The most recent xkcd:

(As usual, click on the image for a larger version.)

I'm not sure that Randall is right about the first panel. It's true that "up to X or more", for some number X, seems by simple compositional semantics to cover the entire number line. But if a store were to raise its prices by 10% across the board, I believe that all rational people would regard the statement "You will save up to 15% or more on your purchases" as false, despite the fact that -10 is on the real line (and on Randall's representation of it).

And lawyers also use "false" differently from logicians, I think. Thus NY Code – Section 350-A says that

In this case, then, the law and ordinary language might be in reasonable alignment.

I say "might" because phrases of the form "up to * or more" are very common in advertising. And I know from experience that the named percentage or amount is often associated with a small number of options in weird sizes and styles, but I've never heard of a regulatory complaint about such ads.

Analogous phrases occasionally occur in journalistic writing, e.g. this Consumer Reports article on kitchen remodeling:

Whether it's a permanent unit with furniture-style looks or a small portable one on casters, an island adds work and storage space. The style should complement your cabinets, but it needn't match exactly. A wood island can add warmth to a white kitchen, and a painted finish can inject color into an all-wood one. Be sure that there is at least a 42-inch clearance around the island to ensure easy access and smooth traffic flow. Prices start around $150 and go up to$1,000 or more.

In such cases — and perhaps in the more honest ads as well — something really is being communicated, namely an approximate statement about a distribution.  I conclude from the last sentence of that Consumer Reports paragraph that if I shop for kitchen islands, I should expect to find plenty of options in the range from $150 to$1,000, with a perhaps-long tail of higher-priced offerings.  And indeed, this is exactly what I find:

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1. ### Steve Kass said,

March 10, 2011 @ 1:20 pm

The Federal Trade Commission mentions “up to” claims in this letter.

- Use Care When Making "Up to" Claims. In substantiating maximum or "up to…" claims (e.g., "save up to 40%"), you should not rely on an unusual or "outlier" test result. The advertised result should be one that the average consumer can reasonably expect to achieve.

I agree with Randall that these ads are “mathematically annoying,” but I’m glad to see that the FTC doesn’t consider the mathematics in this example a valid excuse for businesses to mislead.

2. ### John Cowan said,

March 10, 2011 @ 1:23 pm

As usual, Dorothy Sayers nailed the question long ago. Murder Must Advertise:

"[...] Now Mr. Pym [head of an advertising agency] is a man of rigid morality — except, of course, as regards his profession, whose essence is to tell plausible lies for money —"

(In fact "the more you spend the more you save" is actually true in the Pickwickian sense that the more of a discounted item you buy, the more money you save — relative to the undiscounted price. This assumes that you actually want the items.)

3. ### Jocelyn Hamilton said,

March 10, 2011 @ 1:25 pm

I believe that all rational people would regard the statement "You will save up to 15% or more on your purchases" as false…

I suggest that in normal speech, the word "save" implies that the amount concerned is a non-negative number, hence this very rational intuition.

4. ### Keith said,

March 10, 2011 @ 1:30 pm

For me, the first panel is almost exactly correct, and is what I have found misleading and annoying about advertising for years.

I say that it is "almost exactly correct", because the range of values does not usually reach into negative numbers.

Most often, this pseudo-mathematics is used in phrases such as "you could save up to 15% or more", or "new pack contains up to 15% or more extra [stuff]".

These statements only make sense where n is between 0 and 15… since if n is a negative number I would in fact be paying more, or getting less [stuff] in the pack.

