The (almost really) Complete Works of Lewis Carroll

In attempting any of these Problems, the reader is requested to bear in mind the following assumptions, which I have made throughout:—

(1) That the proposition “all x are y” is the sum total of the two propositions “some x are y” and “no x are not y”.

(2) That the proposition “no x exist” is the sum total of the two propositions “no x are y” and “no x are not y”.

(3) That, in setting a Problem, I hold myself free to demand proof of a less conclusion than might be logically deduced from the premisses: e. g. if the full logical conclusion were “all x are y”, I should hold myself free to set, as the Problem, “prove that some x are y”; or, if the full logical conclusion were “no x exist”, I should hold myself free to set, as the Problem, “prove that no x are y”. Thus, in Paper 8, Problem 7, if it were possible to prove, from the premisses, that “no magistrates exist”, I should hold myself free to set, as the Problem, “prove that no magistrates take snuff”.

(4) That, in setting a Problem of the form adopted in the first four of Paper 8, where the premisses and the conclusion assert certain sequences among certain sub-propositions, but neither assert nor deny any such sub-proposition taken by itself, I hold myself free to demand proof of a less result than might be logically deduced from the premisses: e. g. if the full logical conclusion were “if all a are b, and if some c are d, all x are y”, I should hold myself free to set, as the Problem, “prove that, if all a are b, and if some c are d, some x are y”; or, if the full logical conclusion were “if all a are b, and if some c are d, no x exist”, I should hold myself free to set, as the Problem, “prove that, if all a are b, and if some c are d, no x are y”.

[It may be worth while to remark, in reference to the case last mentioned, that it would not be correct to say that the Problem, whose full conclusion is “if all a are b, and if some c are d, no x exist”, is only soluble, (i. e. that its conclusion is only true), if one of the terms be non-existent; for this conclusion merely asserts a sequence among the three sub-propositions, “all a are b”, “some c are d”, “no x exist”, and the validity of this sequence is entirely independent of the truth or falsehood of these sub-propositions taken by themselves. Thus, it would be quite logical, in the case contemplated, to give an additional premiss, viz. “some x exist”, without making the Problem at all less soluble.

Supposing that this additional premiss were given, the full reply to this Problem, if set in the form “prove that, if all a are b, and if some c are d, no x exist”, would be “from the given premisses, omitting the last, we have proved the required sequence, viz. “if all a are b, and if some c are d, no x exist”: but it is also given that “some x exist”: hence, if the two propositions, “all a are b” and “some c are d”, were simultaneously true, the two propositions, “no x exist” and “some x exist”, would be simultaneously true; which is absurd: hence the two propositions, “all a are b” and “some c are d”, cannot be simultaneously true”.

And the full reply to this same Problem, if set in the form “prove that, if all a are b, and if some c are d, no x are y”, would be “from the given premisses, omitting the last, we have proved the sequence “if all a are b, and if some c are d, no x exist”, which contains, as a portion of itself, the required sequence “if all a are b, and if some c are d, no x are y”: but it is also given that “some x exist”: hence two results follow:—first, combining the two propositions, “no x are y” and “some x exist”, into the single proposition “all x are not y”, we prove the sequence “if all a are b, and if some c are d, all x are not y”; secondly, we see that, if the two propositions, “all a are b” and “some c are d”, were simultaneously true, the two propositions, “no x exist” and “some x exist”, would be simultaneously true; which is absurd: hence the two propositions, “all a are b” and “some c are d”, cannot be simultaneously true”.

To make my meaning yet more clear, let me add a “concrete” illustration. Supposing I had set a Problem, having, as its premisses, the rules enacted by an eccentric school-master as to the daily dinner, and, as its conclusion, “prove that, when there is beef and spinach, there are no potatoes”; and supposing it were found that some of the rules led to the conclusion “when there is beef, there are no boiled potatoes”, and that the others led to the conclusion “when there is spinach, there are no unboiled potatoes”, so that the whole set of rules led to the conclusion “when there is beef and spinach, there are no potatoes”: it would not be correct to say that this Problem is only soluble (i. e. that its conclusion is only true) if potatoes never appear on the table: for this conclusion merely asserts a sequence among the three propositions, “there is beef”, “there is spinach”, “there are no potatoes”: and, if we were told “there always are potatoes on the table”, this fact would not in the least affect the validity of this sequence, but would merely prove that the dinner never includes both beef and spinach, but that one or the other of them must always be absent.]

(5) That the proposition “A is true or B is true” is to be regarded as the contradictory of “A is false and B is false,” and that this is the only state of things which it is meant to exclude. Hence it is to be regarded as compatible with any one of the other three possible states of things, viz.:—

(6) That two phrases, which merely differ slightly in form, are to be regarded as identical: e. g. “sun-shine always brings on fog” is to be regarded as identical with “when the sun shines it is foggy”: similarly, “I light my fire” and “I have a fire” are to be regarded as identical.

(7) That predicates, one of which denies the other, are to be regarded as contradictory, and as constituting an exhaustive division of the things of which they are predicated: e. g. in using the predicates, “popular” and “unpopular”, of men, I assume that every man is one or the other, so that the two classes, “popular men” and “unpopular men”, constitute an exhaustive division of “men”.

(8) That the proposition “B is greater than C” must be assumed to mean that the magnitude, residing in B, is greater than that residing in C, and would continue so to be, even if it resided in something other than B. Hence it must be assumed to be equivalent to “every thing, that is not less than B, is greater than C”.

[N.B. If this assumption be not made; i. e. if it be assumed that the proposition “B is greater than C” predicates the quality “greater than the magnitude residing in C”, of the magnitude residing in B, merely when so residing, and does not predicate any quality of it, when residing in something other than B; it does not seem possible to prove the required conclusion from the data.]