Using regression to prove that X is a necessary / condition condition for Y I have been thinking about how to use regression to prove whether X factor is a necessary or a sufficient condition for Y. I am employing these terms in the traditionally logical sense, i.e.


*

*If no X, then no Y --> X is necessary for Y

*If X, then Y --> X is sufficient for Y


The reason I'm asking is because in social sciences theories we talk about necessary and sufficient conditions all the times (e.g., "(If) No bourgeoisie, (then) no democracy"). At the same time, the most prevalent tool is regression, yet it seems ill-fitted to investigate necessary / sufficient causes.
Indeed, rarely do we talk about necessity and sufficiency in the context of regression. The usual interpretation is that X "causes" Y in an additive sense -- i.e. a certain change in X is associated / caused a certain change in Y.
So, is it possible to interpret regression results as evidence of necessity versus sufficiency? In contrast, a simple 2-by-2 table of X and Y would make the necessity / sufficiency relationship very clear.
P/S: the motivation needs not be restricted to social sciences. For example, if I have a dataset with DV as "Fire occurring", and IVs as presence of fuel, presence of match, presence of lightning, etc. Can we use regression to prove that fuel is necessary but insufficient for fire?
 A: Qualitative Response models, logistic regression being the most widely known, could in principle provide you with a way to arrive at a conclusion on these matters, in a regression framework.  
The usual logistic regression model is specified as
$$P(Y=1\mid X=1) = \frac 1{1+\exp\{-\alpha-\beta X\}}$$
where both $Y$ and $X$ should be binary, to fit our purpose of checking sufficiency and /or necessity. 
One can easily see that
$$\frac {\partial P(Y=1\mid X=1)}{\partial \beta} >0$$
So if the estimated $\hat b$ is found to be "very large", the estimated probability will tend to $1$. But $P(Y=1\mid X=1) = 1$ essentially says "the appearance/existence of $X$ is a sufficient condition for the appearance/existence of $Y$".  
For necessity of $X$ for $Y$, swap the variables' roles and estimate
$$P(X=1\mid Y=1) = \frac 1{1+\exp\{-\gamma-\delta Y\}}$$
If this probability tends to $1$, it conceptually approaches the statement "$X$ is necessary for $Y$".
But see the comments below for why you should not try this at home.
A: Articial Neural Networks developed in the machine learning community can learn highly complex boolean functions (and necessary/sufficient statements are very simple boolean functions). See for example chapter 11 of Hastie, Trevor, et al. The elements of statistical learning. New York: Springer, 2009.
A: Charles Ragin has created a statistical procedure (QCA, or Qualitative Comparative Analysis) for the analysis of necessity and sufficiency in his 2008 book, 'Redesigning Social Inquiry'. But it relies on set relationships, not regression. Roughly put, it uses two parameters to identify sufficient causes: consistency (the proportion of cases that have the purported causal condition that also have the outcome); and coverage (the proportion of cases that have the outcome that also have the purported causal condition). This can be applied to datasets with continuous variables through calibrating these variables into fuzzy set scores and following the procedure that Ragin describes. He has also provided software to carry out this type of analysis, while others have written a QCA package for use in R. See http://www.compasss.org/ for more information.
