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In ordinary language, logic is the reasoning used to reach a
conclusion from a set of assumptions. More formally, logic is the study of valid inference, that is the process of reaching a conclusion from a set of assumptions in a systematic and valid way. Roughly speaking validity means that whenever a given set of
assumptions is true, the conclusion is also true.
As a byproduct, logic provides prescriptions for reasoning, that is, how people—as well as other intelligent beings,
machines, and systems—ought to reason. However, such prescriptions are not essential to logic itself, rather are an
application. How people actually reason is usually studied in other fields, including cognitive psychology.
Traditionally, logic is studied as a branch of philosophy. Since the
mid-1800s logic has been commonly studied in mathematics, and, even more recently, in Computer
Science. 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 and
paradoxes.
Scope of logic
As it has developed, many distinctions have been introduced into logic. These distinctions serve to help formalize different
forms of logic as a science. Here are some of the more important distinctions.
Deductive and inductive reasoning
Originally, logic consisted only of deductive reasoning
which concerns what follows from given premises. However it is important to note that inductive reasoning—the study of deriving a reliable generalization from observations—has
sometimes been included in the study of logic. Correspondingly, we must distinguish between deductive validity and inductive
validity. An inference is deductively valid if and only if there is no possible situation in which all the premises are true and
the conclusion false. The notion of deductive validity can be rigorously stated for systems of formal logic in terms well-understood notions of semantics. Inductive validity on the other hand requires us to define reliable generalization of some
set of observations. The task of providing this definition may be approached in various ways, some less formal than others; some
of these definitions may use mathematical models of probability. For the most part our discussion of logic deals only with
deductive logic.
Formal and informal logic
Somewhat arbitrarily, study of logic is divided 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.
Following are more specific discussions of some systems of logic. See also: list of topics in logic.
Types of logic
Throughout history, logic has been studied in many ways. Most of the methods of study share many concepts. Some of the main
distinctions lie in their formality. In addition, different types of logic are used to claim different types of things. For
example, to prove things about logic or math, mathematical
logic is most often used. To propose simple logical arguments, Aristotelian logic is used.
Aristotelian logic
Main article: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.
In the tradition of aristotelian logic is also term logic.
Mathematical logic
Main article: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, Paul Cohen's proof of the independence of the continuum hypothesis from the axioms of Zermelo-Fraenkel set
theory.
Philosophical logic
Main article 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. One such example is ternary
logic with deals with "three-valued" logic.
In the early 20th century Jan Łukasiewicz investigated the extension of the traditional true/false values to include a third
value, "possible". Modal logic explores logics with values of
possibility.
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.
Closely related fields
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.
See also
Concepts of logic
Techniques and rules
Related Topics
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