Modern Logic: From Frege to Gödel: Frege
MODERN LOGIC: FROM FREGE TO GÖDEL: FREGE
Modern logic began with the publication in 1879 of the Begriffsschrift of Gottlob Frege (1848–1925). In the Begriffsschrift we find for the first time a comprehensive treatment of the ideas of generality and existence, because sentence forms which were hitherto accommodated only by complicated ad hoc theories are here provided with an adequate symbolization by the device of quantification, rules for which are adjoined to the first complete formalization of the classical propositional calculus. The result closely approximates a modern formal axiomatic theory. It meets Frege's aim of a codification of the logical principles used in mathematical reasoning, although the rules of inference (substitution and modus ponens ) and the definition of other logical constants in terms of the primitives (negation, implication, the universal quantifier, and identity) are not explicitly formalized but are mentioned as obviously justified by reference to the intended interpretation. A proof of completeness was not to be had in Frege's day, but he demonstrated the power of his system by deriving a large number of logical principles from his basic postulates and took an important step toward the formulation of arithmetical principles by showing, with the aid of second-order quantification, how the notion of serial order may be formalized.
After the Begriffsschrift, Frege's next major work was Die Grundlagen der Arithmetik (Breslau, 1884), an analysis of the concept of cardinal number presented largely in nontechnical terms. It opens the way for Frege's theories with a devastating criticism of the views of various writers on the nature of numbers and the laws of arithmetic. Difficulties encountered in the analyses of number find explanation and resolution in the celebrated claim that a statement of number contains an assertion about a concept. To say, for instance, that there are three letters in the word but is not, on Frege's view, to attribute a property to the actual letters; it is to assign the number 3 to the concept "letter in the word 'but'." If we now say that two concepts F and G are numerically equivalent (gleichzahlig ) if and only if there is a one-to-one correspondence between those things which fall under F and those which fall under G, we can define the number that belongs to a concept F as the extension of the concept "numerically equivalent to the concept F."
In terms of this definition any two numerically equivalent concepts, such as "letter in the word 'but'" and "letter in the word 'big'," can be seen to determine the same extension, and therefore the same number, and it remains only to specify concepts to which the individual numbers belong. In sketching this and subsequent developments Frege found that the notions used appear to allow of resolution into purely logical terms. He concluded that it is probable that arithmetic has an a priori, analytic status, a view that places him in opposition to Immanuel Kant, who held that propositions of arithmetic were synthetic a priori, and to J. S. Mill, who regarded them as inductive generalizations.
In papers published after the Grundlagen, Frege turned his attention to problems of a more general philosophical nature, and the development of his thought in this period led to a revised account of his logic, which is incorporated in his most ambitious work, Die Grundgesetze der Arithmetik (2 vols., Jena, Germany, 1893–1903), in which he extended and formalized the theory of number adumbrated in the Grundlagen. In the Begriffsschrift he had rejected the traditional subject-predicate distinction but had retained one predicate, "is a fact" (symbolized "⊦"), which indicated that the judgment which it prefaced was being asserted. In his essay "Über Sinn und Bedeutung" this view was abandoned on the ground that the addition of such a sign, conceived as a predicate, merely results in a reformulation of the same thought, a reformulation which in turn may or may not be asserted.
The logic of the Grundgesetze is based on Frege's theory of sense and reference, the interpretation of the symbolism of the Begriffsschrift being modified accordingly. The formal system of the Begriffsschrift is further changed by replacing certain of the axioms with transformation rules, but a more important innovation is the extension of the earlier symbols to cover classes. Corresponding to any well-defined function Φ(ξ ) is the range, or course of values (Wertverlauf ), of that function, written ἐ Φ(ε ), which Frege introduced via an axiom stipulating that ἐ Φ(ε ) is identical with εψ (ε ) if and only if the two associated functions Φ(ξ ) and ψ (ξ ) agree in the values which they take on for all possible arguments ξ. In particular, this axiom licenses the passage from a concept to its extension, the course-of-values notation providing a means of representing classes and foreshadowing Bertrand Russell's class-abstraction operator, ẑ (ϕz ). Another device that found a close analogue in Russell's logic is Frege's symbol ∖ξ. If a course of values ξ has a unique member, then ∖ξ is this member; otherwise ∖ξ is the course of values ξ itself. In the first case ∖ξ provides a translation of expressions of the form "the F " and so corresponds to Russell's description operator, (℩x )(ϕx ); the second case ensures that when ξ has no unique member, ∖ξ is nevertheless well defined.
The preliminary development of logic and the theory of classes is followed by the main subject of the Grundgesetze, the theory of cardinal number, developed with respect to both finite and infinite cardinals. The theory of real numbers is begun in the second volume but the treatment is incomplete, and Frege was probably loath to advance further in this direction after learning, while the second volume was in the press, that the very beginnings of his theory harbored a contradiction. This contradiction, discovered by Russell, resulted from the axiom allowing the transition from concept to class, an axiom in which Frege had not had the fullest confidence. Russell's communication is discussed in an appendix to the second volume, where an emended version of the axiom is put forward. This emendation was not, in fact, satisfactory, and although Frege apparently did not know that a contradiction could still be derived, he eventually abandoned his belief that the program of the Grundgesetze could be carried out successfully and claimed that geometry, not logic, must provide a basis for number theory.
Bede Rundle (1967)