## Exactly n

**An extension of section 16.4.3: Exactly n**

**An extension of section 16.4.3: Exactly n**

Recall that "exactly one" can be broken down into "at least one" and "at most one". Thus (1) should be symbolized as (2):

(1) Exactly one student left.

(2) (∃x) (S^{1}x & L^{1}x & (∀y) ((S^{1}y & L^{1}y) ⊃ x = y))

The same principle applies to all sentences containing "exactly*n*", no matter how big *n* is. To symbolize a sentence like (3), we decompose into an "at least" claim, and an "at most" claim.

(3) Exactly two people are tall.

(3) asserts that *at least* two people are tall, so its symbolization should include "(∃x) (P^{1}x & T^{1}x & (∃y) (P^{1}y & T^{1}y & x ≠y))". (Remember that an "at least *n*" sentence requires *n* existentials, and a claim that none of the existentially quantified variables are identical.) (3) also asserts that *at most* two people are tall, so we need to add that anyone who is tall *must* be one of these two people. Thus (3) is correctly symbolized as (4):

(4) (∃x) (P^{1}x & T^{1}x & (∃y) (P^{1}y & T^{1}y & x ≠y & (∀z) ((P^{1}z & T^{1}z) ⊃ (z = x v z = y))))

In general, a statement of the form "there are exactly *n* ⍺" should be symbolized as follows:

**Exactly n Procedure:**

(∃x

_{1}) (⍺1 & … & (∃x

_{n}) (⍺ x

_{n}& x

_{1}≠x

_{2}& … & x

_{1}≠x

_{n}& x

_{2}≠x

_{3}& …. & x

_{n-1}≠x

_{n}& (∀x

_{n+1}) (⍺x

_{n+1}⊃ (x

_{n+1}= x

_{1}v … v x

_{n+1}= x

_{n})))…)

** **

**Exercises**

- There are exactly two frogs. (F
^{1}: is frog) - Exactly two frogs hopped. (H
^{1}: hopped) - Exactly three students passed. (S
^{1}: is a student; P^{1}: passed) - There are exactly three students and they passed.
- Exactly two boys have turtles. (B
^{1}: is a boy; T^{1}: is a turtle: H^{2}: has) - Every boy has exactly two turtles.

** **

**Solutions**

- (∃x) (F
^{1}x & (∃y) (F^{1}y & x ≠y & (∀z) (F^{1}z ⊃ (z = x v z = y))) - (∃x) (F
^{1}x & H^{1}x & (∃y) (F^{1}y & H^{1}y & x≠y & (∀z) ((F^{1}z & H^{1}z) ⊃ (z = x v z = y)))) - (∃x) (S
^{1}x & P^{1}x & (∃y) (S^{1}y & P^{1}y & (∃z) (S^{1}z & P^{1}z & x≠y & y≠z & x≠z & (∀x_{1}) ((S^{1}x_{1}& P^{1}x_{1}) ⊃ (x_{1}=x v x_{1}=y v x_{1}=z)))) - (∃x) (S
^{1}x & (∃y) (S^{1}y & (∃z) (S^{1}z & (∀x_{1}) (S^{1}x_{1}⊃ (x_{1}=x v x_{1}=y v x_{1}=z) & P^{1}x & P^{1}y & P^{1}z))) - (∃x) (B
^{1}x & (∃x_{1}) (T^{1}x_{1}& H^{2}xx_{1}) & (∃y) (B^{1}y & (∃y_{1}) (T^{1}y_{1}& H^{2}yy_{1}& x≠y & (∀z) ((B^{1}z & (∃z_{1}) (T^{1}z_{1}& H^{2}zz_{1})) ⊃ (z=x v z=y)))) - (∀x) (B
^{1}x ⊃ (∃y) (T^{1}y & H^{2}xy & (∃z) (T^{1}z & H^{2}xz & y≠z & (∀x_{1}) ((T^{1}x_{1}& H^{2}xx_{1}) ⊃ (x_{1}=y v x_{1}=z)))))