Formal Logic 101 – Part 23: Identity and Definite Descriptions (Part 2)

Specifying Numbers:


Consider statements such as “there are two senators from Oregon”, or “there are nine planets”, etc. using identity descriptors. For instance, if look at the phrase “there are two senators from Oregon” we could symbolize it as
 
(∃x)(∃y){[(Sx ^ Ox) ^ (Sy ^ Oy)] ^ x≠y}
Or “there is an x and a y, such that x is a senator and is from Oregon, and y is a senator and is from Oregon, and x and y are not the same person.”

Similarly, if we wanted to say “there are three representatives from New Mexico” we could symbolize that as
(∃x)(∃y)(∃z){[(Rx ^ Nx) ^ (Ry ^ Ny)] ^ (Rz ^ Nz)} ^ [(x≠y ^ x≠z) ^ y≠z]
Or “There is an x, a y, and a z, such that x is a Representative and from New Mexico, y is a Representative and from New Mexico, and z is a Representative and from New Mexico, and x is not y, x is not z, and y is not z”.

Obviously, if we really wanted to, we could say “there are nine planets” but it is worth noting that numbers above 3 or 4 rapidly start becoming unwieldy.

Specifying “At Most” amounts

Moving on, we already know that we can specify that there are “at least” a certain number of people using “(∃x)”, for instance, “at least one person likes football” or “(∃x)Fx”.

On the flip side, we can use (x) to create “at most” statements. Suppose that there were an argument between an atheist and a monotheist, where both grant that there certainly is not more than one god, the point of contention is whether the correct number is 0 or 1. It would be fair to say that these two agree that “there is, at most, one god”. (Do note that to say that there is “at most, one” of something does not imply that one actually exists. It leaves the open the possibility that there are none.) We could symbolize this as:

(x)(y)[(Gx ^ Gy) → x=y]
Or “For whatever you plug in for x and y, if x is a god and y is a god, then x and y must be the same thing.” So any attempt to locate two things that are gods will result in you just listing the same thing twice, which means there is at most one.

If we wanted to say, “there are at most 2 millionaires”, we would say,
(x)(y)(z){[(Mx ^ My) ^ Mz] → [(x=y v x=z) v y=z]}
Or “For whatever you plug in for x, y, and z, if x, y, and z are all millionaires, then either x and y are the same person, x and z are the same person, or y and z are the same person.” Which sets our possible upper limit of millionaires at “two”.

Specifying an Exact Number

Lastly, we can specify an exact number of things. For instance, let's go back to our hypothetical argument between the atheist and the monotheist. The monotheist would presumably not be satisfied with “there is at most one god” but would want to go on to insist that there is EXACTLY one god. To symbolize this statement, we would say:

(∃x)[Gx ^ (y)(Gy → y=x)]
“There is at least one x such that x is a god, and no matter what you plug in for y, if y is a god, then y and x are the same thing.” Or “there is at least, and at most, one god”.

Looking back, you may wonder about our previous example using Oregon’s Senators. After all, we were saying that there are two of them, but we never specified that there are “at most, two”, even though we all know that each state only gets to have two Senators. The previous symbolization:

(∃x)(∃y){[(Sx ^ Ox) ^ (Sy ^ Oy)] ^ x≠y}
leaves open the possibility that there are more than two. Recall that “(∃x)” is our way of saying “some” or “at least one”, so the above symbolization states that there are “at least two” Senators from Oregon. If we wanted to specify “no, no. I mean there are EXACTLY two of them.” We would want to say:
 
(∃x)(∃y)({[(Sx ^ Ox) ^ (Sy ^ Oy)] ^ x≠y} ^ (z)[(Sz ^ Oz) → (z=x v z=y)])
“There is at least one x and y, such that x is a senator and from Oregon, and y is a senator and from Oregon, and x is not y; and for whatever we plug in for z, if z is a senator and from Oregon, then z must either be x or y.” or “there are at least, and at most, two senators from Oregon.

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