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 XII. NOTE. MR. MACCOLL'S QUESTION ON P. 144 OF MfXD FOR JANUARY, 1900. I answer M)'. MacColl's question not for his information but my own. I have much admired his clear and useful system of notation for Symbolic Logic, yet I fear that I must have misunderstood something in his way of vorking it, and I write this answer in the hope that if there is a mistake 1 may discover where it lies. Query. Whether the implication If it in firnbablf that A is i-i'rtaiii it ix fcrtain that A is proliahl/' is always true. It seems to me that while the fact may be so the implication is never valid, that is to say, you might state a case in which it was probable that A was certain and also certain that A was probable, but in order to ensure the truth of the latter statement something more would necessarily be assumed than was contained in the first. It is probable that A is certain is quite consistent with it ix /loxxiiil,- that A ixfalxe and even with it is possible that A is impossible. That a thing is probable means that more than half the chances are in its favour, but the other chances may be of any kind, unless it is expressly stipulated that some chances are excluded. Let ten chances make A certain, one make it false and one impossible, then it is probable that A is certain, but for all that it may be false or impossible. I do not see how you can ever under any circumstances infer certainty from probability. It would clear my thought very much if I could put all possible cases in the Symbolic Notation, but this I cannot do, because as I look at it the two kinds of certainty, probability, etc., ought to be more thoroughly separated than they are by the method of indices. It should be shown that they are always to be kept separate and not mixed in working, and this is not done when they are merely written one after another. A is certain, means either that A is certain in itself, that is to say, is determined by its causes, so that A, if it is an event, either has happened already and so is determined, or if still future is certain to happen, or else it means that the data at my disposal make me certain of it. Now suppose for the moment we put the indices which mark these certainties on opposite sides. If there is to be an eclipse to-morrow, then it is certain, that is to say causes already in existence determine the event. Call this A f. If I see the fact stated in a reliable Almanac then I am certain of it. Call this eA<. I am certain that A is certain. But say that a friend of doubtful accuracy tells me that he has seen the statement in the Almanac, then PA<, it is probable that A is certain ; but is it certain that A is probable ? By no means, for knowing the facts I say, A' + Ai, An eclipse to-morrow is either certain or impossible. Other cases might be stated, but they would suggest another alteration in the notation and this might not be admissible. Now all this appears to be so plain and evident that I am afraid I must have in some way or other mistaken Mr. MacColl's meaning, and if so I would be very glad to know where. .1. N. SHEARMAN.