The category boundary paradox

« previous post | next post »

Today's SMBC:

The aftercomic:


  1. Theophylact said,

    September 20, 2017 @ 1:41 pm

    Sorites, I believe.

    [(myl) I do too.]

  2. unekdoud said,

    September 20, 2017 @ 2:41 pm

    If our experience with grated cheese is any guide, it can be 100% ice cream long before being 0% toast.

  3. Scote said,

    September 20, 2017 @ 2:45 pm

    Kind of un cool to post the bonus comic on your site rather than leaving as a bonus that encourages folks to click over to SMBC and give him some traffic for his free site.

    [(myl) My sad experience over the years is that webcomics sites are especially prone to bit rot. But I hereby urge everyone to patronize the SMBC site and buy their new book.]

  4. Zizoz said,

    September 20, 2017 @ 5:30 pm

    Well, there's still the mouseover text.

  5. D.O. said,

    September 20, 2017 @ 5:36 pm

    On the third panel the boy should have said "no". It would have added bonus of being more realistic.

  6. richardelguru said,

    September 20, 2017 @ 5:48 pm

    However, if you have a bowl of high fiber toast (toast in a bowl????) and replace a tiny fraction of it with ice cream…

  7. Scote said,

    September 20, 2017 @ 7:53 pm

    @ Mark Liberman

    I hear you on the bit rot. I'm still surprised at how many websites use Twitter and Instagram embeds and such for reporting rather than screen shots that reliably archive the information, which is why I didn't say anything about the main comic being downloaded and hosted by you rather than hot linked. And if the bonus comic was key to understanding the main one, I wouldn't have said anything. :-)

  8. David Morris said,

    September 20, 2017 @ 8:59 pm

    Replacing the ice cream at the rate of 0.0001% (one-millionth of the total) per second (just assuming) would take eleven and a half days.

  9. Neal Goldfarb said,

    September 21, 2017 @ 1:45 am

    It strikes me that the sorties paradox is sort of topologically similar to Zeno's paradox.

  10. Miles said,

    September 21, 2017 @ 4:01 am

    @Neal Goldfarb

    I'm not sure about that. In the sorites paradox, the boundary point (where we switch from one category to another) is unknown and the increments are fixed in size. In a Zeno paradox, the boundary is known (the arrow reaches target etc) but the increments are decreasing in size.

  11. Ed M said,

    September 21, 2017 @ 5:25 am

    The child says "I am sure there's a logical fallacy there". While sorites is a paradox, perhaps the fallacy the child is looking for is the continuum fallacy — rejecting a claim because it is not precise enough?

  12. Andrew Usher said,

    September 21, 2017 @ 7:49 am

    I suppose that would be it (the fallacy), though as people don't seriously use arguments like this (which I suppose is the intended humor), unlike many other fallacies which are encountered often, people are unlikely need a name for it. If asked to provide a word or phrase for this substitution I would answer 'lying'.

    And indeed, that's where we can compare this to the '100% grated cheese' matter. The person selling, preparing, serving (etc.) food is making a warranty by its description. Without that assumption we'd have no way of finding food adulteration to be wrong, or even of sensibly defining it. The predicate 'is ice cream' may be a sorites in itself, when used by someone in describing the food that they were served, for example – but, when 'warranted', is a normative claim that the stuff has been made to be, to the best of that person's knowledge, within common acceptation of what is 'ice cream'. This excludes the deliberate addition of any ingredients that are not a part of ice cream; what amount of unintentional contamination would trigger that threshold is moot, I would say, as I imagine one would refuse to eat it before it was reached!

    The lesson of the sorites paradox is that some ascriptions are too vague to be treated by strict logic and this sort of thing is one of them. Zeno's paradoxes, on the other hand, are never paradoxes at all if you accept mathematics.

    k_over_hbarc at

  13. Chandra said,

    September 21, 2017 @ 2:42 pm

    @Andrew Usher: "as people don't seriously use arguments like this"

    Sure they do. I can think of two examples I've heard off the top of my head – the argument that race doesn't exist (so therefore we can all be "colourblind" and ignore it), and likewise for virginity (since there is disagreement on what constitutes sex, the concept of virginity must be meaningless).

  14. mg said,

    September 21, 2017 @ 4:04 pm

    It's a problem when sharing comics. I usually provide the mouse-over text for xkcd because many of my non-tech friends wouldn't know to try hovering their mouse even if I just gave them the link. Omitting it wouldn't provide any incentive to actually go visit the comic. My hope is that occasionally posting xkcd or SMBC will provide incentive for them to start visiting those sites – which is more likely to happen if they also see the extra goodies, IMHO.

  15. Don Sample said,

    September 21, 2017 @ 6:55 pm

    I took it to be a combination of both Sorites and Zeno, with .0001% of the remaining ice cream being transformed into toast each time.

    I'm also surprised that the mother didn't claim to be eating a bowl of toast at the end.

