Why does pressure seem to be an extensive variable considering Dalton's law? I learned that in thermodynamics, pressure is regarded as an intensive variable, while properties that you can add up such as mass or volume, are considered extensive. 
However, to me, Dalton's law seems to contradict this idea. Dalton's law states that the total pressure of a mixture of gases is the sum of the partial pressures of each gas. 
Given this, my question is: if the pressure can be added up, why is pressure intensive? What am I understanding wrong? Why is Dalton's law not a valid argument for pressure being an extensive property? 
To give a simple example, let's say that I have a "mixture" of two same gases (i.e. oxygen gas), at a total pressure of 1 bar. Now, if I "divide" the gas into two mole fractions and determine their partial pressures, the sum of those partial pressures would also be 1 bar. So clearly, in this example, I'm adding the two pressures to get a larger, total pressure. So, to me, this makes it appear that pressure can be an extensive property, since it can be added up. However, I know that I'm wrong somewhere but don't know where, so can someone explain why my example would not prove that pressure is extensive?
 A: 
Given this, my question is: if the pressure can be added up, why is
  pressure intensive? What am I understanding wrong? Why is Dalton's law
  not a valid argument for pressure being an extensive property?

I think it's not valid because what is really being "added up" in Dalton's law is the effect of adding the masses of the individual gases that comprise the mixture. The partial pressure of each gas is the pressure that it would exert, for a fixed volume and temperature, if all the other gases were removed. The greater the number of gas molecules (mass) that occupy a fixed volume the greater the number of collisions per unit area of the walls of the container and thus the greater the pressure. 
Thus, in my view at least, it is the extensive property of mass that is responsible for the "adding up" of the partial pressures per Dalton's Law, as opposed to pressure itself being an extensive property. We know this has to be the case because if pressure were an extensive property, the pressure a gas in a room would be cut in half if we simply divided the room in half.
Hope this helps.
A: I think your idea of what extensive and intensive means might be confusing the question a little. Specifically it's this idea of what you mean by "adding up" a property. If you have a balloon full of gas there's a difference between pumping in a second helping of gas (as with your line of thinking using Dalton's Law) and imagining a second, identical balloon of gas appearing out of thin air. With two balloons things like the temperature, density, and pressure stays the same (intrinsic quantities) while things like the number of particles, mass, energy, and entropy double (extrinsic quantities). 
