Journal of the History of Philosophy

Definability and Logical Structure in Frege J. M. VICKERS I. Truth, Definability, and Structure Logical structure, definability, and the Begriffsschrift--these all work together in Frege's thought. This paper is an attempt to understand some of that working, in particular to understand logical simplicity and its relation to indefinability. The present section has mainly to do with the indefinability of truth. Section II treats the views of some of Frege's predecessors on matters of definability and logical structure. Section III discusses Frege's relations to these predecessors. Sections IV and V are about the effect of the doctrine of implicit definition on Frege's views. Frege makes frequent and important use of the notion of logical simplicity, but he does not, as far as I can discover, provide an account or explication of it. Logical simplicity is a ground for indefinability: "Only what is logically complex can be defined. What is simple can only be pointed to."~ For this reason object, concept, and the distinction of complete from incomplete cannot be defined. 2 Truth cannot be defined either, since any attempt to define it would involve vicious circularity. 3 Truth may also be logically simple, though Frege does not say this. (He does say that it is unique [einzigartig; "Der Gedanke," p. 50].) Although he does not say it, it will help us in understanding logical simplicity to ask what Frege could have meant in saying that truth is logically simple. The reason it will help is this: We shall be able to see something that logical simplicity does not mean, and that will be of general interest. This paper grew out of a talk givenat the Universityof California, Los Angeles, in November, 1977. Thanks to Terence Parsons for helpful comments on an earlier version. ' Gottlob Frege, Grundgesetze der Arithmetik, begriffsschriftlich abgeleitet, 2 vols. in 1 (1893, 1903; reprint ed., Hildesheim: Georg Olms, 1962),vol. 2, sec. 147n.(hereafter cited as Gg followedby vol. and sec. nos.). Parts are translated by Montgomery Furth as The Basic Laws of Arithmetic (Berkeleyand Los Angeles:Universityof California Press, 1964)and by P. E. B. Jourdain and J. Stachelroth in Translations from the Philosophical Writings of Gottlob Frege, ed. Peter Geach and Max Black (Oxford: Blackwell, 1960). 2Frege, Funktion und Begri.ff(Jena, 1891)and "Ober Begriffund Gegenstand," Vierteljahrsschriftfar wissenschaftliche Philosophic 16(1892):192-205. Both are reprinted in Gottlob Frege:Kleine Schriften, ed. Ignacio Angelelli(Hildesheim: Georg Olms, 1967).Both are translated in Geach and Black. "In a definition [of truth] certain characteristics would haveto be stated. And in application to any particular case the question would alwaysarise whether it were true that the characteristics werepresent. So one goes round in a circle" (Frege, "Der Gedanke: Eine IogischeUntersuchung," Beitr~gezur Philosophie des deutschen Idealismus 1 (1918):60. Reprinted in Kleine Schriften. Translated by A. and M. Quinton in Mind 65 (1956):289-311.See also Gottlob Frege: Nachgelassene Schriften, ed. Hans Hermes, Friedrich Kambartel, and Friedrich Kaulbach (Hamburg: FelixMeiner, 1969),p. 146 (in the "Logik") where the attempt to definetruth would put us in the "situation of a man in treadmill." The argument is similar to that which shows that identity cannot be defined. See Frege's letter to Peano, Kleine Schriften, p. 235, and the reviewof Husserl, Kleine Schriften, p. 184. [291] 292 HISTORY OF PHILOSOPHY First, a few remarks on the logical grammar of truth. There are two clearly distinct concepts of truth in Frege: the concept under which falls the true and only the true, and the concept under which fall all and only true thoughts. They are obviously distinct , since only one entity falls under the first, and many entities fall under the second. The first concept is referred to' by the content stroke, or horizontal (Gg 1:5,6). Thus, 1. It is true that snow is white is written 2. --Snow is white. There is in Grundgesetze a rule (amalgamation of horizontals) that allows the transition from "m --/X" to "--A." Frege does not say that this rule preserves sense as well as reference, but it is consistent and natural to treat it as sense preserving . The effect of the amalgamation of horizontals would then...