  16. Michael Watts said,

    September 21, 2017 @ 8:29 pm

    I hear you on the bit rot. I'm still surprised at how many websites use Twitter and Instagram embeds and such for reporting rather than screen shots that reliably archive the information

    Well, screenshots archive information "reliably" in the sense that you can still read the screenshot after the putatively screenshotted page has gone away. They're not "reliable" in the sense of accurately reflecting what a page says or said; you can take a screenshot of a web page saying anything you want. An embed at least shows you what the page currently says rather than being an unverifiable claim about what it says.

  17. Neal Goldfarb said,

    September 21, 2017 @ 11:58 pm

    @miles: "I'm not sure about that. In the sorites paradox, the boundary point (where we switch from one category to another) is unknown and the increments are fixed in size. In a Zeno paradox, the boundary is known (the arrow reaches target etc) but the increments are decreasing in size."

    But that doesn't establish that there's no *topological* similarity (not to be confused with typological similarity). Topologically, a bagel has more similarity to a drinking straw than it does to a bialy, which in turn is topologically similar to the plate it is served on.

    And if that's too surface-y for you, or stretches things too much, just keep in mind that both the sorties paradox and Zeno's paradox can be summed up this way: You can't get there from here.

  18. Miles said,

    September 22, 2017 @ 3:37 am

    @Neal Goldfarb

    I'm still not convinced! (By the way, I have a degree in Maths and understand the concept of topology – you are asserting some kind of continuous mapping that would get us from Zeno to sorites).

    While Zeno has the flavour of "you can't there from here", surely sorites is more "we know we get there from here, but we can't work out where we cross the boundary from there to here".

  19. Andrew Usher said,

    September 22, 2017 @ 7:23 am

    No, I think Zeno's paradoxes were also 'we know we get there from here', so I do see the relation. But of course the resolution of the two is quite different.

    Neither of those two examples (which I do know about) are instances of the fallacy being used 'seriously' in my sense. The first, about race, is the more interesting. Obviously, as is usual with political arguments, most people that talk about the matter are not being original and just parroting what they've heard from those they 'agree with', but someone has to originate the claim.

    In this case I think a major reason is blurring the 'is/ought' distinction, a real and very common fallacy we don't have an agreed name for. It is my observation that many disagreements between well-meaning people on apparently factual issues can be traced to difference between 'is' claims and 'ought' claims (or, rather, beliefs). Here the latter is that 'we should be color-blind' (not 'can' as you quoted – obviously it's logically possible) and the former that we are. Anyone claiming that i.e. there are no races independently of our minds, explicitly based on the moral claim, is committing that fallacy; and if not so based it's just lying or bullshitting.

    The second, about virginity, is basically used to my knowledge as a kind of lame rationalisation rather than an argument; but in any case it's different in kind because for any given definition there can't be a continuum between virginity and the opposite, it's merely that there isn't (as the speaker imagines) one single definition. That doesn't really fall under what I was thinking of, and with Ed calling it 'the continuum fallacy' I suppose it isn't. But I can see the alternative interpretation, naturally.

  20. Helen said,

    September 22, 2017 @ 10:24 am

    For what it's worth, many months ago I did start visiting SMBC because of your posts. Funny stuff.

  21. Bev Rowe said,

    September 23, 2017 @ 10:50 am

    Isn't sorites simply the misuse of a tool? If you were asked to shift a pile of sand you wouldn't get out a pair of tweezers and shift it grain by grain. You'd use a shovel. So, similarly, to ask an observer to make a judgement about a pile of sand by observing single grains being added is misuse of a linguistic tool. Ask them to judge as you add shovelfuls and you get a more meaningful response.

    We are always ready to use words with a carelessness with which we would never use metal tools.

  22. Jim said,

    September 23, 2017 @ 11:47 am

    This is akin to the museum conservator's "George Washington's Hatchet". GW's original hatchet is on display, but due to corrosion a replica head was made from steel forged on site in the manner of 1700's metalworking, and due to rot a replica handle was made from a tree on the Mount Vernon property.

  23. Andrew Usher said,

    September 23, 2017 @ 1:46 pm

    Again, that example has no continuum involved, and is not a sorites. That question is, I believe, known as the 'ship of Theseus' paradox, and while it's obviously false for something like an ax/hatchet, the more complex it is the harder it is to deny continuity. We must conclude likewise that 'the same' can also be vague.

  24. Jaime said,

    September 25, 2017 @ 3:19 pm

    I don't think the setup in the comics really works to create a paradox. Sorites paradox relies on the fact that a heap is made of many individual grains, and how many grains are necessary to make a heap is not well defined. In the comic, however, the use of percentage figures clearly makes it sound as if ice-cream is a continuum. Therefore, the answer is clear: if the ice cream percentage is 100% it is *ice cream*, if the percentage is between 0% and 100% it is *ice cream and toast*, if the percentage is 0% it is *toast*.

  25. BZ said,

    September 26, 2017 @ 4:25 pm

    And if it's less than 10% milk fat it's Frozen Dairy Dessert

RSS feed for comments on this post