# The word 'not' often doesn't mean total negation in mathematical sense?

Consider the following conversations:

X1: I paid \$10 for that hamburger.
Y1: That's not cheap!

X2: I pay \$1 for broadband Internet access.
Y2: That's cheap!

X3: I paid \$1 for a hot dog.
Y3: Seems like a reasonable price.

In conversation 1, "not cheap" means expensive or overpriced.

In conversation 2, "cheap" means lower than the expected or average price.

In conversation 3, the hot dog is neither overpriced not lower than the expected price.

Therefore, the hot dog is neither "cheap", nor is it "not cheap".

Is this a special case, or does 'not' in the English language often not translate to mathematical negation?

This post inspired by a MENSA book stating the 2 sentences below are contrapositives (and thus have the same meaning), even though most people interpret them very differently:

Good food is not cheap.
Cheap food is not good.

• "Not cheap" does not only mean "overpriced". It also literally means "not cheap". – Kosmonaut Dec 29 '10 at 4:36

I think what you have in statement 1 is merely a case of rhetorical understatement, also known as litotes.

In rhetoric, litotes is a figure of speech in which understatement is employed for rhetorical effect. It is most often used to describe the expression of an idea by a denial of its opposite, principally via double negatives. For example, rather than saying that something is attractive (or even very attractive), one might merely say it is "not unattractive."

Litotes is a form of understatement, always deliberate and with the intention of emphasis. However, the interpretation of litotes can depend on context, including cultural context. In speech, it may also depend on intonation and emphasis; for example, the phrase "not bad" can be said in such a way as to mean anything from "mediocre" to "excellent."

• This is a quote from Wikipedia on "litotes". It is indeed to the point: this is the context that leads readers to interpret "not cheap" as the opposite of cheap. // Note that the first sentence is misleading, because it looks like a definition, which it is not, since litotes is only about negating opposites, not merely about understatement. "Most often" is wrong, it should be left out. Lastly, the interpretation of litotes is never "mediocre": that is when an apparent litotes turns out to be no litotes at all. – Cerberus_Reinstate_Monica Dec 29 '10 at 4:11
• @Cerberus: It's better to edit Wikipedia. :-) ( Be bold! and all that.) – ShreevatsaR Dec 29 '10 at 4:38
• @ShreevatsaR: Hmm yeah, perhaps I should have. I used to edit a lot in some language versions of Wikipedia, but I haven't been in the mood for a long while. I do still like it, just not so much editing it. Part of it might be the lack of freedom owing to extensive regulations and political correctness. I was also too lazy to read the entire article, as one should before editing it. But actually I think I will; thanks for your encouragement. – Cerberus_Reinstate_Monica Dec 29 '10 at 4:51
• Done. Now let's hope it won't be reverted. – Cerberus_Reinstate_Monica Dec 29 '10 at 14:35
• @Rahul Thanks! I think this is exactly what I was seeing. The use of "not" as an understatement to indicate a strongly opposite quality, not merely negating an adjective. – barrycarter Jan 3 '11 at 6:12

The main problem in both cases is this: how to convert real language into logical language. In order to do this, one must choose how to interpret real language. This is often difficult, because real language deals with information that is often not apparent, such as implicit premises and the wider context.

X1: I paid \$10 for that hamburger.

Y1: That's not cheap!

The logician wants to convert Y1 into `not[equals(\$10,cheap)]`. However, real language is not that simple. While sometimes "not x" can mean the logical/mathematical negation of x, at other times it has a different meaning. If x is a value in a spectrum, it often means "on the side of the spectrum farthest away from x", just as it means in Y1 `on the side of the price spectrum farthest away from cheap`, i.e. expensive. I took several pieces of information from context for this interpretation: that "cheap" is on a spectrum, that it is on one side of it, the exclamation mark, the price mentioned, my knowledge of prices in the real world - and probably more that I am not aware of.

Consider this example:

X4: I paid \$2 for my hamburger: is that cheap?

Y4: Well, that is not cheap, but it isn't expensive either.

Here "that is not cheap" does not mean the same as it did in Y1. It means "\$2 is not at the cheap side of the spectrum, but it might be anywhere else in the spectrum". This is then refined later by "it isn't expensive either", so that we are left with the middle area of the spectrum.

The reason for this change in meaning is that Y1 and Y4 have different contexts and thus have different logical meanings. You cannot convert real language into logical language without looking at the whole picture, the wider context, i.e. everything you know about the real world and language.

