Page:Mind (New Series) Volume 12.djvu/372

358 HUGH MACCOLL : Paper in the Proceedings of the London Mathematical Society, I employed the symbol A B to assert, not (as here) that a certain unnamed individual of the class A belongs also to the class B, but that every A belongs to the class B. Subsequent experience however taught me that this convention was inconvenient ; so I abandoned it. Let us now consider point No. 4. In their logic of class inclusion their symbol (A = B) asserts that every individual of the class A is included in the class B, and every individual of the class B in the class A. In their logic of propositions this same symbol (A = B) asserts that the propositions A and B are either both true or both false, which is quite a different defini- tion. In my system the symbol (A = B) has neither of those meanings ; it always asserts that it is certain that either A and B are both true or both false. Thus, when A and B denote each a single proposition, if we put (A = B) a for the symbol (A = B) when the latter has their interpretation, and (A = B)0 for the same symbol when it has my interpretation, we get the following comparison and definitions : (A = B). = AB + A'B' (A = B),, = (AB + A'B') e (A - B), - (A - B)l ; so that my symbol (A = B) is formally stronger than their symbol (A = B), just as A* is formally stronger than A T . The symbol A T asserts that A is true (true at least in the case considered) ; whereas A e asserts that A is certain (that is to say, true in all circumstances consistent with our data and definitions). In their logic of class inclusion they use the symbol AB (or its synonym A x B) to denote the class of individuals common to the classes A and B. With irrefutable logic they then infer that their proposition A -< B is equivalent to their proposi- tion A = AB. But consistency of notation demands that this convention as to the meaning of AB should hold good also as regards the classes and 1, which (with them) denote false and true propositions respectively. Now, with this interpretation of their symbols, the class we know, and the class 1 we know, but what is the class Oxl common to both ? Where can we find an intelligible and unambiguous proposi- tion that can be described as both true and false ? False pro- positions are numerous enough, as we often learn to our cost, and they are usually quite clear and unambiguous ; but I have never yet come across an intelligible proposition that could be classed as both true and false. Such propositions I denote in my system, not by the symbol i, which denotes a false but intelligible proposition, nor by the symbol rj, which