Page 301 - Encyclopedia of Philosophy of Language

utterance, its expressing a sense and referring to a truth value, from a third function, which is to raise 'associated representations,' or provide a specific col- oring for the thought. This function does not properly concern logic, but helps expressing emotions or high- lights pragmatic factors relevant only for specific com- municative purposes.
7.3 Logic,Mathematics,andParadox
The major logical work of Frege is his Basic Laws of Arithmetic, in which he develops a 'system of logical laws' from which the theorems of arithmetic can be derived with the help of suitable definitions. Such a system includes seven axioms which give the con- ditions of use for basic signs such as implication, negation, identity, and quantification over first and second order functions. Law V allows transformations between one function and another when their ranges (i.e., the mappings associating a value to each argu- ment) are identical. That law is considered by Frege as 'essential in logic when concept extensions are being dealt with.' Unfortunately, it was that very principle which was responsible for the paradox which Russell noted.
Notational innovations include replacing the equiv- alence sign with an identity sign (as has been noted in the case of proper names), and introducing a set of new signs to express the range of a function. It is characteristic of Frege's idea of a formal system that syntactical rules should not be divorced from semantic interpretation. What he calls 'the highest principle for definitions' requires that 'well-formed names always have a reference.' Frege uses a set of rules to make sure that the principle will be correctly applied while introducing any new expression in the system. Taking his starting point in the fact that the names for truth values do refer, he shows that the various basic terms of his logical symbolism also have a determinate ref- erence.
In a letter dated June 16, 1902, Russell indicated to Frege that a contradiction could be derived in the system of his Basic Laws. Frege's ideography makes it possible to express the fact that a class does not belong to itself; it is true in the system for example, that the class of men is not a man. But one may then define a concept K as 'concept such that its extension does not belong to itself,' and show that the extension of K both belongs and does not belong to itself. Given the importance of Law V for Frege, and his insistence on every expression having determinate reference, this was disastrous.
Various solutions were explored by Frege, Russell, and later logicians to remedy Russell's paradox. Frege himself finally rejected all the available solutions, such as Russell's type theory, insofar as it led to what appeared to him to be unnatural ways of representing logical laws. In spite of Frege's own sense of having failed to achieve a logical reduction of arithmetic
truths, the very contradiction shown to be infecting his system helped later logicians to come to under- stand some of the properties of formal systems revealed through various puzzles and antinomies. Research done later by Tarski, G6del, and Carnap led to the idea of clearly distinguishing between claims made inside a language and the rules being expressed about the language. This is the distinction so fun- damental now to logic between object and meta- languages. A powerful method, first devised by Hilbert, named arithmetization, allowed Godel to show that there were severe, principled limitations to what could be proven inside a formal system including a representation of arithmetic such as Frege's.
See: Paradoxes, Semantic; Russell, Bertrand; Tarski, Alfred.
8. After Frege
Frege's work initiated a complete renaissance in logi- cal studies, and deepened the understanding of the structure of natural language. Although logicism understood as the project of reducing mathematics to logic would only survive in a weakened form, Frege's effort to develop a formal symbolism adequate for expressing the relations of propositions in a system broke new theoretical ground, for logicians, math- ematicians, linguists, and philosophers. Two influ- ential schools in philosophy derived their basic claims from Frege's lessons, and it cannot be chance that they were headed by two former students or admirers of Frege: Carnap and Wittgenstein. The logical pos- itivists, including Carnap, continued the exploration of the properties of formal systems and their foun- dational relevance for science in general. And the con- tinuing tradition of analytic philosophy extended Frege's effort at elucidating the logical structure of ordinary language and dissolving, through logical scrutiny, some of the traditional dialectical illusions, now seen as originating in natural language. With Frege, not only logic itself, but the theoretical back- ground to logic—its connection to philosophy and language studies generally—changed out of all rec- ognition.
There are three directions in which logic has developed since Frege. On the one hand, the for- malization of logic and the study of such formal sys- tems has led to a major branch of mathematics. This work began with Frege, but continued with the pub- lication in 1910-13 of Russell's and Whitehead's Prin- cipia Mathematica. Perhaps the two most important names in this field are Kurt Godel and Alfred Tarski, but the technical details of their work lies beyond this survey.
The second area of research is that of computer science. In a slightly different idiom, some of the results of G6del and Tarski have proven important to
Logic: Historical Survey
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