We can now clearly distinguish between the probabilities of exclusive and unexclusive events. Thus, if A and B are events which may happen together like rain and high tide, or an earthquake and a storm, the probability of A or B happening is not the sum of their separate probabilities. For by the Laws of Thought we develop A ꖌ  B into AB ꖌ A*b* ꖌ *a*B, and substituting α and β, the probabilities of A and B respectively, we obtain α . β + α . (1 - β) + (1 - α) . β or α + β - α . β. But if events are *incompossible* or incapable of happening together, like a clear sky and rain, or a new moon and a full moon, then the events are not really A or B, but A not-B, or B not-A, or in symbols A*b* ꖌ *a*B. Now if we take μ = probability of A*b* and ν = probability of *a*B, then we may add simply, and the probability of A*b* ꖌ *a*B is μ + ν. Let the reader carefully observe that if the combination AB cannot exist, the probability of A*b* is not the product of the probabilities of A and *b*. When certain combinations are logically impossible, it is no longer allowable to substitute the probability of each term for the term, because the multiplication of probabilities presupposes the independence of the events. A large part of Boole’s Laws of Thought is devoted to an attempt to overcome this difficulty and to produce a General Method in Probabilities by which from certain logical conditions and certain given probabilities it would be possible to deduce the probability of any other combinations of events under those conditions. Boole pursued his task with wonderful ingenuity and power, but after spending much study on his work, I am compelled to adopt the conclusion that his method is fundamentally erroneous. As pointed out by Mr. Wilbraham,[113] Boole obtained his results by an arbitrary assumption, which is only the most probable, and not the only possible assumption. The answer obtained is therefore not the real probability, which is usually indeterminate, but only, as it were, the most probable probability. Certain problems solved by Boole are free from logical conditions and therefore may admit of valid answers. These, as I have shown,[114] may be solved by the combinations of the Logical Alphabet, but the rest of the problems do not admit of a determinate answer, at least by Boole’s method. [113] *Philosophical Magazine*, 4th Series, vol. vii. p. 465; vol. viii. p. 91. [114] *Memoirs of the Manchester Literary and Philosophical Society*, 3rd Series, vol. iv. p. 347. *Comparison of the Theory with Experience.*