What Lewis Carroll Says Exists That I Don’t


I borrowed from the library Symbolic Logic, a collection of an elementary textbook — intended for children, and more fun than usual because of that — on logic by Lewis Carroll, combined with notes and manuscript pages which William Warren Bartley III found toward the second volume in the series. The first part is particularly nice since it’s text that not only was finished in Carroll’s life but went through several editions so he could improve the unclear parts. In case I do get to teaching a new logic course I’ll have to plunder it for examples as well as for this rather nice visual representation Carroll used for sorting out what was implied by a set of propositions regard “All (something) are (something else)” and “Some (something) are (this)” and “No (something) are (whatnot)”. It’s not quite Venn diagrams, although you can see them from there. Oddly, Carroll apparently couldn’t; there’s a rather amusing bit in the second volume where Carroll makes Venn diagrams out to be silly because you can make them terribly complicated.

The book also captures, in Book X of the second volume, an argument that’s been, from our point of view, settled sufficiently long ago that it’s easy to forget it was ever controversial, unless you are teaching a logic course and wondering why the students aren’t following you. The question is whether putting forth the proposition — supposing to be true the statement — that “All (something) are (this)” is also making the statement that “(Something) exists”.

I would come down on the side of no, just asserting that all of something has some property doesn’t assert that the thing exists. For example, if I assert, “all unicorns are one-horned animals”, I’m perfectly clear in what I’m talking about, and I might go on to make deductions about what unicorns are like (probably rather good at spindling things) without asserting that there are any unicorns. I apologize to any unicorns who wish to object. But this is more or less the standard position in the mathematical side of things, that putting forth the proposition that all examples of some collection of things has a particular property isn’t by itself an assertion that any of those things exist.

(There is also the graduate student urban legend of the senior ABD who had written a massive thesis on the interesting properties of a class of numbers; at the defense his advisor finally pays enough attention to the student to start working it out and discovers that the set of numbers with that property is — in one variant — empty or — in another variant — the lone number zero, and all that thesis work is for, pardon me, nought.)

Carroll comes down soundly on the “you are too asserting that the thing exists”, and he puts forth this argument in a letter to one disputant (The Elements Of Inductive Logic author Thomas Fowler):

Suppose my (empty) purse to be lying on the table, and that I say

     All the sovereigns in that purse are made of gold;
     All the soverigns in that purse are my property;
         ∴ Some of my property is made of gold.”

That is (according to your interpretation of the copula),

     “If there are sovereigns in that purse, they are all made of gold;
     If there are sovereigns in that purse, they are all my property;
         ∴ If I have any property, some of it is made of gold.”

It seems to me that, though these two premisses are true, the conclusion may very easily be false: it might easily happen that I had much “property”, but that none of it was “made of gold”.

I don’t like Carroll’s claim that the recasting of the argument doesn’t change its content.

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Author: Joseph Nebus

I was born 198 years to the day after Johnny Appleseed. The differences between us do not end there.

6 thoughts on “What Lewis Carroll Says Exists That I Don’t”

  1. You really have to deny existential import to those statements to make the rest of everything work sensibly. The fact that such statements are interpreted as a type of conditional in modern symbolic logic bears this out.

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    1. Yes, and that’s the way it’s interpreted in mathematical logic. Bartley, the editor for this compilation, points out that it captures a moment in the history of mathematics where the question of existential import was not settled, or at least hadn’t been settled on a particular convention, and it’s interesting seeing a person who’s really quite good in logic arguing for the side that lost out. It’s a little moment showing mathematics — and philosophy — as an alive thing.

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  2. Of course, using the “all unicorns are one-horned animals” example sneakily sidesteps the real problem for ordinary intuition: that if existential import is not attributed to all/no statements, the straightforward result[1] is that any such statement about an empty class is (trivially) true.

    [1] The less straightforward way of dealing with this is to start positing imaginary objects or possible worlds.

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    1. And though I mention it on Tuesday, and will in the follow-up to this article that I mean to write, I should point out to readers who stumble across this page alone that you’re absolutely right. I threw in an example that’s intuitively obvious and skipped that it leads also to some big, intuitively not obvious conclusions. In fact, it leads to conclusions that most people would intuitively say are wrong, but if we don’t accept those we have worse problems.

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