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502 HUGH MACCOLL : Venn in his Symbolic Logic (second edition, p. 442) calls "Alice's Problem," and which I understand has (under another form) been already discussed by logicians, but with varying conclusions. Given the statement A B (C : A B '), can C be true ? My answer is (1) that the data are not sufficient to justify the conclusion that C is possible ; and (2) that the data are not sufficient to justify the conclusion that C is impossible. This I prove as follows : Putting < for A B (C : A B '}, we get, by a process explained in my fifth paper in the Proceedings of the Mathematical Society, W0 = A" + B'C" S< = W< + A'B'O = A* + B<O + A*B*O. Now, if C^S^) could be proved = 77, we should have S< : O, and then O would be a legitimate conclusion from <. But C"S< = C'(A" + B e + A e B) = C*S(A B ), which, without further data, can not be proved = 77. Hence, we cannot from < draw the conclusion that C is possible. Again, if O 1 S< could be proved = 77, we should have S<f> : O, and C 71 would then be a legitimate conclusion from <f>. But OS< = A'O, which, again, without further data, cannot be proved = 77. Hence, we cannot from </> conclude that C -is impossible. Thus we have not sufficient data from which to infer either O or its denial O. In other words, though C must either be possible or impossible, we cannot from the given statement A B (C : A B '), without some additional premise, ascertain which alternative is the true one. In obtaining the results W< and S< I assumed a proposi- tion which is not quite self-evident, namely, that the data A B (C : A B ') constitute an impossibility when A, B, C are all three variables. This proposition I now proceed to prove. No implication a : ft can be a variable l when (as in the data of this problem and throughout the preceding argu- ment) its antecedent a and consequent /S are both singulars ; that is to say, when each letter denotes only one statement, and always the same statement, be it of the class e or 77 or 1 This is a point which I did not make sufficiently clear in my recent paper in the Proceedings of the Mathematical Society when discussing W (a : /3)* and S (a : ) e . The implication a : /3 would be a variable if a atid /3 were taken at random repeatedly out of statements belonging some to the class (, some to the class ij, and some to the class 6. For another case in which a : would be a variable, see the concluding paragraph of this paper, preceding the note on a logic of 3" dimensions.