When an edge is dull it takes a lot of force for it to generate the necessary pressure on a material to cause it to rupture, when it is very sharp this force drops down very low levels. Thus the two competing actions are how much force can the material take before you see it deform macroscopically (break) or just move out of the way of the blade vs how much to break microscopically (cut). Just think of a ballon, if a pin is sharp enough you can jab a ballon with it and the ballon will be "cut" before it will move under the impact of the needle. However if the pin is blunt you just push the ballon around.
What does this have to do with the question ?
When you slice with a blade you pull the paper towards you, this puts the paper right under the edge under tension effectively making it more rigid and this means it will take more force for it to be compressed down as a whole and thus it will break at a microscopic level easier. This is why ballons which are very tightly filled are much easier to pop than slack ones because the slack ones will just deform under the pin. This is also why it would be easier to pop a ballon with a blunt pin if someone held it in place. You can see this directly if you try to push cut newsprint vs photocopy paper. It is *much* easier to push cut the photocopy paper as it is more rigid. You can also see it directly on the newsprint if you just pull it tight and then it will enable a push cut much more readily.
The second reason is that when you do a slice you are essentially smearing the total amount of work done over a long distance. The longer the edge you are willing to draw to make a cut the less force you need to apply because the total work (force * distance) will still be high. Because you are using less force this means the paper needs to be less rigid to not bend and thus knives can cut it with duller edges. You can see this if you use a saw to cut a 2x4 for example and see how much force it takes to cut say one inch deep for a two foot draw vs a six inch draw. The more you are willing to smear out the force the lower it will go.
There are other reasons as well such as when slicing the blade could be very sharp in some regions but dull in others but still make a cut on a slice it just won't do very much on the dull regions. However if you try to push cut through using one of the dull spots it fails readily. Thus on a slice the material will see in effect an "average" sharpess. Slicing also allows does other things which reduce the force needed to apply like haul the cut debris out of the way, this all gets compacted in the cut on a push. You notice this obviously when you use a saw vs axe for example. An efficient saw hauls all the cut wood out of the way of the edge which keeps it cutting efficiently.
In regards to fine/coarse edges, this is a different question and coarse edges can shave readily and they don't uniformly break down faster in use. This is a myth which Swaim debunked on rec.knives. Coarse edges are both sharper and have better edge holding when slicing. Polished edges are sharper and have better edge holding on push cuts. The same general principle extends to steels which have a high vs low carbide fraction as noted by Johnston, Landes and Verhoeven (fine structure for push cuts) and Boye and Dozier (really coarse structure for slicing).
-Cliff
