Example: I think $0 is a "cheap" price for a particular laptop. I also think $1 is cheap. And $2,$3,$4 as well. I think $1 million is an "expensive" price. So is $999,999, $999,998, $999,997.

At which point does it switches from cheap to expensive, and vice versa? How should I reason about this?

Edit: I am interested in reasoning about the switch from cheap to expensive, rather than why something is cheap/expensive relative to others.

Another, possibly better, example: 0 degrees Celsius is "cold" and 100 degrees Celsius is "hot". When does it stop being cold and start being hot?

  • Note: awareness of hot/cold distinctions predates human human evolution, and a philosophical understanding of the paradoxes of their vagueness is at least as old as Aristotle. The concept of metrical tempurature scales measurable by thermometers (circa 16th century, Galileo) is our solution to the problem of determining where hot becomes cold.
    – David H
    Commented Jul 10, 2013 at 9:06

6 Answers 6


What you are looking for is the Sorites paradox.

A typical formulation involves a heap of sand, from which grains are individually removed. Under the assumption that removing a single grain does not turn a heap into a non-heap, the paradox is to consider what happens when the process is repeated enough times: is a single remaining grain still a heap? (Or are even no grains at all a heap?) If not, when did it change from a heap to a non-heap?

Much has been written about this. Wikipedia offers a good and accessible summary of proposed resolutions. In case you are interested in a more academic and thorough approach, the SEP-article on the Sorites-paradox is what you're looking for.

  • 2
    Actually what I am looking for. Thanks for helping me identify the topic! +1 and accepted.
    – Legendre
    Commented Jul 10, 2013 at 0:47

This is something which is (in my opinion) formalized quite nicely in Vopěnka's Semiset theory . See this stub on Wikipedia and the first reference there, which can be found on the internet quite easily.

(I personally understand that the motivation behind semiset theory is to introduce tools for dealing with such "fuzzy" phenomena in mathematics, thus "adjusting" it little more to the real world.)

(I would rather post this as a comment, however, I am afraid that I do not have rights for this yet.)

  • Hi @PavelC, I can't find a file for the first reference (a book). And, with regards to fuzzy sets, I don't see them being useful here. Wouldn't that come down to something like: "A $5,000 laptop is 99.5% cheap and 0.5% expensive.", which... is a bit awkward?
    – user3164
    Commented Jul 10, 2013 at 13:06
  • @Gugg: Hi, I believe the book can be found here. As for the fuzzy logic, I agree that simple quantifying does not capture the essence of the problem. The concept of semisets is different and maybe more closely related to non-standard natural numbers. (My knowledge of this theory is, however, very limited; my intension was to provide a possibly useful reference.)
    – PavelC
    Commented Jul 10, 2013 at 14:52

"cheap" and "expensive" is a matter of taste most of the time. Other times they are very clearly defined, as in the price of stocks. Stocks are considered "cheap" if they appreciate and pay out enough dividends in less than 6 years to recover the capital expenditure. 6 to 10 years is fair, and anything above that is considered expensive.

With the price of a laptop, "cheap" can be defined in terms of

  1. Whether you believe you would be willing to pay more for the specifications and features
  2. Whether it costs less than similarly spec'ed laptops.
  3. Whether you should be able to use the laptop in a way that you could recover the cost quite easily (like doing development or design work).
  4. Whether the amount of work you'd have to do to save the money to buy it is relatively little. For instance, I view working five days to pay for a laptop as good value, but six days aren't. But that's my subjective opinion on it.

What I'm getting at is that you first need to decide WHAT you mean by "cheap" before we can give a good answer to it. For instance, many people think 99c for a certain piece of candy is cheap, but $1 is expensive, so it's very much a matter of cognitive dissonance in that case.

In the case of hot and cold, anything that can cause burns is considered hot, and anything that can cause pain is considered cold. these temeratures can be pinpointed scientifically. Everything in-between is a gradient.

It really is a matter of what the situation is.

  • Look at the top rated and accepted answer. While you answer might be appropriate for another question, it seems to misunderstand the OP's concerns.
    – Dennis
    Commented Jul 10, 2013 at 7:28
  • It is necessary to decide on a definition before one can start talking about what when it applies. Commented Jul 10, 2013 at 8:02
  • I think you're focusing on the wrong aspects of the question. The particular examples aren't important, it's more about borderline cases with vague concepts. Hence why the Sorites Paradox is appropriate here.
    – Dennis
    Commented Jul 10, 2013 at 8:05
  • Point taken. but was it necessary to downvote my answer? Commented Jul 10, 2013 at 9:05
  • 2
    Yes, that's the whole point of downvoting. To indicate to visitors of the site when an answer is not a useful answer to the question (it says as much when you hover over the button). It's nothing personal and it is reversible if you edit the question.
    – Dennis
    Commented Jul 10, 2013 at 16:05

Both "cheap"/"expensive" and "hot"/"cold" imply a moderate point or range. For example, you might say that the air temperature is hot if it is hotter than you are comfortable with. Whatever range you are comfortable with is the moderate range. Everthing cooler than the moderate range is "cold", and everything hotter than the moderate range is "hot".

With "cheap"/"expensive", the moderate range's lower limit is the point below which you think you are getting more than what you paid for. It's upper limit is the point above which you deem to be an unacceptable price. Everything cheaper than the moderate range is "cheap", and everything more expensive than the moderate range is "expensive".

Sometimes values within the moderate range are also perceived as "hot"/"cold" or "cheap"/"expensive". This is either based on being above/below the mean of the range, or it signifies that there is some more specific moderate point (or subrange) that is most desirable.

  • Thanks for the answer. But at which point do we switch from "cheap" to "inside the moderate range"? i.e. where are the boundaries of the moderate range?
    – Legendre
    Commented Jul 10, 2013 at 0:47

The answer is for at least this domain of problem is that prices are quantised and not continuous. That is historically speaking laptops would have been introduced with price differentials to attract buyers looking for cheap basic models and those looking for expensive feature-packed models. Then as the market matures and there are more entrants into the market and further gradations of price, the original division into cheap and expensive still persists.

Of course for continuous or finely graded variations in nature - such as colour - this doesn't apply. The usual name for this problem, as already pointed out, is the Sorites paradox.


Well if you put it that way,the i believe the terms cheap and expensive are described in comparison with other products.So,a 1$ laptop is "cheaper" than a 0$ laptop,which is (the 1$ laptop) a "cheap" one.

  • Thanks for the answer. I am more interested in reasoning about the "switch" than actually determining if something is cheap/expensive. E.g. when does "cold" become "hot"? Will edit to reflect this.
    – Legendre
    Commented Jul 9, 2013 at 21:14

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