Implicit Definitions, Part 3: Contextual Definition
I've said that an explicit definition introduces a new expression by stipulating that it be semantically equivalent to an old expression. If there are no non-explicit definitions, this means that you can only define expressions that are in principle redundant. Aren't there counterexamples to this claim?
Consider the definition of the propositional connectives. We can
explicitly define some of them with the help of others, but what if we want
to define all of them from scratch? The common strategy here is to
recursively provide necessary and sufficient conditions for the truth of a
sentence governed by the connective: A 
B is true iff
A is true and B is true.
This definition does not quite provide an explicit equivalent for ''
in old vocabulary. Nevertheless it shows how to translate any sentence
containing '' into an equivalent sentence not containing ''.
For example, "it rains it snows" can be translated into "'it rains'
is true and 'it snows' is true".
In this respect, the definition resembles typical contextual definitions, like any definition of "the average man" and Russell's definition of "the". Even though these definitions introduce only a small bit of vocabulary, they only provide translation rules into old vocabulary for much larger items. I'm inclined to say that they explicitly define these larger items. That's why from "nobody is 40 years old" and "the average man is 40 years old" you can't infer that there is somebody who is 40 years old: These sentence only appear to be of the form "a is F", with "a" being a newly defined expression. In fact, the new expression is the unstructured "nobody is ..." or "the average man is...". Similarly, if "the direction of" is defined by the abstraction principle "the direction of A = the direction of B iff A and B are parallel", then from "the direction of A = the direction of B" you can't infer that there is something which is the direction of B. For what has really been defined (and explicitly defined) is the unstructured new expression "the direction of ... = the direction of ...".
Here I should say something about definitions via introduction and elimination rules. If these rules match, they are just necessary and sufficient conditions (introduction rules expressing the sufficient ones and elimination rules the necessary ones) and can be read as instructions for translation into old vocabulary. I'm not so sure about what to say when this is not the case, as with the quantifier rules.