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Complete Theory
In mathematical logic, a theory is complete if, for every formula in the theory's language, that formula or its negation is demonstrable. Recursively axiomatizable first-order theories that are rich enough to allow general mathematical reasoning to be formulated cannot be complete, as demonstrated by Gödel's first incompleteness theorem. This sense of complete is distinct from the notion of a complete logic, which asserts that for every theory that can be formulated in the logic, all semantically valid statements are provable theorems (for an appropriate sense of "semantically valid")
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Mathematical Logic
Mathematical logic is a subfield of mathematics exploring the applications of formal logic to mathematics. It bears close connections to metamathematics, the foundations of mathematics, and theoretical computer science.[1] The unifying themes in mathematical logic include the study of the expressive power of formal systems and the deductive power of formal proof systems. Mathematical logic is often divided into the fields of set theory, model theory, recursion theory, and proof theory. These areas share basic results on logic, particularly first-order logic, and definability. In computer science (particularly in the ACM Classification) mathematical logic encompasses additional topics not detailed in this article; see Logic
Logic
in computer science for those. Since its inception, mathematical logic has both contributed to, and has been motivated by, the study of foundations of mathematics
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A Priori And A Posteriori
The Latin
Latin
phrases a priori (lit. "from the earlier") and a posteriori (lit. "from the latter") are philosophical terms of art popularized by Immanuel Kant's Critique of Pure Reason
Critique of Pure Reason
(first published in 1781, second edition in 1787), one of the most influential works in the history of philosophy.[1] However, in their Latin
Latin
forms they appear in Latin
Latin
translations of Euclid's Elements, of about 300 BCE, a work widely considered during the early European modern period as the model for precise thinking. These terms are used with respect to reasoning (epistemology) to distinguish "necessary conclusions from first premises" (i.e., what must come before sense observation) from "conclusions based on sense observation" (which must follow it)
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Logic In Computer Science
Logic
Logic
in computer science covers the overlap between the field of logic and that of computer science. The topic can essentially be divided into three main areas:Theoretical foundations and analysis Use of computer technology to aid logicians Use of concepts from logic for computer applicationsContents1 Theoretical foundations and analysis 2 Computers to assist logicians 3 Logic
Logic
applications for computers 4 See also 5 References 6 Further reading 7 External linksTheoretical foundations and analysis[edit] Logic
Logic
plays a fundamental role in computer science. Some of the key areas of logic that are particularly significant are computability theory (formerly called recursion theory), modal logic and category theory
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Metalogic
Metalogic is the study of the metatheory of logic. Whereas logic studies how logical systems can be used to construct valid and sound arguments, metalogic studies the properties of logical systems.[1] Logic
Logic
concerns the truths that may be derived using a logical system; metalogic concerns the truths that may be derived about the languages and systems that are used to express truths.[2] The basic objects of metalogical study are formal languages, formal systems, and their interpretations
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Metamathematics
Metamathematics
Metamathematics
is the study of mathematics itself using mathematical methods. This study produces metatheories, which are mathematical theories about other mathematical theories. Emphasis on metamathematics (and perhaps the creation of the term itself) owes itself to David Hilbert's attempt to secure the foundations of mathematics in the early part of the 20th century. Metamathematics provides "a rigorous mathematical technique for investigating a great variety of foundation problems for mathematics and logic" (Kleene 1952, p. 59). An important feature of metamathematics is its emphasis on differentiating between reasoning from inside a system and from outside a system
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Non-classical Logic
Non-classical logics (and sometimes alternative logics) are formal systems that differ in a significant way from standard logical systems such as propositional and predicate logic. There are several ways in which this is done, including by way of extensions, deviations, and variations. The aim of these departures is to make it possible to construct different models of logical consequence and logical truth.[1] Philosophical logic is understood to encompass and focus on non-classical logics, although the term has other meanings as well.[2] In addition, some parts of theoretical computer science can be thought of as using non-classical reasoning, although this varies according to the subject area. For example, the basic boolean functions (e.g. AND, OR, NOT, etc) in computer science are very much classical in nature, as is clearly the case given the fact that they can be fully described by classical truth tables
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Philosophical Logic
