Gottlob (1848–1925), German mathematician and philosopher. A founder of modern mathematical logic, an advocate of logicism, and a major source of twentieth-century analytic philosophy, he directly influenced Russell, Wittgenstein, and Carnap. Frege’s distinction between the sense and the reference of linguistic expressions continues to be debated.
His first publication in logic was his strikingly original 1879 Begriffsschrift (Concept-notation). Here he devised a formal language whose central innovation is the quantifier-variable notation to express generality; he set forth in this language a version of second-order quantificational logic that he used to develop a logical definition of the ancestral of a relation. Frege invented his Begriffsschrift in order to circumvent drawbacks of the use of colloquial language to state proofs. Colloquial language is irregular, unperspicuous, and ambiguous in its expression of logical relationships. Moreover, logically crucial features of the content of statements may remain tacit and unspoken. It is thus impossible to determine exhaustively the premises on which the conclusion of any proof conducted within ordinary language depends. Frege’s Begriffsschrift is to force the explicit statement of the logically relevant features of any assertion. Proofs in the system are limited to what can be obtained from a body of evidently true logical axioms by means of a small number of truth-preserving notational manipulations (inference rules). Here is the first hallmark of Frege’s view of logic: his formulation of logic as a formal system and the ideal of explicitness and rigor that this presentation subserves. Although the formal exactitude with which he formulates logic makes possible the metamathematical investigation of formalized theories, he showed almost no interest in metamathematical questions. He intended the Begriffsschrift to be used. How though does Frege conceive of the subject matter of logic? His orientation in logic is shaped by his anti-psychologism, his conviction that psychology has nothing to do with logic. He took his notation to be a full-fledged language in its own right. The logical axioms do not mention objects or properties whose investigation pertains to some special science; and Frege’s quantifiers are unrestricted. Laws of logic are, as he says, the laws of truth, and these are the most general truths. He envisioned the supplementation of the logical vocabulary of the Begriffsschrift with the basic vocabulary of the special sciences. In this way the Begriffsschrift affords a framework for the completely rigorous deductive development of any science whatsoever. This resolutely nonpsychological universalist view of logic as the most general science is the second hallmark of Frege’s view of logic. This universalist view distinguishes his approach sharply from the coeval algebra of logic approach of George Boole and Ernst Schröder. Wittgenstein, both in the Tractatus Logico-Philosophicus (1921) and in later writings, is very critical of Frege’s universalist view. Logical positivism – most notably Carnap in The Logical Syntax of Language (1934) – rejected it as well. Frege’s universalist view is also distinct from more contemporary views. With his view of quantifiers as intrinsically unrestricted, he saw little point in talking of varying interpretations of a language, believing that such talk is a confused way of getting at what is properly said by means of second-order generalizations. In particular, the semantical conception of logical consequences that becomes prominent in logic after Kurt Gödel’s and Tarski’s work is foreign to Frege. Frege’s work in logic was prompted by an inquiry after the ultimate foundation for arithmetic truths. He criticized J. S. Mill’s empiricist attempt to ground knowledge of the arithmetic of the positive integers inductively in our manipulations of small collections of things. He also rejected crudely formalist views that take pure mathematics to be a sort of notational game. In contrast to these views and Kant’s, he hoped to use his Begriffsschrift to define explicitly the basic notions of arithmetic in logical terms and to deduce the basic principles of arithmetic from logical axioms and these definitions. The explicitness and rigor of his formulation of logic will guarantee that there are no implicit extralogical premises on which the arithmetical conclusions depend. Such proofs, he believed, would show arithmetic to be analytic, not synthetic as Kant had claimed. However, Frege redefined ‘analytic’ to mean ‘provable from logical laws’ (in his rather un-Kantian sense of ‘logic’) and definitions.
