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Specifying how many things are F

There are at least ___ F's

One: (x) Fx
Two: (x) (y) (Fx & Fy & x y)
Three: (x) (y) (z) (Fx & Fy & Fz & x y & x z & y z)
Four: (x) (y) (z) (w) (Fx & Fy & Fz & Fw & x y & x z & x w & y z & y w & z w)
n: (x₁) ... (x_n) (Fx₁ & ... & Fx_n & x₁ x₂ & ... & x_n-1 x_n)

There are at most ___ F's

One way to say "at most n things are F" is to say

~ "at least n+1 things are F"

by using the symbolization above. Another way is to say it directly.

One: (x) (y) [(Fx & Fy) x = y]
Two: (x) (y) (z) [(Fx & Fy & Fz) (x = y v x = z v y = z )]
Three: (x) (y) (z) (w) [(Fx & Fy & Fz & Fw) (x = y v x = z v x = w v y = z v y = w v z = w)]
n: (x₁) ... (x_n+1) [(Fx₁ & ... & Fx_n+1) (x₁ = x₂ v x₁ = x₃ v ... v x₁ = x_n+1 v x₂ = x₃ v ... v x₂ = x_n+1 v ... v x_n = x_n+1)]

There are exactly ___ F's

One way is to say "exactly n things are F" is to say

"at least n things are F" & "at most n things are F"

by using the symbolizations above. Another way is to say it directly:

Zero: (x) (~Fx)
One: (x) (y) [Fx & (Fy y = x)]
Two: (x) (y) (z) {Fx & Fy & [Fz (z = x v z = y)] & x y}
Three: (x) (y) (z) (w) {Fx & Fy & Fz & [Fw (w = x v w = y v w = z)] & x y & x z & y z}
n: (x₁) ... (x_n) (y) {Fx₁ & ... & F_n & [Fy (y = x₁ v ... v y = x_n)] & x₁ x₂ & ... & x_n-1 x_n}

Specifying how many things there are

This can be done using the symbolizations above, but leaving out the F's. For instance, "There are at least 2 things [in the UD]" may be symbolized

(x) (y) (x y)