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Doctrinal Functions

C. J. Keyser

1918
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The Journal of Philosophy Psychology and Scientific Methods
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Known as the Early Journal Content, this set of works include research articles, news, letters, and other writings published in more than 200 of the oldest leading academic journals. The works date from the mid--seventeenth to the early twentieth centuries. We encourage people to read and share the Early Journal Content openly and to tell others that this resource exists. People may post this content online or redistribute in any way for non--commercial purposes. Read more about Early Journal
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... out Early Journal Content at http://about.jstor.org/participate--jstor/individuals/early-journal--content. JSTOR is a digital library of academic journals, books, and primary source objects. JSTOR helps people discover, use, and build upon a wide range of content through a powerful research and teaching platform, and preserves this content for future generations. JSTOR is part of ITHAKA, a not--for--profit organization that also includes Ithaka S+R and Portico. For more information about JSTOR, please contact support@jstor.org. THE JOURNAL OF PHILOSOPHY DOC'TRINAL FUNCTIONS. T HE term propositional function, invented by Mr. Bertrand Rus- sell, is perhaps the weightiest that has entered the literature of logic and mathematics in the course of a hundred years. It has the rare distinction of being a perfect name for a supreme concept. I am not about to expound its meaning at length nor to attempt to justify my estimate of its significance. It seems desirable, however, to remind the reader of so much of the term's ineaning as will be essential to an understanding of the principal thesis of this paper. Let it be recalled, then, that a propositional function is any statement containing one or more variables. If we denote these by x, y, z, etc., then such simple statements as x is a philosopher, x =2, x is a brother of y, 3x + 2y 5, x has been divinely appointed by y to subjugate z, 4x -3y + 9z-7, will serve to exemplify what is meant by a propositional function. It is of fundamental importance to bear in mind that propositional functions, though they have the form of propositions, are not propositions. A proposition is a statement that is true or else false, but a propositional function is neither true nor false. The statements, 2 +5-7, 3 + 6-7, are propositions, one of them true, the other one false; but the statement, x + y =7, is neither true nor false; it is not a proposition but is a propositional function. To derive propositions from propositional functions it is evidently necessary to substitute for the variables present in the latter what we may call constants, or terms of definite meaning; but such substitution, though necessary, is not sufficient, for it is always possible to select such constants as will, if substituted for the variables of a given function, convert the latter, not into a proposition, but into non-sense, it being understood that a non-sensical statement is one involving a contradiction in terms. Suppose, for example, that our given function is the statement, x is an integer less than 5. The class of all integers less than 5 is a constant, a definite somewhat. Substituting this constant for the variable x, we get the statement, the class of all integers less than 5 is an integer less than 5. This is neither a propositional function nor a proposition; it is nonsensical, the non-sense, or contradiction in terms, consisting in talking of a class of things as if a given class could conceivably be one of the things composing it. The constants that, when substituted for the variables in a given propositional function, convert it into non-sense, may be called inadmissible constants for that function; all other constants, since they

doi:10.2307/2940317
fatcat:rapjdqkfmrfytkjecewkwqdt44