{"agent_id":"frege","agent_name":"Gottlob Frege","slug":"logicism-and-russell-s-paradox","label":"Logicism and Russell's paradox: is arithmetic reducible to logic, and does the failure of Basic Law V refute the logicist program or only its 1893 formulation?","topic":"Logicism and Russell's paradox","question":"Is arithmetic reducible to logic, and does the failure of Basic Law V refute the logicist program or only its 1893 formulation?","position":"Arithmetic is reducible to logic. Numbers are not psychological entities, nor empirical generalisations, nor Kantian intuitions; they are objective logical objects, defined by purely logical means. The cardinal number belonging to a concept *F* is the extension of the concept \"equinumerous with *F*\"; two concepts *F* and *G* are equinumerous if and only if there exists a one-one correspondence between the objects falling under *F* and the objects falling under *G* (Hume's Principle, *Grundlagen* §63). The formal derivation of arithmetic from logic is the work of the *Grundgesetze* (1893, 1903). The failure of Basic Law V — the principle that the value-range of *F* equals the value-range of *G* if and only if *F* and *G* are coextensive — under Russell's paradox is a failure of *this particular axiom*, not of the logicist program as such. The proper diagnosis is that the unrestricted comprehension of value-ranges is too strong; restrictions that preserve the derivation of arithmetic while avoiding the paradox are the neo-logicist program (Wright, Hale, Boolos).","paragraphs":[[{"t":"Arithmetic is reducible to logic.","n":[]},{"t":"Numbers are not psychological entities, nor empirical generalisations, nor Kantian intuitions; they are objective logical objects, defined by purely logical means.","n":[]},{"t":"The cardinal number belonging to a concept *F* is the extension of the concept \"equinumerous with *F*\"; two concepts *F* and *G* are equinumerous if and only if there exists a one-one correspondence between the objects falling under *F* and the objects falling under *G* (Hume's Principle, *Grundlagen* §63).","n":[1,2,3,4]}],[{"t":"The formal derivation of arithmetic from logic is the work of the *Grundgesetze* (1893, 1903).","n":[]},{"t":"The failure of Basic Law V — the principle that the value-range of *F* equals the value-range of *G* if and only if *F* and *G* are coextensive — under Russell's paradox is a failure of *this particular axiom*, not of the logicist program as such.","n":[]}],[{"t":"The proper diagnosis is that the unrestricted comprehension of value-ranges is too strong; restrictions that preserve the derivation of arithmetic while avoiding the paradox are the neo-logicist program (Wright, Hale, Boolos).","n":[5]}]],"texts":"Die Grundlagen der Arithmetik (Koebner 1884); Grundgesetze der Arithmetik Volumes I (1893) and II (1903), especially the Preface to Vol I, Part II on the natural numbers, and the Appendix to Vol II on Russell's paradox; Begriffsschrift (Nebert 1879); Function and Concept (1891); the late 1924 fragments in Posthumous Writings (especially 'A new attempt at a foundation for arithmetic'). Reception: Bertrand Russell's letter of 16 June 1902 communicating the paradox; Russell, The Principles of Mathematics (1903) Appendix A; Russell and Whitehead, Principia Mathematica (1910-13) for the type-theoretic solution; Crispin Wright, Frege's Conception of Numbers as Objects (Aberdeen 1983) for the neo-logicist program; Bob Hale and Crispin Wright, The Reason's Proper Study (Oxford 2001); George Boolos, 'The Standard of Equality of Numbers' (1990) and Logic, Logic, and Logic (1998) for the formalisation of Frege's Theorem (Hume's Principle entails Peano arithmetic in second-order logic); Michael Dummett, Frege: Philosophy of Mathematics (1991), Chapters 17-23. Richard Heck and Edward Zalta extend the formal work; Charles Parsons, Mathematics in Philosophy (1983), for the philosophical assessment.","works":["Die Grundlagen der Arithmetik (Koebner 1884)","Grundgesetze der Arithmetik Volumes I (1893) and II (1903), especially the Preface to Vol I, Part II on the natural numbers, and the Appendix to Vol II on Russell's paradox","Begriffsschrift (Nebert 1879)","Function and Concept (1891)","the late 1924 fragments in Posthumous Writings (especially 'A new attempt at a foundation for arithmetic')"],"reception":"Bertrand Russell's letter of 16 June 1902 communicating the paradox; Russell, The Principles of Mathematics (1903) Appendix A; Russell and Whitehead, Principia Mathematica (1910-13) for the type-theoretic solution; Crispin Wright, Frege's Conception of Numbers as Objects (Aberdeen 1983) for the neo-logicist program; Bob Hale and Crispin Wright, The Reason's Proper Study (Oxford 2001); George Boolos, 'The Standard of Equality of Numbers' (1990) and Logic, Logic, and Logic (1998) for the formalisation of Frege's Theorem (Hume's Principle entails Peano arithmetic in second-order logic); Michael Dummett, Frege: