Monday, September 10, 2007

Lewis Carroll on Existential Import

Lewis Carroll has a fascinating argument about the existential import of propositions in the notes to what would have been the second part to the Symbolic Logic. (Only Part I was published in Carroll's lifetime. Part II seems to have gotten as far as galley proofs, but most of these were lost.) First Carroll has some salutary advice against dogmatism on the matter:

...I maintain that any writer of a book is fully authorised in attaching any meaning he likes to any word or phrase he intends to use. If I find an author saying, at the beginning of his book, "Let it be understood that by the word black I shall always mean white, and that by the word white I shall always mean black," I meekly accept his ruling, however injudicious I may think it. (232)

The question of existential import really boils down to two questions: what views may logically be held and what views may conveniently be held. The three types of propositions we are considering are those that typically are indicated by 'some' (I), by 'no' (E), and by 'all' (A). (If you are wondering where O is, Carroll treats "Some x are not y" as the I proposition, "Some x are non-y".) He identifies two views he regards as logically consistent.

(1) I and A have existential import, but E does not.
(2) E and A have existential import, but I does not.

His reasoning is as follows.

(1) Suppose that I asserts the existence of its subject. We must then regard A as having existential import as well, since A necessarily contains a proposition in I. Thus if I has existential import, so does A. Are we forced on this view to a particular view of the status of E? Yes. Assume that E has existential import. Then, if "No xy are z" is true, it follows that some things exist that are x and y. But since I has existential import, the same thing follows from "Some xy are z." "No xy are z" and "Some xy are z", however, are contradictories; whence it follows that if I, A, and E all have existential import that it is a necessary truth that some things exist that are x and y (for any x and y, of course), which is absurd. Thus, if I has existential import, so does A; and if I and A have existential import, E cannot have it.

(2) Let us suppose that I does not assert the existence of its subject. Suppose, given this, that E does have existential import. Then "No x are y" indicates that some x exist and none of them are y, i.e., all x are non-y, which is an A proposition. Further, "All x are non-y" proves "No x are y"; the two are equivalent. Thus every proposition in A is equivalent to one in E, and has existential import. So if I does not have existential import and E does, A must have existential import.

(3) But what of the third option, that neither I nor E has existential import? If neither I nor E has existential import, this implies that A doesn't have it, either. But this view wreaks havoc with logical facts. Take the valid syllogism Darapti:

All m are x; All m are y; Therefore, Some y are x.

Since this view requires that we take all these statements as hypothetical, it is equivalent to the following argument:

If there were any m in existence, all of them would be x;
If there were any m in existence, all of them would be y;
Therefore, if there were any y in existence, some of them would be x.

But this is invalid. For there could be a case consistent with the premises in which the conclusion would be false, namely, if y existed but x and m did not. The same problems arise with the syllogisms Disamis, Datisi, Felapton, and Fresison.

But even if you were willing to sacrifice Darapti, Disamis, Datisi, Felapton, and Fresison together just to save the no-existential-import view, this view has other problems. I is taken by virtually everyone to allow conversion, i.e., from "Some x is y" you can infer "Some y is x" and vice versa. But if we make these hypotheticals, this simply cannot be done.

Thus only (1) and (2) are viable approaches. (2), however, has the problem (which Carroll brings out very humorously) of diverging quite considerably from ordinary language. (It requires us to say that, "Some millionaires are in my club" can be true without implying that there are any millionaires at all; and that "No one convicted seven times of forgery are allowed" implies that there are people who have been convicted seven times of forgery.) This is not a logical problem, but makes translation into abstract form extraordinarily tricky. Thus he prefers (1).

(Carroll does recognize that there are other possible views that do not assert straightforwardly one way or another. For instance, one could say that "Some x are y" merely means that x and y are compatible. But this would run into a great many practical inconveniences as well. Further, one could hold that A propositions sometimes have existential import and sometimes don't, which does confessedly appear to have some support in common usage.)

The sharp eye will note that Carroll assumes that A propositions contain I propositions. (Indeed, he holds that all A propositions are really compound propositions. "All dogs are canines" is really the double proposition "Some canines are dogs and no dogs are noncanines.") In his editorial notes, William Warren Bartley III says that Carroll is here begging the question, "which is precisely whether it is indeed convenient to regard propositions in A as 'necessarily containing' propositions in I" (233n). This, of course, is not true. The question is not whether A contains I but whether A and I have existential import, and they are not the same. In modern logic it is always assumed that I must have existential import. But Carroll makes no such assumption, nor would any of the algebraic logicians of the time have taken it to be true without argument. (There's a lovely little summary of the history of this point at Buckner's Logic Museum). Thus in assuming that A contains I he is not assuming that A has existential import, because he does not assume that I has existential import. He thinks, of course, that it is more convenient to take I as having existential import than not, but you will notice that one of his logically viable options is the view on which I does not assert the existence of its subject. Because modern logicians are willing to throw out conversion per accidens and subalternation,, i.e., the view that A propositions contain I propositions in some way, they can hold that A and E lack existential import while I has it; I's having existential import is the point they take to be non-negotiable. But Carroll takes this to be entirely negotiable; what he takes to be non-negotiable is the containment principle. And thus we have a different view.

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References are to Lewis Carroll, Symbolic Logic, Bartley, ed. Clarkson N. Potter (New York: 1977).