Page 299 - Encyclopedia of Philosophy of Language

to extend the analogy in the natural way to disjunc- tion, i.e., by treating disjunction as ' + .' For, the sum of two 1's is 2. However, Boole proposed that one think of the 'or' relationship as following slightly different rules from those obtaining in the number series. What results is an algebra in which 1 + 1 = 1.
By thinking in this way, Boole not only invented a special class of algebras, but he encouraged the idea that there could be a connection between mathematics and logic at a level above that of specific content (numbers or propositions). This way of thinking was to have profound consequences for the development of logic as well as mathematics.
7. Frege
Gottlob Frege (1848-1925), was a German math- ematician who, during his lifetime, remained an obscure professor in Jena. Nonetheless, it could be argued that he achieved more important break- throughs in logic than any other single individual with the possible exception of Aristotle. He created modern logic through the use of an artificial language, named 'conceptual writing' (or ideography), which he used to make extensive investigations into the logical struc- ture of natural as well as mathematical languages; he was the first to construct a formal deductive logical system, and also to investigate the logical foundations of arithmetic.
His first published work, entitled Conceptual Notation, a formula language, modeled upon that of Arithmetic,for pure thought (1879) (commonly known as Begriffsschrift), was meant to provide mathematics with an expressive tool capable of displaying the full rigor needed for a precise notion of mathematical proof. The specific goal Frege had in mind in develop- ing such a tool was to show that ultimately math- ematical notions—such as 'number' or 'hereditary property in a sequence'—could be defined using only logical notions and, furthermore, that mathematical truths could be derived as theorems from a system whose axioms included only 'logical laws.' In essence, what Frege wanted to show was that mathematical notions were in effect extensions of logical concepts, and that these latter concepts could be given a precise axiomatic basis. This project, attempted in more detail in 1884 in The Foundations of Arithmetic, and in the
1893-1903 Basic Laws of Arithmetic, has since been named 'the logicist stance' in the philosophy of math- ematics. However, it became quite apparent to those who did read Frege (and here the English philosopher and logician Bertrand Russell figures importantly), that the conceptual notation provided the basis for a more general account of reasoning and proof than that connected solely with mathematics.
See: Frege, Gottlob.
7.1 Conceptual Notation
According to Frege, using a purely logical notation allows one to represent inferential transitions in a precise way and without undetected appeals to intuition. It also allows the concepts used to be dis- played in a more determinate and rigorous way than is permitted by ordinary language. As is indicated in the subtitle of the work, Frege's ideography is inspired by arithmetical formulas, and this puts one in mind of the work of Leibniz as well as Boole. But the analogy between both symbol systems (arithmetic and logic) is not grounded, as was the case for Leibniz and Boole, on arithmetical operations themselves being chosen as basic relations. Instead of using arithmetical ideas as the basis for logic, Frege's conceptual notation uses certain notions implicit in logical structures as would be found in certain natural language constructions. Thus, the conceptual notation is not a mathem- atization of logic so much as a genuine formalization of logical notions.
The logical ideography is essentially a notation for representing the detailed logical form of sentences such as might be found in natural language, though it is not conceived of as a notation tied to natural lang- uage. Frege thought, and he was not alone in thinking, that natural language was too vague and confused to serve as the basis of logical inference. In the notation, constant terms are distinguished from variable signs, and these latter are seen as possible fillings for 'func- tion' terms. Frege conceived of a judgment as a struc- ture resembling a mathematical function (such as the function 'square root') filled out with argument places (as in 'square root of 2,' in which '2' is the argument). A full-blown judgment such as might be expressed in a complete sentence obtains when the empty space in a function term is filled by an argument term of the correct kind, for example, a proper name. Thus, 'is a horse' can be considered a one-place function and 'Desert Song' as an argument to get the judgment 'Desert Song is a horse,' though this would be written in something more like this form: 'Is a horse (Desert Song),' following the practice used bymathematicians in writing functions. This type of symbolism proves muchmoreflexiblethantheclassical subject-predicate analysis of the proposition for two reasons. First, since it is possible for functions to have more than one place, one can deal with relational sentences. For the first time, logical notation had moved beyond the restrictive subject-predicate structure bequeathed to us by Aristotle. Second, one could use variables in argument places and achieve judgments by attaching quantifiers to these variables. This was an enormous advance which, together with the possibility of multiple-place functions, gives the logic a great deal of expressive power.
The goal of the notation is not only to solve prob- lems: it is not only meant as a calculus ratiocinator, it also aims at elucidating the conceptual contents of
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