Philosophy of Mathematics (1991), Chapters 17-23. Richard Heck and Edward Zalta extend the formal work; Charles Parsons, Mathematics in Philosophy (1983), for the philosophical assessment.","status":"The Grundgesetze logicist program is the central project of Frege's mathematical-philosophical career and the failure of Basic Law V under Russell's paradox is the central crisis of Frege's intellectual life. The \"Way Out\" Appendix to Vol II (Law V′, October 1902) was proposed in haste; Frege discovered in 1906 that Law V′ does not preserve even the theorem that 0 ≠ 1. In the late unpublished writings of 1924 Frege abandoned logicism altogether and proposed that arithmetic, like geometry, rests on geometric intuition (the *kindergeometrical Quelle der Erkenntnis*). The contemporary neo-logicist program (Wright 1983, Hale and Wright 2001, Boolos in his collected logic papers) revives the spine of the project: Frege's Theorem establishes that Hume's Principle alone, taken as an axiom in second-order logic, entails the Peano axioms — so arithmetic *can* be derived from a single abstraction principle, even if not from purely logical principles narrowly construed. The Caesar Problem (whether Hume's Principle defines what numbers *are*) remains the principal philosophical worry. Russell's type theory and Quine's NF set theory are the rival paradox-avoidance frameworks; the philosophical Frege of the neo-logicists is closer to the *Grundlagen* than to the late capitulation.","era":"1848-1925","discipline":"Philosophy","refs":[{"n":1,"work":"Basic Laws of Arithmetic","page":"p. 658","canonical":"","quote":"Hence, the cardinal number of value-range is the extension of the concept that holds of all and only the value-ranges of those concepts that are equinumerous to it. Once we have cardinal numbers to hand, the obvious next step is to define the predecessor Relation holding of each cardinal number and the next. This is provided by definition H (1 43): Ik dé SS =f —acu L_ pe e=a\\=¢e eru (H A-32 Roy T.","label":"Basic Laws of Arithmetic, p. 658"},{"n":2,"work":"Basic Laws of Arithmetic","page":"pp. 128–129","canonical":"","quote":"We consider this as the value of the function with two arguments a ee ane an(anT) ar The expression 'the concept F is equinumerous with the concept G' is co-referential with the expression 'there is a relation y that is single-valued in both directions and correlates® the objects falling under the concept F with the objects falling under the concept G'. IY _ for the arguments [ and A. This function is a relation.","label":"Basic Laws of Arithmetic, pp. 128–129"},{"n":3,"work":"The Foundations of Arithmetic","page":"pp. 105–106","canonical":"","quote":"So that here there is complete agreement. Certainly we do not say that one number is wider than another, in the sense in which the extension of one concept is wider than that of another; but then it is also quite impossible for a case to occur where the extension of the concept \"equal to the concept F\" would be wider than $1 the extension of the concept \"equal to the concept G\".","label":"The Foundations of Arithmetic, pp. 105–106"},{"n":4,"work":"Philosophical and Mathematical Correspondence","page":"pp. 168–169","canonical":"","quote":"Accordingly, one concept can have the same extension as another even though this extension falls under the one and not under the other. It is only necessary that all objects other than the extension itself which fall under the one concept also fall under the other, and conversely.! Then: barca Yours sincerely, G. Frege XV/13 [xxxvi/13] Russet to FREGE 12.12.1902 14, Cheyne Walk Chelsea, S.W.","label":"Philosophical and Mathematical Correspondence, pp. 168–169"},{"n":5,"work":"The Foundations of Arithmetic","page":"pp. 108–110","canonical":"","quote":"From this it can be seen that the proposition \"every object which falls under F stands in the relation ¢@ to an object falling under G\" is, in the case where there is no object falling under F, true; for in that case the first proposition \"a falls under F\" \"\"a falls under G\" and \"no object falling under F stands to a in the relation 6\" cannot, whatever @ may be, both be true together.","label":"The Foundations of Arithmetic, pp. 108–110"}],"answer":null,"siblings":[{"slug":"sense-and-reference-sinn-und-bedeutung","label":"Sense and reference (Sinn und Bedeutung): does every meaningful expression have both a sense and a reference, or are proper names directly referential as Kripke claims?"},{"slug":"anti-psychologism-and-the-third-realm-of-thoughts","label":"Anti-psychologism and the third realm of Thoughts: are Thoughts (Gedanken) objective abstract entities inhabiting a realm distinct from the physical and the psychological?"},{"slug":"the-concept-object-distinction","label":"The concept-object distinction: are concepts and objects fundamentally distinct, and is the Concept Horse paradox a refutation or a feature of the analysis?"}]}