TITLE: Why do we prefer classical logic over non-classical logic?
QUESTION [12 upvotes]: In classical logic, we have paradoxes like paradoxes of material implication. If non-classical logic like relevance logic fixes those problems, why do we still continue to use classical logic?

REPLY [4 votes]: A very common opinion of mathematics is "mathematical platonism", which holds that mathematical obejcts exist in some sense and that mathematics is the study of these objects. The rules of classical logic are closely tied to this viewpoiont. 
The "paradoxical" formulas of material implication are verified when we interpret them as talking about truth values in a fixed model. For example, if we know that $p$ and $q$ are statements about a fixed model, we know that $p \to (q \lor \lnot q)$ will be true in the model, by reasoning by cases about the truth values of $p$ and $q$ in the model. 
The completeness theorem for first-order logic says that a formula is provable in first-order logic if and only if it is true in every model. This statement has two parts:

If we can prove a formula in first-order logic, the formula is true in all models. To a platonist, this means that if we already have a fixed model in mind, and we prove a formula, we know the formula will be true in that model. 
A formula that is true in every model is already provable in first-order logic, so we cannot extend first-order logic to a properly stronger logic (that is, one that proves more sentences) while simultaneously maintaining (1).  

The completeness theorem thus says that first-order logic is the strongest logic possible (in terms of proving the most sentences) whose results are sound when applied to an arbitary model that we have already fixed. That is exactly the sort of logic that we would want, as mathematical platonists, in order to study a collection of pre-existing structures. 
Relevance logic, for example, makes fewer implication formulas provable. That is of interest if we are trying to study the natural-language "implies" relationship, but it is less interesting if we are trying to study the field of real numbers and we want to generate as many formulas as possible that are true in that structure.