
> 
> **Motivation**
> 
> 
> Consider the situation: You know that
>  every $x$ that has property $P$ must have property $Q$. $Q$ is a
>  rather strong condition but not strong
>  enough to fulfill $P$. What is *missing*?
> 
> 
> 


Consider the formulas of a first-order language, a distinguished set of axioms $S$, and the formulas with exactly one free variable, factored out by provable equivalence $P(x) \sim\_S Q(x)$ iff $S \vdash P(x) \leftrightarrow Q(x)$ – for short: *properties*.


There is a partial order on the set of properties by $[P(x)] \Rightarrow\_S [Q(x)]$ iff $S\vdash P(x) \rightarrow Q(x)$ – for short: $P \Rightarrow Q$.


Let $P \Rightarrow Q$, but $Q \not\Rightarrow P$, read: $Q$ *is a proper necessary condition of* $P$.



> 
> **Definition**
> 
> 
> A property $D(x)$ may be
>  called a *defect* of $Q(x)$ with
>  respect to $P(x)$ if 
> 
> 
> * $D \not\Rightarrow P$
> * $Q \wedge D \Rightarrow P$
> 
> 
> A defect $D(x)$ may be called *minimal*
>  if there is no other
>  defect $D'(x)$ with $D(x) \Rightarrow D'(x)$.
> 
> 
> 




---



> 
> **Question #1:** Is the search for *(minimal) defects*
>  so manifest – in the working mathematician's life – that it is performed every day without needing a proper name? And is the notion ´– thus – of no (meta-)mathematical importance?
> 
> 
> 




---



> 
> **Question #2:** ... or is there already a proper –
>  and more common – name?
> 
> 
> 




---



> 
> **Question #3:** ... or is the definition above and its presuppositions flawed?
> 
> 
> 




---


**Addendum**


The common divisor graph on the natural numbers shares one strong property $Q$ with the [Rado (= random) graph](http://en.wikipedia.org/wiki/Rado_graph): it contains any finite graph as an induced subgraph (which I guess is a quite general necessary condition for randomness). But it's not isomorphic to the Rado graph, i.e. does not fulfill its defining property $P$. I am now interested in the "defect" of the common divisor graph: which property $D$ - as minimal as possible - prevents the common divisor graph from being truly random?



Deficiency of necessary conditions
Normally your "defect" is called an "additional assumption"/"extra condition"/... and the typical phrase is "the inverse implication also holds under the additional assumption that...". Yes, the search for such things is something that mathematicians do on an everyday basis trying to bridge the gap between what is necessary and what is sufficient. 


The thing you should understand is that in real life it is not necessarily the first priority to have $D$ as weak as possible from the logical standpoint. What often matters much more is that $D$ is easy to verify, holds in many interesting cases, allows one to give an easy and elegant proof, etc. Actually, Joel has already brought the idea of pure logical minimality to its absurd extreme form, so I hardly need to comment more on this issue.


By the way, the assumption that $P\implies Q$ is completely unnecessary in your definition of the "defect"; it makes just as much sense without it. Indeed, the usual story is that we know something ($Q$), we want to conclude something else ($P$), we suspect that the implication $Q\implies P$ is (may be) not always true, but we want this implication not for its own sake but to figure out something about some object $X$, so we ask what other property $D$ $X$ possesses that together with $Q$ will give us $P$. 


