Can definitions become statements?

Question

Can we assign a truth value to a definition after we have define the term? For example, suppose I define $αβγ=\frac{df}{dx}$. Now if after I have define what $αβγ$ means state that $αβγ=\frac{df}{dx}$ isn't this a true a statement? But now the equality has become an equality about a statement and not for a definition?

Even in non-mathematical contexts we could define something. For example, "Man is a rational animal" is a definition. Again if after defining what "man" means if I state "Man is not a rational animal" isn't this a false statement?

So all definitions can be statements?

Answer 1

Indeed, after defining an object or notion by certain properties you can then make the true statement that the defined object or notion has these properties. The statement is then said to be "true by definition".

However, you asked "Can definitions become statements?", to which the answer is no: the definition stays a definition. A statement, even if it coincides almost exactly with some definition, is not that definition. You have to give the definitions of the involved notions and objects first to then make statements about them. So instead of the definition becoming a statement, you now have two things: a definition and a statement.

Answer 2

Yes, definitions can be statements.

For example, in the study of propositional calculi, a definition like

C $\delta$ Np $\delta$ Cp0 could get used.

In the above "C" gets used for material implication, "N" for negation, and '0" for "falsum", and $\delta$ is a functional variable of one argument... or in other words $\delta$ is a variable for the four truth functions which have one argument. The above definition that Np can get defined as Cp0 (that the negation of a proposition can get defined as that proposition implying falsum).

C $\delta$ Np $\delta$ Cp0 is also a tautology, in that all for all values of its variables, and thus we could declare it true, rendering it a statement also.