What I would find more annoying, if I did not find them amusingly idiotic, are phrases such as "get up to x [stuff] from as little as $y"… where x represents the biggest or best example in the range and y is the smallest value in the range. If this were an advertisement for animals, it would be "get up to a blue whale from as little as$2" (assuming that you can still get a goldfish for $2). K. 5. ### Yuval said, March 10, 2011 @ 1:34 pm Though not quite as polite as yours, xkcdsucks's critique of this comic appears to find reasonable fault in Randall's other claims as well. 6. ### army1987 said, March 10, 2011 @ 1:43 pm Panel 3 assumes that the quantity being kept constant when doing the partial derivative of the amount you save wrt the amount you spend is the amount of stuff you buy, while I'd take it to be the unitary price. And it *is* true that if a widget which usually costs €1.50 now costs €1.00, I would save €5 if I spend €10 (i.e. if I buy ten widgets) and €6 if I spend €12 (i.e. if I buy twelve). 7. ### richard howland-bolton said, March 10, 2011 @ 2:03 pm @Yuval I couldn't get any further than xkcdsucks's unreasonable first point. "nobody actually uses [the phrase from the first panel]. He's combined 'up to 15% off all items!' and 'save 15% or more!'" I did a search for: +"up to" +% +"or more" and got 738,000,000 results (which is of course no more true than his statement, so just say a lot. I didn't bother with it further. 8. ### Will said, March 10, 2011 @ 2:47 pm @richard, I agree that xkcdsucks's point about the first panel is absurdly wrong. But part of what he says about the second is valid. He first makes another absurd statement by saying the second panel it is not about mathematics, but rather game theory (I guess ignorant of the fact that game theory is a branch of mathematics?). But then he correctly states that the panel makes a false assumption that the only form of expected value is monetary. It's absolutely true that people sometimes spend money on advertising with no expectation of recuperating the money — perhaps what they want recognition, or to get a message out, etc. Maybe what the customers get out of it really is free. That being said, and despite it, I actually liked panel 2 a lot because the idea behind it rings true. When an advertisement says "FREE" be wary, because chances are, it's not really. And I especially liked panel 3. The idea behind that is spot on. But I have an intense dislike for panel 1. I've heard this argument from so many people so many times, and it always strikes me as being willfully obtuse. It's always seemed completely clear to me that this sort of expression is, as Professor Liberman points out, a statement about distribution, not a mathematically absolute statement. And maybe I'm being thickly egocentric, but I have a hard time believing the people that complain about this are actually interpreting the statement much differently. Seems like are just making a fuss about it because they can. 9. ### Mr Fnortner said, March 10, 2011 @ 3:02 pm The phrase "up to * or more" on Google returns 6.18MM hits for me. Annoying though it is, it's well beyond an anomaly having become a standard commercial expression. My interpretation is that the typical savings is at or just below the named amount, and perhaps higher for the odd item. The problem with the word "free" is the reason that many regulatory bodies stipulate use of "no extra charge" for many situations formerly called "free". This does not preclude the merchant from actually offering free things, with or without the expectation of tangible reward. And finally the third panel misses the point of the advertisement. It is not about how much one pays for a single item, but about how much savings accumulate for quantities greater that 1. A Blondie comic I remember from decades ago had her announcing to Dagwood that she saved him a large amount of money by shopping at a sale at Tudbury's, whereupon he requested that she give him the savings. Her reply was that he was silly and certainly kidding. Therein is the misleading advertising element in panel three. It sounds to me as though the