vacuously (adverb): from Latin vacuus “empty, clear, devoid of.” The Indo-European root is eu- “lacking, empty.” Related borrowings from Latin include vain and vacant, and native English want (as in the phrase “to be wanting”) is a cognate. In mathematics a proposition is said to be vacuously true when there is nothing to contradict it. For example, in combinatorics, if 0 objects were chosen from a group of objects, the choosing may be considered to have taken place vacuously with replacement. [58, 117]

valence (noun): via French, from Latin valentia “capacity,” from the present participle of valere “to be strong, vigorous, healthy, worthy.” The Indo-European root is wal- “to be strong.” Related borrowings from French include prevail and avail. In mathematics the valence of a node in a network is the number of paths that meet at that node. [241, 146]

valid (adjective), validity (noun): from Latin valere “to be strong, vigorous, healthy, worthy.” The Indo-European root is wal- “to be strong.” A native English cognate is wield (= to use power). Related borrowings from Latin are valor and valiant. In mathematics an argument is said to be valid as long as the rules of logic are followed, even if the statements involved are ludicrous. [241]

value (noun): from Latin valere “to be strong, vigorous, healthy, worthy.” The Indo-European root is wal- “to be strong.” The Latin verb evolved into French valoir “to be worth,” with feminine past participle value.