It follows from what has been said that a vague thought has more likelihood of being true than a precise one. To try and hit an object with a vague thought is like trying to hit the bull's eye with a lump of putty: when the putty reaches the target, it flattens out all over it, and probably covers the bull's eye along with the rest. To try and hit an object with a precise thought is like trying to hit the bull's eye with a bullet. The advantage of the precise thought is that it distinguishes between the bull's eye and the rest of the target. For example, if the whole target is represented by the fungus family and the bull's eye by mushrooms, a vague thought which can only hit the target as a whole is not much use from a culinary point of view. And when I merely remember that I met a man, my memory may be very inadequate to my practical requirements, since it may make a great difference whether I met Brown or Jones. The memory "I met Jones" is relatively precise. It is accurate if I met Jones, inaccurate if I met Brown, but precise in either case as against the mere recollection that I met a man. The distinction between accuracy and precision is however, not fundamental. We may omit precision from out thoughts and confine ourselves to the distinction between accuracy and vagueness. We may then set up the following definitions: An instrument is "reliable" with respect to a given set of stimuli when to stimuli which are not relevantly different it gives always responses which are not relevantly different. An instrument is a "measure" of a set of stimuli which are serially ordered when its responses, in all cases where they are relevantly different, are arranged in a series in the same order. The "degree of accuracy" of an instrument which is a reliable measurer is the ratio of the difference of response to the difference of stimulus in cases where the difference of stimulus is small.* That is to say, if a small difference of stimulus produces a great difference of response, the instrument is very accurate; in the contrary case, very inaccurate. * Strictly speaking, the limit of this, i.e. the derivative of the response with respect to the stimulus. A mental response is called "vague" in proportion to its lack of accuracy, or rather precision. These definitions will be found useful, not only in the case of memory, but in almost all questions concerned with knowledge. It should be observed that vague beliefs, so far from being necessarily false, have a better chance of truth than precise ones, though their truth is less valuable than that of precise beliefs, since they do not distinguish between occurrences which may differ in important ways.