writer has some mathematical peeves not unlike the sort of grammar peeves many language folks have. 10. ### Ron Stack said, March 10, 2011 @ 3:05 pm I'm afraid I'm going to out myself as a purveyor of both Panel 1 and a modified version of Panel 3. I run a loyalty program that enables local merchants to provide cash back rewards to participating consumers. Merchants establish both the eligibility criteria and the percentage of reward offers. Merchants commonly establish, and consumers may be eligible for, multiple offers. So Consumer A may earn 10% cash back from Merchant X just for making a purchase but Consumer B may earn an additional 15% because it's her birthday, because she lives in a specific zip code and because she is over 60. In our marketing we occasionally use "up to x% or more" in the long tail sense Mark describes. We couldn't use the simple "up to" construction without referring to an outlier (one of our merchants offers 95% cash back under limited circumstances), which as @Steve Kass points out is a bad idea. So, we say that consumers can earn "up to" a maximum reward percentage that is common among our merchants (say, 20%) "or more". The intention is that consumers correctly understand that it is very likely (but not certain) that they can earn 20% cash back and less likely (but still possible) that they can earn more. Is this misleading? I really don't think so. We don't say, "The more you spend, the more you save" but we do say, "The more you shop, the more you earn" (i.e., in cash back rewards). This is clearly true in our program. Is the phrasing that Randall objects to very different in meaning or am I just – as usual – not keeping up with the math? 11. ### Ran Ari-Gur said, March 10, 2011 @ 3:06 pm @Will: Ads are frequently (usually?) disingenuous in some fashion or other. This makes it hard to follow a Gricean approach to interpreting them; if I heard a normal person use "up to __% or more" in a cooperative conversational context, I would interpret it as a vague statement about the approximate distribution (per the Maxim of Quantity: presumably they're trying to convey something, and that's the best bet), but when I hear an advertisement use it, I feel like it's trying to pull a fast one. It's trying to exploit my expectation of cooperation to create an implication that may or may not be realistic. [(myl) On the face of it, advertisements and political texts ought to be interpreted in a sort of anti-Gricean manner, based on the assumption that the author wishes to push the reader as far as practicable towards some state or action, with the only restraints being common sense ("Do not say what the hearer will never believe"), law or regulatory policy ("Do not say what the FTC will make you take back"), and reputational risk ("Avoid annoying people whose future business you care about"). For some reason, the formal pragmatics of anti-Gricean implicature haven't gotten much recent attention from linguists and philosophers of language. Or at least I haven't seen it.] 12. ### Adrian Bailey (UK) said, March 10, 2011 @ 3:40 pm @ Will: recuperating>recouping 13. ### KevinM said, March 10, 2011 @ 3:44 pm "The more you spend, the more you save." Sorry, doesn't bother me. No one would take that to mean that spending more of the money in your pocket results in larger amount of money in your pocket. For that to happen, the merchant would have to pay you. The advertiser is using "savings" in the commonly understood sense of a discount below what you'd otherwise pay for an item. He or she is saying that the discount becomes larger in absolute terms if you buy in quantity. If the list price of an item is$100 and the discount ("savings") is 15%, I "save" $15. Plus I now possess an item that is, ex hypothesii, worth$100. If I buy two, I "save" $30, and also own$200 worth of merchandise. Indeed, it wouldn't be uncommon for the discount rate to increase as you buy a larger quantity.
"Savings" as used here are defined as a reduction of what you'd otherwise have to pay, not as an increase in the cash you possess; that could happen only if the merchant paid you, which nobody expects.

