Since $p=F/A$, sharp blades with smaller areas exert higher pressure and can cut vegetables easily whereas paper can't although it has a smaller area. I tried to cut a vegetable using newspaper (which has a small area) but was unable to.
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In short, the force that you apply on one section of the paper is not necessarily the same as the force that the other end of the paper applies on the vegetable.
When you're calculating the pressure exerted on the vegetable, the relevant quantity is the force being exerted by the paper on the vegetable. This force has to be transmitted from one end of the paper to the other, because you're pushing on the end not directly in contact with the vegetable.
If a material is sufficiently rigid, then force can be transmitted through it without modification. If not, the material deforms (bends, squishes, buckles, etc.), and the force used for that deformation cannot be transmitted through the material. Paper is not very rigid (even less so when wet), and so not much force can be transmitted through it, normally. When you push on one end of the paper, it easily bends or buckles instead of transmitting the force.
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