Now the "good food" example:

Good food is not cheap.

Cheap food is not good.

It all depends on how you interpret the real language and how you thus convert it into logical language. "Good food is not cheap" you could interpret as `not[equals(good food,cheap)]`. But as you can see in the answers of others, this gives problems.

You could also interpret it as `if food x is good, then food x is not cheap`. I believe this comes much closer to what most speakers would mean if they made this statement in real language. "Cheap food is not good" I would accordingly interpret as `if food x is cheap, food x is not good`.

I chose these interpretations because the real grammatical construction noun + copula + adjective does not express a relation of equality, but rather that of the noun's having the property expressed by the adjective. The word "to be" can express equality (ten is two times five), having a property (the house is yellow), belonging to a class (Socrates is a man), or existence (there is a God) - and possibly some more. Only when it expresses equality are the two variables of the same kind and in a reciprocal relation: otherwise, the relation is not reversible and the variables not of the same kind.

Every time you convert real into logical language, you just need to choose the interpretation that best fits the context. There is no rule for which one to pick, except that the logical statement must give the same false and true values as the real statement if you enter the same parameters; you will probably judge this for a good deal by common sense. Do you know how to use truth tables? They are what you can use to test logical equivalence between two logical statements.

As an added illustration, I will give a classic example of interpretation.

The King of France is bald.

France has no King, it is a republic. How do we interpret this? A few options:

1. `HasProperty(KoF, bald)`
2. `Equals(KoF, bald)`
3. `Exists(KoF) and HasProperty(KoF, bald)`

Number 1 would give us neither true nor false, because the input KoF would remain empty. If we choose this, then we assume that a real answer to the real statement would be "your question is moot", because the question presumes that there is a King of France. (I have heard that Japanese has a special answer to this kind of question, "mu".)

Number 2 is problematic, among other things because "bald" is a property and "KoF" is a person, and a property and a person cannot be identical. A real answer might be "your question is a tautology and moot".

To number 3, I'd give the real answer "no, your question is a tautology", since it is always not true, as there exists no King of France.

Which one do you prefer? I believe both 1 and 3 could be valid conversions. It just depends on your choice; there is no single reasonable interpretation in this case.

• The contrapositive of "if p then q" is "if not q then not p". It is NOT "if q then p". – Marthaª Dec 29 '10 at 14:47
• I believe you, but "if not p, then not q" is in none of the "good food" statements from the OP, so I did not discuss it. – Cerberus_Reinstate_Monica Dec 29 '10 at 15:25
• The contrapositive is in the OP's question: the contrapositive of "If p then not q" is "If q then not p". This is precisely the good-cheap example. (In other words, I think your answer is great except that you should remove the "In real life, it could be said that "if p, then q" and "if q, then p" are in some cases equivalent" paragraph. Those two statements are logically inequivalent, and more often than not, they're inequivalent even in real life. That paragraph is just confusing given the context.) – ShreevatsaR Dec 29 '10 at 20:29
• @ShreevatsaR: Hmm what you are saying about their being inequivalent is exactly what I tried to convey there, but when I read back I see that the paragraph was not so clear or useful, so okay poof it is gone. // I merely said that Martha's "if not q then not p" was not in the OP's question, which I believe is true. I did not discuss the word "contrapositive" at all, because, frankly, I forgot its definition and I didn't really need the term or the concept - i.e. I was too lazy to look it up. I have a hunch it might mean opposite truth output values if same input values... still too lazy. – Cerberus_Reinstate_Monica Dec 31 '10 at 1:44
• All clear now, upvoted. :-) The contrapositive is just another way of saying a statement. Its definition is that the contrapositive of "If p then q" is "If not q then not p" (and so the contrapositive of "If p then not q" is "If q then not p", etc). The contrapositive of the contrapositive of a statement is the statement itself. A statement and its contrapositive are always logically equivalent, and the contrapositive of "If food is good then it is not cheap" is "If food is cheap then it is not good". – ShreevatsaR Dec 31 '10 at 6:47

That MENSA book needs to review its mathematical concepts. If one statement is the contrapositive of the other, it DOESN'T mean the same thing, it just follows logically from it: if one of the statements is true, then so is the other.

Now, it's certainly true that everyday language often plays fast and loose with logic — witness the number of people who see nothing wrong with saying "I could care less" to mean its opposite. But Good food is not cheap vs. Cheap food is not good is not an example of fuzzy logic, just two related statements where if you believe one, then logically you must believe the other.

