We frequently define an acid or a base as a substance whose aqueous solution gives, respectively, a higher concentration of hydrogen ion or of hydroxide ion than that furnished by pure water. This is a very one sided definition.
Since hydrogen is a constituent of most of our electrolytic solvents, the definition of an acid or base as a substance which gives up or takes up hydrogen ion would be more general than the one we used before, but it would not be universal.
When we discuss aqueous solutions of substances which do not contain hydroxyl, it is simplest to define a base as a substance which adds hydrogen ion.