Question: This is a sort of paradox that has me confused. i just need some help clearing things up. 

Let $x,y$ be two discrete, finite variables, where $x$ can take on $N$ values and $y$ can take $M$ values. Suppose someone hands you a table of the numbers $P(x|y)$. Can you determine $P(x,y)$? Intuitively, I think the answer is No. However, consider the following system of equations:

$$P(x|y)\sum\_\xi P(\xi,y) = P(x,y)$$

(This is just another way to write the product rule $P(x|y)P(y)=P(x,y)$.)

Here you have $M\times N$ linear equations in the $M\times N$ unknowns $P(x,y)$. So it seems that $P(x,y)$ could be determined from this system of equations. If this is true, then knowledge of the $M\times N$ numbers $P(x|y)$ are enough to determine the $M\times N$ numbers $P(x,y)$, but this doesn't feel right. What am I doing wrong?

Answer: 
You use a method designed for the type of Y variable you have. If it is a dichotomy, then logistic regression is a place to start. If ordinal, ordinal logistic regression; if nominal, multinomial logistic.
According to Gujarati (2007) the most appropriate econometric method to be used for a study in which the dependent variable is qualitatively captured is qualitative response regression model (probit, tobit, logit [multinomial logit, ordinal logit and nested logit).