Final answer:
The formula ∀x∀y((D(x)&D(y)) → x=y) only ensures there are not two different dogs but does not necessarily imply the existence of any dog. To express 'there is exactly one dog,' we must use ∃x(D(x) & ∀y(D(y) → y=x)), combining existence and uniqueness. Without existence, the expression could be true even if no dogs existed.
Step-by-step explanation:
You are correct that the formula ∀x∀y((D(x)&D(y)) → x=y) does indeed express that there is exactly one dog, but it is missing a crucial element. It guarantees that there cannot be two different dogs, but it does not ensure that there is at least one dog. For a complete expression that there is exactly one dog, we would use the formula ∃x(D(x) & ∀y(D(y) → y=x)). This formula states that there exists an x such that x is a dog, and for all y, if y is a dog, then y must be the same as x. This ensures both the existence of at least one dog and that there is no more than one dog.
Using our understanding of logical constructs, we can see that to formally express 'there is exactly one,' we need two components: existence (∃) and uniqueness (∀). The existence of at least one dog is shown by the part ∃x(D(x)), and the uniqueness is guaranteed by the part ∀y(D(y) → y=x). Without the existential quantifier, the statement would not necessarily imply that any dog exists at all; it would only preclude the possibility of more than one dog. Indeed, without asserting that there is at least one dog, the formula could be true in a world with no dogs, which would not fit the intention of stating that 'there is exactly one dog.'