the study of propositions and their use in argumentation. The major task of logic is to establish a systematic way of deducing the logical consequences of a set of sentences. In order to accomplish this, it is necessary first to identify or characterize the logical consequences of a set of sentences. The procedures for deriving conclusions from a set of sentences then need to be examined to verify that all logical consequences, and only those, are deducible from that set. Finally, in recent times, the question has been raised whether all the truths regarding some domain of interest can be contained in a specifiable deductive system. From its very beginning, the field of logic has been occupied with arguments, in which certain statements, the premises, are asserted in order to support some other statement, the conclusion. If the premises are intended to provide conclusive support for the conclusion, the argument is a deductive one. If the premises are intended to support the conclusion only to a lesser degree, the argument is called inductive. A logically correct deductive argument is termed valid, while an acceptable inductive argument is called cogent. The notion of support is further elucidated by the observation that the truth of the premises of a valid deductive argument necessitates the truth of the conclusion: it is impossible for the premises to be true and the conclusion false. The truth of the premises of a cogent inductive argument, on the other hand, confers only a probability of truth on its conclusion: it is possible for the premises to be true while the conclusion is false. Logic is not concerned to discover premises that persuade an audience to accept, or to believe, the conclusion. This is the subject of rhetoric. The notion of rational persuasion is sometimes used by logicians in the sense that, if one were to accept the premises of a valid deductive argument, it would not be rational to reject the conclusion; one would in effect be contradicting oneself in practice. The case of inductive logic will be considered below. From the above characterization of arguments, it is evident that they are always advanced in some language, either a natural language such as English or Chinese or, possibly, a specialized technical language such as mathematics. To develop rules for determining the validity of deductive arguments, the statements comprising the argument must be analyzed in order to see how they relate to one another. The analysis of the logical forms of arguments can be accomplished most perspicuously if the statements of the argument are framed in some canonical form. Additionally, when stated in a regimented format, various ambiguities or other defects of the original statements can be avoided. When they are stated in a natural language, some arguments appear to give support to their conclusions or to confute a thesis. Such a defective, although apparently correct, argument is called a fallacy. Some of these errors in argument occur often enough that types of such fallacies are given special names. For example, if one were to attack the premises of an argument by casting aspersions on the character of the proponent of the argument, this would be characterized as committing an ad hominem fallacy. The character of the proponent of an argument has no relevance to the validity of the argument. There are several other fallacies of relevance, such as threatening the audience (argumentum ad baculum) or appealing to their feelings of pity (argumentum ad misericordiam). The other major grouping of fallacies concerns those apparently correct arguments whose plausibility depends on some ambiguity. For an argument to be valid it is required that the terms occurring in the argument retain one meaning throughout. Subtle shifts of meaning that destroy the correctness of any argument can occur in natural language expressions: Today chain-smokers are rapidly disappearing. Karen is a chain-smoker. Therefore, today Karen is rapidly disappearing. Clearly what is intended in the first premise is that the class of chain-smokers is becoming a smaller class, not that the individuals in the class are undergoing any change. A well-known, classic example of incorrect reasoning based on an ambiguity arising from the grammatical construction employed, the so-called amphiboly, is the case of Croesus, king of Lydia in the 6th century BC, who was considering invading Persia. When he consulted the oracle at Delphi, he is reported to have received the following reply: If Croesus goes to war with Cyrus (the king of Persia), he will destroy a mighty kingdom. Croesus inferred that his campaign would be successful, but in fact he lost, and consequently his own mighty kingdom was destroyed. 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 valuestrue or falseof 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? Additional reading The best starting point for exploring any of the topics in logic is D. Gabbay and F. Guenthner (eds.), Handbook of Philosophical Logic, 4 vol. (198389), a comprehensive reference work. See also Gerald J. Massey, Understanding Symbolic Logic (1970), an introductory text; and Robert E. Butts and Jaakko Hintikka, Logic, Foundations of Mathematics, and Computability Theory (1977), a collection of conference papers.

# LOGIC

## Meaning of LOGIC in English

Britannica English vocabulary. Английский словарь Британика. 2012