Do standard definitions of substitute goods and complementary goods assume the Law of Demand? It seems to me that they do. Suppose $x$ is a Giffen good (that is, it violates the law of demand). Then when $p_x$ increases, the quantity of $x$ demanded increases. Suppose also that this increase of $p_x$ decreases the demand for $y$. Now, the standard definition for complements suggests that $x$ and $y$ are complements because the demand for $y$ decreased when $p_x$ increased. I would argue, however, that they are, in fact, better thought of as substitutes as the change in $p_x$ leads to the quantity demanded of $x$ increasing as the demand for $y$ decreases, i.e., the consumer is substituting from $y$ to $x$.

By "standard definitions" of substitutes and complements, I'm referring to:


$x$ and $y$ are substitutes if demand for $y$ increases when $p_x$ increases.
$x$ and $y$ are complements if demand for $y$ decreases when $p_x$ increases.


Is there an alternative definition out there that is robust to Giffen goods?
A quick response is that if the law of demand is violated, then the standard definition for substitutes and complements may or may not apply.


  Consider buying fast food burgers as your only choice of meal. When the price of the meal drops, you might want to consume something healthier like a bowl of salads. You consume less fast food and may increase consumption of, say, broccoli. Then, this is your example. $\frac{\partial X_F}{\partial P_F}&gt;0$. But, $\frac{\partial X_B}{\partial P_F}&lt;0$ where P and X are price and quantity and subscripts F and B represent Fast Food and Broccoli respectively.


Consider another example:


  Consider buying a game CD for Xbox One that is a special and limited edition. As marketing strategy, the manufacturer emails you there are only few left and price has increased 5%. You are determined to get the game now for one collection purpose and another for actual game use. But then, you were gonna get a limited edition version of wireless controller to enhance your gaming experience. It is reasonable to consider they are complements. But because you decided to buy two CDs, you buy no wireless controller. In this example, we still have $\frac{\partial X_L}{\partial P_L}&gt;0$. But, $\frac{\partial X_C}{\partial P_L}&lt;0$ where P and X are price and quantity and subscripts L and C represent the limited game cd and wireless controller respectively.


Could you still argue the limited special edition CD is a substitute for the wireless controller? Maybe. But notice you still desire the wireless controller given it is wireless and limited edition. One of the reasons why you forgo the chance to buy this controller is you are effectively poorer than before. And this is an important distinction to make, because the decrease in the quantity of the controller may be from the direct impact of the increase in the price of the game CD but also the induced effect through your wealth (i.e. thinner wallet).

If you want to investigate the true relationship between two goods, you have to take both wealth and price effects into account on top of the given information, say whether the good is Giffen or inferior or both or something else.

Hope this helps.