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

symbolic logic or mathematical logic, formalized system of deductive logic, employing abstract symbols for the various aspects of natural language. Symbolic logic draws on the concepts and techniques of mathematics, notably set theory, and in turn has contributed to the development of the foundations of mathematics. Symbolic logic dates from the work of Augustus De Morgan and George Boole in the mid-19th cent. and was further developed by W. S. Jevons, C. S. Peirce, Ernst Schröder, Gottlob Frege, Giuseppe Peano, Bertrand Russell, A. N. Whitehead, David Hilbert, and others.

Truth-functional Analysis

The first part of symbolic logic is known as truth-functional analysis, the propositional calculus, or the sentential calculus; it deals with statements that can be assigned truth values (true or false). Combinations of these statements are called truth functions, and their truth values can be determined from the truth values of their components.

The basic connectives in truth-functional analysis are usually negation, conjunction, and alternation. The negation of a statement is false if the original statement is true and true if the original statement is false; negation corresponds to "it is not the case that," or simply "not" in ordinary language. The conjunction of two statements is true only if both are true; it is false in all other instances. Conjunction corresponds to "and" in ordinary language. The alternation, or disjunction, of two statements is false only if both are false and is true in all other instances; alternation corresponds to the nonexclusive sense of "or" in ordinary language (Lat. vel ), as opposed to the exclusive "either … or … but not both" (Lat. aut ).

Other connectives commonly used in truth-functional analysis are the conditional and the biconditional. The conditional, or implication, corresponds to "if … then" or "implies" in ordinary language, but only in a weak sense. The conditional is false only if the antecedent is true and the consequent is false; it is true in all other instances. This kind of implication, in which the connection between the antecedent and the consequent is merely formal, is known as material implication. The biconditional, or double implication, is the equivalence relation and is true only if the two statements have the same truth value, either true or false. In any truth function one may substitute an equivalent expression for all or any part of the function. The validity of arguments may be analyzed by assigning all possible combinations of truth values to the component statements; such an array of truth values is called a truth table.

The Predicate Calculus

There are many valid argument forms, however, that cannot be analyzed by truth-functional methods, e.g., the classic syllogism : "All men are mortal. Socrates is a man. Therefore Socrates is mortal." The syllogism and many other more complicated arguments are the subject of the predicate calculus, or quantification theory, which is based on the calculus of classes. The predicate calculus of monadic (one-variable) predicates, also called uniform quantification theory, has been shown to be complete and has a decision procedure, analogous to truth tables for truth-functional analysis, whereby the validity or invalidity of any statement can be determined. The general predicate calculus, or quantification theory, was also shown to be complete by Kurt Gödel, but Alonso Church subsequently proved (1936) that it has no possible decision procedure.

Analysis of the Foundations of Mathematics

Symbolic logic has been extended to a description and analysis of the foundations of mathematics, particularly number theory. Gödel also made (1931) the surprising discovery that number theory cannot be complete, i.e., that no matter what axioms are chosen as a basis for number theory, there will always be some true statements that cannot be deducted from them, although they can be proved within the larger context of symbolic logic. Since many branches of mathematics are ultimately based on number theory, this result has been interpreted by some as affirming that mathematics is an open, creative discipline whose possibilities cannot be delineated. The work of Gödel, Church, and others has led to the development of proof theory, or metamathematics, which deals with the nature of mathematics itself.

Bibliography

See D. Hilbert and W. Ackermann, Principles of Mathematical Logic (tr. of 2d ed. 1950); W. V. Quine, Mathematical Logic (1968) and Methods of Logic (3d ed. 1972).

Symbolic logic

Symbolic logic is the branch of mathematics that makes use of symbols to express logical ideas. This method makes it possible to manipulate ideas mathematically in much the same way that numbers are manipulated.

Most people are already familiar with the use of letters and other symbols to represent both numbers and concepts. For example, many solutions to algebraic problems begin with the statement, "Let x represent.…" That is, the letter x can be used to represent the number of boxes of nails, the number of sheep in a flock, or the number of hours traveled by a car. Similarly, the letter p is often used in geometry to represent a point. P can then be used to describe line segments, intersections, and other geometric concepts.

In symbolic logic, a letter such as p can be used to represent a complete statement. It may, for example, represent the statement: "A triangle has three sides."

Mathematical operations in symbolic logic

Consider the two possible statements:

"I will be home tonight" and "I will be home tomorrow."

Let p represent the first statement and q represent the second statement. Then it is possible to investigate various combinations of these two statements by mathematical means. The simplest mathematical possibilities are to ask what happens when both statements are true (an AND operation) or when only one statement is true (an OR operation).

One method for performing this kind of analysis is with a truth table. A truth table is an organized way of considering all possible relationships between two logical statements, in this case, between p and q. An example of the truth table for the two statements given above is shown below. Notice in the table that the symbol ∧ is used to represent an AND operation and the symbol ∨ to represent an OR operation:

p q p∧q p∨q

T T T T

T F F T

F T F T

F F F F

Notice what the table tells you. First, if "I will be home tonight" (p) and "I will be home tomorrow" (q) are both true, then the statement "I will be home tonight and I will be home tomorrow"—(p) and (q)— also must be true. In contrast, look at line 3 of the chart. According to this line, the statement "I will be home tonight" (p) is false, but the statement "I will be home tomorrow" (q) is true. What does this tell you about p∧q and p∨q?

First, p∧q means that "I will be home tonight" (p), and "I will be home tomorrow" (q). But line 3 says that the first of these statements (p) is false. Therefore, the statement "I will be home tonight and I will be home tomorrow" must be false. On the other hand, the condition p∨q means that "I will be home tonight or I will be home tomorrow." But this statement can be true since the second statement—"I will be home tomorrow"—is true.

The mathematics of symbolic logic is far more complex than can be shown in this book. Its most important applications have been in the field of computer design. When an engineer lays out the electrical circuits that make up a computer, or when a programmer writes a program for using the computer, many kinds of AND and OR decisions (along with other kinds of decisions) have to be made. Symbolic logic provides a precise method for making those decisions.

logic symbols A set of graphical symbols that express the function of individual logic gates in a logic diagram. The most common symbols are those for the simple Boolean functions and for flip-flops, as shown in the diagram.

symbolic logic The treatment of formal logic involving the setting up of a formalized language. The propositional calculus and predicate calculus are two of the more common areas of interest.