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Logic

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

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

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

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

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

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

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 of 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

• analytic proposition
• argument form
• cogency
• college logic
• computability logic
• hybrid logic
• interpretability logic
• provability logic
• soundness
• validity

• affirming the consequent
• deduction and induction
• disjunctive syllogism
• modus ponens
• modus tollens

• faith
• fuzzy logic
• game semantics
• history of logic
• lambda calculus
• set theory
• Important publications in logic

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