, and White has Tea ticked, then you can tick White and Tea, and all other in those rows and columns can be crossed.

This method has everything the other method has, and more.

Actually the solutions are pretty straight forward.

1) You want to set one row fix. To choose one count the number of green leads (Filter positive + row) and choose the one with the highest amount. In case of tie do the same for negative hints. Form the experience this row is mostly in the lower area.

2) You can fully eliminate all hints for the row from 1). Keep the filter on this row and eliminate all hints until it says no lead left, filter resets automatically at this point.

3) Keep eliminating leads for all rows now just as you did in 2)

4) synchronize green leads. The Highlights in the grid must be in both rows of one Column. Die example you can reduce

GGGX XGGG

to

XGGX XGGX

5) Repeat 3 and 4 whenever possible. Look out for Icons which can be only in one Column left and have an eye for passively fixed icons since the don't remove the Icon passively from other Column. Keep it clean.

6) In most Cases repeating 3-5 already leads you to the solution. If not , then not much is left. I don't why but I went very well by guessing the Icon with the least obvious impact. For example you have

XRRX RRRX

then I guess

XRRX RXXX

Especially if several hints would lead to this same guess. Obviously repeat 3-5 after.

One method I use is something I learned from Sudoku. If there is a pair of the same 2 items in two different boxes, you can eliminate those two items from all other boxes in that line. It even works for trios and quartets, even if all 3 or 4 boxes don't have all of the items. As long as all the boxes don't have anything else. For a trio, it might look like this: ABCDEFGH, AB, AC, ABC, ABCEF, ABCGH You can reduce down to DEFGH, AB, AC, ABC, EF, GH.