Philosophical logic refers to those areas of philosophy in which recognized methods of logic have traditionally been used to solve or advance the discussion of philosophical problems.[1] Among these, Sybil Wolfram highlights the study of argument, meaning, and truth,[2] while Colin McGinn presents identity, existence, predication, necessity and truth as the main topics of his book on the subject.[3] Philosophical logic also addresses extensions and alternatives to traditional, "classical" logic known as "non-classical" logics. These receive more attention in texts such as John P. Burgess's Philosophical Logic,[4] the Blackwell Companion to Philosophical Logic,[5] or the multi-volume Handbook of Philosophical Logic[6] edited by Dov M
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Philosophy Of Logic
Following the developments in formal logic with symbolic logic in the late nineteenth century and mathematical logic in the twentieth, topics traditionally treated by logic not being part of formal logic have tended to be termed either philosophy of logic or philosophical logic if no longer simply logic. Compared to the history of logic the demarcation between philosophy of logic and philosophical logic is of recent coinage and not always entirely clear. Characterisations include Philosophy
Philosophy
of logic is the area of philosophy devoted to examining the scope and nature of logic.[1] Philosophy
Philosophy
of logic is the investigation, critical analysis and intellectual reflection on issues arising in logic
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Set Theory
Set theory
Set theory
is a branch of mathematical logic that studies sets, which informally are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics. The language of set theory can be used in the definitions of nearly all mathematical objects. The modern study of set theory was initiated by Georg Cantor
Georg Cantor
and Richard Dedekind
Richard Dedekind
in the 1870s
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Abductive Reasoning
Abductive reasoning (also called abduction,[1] abductive inference,[1] or retroduction[2]) is a form of logical inference which starts with an observation or set of observations then seeks to find the simplest and most likely explanation. In abductive reasoning, unlike in deductive reasoning, the premises do not guarantee the conclusion
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Analytic–synthetic Distinction
The analytic–synthetic distinction (also called the analytic–synthetic dichotomy) is a semantic distinction, used primarily in philosophy to distinguish propositions (in particular, statements that are affirmative subject–predicate judgments) into two types: analytic propositions and synthetic propositions
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Antinomy
Antinomy (Greek ἀντί, antí, "against, in opposition to", and νόμος, nómos, "law") refers to a real or apparent mutual incompatibility of two laws.[1] It is a term used in logic and epistemology, particularly in the philosophy of Kant
Kant
and Roberto Unger. There are many examples of antinomy. A self-contradictory phrase such as "There is no absolute truth" can be considered an antinomy because this statement is suggesting in itself to be an absolute truth, and therefore denies itself any truth in its statement
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Deductive Reasoning
Deductive reasoning, also deductive logic, logical deduction is the process of reasoning from one or more statements (premises) to reach a logically certain conclusion.[1] Deductive reasoning
Deductive reasoning
goes in the same direction as that of the conditionals, and links premises with conclusions. If all premises are true, the terms are clear, and the rules of deductive logic are followed, then the conclusion reached is necessarily true. Deductive reasoning
Deductive reasoning
("top-down logic") contrasts with inductive reasoning ("bottom-up logic") in the following way; in deductive reasoning, a conclusion is reached reductively by applying general rules which hold over the entirety of a closed domain of discourse, narrowing the range under consideration until only the conclusion(s) is left
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Axiology
Axiology (from Greek ἀξία, axia, "value, worth"; and -λογία, -logia) is the philosophical study of value. It is either the collective term for ethics and aesthetics[1], philosophical fields that depend crucially on notions of worth, or the foundation for these fields, and thus similar to value theory and meta-ethics. The term was first used by Paul Lapie, in 1902,[2][3] and Eduard von Hartmann, in 1908.[4][5] Axiology studies mainly two kinds of values: ethics and aesthetics. Ethics
Ethics
investigates the concepts of "right" and "good" in individual and social conduct
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Definition
A definition is a statement of the meaning of a term (a word, phrase, or other set of symbols).[1] Definitions can be classified into two large categories, intensional definitions (which try to give the essence of a term) and extensional definitions (which proceed by listing the objects that a term describes).[2] Another important category of definitions is the class of ostensive definitions, which convey the meaning of a term by pointing out examples. A term may have many different senses and multiple meanings, and thus require multiple definitions.[3][a] In mathematics, a definition is used to give a precise meaning to a new term, instead of describing a pre-existing term
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