Page 302 - Encyclopedia of Philosophy of Language

Language and Logic
the development of the theory of computation. The name of Alan Turing, the English mathematician, is most well-knownhere.
Finally, the development of complex formal sys- tems capable of expressing many of the nuances of natural language has made logic into one of the most important philosophical and linguistic tools in the current literature. One only has to look at journals in these areas to see this. So, after a very checkered history, the various strands of logic have now come together: it is now a fundamental part of our under- standing of language and the mind in the way that it always promised to be in its earliest stages.
See also: Natural Deduction; Reasoning; Entailment; Deviant Logics.
Bibliography
Bochenski J I M 1970 A History of Formal Logic. Chelsea Publishing, New York
Kneale W, Kneale M 1962 The Development of Logic. Oxford University Press, Oxford
Moody E A 1953 Truth and Consequence in Medieval Logic. Amsterdam
Nagel E, Newman J R 1958 Gddel 's Proof. New York Uni- versity Press, New York
Van Heijenoort J (ed.) 1967 From Frege to Gddel: A Source Book in Mathematical Logic. Harvard University Press, Cambridge, MA
Logic is the study of form rather than content. This simple observation gives little clue to the difficulties inherent in giving a satisfactory account of logical form. Although the formulae of truth-functional, or sentential, logic seem to provide a skeletal form for the corresponding sentences of natural language, the structures of the logic of quantifiers are somewhat remote from their natural language counterparts. It can be helpful to think of logic as providing a trans- lation of sentences of natural language into those of a formal (that is, symbolic) language, rather than as revealing anything about the form (logical, gram- matical, or semantic) of sentences in natural languages (seeGuttenplan 1986).
1. Logical Form and Natural Language
The exploration of logical form, however, has been an important stimulus in both philosophy and linguistics. Gottlob Frege tried to give an account of the sem- antics of natural languages based on the assimilation of their features to those of formal languages. Ludwig Wittgenstein regarded logic, in his early period, as 'the great mirror,' its forms revealing not only something about the forms of natural language sentences, but also something about the forms of objects themselves and the possibilities of their relations with other objects. Logical form, then, for Wittgenstein, was critical both to semantics and to ontology, and a major concern of his Tractatus Logico-Philosophicus is the search for the general form of the sentence. The search was flawed, however, since Wittgenstein assumed wrongly that the form of all natural language sen- tences could be given using only the apparatus of truth-functions.
1.1 Sentential Connectives
To grasp the central idea of logical form, it is helpful to distinguish between certain 'logical' words, such as //, and, or, not, all, and some, and other, 'nonlogical' words. Thus, the sentence // is raining and it is wet is analyzed as consisting of two simpler sentences con- nected using the logical word and. Using letters to replace the sentences, and the sign ' A ' for and, the logical counterpart of the sentence reads p A q. This is held to reveal the logical form of the original sentence in that the substitution of any other sentences for p and q will change only the content, not the form, of the original. For the truth-functions, the interpretation of each sentence consists simply in assigning to it one of the two truth-values, 'true' or 'false.' Where a sentence is a compound of simple sentences, the method of truth tables gives a vivid representation of how the interpretation (truth-value) of the whole sentence depends systematically on the truth-values of its parts.
1.2 Predicate Logic
The introduction of the quantifiers all and some involves considerable rephrasing of the natural language counterparts before reaching the logical forms. Classical logic, as expounded in the nineteenth century, notably by Frege, treats universal sentences as conditionals and existential sentences as conjunc- tions. Hence, the logical form of (1):
280
Logical Form A. A. Brennan
All fish swim is (2):
(1)
All things, x, are such that if x is a fish, (2) then x swims