• If A implies B and B implies A, each follows from the other; they are equivalent. So when you say "it DOESN'T mean the same thing", you mean only that the emphasis is different, since the implication is the same. Right? – ShreevatsaR Dec 29 '10 at 3:24

In the 'not cheap' example, you are negating the fact that the hamburger is cheap. However, 'cheap' by itself is subjective; what is cheap to one person is not necessarily cheap to someone else. Therefore, you can't assign a strict mathematical negation to the word 'not', as it is negating something that is subjective.

There are several things going on here.

The MENSA test may be confusing if we interpret the statements as being about the real world.

First, let's show that the statements are equivalent in the narrow logician's sense. Let A stand for "good food" and let B stand for "cheap food".

Logically we interpret the sentences as

```Good food is not cheap (food)
Cheap food is not good (food)
```

Then we can substitute A and B into the sentences, to get

```A is not B
B is not A
```

And we can see they are equivalent, however...

In the real world, "good" and "cheap" are orthogonal measures i.e. not necessarily correlated. Good is a measure of quality, cheap is a measure of cost.

Indeed, one can find examples of

```cheap food that is good
cheap food that is not good
Good food that is cheap
Good food that is not cheap.
```

Now, if we let "good" = "nutritious" and let "cheap" = "inexpensive", we can rewrite the examples above as:

```inexpensive food that is nutritious - home cooked perhaps?
inexpensive food that is not nutritious - typical 'fast food'
Nutritious food that is inexpensive - fruit, vegetables?
Nutritious food that is not inexpensive (costly) - upscale restaurant
```

In all cases, "not" means the logical inverse, and not all "logical" statements are true.

For example: "Good food is not cheap" is demonstrably false.

Also, in the real world,
"Good" does not mean "expensive"

• I might be misunderstanding, but MENSA isn't claiming the sentences are true. They're only claiming the sentences are equivalent: if you believe one, you must believe the other. Are you saying the falseness of the statement makes a difference? I suppose you could mathematically argue that two false sentences are equivalent, but that seems like a copout. – barrycarter Dec 28 '10 at 19:25
• @barrycarter - Sorry, I'm not sure what MENSA is saying about the truth of the statements. Nor am I saying that the truthfulness of the statements affects their equivalency. I am saying in the narrow "logical" context of the MENSA example the two statements are equivalent because the statements equate "good food" with "not cheap (food)". In that context, the statements are equivalent regardless of their truth or falsity. I'll try to clarify the answer. – John Satta Dec 28 '10 at 21:23
• Contrapositive merely says that if you believe the statement "Good food is not cheap" to be false, then logically you must also believe the statement "Cheap food is not good" to be false. I don't see any difference between real world and mathematical usage of "not" here. – Marthaª Dec 28 '10 at 22:29
• @Martha - I agree, as stated above "In all cases, 'not' means the logical inverse". I interpreted the OP's question to indicate some confusion ("Is this a special case...?"). I think the source of the confusion is the MENSA example explicitly equates the descriptions "good" and "not cheap", while in the real world those descriptions mean different things. So to me, the two MENSA statements in context are equivalent by definition and false in the real world. In the real world the two MENSA statements are not equivalent (and still false). And "not" means the same thing everywhere. – John Satta Dec 28 '10 at 23:04
• @barrycarter, @Martha, I changed "contrapositive" in my answer to "equivalent" to avoid any misunderstanding. – John Satta Dec 28 '10 at 23:07

'Not' in a mathematical sense refers to binary logic. This is not always the case in communication.

'Fred is not in Demark' would be a binary situation; Fred is either in Demark or he is not.

'The hamburger is not cheap' refers to a pricing scale, say: 'really cheap', 'cheap', 'reasonably priced', 'a bit expensive', 'expensive' to 'ridiculously overpriced'. As a real world example: 'Chery', 'Kia', 'Mazda', 'Subaru', 'BMW', 'Rolls Royce'.

Just because a car is not a Kia does not make it a Rolls Royce. In the case of 'not a cheap hamburger' the not merely removes the 'cheap' qualifier, leaving you with just a hamburger.

• If you're saying "not a cheap hamburger" and "hamburger" have the same meaning, I disagree. The former suggests the hamburger is expensive or overpriced. – barrycarter Dec 28 '10 at 22:26
• True, but it does not indicate where in the hamburger pricing scale the hamburger sits. It strongly implies that it is reasonably priced or greater. – dave Dec 28 '10 at 22:40