> **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 shares one strong property $Q$ with the [Rado (= random) graph][1]: 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?


  [1]: http://en.wikipedia.org/wiki/Rado_graph