Logic is the study of valid argument. Valid argument (or equivalently valid reasoning) is the process of reaching a conclusion from a set of assumptions, in such a way that the conclusion is true whenever the assumptions are true. As a byproduct, logic provides prescriptions for reasoning, that is, how people (as well, perhaps, as other intelligent beings/machines/systems) ought to reason. This prescriptive aspect of logic, however, is not an essential part of logic, any more than prescriptions for not walking off cliffs are an essential part of scientific investigations of gravity.
Traditionally, logic is studied as a branch of Philosophy, but some parts of it are systematically studied in mathematics and Computer Science. How people actually reason is usually studied under other headings, including Cognitive Psychology. Logic is traditionally divided into Deductive Reasoning, concerned with what follows logically from given premises, and Inductive Reasoning, concerned with how we can go from some number of observed events to a reliable generalization.
As a science, logic investigates and classifies the structure of statements and arguments and devises schemata by which these are codified. The scope of logic can therefore be very large, including reasoning about probability and causality. Also studied in logic are the structure of fallacious arguments
__Formal vs informal logic__
Somewhat arbitrarily, we divide the study of logic into formal and informal logic.
Formal Logic (sometimes called symbolic logic) approaches logic and in particular logical argument as a set of rules for manipulating symbols. There are two kinds of rules in any system of formal logic: Syntax rules and rules of inference. Syntax says how to build meaningful expressions; rules of inference say how to obtain true formulas from other true formulas. Logic also needs semantics, which says how to assign meaning to expressions. Formal logic encompasses a wide variety of logical systems. For instance, propositional logic and predicate logic are a kind of formal logic, as well as temporal logic, modal logic, Hoare logic, the calculus of constructions etc. Higher order logics refer to logical systems based on a hierarchy of types.
Informal Logic is the study of logic as used in natural language arguments. Informal logic is complicated by the fact that it may be very hard to tease out the formal logical structure imbedded in an argument. Informal logic is also more difficult because the semantics of natural language assertions is much more complicated than the semantics of formal logical systems.
__Aristotelian Logic__
The Prior Analytics was Aristotle's pioneering work establishing a system of logic and inference based on the forms of the premises and the conclusion. These rules were codified into various forms of syllogisms which, until recently at least, were part of the standard high school curriculum in the West, much like euclidean plane geometry. Aristotelian logic is sometimes referred to as formal logic because it specifically deals with forms of reasoning, but is not formal in the sense we use it here or as is common in current usage. It can be considered as a precursor to formal logic.
__Mathematical Logic__
Mathematical Logic refers to two distinct areas of research: The first, primarily of historical interest, is the use of formal logic to study mathematical reasoning, and the second, in the other direction, the application of mathematics to the study of formal logic. At the beginning of the twentieth century, philosophical logicians including (Frege, Russell) attempted to prove that mathematics could be entirely reduced to logic. The reduction had limited success (for reasons which are well beyond the scope of this article) but in the process, logic took on much of the notation and methodology of Mathematics. In the other direction, in the early 1930s, Kurt Gödel embarked on an ambitious program of considering logic and proof as an object of mathematical study, leading him to state far reaching results on provability and model theory such as the incompleteness theorems of first order arithmetic. This line of research has continued to the present time, leading to various stunning results such as for example, Paul Cohen's proof of the independence of the continuum hypothesis from the axioms of Zermelo-Fraenkel set theory.
__Philosophical Logic__
Philosophical Logic deals with formal descriptions of natural language. Most philosophers assume that the bulk of "normal" proper reasoning can be captured by logic, if one can find the right method for translating ordinary language into that logic. Philosophical logic is essentially a continuation of the traditional discipline that was called "Logic" before it was supplanted by the invention of Mathematical logic. Philosophical logic has a much greater concern with the connection between natural language and logic. As a result, philosophical logicians have contributed a great deal to the development of non-standard logics (e.g., free logics, tense logics) as well as various extensions of classical logic (e.g., modal logics), and non-standard semantics for such logics (e.g., supervaluation semantics).
__Multi-valued logic__
The logics discussed above are all "bivalent" or "two-valued"; that is, the semantics for each of these languages will assign to every sentence either the value "True" or the value "False." Systems which do not always make this distinction are known as non-Aristotelian logics, or multi-valued logics.
In the early 20th century Jan Lukasiewicz investigated the extension of the traditional true/false values to include a third value, "possible".
Logics such as fuzzy logic have since been devised with an infinite number of "degrees of truth", e.g., represented by a real number between 0 and 1. Bayesian probability can be interpreted as a system of logic where probability is the subjective truth value.
__Logic and computation__
Logic is extensively used in the fields of Artificial Intelligence, and Computer Science.
In the 1950s and 1960s, researchers predicted that when human Knowledge could be expressed using logic with mathematical notation, it would be possible to create a machine that reasons, or artificial intelligence. This turned out to be more difficult than expected because of the complexity of human reasoning. Logic programming is an attempt to make computers do logical reasoning and Prolog programming language is commonly used for it.
In symbolic logic and mathematical logic, proofs by humans can be computer-assisted. Using automated theorem proving the machines can find and check proofs, as well as work with proofs too lengthy to be written out by hand.
In computer science, Boolean algebra is the basis of hardware design, as well as much software design.
There are also various systems for reasoning about computer programs. Hoare logic is one the earliest of such systems. Other systems are CSP, CCS, pi-calculus for reasoning about concurrent processes or mobile proceses. See also Computability Logic; this is a formal theory of computability in the same sense as Classical Logic is a formal theory of truth.
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