3 - Set relations

Summary

Sufficient conditions

Crisp sets

Basic logic of sufficiency

Given some plausible theoretical arguments, a condition can be considered sufficient if, whenever it is present across cases, the outcome is also present in these cases. In other words, there should not be a single case that shows the condition but not the outcome. Say, for example, we claim that being a Western European country (X) is a sufficient condition for being a democracy (the outcome Y). If this claim is true, all countries in Western Europe would also have to be democracies; no Western European country can be a non-democracy. As shown in section 2.5, this can be expressed as follows:

X → Y.

This statement should be read: “if X, then Y,” or “X implies Y,” or “X is a subset of Y.” Based on this statement, what do we know about the value of Y in cases that do not show a positive value for X? Asked another way, what expectations about the outcome value do we have for countries that are not located in Western Europe (~X)? Does our claim that X is sufficient for Y automatically mean that ~X implies ~Y? The answer is no! But why?

The statement that X is sufficient for Y generates expectations on the value of Y only for cases that display X. All cases that are not members of X are not relevant for the statement of sufficiency. That is to say, they neither help to verify nor falsify our claim, independently of whatever value of Y these cases might display. While counterintuitive at first sight – especially for anybody with thorough training in correlational methods – the statement “if X, then Y” creates expectations for values on Y only when X is present. It does not generate any such expectation, or any expectation at all, in cases where ~X is present. It follows that countries in places other than Western Europe (~X) can be stable democracies (Y) or not stable democracies (~Y) – and indeed there are plenty of both types – neither of which confirms or contradicts the statement that X is sufficient for Y.