# Does a definition have a truth value?

Can we speak of true and false definitions, and if so, in what context? Is saying what x is different from saying the definition of x? Would it perhaps be better to say that a statement which affirms the content of a definition is what has truth value, rather than it being the definition itself?

• In a logic context, a definition is an axiom; thus, it is true (in the interpretation of the theory). – Mauro ALLEGRANZA Oct 17 '17 at 6:12
• @MauroALLEGRANZA Would you be able to expand on that? – Niwilger Oct 18 '17 at 21:58

For the logical analysis of definitions, we can see The logic of definitions.

According to a view, definitions are a sort of "typographical conventions". We introduce in the meta-theory a new symbol that stands for a "complex" expression formed with the primitive symbols of the language.

For example, in first order arithmetic, we define the binary predicate n < m as an abbreviation for the formula ∃k (m=n+(k+1)).

According to a more subtle analysis, a new symbol (name, predicate) is introduced through an expansion of the language L of the theory.

This expansion L+ must be conservative, in the sense that no new formulas expressed in the original language L must be provable by the introduction of the new symbol.

In addition, we want that the new symbol (the definiendum) can be eliminable replacing in a context its occurrence with the "original" expression (the definiens).

The Conservativeness criterion can be made precise as follows.

Any formula of L that is provable in L+ is provable in L.

That is, the definition does not enable us to prove anything new in L.

The Eliminability criterion can be made precise thus:

For any formula A of L+, there is a formula of L that is provably equivalent in L+ to A.

A simple example can help: we can introduce a new "name" (an individual constant) a through the definition:

Def: a=x ≡ ψ(x)

provided that we can prove that there exists a unique ψ(x).

This amounts to proving in the theory that

∃x ψ(x) ∧ ∀y (ψ(y) → y=x).

To say that the above formula is provable in the theory means that it is true in every model of the theory (i.e. in every interpretation that satisfies the axioms of the theory).

Thus, the said formula guarantee that in every model of the theory there is "a unique object such that ψ holds of it".

Adding the stipulation expressed by the definition, we call that object a.

In this sense, we can say that "the existence of a is true by definition".

I would say that a definition that is internally contradictory is "false". Application of such a definition would always yield a false statement. "A triangle has three sides and four internal angles." Such a definition would produce a series of mutually contradictory answers when applied in trigonometry.

You wouldn't define something to be false unless your intentions was to define it as such. But in the norm , definitions are ground rules on which you build your case. If you are to question the definition itself on its validity you would at the end use definitions in return to make a case on its validity. So you do need to have a particular ground on which you can agree on its trueness. So yeah, definitions you make are true by default in a subjective universe you build until the definition you made is questioned in which case , your definition is replaced, altered or stays the same. Whether your definitions are objetively true in any context is another case.