8.3.3. Exactly n
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:
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:
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:
And, in general, Exactly 1 C is such that (… it …) can be analyzed as
The analogous variation on the first pattern would be
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 [ _ is] can be dropped to yield the analysis
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 8.4.2, 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:
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 [… x …]x and ρ for [ _ is a C]):
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.