There is a general principle in mathematics that
A trivial object is too simple to be simple.
Quite often, classical references will define ‘simple’ (or an analogous term) in naïve way, so that a ‘trivial’ object is simple, but later it will become clear that more sophisticated theorems (especially classification theorems) work better if the definition is changed so that the trivial object is not simple. Usually this can be done by changing ‘if’ to ‘iff’ (or sometimes changing ‘or’ to ‘xor’) in the classical definition.
But of course one may still find definitions used that disagree.
Not that anybody would be naïve enough to believe otherwise, but perhaps the basic example is that
The point is that, in many cases, the naïve definition imposes only a uniqueness requirement (so that some set of possibilities —such as the set of proper divisors of a prime number, or the set of non-invertible elements of a field— must be a subsingleton) when it should in fact impose an existence and uniqueness requirement (so that the set of possibilities must be a singleton). With truth values, uniqueness is automatic, so existence is easier to notice.
The general pattern is a progression of definitions (of ‘simple’) from more to less naïve:
In many of the above examples one can obtain the sophisticated definition from the naïve definition by replacing a 2-ary function by a function of arbitrary (finite) arity. For example we would replace
Then is not a prime because it is equal to the empty product () but not equal to any of the (because there aren’t any)! Similarly we have:
This illustrates one advantage of using unbiased rather than biased definitions: if one has replaced “for all ” with “for both and ” then it is very easy to forget the “” case and end up with a definition that fails for the trivial case.
In a similar vein we can define path connected by
Then the empty space is not path connected because it has no paths at all and hence no path through the empty subset.