Is a "Definition" a statement or a sentence?

Question

I am learning the basics to proof construction. The author of my book states that a "sentence" is a comment that is "too ambiguous" to be evaluated as being true or false. In contrast, "statements" are described as either true or (mutually exclusive) false.

An example of a sentence is: "$x$ is even" ... The rationale for this being "ambiguous" is that we do not know what "$x$" is referring to.

The author claims that by adding a quantifier, we can strip away the ambiguity and render the sentence "able to be evaluated" as either true or (mutually exclusive) false. In this way, quantifiers promote sentences to statements. An example of a statement is "There exists some integer that is even". The way I understand it, the ambiguity is stripped away because we are now talking about a specific collection of elements...and we can evaluate whether or not this collection of elements has the property in question: if they do, then we say TRUE and if they do not, we say FALSE.

With these out of the way...what exactly is a "Definition"? From what I have seen on different forums...definitions are tautologies, and therefore, because they possess "truth" (and therefore must be able to be evaluated) they are presumably a type of statement. However the book's following example makes me wonder.

The book's example of a 'definition' is, "An integer $x$ is even if there exists an integer $y$ such that $x = 2*y$ "

This sounds like a conditional statement of the form "$q$ if $p$ "...but the $q$ is structured like a sentence (i.e. it lacks a quantifier). Although the $p$ has a quantifier (in the form of there exists), the $q$ does not. Therefore this definition is composed of both a sentence and a statement...which makes me wonder if a definition is, in fact, a statement. Could anyone offer some insight?

Would it have been more pedantically correct if the author had, instead, written:

"There exists even integers if there exists an integer $y$ such that $x=2*y$"?

Edit: Moreover, can you even define a "definition"? Wouldn't that be the most circular attempt ever?

Answer 1

Here's how I teach this issue in my mathematics classes.

A definition has three components: (1) The new term that is being defined; (2) Certain objects that are given; and (3) A statement of equivalence that one hereafter takes as a tautology. One side of that equivalence is a statement with the new term and the given objects used together in their correct grammatical setting. The other side of the equivalence is a statement without the new term but with the given objects.

For example:

Definition of an (1) even integer: (2) Given an integer $x$, we say that (3) $x$ is even if and only if there exists an integer $y$ such that $x=2y$.