14. ### Capn said,

March 10, 2011 @ 4:00 pm

@rhb

738,000,000 results

Sounds like a lot of false positives …

+"up to" +% +"or more" == +"up to" +"or more"

lrn2google and stop feeling so high and mighty about defeating an absolute statement by example that is also an obvious strawman argument with piss poor google-foo.

I hope I can do html code, there is no preview button.

15. ### Boris said,

March 10, 2011 @ 4:01 pm

"The more you spend the more you save" is almost always false. Here are the possible meanings:

1. The more you spend (as opposed to spending less in the same store) the more you save (on those extra items you'll be buying) – This is not useful. If you don't buy the items, you obviously save more overall.

2. The more you spend (here as opposed to elsewhere on the same purchases) the more you save. This is a good deal, but is very unlikely to use this phrase. "Shop here and save" is much shorter and gets the same point across.

3. The more you spend (now, when we have this special sale) the more you save (if you wait until the sale ends). Again, this is a good deal, but simpler would be "act now before the offer runs out"

4. You will actually pay less total if you buy more. This is rare, but it does happen (save 10% on purchases of $100 or more). But this is false for a different reason, as you're not actually spending more. Better would be "the more you buy the more you save". 5. The more you spend (now) the more you save (on your next visit). This is a blend of 3 and 4 and is the only one that is likely, truthful, and useful. Because you actually are spending more now, but save money in the long run. I did some googling of the phrase to see which sense is actually the most common, but I am mostly getting pages talking about the hypocricy and talking about government spending in a dubious way. The first genuine example appears to be #5: "Grandma Lena's BBQ -Come to Grandma Lena's and Hullihen's and place your order -While you wait give the person at the window your phone number from your sign-up form (see below) and we will record your transaction. -For every$25 you spend on food, drink, or ice cream, you earn 1% off a future purchase. "

So is the second and third.

The fourth example is #4:
"make a purchase of $50.00 and save 10% make a purchase of$75.00 and save 15%
make a purchase of $100.00 and save 20% make a purchase of$150.00 and save 25% "

It's not until the fifth example that you get an example of #1, something useless:

29. ### chris said,

March 11, 2011 @ 9:23 am

Randall isn't really engaging with the actual claim behind panel 3-type ads. The diagram should have "amount of product you buy" on the x-axis, "amount you pay" on the y-axis, and two different lines with positive slope passing through the origin, "our prices" (lower slope) and "competition's prices" (higher slope). Then "amount you save" is the gap between the lines, and actually does increase proportionally to "amount you spend" (since both are linear functions of "amount of product you buy").

So the claim is correct, in a sense (assuming their prices really are lower than the competition's), but may be somewhat misleading if you don't actually *need* 500 rolls of toilet paper (or whatever).

30. ### cameron said,

March 11, 2011 @ 9:29 am

I associate the phrase "up to * or more" with the TV ads made by baseball legend Phil Rizzuto for The Money Store back in the 80s. These were widely shown on TV in the New York area at the time, I don't know if the The Money Store aired the spots elsewhere.

If I recall the sequence, early on, they started with the catch-line "You can borrow up to $15,000!" – or whatever the value was. A few months later it changed, of course, to "Up to$15,000 – Or More!", and then reached complete absurdity with "Up to No Limit!!"

31. ### Jerry Friedman said,

March 11, 2011 @ 2:20 pm

@chris: That graph or one with "difference between your unit price and the price of a single unit" on the y-axis.

32. ### Josh Bowles said,

March 11, 2011 @ 9:53 pm

"the real number line goes up to infinity, or more"
I don't see anything mathematically false about "up to" including any and all real numbers… similar to an upper bound.

It is the contextual pragmatics that's going to tease out the expectations and presuppositions that will define where we start going "up" from… and it seems, whether Gricean or not, the expectation starts at 0, just as we do not expect to save .1543445677890 on an item. That is, there is some kind of mathematical expectation to be met in a consumer context… (for example, do we really expect to pay $0.99 for anything??, or houses advertised in the$160's — only someone who knows that houses don't sell for $160.00 will get the intended interpretation… and for those who understand the intent don't seem to see this as any kind of manipulation—but simply an expectation on the part of the advertiser that the reader be sufficiently "common sensical".) And interestingly enough, advertising something as$0.99 may not turn out to be any more "devious" than asking "Can you pass me the salt?" and intending "pass me the salt". Without further research we won't know, but I would bet that in many cases there is a disconnect between the advertiser's expectation for consumer "common sense" and the consumer's interpretation of the advertiser intent….

33. ### Ben said,

March 11, 2011 @ 10:22 pm

@Ron: "The more you shop, the more you earn" can only be true for shopping critics, because "earn" means to expend effort producing something which is of value to society. Except for the critics doing research for a review, shoppers at your store aren't producing anything, and therefore cannot be said to "earn" anything.

34. ### John Cowan said,

March 13, 2011 @ 1:54 am

Ben: Your definition of earn is far too narrow to fit the facts of usage. Criminals are said to earn their punishments, batters to earn their runs, and we are even told that the wages of sin is death.

35. ### Will said,

March 16, 2011 @ 12:54 pm

Adrian Bailey (UK) said "@ Will: recuperating>recouping"