On Wednesday, September 18, 2013 6:11:53 PM UTC-5, Rotwang wrote: > Definitions cannot give rise to contradictions.
A definition is generally based on premises (called "preconditions") and those premises can be false, in which case the definition gives rise to contradiction. For instance, the very statement of a definition may make reference to an object in the singular that itself happens (in fact) to not be uniquely defined, such as a "function" that is not a function.
In most formal math treatments, before a definition can be posed, in many cases, one first needs to prove an "enabling" lemma that assures that the definition in fact does not give rise to contradiction -- before even stating the definition.
Patrick Suppes in his book on Set Theory laid out the foundation for formulating definitions, specifically addressing this issue, as well as others.