Frege’s strategy for these proofs rests on an analysis of the concept of cardinal number that he presented in his nontechnical 1884 book, The Foundations of Arithmetic. Frege, attending to the use of numerals in statements like ‘Mars has two moons’, argued that it contains an assertion about a concept, that it asserts that there are exactly two things falling under the concept ‘Martian moon’. He also noted that both numerals in these statements and those of pure arithmetic play the logical role of singular terms, his proper names. He concluded that numbers are objects so that a definition of the concept of number must then specify what objects numbers are. He observed that (1) the number of F % the number of G just in case there is a one-to-one correspondence between the objects that are F and those that are G. The right-hand side of (1) is statable in purely logical terms. As Frege recognized, thanks to the definition of the ancestral of a relation, (1) suffices in the second-order setting of the Begriffsschrift for the derivation of elementary arithmetic. The vindication of his logicism requires, however, the logical definition of the expression ‘the number of’. He sharply criticized the use in mathematics of any notion of set or collection that views a set as built up from its elements. However, he assumed that, corresponding to each concept, there is an object, the extension of the concept. He took the notion of an extension to be a logical one, although one to which the notion of a concept is prior. He adopted as a fundamental logical principle the ill-fated biconditional: the extension of F % the extension of G just in case every F is G, and vice versa. If this principle were valid, he could exploit the equivalence relation over concepts that figures in the right-hand side of (1) to identify the number of F with a certain extension and thus obtain (1) as a theorem. In The Basic Laws of Arithmetic (vol. 1, 1893; vol. 2, 1903) he formalized putative proofs of basic arithmetical laws within a modified version of the Begriffsschrift that included a generalization of the law of extensions. However, Frege’s law of extensions, in the context of his logic, is inconsistent, leading to Russell’s paradox, as Russell communicated to Frege in 1902. Frege’s attempt to establish logicism was thus, on its own terms, unsuccessful.
In Begriffsschrift Frege rejected the thesis that every uncompound sentence is logically segmented into a subject and a predicate. Subsequently, he said that his approach in logic was distinctive in starting not from the synthesis of concepts into judgments, but with the notion of truth and that to which this notion is applicable, the judgeable contents or thoughts that are expressed by statements. Although he said that truth is the goal of logic, he did not think that we have a grasp of the notion of truth that is independent of logic. He eschewed a correspondence theory of truth, embracing instead a redundancy view of the truth-predicate. For Frege, to call truth the goal of logic points toward logic’s concern with inference, with the recognition-of-thetruth (judging) of one thought on the basis of the recognition-of-the-truth of another. This recognition-of-the-truth-of is not verbally expressed by a predicate, but rather in the assertive force with which a sentence is uttered. The starting point for logic is then reflection on elementary inference patterns that analyze thoughts and reveal a logical segmentation in language. This starting point, and the fusion of logical and ontological categories it engenders, is arguably what Frege is pointing toward by his enigmatic context principle in Foundations: only in the context of a sentence does a word have a meaning. He views sentences as having a function-argument segmentation like that manifest in the terms of arithmetic, e.g., (3 $ 4) ! 2. Truth-functional inference patterns, like modus ponens, isolate sentences as logical units in compound sentences. Leibniz’s law – the substitution of one name for another in a sentence on the basis of an equation – isolates proper names. Proper names designate objects. Predicates, obtainable by removing proper names from sentences, designate concepts. The removal of a predicate from a sentence leaves a higher level predicate that signifies a second-level concept under which first-level concepts fall. An example is the universal quantifier over objects: it designates a second-level concept under which a first-level concept falls, if every object falls under it. Frege takes each first-level concept to be determinately true or false of each object. Vague predicates, like ‘is bald’, thus fail to signify concepts. This requirement of concept determinacy is a product of Frege’s construal of quantification over objects as intrinsically unrestricted. Thus, concept determinacy is simply a form of the law of the excluded middle: for any concept F and any object x, either x is F or x is not F. Frege elaborates and modifies his basic logical ideas in three seminal papers from 1891–92, ‘Function and Concept,’ ‘On Concept and Object,’ and ‘On Sense and Meaning.’ In ‘Function and Concept,’ Frege sharpens his conception of the function-argument structure of language. He introduces the two truth-values, the True and the False, and maintains that sentences are proper names of these objects. Concepts become functions that map objects to either the True or the False. The course-of-values of a function is introduced as a generalization of the notion