backer board over particle board?

I'm going to be tiling my kitchen, and having looked under the current two layers of vinyl, I've discovered that the subfloor is made of particle board. Can I put quarter inch backerboard directly onto the particle board? I wouldn't mind replacing the subfloor except for the fact that all the cabinets are built ontop of the particle board. Is backer board over particle board an acceptable tiling surface, or do I need to rip out the particle board and put in plywood? If I do need to put in new plywood, would I just cut the particle board as close to the cabinets as possible?
Thanks for the help!

 

Yes, replace the particle board with 3/4" ply and yes cut around thr cabinets. What is the size and spacing of the joist under the floor and the length of its unsupported span? Need to know because you may need an additional layer of ply to stabilize the floor, or some additional supports under the joists.

 

Are you sure it's particle board and not OSB.

 

I went under the house today and found that there is a 15 inch space between joists. Will the 3/4" plywood be sturdy enough? Also, should I just use a circular saw set to the right depth to cut around the cabinets? That would leave about 2 inches of particle board jutting out from under the cabinets.. I think the joists are 2X8 (didn't measure, what's the dividing line between strong enough and not strong enough?) (Also didn't measure the unsuported span) Please let me know what is too wide an unsupported space and what to do if it is too wide. Thanks a lot. This is extremely helpful.