Topics Logic

Question If I am correct, the opposite of 'A' is not 'B', 'C', 'D', etc., but rather, the opposite of 'A' is 'not-A.' Likewise, the opposite of 'Green' is not 'Blue', 'Orange', etc., but rather, the opposite of 'Green' is 'not-Green.' And the opposite of 'Dog,' is not 'Cat' or 'Whale,' but rather, the opposite of 'Dog' is 'not-Dog.' And so on. However, each letter 'B' through 'Z' is not 'A' (after all, it seems, 'B' is not 'A', 'C' is not 'A'. and so on). Does that mean that 'not-A' is, or can be, or includes 'B' through 'Z'? Thus, does that mean that the opposite of 'A' is or can be 'B', 'C', etc.? Logically, I suppose, letters can stand for anything -- so perhaps 'A' is or can be equal to, say, 'B' and, therefore, 'not-A' would be equal to 'not-B,' so the opposite of 'A' might be 'not-B'. But what about objects that are not logical symbols? Cats and whales or not dogs. So, if the opposite of 'dog' is 'not-dog', and if cats and whales are not dogs, then are cats and whales the opposite of dogs? Am I missing something altogether?

Accepted:
November 15, 2008

Comments

The idea of an "opposite" isn't really well-defined. What you're calling the opposite (e.g., "not-dog" as the opposite of "dog") is what a logician might call the contradictory. But even though "opposite"' doesn't have a precise meaning, it's clear from the way that people use the term that it doesn't just mean the contradictory.

If we want to figure out what a term means, we're well-advised to attend to how competent speakers use it. Ask any competent speaker for the opposite of "white" and she'll say "black." Ask any competent speaker for the opposite of "tall" and he'll say "short." But what can we gather from this? First, that a term and its opposite can't both apply to the same thing. Opposites are contraries. A bit more precisely, if the term "Y" is the opposite of a term "X," then "a is an X" and "a is a Y" can't both be true. However, in typical cases of opposites, they could both be false. (My pen is neither white nor black, for instance.) Still, that isn't enough. After all, "white" and "green" are contraries, but "white" and "green" aren't opposites. So what more is needed?

Here things get mushy. Intuitively we might think that opposites lie at distinct ends of some common scale. White and black are at opposite ends of the grayscale, for example. But that won't quite do. "Tall" and "short" are opposites, but once we've settled on a standard (tall for a person, for example, where the middle of the scale is more-or-less average height) "tall" and "short" don't pick out precise points at opposite ends of the height scale. (Given a specific collection that's at issue, however, "tallest" and "shortest" might.)

So there's no nice logical story about opposites. We can say that they are almost always contraries but not contradictories, and we can say that they often lie across some more or less vague midpoint from one another of some more or less well-defined scale, at more or less equal distances. Not all terms have opposites. (What's the opposite of "driveway?" Or of "sarsparilla?") Some opposites have a sort of "naturalness" about them ("minus" and "plus," for example) while others are pretty clearly dependent on more arbitrary conventions ("dog" and "cat" for example.) But language is like that. If we come to it with a logician's expectations, we'll often be disappointed. Or delighted.