It is also possible to give a somewhat simpler symbolic representations of the quantifier phrase exactly n Cs than we get by way of truth-functional compounds of at least-m forms. Here are a couple of approaches for the case of exactly 1:
I forgot just one thing
Something is such that (I forgot it and nothing else)
∃x I forgot x and nothing else
∃x (I forgot x ∧ I forgot nothing other than x)
∃x (Fix ∧ nothing other than x is such that (I forgot it))
∃x (Fix ∧ (∀y: y is other than x) ¬ I forgot y)
∃x (Fix ∧ (∀y: ¬ y = x) ¬ Fiy)
∃x (Fix ∧ ∀y (¬ y = x → ¬ Fiy))
I forgot just one thing
Something is such that (I forgot it and it was all I forgot)
∃x I forgot x and x was all I forgot
∃x (I forgot x ∧ x was all I forgot)
∃x (Fix ∧ everything I forgot is such that ( x was it))
∃x (Fix ∧ (∀y: I forgot y) x was y)
∃x (Fix ∧ (∀y: Fiy) x = y)
∃x (Fix ∧ ∀y (Fiy → x = y))
[F: λxy (x forgot y); i: me]
And, in general, Exactly one thing is such that (... it ...) can be analyzed as any of the following (where θx abbreviates ... x ...):
| ∃x (θx ∧ (∀y: ¬ y = x) ¬ θy) | ∃x (θx ∧ ∀y (¬ y = x → ¬ θy)) |
| ∃x (θx ∧ (∀y: θy) x = y) | ∃x (θx ∧ ∀y (θy → x = y)) |
The forms in columns are equivalent by the symmetry of identity and the following equivalences:
(∀x: ρx) θx ⇔ (∀x: θx) ρx
φ → ψ ⇔ ψ → φ
The first of these is traditionally called contraposition and that name is sometimes used for the second also. The first licenses the restatement of Only dogs barked by Everything that barked was a dog. The second would apply to the same pair of sentences when they are represented using unrestricted quantifiers and also to the restatement of The match burned only if oxygen was present by If the match burned, then oxygen was present.
The initial unrestricted quantifier in the above analyses of exactly 1 thing can also be replaced by a restricted quantifier. The following analysis of a slightly more complex example uses this sort of variation on the second pattern above:
I forgot just one number
Some number I forgot is such that (it was all the numbers I forgot)
(∃x: x is a number I forgot) x was all the numbers I forgot
(∃x: x is a number ∧ I forgot x) every number I forgot is such that (x was it)
(∃x: x is a number ∧ I forgot x) (∀y: y is a number I forgot ) x was y
(∃x: Nx ∧ Fix) (∀y: y is a number ∧ I forgot y) x was y
(∃x: Nx ∧ Fix) (∀y: Ny ∧ Fiy) x = y
And, in general, Exactly 1 C is such that (... it ...) can be analyzed as
(∃x: x is a C ∧ ... x ...) (∀y: y is a C ∧ ... y ...) x = y
The analogous variation on the first pattern would be
(∃x: x is a C ∧ ... x ...) (∀y: y is a C ∧ ¬ y = x) ¬ ... y ...
In the case of, I forgot just one number, this pattern would amount to saying Some number that I forgot is such that I forgot no other number.
The sentence There is exactly 1 C can be understood as Exactly 1 C is such that (it is) and the dummy predicate λx (x is) can be dropped to yield the analysis
(∃x: x is a C) (∀y: y is a C) x = y
which can be understood to say Some C is such that (it is all the Cs there are).
This sort of pattern will be important for the analysis of definite descriptions in 8.4.1, but the first approach (i.e., by way of nothing else) is probably the more natural way of extending the analysis to claims of exactly n for numbers n > 1—as in the following example:
Exactly 2 things are in the room
2 things are such that (they are in the room but and nothing else is)
∃x (∃y: ¬ y = x) x and y are in the room but and nothing else is
∃x (∃y: ¬ y = x) ((x is in the room ∧ y is in the room) ∧ nothing other than x and y is in the room)
∃x (∃y: ¬ y = x) ((Nxr ∧ Nyr) ∧ (∀z: z is other than x and y) ¬ z is in the room)
∃x (∃y: ¬ y = x) ((Nxr ∧ Nyr) ∧ (∀z: z is other than x ∧ z is other than y) ¬ Nzr)
∃x (∃y: ¬ y = x) ((Nxr ∧ Nyr) ∧ (∀z: ¬ z = x ∧ ¬ z = y) ¬ Nzr)
[N: λxy (x is in y); r: the room]
The general forms for exactly 2 things are such that (... they ...) and exactly 2 Cs are such that (... they ...) along these lines are the following (using θ for λx (... x ...) and ρ for λx (x is a C)):
∃x (∃y: ¬ y = x) ((θx ∧ θy) ∧ (∀z: ¬ z = x ∧ ¬ z = y) ¬ θz)
(∃x: ρx) (∃y: ρy ∧ ¬ y = x) ((θx ∧ θy) ∧ (∀z: ρz ∧ ¬ z = x ∧ ¬ z = y) ¬ θz)
Notice that the restricting predicate ρ is added to each of the three quantifiers in the second. In particular, Exactly 2 boxes are in the room means 2 boxes are such that (they are in the room and no other boxes are) rather than 2 boxes are such that (they are in the room and nothing else is), which says that two boxes are the only things in the room.