A Double-Negative does not Equal a Positive




It is a common misconception among people who have only studied a little bit about logic that "a double-negative equals a positive."  By double-negative they mean a proposition with two negatives, like "not" or "no" or "un-" and so on. We often hear this from nagging people who want us to clear up our language, getting rid of unnecessary words.  According to this theory, any sentence with an even number of negatives (0, 2, 4, 6, 8, ...) is equivalent, and any sentence with an odd number of negatives (1, 3, 5, 7, ...) is also equivalent, so in order to write most clearly, we should reduce the number of negatives in a sentence to either 0 or 1 (you might call this the binary theory of logical grammar).  

But there's a problem.  Sometimes, in changing the number of negatives from any even number to zero, we change the meaning of the sentence.

It's true that sometimes a double-negative can be logically equivalent to a positive, as in this pair of sentences:

1. The process wasn't unfinished.

2. The process was finished.

But this is not always the case.  Here's an example of two sentences in which a double-negative is not logically equivalent to a positive:

1. I don't know how to untie this knot.

2. I know how to tie this knot.

These sentences clearly mean different things.  And they are not logically equivalent - it's quite possible to imagine circumstances in which one of them is true and the other is false.  (In fact, it seems most probable that if you knew how to tie a knot, you would be more likely to know how to untie it, doesn't it?)

Or, here's another example:

1. I didn't see the bar's invisible magnetic field.

2. I saw the bar's visible magnetic field.

These two sentences seem to mean the opposite of each other, don't they?  Yet, if "a double-negative equals a positive," then they should be logically equivalent. 

How about another example?

1. I didn't see him unlocking the door.

2. I saw him locking the door.

Clearly, those mean two different things.  They're not logically equivalent.  It's quite possible that you saw him when he locked it, and also when he unlocked it, in which case the second sentence would be true, but the first sentence would be false.  Alternately, it's possible that you missed both the locking and the unlocking.


Here's my favorite example:

1. You shouldn't punch unmarried men.

2. You should punch married men.

There we go!  We got rid of those nasty extra negatives, and the sentence means exactly the same thing, right?



1. I don't think red-haired people are necessarily unhappy.

2. I think red-haired people are necessarily happy.

The first sentence there is perfectly reasonable.  The second is ridiculous.



1. I don't think aliens are unseen among us.

2. I think aliens are seen among us.

Either way, you just can't get rid of those pesky aliens.


Here's an example that will be easy to think through for those who have studied a bit more logic:

1. No invertebrates are plants.

2. All vertebrates are plants.

These sentences certainly are not logically equivalent.  The first statement here is true.  The second is false.  (I, for one, am a vertebrate, but not a plant.)

I could go on all day.  Indeed, it's much easier to come up with examples where a double-negative is not the same as a positive than it is to come up with examples where a double-negative is logically equivalent to a positive.  (I spent most of my time writing this article trying to come up with the first pair of sentences, today, about the "process" being "finished."  I... think... those two sentences are logically equivalent, but to be honest I'm not entirely sure.  Maybe someone in the comments can point out to me how they are not equivalent.)

So: what's going on here?  Several things.  First of all, some of these examples involve something like set theory ("vertebrates" and "invertebrates" are both subsets of "animals," which is logically distinct from "plants"; "red-haired people" and "happy people" are overlapping sets, but not co-extensive, etc.), and some of them involve modal logic (notice the word "necessarily" in the example about redheads).  Others involve modal verbs ("should"), which are quite tricky in logic.  Still others involve dependent or independent clauses, and I could easily come up with much more complex examples.

But we can sum up all of these issues this way: in normal, natural language, like English, negatives do not simply "cancel out" the way you can cancel out minus signs on either side of a mathematical equation, because these negatives are often negating different things.  Yes, "not not x" means the same thing as "x," because in that case, both of the nots are modifying the same x.  But, for instance, "I don't believe that [x]" does not necessarily mean the same thing as "I believe that [not x]" - you might simply have no opinion on the subject.  (For example - I can't help it, one more example - "I don't believe that Pythagoras and Theano were married" is not logically equivalent to "I believe that Pythagoras and Theano were unmarried".  It's quite possible that, like the rest of us, you simply don't know whether they were married, or whether Theano even existed, or for that matter whether Pythagoras did, and so you have no relevant beliefs on these subjects.)  You can't simply move the word "not" anywhere you like in a sentence and expect the sentence to mean the same thing.

This is not to say that "logic doesn't apply" to the sentences in the examples above.  It does.  But logic turns out to be much more complex, much harder work, than it may first appear, especially when applied to natural language.  "A double-negative equals a positive" may be a perfectly good rule for the simplistic kinds of propositional logic that you learn in high school, but real language and the real world are much more complicated.  That's why the examples in your high school math textbook were so torturously abstract and far-fetched: because they were absurdly simplistic.  After you learn that stuff, you can get to predicate logic and modal logic and so on and so on - and that's precisely why we pay logicians, to figure out some of these more complex puzzles.  And when you get there, you realize that "a double-negative equals a positive" is so ridiculously simplistic that it's often flat-out wrong.


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