Logic, Symbolic

Logic, Symbolic

BIBLIOGRAPHY

Symbolic logic is sited at the intersection of philosophy, mathematics, linguistics, and computer science. It deals with the structure of reasoning and the formal features of information. Work in symbolic logic has almost exclusively treated the deductive validity of arguments: those arguments for which it is impossible for the premises to be true and the conclusion false. However, techniques from twentieth-century logic have found a place in the study of inductive or probabilistic reasoning, in which premises need not render their conclusions certain.

The historical roots of logic go back to the work of Aristotle (384–322 BCE), whose syllogistic reasoning was the standard account of the validity of arguments. Syllogistic reasoning treats arguments of a limited form: They have two premises and a single conclusion, and each judgment has a form like “all people are mortal,” “some Australian is poor,” or “no politician is popular.”

The discipline of symbolic logic exploded in complexity as techniques of algebra were applied to issues of logic in the work of George Boole (1815–1864), Augustus de Morgan (1806–1871), Charles Sanders Peirce (1839–1914), and Ernst Schröder (1841–1902) in the nineteenth century (see Ewald 1996). They applied the techniques of mathematics to represent propositions in arguments algebraically, treating the validity of arguments like equations in applied mathematics. This tradition survives in the work of contemporary algebraic logicians.

Connections between mathematics and logic developed into the twentieth century with the work of Gottlob Frege (1848–1925) and Bertrand Russell (1872–1970), who used techniques in logic to study mathematics. Their goals were to use the newfound precision in logical vocabulary to give detailed accounts of the structure of mathematical reasoning, in such a way as to clarify the definitions that are used, and to make fully explicit the commitments of mathematical reasoning. Russell and Alfred North Whitehead’s (1861–1947) Principia Mathematica (1912) is the apogee of this project of logicism.

With the development of these logical tools came the desire to use them in different fields. In the early part of the twentieth century, the logical positivists attempted to put all of science on a firm foundation by formalizing it: by showing how rich theoretical claims bear on the simple observations of experience. The best example of this is the project of Rudolf Carnap (1891–1970), who attempted to show how the logical structure of experience and physical, psychological, and social theory could be built up out of an elementary field of perception (Carnap 1967). This revival of empiricism was made possible by developments in logic, which allowed a richer repertoire of modes of construction or composition of conceptual content. On an Aristotelian picture, all judgments have a particularly simple form. The new logic of Frege and Russell was able to encompass much more complex kinds of logical structure, and so with it, theorists were able to attempt much more (Coffa 1991).

However, the work of the logical positivists is not the enduring success of the work in logic in the twentieth century. The radical empiricism of the logical positivists failed, not because of external criticism, but because logic itself is more subtle than the positivists had expected. We see this in the work of the two great logicians of the mid-twentieth century. Alfred Tarski (1902–1983) clarified our view of logic by showing that we can understand logic by means of describing the language of logic and the valid arguments by giving an account of proofs. However, we view logic by viewing the models of a logical language, and taking a valid argument as one for which there is no model in which the premises are true and the conclusion false. Tarski clarified the notion of a model and he showed how one could rigorously define the notion of truth in a language, relative to these models (Tarski 1956). The other great logician of the twentieth century, Kurt Gödel (1906–1978), showed that these two views of logic (proof theory and model theory) can agree. He showed that in the standard picture of logic, validity defined with proofs and validity defined by models agree (see von Heijenhoort 1967).

Gödel’s most famous and most misunderstood result is his incompleteness theorem: This result showed that any account of proof for mathematical theories, such as arithmetic, must either be completely intractable (we can never list all of the rules of proof) or incomplete (it does not provide an answer for every mathematical proposition in the domain of a theory), or the theory is inconsistent. This result brought an end of the logicist program as applied to mathematics and the other sciences. We cannot view the truths of mathematics as the consequences of a particular theory, and the same holds for the other sciences (see von Heijenhoort 1967).

Regardless, logic thrives. Proof theory and model theory are rich mathematical traditions, their techniques have been applied to many different domains of reasoning, and connections with linguistics and computer science have strengthened the discipline and brought it new applications.

Logical techniques are tools that may be used whenever it is important to understand the structure of the claims we make and the ways they bear upon each other. These tools have been applied in clarifying arguments and analyzing reasoning, and they feature centrally in the development of allied tools, such as statistical reasoning.

One contemporary debate over our understanding of logic also bears on the social sciences. We grant that using languages is a social phenomenon. How does the socially mediated fact of language-use relate to the structure of the information we are able to present with that use of language? Should we understand language as primarily representational, with inference valid when what is represented by the premises includes the representation of the conclusion, or should we see the social role of assertion in terms of its inferential relations? We may think of assertion as a social practice in which the logical relations of compatibility and reason-giving are fundamental. Once we can speak with each other, my assertions have a bearing on yours, and so logic finds its home in the social practice of expressing thought in word (Brandom 2000).

SEE ALSO Aristotle; Empiricism; Logic; Models and Modeling; Philosophy; Social Science; Statistics in the Social Sciences

BIBLIOGRAPHY

Brandom, Robert. 2000. Articulating Reasons: An Introduction to Inferentialism. Cambridge, MA: Harvard University Press.

Carnap, Rudolf. 1967. The Logical Structure of the World, and Pseudoproblems in Philosophy. Trans. Rolf A. George. New York: Routledge and Kegan Paul.

Coffa, J. Alberto. 1991. The Semantic Tradition from Kant to Carnap: To the Vienna Station, ed. Linda Wessels. Cambridge, U.K.: Cambridge University Press.

Ewald, William, ed. 1996. From Kant to Hilbert: A Source Book in the Foundations of Mathematics. Oxford: Oxford University Press.

Russell, Bertrand, and Alfred North Whitehead. 1912. Principia Mathematica. Cambridge, U.K.: Cambridge University Press.

Tarski, Alfred. 1956. Logic, Semantics, Metamathematics: Papers from 1923 to 1938. Trans. J. H. Woodger. Oxford: Clarendon.

von Heijenhoort, Jan. 1967. From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931. Cambridge, MA: Harvard University Press.

Greg Restall