Logic is the the study of propositions and of their use in argumentation. This study may be carried on at a very abstract level, as in formal logic, or it may focus on the practical art of right reasoning, as in applied logic.

Valid arguments have two basic forms. Those that draw some new proposition (the conclusion) from a given proposition or set of propositions (the premises) in which it may be thought to lie latent are called deductive. These arguments make the strong claim that the conclusion follows by strict necessity from the premises, or in other words that to assert the premises but deny the conclusion would be inconsistent and self-contradictory. Arguments that venture general conclusions from particular facts that appear to serve as evidence for them are called inductive. These arguments make the weaker claim that the premises lend a certain degree of probability or reasonableness to the conclusion. The logic of inductive argumentation has become virtually synonymous with the methodology of the physical, social, and historical sciences and is no longer treated under logic. Logic as currently understood concerns itself with deductive processes. As such it encompasses the principles by which propositions are related to one another and the techniques of thought by which these relationships can be explored and valid statements made about them.

In its narrowest sense deductive logic divides into the logic of propositions (also called sentential logic) and the logic of predicates (or noun expressions). In its widest sense it embraces various theories of language (such as logical syntax and semantics), metalogic (the methodology of formal systems), theories of modalities (the analyses of the notions of necessity, possibility, impossibility, and contingency), and the study of paradoxes and logical fallacies. Both of these senses may be called formal or pure logic, in that they construct and analyze an abstract body of symbols, rules for stringing these symbols together into formulas, and rules for manipulating these formulas. When certain meanings are attached to these symbols and formulas, and this machinery is adapted and deployed over the concrete issues of a certain range of special subjects, logic is said to be applied. The analysis of questions that transcend the formal concerns of either pure or applied logic, such as the examination of the meaning and implications of the concepts and assumptions of either discipline, is the domain of the philosophy of logic.

Logic was developed independently and brought to some degree of systematization in China (5th to 3rd century BC) and India (from the 5th century BC through the 16th and 17th centuries AD). Logic as it is known in the West comes from Greece. Building on an important tradition of mathematics and rhetorical and philosophical argumentation, Aristotle in the 4th century BC worked out the first system of the logic of noun expressions. The logic of propositions originated in the work of Aristotle's pupil Theophrastus and in that of the 4th-century Megarian school of dialecticians and logicians and the school of the Stoics. After the decline of Greek culture, logic reemerged first among Arab scholars in the 10th century. Medieval interest in logic dated from the work of St. Anselm of Canterbury and Peter Abelard. Its high point was the 14th century, when the Scholastics developed logic, especially the analysis of propositions, well beyond what was known to the ancients. Rhetoric and natural science largely eclipsed logic during the Renaissance. Modern logic began to develop with the work of the mathematician G.W. Leibniz, who attempted to create a universal calculus of reason. Great strides were made in the 19th century in the development of symbolic logic, leading to the highly fruitful merging of logic and mathematics in formal analysis.

Modern formal logic is the study of inference and proposition forms. Its simplest and most basic branch is that of the propositional calculus (or PC). In this logic, propositions or sentences form the only semantic category. These are dealt with as simple and remain unanalyzed; attention is focused on how they are related to other propositions by propositional connectives (such as "if . . . then," "and," "or," "it is not the case that," etc.) and thus formed into arguments. By representing propositions with symbols called variables and connectives with symbolic operators, and by deciding on a set of transformation rules (axioms that define validity and provide starting points for the derivation of further rules called theorems), it is possible to model and study the abstract characteristics and consequences of this formal system in a way similar to the investigations of pure mathematics. When the variables refer not to whole propositions but to noun expressions (or predicates) within propositions, the resulting formal system is known as a lower predicate calculus (or LPC).

Changing the operators, variables, or rules of such formal systems yields different logics. Certain systems of PC, for example, add a third "neuter" value to the two traditional possible values--true or false--of propositions. A major step in modern logic is the discovery that it is possible to examine and characterize other formal systems in terms of the logic resulting from their elements, operations, and rules of formation; such is the study of the logical foundations of mathematics, set theory, and logic itself.

Logic is said to be applied when it systematizes the forms of sound reasoning or a body of universal truths in some restricted field of thought or discourse. Usually this is done by adding extra axioms and special constants to some preestablished pure logic such as PC or LPC. Examples of applied logics are practical logic, which is concerned with the logic of choices, commands, and values; epistemic logic, which analyzes the logic of belief, knowing, and questions; the logics of physical application, such as temporal logic and mereology; and the logics of correct argumentation, fallacies, hypothetical reasoning, and so on.

Varieties of logical semantics have become the central area of study in the philosophy of logic. Some of the more important contemporary philosophical issues concerning logic are the following: What is the relation between logical systems and the real world? What are the limitations of logic, especially with regard to some of the assumptions of its wider senses and the incompleteness of first-order logic? What consequences stem from the nonrecursive nature of many mathematical functions?

Excerpt from the Encyclopedia Britannica without permission.