First order logic

jamborta

hi,

could someone explain to me why the sentence - There are exactly two purple mushrooms is represented in FOL like this:
(Ex)(Ey) mushroom(x) ^ purple(x) ^ mushroom(y) ^ purple(y) ^ ~(x=y) ^ (Az) (mushroom(z) ^ purple(z)) => ((x=z) v (y=z))

especially the last part i have problem with. i assume that i misunderstood some of the definitions which are the basis of FOL, that might be the source of the confusion.

thanks for your help

MathematicalPhysicist

well without the last part with z, you actually stating that there are at least 2 purple mushrooms, while with the last part you stating that there are exactly two.

jamborta

thanks. but i dont understand why it's and OR ((x=z) v (y=z)) which would allow either of them being equal to z.

HallsofIvy

Yes, that's the whole point. If you posit three purple mushrooms, x, y, and z, saying that there are, in fact, only two, the last of them, z, must be the same as either one of the first two. If you start of with two purple mushrooms, x and y, and state they are not the same mushroom, then any third purple mushroom must be the same mushroom as x or y but you don't know which. Yes, definitely, either of them could be equal to z.