Writings of Charles S. Peirce: A Chronological Edition, Volume 8. Charles S. Peirce

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СКАЧАТЬ algebra immediately subserves, we are to study the modes of necessary inference. A proposition or propositions, called PREMISES, being taken for granted, the question is what other propositions, called CONCLUSIONS, these premises entitle us to affirm. The truth of the premises is not now to be examined, for that is assumed to have been satisfactorily determined, already; and any process of inference (i.e., formation of a conclusion from premises) will be satisfactory, provided it be such that the conclusion is certainly true unless the premises are false. That is to say, if P signifies the premises and C the conclusion, the condition of the validity of the inference is that either P is false or C is true. For human reason cannot undertake to guarantee that the conclusion shall be true if the premises on which it depends are false. It is true that we can imagine inferences which satisfy this condition and yet are illogical. Such are the following:—

      P true, C true. The world is round; therefore, the sun is hot.

      P false, C true. The world is square; therefore, the sun is hot.

      P false, C false. The world is square; therefore, the sun is cold.

      The reason why such inferences would be bad is that nobody could, in such cases, know that either P is false or C true, unless he knew already that P was false (when it would not properly be a premise), or else knew independently that C was true (when it would not properly be a conclusion drawn from P). But if the proverbial Angel Gabriel, who has been imagined as making so many extraordinary utterances, were to descend and tell me “Either the earth is not round, or the further side of the moon is blue,” it would be perfectly logical for me, from the known fact that the earth is round, to conclude that the other side of the moon is blue. It is true that the inference would not be what is called a complete or logical one; that is to say, the principle that either P is false or C true could not be known from the study of reasonings in general; but it would be a perfectly sound or valid inference.

      From what has been said it is plain that that relation between two propositions which consists in our knowing that either the one is true or the other false is of prime importance as warranting an inference from the former to the latter. It is, therefore, desirable to have an abbreviation to express this relation. The sign is to be used in such a sense that x y means that x is at least as low on the scale of truth as y. The sign is to be called the COPULA, and for the sake of brevity it may be read “gives,” that is, warrants the inference of. A proposition like x y will be called a HYPOTHETICAL, the proposition x preceding the copula will be called the ANTECEDENT, and the proposition y following the copula will be called the CONSEQUENT. The meaning of may be more explicitly stated in the following propositions, which, for convenience of reference, I mark A, B, C.

      A. If x is false, x y.

      B. If y is true, x y.

      C. If x y, either x is false or y is true.¹

      Rules of the Copula

      The sign is subject to three algebraical rules,² as follows:—

      RULE I. If x y and y z, then x z. This is called the principle of the transitiveness of the copula.

      RULE II. Either x y or y z.

      RULE III. There are two propositions, u and v, such that v u is false.

      To these is to be added the following:

      RULE OF INTERPRETATION. If y is true, v y; and if y is false, y u.

      Rule I can be proved from propositions, A, B, C. For by C, if x y either x is false or y is true. In the statement of A, substitute z for y. It then reads that if x is false, x z. Hence, if x y, either x z or y is true. Call this proposition P. In the statement of C, substitute y for x, and z for y. It then reads that if y z, either y is false or z is true. Combining this with P, we see that if x y and y z, either x z or z is true. Call this proposition Q. In the statement of B, substitute z for y. It then reads that if z is true, x z. Combining this with Q, we conclude that if x y and y z, x z. Q.E.D.

      Rule II can be proved from propositions A and B alone. For in the statement of A, substitute y for x and z for y. It then reads that if y is false, y z. But by B, if y is true, x y. Hence, either x y or y СКАЧАТЬ