Writings of Charles S. Peirce: A Chronological Edition, Volume 2. Charles S. Peirce

      On the other hand Some man is not completely defined in extension since it is disjunctive alternative while any man applies definitely to certain things.

      Note the meaning of a particular in the predicate. Some S means Either S′ S″ S‴ &c. select which I please.

      Some S is P, that is let me take as my subject one what one I please of S’ S” S’” &c. and I can make the proposition true.

      Now this does not hold for M is some P unless M is completely determined in comprehension just as Any S is P only holds if P has a very wide extension.

      We may therefore

      Oct. 2

      §2. Of the Effect of a Change of Information

      Suppose it is learned that

      Any S is P

      Then S receives an addition to its comprehension.

      P an addition to its extension.

      If we looking at an S find it to be P

      Some S is P

      This adds to the extension of P—supposing we know what S.

      Any S is not-P

      This adds to the comprehension of S—supposing we know something of not-P.

      1867 Nov. 24

      I wish to investigate the nature of a simple concept. Such a concept first arises as predicated of some object (occasion of experience)

      S is M

      On the ground of some previous representation of the object. (Not immediate)

      The predication of the concept is virtually contained in this previous representation.

      To say that a simple concept is the immediate apprehension of a quality is but a mode of saying that its meaning is given in the representation which gives rise to it in as much as it is as much as to say that that quality is contained in that representation.

      1867 Dec. 7

      When I conceive a thing as say ‘three’ or say ‘necessary’ I necessarily have some concrete object in my imagination. I have some concrete object—‘the necessary’. By saying that I have the necessary in my mind, it is not meant that I have all necessary things in mind. Nor that I have simply the character of necessity. For what I am thinking is not necessity but the necessary. Then I must have something which I recognize as a general sign of the necessary. But why should that particular feeling which is a sign of the necessary be a sign of that any more than of anything else? Because such is my constitution. Very true.

      When I conceive as say “necessary,” I have some singular object present to my imagination. I have not all necessary things separately imaged.

      Doubted whether I ever have an absolutely singular object—

      1. entirely knowable. Intuitional. Capable of external existence.

      [THE AMERICAN ACADEMY SERIES]

       On an Improvement in Boole’s Calculus of Logic

       P 30: Presented 12 March 1867

      The principal use of Boole’s Calculus of Logic lies in its application to problems concerning probability. It consists, essentially, in a system of signs to denote the logical relations of classes. The data of any problem may be expressed by means of these signs, if the letters of the alphabet are allowed to stand for the classes themselves. From such expressions, by means of certain rules for transformation, expressions can be obtained for the classes (of events or things) whose frequency is sought in terms of those whose frequency is known. Lastly, if certain relations are known between the logical relations and arithmetical operations, these expressions for events can be converted into expressions for their probability.

      It is proposed, first, to exhibit Boole’s system in a modified form, and second, to examine the difference between this form and that given by Boole himself.

      Let the letters of the alphabet denote classes whether of things or of occurrences. It is obvious that an event may either be singular, as “this sunrise,” or general, as “all sunrises.” Let the sign of equality with a comma beneath it express numerical identity. Thus is to mean that a and b denote the same class,—the same collection of individuals.

      Let denote all the individuals contained under a and b together. The operation here performed will differ from arithmetical addition in two respects: 1st, that it has reference to identity, not to equality; and 2d, that what is common to a and b is not taken into account twice over, as it would be in arithmetic. The first of these differences, however, amounts to nothing, in as much as the sign of identity would indicate the distinction in which it is founded; and therefore we may say that

      It is plain that

      and also, that the process denoted by and which I shall call the process of logical addition, is both commutative and associative. That is to say

      and

      Let a, b denote the individuals contained at once under the classes a and b; those of which a and b are the common species. If a and b were independent events, a, b would denote the event whose probability is the product of the probabilities of each. On the strength of this analogy (to speak of no other), the operation indicated by the comma may be called logical multiplication. It is plain that

      Logical multiplication is evidently a commutative and associative process. That is,

      Logical addition and logical multiplication are doubly distributive, so that

      and

      where any of these letters may vanish. These formulæ comprehend every possible relation of a,b, and c; and it follows from them, that

      But