THE JOURNAL OF PHILOSOPHY - Vol. LXI - Implicit Definition Sustained

Sep 21, 2026

Uploaded by Skyler

A philosophical note develops how axioms characterized as implicit definitions can be defended through a mechanical procedure. Starting from interpreted undefined predicates governed by true axioms, it switches to an equally economical predicate set with auxiliary arithmetical notations so that old axioms become arithmetical truths. The paper links the routine to implicit definition doctrine and improves a 1940 result by focusing on predicate-level theory, using elementary number theory, and ensuring definitional findability via a strengthened Löwenheim-style theorem.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages32

Reading Time1 h 21 min

AccessFree to download

Implicit Definition Sustained

Aims and background

Mechanical routine for implicit definitions

Relation to the 1940 result

Strengthened Löwenheim-style theorem

DETROIT. MICHIGAN THE EASE DO NOT REMOVe JOURNAL OF PHILOSOPHY VoL.LXI, No.2:JANUARY 16,1964 Implicit Definition Sustained:W.V.QUINE ………………………71 Comments and Criticism Ayer on Negation: SIDNE AXIN ………………………………………74 Book Reviews ERNESTNAGEI,PATRICK SUPPES, and ALFRED TARSKI, eds.: Logic, Methodology,and the Philosophy of Science. Three reviews: …………………………76 ELLIOTT MENDELSON ……………… …………………………79 RICHARD C. JEFFREY …… …………………………89 ERNEST ADAMS ……………………… …………………………94 Notes and News ………………… ………………………………95 New Books ………………………… PUBLISHED FORTNIGHTLY BY THE JOURNAL OF PHILOSOPHY, INC. THE JOURNAL OF PHILOSOPHY FoUNDED BY FREDERICK J.E.WoODBRIDGE AND WENDELLT.BUSH EDITORS:John H.Randall,Jr.,Robert D.Cumming,George L.Kline,Sidney Morgenbesser,Arthur Danto,and James j.Walsh. CoNSULTING EDITORS:Herbert w.Schneider,Alexandre Koyré,andJosé Ferrater Mora. MANAGING EDITOR:Leigh S.Cauman. Communieations to the editors should be addressed to the Managing Edi-tor,720 Philosophy Hall,Columbia University,New York,N.Y,10027.Subseriptions and advertisements should be sent to THE JOURNALOF PHILOSOPHY,720 Philosophy Hall,Columbia University,NewYork,N.Y.,10027.The annual subseription price(24 numbers)is $7.50;to students,$6.00.The price of single numbers is $0.50;double numbers,$1.00;complete volumes(from 1960 on),$12.00. THE JOURNAL OF PHILOSOPHY is owned by the Journal ofPhilosophy,Ince.,and published bi-weekly(every four weeks in Julyand August)by Lancaster Press,Ine.,Prince and Lemon Sts.,Lan-easter,Pa.,17604.Editorial Offce:720 Philosophy Hall,ColumbiaUniversity,New_York,N.Y.,10027.President,Albert G.Redpath;Vice President,John H.Randall,Jr.;Treasurer,Frederick A.Dewey;Secretary,Corliss Lamont. Second-class postage paid at Lancaster,Pa. Stechert-Hafner,Inc.,New York-Atlas Publ.&Distr.Co.,Ltd.Lon-don-Hermann&Cle,Paris-Nicola Zanichell,Bologna-H.Bouvler u.Co.,Bonn a/Rh.-Librerla Editorlal_Augustinus,Madrid-F.Machado&Cia,Porto-Gerold &Co.,Wien-Librafrle Payot,S.A,Lausanne-TheMaruzen Company,Tokyo. # INTERNATIONAL REVIEW OF SCIENTIFICSYNTHESIS(A REVIEW COVERING ALL SCIENCES)(1964—58th year)Chief-Editor:P.BONETTI Scientifc Committee:G.ABETTI-G.B.BONINO -P.CALDIROLA L.CALIFANO-R.CANESTRARI -G.COLONNETTI-B.DE FI-NETTI-M.GORTANI-G.LEVI DELLA VIDA-G.MONTALENTI E.PADOA-V.SOMENZI-S.TONZIG “SCIENTIA”is the only review of its type which:has a world wide cir-culation·deals with the most recent and fundamental problems of allbranches of science·can boast among its contributors of the most illus-trious men of science in the whole world·publishes the articles in thenative language of ther Authors(English,French,Italian,German,Spanish).Each issue includes a,Supplement containing the complete Frenchtranslation of the articles which in the text are published in language other thanFrench. “SCIENTIA”has therefore a very strong appeal to the scientific-mindedreader all over the world. Fuli details and a free back copy will be sent by appiying to“SCIENTIA,,一ASSo(Como,Italy)sending 80.25,or equivalent amount in other currency,preferabiy inalr-mail postage stamps of your country,merely to cover_packing andpostage.For a number of the current year,piease send $1.50,or equiva-lent amount in other currency,which wli be deducted from the sub-scription price.ANNUAL SUBSCRIPTION:$15.50 U.S.A. # THE JOURNAL OF PHILOSOPHY IMPLICIT DEFINITION SUSTAINED THE characterization of axioms as implicit definitions can befound as far back as 1818,in Gergonne,¹and it was stillvigorous thirty years ago.What is exasperating about the doc-trine is its facility,or cheapness,as a way of endowing statementswith the security of analytic truths without ever having to showthat they follow from definitions properly so called,definitionswith eliminable dcfinienda. Russell gave the doctrine its due,I felt,though he did notmention it by name,when he wrote in 1919 that'the method of‘postulating'what we want has many a

FAQs

What does the note aim to accomplish?

It states a purpose to discharge a “somber duty” by defending axioms as implicit definitions with detailed technical literalness.

How does the mechanical routine work at a high level?

Given interpreted undefined predicates governed by true axioms, it produces new undefined predicates and definitions plus auxiliary arithmetical notations so the old axioms become true by arithmetic.

What improvements does the paper claim over the 1940 result?

It is geared to predicates rather than singular terms, draws specifically on arithmetic (elementary number theory) instead of set theory, and provides a mechanical routine for finding the definitions.

Sep 21, 2026

Uploaded by Skyler