But my chief object in writing to you to-day is to bring specially to your notice my ideas on the nature of so-called “necessary truths.” I am not quite clear how far you will find my views harmonise with your own. To a great extent I am inclined to think they are simply a further analysis of the views you express in _The Monist_ and in your _Fundamental Problems_. I will briefly sketch my own ideas and you can then judge whether they are yours also or not. In my _Essay on Reasoning_ I classify assertions as Truisms (assertions whose truth depends solely on the definitions of their terms) and Real Assertions, which convey some real subjective or objective information. I show that the validity of all purely formal knowledge depends on the fact that it is deduced from definitions alone, which are laid down _arbitrarily_ and that the supposed peculiar certainty of the theorems of pure mathematics is merely due to the fact that they are all truisms. Thus, I think it a misnomer to call such theorems “necessary” truths. It would be nearer the mark to call them “arbitrary” truths. There is no _necessity_ whatever about the theorem “twice two is four.” “Two” is defined as 1 + 1; “twice,” as the operation of adding a thing to itself. It follows from this that “twice two” is 1 + 1 + 1 + 1; and this, by _definition_, is “four.” If “four” were defined as 1 + 1 + 1, (and there is no “necessary” reason why it should not be,) then “twice two” would _not_ be “four.” The assertion “twice two is four” conveys no real information whatever—at best it could only tell us what one of its terms meant if they had not all been previously defined. I cannot insist too strongly on the importance of a proper understanding and use of logical definition. If you desire to know whether a given assertion is true or false, _a priori_ or _a posteriori_, the first step in the investigation MUST be to find out how its terms were defined. If it turns out that the truth (or falsehood) of the assertion can be formally deduced from these definitions, then the assertion is a truism (or contradiction in terms): in either case it can give no real information, and even if true cannot be a “necessary” truth. Only if the definitions of the terms are both independent and consistent is it open to discussion how we might come to a knowledge of the fact it expresses.